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SEMIRA PUSKAR
DEFINED BENEFIT VERSUS DEFINED CONTRIBUTION PENSION PLANS:
HOW THEY COMPARE FOR DIFFERENT WORKING HISTORIES
Mémoire présenté à la Faculté des études supérieures de l'Université Laval
dans le cadre du programme de maîtrise en mathématiques pour l'obtention du grade de maître es sciences (M.Se.)
DEPARTEMENT DE MATHEMATIQUES ET DE STATISTIQUE FACULTÉ DES SCIENCES ET DE GÉNIE
UNIVERSITÉ LAVAL QUÉBEC
2008
© Semira Puskar, 2008
Résumé Le système de retraite canadien a trois composantes : les programmes de pension
administrés par le gouvernement, les régimes de retraite offerts par les employeurs et les
épargnes personnelles. Depuis environ dix ans, deux tendances affectant les régimes offerts
par les employeurs émergent : la diminution du taux d'adhésion aux régimes de pension
agréés et une transition des régimes à prestations déterminées aux régimes à cotisation
déterminée.
Dans cette étude, après avoir survolé les raisons pouvant expliquer ces deux
tendances, nous utilisons un modèle stochastique pour projeter l'inflation, l'augmentation
salariale et le rendement sur les investissements afin de calculer les prestations de retraite
pour différents historiques de travail. Nous analysons la distribution de ces prestations,
exprimées en pourcentage du salaire final, et ce, pour quatre régimes de retraite différents :
le régime de service total, le régime salaire final, le régime salaires de carrière et le régime
à cotisation déterminée.
Abstract The Canadian retirement System has three components: government-administered
pension programs, employer-sponsored pension plans and personal savings. Over the last
décade or so, two trends affecting employer-sponsored pension plans hâve become
noticeable: a décline in registered pension plan coverage and a shift from defined benefit
pension plans to defined contribution pension plans.
In this study, after surveying the reasons found to explain those two trends, we use a
stochastic model for inflation, wage growth and investment returns in order to calculate the
pension benefits for several différent working historiés. We analyse the distribution of
pension benefits, as a percentage of final earnings, under the following four pension
arrangements: total service pension plan, final salary pension plan, career average pension
plan and defined contribution pension plan.
Acknowledgments I would like to warmly thank my thesis supervisor, Professor Claire Bilodeau, for
accepting to be my supervisor and joining me on this challenging journey. I would like to
express my sincère gratitude for her invaluable support and encouragement, enthusiasm and
insights in preparing and writing this thesis. Without her assistance, this thesis would not
hâve been possible. Her guidance, patience, time and kindness throughout this thesis are
greatly appreciated. Merci, Claire.
I would like to thank Professors Michel Jacques and Louis Adam from the École
d'actuariat for having accepted to read my thesis and for their valuable and constructive
comments and suggestions that allowed me to improve the quality of this thesis.
I would like to extend my gratitude and appréciation to the Chaire d'actuariat for
providing the bourse Chaire d'actuariat, as well as to ail Professors in the École d'actuariat
for their help and support.
I would also like tp thank Mme Sylvie Drolet from the Département de
mathématiques et de statistique and Mme Rachel Asselin from the École d'actuariat for
their help with registration issues.
Thank you to ail my friends, old and new, you enrich my life in so many ways.
Thank you to Christian's family for their support and understanding.
I am sincerely and profoundly grateful to my parents, my heroes - thank you for
your wisdom - and my family. Every single one of you was a source of inspiration for this
thesis. Your unfailing love, support and encouragement provided me the strength I needed
throughout this journey. Thank you for being there for me. My dear nièces and nephew,
we made it, yay!
Last and by no means least, I wish to express my sincère and deepest gratitude to
Christian for his encouragement to pursue this degree and for his support, both emotional
and financial. Thank you Christian for making it possible.
Mojim roditeljîma, s bezgranicnom Ijubavlju i postovanjem
To my parents, with endless love and gratitude
To myfamily
Contents Résumé i Abstract ii Acknowledgments iii Contents v List of Tables vii List of Figures viii Chapter 1 Introduction 1 Chapter 2 Canada's Retirement Income System 7
2.1 How It Ail Started: A Walk Through History 7 2.2 Retirement Income System Today 11
2.2.1 Government Programs 11 2.2.1.1 Old Age Security 12 2.2.1.2 Canada and Québec Pension Plans 13
2.2.2 Employer-Sponsored Pension Plans 13 2.2.3 Personal Savings 14
Chapter 3 Employer-Sponsored Pension Plans: A Closer Look 16 3.1 Types of Pension Plans 17
3.1.1 Defined Contribution Plans 17 3.1.2 Defined Benefit Plans 19
3.1.2.1 Unit Benefit Formula 19 3.1.2.1.1 Career Average Formula 20 3.1.2.1.2 Final Average Formula 21 3.1.2.1.3 Modified Career Average Formula 23
3.1.2.2 Fiat Benefit Formula 23 3.2 Which Plan Is Better, DB or DC? 24 3.3 When to Retire? 25 3.4 Benefit Payments 27 3.5 Intégration with Canada/Québec Pension Plans (C/QPP) 28 3.6 Quitting Before Retiring 30 3.7 A Word on Régulations 32
Chapter 4 Literature Review 34 4.1 Observations in the U.S., the U.K. and Canada 35 4.2 Reasons Underlying the Switch 37 4.3 Research Results Related to the Trend and its Impact .....40
Chapter 5 Model Used to Simulate Economie Variables 50 5.1 General Description of the Model 50 5.2 Equations of the Model 52 5.3 Estimation of the Parameters of the Model 55 5.4 Economie Scénarios Generated 59
Chapter 6 Calculation of Pension Benefits 69 6.1 Working Historiés 69 6.2 Pension Plans 72 6.3 Assumptions 74
VI
6.3.1 Service 74 6.3.2 Earnings and Salary Scale 75 6.3.3 Mortality 76 6.3.4 Annuity 76 6.3.5 Contribution Rate 78
6.4 Pension Benefits 79 6.4.1 Total Service Pension 80 6.4.2 Final Salary Pension 80 6.4.3 Career Average Pension 82 6.4.4 Defined Contribution Pension 82
Chapter 7 Results 84 7.1 Age-Based Merit 85
7.1.1 Total Service Pension 88 7.1.2 Final Salary Pension 88 7.1.3 Career Average Pension 91 7.1.4 Defined Contribution Plan 91
7.2 Experience-Based Merit 93 7.2.1 Total Service Pension 97 7.2.2 Final Salary Pension 97 7.2.3 Career Average Pension 97 7.2.4 Defined Contribution Plan 98
7.3 Summary of Results 99 Chapter 8 Conclusion 101 Bibliography 105 Appendix A Economie Variables: Data 113
A.l Inflation 113 A.2 Wage Growth 113 A.3 Treasury Bills 114 A.4 Bonds 115 A.5 Common Stock 115
Appendix B Economie Variables: Model 119 Appendix C Working Historiés 120 Appendix D Salary Scale 121 Appendix E Mortality 122
List of Tables Table 1.1 Percentage of total income of people aged 65 and older by source of income 2 Table 1.2 Number of members and proportion of paid workers covered by an RPP 3 Table 1.3 Total number of RPPs and members by type of plan 4 Table 3.1 Total RPPs and members by type of plan 17 Table 4.1 Number of private pension plans and active participants in the U.S 35 Table 4.2 Number of private sector pension plans and active members in the U.K 36 Table 4.3 Number of pension plans and members in Canada 37 Table 5.1 Historical statistics 68 Table 5.2 Simulated statistics 68 Table 7.1 Average and standard déviation of replacement ratio (age-based merit) 85 Table 7.2 Intervais for the replacement ratio (age-based merit) 87 Table 7.3 Probabilities of DC plan faring better than DB plan (age-based merit) 88 Table 7.4 Average and standard déviation of replacement ratio (experience-based merit) .94 Table 7.5 Intervais for the replacement ratio (experience-based merit) 96 Table 7.6 Probabilities of DC plan faring better than DB plan (experience-based merit) ...96 Table A.l Historical rates of inflation and wage growth 117 Table A.2 Historical rates of inflation and return on différent assets 118 Table B.l Parameters of the multivariate model 119 Table C l Employment pattern of différent catégories of employées 120 Table D.l Salary scale 121 Table E.l Mortality table: UP-94 projected to 2015 using AA scale 123
List of Figures Figure 5.1 Inflation 62 Figure 5.2 Wage growth 63 Figure 5.3 Return on Treasury bills 63 Figure 5.4 Yield to maturity on long-term bonds 64 Figure 5.5 Return on common stocks 64 Figure 5.6 Return on long-term bonds 66 Figure 5.7 Investment return (50% bonds and 50% stocks) 67 Figure 7.1 Visual représentation of average and standard déviation of replacement ratio
(age-based merit) 86 Figure 7.2 Visual représentation of average and standard déviation of replacement ratio
(experience-based merit) 95
Chapter 1 Introduction Retirement from a workforce represents a major change in one's life. For some
people it is something to look forward to but for others not a pleasant thought at ail.
Regardless of their attitude towards retirement, people should prépare themselves for a new
lifestyle in their golden years. It is very important, today more than ever, that employées
plan their retirement carefully and know what they are entitled to if they want to maintain a
certain standard of living after retirement.
Times hâve changed, retirement has changed too. People enter their working career
at a later âge due to more years in school. People retire earlier (although that trend seems
to be reversing slowly). People live longer. Public pensions are getting smaller. (For
instance, in Canada, a previously universal benefït now is reduced for high-income
earners.) Employer benefits are neither as generous nor as common anymore. Hence,
people need to save more for their retirement over a shorter period of time. How is it
possible to save more in changing social, économie and workplace environments?
The most striking change that has taken place over the past 20 years or so with
employer-sponsored plans is the shift away from traditional defined benefït (DB) pension
plans toward defined contribution (DC) pension plans. In Canada, this shift has not been as
pronounced as it has been in the United States (U.S.), the United Kingdom (U.K.) and
Australia.
Canada's retirement system has been subject to change over the last few décades in
order to meet its main objectives: fight against poverty (provide a basic income to ail
elderly) and income replacement (maintain a certain standard of living after retirement) (Li
2006). Even though Canada's retirement system ranks among the best in the world (OECD
2001), critics say that it still does not meet the basic objectives. They point out that there
are many unattached women, with income below poverty level (Townson 2006) and that
low- and middle-income earners do not receive sufficient pensions to maintain their
standard of living (Baldwin 2006).
2
Employer-sponsored pensions are henceforth also referred to as private pension
plans to distinguish them from the government pension plans, also referred to as public
pension plans. Private pension plans hâve undergone many changes since their first
establishment in order to meet the objectives of government, workers and employers.
In the 1940s, employer-sponsored plans grew rapidly due to several reasons such as
a stronger economy and increased tax déductions for contributions made to private pension
plans. From the 1970s to the 1990s, the fédéral and provincial governments hâve reformed
their législative standards governing employer pension plans. Thèse reforms addressed
long-time issues such as membership eligibility, benefits portability, early retirement,
vesting provisions and survivor benefits. Private pensions hâve gradually become a
growing source of income for retired seniors.
As an illustration, in 1990, income from private pension plans was 18.4% of the
total income of people aged 65 and older, while public pensions and Canada/Québec
Pension Plans (C/QPP) were 29.8% and 16.3% respectively. In 1999, income from private
pension plans amounted to 29.2% for the same group, while public pensions and C/QPP
represented 26.9% and 20.1% respectively. Thèse figures are summarized in Table 1.1.
(Source: Table 1-4 Number and income of persons aged 65 and older, by income source,
Statistics Canada 2000.)
Table 1.1 Percentage of total income of people aged 65 and older by source of income
Year Public C/QPP Private 1990 29.8% 16.3% 18.4% 1999 26.9% 20.1% 29.2%
Despite the fact that employer-sponsored pensions helped improve the well-being of
current seniors, they are going through changes that are creating concern and uneasiness for
new générations of retirées to corne. There are two noticeable trends that employer-
sponsored pensions exhibit: declining coverage and shift from defined benefit pension
plans to defined contribution pension plans.
The proportion of ail paid workers covered by a registered pension plan (RPP) in
1985 was 44.2%. Then it grew to 45.3% in 1991. In 1995, RPP coverage dropped to
3
42.8%, in 2000 it dropped to 40.8%, and in 2005 it dropped again to 38.5%. Although the
period from 1991 to 2000 has been characterized by a décline in membership for both
women (40.8% to 39.5%) and men (49.1% to 41.9%), it is worth mentioning that coverage
of women has not declined as much as that of men. As a resuit, the gender gap in terms of
coverage has been reduced over that period. In 2005, 38.7% of female paid workers and
38.3% of maie paid workers were covered by RPPs. By then, coverage had lowered again
while the gender gap had basically disappeared. (Source: Statistics Canada, Pension Plan
in Canada and Labour Force Survey.)
In 1991, men accounted for 58.8% of total RPP membership and women for 41.2%;
it is a différence of almost 18 percentage points. In 2000, the différence was about 10
percentage points; the membership was 54.9% men and 49.1% women. It is important to
add that it is more difficult for women to maintain RPP coverage throughout their
employment history (they, as caregivers, often take a break to take care of children or
elderly members of the family) due to rigid pension plan rules. (Source: Statistics Canada,
Pension Plan in Canada and Labour Force Survey.)
RPP membership and coverage ofpaid workers hâve been summarized in Table 1.2.
Table 1.2 Number of members and proportion ofpaid workers covered by an RPP
Year
Total Men Women
Year
Number ofRPP
members
Percentage ofpaid workers
Number ofRPP
members
Percentage ofpaid workers
Number ofRPP
members
Percentage ofpaid workers
1985 4,668,381 44.2% 3,047,160 50.5% 1,621,221 35.7% 1991 5,318,090 45.3% 3,129,263 49.1% 2,188,827 40.8% 1995 5,149,912 42.8% 2,894,564 44.5% 2,255,348 40.9% 2000 5,431,578 40.8% 2,984,444 41.9% 2,447,134 39.5% 2005 5,690,580 38.5% 2,977,758 38.3% 2,712,822 38.7%
Some of the reasons that may hâve caused a décline in coverage are: the shift in
employment type from the manufacturing sector to the service sector and the associated
décline in unionization; the reduced number of public sector employment; the increased
number of self-employed who are not eligible for RPP membership; the increased number
of other pension arrangements such as group Registered Retirement Saving Plans (group
RRSPs); the compétition among employers (some of them terminate pension plans in order
4
to reduce labour costs); the disappearance of small plans; and the high administrative costs
related to defined benefit pension plans.
Most pension plans are defined contribution plans but most pension plan members
are covered by a defined benefit plan. However, statistics show that there is increased
coverage in DC pension plans. In 1990, there were 5,109,363 RPP members, 8.4% of
which belonged to DC plans and 90.7% of which belonged to DB plans. In 2000, there
were 5,267,894 RPP members, 13.6% of which belonged to DC plans and 84.6% of which
belonged to a DB plan. In 2005, there were 5,669,858 RPP members, 15.6% of which
belonged to DC plans and 81.2% of which belonged to a DB plan. (In each year, the
percentages do not add up to 100%. The remaining percentage of members belonged to
hybrid and other types of plans. Besides, some numbers differ from those given in Table
1.2, not because of errors of transcription, but because of disagreeing sources providing
overlapping information.) We summarized this and some additional information in Table
1.3. (Source: Statistics Canada, CANSIM database, Table 280-0016.)
Table 1.3 Total number of RPPs and members by type of plan
Year Number of p ans Number of members
Year Total DB DC Total DB DC 1980 14,586 8,035 6,170 4,475,429 4,194,283 231,275 1986 21,094 8,215 12,637 4,668,381 4,295,691 325,320 1990 19,956 8,284 11,443 5,109,363 4,633,587 430,561 1995 15,845 6,990 8,609 5,169,644 4,582,154 518,669 2000 15,557 7,108 8,152 5,267,894 4,456,034 716,646 2005 15,336 7,561 7,485 5,669,858 4,604,775 885,840
There is no firm évidence as to what caused this trend (increased coverage in DC
plans) but some of the reasons mentioned earlier may apply hère too. The most important
could be the décline in employment in unionized industries and the public sector where DB
plans were dominant. Some possible reasons could be also that employers were in favour
of DC plans because they hâve more prédictive costs than DB plans and are less complex to
administer. Later, in the period from 2000 to 2003, the low interest rates and the stock
market décline led to déficits in DB plans that required additional payments from
employers and this likely further drew employers away from DB plans.
5
The traditional type of employment, where one's entire working history is tied to
one employer, is slowly disappearing. In today's compétitive job market (and job
globalization), employées tend to change jobs throughout their working career and they
want their retirement benefits to be portable from job to job. There are more part-time and
temporary types of employment (the latter are usually excluded from an employer's
pension plan). A DC pension plan might be more suitable to meet thèse employées' needs
since it is more flexible than a DB plan.
Improvements in médical science and improvements in standard of living hâve led
to increased longevity in many countries. This means that not only will people survive to
their retirement âge, but also that they will live longer during their retirement. The ratio of
retirées to the overall population is increasing and this puts financial pressure on both
private and public pension plans.
Volatile financial markets and the growing cost of funding requirements make it
less likely for employers to commit to a DB plan for new employées. Also, low interest
rates make deferred annuities more expensive.
Many countries modified their social security System. Some countries switched
away from the established DB model to a DC model, while others modified their DB model
by reducing the social security benefit level, delaying the âge at which workers are eligible
to receive benefits, increasing contribution rates, etc. Thèse changes in the social security
System put more emphasis on retirement benefits from employer-sponsored pension plans
and personal savings. And this is the reason why more people worry about the future of
their employer-sponsored plans, especially if of the DB type.
In this thesis, we focus on employer-sponsored pension plans and we examine
retirement benefits provided by différent types of pension plans to employées with différent
working historiés. Benefits from différent pension plans for each working history are
calculated and compared to the last salary that person received just before retirement; in
other words, the focus is on replacement ratios. We use two approaches to project the
wages: using a salary scale tied to âge, and using a salary scale tied to expérience. That has
an impact for people who hâve breaks in employment for, while they keep on aging, they
6
do not always gain expérience. One area of spécial attention is on benefits for employées
who change jobs frequently under a DB or a DC pension plan.
The thesis is divided into eight chapters, the first one being this introduction. In the
next chapter, we introduce the Canadian retirement income System, from its early âge to
today's setting. In the third chapter, we look at employer-sponsored pension plans in more
détails. In the fourth chapter, we discuss other authors' work that has served as the basis of
this thesis. The fifth chapter contains a description of the model used to simulate the
économie variables. The sixth chapter describes the équations used to calculate benefits
under différent types of pension plans. We discuss results in the seventh chapter and
complète this thesis with a conclusion.
Chapter 2 Canada's Retirement Income System In order to understand employer pension plans, the importance of thèse plans in the
retirement system and the effect of the switch to a DC type on employées, we fîrst need to
look at how the whole retirement System works and what it provides.
In this chapter, we focus on Canada's retirement income system. In Section 2.1, we
give a summary of the history of the retirement income system in Canada. This summary is
based on information gleaned from the ll t h édition of the Mercer Handbook of Canadian
Pension and Benefit Plans, published in 1996, and The History of Canada's Public
Pensions from the Civilization.ca web site. Then, in Section 2.2, we discuss the system as
it exists today. The description of the system is adapted from the 13th édition of the
Morneau Sobeco Handbook of Canadian Pension and Benefit Plans, published in 2005,
while actual figures (effective at the time of writing) were obtained from the Human
Resources and Social Development Canada (Service Canada) web sites.
2.1 How It Ail Started: A Walk Through History The first traces of the Canadian retirement income system reach ail the way back to
the late 1800s. Ever since, this system has been changing to meet the needs and demands
of government, workers and employers.
Before the industrialization era, most families lived and worked on their farm or
owned a small craft business. Older family members were usually able to do some light
tasks around the house or farm. Care and support for elderly members were the
responsibility of the family. Elderly that did not hâve family to support them had to live on
scarce, and very difficult to obtain, public assistance. Poor seniors had to do hard physical
work until they died unless they had a médical proof for work exemption, in order to obtain
this minimum assistance. With urbanization and other social and économie changes, the
traditional family unit changed, family needs changed and so did care and support for the
elderly. Many people left their farm and moved to the city to work in shops. A whole new
era had begun.
8
The period of the late 1800s and early 1900s was characterized by the
industrialization, the growth of the labour force and the increased population of elderly.
Ail thèse factors brought higher demands for pensions and care for the elderly. Only a
minority was covered by employer-sponsored pensions. Those workers who worked in
industry (a majority in those days), especially older workers, faced several problems related
to their retirement and pensions. The mechanization and the préférence for a younger
workforce threatened their job security. Layoffs and unemployment were common among
older workers. Thèse workers had to provide for their parents and their own old âge but the
layoff made that impossible because their pension benefits were lost upon layoff.
The fédéral government created programs and offered tax législations to stimulate
the establishment of pension plans by individuals or by employers and to encourage
individuals to save on their own. By offering tax incentives, the government tried to reduce
the pressure to increase the minimal income provided to elderly.
One of the first employer-sponsored pension plans dates back to 1874 when the
Grand Trunk Railway introduced a plan for its workers. Other employers such as the
fédéral public service and some banks also offered pension plans to their employées. The
problem with thèse early employer pension plans was that they generally were neither
affordable nor available to ail workers.
In 1887, the fédéral government established the Pension Funds Societies Act which
gave the right to employées to set up a pension fund to which an employer might or might
not contribute. Then, in 1908, the Government Annuities Act was established to encourage
individuals to save for their retirement through the purchase of government annuities.
However, few could afford to finance their retirement be it with their employer or on their
own.
In 1917, the fédéral government introduced income taxes through the Income War
Tax Act program (later known as the Income Tax Act (ITA)). Taxes then became a source
of revenue for the government to develop national social programs. A year later, the
Pension Act was passed to provide compensatory benefits to the survivors of the soldiers
killed in the war. In 1919, the ITA allowed employée contributions to be tax-deductible.
9
Then in 1927, the government passed the Old Age Pensions Act that provided
income to persons 70 and older, subject to 20 years of residency. This program was subject
to a means test. This test was used to détermine the senior's income (or means) from ail
possible sources of income even if no actual income was received from a particular source.
For example, income from property was assumed in the calculation even if, in reality, the
property did not generate any income. If the assumed income exceeded a certain amount, a
senior would not get any assistance, even if much needed in actual fact.
In the 1930s, the Great Dépression hit Canada, bringing with it severe
unemployment and poverty that affected the entire country. World War II helped the
recovery of the economy; as a resuit, employment increased, income increased, but so did
priées and inflation.
The period from 1940 to 1952 was a harsh period for seniors who received income
from either Old Age Pension, employer pension or government annuities because inflation
was eroding the purchasing power of their pensions. This was a time when many sought
more active government involvement in the social security System that would give
protection against poverty for everyone and greater government intervention in the
economy. The government slowly responded by introducing Unemployment Insurance in
1940 (that provided benefits to unemployed workers) and Family Allowances in 1945.
The period after the war was characterized by a stable economy, growing service
sector jobs which offered higher salaries than industry jobs, a slower industrialization
process, stronger and more influential unions, the baby boom and, as a result of the baby
boom, a lower percentage of older people.
In the late 1950s, however, the economy slowed down. Again, many were
demanding national social programs that would provide a basic, guaranteed level of
économie security in old âge. As a resuit, in 1952, the Old Age Security (OAS) Act
replaced the Old Age Pensions Act. OAS was not a measure of income replacement but a
source of basic income that provided benefits to ail men and women 70 years of âge and
older who had lived in Canada for 20 years. Eligibility for benefits was not means-tested.
Employer-sponsored pensions and private savings were supposed to supplément this
10
amount. Some workers had employer-sponsored pension plans, but thèse plans were not
flexible. In order to be eligible to receive a pension from an employer, a worker had to
spend his or her entire career with the same employer, working full-time, and, if the plan
was contributory, making contributions to the plan throughout his or her employment years.
Employer-sponsored plans usually did not provide survivor benefits or médical benefïts
even if the death or injury was job-related (caused by or occurring at work). Unions asked
for a public pension plan that would be portable from job to job, which would help provide
replacement income and maintain a standard of living at retirement, and which would also
hâve survivor and disability benefits.
To improve retirement replacement income, in 1957, the government introduced
Registered Retirement Savings Plans (RRSPs), arrangements for self-employed people and
employées without an employer-sponsored pension plan to save for their retirement. But
poverty was still the main problem for old people and the standard of living of many
retirées was low.
In 1966, the Canada Pension Plan and Québec Pension Plan (C/QPP) were
established, partly as a resuit of union demands. Thèse plans were compulsory,
contributory plans for ail workers in the workforce between the âges of 18 and 70. At
retirement, every contributor was eligible to receive C/QPP benefits (whether or not his or
her working career was tied to only one employer).
Despite the économie growth and low unemployment of the 1960s, seniors were
still poor. To address this problem, in 1967, the Guaranteed Income Supplément (GIS) was
established to help low-income récipients of OAS. Then, in 1975, the Spouse's Allowance
(now Allowance) and, in 1985, the Widowed Spouse's Allowance (now Allowance for
Survivors) were established to provide benefits for spouses and widowed spouses,
respectively, of OAS (and GIS) récipients and deceased OAS (and GIS) récipients. In
1989, the ITA imposed a tax, known as a clawback, on the OAS benefits. The period going
from the mid 1960s to the late 1980s was characterized by changes in the fédéral
government's retirement income programs recognizing social changes that marked that
period such as gender equality and women's libération. Economy had its downs (early
1980s) and its ups (mid 1980s) over that period.
11
High unemployment characterized the early 1990s (with a peak of 12% in 1992).
By 1999, the unemployment rate had gone down to about 8%.
By 1997, the social security System, which includes the public pension System, had
substantially improved the seniors' situation. In 1997, 19 percent of seniors had low
income compared to 34 percent in 1980.
But then a new type of poverty had emerged in the society: poverty among single-
parent families. Also, a new question had arisen: is public pension going to be strong
enough to support seniors given that life expectancy is increasing, the population of seniors
is growing, the baby boom génération is slowly but surely approaching retirement, and, in a
not so distant future, the number of workers contributing to C/QPP will be decreasing? The
government responded by increasing contributions to the C/QPP in 1998 (some speculate
that the later retirement âge eligibility will follow soon due to an increase in life
expectancy) and pointing out the importance of private pensions (as it had always had) such
as employer-sponsored pensions, RRSPs and private savings.
2.2 Retirement Income System Today Today's retirement System has three layers:
1. government programs,
2. employer-sponsored pension plans, and
3. personal savings.
As was mentioned earlier, retirement income from employer-sponsored pension plans can
be referred to as private pensions to distinguish it from income from the government
referred to as public pensions.
2.2.1 Government Programs
Government sources of retirement income are Old Age Security (with Guaranteed
Income Supplément, Allowance and Allowance for Survivors) and Canada and Québec
Pension Plans.
12
2.2.1.1 Old Age Security
Old Age Security (OAS) provides minimum retirement benefits to ail people aged
65 and older who hâve lived in Canada for at least 10 years after their 18th birthday. No
contribution (from a potential récipient) is required; benefits are paid from the fédéral
government's revenue fund. OAS can be received outside of Canada, for a period of six
months, if the person lived in Canada for 20 years after his or her 18th birthday. OAS
benefits ($491.93 per month for the January-March 2007 period) are reduced for those
whose net income (before adjustment) exceeds a certain amount. The réduction is 15% of
the différence between the net income and that threshold ($63,511 per year for the January-
March 2007 period). If the net income for the January-March 2007 period was $102,865 or
more per year, OAS benefits were lost completely. Income from the OAS pension is
taxable and indexed quarterly to the cost of living.
The Guaranteed Income Supplément (GIS) provides benefits to OAS récipients with
low income. This is an income-tested benefit, so GIS benefits are reduced by $1 for each
full $2 of income from sources other than OAS. GIS is not paid if the annual income
exceeds a certain amount ($35,712 for the January-March 2007 period). The maximum
monthly GIS benefits are the same for a single person and a person whose spouse or
common-law partner does not receive either an OAS pension or an Allowance benefit
($620.91 for the January-March period 2007). The maximum monthly GIS benefits
received by a person whose spouse or common-law partner receives either an OAS pension
or an Allowance benefit are lower ($410.04 for the January-March 2007 period).
The Allowance provides benefits to the spouse or common-law partner aged 60 to
64 of an OAS and GIS pensioner with low income. No Allowance is paid if the annual
income exceeds a certain amount ($27,600 for the January-March 2007 period). The
Allowance benefits are reduced by $3 for every $4 of the combined couple's income from
sources other than OAS until the amount of the réduction is equal to the OAS pension;
further on, the réduction is $1 for every $4. (The maximum monthly Allowance benefit for
the January-March 2007 period was $901.97.)
13
The Allowance for the Survivor provides income to a widowed spouse or comraon-
law partner aged 60 to 64 of a low-income OAS and GIS pensioner. A widowed spouse or
common-law partner is not eligible for Survivor's Allowance benefits if the individual's
maximum annual income exceeds a certain amount ($20,064 for the January-March 2007
period). (The maximum monthly Survivor Allowance benefit for the January-March 2007
periodwas $999.81.)
The eligibility condition for GIS and Allowance and Allowance for the Survivor is a
minimum of 10 years of residency. Unlike OAS, GIS and Allowance and Allowance for
the Survivor are tax-exempt. However, they too, like OAS, are indexed quarterly to the
cost of living.
2.2.1.2 Canada and Québec Pension Plans
The Canada and Québec Pension Plans (C/QPP) are contributory, earnings-related
pension plans. Contributions are compulsory. Both employée (aged 18 or over and in a
workforce) and employer contribute to C/QPP; someone who is self-employed contributes
both as employée and as employer. C/QPP provides retirement benefits based on the
contributory period and amount contributed to C/QPP. C/QPP benefits normally start at
âge 65 in which case the pensioner receives his or her full benefits. A pensioner may elect
to receive his or her C/QPP benefits starting as early as âge 60 but the benefits will be
reduced by 0.5% for each month before his or her 65* birthday. If a pensioner starts to
receive C/QPP benefits after he or she turns 65, then the benefits will be increased by 0.5%
for each month after his or her 65th birthday. C/QPP also provides disability benefits,
survivor benefits, and death benefits.
2.2.2 Employer-Sponsored Pension Plans
There are two catégories of employer-sponsored pension plans: registered and non-
registered. Registered pension plans (RPPs) enjoy a spécial tax treatment. They are
registered with the Canada Revenue Agency as well as fédéral or provincial pension
authorities. As such, they must comply with the Income Tax Act and other pension
régulations. Non-registered plans (also called supplementary pension arrangements) do not
enjoy the tax benefits reserved for registered plans and they are not as regulated as RPPs.
14
The most common employer-sponsored pension plan is an RPP. RPPs require
employers to contribute. If the plan is contributory, employées make contributions as well.
(A plan is said to be contributory if employées are required to contribute a portion of their
salary for their benefit accrual.) RPPs provide regular income to RPP members once they
are retired.
Since employer-sponsored pension plans are the focus of this thesis, we will say
more about them in the next chapter, entirely devoted to such plans.
2.2.3 Personal Savings
There are plans that allow individuals to save for their retirement on their own.
Registered Retirement Savings Plans (RRSPs) are such plans. Thèse plans were
established for people who did not hâve pension plans. Today, even if a person participâtes
in an employer-sponsored plan, she or he still can hâve an RRSP in addition to the
employer's plan.
RRSPs really became popular and interesting investment vehicles only in the 1990s,
when new rules increased the contribution limit. An individual can contribute up to 18% of
his or her earned income from the previous year to his or her RRSP (up to $19,000 for
2007). Contributions are tax-deductible and ail invested monies accrue tax-free. The
RRSP has to be converted into a stream of income by the end of the year in which a person
turns 71 (it changed from 69 to 71 in 2007). The RRSP can be converted into an annuity, it
can be commuted into a Registered Retirement Income Fund (RRIF), it can be withdrawn
as a lump sum, or any combination of the three. When the RRSP is converted into an
income, a person has to pay income tax on the amounts that are withdrawn. The marginal
tax rate then applicable is typically lower than that applicable when the contributions were
made (income is generally lower at retirement; hence a lower tax is applicable on that
income).
People can also choose to invest in bonds, stocks and other investment instruments
outside RRSPs. The différence between RRSPs and this type of savings is that
contributions are not tax-deductible and tax is paid on investment income as it is realized.
There is however no tax payable on the amounts that are withdrawn.
15
The ownership of a house also counts as a major investment for many people. It
benefits from a spécial tax treatment and can represent a significant portion of retirées'
assets. There even are financial products, called reverse mortgages, which are designed to
turn part of the equity of the house in the home into an annuity.
Chapter 3 Employer-Sponsored Pension Plans: A Closer Look
In this chapter we discuss employer-sponsored pension plans in more détail.
The fundamental objective of a pension plan is to provide an income, namely a
pension, to the plan member or participant at retirement. In addition, the plan may, for
instance, provide benefits payable in the event of total disability before retirement, or in the
event of death before or after retirement. Those benefits are called ancillary benefits, as
they are meant to be secondary to the pension benefits.
Some employers establish pension plans because their competitors provide pension
plans or as a resuit of collective bargaining. Establishing a (good) pension plan improves
the employer's compétitive position because it makes it easier to recruit and retain skilled
employées and it facilitâtes the retirement of employées who reach retirement âge (which is
désirable insofar as they also are less productive or redundant employées).
If the employer's contributions to a pension plan are based on the company's profits
(such as in profit-sharing plans), this may encourage employées and lead to increased
productivity since they benefit from the company's profits, hence enhancing the company's
productivity, profits and competitiveness. So both the employer's and the employée's
needs are met.
There naturally is a cost associated with establishing a pension plan, but if the plan
compiles with regulatory requirements, there are advantages of having such a plan. Some
of the advantages are tax-free accumulation of investment earnings; also, if the plan is
contributory, contributions by members reduce the cost of pensions for the employer.
For an employée, a good pension plan (registered and funded) gives the peace of
mind of having a secure income at retirement. The pension offered by the employer
together with the pensions received from public programs should be adéquate for a retirée
to enjoy a standard of living that is commensurate with what she or he was enjoying prior
17
to retirement. One advantage of participating in an employer's pension plan is that the
pension benefits are not taxable until they are actually received.
3.1 Types of Pension Plans There are two basic types of pension plans:
1. defined contribution (DC) plans, and
2. defined benefit (DB) plans.
In Canada, the number of plans of either type is about equal. However, most
pension plan members are covered by a defined benefit plan. As can be seen in Table 3.1,
according to Statistics Canada (CANSIM database, Table 280-0016), in 2005, 81.2% of ail
pension plan participants were covered by DB plans. The number of defined contribution
plans and defined benefit plans at that time was 7,485 and 7,561 respectively. From 1995
to 2005, increased defined contribution plan membership is observed, while defined benefit
plan membership is more or less stagnant. Over the same period, it may be surprising to
realize that, while the number of DC plans has decreased, the number of DB plans has
increased. (A look at Table 1.3 confirms that this trend differs from trends observed
earlier.)
Table 3.1 Total RPPs and members by type of plan
Year Number of plans Number of mem ?ers
Year Total DB DC Total DB (%) DC (%) 1995 15,845 6,990 8,609 5,169,644 4,582,154 88.6 518,669 10.0 2000 15,557 7,108 8,152 5,267,894 4,456,034 84.6 716,646 13.6 2005 15,336 7,561 7,485 5,669,858 4,604,775 81.2 885,840 15.6
3.1.1 Defined Contribution Plans
Defined contribution plans define in advance what the employer (and the employée,
if the plan is contributory) will contribute each year. An individual account is maintained
for each plan participant and the amount in the account, including investment income, is
used to provide income at retirement. Plan members bear the investment risk. That is, plan
members individually bear the risk of low investment returns on their assets and, as a
conséquence, the risk of an inadéquate pension at retirement.
There are two types of DC plan:
18
• Money purchase plans - Such plans hâve a formula that predefines required
employer and employée contributions. Thèse plans may be non-contributory, in
which case employer contributions are a fixed percentage of the employée's
earnings or some fixed amount per unit of time worked, or they may be
contributory, in which case both employer and employée contributions are
defined as a percentage of earnings or an amount per unit of time worked.
• Profit-sharing plans - In such plans, only the employer makes contributions that
are a function of the company's profits in proportion to the employée's earnings.
In defined contribution plans, pension benefits dépend on the amount accumulated
in the individual account and annuity rates available just before retirement (at the time of
conversion). The fact that annuity rates vary over time adds to the risk faced by DC plan
participants.
There is a kind of plan which, without promising any spécifie benefit at retirement,
sets the annual contribution rate so that, as per assumptions, at retirement, the accumulated
fund will provide for a projected retirement benefit. Such plans are called target benefit
plans. Contribution rates change as assumptions are revised. However, there is no
guarantee made as to the actual retirement benefit that the accumulated fund will afford.
Hence, the key différence between target benefit plans and money purchase plans is the
way the contribution rate is set.
There is yet another kind of plan which, without being a pure defined contribution
pension plan, has more in common with DC plans than with DB plans: the cash balance
plan. Cash balance plans eliminate ail or part of the investment risk during the
accumulation phase by guaranteeing the rate of return on the fund, either as a fixed rate or
based on an index.
19
3.1.2 Defined Benefït Plans
Defined benefït plans use a formula to détermine the retirement benefits (at the time
of retirement). The plan sponsor's contributions are a function of the cost of providing the
promised benefits, taking employée contributions into account if the plan is contributory.
Employer contributions vary from year to year. The employer often bears ail the
investment risk (in some plans, however, the employée contribution rate is variable and
thus employées also bear part of the investment risk).
A formula that is used to calculate benefits in defined benefit plans is called a
benefït formula. Benefït formulas may be classified in two catégories:
1. unit benefit,
2. fiât benefit.
3.1.2.1 Unit Benefit Formula
Under the unit benefit formula, an explicit unit of benefit is credited for each year of
service with the employer, expressed as a percentage of compensation for each year of
service, such as 1.5%, or as a spécifie dollar amount for each year of service, such as $30
per month. The latter approach is popular with unions and this type of unit benefit is
subject to union negotiation and upgrades in formula (dollar amount). Because this flat unit
benefit ignores any différence in earnings, upgrades must take place in the benefit formula
in order to reflect inflation as well as wage increases.
For example, assume Joey has 30 years of service, a career average compensation of
$30,000, and a pension benefit accrual of 1.5% of his compensation. He then would
receive an annual pension of
1.5% x $30,000x30 = $13,500. (3.1)
If Joey's plan uses a spécifie dollar amount, say $37.50 per month for each year of
his 30-year service, he would receive an annual pension of
$37.50x30x12 = $13,500. (3.2)
It may seem as though there is no différence between the specified percentage
benefit formula and the specified dollar benefit formula based on this particular case but
20
this is not the case in gênerai. Any of the components determining a pension, a percentage
of compensation, a spécifie dollar amount, a compensation amount, or years of service, can
vary and any variation in the components will lead to a différent pension.
For example, assume Joey's friend Amy is a member of a plan that offers $40 per
month for each year of service. If Amy also worked for 30 years, she would receive an
annual pension of
$40x30x12 = $14,400. (3.3)
When benefits are expressed in ternis of compensation, such as under the unit
benefit formula with a spécifie percentage rather than a spécifie dollar amount, it is
important that the plan clearly indicates which parts of compensation will be included in
the earnings base. This means that the plan must indicate whether or not commissions,
bonuses, overtime pay, sick pay are included in the earnings base. Because the earnings
base for the calculation of benefits under a unit benefit type formula varies, the unit benefit
formula is subdivided into two classes according to the base used for the calculation of
benefits. They are
1. career average formula, and
2. final average formula.
3.1.2.1.1 Career Average Formula
Under the career average formula, the unit of benefit credited during any particular
year of employment is based on the employée's compensation during that year.
More generally, if the individual's pay at âge x is Payx, and the accrual rate crédit
for âge x is ax, then the total pension at retirement for an individual hired at âge H and
retiring at âge R is
^xPayx. (3.4)
If ax is a constant, a, for ail âges (as it was in Joey's case), then the expression can
be rewritten as
21
«-1 R-\ ï>^ ^a-Pay^a^Pay^a-iR-H)-^—-. (3.5) x=H x=H (K-H)
We see that the retirement benefit under the career average formula is the product of an
accrual rate a, the number of years of service(R-H), and the average pay over the
individual's entire career a-]
^ . (3.6) (R - H)
3.1.2.1.2 Final Average Formula
Under the final average formula, benefits accrue on the basis of the participant's
average compensation during a specified period, such as 3, 5, or 10 years immediately
before retirement (the formula is then called a final 3-year average, a final 5-year average
or a final 10-year average, respectively). Such a formula is better designed to maintain a
certain standard of living than a career average formula, since early salaries are typically
much lower.
Focusing on the last few years before retirement is done on the premise that
compensation typically rises over time and on the additional assumption that the best
earnings are the final ones. That is not always the case however. An alternative formula
uses the years (or months, as specified in the plan document) producing an individual's
highest average compensation. The years (or months) entering the average may be taken
from the entire career or only over a specified period, such as the best five years during the
10 years before retirement. Moreover, the years (or months) may or may not hâve to be
consécutive. This alternative formula is called the best average formula.
Say a plan uses a final five-year average formula. Then, at retirement, the final five-
year average pay is the sum of the last five years' earnings divided by five. The pension
under a final five-year average plan for an individual with an accrual rate bx in year x is
calculated as follows:
22
A-, I>^ I V ^ — (3.7)
If è* is a constant, è, this can be written as follows: R-\
b-(R-H)-^^ . (3.8)
We see that the retirement benefit under the final five-year average formula is the product
of an accrual rate b, the number of years of service (R - H), and the final five-year average
pay,
What would Joey's pension be if his plan were using such a final average formula?
Let us assume that Joey's final average compensation (the final five) is $50,400.
With his 30 years of service and a pension benefit accrual of 1.5% of compensation
regardless of âge, he would receive an annual pension of
1.5% x $50,400x30 = $22,680. (3.10)
Similarly, we can calculate benefits for a pension plan using a best average formula.
Assume the plan uses the best average five years formula (no specified référence period, no
consecutiveness requirement). Then, ail we need to do is to identify the five highest
earnings in his career, average them, then multiply by the accrual rate and the number of
years of service.
The average earnings used to détermine the pension benefit are higher under the
final average formula than under the career average formula. A pension calculated using
the career average formula for an employée who has been promoted during her or his
working career is likely to be particularly déficient with respect to her or his retirement
needs since this formula gives equal weight to ail of her or his earnings received over time
(even though the earnings hâve changed appreciably over the course of her or his career,
23
each step 'up' leading to higher earnings). A pension calculated using the career average
formula will not be sufficient either for those employées whose career was marked by a
long period of inflation. To address this problem, plan sponsors can use yet another benefit
formula, the modified career average formula.
3.1.2.1.3 Modified Career Average Formula
Under the modified career average formula, the benefïts are based on career average
earnings, but they are reviewed to détermine whether salary increases hâve been such that
the pay base for past service benefïts is out of date, and if the plan sponsor can afford to
update past service benefïts.
This type of formula gives the same total benefit as that for the final average
formula if the modified career average formula is continually updated, but plan sponsors
typically do not practice continuous updates. Most plan sponsors provide ad hoc past
service updates (and do not commit to future updates).
So, there is hope that Joey ends up with an annual pension that is greater than
$13,500, though perhaps not reaching $22,680, whether his plan uses a career average
formula or a specified dollar amount, insofar as his employer may provide ad hoc increases.
3.1.2.2 Fiat Benefit Formula
The other category of benefit formula, the flat benefit formula, disregards the length
of service, so long as the minimum period of service specified under the plan has been
completed. One example of such a formula provides a specified percentage of
compensation, such as 60%, at retirement. This type of formula typically is used to provide
a minimum guaranteed benefit based on a minimum service period.
In this case, assuming Joey satisfies a minimum period of service, the plan may
stipulate that the annual pension will be 60% of his career average compensation (which we
hâve already assumed to be $30,000), so he would receive an annual pension of
60% x $30,000 = $18,000. (3.11)
24
The fixed benefit, a type of flat benefit formula, is not a function of compensation or
service. This type of formula provides a flat dollar benefit to ail employées who satisfy a
minimum period of credited service, such as $30,000 per year. A fixed benefit is rarely
used today because it has an arbitrary accrual of benefit, and when it is used, it serves to
provide a minimum guaranteed benefit on a minimum service period when an alternative
benefit formula would provide less.
For example, the plan in which Joey participâtes may specify that Joey's annual
pension benefit is to be $10,000 or 1.5% of career average compensation for each year of
service, whichever is greater. Hence, the flat fixed benefit guarantees $10,000. Using the
unit benefit, the pension amount is
1.5% x $30,000x30 = $13,500. (3.12)
Hence, because the career average formula yields an amount greater than the guaranteed
minimum, Joey would receive an annual pension of $13,500.
Now let us assume that Joey has worked for only 6 years, which is taken to be more
than the minimum period of service as determined in the plan document. Again, the flat
fixed benefit guarantees $10,000. Using the unit benefit, the pension amount is
1.5%x $30,000x6 = $2,700. (3.13)
Because of the minimum guaranteed pension of $10,000, Joey ends up getting a "better
deal", $10,000, instead of $2,700.
Some plan sponsors create defined benefit plans in which the employer pays for the
basic benefit pension plan and employées pay for additional ancillary benefits, such as
unreduced early retirement and automatic indexation. Thèse plans are called flexible
pension plans. They are more complex and beyond the scope of this thesis.
3.2 Which Plan Is Better, DB or DC? Whether a DB plan or a DC plan is better dépends on the employée's âge, years of
service and career path. For younger employées who are more likely to frequently change
jobs, a defined contribution plan could be a better choice because the amount of benefits
that is in the account, at the time of termination, belongs to the employée. For employées
25
with long service with the same employer and close to retirement, a defined benefit plan
likely would be a better choice. Employées close to retirement are more protected with the
defined benefit plan because their employer guarantees their pension and the benefits are
often a function of final average earnings.
One of the key différences between the two types of plans is investment risk. For
those employées who are risk averse, a DB plan might be a better choice because the
employer retains the liability for paying promised benefits and ail financial risk stays with
the employer. In DC plans, this risk is transferred from the employer to the employée.
Those employées who can make sound investment décisions and who are ready to accept
investment risk might do as well or even better in a DC plan than a DB plan.
Another risk that is transferred from the employer to the employée is longevity risk.
This risk is becoming more important because of increasing life expectancy. Under a DC
plan, employers do not promise retirement benefits for life, as it is the case under a DB plan
(which is one of the reasons that employers switch their plans from DB to DC). Therefore,
DC plan retirées will bear greater risk of outliving their savings if they do not buy an
annuity at retirement or of having to pay more than anticipated for the annuity if they buy
one.
The différence between DB and DC types of plans and the question as to which of
the two is better will be further addressed in the rest of the thesis, as the main focus of this
thesis is on the distribution of retirement benefits, as a percentage of final salary, under the
two types of plans.
3.3 When to Retire? The retirement âge is specified in the pension plan, and there essentially are three
retirement âges: normal, early and postponed.
At normal retirement, the employée has the right to retire and receive a full,
unreduced pension, as determined by the benefit formula. (Ail the pensions calculated in
the previous subsection were pensions payable starting at the normal retirement âge.) The
normal retirement âge typically is 65. Some plans also offer a full pension when the sum of
26
the number of years of service and the âge reaches a certain number, say 80 (attained, for
example, with 60 years of âge and 20 years of service), or after a certain number of years of
service (30 for example), regardless of âge.
The early retirement is the time when the employée can retire before normal
retirement âge, usually within 10 years of the normal retirement âge. The employée then
has the right to a reduced pension. The réduction is justified because the payments start
earlier and are going to last longer compared to retirement benefits starting at normal
retirement âge. The early retirement pension often is the normal retirement pension
reduced by some factor, such as 6%, for each year of early retirement. The early retirement
pension could also be calculated on the basis of the actuarial équivalent of the pension the
employée has earned up to the early retirement date and that would be payable starting at
normal retirement âge. Actuarial équivalence often would resuit in much larger réductions
than the factors set out in the plan document.
For example, say Joey, who is 60, has accrued a pension of $1,500 a month starting
at 65, but he wants to retire now. Assuming the early retirement factor set out in the
pension plan is 6% per year, he would receive a pension of 70% of the full amount (6%
réduction x 5 years before normal retirement = 30%), that is, 70% x $1,500 = $1,050 per
month.
The employée can choose to retire after the normal retirement âge, that is, she or he
can choose to postpone retirement, up to some maximum âge which may be stated in the
plan document or in accordance with governing pension and tax régulations. Pension plans
may allow benefits to continue to accrue after normal retirement âge. If this is the case, the
pension as determined using the formula defining the normal retirement benefit will be
higher when retirement actually takes place than it would hâve been at the time of normal
retirement, assuming service keeps accruing or référence pay increases. Alternatively,
because a participant who postpones retirement will receive benefits over a shorter period
of time, an adjustment may be made, thus increasing the pension to be received. Not unlike
early retirement factors, the adjustment for postponed retirement may be defined as a
percentage per year of postponement or obtained by actuarial équivalence.
27
3.4 Benefit Payments The main purpose of a pension plan is to provide a life income during retirement.
The payment of the pension benefits is usually provided in the form of a life annuity
if the benefits are guaranteed (or underwritten) by a life insurance company, or directly
from the trust fund if the plan is funded through a trust (although an annuity contract from
an insurance company can be purchased by the trust fund at the time of the employee's
retirement).
The most common form of life annuity in non-contributory plans is the straight life
annuity (which provides monthly payments for as long as the annuitant is alive) (McGill et
al. 1996). Contributory plans often offer some kind of guarantee so as to avoid, or at least
minimize, the impression some may hâve that the plan has ripped off a participant who
happens to die shortly after retiring. Under a modified cash refund annuity, if the
participant dies before receiving benefits equal to the accumulated value of her or his
contributions, with or without interest, the différence between the benefits received and the
accumulated value will be given in a lump sum to the participant's beneficiary or estate. It
is more common, however, that contributory plans offer a life annuity with payments
guaranteed for a certain number of years, say 5 or 10, because it is easier to understand and
this will basically ensure that the participant's accumulated contributions are returned
(Whiston and Gottlieb 2005).
For a member who has an eligible spouse, every pension plan must provide benefits
payable in the form of a joint and survivor annuity continuing to the spouse, unless the
member chooses some other form with the spouse's written consent. In most jurisdictions
that require a joint and survivor annuity, the minimum pension payable to the surviving
spouse is 50% or 60% of the pension previously payable to the retired member. The
gênerai rule is that the pension payable after the first death of the member and spouse
should not be less than 60% of the pension payable before the first death. An exception to
this rule is Manitoba where the joint and survivor annuity is a minimum of 66%% of the
member's pension. Note that, while pension plans must make joint annuities available to
members that hâve a spouse, that annuity form needs not be the default one for which the
28
benefit formula is meant to be used. Hence, if the default annuity form is the single life
annuity, the pension payable in the form of a joint annuity will not be equal to the benefit
derived using the benefit formula; rather, it will be determined by actuarial équivalence
(and be less than that payable in the form of a single life annuity).
The amount calculated using the benefit formula, along with any adjustments
needed for retirement at an âge other than the normal retirement âge or for an annuity form
différent from the default one, often is an amount per year. Since the usual payment
frequency is monthly, the first amount a pensioner receives is that amount divided by 12.
Besides, that amount does not necessarily remain the same over the whole life of the
pensioner. In periods of high inflation, a constant pension would make the pensioner
particularly vulnérable to an érosion of her or his purchasing power.
As was mentioned earlier, ad hoc increases may be granted by the employer, and
those ad hoc increases typically would benefit both the current employées and the ones who
hâve retired already.
As an alternative, a plan document may allow for systematic indexation of the
benefits. When offered by a plan, the indexation usually is with respect to the Consumer
Price Index (CPI), even though it could be with respect to a wage index or be some fixed
percentage. Moreover, the indexation is not necessarily full; a fully indexed pension
increases exactly in line with the CPI. There are several ways of offering partial
indexation: the CPI increase above a minimum percentage; the CPI increase up to a
maximum percentage; a fraction of the CPI increase; or any combination thereof. When
provided for in the plan document, indexation typically is done on an annual basis.
Pension plans, although only under rare circumstances (such as reduced life
expectancy), may also pay the pension benefits in a lump sum.
3.5 Intégration with Canada/Québec Pension Plans (C/QPP) Pensions from private plans are usually integrated with pensions from government
sources. This is because the benefits from the Canada/Québec Pension Plans (C/QPP) play
29
a significant rôle in achieving the 70% income replacement ratio, which is commonly
assumed to be sufficient income replacement at retirement.
The most common method of integrating a pension plan with C/QPP is the so-called
"step rate" method. Under this method, there are two accrual rates for benefits and
contributions:
• lower rates apply to annual earnings that are covered by C/QPP (up to the
Year's Maximum Pensionable Earnings (YMPE)), such as 1.4% for benefits and
4.2% for contributions; and
• higher rates apply to annual earnings above the YMPE, such as 2% for benefits
and 6% for contributions.
For example, assume Joey is 65 and has 35 years of service. We assume that he is
eligible for normal retirement as defined in the pension plan, which is a final salary plan.
We also assume that
• his last salary was $80,000;
• the last YMPE was $43,700 (which is the YMPE for 2007); and
• he qualifies for the C/QPP maximum benefit, $10,365 per year (as of 2007).
Hère is the détail of the calculation of the annual pension provided by the pension
plan:
Pension on earnings up to YMPE 1.4% x $43,700 x 35 = $21,413
Pension on earnings above the YMPE 2% x $36,300 x 35 = $25,410
Total pension $46,823 (3.14)
As he is also eligible for the C/QPP maximum benefit of $10,365 per year, his total annual
pension based on his career earnings is
$46,823+ $10,365 = $57,188. (3.15)
Hence, Joey gets $57,188 per year, which represents 71.5% of his pre-retirement earnings
($80,000), which is slightly more than 70%.
30
If he were in a final salary plan with a 2% accrual rate and no intégration, the total
annual pension he would get based on his career earnings (sum of pensions from the
employer's plan and C/QPP) would be determined as follows:
Pension plan benefits 2% x $80,000 x 35 = $56,000
C/QPP maximum $10,365
Total $66,365 (3.16)
This represents 83.0% of $80,000, which is much more than 70%. Very few employers are
so generous. They integrate their pensions with public pensions in order not to commit to
more than needed.
Ail provincial pension régulations prohibit intégration with Old Age Security and
hâve done so for a certain number of years, but this intégration is still permitted for the
years of service prior to the year in which the régulation came into effect.
3.6 Quitting Before Retiring None of the examples presented so far hâve dealt with the situation in which an
employée does not start receiving a pension immediately after leaving his or her employer.
It was more or less assumed that the employée was ceasing to work at some âge past the
earliest possible retirement âge in order to retire and start collecting a pension.
Let us consider the case when the employée leaves the employer before being
eligible for early retirement. What the employée is entitled to dépends on the vesting rules.
Full vesting (complète right to pension benefits accrued thus far) may be immédiate or be
granted after a certain number of years or after a certain âge, or a combination of the two.
In Canada, currently, vesting after two years of plan participation is quite common; in
Québec, immédiate vesting has been the norm for a few years. Partial vesting (right to a
portion of the pension benefits accrued thus far) may be in effect before full vesting is
achieved if vesting is graduai. With cliff vesting, however, if a participant leaves the plan
before having met the requirements for full vesting, ail rights are forfeited.
If the plan was a DC plan, what the participant is entitled to if fully vested is the
value of her or his contributions, as well as the value of the contributions the employer has
31
made in her or his name, accumulated with interest. The participant could leave her or his
money in the plan until actually retiring. Meanwhile, the money would keep on growing
thanks to the returns realized as per the fund's investment policy or, if the plan is so
designed, as per her or his asset picks. The participant could also withdraw the money that
has accumulated so far in her or his name and take it out as a lump sum. That is the option
of choice for many, since it gives them more latitude in their investment choices and it also
avoids having to keep track of one more plan until retirement. That lump sum still has to
be used to provide retirement income though, in keeping with the fundamental objective of
a pension plan. To ensure that indeed the money will be used to that end, a lump sum has
to be transferred to a locked-in retirement account. Such an account has strict rules as to
when money can be withdrawn and limits (both lower and upper) on the amount that can be
withdrawn in any given year.
In the case of a DB plan, if fully vested, the participant is entitled to a pension
deferred to the normal retirement âge. The amount then payable is calculated using the
benefit formula, based on the service and earnings with that employer. In the gênerai
formulas that we hâve seen, we would replace âges H and R by âges B and E, respectively.
Age B would be the âge at which the employée began working for that particular employer.
Age E would be the âge at which the employée stopped working for that particular
employer. For example, for a career average plan with constant accrual rate a, the (initial)
pension benefit would be given by £-1
J>/V% ^P^y^a-iE-B)-^-—, (3.17)
while, for a final «-year average plan with constant accrual rate b, the (initial) pension
benefit would be given by
£*% b-(E-B)- ' = "■»<*•*•-"> . (3.18)
min(E-B,n) As always, the benefit formula yields an amount that is to be paid provided that the annuity
form is the default one. Any other annuity form will require the calculation of the benefit
by actuarial équivalence.
32
With a final average plan or best average plan, a participant that leaves the plan
before retiring foregoes any benefit increase not only through longer service but also
through higher average earnings effectively affecting the entire service. That may represent
an important loss of value. As a way to mitigate that loss, some plans revalue the pension
in deferment using, for example, a wage index. The same benefit formula is used as a
starting point, but the resulting amount is increased to reflect increases in the chosen index
between âge E and normal retirement âge.
If the plan allows, the employée may ask that her or his pension start being paid at
some other âge among the possible retirement âges. The pension amount is then adjusted
and the applicable adjustment may be the same as if working up until retirement or it may
be less generous.
The plan may also allow the participant who is leaving to take with her or him the
actuarial présent value of the pension benefits she or he otherwise would be entitled to start
receiving at the normal retirement âge. That value may be transferred to another pension
plan or, as in the case of DC plan, to a locked-in retirement account.
In theory, even if an employée quits her or his job after the earliest possible
retirement âge, she or he might prefer to begin receiving pension benefits at a later âge. If
allowed by the pension plan, that case is comparable to the case of the employée leaving
before having attained the earliest possible retirement âge. However, it is not uncommon
for DB plans to begin paying benefits automatically, right after an employée leaves her or
his job past the earliest retirement âge. (Surely, pension benefits will begin automatically,
even if the employée keeps working, at the latest possible retirement âge.)
3.7 A Word on Régulations In order to become a registered pension plan and enjoy a spécial tax treatment, a
pension plan must comply with certain régulations. Many of thèse régulations are created
to protect plan members. There are two forms of régulation: régulations on pension plan
standards and its opération, and limits on the tax assistance for retirement savings.
33
Pension plan régulations are under provincial jurisdiction (unless the plan is subject
to fédéral authority) and each province (except Prince Edward Island) has its own
législation. This régulation governs the terms such as eligibility for membership,
portability of benefïts, vesting and death benefits (so called minimum benefit standards).
The fédéral government limits the tax assistance provided to a pension plan through
the Income Tax Act (ITA). The Canada Revenue Agency (CRA) sets limits on
contributions that can be déductible in order to ensure that not too much tax revenue is lost.
It also sets limits on benefits that can be paid.
Over the years, pension plan régulations hâve been subject to reforms in many
provinces in order to protect plan participants (by further improving the minimum benefit
standards). More information about pension and benefit plans can be found in the 13'
édition of the Morneau Sobeco Handbook of Canadian Pension and Benefit Plans,
published in 2005. Another excellent référence is the 7t1 édition of Fundamentals of
Private Pensions, published in 1996.
In this chapter, we provided an overview of the theory concerning employer pension
plans. In the chapters that follow, our main focus will be on comparing certain DB plans
with a money purchase DC plan. None of the plans considered will be indexed or
integrated with C/QPP. Furthermore, we will assume that ail pensions start at normal
retirement âge, regardless of when the person chooses to cease working. Moreover, we will
consider that the default annuity form is the single life annuity and that that is the form
elected by the participant upon retiring (and approved by her or his spouse if applicable).
Chapter 4 Literature Review In this chapter, we discuss the ongoing trend and impact of switching from DB
pension plans to DC pension plans. We start with a gênerai discussion. We then look at
the évolution of plan count and membership in some countries and put forward possible
reasons for the occurrence of the observed trend as pointed out by many researchers. Then
we look at some of the research results related to this trend.
Employer-sponsored pension plans were introduced to satisfy the différent needs
and demands of employers, employées and the government. Employers wanted to attract
and retain workers and to ensure the orderly exit of workers from the labour force.
Employées wanted to receive an income during their retirement in order to cover the basic
living expenses for them and their families. The government wanted to promote self-
reliance and to reduce the pressure to increase social benefits. Over the years, pension
plans hâve changed and they still are subject to change. Accommodating for différent
social, économie and political issues, pension plans hâve become more complex and more
expensive to offer.
Employer-sponsored pension plans historically hâve provided retirement benefits
through a DB plan which provided a pension in retirement based on years of service with
the employer and the salary received in year(s) just before retirement. With such a plan,
employers are responsible to ensure that the promised benefits will be delivered. DB
pension plans are designed to provide safe and sufficient pensions for employées with long
service but not for those who change jobs. Workers who change employers hâve their
benefits from previous employers calculated with wages that were earned at the time of job
termination, not wages earned at the time of retirement; as a resuit, their pensions are
devalued. DB plans are often criticised for this reason (poor portability). Another criticism
of DB plans is that they are complicated for workers to understand and it is difficult for
them to evaluate the accrual of their pension benefits. During the 1980s and later, there has
been a noticeable shift from DB plans towards DC plans in many countries. This means
that risks associated with providing pension benefits shifted from employers to individual
members.
35
4.1 Observations in the U.S., the U.K. and Canada The most pronounced shift has been in the U.S. The number of active participants in the
private sector covered by a DB plan decreased from 30.1 million in 1980 to 20.3 million in
2005. Over the same period the number of active participants covered by a DC plan
increased from 18.9 million to 62.4 million. The phenomenon is identical in tenus of the
number of plans. Indeed, from 1980 to 2005, the number of DC plans increased from
340,805 to 631,481 while the number of DB plans decreased from 148,096 to 47,614. How
thèse numbers evolved over that period can be seen in Table 4.1. (Source: U.S. Department
of Labor, Employée Benefits Security Administration 2008.)
Table 4.1 Number of private pension plans and active participants in the U.S.
Year Number of plans Number of members (millions)
Year Total DB DC Total DB (%) DC (%) 1980 488,901 148,096 340,805 49.0 30.1 61.4 18.9 38.6 1985 632,135 170,172 461,963 62.1 28.9 46.6 33.2 53.4 1990 712,308 113,062 599,245 61.5 26.2 42.6 35.3 57.4 1995 693,404 69,492 623,912 65.6 23.4 35.7 42.2 64.3 2000 735,651 48,773 686,878 73.1 22.2 30.4 50.9 69.6 2005 679,095 47,614 631,481 82.7 20.3 24.6 62.4 75.4
In other parts of the world, the shift toward DC plans has not been as fast as in the
U.S. but it has been steady. For instance, in the U.K., in 1995, the number of pension plans
in the private sector (known in the U.K. as occupational pension schemes) was about
151,000. The total number of active members in private sector pension plans was 6.1
million. DC plans covered 1.1 million members (18.0% of ail active members in the
private sector) and DB covered 4.7 million members (77.0% of ail active members in the
private sector). The rest of the plans were hybrid plans. (The number of pension plan
members that belonged to the public sector was about 4.1 million and they were ail covered
by a DB pension plan.) In 2000, there were approximately 105,000 pension plans in the
private sector. The total number of active members in private sector pension plans was
about 5.7 million; 0.9 million members (15.8% of ail active members in the private sector)
were covered by DC plans and 4.6 million members (80.7% of ail active members in the
private sector) were covered by DB plans. The rest of the plans were hybrid plans. (The
number of pension plan members that belonged to the public sector was about 4.5 million
36
and they were ail covered by a DB pension plan.) In 2005, the number of pension plans in
the private sector was about 69,000, covering about 4.7 million active members. About 1.0
million members were covered by a DC plan (21.3% of ail active members in the private
sector) and 3.7 million by a DB plan (78.7% of ail active members in the private sector).
(Information regarding plan count and membership in the public sector in 2005 was not
found. Besides, for that year, active members seem to hâve been fully split between DB
and DC plans.) (Source: Government Actuary's Department 2001, 2003, 2006.) That
information, for the private sector only, has been summarized in Table 4.2.
Table 4.2 Number of private sector pension plans and active members in the U.K.
Year Number of plans Number of members (millions) ;
Year Total DB DC Total DB (%) DC (%) 1995 151,000 38,000 109,500 6.1 4.7 77.0 1.1 18.0 2000 105,000 39,300 64,100 5.7 4.6 80.7 0.9 15.8 2005 69,000 12,000 53,500 4.7 3.7 78.7 1.0 21.3
Whereas statistics for the U.K. dating as far back as 1980 were not available, two
comments found on page xii of the 2006 édition of the Occupational Pension Schemes
Annual Report attest to the décline of DB plans in the U.K.:
The number of open single section private sector defined benefit schemes
has fallen sharply in récent years, from 18,350 in 2000 to 3,470 in 2006.
Over the last 15 years, the décline in private sector membership reflects a
fall in membership of defined benefit schemes, from 5.6 million in 1991 to
3.3 million in 2006.
Note that open schemes are those schemes that admit new members. The figures in
Table 4.2 include not only open schemes, but also schemes that are closed, frozen or
winding up. Closed schemes are schemes that do not admit new members but accept
contributions from existing members. Frozen schemes are schemes that no longer accept
contributions at ail. Schemes that are winding up are schemes that are in the process of
being terminated, which means that the plan assets and liabilities are being transferred to
another plan or to an insurance company.
37
In Canada, this trend has not been as pronounced as in other countries, but changes
hâve been observed nonetheless. In 1980, there were 14,586 pension plans covering 4.5
million pension plan members. DC plans covered 5.2% of ail plan members and DB plans
covered 93.7% of ail plan members. In 2005 there were 15,336 pension plans covering
almost 5.7 million pension plan members. DC plans covered 15.6% of ail plan members
and DB plans covered 81.2% of ail plan members. The plan members who were covered
neither by a DC plan nor by a DB plan were covered by some other kind of plan such as a
hybrid plan. As can be seen in Table 4.3, the number of DB plans has been varying over
that period, while the number of DC plans has essentially increased then decreased. In
terms of members, the absolute numbers hâve been fairly constant over the period for DB
plans while they hâve greatly increased for DC plans. In relative terms, this means that,
from 1980 to 2005, the percentage of members in a DC plan has increased significantly at
the expense of the percentage of members in a DB plan. (Source: Statistics Canada,
CANSIM database, Table 280-0016.)
Table 4.3 Number of pension plans and members in Canada
Year Number of p ans Number of members
Year Total DB DC Total DB (%) DC (%) 1980 14,586 8,035 6,170 4,475,429 4,194,283 93.7 231,275 5.2 1986 21,094 8,215 12,637 4,668,381 4,295,691 92.0 325,320 7.0 1990 19,956 8,284 11,443 5,109,363 4,633,587 90.7 430,561 8.4 1995 15,845 6,990 8,609 5,169,644 4,582,154 88.6 518,669 10.0 2000 15,557 7,108 8,152 5,267,894 4,456,034 84.6 716,646 13.6 2005 15,336 7,561 7,485 5,669,858 4,604,775 81.2 885,840 15.6
4.2 Reasons Underlying the Switch
The reasons for the observed switch from DB plans to DC plans are différent for
différent countries. The most common reasons cited are employers' désire to reduce the
risk associated with providing a DB plan in a volatile financial market; présence of a
surplus in DB plans; increased législation and régulation which made DB plans more
expensive and complex to administer; changes in work pattern and increased longevity
(Bharmal 1988 and Ostaszewski 2001). We look more closely at thèse reasons in the
following paragraphs.
38
From the 1980s onwards, financial markets hâve been volatile. Volatility in the
financial markets prevents employers from predicting the exact cost associated with
funding retirement benefits. Improvements in life expectancy also contributed to the higher
cost of providing adéquate pension benefits. More employers wanted to pass on some of
the costs to employées either by increasing employée contributions or by reducing some of
the promised benefits (say health benefits or early retirement benefits). What a better way
to shift some of the responsibility to employées but to offer DC plans to new employées
and encourage older employées to join the new plans. At the time, high rates of return
made DC plans more attractive to employées anyway. The asymmetric risk is yet another
reason for employers to switch to DC plans. The asymmetry cornes from the fact that
employers typically bear ail of the financial risk when the plan is underfunded but they do
not benefit fully, if at ail, from overfunding (as a gênerai rule, any surplus goes to the
employées, at least partially if not fully).
Whereas little if any of the surplus may hâve been available to plan sponsors in
some jurisdictions, this was not true for every single plan, depending on both the terms of
the plan document and the rules applicable in the given jurisdiction. In instances where the
sponsor was entitled to a significant portion of the surplus, any large surplus would hâve
served as a further enticement to switch to a DC plan. That would be particularly true if the
whole surplus was released only on plan termination, or if légal restrictions on the use of
plan surplus were known to be impending.
In order to ensure that contributions made to pension plans are protected, many
governments set strict légal, funding and solvency laws. So as to limit the tax revenue that
is lost or delayed, tax laws on deductibility and limitations on contributions also exist.
There are other various régulations concerning contributions (as to who can make
contributions and on protecting those on whose behalf contributions are made). Additional
régulations on contributions and benefits, introduced by reforms that took place mostly in
the 1980s and 1990s, made DB plans less attractive for employers. New accounting
standards imposed additional costs on employers. Ail thèse changes made DB plans more
costly and more complex.
39
The work pattern has changed. Workers are more mobile, lengthy careers with the
same employer are less common; hence retiring with one long period of service with the
same employer is less common. There are more part-time, contract and temporary types of
employment. Employées want more flexible plans and portability of their benefits.
Participation of women in the workforce is increasing. (As can be derived from Table 1.2,
in Canada, in 2005, 47.7% of RPP members were women, compared to 43.8% in 1995 and
34.7% in 1985.) There are more two-income couples. The birth rate is decreasing. Worth
noticing is the increase in the level of éducation which postpones career entry (without
necessarily leading to an équivalent increase in the âge at retirement). Employers also do
not see their workers as potential lifetime employées. A pension program that originally
was designed for a maie breadwinner that supports his large family no longer is
appropriate. Since employers do prefer DC plans for various reasons (such as less stringent
régulations, less variable costs and no concern about the payment of promised benefits),
they are more likely to provide thèse plans to their employées.
So DC plans may seem to be the solution for catering to the différent needs of
employers and employées, although to différent degrees.
As the trend towards DC pension plans was taking off, scholars increasingly took
notice and the papers on the topic added to the body of research related to pension plans.
The switch was taking place, mainly because everything was crystal clear about DC plans
and they brought about no surprise (from the employer's point of view, but not from the
employée's), but there was little information and évidence about the adequacy of thèse
plans.
DC plans do not hâve the portability problem DB plans hâve but DC plan members
face some other problems and risks. Thèse risks are primarily related to inadéquate
pensions due to any combination of the following factors:
• Low contribution rates - for some employées (such as those who delay joining
their employer's pension plan or those who enter the workforce later in life), the
prescribed contribution rate may be too low to achieve the target replacement
ratio (usually 70% of the final salary).
40
• Periods of unemployment - employées usually do not contribute to the
employer-sponsored pension plans during periods of unemployment so the
benefits do not accrue at the same pace as if they were employed and continued
to make contributions.
• Fall in asset values - an employée's pension may be devalued due to a fall in
asset values.
•
•
Inflation - if a pension is paid as a level annuity, the real value of the pension
decreases due to inflation (inflation also greatly affects the value of DB pensions
which are not indexed).
Low annuity - if interest rates at the time of retirement are low, the annuity that
the accumulated contributions can buy will be lower.
• Longevity - if the accumulated fund is not used to buy an annuity, there is a
possibility of outliving pension payments.
Also, employées may not hâve sufficient knowledge to make adéquate investment choices
and properly manage their investments and their money. Some other problems can be the
variability in future earnings and unfavourable changes in plan régulations.
4.3 Research Results Related to the Trend and its Impact We briefly look at some of the research that has been done with respect to thèse
issues.
Bodie, Marcus and Merton (1985) compared DB plans and DC plans. They
examined the relative advantages of each type of plan. They developed a model for
pension benefits subject to uncertainty in both wage and interest rate.
The authors pointed out that it is possible to design a DC plan that mimics a DB
plan as far the benefit accrual pattern is concerned. By nature, DB plans are backloaded
(that is, benefit accruals are worth much more close to retirement than at the beginning of
the career). With a constant contribution rate, DC plans are frontloaded, but they can be
41
backloaded by choosing a contribution rate that increases with an employée's âge and
service.
Their results showed that, unlike DB plans, DC plans are sensitive to the real rate of
investment return and this makes them inferior to DB plans from the employée's point of
view since, under DC plans, ail the investment risk rests with the plan participant. The
authors argued that DC plans hâve an advantage in periods of variable inflation because
plan participants can reallocate their funds and invest them in inflation-hedged portfolios
but the problem is that not many know how to make their portfolios risk-free. In terms of
portability, DC plans hâve an advantage in that breaks in service do not affect the potential
value of benefits already accrued. In addition, employées can détermine the true présent
value of the retirement benefit earned in any year although the future flow of retirement
income may not be certain. However, if the key goal upon retirement is to hâve a certain
replacement ratio, DB plans are superior to DC plans since their benefits are defined in
those terms. Thus, according to the authors, the main advantage of DB plans is that they
provide a stable replacement ratio of final income.
Knox (1993) examined the distribution of retirement income provided by a DC plan
using a stochastic model for inflation and investment returns (as independent random
variables) based on Australian expérience. He considered the retirement income benefit for
a single man who is in full-time employment for 45 years, contributing at a constant rate
(expressed as a percentage of his salary) throughout his career. As a basic investment
strategy, Knox considered a balanced portfolio of equities and properties. Plan-related
expenses (administration, investment costs, etc.), allowances for taxes on benefits and
social security benefits were excluded. The basic model was compared to various scénarios
such as changes in entry and exit âges, changes in investment stratégies, changes in work
pattern, and changes in inflation and annuity assumptions.
The main resuit was that the defined contribution rate set for ail employées to be
paid for 40 or 45 years would provide various levels of retirement benefits. The prescribed
contribution rate may be satisfactory for some future retirées, but not for ail. A higher
contribution rate is needed for individuals who do not work full-time throughout their
career, for members with dépendent spouses, for females, for individuals who enter the
42
workforce later due to increased éducation or unemployment, for individuals who choose or
are forced to take early retirement, and for individuals who may be subject to any
combination of those.
Variable investment returns over the long periods of employment, possible annuity
rates offered at retirement, and expenses are also to be taken into considération when
defining the contribution rate. Investment returns affect retirement benefits in such a way
that employées with otherwise identical salaries, working history, and contribution rate may
end up with différent pensions (insufficient for many). Knox pointed out that this situation
is most likely when the economy is depressed, causing reduced investment returns, and
affecting ail DC plan members (hence a particular génération of retirées, not just retirées
from a particular plan). Knox also pointed out that switching to a less volatile investment
strategy a few years before retirement can reduce the variability of future retirement
income, but the expected income is then also reduced. He stressed that this outcome must
be acknowledged.
Khorasanee (1995) examined the problem of investment risk in money purchase
plans based on the U.K. expérience using différent stochastic models. He critiqued
modeling equity returns as independent, identically distributed lognormal random variables,
and not taking equity dividend yields into account. Khorasanee argued that this model
overestimates the variability in accumulated funds that are invested using annual
contributions over a long period. He proposed to use a stochastic model that allows for
equity dividend yield (Wilkie's model) and which produces more plausible results
consistent with empirical studies. Even with this model, the variability in projected
retirement fund for a new DC plan member is still high and reduces as the member
approaches retirement.
Khorasanee also examined différent investment stratégies. He found that investing
a signifïcant proportion of the fund in low-risk assets (balanced investment strategy instead
of 100% in equities) for a long time would produce a lower retirement fund. He pointed
out that the time to switch investment strategy dépends on the member's targeted fund
value and prevailing equity dividend yield.
43
Cooper (1997) examined the distribution of retirement benefits under différent
pension plan arrangements for workers with différent working historiés using a
deterministic approach based on U.K. data. She compared benefits provided by différent
types of employer-sponsored pension plans (final salary, revalued career average and
money purchase) with the benefits provided by a total service pension plan. The total
service pension plan is a type of final salary pension plan that ignores changes in type of
employment (part-time and full-time employment), breaks in service, and changes of
employer. Ancillary benefits and public pensions were not taken into account. It was
assumed that full-time employment salaries increase in line with both real wage growth and
price inflation. In periods of part-time employment or during breaks in service, however,
salaries were assumed to increase only with price inflation. She also assumed a
contribution rate to the DC plan that was higher for full-time employées than for part-time
employées.
Her results showed that DB plans provide good benefits for most working historiés.
In the case of early withdrawals from a final salary pension plan, results indicated that
employées receive lower pensions compared to what they would receive under a total
service pension. The relative loss in benefits gets even greater when the real rate of annual
wage growth is higher and the rest of the assumptions remain the same. This is because, in
the U.K., deferred pensions are indexed with price inflation during the period of deferment.
Under the assumptions made in the paper, with strictly positive real wage growth, deferred
pensions are increased at a rate lower than the rate of wage growth, and hence, the greater
real wage growth is, the greater the relative loss in benefits is. Working historiés that
appear to do well despite an early withdrawal are those that hâve a small proportion of full-
time employment; because their earnings are more related to inflation than to wage growth
(as per assumptions), the différence between benefits that they would receive under a total
service plan and their actual deferred benefits is not as important.
Employées who are in full-time employment but who change jobs frequently (every
five or ten years, for example) are likewise disadvantaged in a final salary plan if their
deferred benefits accrue with inflation only.
44
For employées that are in a final salary pension plan and who want to take a break
some time in their career, results showed that it is better to take the break early in the career
than late, because the deferred pension is then based on a shorter period of service.
Revalued career average plans are modified career average plans which
systematically index (or revalue) the career earnings used to calculate the pension benefits.
Each year's earnings are indexed with respect to price inflation or in Une with some wage
index, as specified in the plan document. With such plans, when the revaluation is with
price inflation, results show that employées who do relatively well are those who work
part-time, even if the revaluation of deferred benefits is with inflation only. This is not
surprising since salaries for thèse employées grow with inflation only (as per assumptions
made).
However, if the revaluation is with wage growth, then ail catégories of employées
would do well. In fact, ail catégories other than full-time do better than they would do
under a final salary pension plan. Revaluation with wage growth compensâtes for breaks in
service in the sensé that it projects the salary as though the service had been completed with
no breaks in service, that is, as a full-time service. In effect the pre-retirement salary that
would hâve entered the benefit calculation would be the same as the pre-retirement salary
for a full-time working history. Such revaluation overestimates the actual pre-retirement
salary for any other working history since, as per the assumptions, the salary increases with
inflation only during breaks in service.
When the pension under the defined contribution plan (or money purchase plan) is
compared to the total service pension, results may lead to believe that those employées with
breaks in service do better than a standard full-time employée. That is, the DC pension
seems to be better than a total service pension for most historiés but not that much better for
a full-time history. This is because the total service pension is based on the time spent in
employment. More importantly, this also arises because it is assumed that ail employées
start contributing to DC plans when they begin working and thèse early contributions are
very important for further fund accumulations. (However, when income replacement ratio
is considered, full-time employées do better than those with breaks in service.)
45
Cooper pointed out the importance of the contribution rate. An inadéquate
contribution rate based on wrong assumptions (for example, if the real rate of return and
interest rate are lower than assumed) can huit ail catégories of employées. The most
vulnérable are those with part-time employment and breaks in service because they
contribute based on their employment status (as per the author's assumption). Cooper
argued that DC plans do better than DB plans in two circumstances:
• When salaries increase faster than inflation and deferred benefits are increased
with inflation only.
o In particular, under such circumstances, DC plans may be a better choice for
employées who are in a full-time type of employment but frequently change
jobs.
• In the case of a surplus in a DB plan where questions arise as to who owns this
surplus.
o When actual expérience differs from the assumptions in a favourable way
(for instance, interest rates are higher than increases in salary and than
assumed), this surplus goes to the owner of the account, in the case of a DC
plan. In DB plans, however, surplus ownership is not clear. If employer
contributions are viewed as deferred pay, then the surplus should go to plan
members. Usually employer contributions are not calculated to be member-
specific and they vary not just with respect to the employee's salary but also
with respect to the circumstances as actual expérience is différent from the
set assumptions. Hence one cannot assume that this contribution is
completely deferred pay and some way of sharing any surplus should be in
place.
Samwick and Skinner (1998) addressed the question related to the adequacy of
retirement income provided by DC plans compared to DB plans based on U.S. expérience.
The authors used data from the Survey of Consumer Finances (SCF), which provides a
detailed survey of household wealth in the U.S., as well as data from Pension Provider
46
Surveys (PPS), which obtain the summary plan description for a plan, from the plan
provider, for those SCF respondents who reported being covered by a pension plan.
They sampled DB plans in 1983 (because thèse provided most of the data on DB
plans during the transition) and DC plans in 1995 (because thèse provided the best
description of features of future DC plans) and they compared benefits that could be
expected from thèse plans. To each worker covered by a DB plan in the 1983 survey,
authors assigned a randomly chosen DC plan from the 1995 survey. Simulations showed
that the typical DC plan from the 1995 sample would provide higher retirement benefits, on
average, than the typical DB plan from the 1983 sample.
Samwick and Skinner examined différent possibilities regarding the spending of
lump-sum distributions from DC plans when workers switch jobs and also regarding
workers' contribution négligence. Results show that, despite spending 50% of their lump-
sum between jobs, workers can still hâve an almost équivalent retirement income as DB
plans would offer. Regarding contribution négligence, the authors argued that those
workers who fail to contribute to their DC plans are mostly those for whom DC plans are
additional to their main plan, which usually is a DB plan. (In their 1997 study, they
estimated that between 2% and 4% of ail workers having a DC plan as the only pension
arrangement do not make their contributions.)
Results confïrm that DC plan members are exposed to more risk from investment
rates of return but, because the mean of the benefit distributions is also higher, 1995 DC
plans are still preferred to 1983 DB plans (even for a highly risk averse worker). Results
from this study suggest that DC plans can provide adéquate financial security in retirement.
Brown and Liu (2001) examined the shift toward DC plans in Canada. They tested
if macroeconomic factors contributing to this change, given by Ostaszewski (2001), match
Canadian expérience.
According to Ostaszewski (2001), some of the théories that can explain the shift
toward DC plans are the following:
47
• The New Economy Theory - There is a change in employer-employée
relationship: employers are willing to offer more flexible benefits to a new type
of employées who are mobile, independent, and who désire portability of their
assets and control over their assets.
• The Excessive Régulation Theory - Tax laws and funding régulations that make
DB plans more expensive and complex to administer lead to the démise of DB
plans.
• The Risk Averse Employer Theory - With heightened volatility in financial
markets, the cost of providing retirement benefits becomes less predictable;
employers are aware of this risk and, since they hâve control over which type of
plan they offer, it is normal that DC plans prevail.
Ostaszewski further argued that thèse théories did not completely describe the trend
because none of them does fully explains workers' motivations; none of the théories views
workers as économie decision-makers.
Ostaszewski argued that, if workers were perceived as économie decision-makers,
then participating in pension plans would be like holding a share of stock, security or any
capital asset: they give up today's cash flow (they pay for the security with wage
concessions) in exchange for future cash flows (pension benefits). The main différence
between participating in DB plans and holding other securities is that future cash flows for
DB plans dépend on wages (together with the length of service) while cash flows for other
securities dépend on capital asset priées in the financial markets. Future cash flows from
DC plan participation, unlike those from DB plan participation, actually dépend on the
performance of capital assets held by the plan member. In DB plans, workers hold a long
investment position, not in capital assets, but in a security that dépends on wage growth in
the national economy during their working career (macroeconomic factor). It is argued that
members of DC plans place their assets in low-risk securities such as Treasury bills or
money market instruments, taking little risk with their assets. Still, returns from thèse DC
investments greatly outperform DB plan participation (the derivative security created by
DB plans) that is indexed to wages and salaries. With the récent trends of overall wage
48
décline and overall growth of gross domestic product (GDP) in the national economy, plan
participants, as rational décision makers, would prefer DC plans to DB plans.
This hypothesis can be used to support The New Economy Theory, Ostaszewski
argued, but it gives less credibility to the other two théories. He still found some reasons
that could make thèse théories valid. For The Excessive Régulation Theory, Ostaszewski
relied on govemment involvement and argued that the govemment has influence on
économie décision makers.
To secure the future of DB plans, Ostaszewski proposed two solutions. One is to
change the method of calculation of retirement benefits so as to account for the
performance of capital markets. The other is to design DB plan participation as a derivative
security which, while indexed to wages and salaries, is more attractive to workers.
As for The Risk Averse Employer Theory, Ostaszewski found it a bit hard to
validate, arguing that even investments in low-risk securities outperformed the wage index
investments (hère he was referring to the signifïcant rate of return on T-bills compared to
the rate of wage growth).
Brown and Liu used Canadian data to test Ostaszewski's hypothèses. The authors
argued that the proportion of DB plans (out of ail private plans) in Canada is more stable
than it is in the U.S. Canadian data do not support Ostaszewski's hypothesis of a strong
corrélation between the share of national income that is going to wages and salaries and the
share that is going to DB plans. Neither is there clear évidence to support the last
hypothesis that interprets the shift to DC plans as a rational move to greater security caused
by the décline in the importance of wages in the national economy. Brown and Liu found
pension régulation to be an important factor in the design of an employer pension plan.
Brown and Liu argued that pension and tax législations and régulations are
necessary to set the basic plan design requirements. The purpose of regulatory measures is
to protect the économie security of retirées, encourage and facilitate pension arrangements,
and safeguard against discrimination. The authors listed many différences between the
Canadian and the U.S. régulation and taxation Systems and explained why Ostaszewski's
49
hypothesis does not fit the Canadian framework. Brown and Liu pointed out that
registration rules for DB plans make DB plans advantageous for employers, coverage rules
are less strict (but still must not violate Human Rights Législation) than in the U.S., and
ancillary benefits (such as death or disability before retirement) are provided only under
DB plans. The government policies regarding taxation and funding treat DB and DC plans
equally while the U.S. government implements policies that favour DC plans in many
ways. In Canada, DB and DC benefits must be paid out as life annuities; lump sums are
allowed only in some minor exceptions; loans and cash payments from Registered Pension
Plans are not allowed; and the tax treatment of contributions is more advantageous than in
the U.S. Union participation, although slowly declining, is still higher in Canada than in
the U.S. and this may also be the reason for différences in plan design. The authors also
highlighted the higher risk aversion in Canada and the national financial market situation
which is différent from that in the United States.
Chapter 5 Model Used to Simulate Economie Variables In this thesis, we examine retirement benefits that arise from différent pension plan
designs as they apply to différent work patterns. We investigate which plan design would
be more suitable for a particular working pattern. How well a pension plan does dépends,
among other things, on how the economy has performed and how the wages hâve evolved
over time. We would not get the full picture by looking at just one scénario. To get a
better sensé of how différent plans perform for différent types of workers, we use stochastic
simulation to generate a range of outeomes for the future value of pension plan benefits.
To calculate future benefits we need, among other things, to set assumptions regarding
inflation, wage growth and investment returns. Since it is impossible to know what the true
value of those variables will be over the next 30 or 40 years, we use a stochastic model to
generate différent possibilities.
In this chapter, we describe the stochastic model used to simulate thèse variables.
We first give a gênerai description of the model, explaining the properties we will want the
model to hâve. We then présent the équations of the model and explain how the parameters
of the model were estimated. Lastly, we look at the scénarios that were generated using the
model.
5.1 General Description of the Model This description of the model is based on the December 2005 Society of Actuaries
Course 7 offering.
As indicated above, we want to simulate inflation, wage growth and investment
returns. For investment returns, we actually need to simulate returns on the assets in which
the funds will be invested, long-term government bonds and common stock. Moreover,
because of the structure of the model, we also need to simulate the return on Treasury bills
(T-bills).
In fact, we consider the following five économie variables: inflation, wage growth,
return on T-bills, long-term government bond yield to maturity (YTM) and stock return.
51
The reason we model YTM values instead of long-term bond returns is that we need
some proxy for the interest rate used at the time of retirement to calculate the annuity
factor. Besides, bond returns are more easily obtained from YTMs than the reverse.
Variables are assumed to be interdependent; in other words, variables are not
independent from one another. To be more précise, we assume that ail rates of return
dépend on one another and on inflation, but not on wage growth, whereas wage growth
dépends on inflation alone. We simulate future investment returns, separately from wage
growth, in the following order:
inflation,
return on T-bills,
YTM,
stock return.
That order cornes from an intuitive understanding of which variable drives the others on the
markets.
As wage increases dépend on inflation, we hâve the following order of variables for
modeling wage growth:
inflation,
wage growth.
Moreover, each variable is autoregressive. It means that the value of the variable
for a given year also dépends on the value of that same variable for the previous year. As a
resuit, inflation dépends on last year's inflation only and does not dépend on any other
variable. Wage growth dépends on last year's wage growth as well as current inflation.
The return on T-bills dépends on last year's return on T-bills and current inflation. The
YTM dépends on last year's YTM, current inflation and current return on T-bills. As for
the stock return, it dépends on last year's stock return, current inflation, current return on T-
bills and current YTM.
We also assume ail five économie variables to hâve a lognormal distribution in
order to eliminate the possibility of unrealistically low values as well as to better reflect the
skewness of the distribution.
52
The lognormal distribution has a natural lower bound at 0. However, we know from
historical data that négative returns are possible. Hence, we actually assume variables to
follow a shifted lognormal distribution.
Lastly, except for inflation, we will work in real terms, as opposed to nominal
terms. Since we ultimately want to work in nominal terms and because inflation will be
simulated along with the other variables, we will transform real returns back to nominal
returns.
5.2 Equations of the Model Now that we hâve laid out the properties of the model, we need to lay down its
underlying équations so we may estimate its parameters and then use it to generate
économie scénarios.
Let CPI, be the rate of inflation in year t. Let ccc be the distance by which the
lognormal distribution is shifted to the left. Because CPlt +ac is assumed to be
lognormal,
C, =ln(OY, + a c ) (5.1)
is normally distributed.
Because current inflation dépends on nothing other than last year's inflation, the
équation that describes the évolution of inflation is as follows:
c , = Me + <P,ast(C) (c,-\ - Me ) + ef > (5 -2)
where juc is the value towards which the shifted natural logarithm of the inflation rate
tends to revert, <pjasl(C) captures the extent to which this year's value dépends on last year's
value, and
ef~NQtal). (5.3)
In total, for the rate of inflation alone, four parameters (ac , juc , (p,asl(C), crc) enter
the model. For the other variables, more will be needed because of the hierarchical
dependence between the variables.
53
We continue with the rate of wage growth, which dépends on its last year's value as
well as the current inflation. Let Wage, dénote the rate of wage growth in year t and let
aw be the shift to the left of the associated lognormal distribution. Since we work with the
real rate of wage growth rather than the nominal rate, we are assuming that
Wt = \n(Wage, -CPl,+aw) (5.4)
follows a normal distribution.
Because of the dependence of the real rate of wage growth on other values, the
équation that describes the évolution of wage growth is as follows:
Wt=Mw + <Plast(W)<Wt-\ ~Mw) + <PC(W) (C, - Hc ) + S? , (5.5)
where juw is the value towards which the shifted natural logarithm of the real rate of wage
growth tends to revert, (p,asl(W) captures the extent to which this year's value dépends on
last year's value, <pC(w) captures the extent to which wage growth varies with inflation, and
e?~N<P,<rl). (5.6)
Equations 5.1 to 5.6 capture the dynamics of inflation together with wage growth.
We proceed to capture the dynamics relevant to the assets we are considering:
Treasury bills, long-term bonds and common stocks. Again, Treasury bills enter the picture
not because funds will be invested in Treasury bills but because the structure of the model
assumes that inflation, return on Treasury bills, yield to maturity on long-term bonds and
return on common stocks are ail interdependent.
Let Tbill, be the rate of return on T-bills in year /. Like the rate of wage growth,
the rate of return on T-bills is assumed to dépend on its last year's value as well as current
inflation. Hence, the équations describing its dynamics mimic those for the rate of wage
growth. Let aB be the shift to the left of the associated lognormal distribution.
Once again, we work with the real rate of return rather than the nominal rate.
Hence, we are assuming that
B, = ïnÇTbiU, -CPI, +aB) (5.7)
54
follows a normal distribution. The équation that describes the évolution of the real rate of
return on T-bills is as follows:
B,=MB+ <Picmm(B<-\ ~MB) + <PC(B)(C, ~MC) + S?> (5-8)
where /uB is the value towards which the shifted natural logarithm of the real rate of return
on T-bills tends to revert, <p,asl(B) captures the extent to which this year's value dépends on
last year's value, <pC(B) captures the extent to which return on T-bills varies with inflation,
and
e?~N(V,o2B). (5.9)
We continue with the équations for the yield to maturity on long-term government
bonds, those bonds with a maturity greater than 10 years. Let YTMt be the yield to
maturity on bonds in year /. Also, let aY be the shift to the left of the lognormal
distribution of the real YTM. As a resuit, the variable we are assuming to be normally
distributed is
Y, = \n(YTM, -CPJ, +aY) (5.10)
We assume the real YTM to vary over time according the following équation:
Y, =MY + 0 W ) ( Ï M -MY) + <Pc(Y)(Ct -MC) + <PB(Y)(B, -MB) + £Ï> (5-11)
where juY is the value towards which the shifted natural logarithm of the real YTM on
bonds tends to revert, cplaM(Y) captures the extent to which this year's value dépends on last
year's value, (pC{Y) captures the extent to which the bond yield varies with inflation, <pB{Y)
captures the extent to which it varies with the return on T-bills, and
ej ~N(0,a2Y). (5.12)
The last séries of équations we need is for the return on common stocks. The
central équation will be the most complicated as we assume that variable to dépend not
only on its previous value, but also on current values of inflation, return on T-bills and
YTM.
55
Let Stock, be the return on common stocks in year t. Also, let as be the shift to the
left of the lognormal distribution of the real stock return. It follows that
S, = \n(Stock, -CPI, +as) (5.13)
is normally distributed.
Because of the hierarchical dependence inhérent in the chosen model, we assume
the real stock return to vary over time according the following équation:
S, = Ms + <Plast(S) (5r-l - Ms ) + <Pci.S) (C> ~ Me ) c (5-14)
+ <PB(S)(BI-MB) + Ç)Y(S)(Y,-MY) + S, ,
where jus is the value towards which the shifted natural logarithm of the real return on
stocks tends to revert, (pta!il(S) captures the extent to which this year's value dépends on last
year's value, <pC(S) captures the extent to which the stock return varies with inflation, (pB(S)
captures the extent to which it varies with the return on T-bills, <pY<s) captures the extent to
which it varies with the bond yield, and
s? ~N(0,a2s). (5.15)
Equations 5.1 to 5.3, together with Equations 5.7 to 5.15, describe the behaviour of
the financial markets we are interested in. Ail the équations hâve a similar structure, more
or less complex according to the number of variables that are needed to capture the level of
interdependence.
5.3 Estimation of the Parameters of the Model With Equations 5.1 to 5.15, we would be ready to generate économie scénarios, if
only we knew the values of the parameters. We could hâve used the method of moments to
estimate the parameters, but that method does not prescribe by how much to shift each of
the lognormal distributions. We would hâve had to set the différent values of a based on a
blend of past observations and intuition, along perhaps with some ad hoc adjustments.
Instead, we chose to use the method of maximum likelihood. Hère is a brief
description of how this method works.
56
Let xï>x2,...,xn be n observed values of a random variable X, assumed to follow
theprobability density fonction (pdf) fx(x), whosemparameters are Ox,92,...,0m.
Using maximum likelihood, the estimated values for those parameters are those
values that maximize the likelihood L{6X, 02,..., ôm ) , which is defined by
L(0l,02,...,0m) = flfAxi). (5.16)
Because the natural logarithm is a monotonically increasing transformation, and
also because a product of positive values less than 1 can get inconveniently small, instead
of maximizing the likelihood itself, it is more common to maximize the log-likelihood,
\nL(0x,O2,...,em) = Y\xifx{xi). (5.17)
(Because the pdf takes on values between 0 and 1, the log-likelihood necessarily is
négative. Rather than maximize the log-likelihood itself, some prefer to minimize the
absolute value of the log-likelihood. Ail approaches are équivalent and produce the same
estimâtes.)
Hence, to apply this method, we need observed values of the random variables of
interest and we also need to know the pdf for each of the five économie variables we hâve
been modelling.
In ternis of observed values, we hâve chosen to focus on historical data from 1981
to 2004, essentially to avoid most of the earlier years of high inflation - thus reflecting the
inflation control which the government has been successfully enforcing for a while now
and still seems dedicated to enforcing in the foreseeable future - while allowing for a
period long enough to capture différent business cycles. In fact, the focus really is on the
data from 1982 to 2004, but the 1981 values are needed because of the autoregressive
nature of the model. Ail the historical data used can be found in Appendix A. (The end
year of 2004 reflects the latest data available at the time modelling work began for this
thesis.)
57
In terms of pdf s, let us fïrst consider the basic form of the lognormal distribution,
which is our basic building block. IfXis lognormal, then its pdf is
1 fin x-/j
fx(x)= J - e 2{ ° ' , f o r x > 0 , (5.18) ■Jlnox
where ju and a are the parameters of the lognormal distribution.
If the lognormal distribution is shifted to the left by a value a, which means that it is
R=X+a rather than X itself that is lognormally distributed, then the pdf of R is a
modification of Equation 5.18 as follows:
fR(r) = - = i*[ ° J , f o r r > - a , (5.19) yjln <r{r + a)
Ail five économie variables hâve a pdf similar to that given by Equation 5.19,
except that the parameter fi has to be replaced by a somewhat complex expression that
captures the interdependence inhérent in the model.
The probability density function of the random variable CPI,, the rate of inflation
in year t, is given by
1 \\n(x+ac)-fi' ] i
fen, O) = - 7 = e y , for x > -ac , (5.20) V2;r<Tc(x + flrc)
where
ftf = /ic + <plm(C) (C,., - pc). (5.21 )
The probability density function of the random variable Waget, the rate of wage
growth in year t, is given by
1 r ln(*-«'/,+a,,. )-//,'" V
-J27taw (x - CP/, + aw )
where
and
x > CPI, - aw
58
tf =Mw+ ?iast(W)Wt-\ ~Mw) + <Paw)(c, -Me)- (5-23)
The probability density function of the random variable Tbill,, the rate of return on
T-bills in year t, is given by
\( bi(x-CPl, +aB)-fil> V
fmili(x)= l -e>[ a" K (5.24)
where
x>CPI, -aB
and
M? = MB + <Piast(B)(B<-i ~MB) + 9C(B) (C, ~ Me ) • (5-25)
The probability density function of the random variable YTM,, the yield to maturity
on long-term government bonds in year t, is given by
■ W * ) = g - : nui r**1 av J> (5-26) V 2n aY (x - CPI, +ocY)
where
x > CPI, - a Y
and
M! = MY + <Pias,(Y)(Xt-\ ~MY) + <PC{Y)(Ç, ~MC) + <PB(.Y)(B, -MB)- (5-27)
The probability density function of the random variable Stock,, the rate of return on
common stocks in year /, is given by
l(ln(x-CP/,+tts.)-/jf 1 2
l2ncrs(x-CPIt+as)
where
x>CPI, -as
and
M? = M s + <Plas«S) (5r-l - Ms ) + Pc(S) ( C , " Me ) .
+ ft»(5) ( f l , - / * * ) + 0V<S) Œ - MY )•
59
Technically, under maximum likelihood, estimâtes of the parameters are found by
first differentiating the likelihood (or log-likelihood) with respect to each of the parameters,
and then solving for the parameter values that make ail thèse derivatives equal to zéro. For
our model, that would be rather complex. Instead, we hâve used Microsoft Excel's Solver
function to find the values of the parameters that maximize the log-likelihood. Those
values are found in Appendix B.
Because some équations contain parameters that also appear in earlier équations, it
is important to find the estimâtes of the parameters in the same order that the variables were
presented. Also, some values in Appendix B may look rather small, to the point of being
insignificant, but no test of significance was performed and values were taken as estimated.
5.4 Economie Scénarios Generated With a fully specified model, we now are able to generate économie scénarios. This
section indicates how scénarios were generated and gives some statistics calculated from
those scénarios.
First of ail, we generated 1000 sets of (independent) standard normal random
numbers each covering a 50-year period (thus generating a 1000x50 matrix) for each
économie variable. Standard normal random numbers are normally distributed with mean 0
and variance 1.
Second, using the normal random numbers, sets of future values of the natural
logarithm of the shifted inflation or shifted real rate were generated.
Third and last, in the case of inflation, the values obtained in the second step were
exponentiated and the value of the shift was removed. In every other case, the values
obtained in the second step were exponentiated, and the rate of inflation was added (to go
from real to nominal) while the value of the shift was removed.
Ail thèse steps are illustrated in turn for each of the five économie variables.
60
Let z£, / = 2005,..., 2054, / = 1, ...,1000, be the normal random number to be
used to generate the rate of inflation in year t under the i scénario. The realized value of
C, in that scénario, denoted by ctj, is given by
c,,t = Mc + <Pias,(C) (<Vi,/ - /"c ) + °"c 2l> ■ (5-30)
Note that, regardless of the scénario, c2m, is the actual value this variable took in 2004.
The simulated rate of inflation in year t in the i'h scénario, cpi,,, is then given by
cpitl = ec,J - ac . (5.31)
Let zjj, t = 2005,..., 2054, / = 1,..., 1000, be the normal random number to be
used to generate the rate of wage growth in year t under the i'h scénario. The realized value
of Wt in that scénario, denoted by w,,, is given by
W,,, = Mw + <Pu*t(W) (W,_W - M w ) + <PC{W) (C,,i ~ Me ) + <*W ZU ■ (5-32)
The simulated rate of wage growth in year t in the i' scénario, wage,,, is then given by
wage,, = ew'-' +cpiti -aw . (5.33)
Let zfj, t = 2005,..., 2054, i = 1, ...,1000, be the normal random number to be
used to generate the rate of return on T-bills in year t under the i' scénario. The realized
value of B, in that scénario, denoted by b,,, is given by
K = /"B +<Piast(B)(bt-v -MB) + (PC(B)^U -/UC) + CJBZI, . (5.34)
The simulated rate of return on T-bills in year t in the i' scénario, tbills,,, is then given by
tbill,, = e '■' + cpi,, -aB. (5.35)
Let zj,, t = 2005,..., 2054, / = 1, ...,1000, be the normal random number to be
used to generate the yield to maturity on long term bonds in year / under the ith scénario.
The realized value of Y, in that scénario, denoted by y,,, is given by
y,j =MY+ VtasKY) (y,-u -Mr) , ^ x (5.36)
+ <PC(Y) (C,,, - Me ) + <PB(Y) (Ki ~MR) + CrY *,J ■
61
(5.38)
The simulated yield to maturity on long-term bonds in year t in the i scénario, ytm,,, is
then given by
ytm,j = ey'-' + cpiti - aY. (5.37)
Let z^, t = 2005,..., 2054, i = 1,..., 1000, be the normal random number to be
used to generate the rate of return on common stocks in year t under the i'h scénario. The
realized value of S, in that scénario, denoted by s,,, is given by
s,j = M* + <Pias«s)0,-u ~Ms) + <Pc(S)(P,j -H c)
+ <PB(S)(b,j ~MB) + 9nS)(y,,t ~Mr) + <*SA
The simulated rate of return on common stocks in year t in the ith scénario, stock, t, is then
given by
stocku =es'-' +cpiu -as. (5.39)
Figures 5.1 to 5.5 represent historical and generated économie variables. By
examining thèse graphs, we can see that the future behaviour of ail variables more or less
covers the same range as the historical values on which they hâve been based.
Figure 5.1, for the rate of inflation, looks good overall, except perhaps in terms of
the lst percentile. Négative inflation has not been observed for a long while. However, the
5th percentile lies above 0 and will motivate us to focus on the 5th to 95th percentiles when
we look at the results obtained under the différent scénarios in Chapter 7. Incidentally, it is
important to realize that the set of curves starting at 2004 are not scénarios. For instance, it
should not at ail be construed that the médian value simulated in 2010 necessarily is
followed by the médian value simulated in 2011.
Figure 5.2, for the rate of wage growth, raises concerns similar to those of Figure
5.1. Indeed, whereas wage freezes and even wage decreases can occur at the company
level or even within some industry, it would seem rather exceptional that wages decrease
economy-wide. Once more, the scénarios encourage us to focus on the 5l to 95'
percentiles. Just as given percentiles do not form one scénario, it is important to realize
62
that a given percentile for the rate of wage growth in a given year is not necessarily
associated with the same percentile for the rate of inflation in the same year.
Figure 5.3, for the rate of return on T-bills, raises questions because of the curve for
the 99th percentile. We were concerned that that would impact the values we would obtain
for the variables that dépend on it, namely the yield to maturity on bonds and the return on
common stocks.
It turns out that Figure 5.4, for the yield to maturity on long-term government bonds
looks rather reasonable, except perhaps for the fact that the range of simulated values is
much wider than the range of historical values.
As for Figure 5.5, for the rate of return on common stocks, the same comment that
was made on Figure 5.4 would apply.
Overall, we are comfortable with the scénarios generated, especially if we restrict
ourselves to the range between the 5th and 95th percentiles.
Figure 5.1 Inflation
1980 1990 2000 2010 2020 2030 2040 2050 2060 Year
— Historical values 1st percentile ' 5th percentile - * - 25th percentile - * - 50th percentile -+- 75th percentile —95th percentile —99th percentile
Figure 5.2 Wage growth
14%
-2%
-++f+*4 t^4H—f
4*** \ 7 \ 7 ' / \ H I < j l H * * » * * - « «-«-»»« »„«»,,..■■»■.«■■ ««««-«.«'«(««'«««««'««««imKKiinKlKit
1980 1990 2000 2010 2020 Year
2030 2040 2050 2060
-Historical values 1st percentile 5th percentile - * - 25th percentile - 50th percentile -+~ 75th percentile^ — 95th percentile — 99th percentile
Figure 5.3 Return on Treasury bills
30%
25%
20%
15%
10%
5%
0%
***********
1980 1990 2000 2010 2020 2030 2040 2050 2060 Year
— Historical values 1st percentile 5th percentile -*-25th percentile - * - 50th percentile -t-75th percentile —95th percentile — 99th percentile
Figure 5.4 Yield to maturity on long-term bonds
S. ■ j j H t | H y , y + i
*»^*i l«K**JWHlHlt**^^
2030 2040 2050
— Historical values 1st percentile ■■■*5th percentile - * - 25th percentile -«-50th percentile -+- 75th percentile — 95th percentile ■■■- 99th percentile
Figure 5.5 Return on common stocks
60%
-40% 1980 1990 2000 2010 2020
Year 2030 2040 2050
— Historical values 1st percentile « 5th percentile -*-25th percentile -»-50th percentile -+- 75th percentile —95th percentile —99th percentile
65
We might think that we hâve everything we need in terms of économie scénarios,
but that is not quite the case. We hâve indicated earlier that we generated yields to
maturity, instead of rates of return, on long-term bonds, because we need both and also
because it is easier to go from yields to returns than from returns to yields.
We actually need rates of return on long-term bonds because we assume that the
pension fund is invested in stocks and bonds. We then need to figure out how to get the
rate of return from the YTMs.
To calculate the rate of return in year t, we use the same approach as the Canadian
Institute of Actuaries in its annual Report on Canadian Economie Statistics. That is, we
assume that a newly issued bond that matures in 18 years is purchased in December of year
t - 1 at its face value FV, with a semi-annual coupon rate of YTMt_x. In other words, we
find and buy an 18-year bond at par, a JTM,_,% bond with a semi-annual coupon of
\YTMt_y x FV every six months.
After six months, in June of year t, we use the collected coupon to buy a fraction of
a new 18-year maturity bond, again having face value FV and sold at par, offering the
average yield of the year, \ {YTMt_x + YTMt ) .
The following December, December of year /, after receiving the second coupon,
the fraction of the bond purchased with the first coupon and the original bond purchased a
year earlier are both sold at the price corresponding to the current yield YTM,.
We dénote by Vt the sum of the second coupon and the proceeds of the sale of the
bonds in December of year /. The value of V, is given by the following équation:
Vt =vMFV + ±YTM,_tFV f v +\(YTM,_, + YTM,)fjvk +v35
\k=0 k=0 (5.40)
where
v = (\ + \YTM,y\ (5.41)
66
We can calculate the rate of return realized in year t, denoted by g,, from the
following:
FVx(\ + gt) = V,, (5.42)
and obtain
8> FV (5.43)
We assume that funds are invested in bonds and stocks, in equal proportions. We
also assume that the portfolio allocation remains constant over the accumulation phase, that
is, up until retirement benefits start being paid. This implies an annual rebalancing of the
portfolio to ensure that the proportions in stocks and bonds remain at 50% over time.
Knowing the rates of return on bonds and stocks in year t, we can détermine the investment
return for year t, r,, as follows:
r,=\(g,+stock,). (5.44)
Figure 5.6 Return on long-term bonds
80%
70%
60%
50%
40%
30% ^
20%
10%
0% 1 -10%
-20%
-30%
-40%
-50% 1980 1990 2000 2010 2020 2030 2040 2050 2060
Year
- Historical values 1st percentile < 5th percentile - * - 25th percentile - 50th percentile -+- 75th percentile — 95th percentile - 99th percentile
Figure 5.7 Investment return (50% bonds and 50% stocks)
50%
40%
-10%
-20%
-30% W^ 1980 1990 2000
• * * * ■
2010 2020 Year
2030 2040 2050 2060
- Historical values 1st percentile - * 5 t h percentile -*-25th percentile - 50th percentile -+- 75th percentile — 95th percentile -—• 99th percentile
Figure 5.6 features the past and simulated values for the rate of return on long-term
govemment bonds, while Figure 5.7 shows the past and simulated values for the investment
return on the pension funds assumed to be composed of 50% bonds and 50% stocks. As
was the case for Figures 5.4 and 5.5, Figures 5.6 and 5.7 look reasonable overall, but the
range of simulated values is noticeably larger than the range of historical values.
Another way of comparing values observed in the récent past with simulated futures
is to compare their statistics. Table 5.1 gives différent statistics of the values observed
between 1981 and 2004, while Table 5.2 gives the corresponding statistics derived from the
1000 scénarios generated for the years 2005 to 2054. As our focus is not on the économie
model itself but on the outeomes with différent pension plans and working historiés, we are
comfortable with those scénarios and are confident they will allow us to perform the
comparisons we are interested in.
Table 5.1 Historical statistics
Variable Average Médian Standard déviation Minimum Maximum
Inflation 3.71% 2.74% 2.88% 0.20% 12.40% Wage growth 3.62% 2.69% 2.74% 0.98% 11.94% T-bills 7.80% 6.95% 3.81% 2.55% 17.01% YTM on bonds 8.44% 8.76% 2.76% 4.86% 15.27% Return on bonds 12.38% 13.36% 11.65% -10.46% 42.98% Common stock 10.03% 10.02% 15.20% -14.80% 35.49% Investment return 11.20% 11.88% 10.54% -6.63% 27.71%
Table 5.2 Simulated statistics
Variable Average Médian Standard déviation
lst
percentile 99th
percentile Inflation 2.49% 2.40% 1.45% -0.48% 6.29% Wage growth 2.50% 2.42% 1.50% -0.66% 6.35% T-bills 5.57% 4.85% 3.70% 0.58% 18.84% YTM on bonds 6.70% 6.59% 2.21% 2.05% 12.40% Return on bonds 7.62% 6.40% 20.43% -34.76% 60.38% Common stock 10.55% 10.39% 14.81% -23.23% 45.68% Investment return 9.08% 8.66% 12.17% -17.63% 39.06%
Chapter 6 Calculation of Pension Benefîts In this chapter, we look at the équations used to calculate pension benefîts for
différent work patterns, under différent defmed benefit (DB) and defined contribution (DC)
plans, under each of the économie scénarios generated in the previous chapter.
We start by introducing the différent working patterns we will consider. We then
look at the pension plan designs used in this study. We also présent the simplifying
assumptions that we made. We complète this chapter by deriving the équations necessary
to calculate the retirement benefîts.
6.1 Working Historiés We consider several working patterns for an employée who joins a pension plan at
âge 25 (at the beginning of 2005, the first year for which économie scénarios were
generated) and who does not start collecting benefîts until âge 65 (at the beginning of
2045). We consider the following working patterns, adapted from Cooper (1997):
• Full-time, no breaks in employment
In thèse historiés, we assume the employée works at least full-time from âge 25
to âge 65, without any interruption in the working career.
o Full-time employment throughout the working career;
o Full-time and bonus - we assume the employée receives a 20% bonus for
working overtime until âge 50 and then continues with full-time type of
employment until âge 65.
• Full-time, one break in employment
In thèse historiés, we assume that the employée begins working at âge 25, has a
subséquent break in employment history, then returns to a full-time employment
after the break and keeps working until âge 65, with the following variations:
70
o Early break and full-time - we assume the worker takes a ten-year break
from full-time employment early in the working career, at âge 30;
o Late break and full-time - we assume the worker takes a ten-year break from
full-time employment late in the working career, at âge 35;
o Short break and full-time - we assume the worker takes a five-year break
from full-time employment late in the working career, at âge 35;
o Long break and full-time - we assume the worker takes a fifteen-year break
from full-time employment late in the working career, at âge 35.
• Full-time, multiple breaks in employment
In thèse historiés, we assume the employée has more than one break in
employment between âges 25 and 65. This includes the possibility of a break
starting at âge 25, which actually means that the person does not start working at
âge 25. (Even if the person starts working later, he or she is assumed to be 25
years old at the beginning of the simulations, in 2005.)
o Late break - we assume the employée takes a five-year break from full-time
employment twice in the career, at âges 35 and 50, each time returning to a
full-time employment, then ceases working at âge 60;
o Late start - we assume the employée starts full-time employment at a later
âge, at âge 30, and later takes a five-year break from full-time employment,
at âge 40, then keeps working full-time until âge 65;
o Cyclical - we assume that the employée alternâtes between five-year periods
of full-time employment and five-year breaks, thus starting to work at âge
25 and quitting for the last time at âge 60.
71
• Part-time, at least one break in employment
In thèse historiés, we assume the employée works part-time for part of the career
and takes at least one complète break from employment.
o Break and part-time - we assume the employée begins by working full-time
at âge 25, but has a five-year break in employment history, at âge 35, after
which he or she returns to a part-time employment, working at 40% the
équivalent of full-time for ten years and at 75% the équivalent of full-time
for the remaining 15 years until retiring at âge 65;
o Break and mixed - we assume the employée starts full-time employment at
âge 25, has a first five-year break in employment at âge 35, résumes
working at 40% the équivalent of full-time for five years, takes another five-
year break at âge 45, résumes working at 75% the équivalent of full-time for
five years and then works full-time for five more years, thus ceasing to work
at âge 60.
• Full-time, changes of employer
In addition to the historiés defined by Cooper, we consider benefits that arise for
an employée who changes jobs frequently in her or his career but is always
considered to be in a full-time employment, from âge 25 to 65.
o Full-time, changes every ten years - we assume the employée changes
employers every ten years.
o Full-time, changes every five years - we assume the employée changes
employers every five years.
Ail those historiés reflect working patterns of women because women tend to hâve
breaks in their careers to care for children or aging relatives, but they also can be used to
consider various situations in employment history for both maie and female employées.
For example, a redundant employée who loses his or her job may not be able to find new
employment, or may only find temporary or part-time employment. One can also think of
72
the working pattern of immigrants, or of a two-breadwinner family where one career
dominâtes the other and indirectly dictâtes the pattern of the other.
6.2 Pension Plans We are interested in the retirement benefits that arise under différent pension
arrangements. Regardless of the pension plan, we assume immédiate full vesting. We also
assume there is neither revaluation of the benefits between quitting and commencing
benefits nor indexation of benefits during retirement. Besides, none of the plans considered
features intégration with public pension. Also, because of our focus on retirement benefits,
we implicitly assume that the pension plans offer no other ancillary benefits.
The following pension plan designs are considered:
• total service pension plan;
• final salary pension plan;
• career average pension plan;
• defined contribution pension plan.
We described the last three plan designs in Section 3.1. We add the total service
pension because we will use it as a benchmark. The total service pension is the pension
calculated using the final salary formula ignoring breaks in service and changes of
employer, that is, assuming that the employee's service is completed in consécutive years,
remaining with the same employer throughout the career. Hence, it is another kind of
defined benefit pension plan.
To calculate the total service pension (TSP) for an employée, we need to know the
employee's salary just before retirement and the total number of years spent in the
workforce. If the employée had breaks in her or his working career, we do not keep track
of salaries every time a break occurred as we normally would in order to détermine the final
salary pension (FSP) (especially if the employée changes job after the break). The final
salary pension (FSP) and the total service pension (TSP) are the same when there is no
break in employment.
73
For example, let us calculate FSP and TSP for an employée who is retiring after 40
continuous years of full-time service, has a final salary of $80,000 and a pension benefit of
2% of salary for each year of service.
• FSP: Since there were no breaks in employment, we calculate the employee's
FSP as foliows:
2% x 40 x $80,000 = $64,000. (6.1)
• TSP: Since we know that employée spent 40 years in the workforce and the
employee's final salary was $80,000, we calculate TSP as follows:
2% x 40 x $80,000 = $64,000 . (6.2)
We can see that, in this case, FSP is equal to TSP.
Let us now calculate FSP and TSP for an employée who had a break in
employment. For example, after 10 years in service and a last salary of $40,000, we
assume the employée takes a five-year break and returns to a full-time employment until
retirement âge. The last salary before retirement is $80,000 and the pension benefit is 2%
of salary for each year of service.
• FSP: Since there was a break in employment, we calculate the employee's FSP
as follows, considering each continuous period separately:
2% x 10 x $40,000 + 2% x 25 x $80,000 = $48,000. (6.3)
• TSP: Since we know that employée spent total of 35 years in employment and
the employee's final salary was $80,000, we calculate TSP as follows:
2% x 35 x $80,000 = $56,000. . (6.4)
In this case, we can see that, not only are the two pension amounts différent, but also TSP is
greater than FSP. This results directly from a lower salary being considered for the first ten
years of service under the final salary pension plan.
74
For ail the defined benefit pension plans under considération, we assume the accrual
factor to be 2%. For the defined contribution plan, as was indicated earlier, we assume an
investment policy of 50% in bonds and 50%) in stocks, with annual rebalancing.
6.3 Assumptions In the previous chapter, we covered ail assumptions related to the économie
variables that influence the retirement benefits. In this section, we discuss the other
éléments that influence the retirement benefits accrual, such as earnings, mortality, form of
annuity and contribution rate.
Ail calculations were performed assuming a "genderless" employée, one that is
neither maie nor female. In fact, for ail the components that hâve différent values for maies
and females, such as mortality and salary scale, we hâve used the average of the two values.
6.3.1 Service
For full-time employment, we assume service from âge 25, at the beginning of
2005, to âge 65, at the beginning of 2045. We assign a value of 1 to a year of full-time
employment.
For part-time employment, we assume either 40% or 75% of full-time hours per
week and we accordingly assign a value of 0.4 or 0.75 to a year of part-time employment.
For full-time employment with overtime or bonus considered as salary as per the
terms of the plan document, we assume 120% of full-time hours per week and we assign a
value of 1.2 to each year in which such overtime or bonus is paid.
Years of unemployment are assigned a value of 0.
Pension payments are assumed to start at retirement âge 65 for ail working historiés
under ail pension arrangements. In particular, even if a person ceases working earlier, say
at âge 60, pension benefits only start at âge 65. The working historiés, described in Section
6.1 and used with our simulations, can be found in Appendix C.
75
6.3.2 Earnings and Salary Scale
In addition to the wage increase that is due to wage growth (which we use as the
basis to project salaries, hence pension benefits), we also use two salary scales to account
for merit and project the salary: a salary scale that dépends on âge and another salary scale
that dépends on expérience. This way we can project two types of pension benefits, one
based on the employée's âge and another based on the employée's expérience, and compare
the two.
We obtained age-dependent salary scales for maies and females from Professor
Louis Adam's course "ACT-20765: Coût et financement des régimes de retraite" (Winter
2006 term) web site. We used those two scales to calculate an average age-dependent
salary scale for the genderless employée and then, from that one, we derived an experience-
dependent salary scale. Age-dependent scales can be found in Appendix D. In fact, the
same scale is used for both types of dependency. With age-dependency, the person changes
level every year, whether working or not. With experience-dependency, the person moves
up a level only after having accumulated one full additional year of expérience.
For an employée currently aged x in year t, we assume Sal25 2005 = $30,000 and we
project full-time salaries from year to year using the following équation:
Salx+X,+, = Salx, x - a L x (1 + waget ) , (6.5) • ssx
where
Salx, is the salary at âge x in year t;
Salx+] ,+1 is the salary at âge x + 1 in year t + 1 ;
ssx is the salary scale at âge x;
ssx+] is the salary scale at âge x + 1 ;
wage, is the rate of wage growth in year /.
We used Equation 6.5 to project ail the pre-retirement salaries. Whereas values of
the salary scale do not vary with time but only dépend on âge (or expérience), the rate of
wage growth varies over time and is différent in each of the 1000 économie scénarios.
76
As has been indicated earlier, under the age-based salary scale, we assume that
breaks in service hâve no effect on subséquent salaries in the sensé that current salaries are
not impacted in any way by previous breaks in service. So when the employée re-enters
employment, she or he will receive the same salary she or he would hâve received if there
had been no break.
Under the experience-based salary scale, breaks do hâve an impact on future salaries
in the sensé that, after returning from a break, the worker will enjoy increases due to wage
growth that were observed over that period of time but will not be given any crédit for the
years not worked as far as the salary scale goes.
For example, assume we are in the year 2000 and are interested in the salary of an
employée now aged 50 that has worked full-time since âge 25 except for a 5-year break in
service between âges 35 and 40. Under the age-based salary scale, he would received the
same salary as an employée now aged 50 that has worked full-time since âge 25 without
any break in service. Under the experience-based salary scale, however, he would receive
the same salary as an employée now aged 45 that has worked full-time since âge 25 without
any break in service.
6.3.3 Mortality
Mortality is ignored throughout the analysis except to calculate the annuity factor at
retirement. In other words, we assume mortality applies only after retirement. For this
purpose, we used the UP-94 mortality table and projected it to 2015 using the AA scale
(Society of Actuaries UP-94 Task Force 1995). We calculated a genderless mortality table
as the average of maie and female mortality tables. The actual mortality table used can be
found in Appendix E.
Other décréments that can affect pension plan benefits are also ignored.
6.3.4 Annuity
We only consider the case of the single life annuity. Thus, we implicitly assume
that, at the time of retirement, either the worker has no spouse or the spouse agrées in
writing that the worker will take a single life annuity instead of a joint life annuity.
77
Moreover, we assume that the worker does not elect any form of guarantee on the annuity.
Hence, retirement benefits are paid only as long as the worker is alive. Also, although
annuity payments typically are made on a monthly basis, we make the simplifying
assumption that they are annual.
We calculate annuity factors based on the assumed mortality table and interest rates
obtained from simulated YTM values. For an employée who begins working at âge 25 in
year t, the annuity factor effective when benefits begin at âge R under scénario i is
àR = V k=0
1 kpR,fmi = 1,2,. ..,1000, (6.6)
where
l + ytmt+(R_25)i)
y*mt+(R-2ï\i + y*mi+(R-25)-l,i + y*mt+(R-2S)-2,i ytmtHR-25)j = ■ 3 ( 6 - 7 )
is the average of the last 3 yields to maturity known at the time that pension benefits start,
and
*/>*-fia-**♦,) (6-8) is the probability that a person R years old will be alive at âge R + k.
An average of yields to maturity is taken to reflect the fact that insurance companies
consider current bond market conditions in setting their annuity rates but probably do some
kind of smoothing so as to dampen changes over time.
Because we hâve assumed that retirement benefits would begin at âge 65 regardless
of when the person would stop working, we effectively hâve that R is equal to 65 for ail
working historiés. Hence the yields to maturity that will enter the calculation are those that
will be effective in 40, 39 and 38 years from now, respectively, according to Equation 6.7.
On the Immédiate Annuities web site, we found that the average annuity factor, for
an accumulated fund of $100,000, was about 14.0. However, the average of the latest three
yields to maturity we could observe was a bit too high to match that annuity factor. We
found that subtracting 1.5% from the YTM values brought us closer to current annuity
78
priées. Thus, the actual formula used to ealculate annuity faetors (modified Equation 6.6)
is
"l+(fl-25),i 1—1 k=0
!
Kl + ytmtHR_25)l,- 0.015 j
along with Equations 6.7 and 6.8.
pR, for Z = l ,2, . . . , 1000, (6.9)
Using this modification, we better reproduced the current annuity factor. Also, over
the 1000 scénarios we generated, we obtained an average annuity factor (at âge 65, in 40
years' time) of about 12.3.
6.3.5 Contribution Rate
To be able to compare results from DB plans with results from DC plans, we need
to find a contribution rate to the DC plan that will make the comparison meaningful.
In a first attempt to find such a rate, the contribution rate to the DC pension plan
was calculated assuming that the actuarial présent value of pension benefits (to be received
starting upon the employée's retirement at âge 65) for a full-time employée with 40 years
of service from a DB final salary pension plan with a 2% accrual rate should satisfy the
following equality: 64
^vJ"25x/xSa/,, (+( I-25) = 2%x40xSal64,l+39 xv40 xà6i, (6.10) v=25
where / stands for the contribution rate to be determined and Sal x,t+(x-25) is the average
salary at âge x over the 1000 simulations. The annuity factor (à65) and the discounting
factor (vx~25) are calculated assuming a 6% interest rate. It is about 1% lower than the
average YTM value used to ealculate the annuity factor at âge 65; that is, 1000
1 1000
/ = 1
f 65 \
V *=63 )
- 1 % « 6 % . (6.11)
Hence, there is some conservatism in the assumptions used in the first détermination of the
contribution rate.
Solving for/gives:
2% x 40 x Sal64>(+39 x v40 x a65 / s ~ « l i / o .
/ v Salx,t+(x-2S) x=25
Hence, we initially considered using a 13%) contribution rate.
After preliminary calculations with this contribution rate, we realised that it was
much higher than the contribution rate needed to achieve an average 80%> replacement ratio
for a full-time history. We felt this would make our comparison of replacement ratios
difficult and that it would be préférable to find a contribution rate which would lead to
comparable average replacement ratios for a full-time history.
By trial and error, we found that a contribution rate of 7.5%> would achieve a
replacement ratio of 50% or higher for ail working historiés. Also, for a person working
full-time over the whole 40 years, such a contribution rate would lead to an average
replacement ratio only slightly above 80%>. (A 7% contribution rate would hâve led to an
average replacement ratio below 80% for that person.) Therefore, a 7.5% contribution rate
was used to generate defined contribution benefits (and it was kept constant throughout the
employée's working history).
6.4 Pension Benefits Everything is now in place so we can consider how to calculate pension benefits for
the différent working historiés and pension plans, under the many différent scénarios. We
look at the formula that corresponds to each of the pension plans under considération in the
following subsections.
Our ultimate measure of interest is the replacement ratio, which is the retirement
benefit divided by the employee's final salary. In Canada, the target total replacement ratio
(from ail possible retirement income sources, that is, from both public and private
programs) is generally considered to be about 70% of the pre-retirement salary. (The target
replacement ratio actually is higher for lower incomes.)
79
(6.12)
80
6.4.1 Total Service Pension
The total service pension assumes an annual pension benefit of 2% of the very last
salary rate at the time of retirement. Any break in service and any change of employers are
ignored; this means that ail periods of service are considered consécutive and treated as a
single period of continuous service. The total service pension (TSP) is calculated as
follows: 64
TSP = 2%xSalMMM.25) x ^ X , (6.13) x=25
where
SalM ,+(64_25) is the salary received in the year before retirement;
Hx is the value assigned to employment at âge x.
Note that the values taken by Hx between âges 25 and 64 dépend on the working
history considered and can take on any of the following values: 0, 0.4, 0.75, 1 and 1.2. The
actual values are found in Appendix C.
The value obtained for TSP varies by scénario because the final salary varies
according to the différent rates of wage growth simulated. However, the replacement ratio
is constant because dividing TSP by the last salary yields a constant which dépends only on
the accrual rate, 2%, and the values that describe the working history.
6.4.2 Final Salary Pension
For the final salary pension plan, we assume that the annual pension benefit is 2%
of the final salary for each year of service.
In real life, when there is a break in service and the person résumes service with the
same employer, a question is raised as to whether separate periods of work are considered
as a whole or separately.
If ail the separate periods are treated as a whole, it means the pension benefit will
résume accruing every time the person résumes service and the benefit calculated under the
final salary pension plan turns out to be calculated as a total service pension. This does not
81
mean that breaks in service are counted towards service. It does mean however that
separate periods are bundled together and the very final salary (the final salary in the last of
those separate periods) is used to calculate the benefit for ail of them.
We actually will consider that separate periods of work are considered separately.
We adopt this point of view so as to make further comparisons possible. This also allows
us to account for the fact that the person may not résume service with the same employer
but start working for another. It also is possible that, despite resuming service with the
same employer, separate periods are treated separately in a given pension plan.
Under that assumption (regarding the treatment of the breaks in service), the
pension benefit earned in any given period of work (between any two breaks, whether the
employment is full-time or part-time or a combination of the two in between) is given by
FSPy=2%xSaly„Kl+(y„]_25)xfjHx (6.14)
where
y is the exit âge from a particular employment (or âge at the
beginning of the break in service);
Sal x r+(>,_i_25) is the salary received in the last year of continuous service
(whether full-time or part-time);
w is the entry âge with the particular employer (or âge at the
beginning of that period of continuous service);
Hx is the value assigned to employment at âge x.
The total pension benefit FSP received at retirement then is the sum of the benefits
earned in separate periods of continuous employment. The value obtained for FSP varies
by scénario because the final salaries that enter the calculation vary according to the
différent rates of wage growth simulated. Moreover, insofar as more than one separate
employment period is considered, the replacement ratio is variable because dividing FSP
by the last salary générâtes a weighted sum of ratios of différent salaries.
82
6.4.3 Career Average Pension
For the career average pension plan, we again assume a constant accrual rate of 2%
and we use the career average benefit formula: 1 64
CAP = 2% x (65 - 25) x — — £ Hx x SalXJ+(x_25) 6 5 ~25x-25 (6.15)
= 2%x^HxxSalxMx.2 x=25
where
Hx is the value assigned to employment at âge x;
Salx ,+(x_25) is the salary received at âge x, in year t + (x - 25).
As was noted earlier, the person does not necessarily begin working at âge 25, but
ail the simulations are started assuming the person is âge 25.
Using a reasoning similar to that used earlier for the final salary pension, both the
values obtained for CAP and the corresponding replacement ratios vary by scénario.
Indeed, the career average pension is équivalent to calculating a final salary pension for
someone who would work with a new employer every year.
6.4.4 Defïned Contribution Pension
Our last pension plan under considération is the defïned contribution pension plan.
Since such plan defines the contribution rather than the benefit, we first need to find out
how funds accumulate over time before we can calculate the benefits we can buy with those
funds at retirement.
We do not take into account any expenses related to the pension plan administration
or the management of the funds; that is, no fées are subtracted from the investment returns
we hâve generated. Thus, we implicitly assume that the employer pays for the pension plan
administration and that management fées are negligible.
We also assume that contributions are paid in the middle of each year during the
employee's working lifetime and that investment income is generated until the employee's
83
retirement âge. Then the value DCA of the DC account at the time of buying the annuity, at
âge 65, can be expressed as follows: 64 , 64
DCA = fxJjHxSalxMx_25)(l + rlHx_25)yY\(l + rl+{lt_25)) (6.16) x=25 u=x+\
where / is the contribution rate;
Hx is the value assigned to employment at âge x;
Salx t+(x_25) is the salary received at âge x, in year t + (x- 25);
rt+(x__25) is the investment return in year t + (x - 25).
The DC pension, denoted by DCP, is calculated simply by dividing the accumulated
DC fund, DCA, by the annuity factor â65 (both components are simulation-specific):
DCA DCP = ^ - . (6.17)
«65
The value obtained for DCP varies by scénario because the salaries that enter the
calculation vary according to the différent rates of wage growth simulated and also because
the investment returns vary by scénario. The replacement ratio also is variable over the
scénarios for the same reasons.
Chapter 7 Results In this chapter, for each of the thirteen working historiés introduced in chapter 6, we
look at the results of the calculation of the pension benefits under the différent scénarios
generated in Chapter 5, as provided under the foliowing pension plan arrangements:
• total service (TS) pension plan;
• final salary (FS) pension plan;
• career average (CA) pension plan;
• defined contribution (DC) pension plan.
This will allow us to study how différent working historiés are affected by différent
retirement Systems.
In most of the pension plan arrangements under considération, it matters little
whether the person is always with the same employer or changes employers at times other
than after returning to work after a break. Indeed, for a career average pension plan, ail
salaries are considered and it matters little whether ail service pertains to one plan or many
(so long as benefits are ail vested). Likewise, for a defined contribution pension plan, ail
other things being equal, it does not matter whether ail contributions accumulate in only
one fund or in many separate ones.
However, in a final salary pension plan, changing employers can be very hurtful as
this affects the salary on which the pension benefit is based. To illustrate and quantify the
impact, retirement benefits for an employée who always is in a full-time (FT) employment
but who changes jobs frequently are compared with the retirement benefits for an employée
who always is in a full-time employment and never changes jobs.
We start by presenting the results based on age-based merit and investigating how
well the différent pension arrangements compare to the total service pension (and to each
other). The results based on experience-based merit are presented in the second section of
this chapter.
85
7.1 Age-Based Merit The results using age-based merit under différent pension arrangements are
summarised in Tables 7.1 to 7.3, as well as Figure 7.1.
In Table 7.1, the main entry is the average replacement ratio (benefit as a percentage
of the last salary rate). The value in parenthèses is the standard déviation of the
replacement ratio. Under the total service pension plan, the standard déviation is nil since,
given the working history and final salary rate, the benefit amounts to a constant
replacement ratio.
Table 7.1 Average and standard déviation of replacement ratio (age-based merit)
History Last
âge
wor
ked
Yea
rs o
f se
rvic
e
DB plan
DC plan History Last
âge
wor
ked
Yea
rs o
f se
rvic
e
TS FS CA DC plan Full-Time and Bonus (FTB) 64 45 90% (0%) 90% (0%) 50% (4%) 99% (43%)
Full-Time (FT) 64 40 80% (0%) 80% (0%) 46% (4%) 85% (36%)
Full-Time, Changes Every 10 Years (FT 10) 64 40 80% (0%) 52% (3%) 46% (4%) 85% (36%)
Full-Time, Changes Every 5 Years (FT5) 64 40 80% (0%) 49% (3%) 46% (4%) 85% (36%)
Short Break and Full-Time (SBFT) 64 35 70% (0%) 57% (1%) 42% (3%) 71% (30%)
Early Break and Full-Time (EBFT) 64 30 60% (0%) 53% (0%) 38% (3%) 55% (21%)
Late Break and Full-Time (LBFT) 64 30 60% (0%) 47% (1%) 37% (3%) 61% (25%)
Late Start (LS) 64 30 60% (0%) 48% (1%) 38% (3%) 54% (21%)
Break and Part-Time (BPT) 64 25% 50% (0%) 38% (1%) 29% (2%) 56% (25%)
Long Break and Full-Time (LLBFT) 64 25 50% (0%) 37% (1%) 31% (2%) 52% (23%)
Late Break (LB) 59 25 50% (0%) 33% (2%) 29% (2%) 70% (31%)
Break and Mixed (BM) 59 203/4 42% (0%) 28% (1%) 25% (2%) 58% (26%)
Cyclical (C) 59 20 40% (0%) 25% (1%) 24% (2%) 54% (24%)
86
To help visualize the results of Table 7.1, we generated Figure 7.1. Each circle is
centered at the average replacement ratio, while the area of the circle is proportional to the
standard déviation of the replacement ratio. (In cases where the standard déviation is nil,
the area of the circle is made nonzero so as to be visible.)
Figure 7.1 Visual représentation of average and standard déviation
of replacement ratio (age-based merit)
120% i - - - • ,
100%
tio
80% re Q£
+■»
c F 60% 0) o <U Q.
a. 40%
20%
0%
çP? 8 8 o
FTB FT FT10 FT5 SBFT EBFT LBFT LS BPT LLBFT LB BM C
History
®Defined Contribution • Total Service «Final Salary OCareer Average
Whereas the average replacement ratio is an interesting value in its own right, the
standard déviation does not provide an accurate idea of the variability of the replacement
ratio, because its distribution is skewed.
So as to get a better notion of how variable the replacement ratio can be, we
generated Table 7.2 which contains the intervais spanned by the 5l and 95n percentiles
obtained over the 1000 scénarios for each of the working history and pension plan
considered.
87
For the total service pension, the interval is only one point since the replacement
ratio is a constant.
Table 7.2 Intervais for the replacement ratio (age-based merit)
History Last
âge
wor
ked
Yea
rs o
f se
rvic
e
DB plan DC plan History La
st âg
e w
orke
d
Yea
rs o
f se
rvic
e TS FS CA DC plan
FTB 64 45 (90%; 90%) (90%; 90%) (43%; 57%) (45%; 180%) FT 64 40 (80%; 80%) (80%; 80%) (40%; 52%) (39%; 153%) FT10 64 40 (80%; 80%) (47%; 58%) (40%; 52%) (39%; 153%) FT5 64 40 (80%; 80%) (43%; 54%) (40%; 52%) (39%; 153%) SBFT 64 35 (70%; 70%) (55%; 59%) (37%; 47%) (34%; 128%) EBFT 64 30 (60%; 60%) (52%; 54%) (34%; 43%) (27%; 95%) LBFT 64 30 (60%; 60%) (45%; 49%) (33%; 41%) (29%; 109%) LS 64 30 (60%; 60%) (47%; 50%) (34%; 42%) (26%; 93%) BPT 64 25% (51%; 51%) (36%; 39%) (26%; 33%) (26%; 105%) LLBFT 64 25 (50%; 50%) (35%; 39%) (28%; 34%) (24%; 96%) LB 59 25 (50%; 50%) (29%; 36%) (25%; 32%) (31%; 130%) BM 59 2oy4 (42%; 42%) (26%; 30%) (22%; 27%) (25%; 111%) C 59 20 (40%; 40%) (23%; 28%) (21%; 27%) (24%; 98%)
Ultimately, when a DC plan is substituted for a DB plan, what the plan member
really cares to know is whether the new plan will provide him or her with better retirement
benefits than the old one, for his or her own working history and for the one and only
économie scénario that will corne to materialize.
Because the future is uncertain, we do not know the exact answer to that question.
However, for each of the 1000 scénarios, we can check whether the benefit under one plan
is better than that under another.
For each of the working historiés and defined benefit pensions plans considered,
Table 7.3 features the probability that the DC plan provides greater benefits than the given
DB plan. We found that probability by dividing the number of scénarios for which the DC
pension turns out to be greater than the given DB pension by 1000.
88
Table 7.3 Probabilities of DC plan faring better than DB plan (age-based merit)
History Last âge worked
Years of service
DBplan History
Last âge worked
Years of service TS FS CA
Full-Time and Bonus (FTB) 64 45 50% 50% 93% Full-Time (FT) 64 40 47% 47% 92% Full-Time, Changes Every 10 Years (FT10) 64 40 47% 85% 92% Full-Time, Changes Every 5 Years (FT5) 64 40 47% 89% 92% Short Break and Full-Time (SBFT) 64 35 42% 65% 89% Early Break and Full-Time (EBFT) 64 30 33% 45% 79% Late Break and Full-Time (LBFT) 64 30 42% 68% 87% Late Start (LS) 64 30 30% 53% 78% Break and Part-Time (BPT) 64 25% 52% 78% 93% Long Break and Full-Time (LLBFT) 64 25 45% 74% 88% Late Break (LB) 59 25 72% 94% 97% Break and Mixed (BM) 59 203/4 72% 94% 96% Cyclical (C) 59 20 69% 94% 96%
7.1.1 Total Service Pension
Ail historiés under the total service pension do relatively well. This is because
service is assumed to be completed in consécutive years; that is, the breaks in service are
ignored and ail that matters is the service. As the benefits are based on service, the final
salary, which is typically the highest, is applied to ail years of service. Under the total
service pension, historiés with the same length of service hâve the same average
replacement ratios. For example, historiés with 30 years of service (EBFT, LBFT and LS)
hâve an average replacement ratio of 60%; similarly, historiés with 25 years of service
(LLBFT and LB) hâve an average replacement ratio of 50%. Again, the standard déviation
is nil for the total service pension as per the définition of the benefit.
7.1.2 Final Salary Pension
Benefits under the final salary pension do not perforai as well compared to the total
service pension because it accounts for breaks in service. Every time there is a break in
employment, benefits are calculated based on the last salary just before the break, which
generally is lower than the final salary which would hâve been used if there had been no
break. This is why we can observe some variability (although, not high) in the results
under the final salary pension plan. The variability cornes from the variability in wages
89
because, in historiés with breaks in service, there are a few more salaries used in the
calculation of benefits.
The impact of fréquent job changes is best illustrated considering employées who
are in full-time employment throughout their career but who change jobs frequently. Their
benefits under the final salary pension are greatly devalued compared to the total service
pension. More fréquent changes resuit in lower benefits.
Changing employers every five years would resuit in an average 49% replacement
ratio compared to 52% if the changes occurred every ten years, a réduction (in relative
terms) of 39% and 34% respectively compared to 80% under both the total service
arrangement, which does not take into account job changes, and the final salary
arrangement, in the case that the service has been completed with one employer. (When
comparing changes every ten years with changes every five years, the longer ten-year
periods spent with a particular employer allowed for higher final earnings to be used for
certain sub-periods and hence higher benefits.)
Historiés that appear to do relatively well under the final salary pension plan, other
than FTB and FT, are those with short and early breaks in employment (SBFT and EBFT)
and LS and LBFT.
We can also see that historiés with the same years of service do not necessarily hâve
the same replacement ratios as would otherwise be the case under the total service plan.
For example, EBFT, LBFT, and LS ail hâve 30 years of service but average replacement
ratios of 53%, 47% and 48% respectively (a relative réduction of 12%, 22% and 19%
respectively compared to their respective ratios under the total service pension). This
différence cornes from the way benefits are calculated.
Thèse historiés illustrate the following principle, observed from the results: historiés
with longer service (especially in the years before retirement) and short breaks in
employment hâve better benefits. As a matter of fact, historiés with early and short breaks
generally hâve better benefits because early, typically low, salaries are applied to a shorter
service.
90
EBFT has the highest replacement ratio of the three because salary at âge 29, which
is relatively low, is applied to only five years of service and salary at âge 64, the highest
salary, is applied to twenty-five years of service. EBFT also has the lowest variability of
the three because there is less variability in earlier salaries compared to later salaries which
inherit variability from previous years' salaries. LS has a somewhat higher replacement
ratio than LBFT because the benefits are calculated using ten years of salary at âge 39 and
twenty years of salary at âge 64 compared to benefits for LBFT which are based on ten
years of salary at âge 34 (which is typically lower than salary at âge 39) and twenty years of
salary at âge 64.
Historiés that do not appear to do so well (either compared to other historiés under
the final salary plan or compared to the total service plan) are those with long or late
breaks. Thèse historiés highlight how important it is to hâve a long period in employment
before retirement under the final salary plan. A long break makes the period just before
retirement relatively smaller. As a conséquence, the final, the larger salary is applied to a
short period. In the case of late breaks, an early, lower salary is applied to more years than
if the break had been earlier. (Historiés with a combination of part-time employment and
break are also affected by a break.)
As it is the case under the total service plan, BPT actually has a little bit higher
average replacement ratio than LLBFT (actually exactly 0.5% larger). This is because BPT
has somewhat longer service (towards the end of career) tied to the same final salary (15.25
years compared to 15 for LLBFT). Thèse historiés hâve the same variability (under the
final salary plan) because, for both historiés, the variability cornes from the same salaries,
salary at âge 34 and salary at âge 64.
LB, BM and C historiés hâve low replacement ratios under the final salary plan
because they hâve shorter service and fewer years of service before retirement tied to the
final salary. LB has the highest replacement ratio of the three but also the highest
variability, because the variability of the salary at âge 34 seems to be higher than the
variability of the salaries at âges 29 and 39 combined.
91
7.1.3 Career Average Pension
Benefits under the career average arrangement produce the lowest average
replacement ratio for ail working historiés and ail defined benefit arrangements considered.
This is because the career average formula gives equal weight to ail of the employee's
earnings received over time (producing small average earnings and hence benefits) and the
final earnings are relatively high (compared to average earnings). In other words, the
benefit accrued in any given year dépends on the salary earned that year, regardless of past
or future service and earnings; besides, the salaries earned earlier in the career are typically
significantly smaller than the salaries earned towards the end of the career; ail that leads to
smaller replacement ratios than those provided under final salary pension plans. The career
average pension has the highest variability factor of ail DB arrangements because more
salaries, ail of which are variable, enter the calculation.
Compared to the total service plan, benefits for FTB and FT under the career
average plan are greatly devalued. The relative réduction is 45% and 43% respectively.
Historiés that do relatively well (not well) under the final salary plan also do
relatively well (not well) under the career average plan. LLBFT does a bit better than BPT
(2% better) under the career average plan (under the final salary plan, BPT actually has a
0.5% higher replacement ratio). This is because LLBFT is in full-time employment in
years before retirement so there is more weight on the later, close to retirement, salaries
compared to the weight that BPT has in those years. We also observe higher variability
with BPT, because more salaries, thus higher variability, enter the calculation.
7.1.4 Defîned Contribution Plan
The defined contribution (DC) plan produces the highest average replacement ratio
for almost ail historiés compared to DB plans. For a DC plan, the early contributions are
very important. It does not matter that the contributions are made on lower salaries because
the fund grows with the investment return, which is on average greater than the salary
increase. However, the variability is also high under a DC plan. The variability arises not
only from the salaries (the coefficient of variation of the final salary is 0.17), but also from
92
the investment returns, which are highly variable (the coefficient of variation is as high as
1.85) and the annuity factors (their coefficient of variation is 0.16).
Historiés that do relatively well under this arrangement are those whose periods of
part-time employment occur later rather than earlier in the career and those whose break in
service occurs later in the career. In particular, thèse historiés are LB, BM and BPT. Their
replacement ratios, on average, are 70%, 58% and 56% respectively (a relative increase of
39%, 41% and 11% compared to their total service ratios, respectively). Ail of thèse
historiés hâve relatively short total service but ail assume full-time employment at the
beginning of the career so the largest proportion of the contributions is made in thèse early
years.
Historiés that do not do that well are EBFT and LS. Thèse historiés are among the
four historiés that do the worst under a DC plan, the other two being C and LLBFT. For
EBFT and LS, this cornes as no surprise because of the effect of early contributions under
DC plans. For LLBFT, the low replacement ratio is due to the fact that no contributions are
made for fifteen years due to unemployment. Cyclical (C) history does not do well because
no contributions are made during the periods of unemployment, especially from âge 30 to
âge 34. For EBFT and LS, participating in a DC plan instead of a total service plan results
in a relative 9% and 11% réduction in the average replacement ratio, respectively.
Even though DC plans produce the highest average replacement ratio for almost ail
historiés, they are not risk-free. Indeed, as can be seen in Table 7.1 or Figure 7.1, the
replacement ratios are highly variable. The results show that historiés that do well under
this arrangement compared to DB plans, namely LB, BM and BPT, also hâve high
variability (sometimes even higher than the average replacement ratios obtained under the
final salary plan). FTB and FT historiés hâve the highest average replacement ratios but
they also hâve the highest standard déviation, 43% and 36% respectively. On the other
hand, however, ail the historiés that do not well under a DC plan (EBFT, LS, LLBFT and
C) hâve a lower variability in replacement ratios (EBFT and LS hâve the lowest) compared
to other historiés that do well under a DC plan.
93
In addition to high variability, the probabihty that DC benefits will be greater than
total service or final salary benefits is relatively low, especially for those historiés that do
relatively well under the total service and final salary plans. For example, the probabihty
that the replacement ratio under a DC plan for FT is greater than 80% is only 47%. Two
other historiés that do not hâve a high probabihty of having a DC replacement ratio higher
than that obtained under another arrangement are EBFT and LS; compared to the total
service plan, the respective probabilities are 33% and 30%. Thèse results lead us to believe
that DB plans are more suitable for employées who stay with the same employer, if not
throughout their career, at least for a large number of years just before retirement.
A higher probabihty (of having a higher DC replacement ratio) is observed for
historiés that do relatively well under a DC plan compared to total service and final salary
plans. When the total service pension is considered, LB and BM historiés hâve the highest
probabihty (of having a higher DC replacement ratio), 72%, and when the final salary
pension is considered, LB, BM and C historiés hâve the highest probabihty (of having a
higher DC replacement ratio), 94%.
Also, the probabihty that a DC plan produces higher replacement ratios than the
final salary plan for a full-time history with fréquent job changes is relatively high: 85%, if
changes occur every ten years, and 89%, if changes occur every five years. Ail thèse
results lead us to believe that DC plans might be more suitable for mobile employées, as
well as for those with periods of part-time employment and a later break in the career.
However, because ail the risk associated with a DC plan, the employées might still prefer a
DB plan.
7.2 Experience-Based Merit The results using experience-based merit under différent pension arrangements are
summarised in Tables 7.4 to 7.6 and Figure 7.2.
In Table 7.5, the main entry is the average replacement ratio (benefit as a percentage
of the last salary rate). The value in parenthèses is the standard déviation of the
replacement ratio. Under the total service plan, the standard déviation is nil, as per the
94
définition of the benefit. In gênerai, the results with experience-based merit are very close
to the results obtained when benefits are calculated using age-based merit.
Table 7.4 Average and standard déviation of replacement ratio (experience-based merit)
History Last
âge
wor
ked
Yea
rs o
f se
rvic
e
DB plan
DCplan History Last
âge
wor
ked
Yea
rs o
f se
rvic
e
TS FS CA DCplan Full-Time and Bonus (FTB) 64 45 90% (0%) 90% (0%) 51% (4%) 101% (43%)
Full-Time (FT) 64 40 80% (0%) 80% (0%) 46% (4%) 85% (36%)
Full-Time, Changes Every 10 Years (FT10) 64 40 80% (0%) 52% (3%) 46% (4%) 85% (36%)
Full-Time, Changes Every 5 Years (FT5) 64 40 80% (0%) 49% (3%) 46% (4%) 85% (36%)
Short Break and Full-Time (SBFT) 64 35 70% (0%) 57% (1%) 40% (3%) 70% (29%)
Early Break and Full-Time (EBFT) 64 30 60% (0%) 53% (1%) 36% (3%) 53% (21%)
Late Break and Full-Time (LBFT) 64 30 60% (0%) 47% (1%) 35% (2%) 61% (26%)
Late Start (LS) 64 30 60% (0%) 48% (1%) 36% (3%) 52% (20%)
Break and Part-Time (BPT) 64 25% 50% (0%) 38% (1%) 28% (2%) 59% (26%)
Long Break and Full-Time (LLBFT) 64 25 50% (0%) 38% (1%) 30% (2%) 54% (24%)
Late Break (LB) 59 25 50% (0%) 33% (2%) 29% (2%) 73% (33%)
Break and Mixed (BM) 59 20% 42% (0%) 29% (1%) 26% (2%) 66% (30%)
Cyclical (C) 59 20 40% (0%) 27% (2%) 25% (2%) 59% (26%)
To help visualize the results of Table 7.4, we generated Figure 7.2. As for Figure
7.1, each circle is centered at the average replacement ratio, while the area of the circle is
proportional to the standard déviation of the replacement ratio. (Again, in cases where the
standard déviation is nil, the area of the circle is made nonzero so as to be visible.)
95
120%
Figure 7.2 Visual représentation of average and standard déviation
of replacement ratio (experience-based merit)
' • \ • i
° o 8 0 n ^ ® ^ u a o"a 0 *' " *
FTB FT FT10 FT5 SBFT EBFT LBFT LS BPT LLBFT LB BM C
History
©Defined Contribution • Total Service «Final Salary OCareer Average|
So as to get a better notion of how variable the replacement ratio can be, because the
replacement ratio does not follow a symmetrical distribution, we also generated Table 7.5.
That table contains the intervais spanned by the 5l and 95l percentiles obtained over the
1000 scénarios for each of the working history and pension plan considered.
To convince oneself of the skewness of the underlying distribution for the
replacement ratio, one simply has to check where the mean falls within each of the
intervais.
For the total service pension, the interval is only one point since the replacement
ratio is a constant.
96
Table 7.5 Intervais for the replacement ratio (experience-based merit)
History Last
âge
wor
ked
Yea
rs o
f se
rvic
e
DB plan
DC plan History Last
âge
wor
ked
Yea
rs o
f se
rvic
e
TS FS CA DC plan FTB 64 45 (90%; 90%) (90%; 90%) (44%; 58%) (46%; 183%) FT 64 40 (80%; 80%) (80%; 80%) (40%; 52%) (39%; 153%) FT10 64 40 (80%; 80%) (47%; 58%) (40%; 52%) (39%; 153%) FT5 64 40 (80%; 80%) (43%; 54%) (40%; 52%) (39%; 153%) SBFT 64 35 (70%; 70%) (55%; 59%) (35%; 45%) (33%; 125%) EBFT 64 30 (60%; 60%) (52%; 54%) (32%; 40%) (26%; 93%) LBFT 64 30 (60%; 60%) (46%; 49%) (31%; 39%) (28%; 110%) LS 64 30 (60%; 60%) (47%; 50%) (32%; 40%) (25%; 90%) BPT 64 25% (51%; 51%) (36%; 40%) (25%; 32%) (27%; 110%) LLBFT 64 25 (50%; 50%) (36%; 40%) (27%; 33%) (24%; 101%) LB 59 25 (50%; 50%) (30%; 37%) (25%; 33%) (32%; 137%) BM 59 203/4 (42%; 42%) (27%; 32%) (23%; 28%) (28%; 126%) C 59 20 (40%; 40%) (24%; 29%) (22%; 28%) (26%; 110%)
Moreover, for each of the working historiés and defined benefit pensions plans
considered, Table 7.6 features the probability that the DC plan provides greater benefits
than the given DB plan.
Table 7.6 Probabilities of DC plan faring better than DB plan (experience-based merit)
History Last âge worked
Years of service
DB plan History
Last âge worked
Years of service TS FS CA
Full-Time and Bonus (FTB) 64 45 52% 52% 93% Full-Time (FT) 64 40 47% 47% 92% Full-Time, Changes Every 10 Years (FT10) 64 40 47% 85% 92% Full-Time, Changes Every 5 Years (FT5) 64 40 47% 89% 92% Short Break and Full-Time (SBFT) 64 35 40% 63% 90% Early Break and Full-Time (EBFT) 64 30 29% 40% 80% Late Break and Full-Time (LBFT) 64 30 41% 67% 89% Late Start (LS) 64 30 28% 48% 80% Break and Part-Time (BPT) 64 25 VA 56% 79% 94% Long Break and Full-Time (LLBFT) 64 25 48% 75% 91% Late Break (LB) 59 25 74% 95% 98% Break and Mixed (BM) 59 203/4 78% 95% 98% Cyclical (C) • 59 20 75% 95% 96%
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7.2.1 Total Service Pension
The average replacement ratios under the total service pension with experience-
based merit are identical to the average replacement ratios for the total service pension with
age-based merit. This is, again, because the final salary is applied to ail years of service.
Hence, ail that really matters in determining the replacement ratio is the length of service.
The historiés that do well under age-based merit also do well under experience-based merit.
7.2.2 Final Salary Pension
The average replacement ratios and their variability under the final salary pension
with experience-based merit appear to be slightly higher than those with age-based merit
with the exception of LS. However, the actual amount of salaries is relatively lower when
benefits are calculated using merit based on expérience, because merit is (generally) lower
when it is experience-based than when it is age-based. Unless they work full-time and
without any break in employment, the employées do not gain as many years of expérience
as they âge. That is, unlike when merit is age-based, the part-time employment and breaks
in employment hâve a négative impact on salary growth when merit is experience-based.
With regard to performance, historiés that do well (not well) under the final salary
arrangement with age-based merit also do well (not well) with experience-based merit.
7.2.3 Career Average Pension
Benefits under the career average arrangement produce the lowest replacement ratio
for ail working historiés and ail DB arrangements considered. In gênerai, historiés that do
relatively well (not well) are the same historiés that do relatively well (not well) with age-
based merit.
The average replacement ratios and the average variability under the career average
pension with experience-based merit appear to be slightly lower than those with age-based
merit with the exception of FTB, BM and C. Lower replacement ratios under experience-
based merit generally are due to a lower merit applied to each year's salary. FTB is an
exception because a higher merit is gained by working 20% more each year for the first
twenty-five years. BM and C historiés seem to do better with experience-based merit
98
because they ail cease employment at âge 59, and salaries, in gênerai, after that âge, do not
increase much due to merit increases alone, whether we use âge or expérience as a base for
merit. In other words, higher merit increases are earned/achieved from âge 25 to âge 50, in
gênerai. From âge 50 to âge 60, merit still increases but not much, and starting with âge
60, merit increases décline to zéro.
7.2.4 Defined Contribution Plan
Again, average benefits under DC plans are the highest. The results suggest that,
generally, historiés that do relatively well with age-based merit do also well with
experience-based merit, sometimes even better (as is the case with FTB, BPT, LLBFT, LB,
BM and C). The results also suggest that historiés with fewer years of total service,
especially those that cease working at âge 59 (LB, BM and C), do better than with age-
based merit, and historiés that do not do well, and worse than with age-based merit, are
those with early breaks and late start. When using expérience to détermine merit, early
contributions look even larger compared to the final salary, since, overall, salaries do not
grow as high as when merit is based on âge. Thus, with experience-based merit, the
importance of early contributions is even greater.
As Table 7.4 indicates, the variability under DC plans is higher than the variability
under final salary and career average plans. Historiés that hâve higher replacement ratios
hâve higher variability and those that hâve lower replacement ratios hâve lower variability.
This confirms once again that DC plans are not risk-free.
We also observe a similar pattern in the distribution of benefits under a DC plan
compared to DB plans, as with the age-based merit. That is, historiés that do relatively well
under this arrangement generally are those whose periods of part-time employment occur
later rather than earlier in the career and those whose break in service occur later in the
career. Once again, the results lead us believe that DC plans are more suitable for mobile
employées. But since ail the risk associated with a DC plan rests with the employées,
employées then, especially those who are more risk averse, might still be in favour of DB
plans (or slightly lower but less variable pensions).
99
7.3 Summary of Results From the above tables, figures, and previous analysis, we can state some gênerai
principles:
• The longer the service, the higher the benefit, regardless of the type of plan.
• For DB plans, the greater the number of years spent in employment immediately
before retirement and the higher the final salary, hence the higher the benefit.
Early and short breaks, if any, are préférable under DB plans because lower
salaries are applied to shorter periods of employment.
• Among DB plans, the total service plan has the highest replacement ratios.
Final salary benefits fall short compared to total service benefits because the
benefits are calculated based on salaries just before the break, which are smaller
than the very last salary. The career average plan produces the lowest benefits
of ail DB plans because the benefit accrued in any given year dépends on the
salary earned that year and also because of the equal weight applied to ail
earnings.
• For DC plans, the earlier contributions are made to the pension fund, the better
the benefit. Even though the early contributions are lower since they are made
from lower salaries, the investment return is typically higher than the wage
growth and thus the early contributions hâve more impact on the ultimate value
of the benefit. DC plans produce higher average replacement ratios for ail
historiés but they are not risk free. The risk arises from the assumptions in
salary increase, investment return, contribution rate and interest rates. The
lower variability of the average replacement ratios of DB plans may make DB
plans more désirable than DC plans.
• It is important to stress that the same defined contribution rate may not achieve
the desired replacement ratio for différent working patterns. For example, if a
certain contribution rate is sufficient to provide the desired replacement ratio
over a longer period in employment, the same contribution rate will not be
100
sufficient to provide the same replacement ratio over a shorter period in
employment. A higher contribution rate is needed in the latter case. The new
contribution rate will dépend on the number of years the contributions hâve to
accumulate.
Chapter 8 Conclusion Employer-sponsored pension plans play an important rôle in the Canadian
retirement System. In fact, they are one of the key components of financial wealth in
gênerai: according to Statistics Canada, employer-sponsored pension plans represented
about 18.5% of ail assets in 2005, an increase from the 1999 figure of 17.2%. Their
importance is even more accentuated with the récent increases in life expectancy.
Increased life expectancy does not affect only the public pension System but private
pension Systems as well, and, hence, employer-sponsored pension plans. Today, employers
are challenged not only with increased life expectancy but with other changes such as
workforce mobility, increased régulations related to operating a pension plan, and volatile
financial markets. Also, early retirement is still relatively popular. Thèse factors coupled
with the growing cost associated with the opération of a defined benefit (DB) plan
prompted many employers to switch from typically offered DB pension plans to defined
contribution (DC) pension plans. Those same reasons make it less likely that employers
setting up a new pension plan will commit to a DB plan. DC plans are easier to administer
and the cost of operating thèse plans is easier to predict. However, any shift to DC plans
involves an enormous transfer of risk from the DB sponsor to the individual DC plan
member.
The trend from DB to DC plans in employer-sponsored pension plans was the
inspiration for this thesis. Our aim was to examine the impact of différent pension schemes
on différent working historiés. To quantify this impact, we calculated retirement benefits
accrued under four pension arrangements (total service pension plan, final salary pension
plan, career average pension plan and defined contribution pension plan) and divided those
benefits by the final salary earned in order to obtain replacement ratios.
In order to calculate retirement benefits, we first needed to set assumptions
regarding économie variables (inflation, wage growth, T-bills, bond yields, bond returns
and stock returns). We used a stochastic model to simulate their future values using
parameters based on historical data.
102
To ease the comparison, our assumptions did not take into account the expenses
related to the plan. Moreover, no intégration was made with public pensions, mortality was
ignored (except to calculate the annuity factor), and ail pension payments were assumed to
start at retirement âge 65 for ail working historiés under ail pension arrangements. Also,
any value (more precisely, mortality rate and salary scale) that normally would vary by
gender was made genderless by averaging the values for maies and females.
The results show that historiés with long service hâve the highest replacement ratios
in ail types of plan, in gênerai. Under DB plans, this is no surprise because the DB design
rewards long service. Good performance under DC plans also cornes naturally, because
longer service allows for a longer contribution (and accumulation) period, thus higher
benefits.
Generally, ail historiés with short and early breaks at the beginning of the career,
but uninterrupted employment for a number of years immediately before retirement (at least
fifteen), do relatively well under final salary plans because lower salaries are applied to a
short period and more years are tied to the final salary, which is typically the highest.
Historiés with late breaks are disadvantaged under final salary plans because the
final salary applicable to each period preceding a break is lower than it would hâve been if
there had been no break. For example, for a history with a late break in employment (LB
history), participating in a final salary plan rather than in a total service plan causes an
average 35% réduction (in relative terms) in benefits and participating in a career average
plan rather than in a total service plan leads to an average 43% réduction in relative terms.
There is some variability associated with DB plans. The variability of the benefits
under DB plans is caused by the variability in salaries, as ail other factors used to détermine
DB benefits are fixed as per the plan design. A career average plan has the highest
variability of ail DB plans. This is because the benefits dépend on each salary earned in
any given year and, as more salaries enter the calculation, the variability is higher under
this plan.
103
In gênerai, DC plans hâve higher average replacement ratios than any of the DB
plans. This is because the DC fund grows with the investment return, unlike DB plans
whose benefits rather increase with wages. Investment returns are typically larger than the
increases in wages. This explains the importance of the early contributions in DC plans;
even though the early contributions are lower since they are based on smaller wages:
typically, the investment return will more than compensate for the low contributions
(calculated on low wages) during the accumulation phase. Missing the early contributions
to a DC plan may reduce the average replacement ratio by as much as 12% (in relative
terms) with age-based merit (and 15% with experience-based merit) for historiés with an
otherwise comparable work pattern. If we compare the average replacement ratio obtained
under the DC plan for a person that enters the workforce at a later âge (after âge 25 as set in
the assumptions) with the ratio obtained under the différent DB plans, a réduction is
observed only when compared to the total service plan (11% with age-based merit, 14%
with experience-based merit). The results show that historiés that miss early contributions
do not do that well under a DC plan, but it still seems that they do better under a DC plan
than under final salary aiid career average plans.
DC plans do not only produce higher replacement ratios, they also entail a higher
risk. In actual fact, the risk arises not only from the variability in salaries, but also from the
variability in investment returns and annuity factors.
The other risk lies in the contribution rate. The same defined contribution rate will
not be sufficient for ail historiés. Higher contribution rates (given that ail other factors -
wage growth, investment returns and annuity factors - used in the calculations remain the
same) are needed for historiés with long and fréquent breaks, as contributions are not made
(according to the assumptions) during the period of unemployment. If the investment
returns were lower than anticipated, or if wage increases were higher than forecast, then a
higher contribution rate would be required for ail historiés considered in this study.
So, do DC plans meet employées' needs? There are two ways in which they do:
they are portable from job to job (unlike DB plans) and flexible (employées are more
involved in making investment décisions). A question arises as to whether the flexibility is
a good offset for a greater risk. This dépends on the personal aversion to risk.
104
Nevertheless, the results obtained in this study lead us to conclude that high variability in
benefits (in absolute terms, or in relative terms compared to the last salary) is the major
disadvantage of DC plans and makes it less of a viable alternative to DB plans, especially
because employées bear ail the risk associated with DC plans.
As the workforce is more mobile, as there are more temporary and contract-type
employment, and as there are fewer employers that offer DB plans, another question arises:
is it fair that employées bear ail the risk that cornes with DC plans? We cannot answer this
question, as it is beyond the scope of this study. One thing is sure, though: as ail the above
changes are taking the place, ail pension stakeholders, that is, government, employers and
employées, should work together to identify and meet the needs of the changing workforce.
Bibliography Adam, L., ACT-20765: Coût et financement des régimes de retraite, Notes de cours,
Québec City, Université Laval, [Online], Winter 2006 term, 2006, https://cours.act.ulaval.ca/2006h/20765 (accessed 5 April 2006).
Aitken, W. H., A Problem-Solving Approach to Pension Funding and Valuation, 2n
édition, Winsted (Conn.), ACTEX Publications, 1996.
Altmann, R., Let's get DC right before it's too late, Journal of Pensions Management, Vol. 7, No. 1 (September 2001), p. 38-45.
Ambachtsheer, K., Taking the D out of DB, In Benefits Canada, BenefitsCanada.com- News-, [Online], (September 2005), http://www.benefitscanada.com/news/article.jsp?content=20060717163341 4012 (accessed 15 February 2006).
Baldwin, B., A shaky third pillar: The vulnerability of retirement incomes, In Stone, L. O. (éd.), New Frontiers of Research on Retirement, Catalogue no. 75-511-XIE, Ottawa, Statistics Canada, 2006.
Baldwin, B., Retirement and pensions in Canada: a workers' perspective, In Canadian Labour Congress/Congrès du travail du Canada, CLC/CTC > Pensions & Retirement, [Online], Research Paper #39, (4 November 2005), http://canadianlabour.ca/updir/Retirement and pensions in Canada-RP39.pdf (accessed 5 June 2006).
Banks, J. and Blundell, R., Private pension arrangements and retirement in Britain, Fiscal Studies [Online], Vol. 26, No.l (March 2005), p. 35-53, http://www.blackwell-synergy.eom/toc/fisc/26/l (accessed 15 February 2006).
Bharmal, S. Y. M., Debate no. 7: Defined benefit pension plans are not well suited to tomorrow's work force and should be replaced by defined contribution plans, Proceedings ofthe Canadian Institute ofActuaries, Vol. 19, No. 2 (1988), p. 117-133.
Blake, D., Does it matter what type of pension Scheme you hâve?, The Economie Journal, Vol. 110, No. 461 (February 2000), p. 46-81.
Blake, D., Cairns, A. J. G. and Dowd, K., Pensionmetrics: stochastic pension plan design and value-at-risk during the accumulation phase, Insurance: Mathematics and Economies, Vol. 29, No. 2 (19 October 2001), p. 187-215.
106
Bodie, Z., Marcus, A. J. and Merton, R. C, Defined benefit versus defined contribution pension plans: What are the real trade-offs?, In National Bureau of Economie Research, NBER Working Paper s, [Online], No. 1719 (October 1985), http://papers.nber.org/papers/wl 719 (accessed 28 March 2005).
Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A. and Nesbitt, C. J., Actuarial Mathematics, 2nd édition, Schaumburg (111.), Society of Actuaries, 1997.
Brown, R. L. and Liu, J., The shift to defined contribution pension plans: Why did It not happen in Canada?, North American Actuarial Journal, Vol. 5, No. 3 (July 2001), p. 65-77.
Canadian Institute of Actuaries, Report on Canadian Economie Statistics 1924-2004 [Online], Ottawa, Canadian Institute of Actuaries, 2005, http://www.actuaries.ca/members/publications/2005/205016e.pdf.
Civilization.ca, The History of Canada 's Public Pensions, Human Resources Development Canada, Muséum of Civilization Corporation, Created September 10, 2002, Last update January 4, 2005, http://www.civilization.ca/hist/pensions/cppl sp e.html (accessed 9 April 2006).
Clark, R. L. and Munzenmaier F. W., Impact of replacing defined benefit pension with a defined contribution plan or a cash balance plan, North American Actuarial Journal, Vol. 5, No. 1 (January 2001), p. 32-56.
Cohen, B. and Fitzgerald, B., The Pension Puzzle: Your Complète Guide to Government Beneflts, RRSPs, and Employer plans, 2nd édition, Etobicoke (Ont.), J. Wiley & Sons Canada, 2004.
Cooper, D. R., Providing pensions for U.K. employées with varied working lives, Journal of Actuarial Practice, Vol. 5, No. 1 (1997), p. 5-45.
Copeland, C, Employment-based retirement plan participation: Géographie différences and trends, 2004, In Employée Benefit Research Institute, [Online], Issue Brief No. 286 (October 2005), http://www.ebri.org/publications/ib/index.cfm?fa=ibDisp&content id=3590 (accessed 15 February 2006).
Côté, J.-D., Pension planning: A paternalistic approach, In Benefits Canada, BenefitsCanada.com-News-, [Online], (September 2005), http://www.benefitscanada.com/news/article.isp?content=20060717_163341_4012 (accessed 15 February 2006).
Daffin, C. (éd.), Occupational Pension Schemes Annual Report, [Online], No. 14, 2006 édition, Office for National Statistics, http://www.statistics.gov.uk/downloads/theme_population/Occ-pension- 2006/OPSS Annual Report 2006.pdf (accessed April 2, 2008).
107
Davis, A., A road map to manage risk, In Benefits Canada, Publications, DC Forum, DC Forum-Issues, [Online], (March 2005), http://www.dcforum.ca/pdf/dcforum mar200S.pdf (accessed 20 September 2005).
Davis, A., Global trends in DC plans: What Canadian plan sponsors can learn from the U.S., Australia, and the U.K., In Benefits Canada, Publications, DC Forum, DC Forum-Issues, [Online], (December 2005), http://www.dcforum.ca/pdf/DCforumQ3_DecQ5.ac.pdf (accessed 15 February 2006).
Davis, A., Making the switch: What it takes to move from a defined benefit (DB) to a defined contribution (DC) plan, In Benefits Canada, Publications, DC Forum, DC Forum-Issues, [Online], (June 2006), http://www.dcforum.ca/pdf/dcforum en june06.pdf (accessed 12 September 2006).
Diamond, P. A., Proposais to Restructure Social Security, The Journal of Economie Perspectives [Online], Vol. 10, No. 3 (Summer 1996), p. 67-88, http://web.ebscohost.com/ehost/detail?vid=3&hid=21&sid=ef5f84be-78ed-44b7-af96-e356d223a44a%40sessionmgr9 (accessed 2 October 2005).
Everness, G., The effect of pensions on increasing life expectancy, Journal of Pensions Management [Online], Vol. 7, No. 2 (December 2001), p. 178-184, http://proquest.umi.com/pqdlink?PMID=49311 &TS= 1182028187&SrchMode=3&S rtM=0&PCID=1575474&VTvpe=POD&VInst=PROD&aid=l&clientId=9268&RQ T=572&VName=PQD&firstIndex=0 (accessed 20 September 2005).
Félix, S., Back stretch: Lessons from a DC world, In Benefits Canada, Publications, DC Forum, DC Forum-Issues, [Online], (June 2006), http://www.dcforum.ca/pdf/dcforum en_june06.pdf (accessed 12 September 2006).
Ginn, J., Street, D. and Arber, S. (éd.), Women, Work and Pensions: International Issues and Prospects, Philadelphia, Open University Press, 2001.
Gordon, M.S., Mitchell, O. S. and Twinney, M.M. (éd.), Positioning Pensions for the Twenty-First Century, Philadelphia, Pension Research Council, Wharton School of the University of Pennsylvania, University of Pennsylvania Press, 1997.
Government Actuary's Department, Occupational pension schemes survey 1995, In Government Actuary's Department, Government Actuary's Department-News [Online], http://www.gad.gov.uk/news.asp?nid=617 (accessed 8 February 2007).
108
Government Actuary's Department, Occupational pension schemes 2000: The eleventh survey by the Government Actuary, In Government Actuary's Department, Pension-Pension scheme survey-results for 2000 [Online], http://www.gad.gov.uk/publications/docs/opss2000 final results final_7april2003. pdf (accessed 8 February 2007).
Government Actuary's Department, Occupational pension schemes 2004: The twelfth survey by the Government Actuary, In Government Actuary's Department, Pension-Pension scheme survey-results for 2004 [Online], http://www.gad.gov.ulc/Publications/docs/Final%20Report%2016Jun2005.pdf (accessed 8 February 2007).
Government Actuary's Department, Occupational pension schemes 2005: The thirteenth survey by the Government Actuary, In Government Actuary's Department, Pension-Pension scheme survey-results for 2004 [Online], http://www.gad.gov.uk/Publications/docs/OPSS Final Report 2006.pdf (accessed 8 February 2007).
Hall, G. M. (éd.), Mercer Handbook ofCanadian Pension and Benefit Plans, 11 édition, North York (Ont.), CCH Canadian Limited, 1996.
Hogg, R. V. and Craig, A. T., Introduction to Mathematical Statistics, 5l édition, Englewood Cliffs (N.J.), Prentice Hall, 1995.
Human Resources and Social Development Canada, Canada Pension Plan and OldAge Security, [Online], http://www.hrsdc.gc.ca/en/oas-cpp/index.shtml (accessed 28 November 2005 and 19December2007).
ImmediateAnnuities.com, Immédiate Annuities - Instant Annuity Quote Calculator [Online], http://www.immediateannuities.com/ (accessed 5 April 2006).
Khorasanee, M. Z., Simulation of investment returns for a money purchase fund, Journal of Actuariat Practice, Vol. 3, No. 1 (1995), p. 93-115.
Knox, D. M., A critique of defined contribution plans using a simulation approach, Journal of Actuariat Practice, Vol. 1, No. 2 (1993), p. 49-66.
Li, J., Séparation, linkage and blurring in the public and private pillars of Canada's retirement income System, In Stone, L. O. (éd.), New Frontiers of Research on Retirement, Catalogue no. 75-511-XIE, Ottawa, Statistics Canada, 2006.
Markham, L, DB: Can it be fixed?, In Benefits Canada, BenefitsCanada.com-News-, [Online], (September 2005) http://www.benefitscanada.com/news/article.jsp?content=20060717 163341 4012 (accessed 15 February 2006).
McGill, D. M. et al., Fundamentals ofPrivate Pensions, 1 édition, Philadelphia, University of Pennsylvania Press, 1996.
Miller, L., The ongoing growth of defined contribution and individual account plans: Issues and implications, In Employée Benefit Research Insiitute, [Online], Issue Brief No. 243 (March 2002), http://www.ebri.org/publications/ib/index.cfm?fa:=ibDisp&content_id:=160 (accessed 2 October 2005).
Morneau Sobeco (Firm), Whiston, B. B. A. and Gottlieb, L. C. (co-ordinators), Morneau Sobeco Handbook ofCanadian Pension and Benefit plans, 13l édition, Toronto, CCH Canadian Limited, 2005.
Organisation de coopération et de développement économiques (OCDE), Vieillissement et revenus : les ressources des retraités dans 9pays de l'OCDE, Paris, OCDE, 2001.
Ostaszewski, K. M., Macroeconomic aspects of private retirement programs, North American Actuarial Journal, Vol. 5, No. 3 (July 2001), p. 52-64.
Panjer, H. H. (éd.), Financial Economies: With Applications to Investments, Insurance, and Pensions, Schaumburg (111.), Society of Actuaries Foundation, 1997.
Perkin, S., Securing the DB future, In Benefits Canada, BenefitsCanada.com-News-, [Online], (September 2005), http://www.benefitscanada.com/news/article.isp?content=20060717_163341_4012 (accessed 15 February 2006).
Rajnes, D., An evolving pension System: Trends in defined benefit and defined contribution pension plans, In Employée Benefit Research Institute, [Online], 2002 Issue Brief, No. 249 (September 2002), http://www.ebri.org/publications/ib/index.cfm?fa=ibDisp&content id=166 (accessed 2 October 2005).
Samwick, A. A. and Skinner, J., How will defined contribution pension plans affect retirement income?, In National Bureau of Economie Research, NBER Working Papers, [Online], No. 6645 (July 1998), http://papers.nber.org/papers/w6645 (accessed 7 March 2005).
Samwick, A. A. and Skinner, J., How will 401(k) pension plans affect retirement income?, The American Economie Review [Online], Vol. 94, No. 1 (March 2004), http://www.atvpon-link.com/AEAP/doi/pdf/10.1257/000282804322970832 (accessed 7 March 2005).
Service Canada, Income Security Programs Information Card (Rate Card), [Online], http://www.hrsdc.gc.ca/en/isp/statistics/rates/infocard.shtml, (accessed 28 February, 2007).
110
Society of Actuaries, Course 7 Applied Modelling: Lectures préliminaires et documents de référence: Cas pratique spécialisé, Seminar in Montréal from December 5 to 8, 2005.
Society of Actuaries UP-94 Task Force, The 1994 Uninsured Pensioner mortality table, Transactions ofthe Society of Actuaries, Vol. 47 (1995), pp. 819-863.
Statistics Canada, n.d. Proportion of labour force andpaid workers covered by a registered pension plan (RPP) (table). Pension Plan in Canada and Labour Force Survey (source). http://www40.statcan.ca/101/cst01/labor26a.htm (accessed 8 March 2008).
Statistics Canada, n.d., Table 176-0043 Financial market statistics, last Wednesday unless otherwise stated, monthly (percent) (table). CANSIM (database). Using E-STAT (distributor). http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CH 1 E.htm (accessed 16 February 2006).
Statistics Canada, n.d. Table 280-0008 Registered pension plan (RPP) members, by area of employment, sector, type ofplan and contributory status, annual (table). CANSIM (database). Using E-STAT (distributor). http://estat2.statcan.ca/cgi-win/CNSMCGI.exe?Lang=F&RootDir=ESTAT/&CANSIMFile=EStat\Francais\CII _l_F.htm (accessed 10 March 2006).
Statistics Canada, n.d. Table 280-0016 Registered pension plans (RPPs) and members, by type ofplan, sector and contributory status, annual (table). CANSIM (database). Using E-STAT (distributor). http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CH 1 E.htm (accessed 27 February 2008).
Statistics Canada, n.d. Table 281-0002 Average weekly earnings of employées, (SEPH), monthly (dollars) Terminated (table). CANSIM (database). Using E-STAT (distributor). http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CII 1 E.htm (accessed 16 February 2006).
Statistics Canada, n.d. Table 281-0021 Average weekly earnings, ail employées, (SEPH), by Standard lndustrial Classification, 1960 (SIC), monthly (dollars) Terminated (table). CANSIM (database). Using E-STAT (distributor). http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CII 1 E.htm (accessed 16 February 2006).
111
Statistics Canada, n.d, Table 281-0028 Average weekly earnings (SEPH), including overtime, seasonally adjusted, for ail employées, by selected industries classified using the North American Industry Classification System (NAICS), monthly (dollars) (table). CANSIM (database). Using E-STAT (distributor). http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CII 1 E.htm (accessed 16 February 2006).
Statistics Canada, n.d. Table 326-0002 Consumer price index (CPI), 2001 basket content, annual (table). CANSIM (database). Using E-STAT (distributor). http://estat.statcan.ca/cgi-win/CNSMCGI.EXE?CANSIMFILE=EStat\English\CII 1 E.htm (accessed 16 February 2006).
Statistics Canada, Canada's RetirementIncome Programs: A Statistical Overview (1990-2000), Statistics Canada Catalogue no. 74-507-XIE, Ottawa, 2003.
Statistics Canada, Pension Plans in Canada, Statistics Canada Catalogue no. 13F0026MIE2004001, Ottawa, 2004. http://www.statcan.ca/cgi-bin/downpub/listpub.cgi?catno=13F0026MIE2004001 (accessed 10 March 2005).
Statistics Canada, New Frontiers of Research on Retirement, Statistics Canada Catalogue no. 75-511-XIE, Ottawa, 2006. http://www.statcan.ca/bsolc/english/bsolc?catno=75-511 -X&CHROPG=l (accessed 6 May 2006).
Statistics Canada, The Wealth of Canadians: An Overview ofthe Results ofthe Survey of Financial Security, Statistics Canada Catalogue no. 13F0026MIE, Ottawa, 2006. http://www.statcan.ca/cgi-bin/downpub/listpub.cgi?catno=13F0026MIE2006001 (accessed 11 January, 2007).
Toronto Stock Exchange, Toronto Stock Exchange Review, Toronto, Toronto Stock Exchange, 1981.
Toronto Stock Exchange, Toronto Stock Exchange E-Review, In Université Laval. Bibliothèque, TSXE-review (Toronto Stock Exchange) [Online], Vol. 71, No. 12E (December 2005, Chapter 5, p. 27), p. 263, http://www.bibl.ulaval.ca/doelec/pes/tsx ereview/20051231 TSX eReview.pdf.
Townson, M., The impact of precarious employment on financial security in retirement, In Stone, L. O. (éd.), New Frontiers of Research on Retirement, Catalogue no. 75-511-XIE, Ottawa, Statistics Canada, 2006.
112
U.S. Department of Labor, Employée Benefits Security Administration, Private pension plan bulletin: Abstract of 1995 Form 5500 Annual Reports, In U.S. Department of Labor, Publications and Reports Main Page, [Online], http://www.dol.g0v/ebsa/publications/main.html#section8 (accessed 4 September 2006).
U.S. Department of Labor, Employée Benefits Security Administration, Private pension plan bulletin: Abstract of 2000 Form 5500 Annual Reports, In U.S. Department of Labor, Publications and Reports Main Page, [Online], http://www.dol.gOv/ebsa/publications/main.html#section8 (accessed 4 September 2006).
U.S. Department of Labor, Employée Benefits Security Administration, Private pension plan bulletin: Abstract of 2003 Form 5500 Annual Reports, In U.S. Department of Labor, Publications and Reports Main Page, [Online], http://www.dol.gOv/ebsa/publications/main.html#section8 (accessed 7 November 2006).
U.S. Department of Labor, Employée Benefits Security Administration, Private Pension Plan Bulletin Historical Tables, [Online], (February 2008), http://www.dol.gov/ebsa/pdf/privatepensionplanbulletinhistoricaltables.pdf (accessed 23 March 2008).
VanDerhei, J. and Copeland, C, The changing face of private retirement plans, In Employée Beneflt Research Institute, [Online], 2001 Issue Brief, No. 232 (April 2001), http://www.ebri.org/publications/ib/index.cfm?fa=main&doc_type=l (accessed 2 October 2005).
VanDerhei, J. and Copeland, C, Can America afford tomorrow's retirées: Results from the EBRI-ERF retirement security projection model, In Employée Benefit Research Institute, [Online], 2003 Issue Brief, No. 263 (November 2003), http://www.ebri.org/publications/ib/index.cfm?fa=main&doc type=l (accessed 2 October 2005).
VanDerhei, J. and Copeland, C, ERISA at 30: The décline of private-sector defined benefit promises and annuity payments? What will it mean?, In Employée Benefit Research Institute, [Online], 2004 Issue Brief, No. 269 (May 2004), http://www.ebri.org/publications/ib/index.cfm?fa=main&doc type=l (accessed 2 October 2005).
Wilkie, A. D., More on a stochastic asset model for actuarial use, British Actuarial Journal, Vol. 1, No. 5 (1994), p. 777-964.
Appendix A Economie Variables: Data Ail of the économie data used to estimate the model parameters are found in Tables A. 1
(inflation and wage growth) and A.2 (inflation and rates of return on différent assets), at the
end of this appendix. The sources used to find those data are identified separately in the
following sections.
A.l Inflation The annual values of the Consumer Price Index (CPI) from 1980 to 2004 hâve been
extracted from the following source:
Statistics Canada, n.d. Table 326-0002 Consumer price index (CPI), 2001 basket content, annual (table). CANSIM (database). Using E-STAT (distributor).
http://estat.statcan.ca/cgi-win/CNSMCGI.EXE?CANSIMFILE=EStat\Enelish\CII 1 E.htm
(accessed 16 February 2006).
The inflation rate in any given year is the percentage increase of that index between the
previous year and that year. For example, the inflation rate for 1990 is obtained as follows:
93 3 — - 1 = 4.83%. (A.l) 89.0
A.2 Wage Growth Because none of the sources of wage data that were found covered the whole period of
interest, three différent sources had to be used. We extracted data for 1980 to 1983 from
the first source, for 1983 to 1991 from the second source, and for 1991 to 2004 from the
third source. Whereas the 1983 datum from the first source is needed to compute the wage
growth for 1983, the 1983 datum from the second source is needed to compute the wage
growth for 1984. A similar comment applies to the two values that were extracted for
1991. The three sources are as follows:
114
Statistics Canada, n.d. Table 281-0021 Average weekly earnings, ail employées, (SEPH), by
Standard Industrial Classification, 1960 (SIC), monthly (dollars) Terminated (table).
CANSIM (database). Using E-STAT (distributor).
http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CII 1 E.htm
(accessed 16 February 2006).
Statistics Canada. Table 281-0002 Average weekly earnings of employées, (SEPH), monthly
(dollars) Terminated (table). CANSIM (database). Using E-STAT (distributor).
http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CH 1 E.htm
(accessed 16 February 2006).
Statistics Canada, n.d, Table 281-0028 Average weekly earnings (SEPH), including
overtime, seasonally adjusted, for ail employées, by selected industries classified using the
North American Industry Classification System (NAICS), monthly (dollars) (table).
CANSIM (database). Using E-STAT (distributor).
http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CII 1 E.htm
(accessed 16 February 2006).
The observant reader will notice that the values extracted are monthly values. The values
reported in Table A. 1 are averages of the monthly values in a given year, which are
computed directly using the interface of the CANSIM database. Those averages are used to
compute wage growth in the same way that price inflation was computed. For instance, the
wage growth for 1990 is obtained as follows:
506.16 484.07
1 = 4.56%. (A.2)
A.3 Treasury Bills We extracted the rates of return on Treasury bills (T-bills) directly from the following
source:
Statistics Canada, n.d., Table 176-0043 Financial market statistics, last Wednesday unless
otherwise stated, monthly (percent) (table). CANSIM (database). Using E-STAT
(distributor).
115
http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CII 1 E.htm
(accessed 16 February 2006).
As may be noted in the détails of the source, rates of return on T-Bills were available on a
monthly basis. The rate of return for a given year is the arithmetic average of the 12
monthly values in that year. The interface of the CANSIM database actually makes those
computations and the values in Table A.2 are those obtained directly from that database.
A.4 Bonds We used the December yields-to-maturity (YTMs) on the Government of Canada long
bond index (over 10-year term). They were available from the following source:
Statistics Canada, n.d., Table 176-0043 Financial market statistics, last Wednesday unless
otherwise stated, monthly (percent) (table). CANSIM (database). Using E-STAT
(distributor).
http://estat.statcan.ca/cgi-in/CNSMCGI.EXE?CANSIMFILE=EStat\English\CH 1 E.htm
(accessed 16 February 2006).
A.5 Common Stock We extracted the values of the Total Return Index (TRI) based on closing indices from the
following sources:
Toronto Stock Exchange, Toronto Stock Exchange Review, Toronto Stock Exchange, 1981.
Toronto Stock Exchange, Toronto Stock Exchange E-Review, In Université Laval.
Bibliothèque, TSX E-review (Toronto Stock Exchange) [Online], Vol. 71, No. 12E
(December 2005, Chapter 5, p. 27), p. 263,
http://www.bibl.ulaval.ca/doelec/pes/tsx_ereview/20051231 TSX eReview.pdf.
The Stock Price Index (SPI) may be better known and more readily available, but would
not be the relevant index for our purposes. Whereas the SPI only tracks changes in stock
price, the TRI assumes reinvestment of dividends in addition to tracking changes in stock
price.
116
The rate of return on common stock was obtained in the same way that price inflation and
wage growth were calculated. For instance, we found the rate of return for 1990 as follows:
5,617.01 _ 1 = _ 1 4 8 Q %
6,592.58
Table A.l Historical rates of inflation and wage growth
Year
Inl lation Wage
Year CPI Inflation rate
Index (0021)
Index (0002)
Index (0028)
Wage increase
1980 52.4 317.39 1981 58.9 12.40% 355.28 J 11.94% 1982 65.3 10.87% 390.79 9.99% 1983 69.1 5.82% 419.51 382.68 7.35% 1984 72.1 4.34% 398.58 4.15% 1985 75.0 4.02% 412.68 3.54% 1986 78.1 4.13% 425.11 3.01% 1987 81.5 4.35% 441.11 3.76% 1988 84.8 4.05% 460.58 4.41% 1989 89.0 4.95% 484.07 5.10% 1990 93.3 4.83% 506.16 4.56% 1991 98.5 5.57% 529.48 553.43 4.61% 1992 100.0 1.52% 572.59 3.46% 1993 101.8 1.80% 583.12 1.84% 1994 102.0 0.20% 593.08 1.71% 1995 104.2 2.16% 598.90 0.98% 1996 105.9 1.63% 611.22 2.06% 1997 107.6 1.61% 623.65 2.03% 1998 108.6 0.93% 632.98 1.50% 1999 110.5 1.75% 640.66 1.21% 2000 113.5 2.71% 655.81 2.36% 2001 116.4 2.56% 667.23 1.74% 2002 119.0 2.23% 680.81 2.04% 2003 122.3 2.77% 690.32 1.40% 2004 124.6 1.88% 705.48 2.20%
Table A.2 Historical rates of inflation and retum on différent assets
118
Year
Ini lation Rate of retum on T-bills
Yield-to-maturity on bonds
Common stock
Year CPI Inflation rate
Rate of retum on T-bills
Yield-to-maturity on bonds TRI
Rate of return
1980 52.4 12.67% 2,705.51 1981 58.9 12.40% 17.01% 15.27% 2,428.29 -10.25% 1982 65.3 10.87% 14.01% 11.69% 2,562.85 5.54% 1983 69.1 5.82% 9.63% 12.02% 3,472.33 35.49% 1984 72.1 4.34% 11.78% 11.66% 3,369.25 -2.97% 1985 75.0 4.02% 9.94% 10.06% 4,238.78 25.81% 1986 78.1 4.13% 9.22% 9.23% 4,618.32 8.95% 1987 81.5 4.35% 8.99% 10.34% 4,889.82 5.88% 1988 84.8 4.05% 10.01% 10.36% 5,431.68 11.08% 1989 89.0 4.95% 11.79% 9.69% 6,592.58 21.37% 1990 93.3 4.83% 12.53% 10.51% 5,617.01 -14.80% 1991 98.5 5.57% 8.80% 8.97% 6,291.90 12.02% 1992 100.0 1.52% 6.67% 8.54% 6,201.72 -1.43% 1993 101.8 1.80% 5.49% 7.12% 8,220.23 32.55% 1994 102.0 0.20% 6.68% 9.16% 8,205.73 -0.18% 1995 104.2 2.16% 7.22% 7.43% 9,397.97 14.53% 1996 105.9 1.63% 4.79% 6.77% 12,061.95 28.35% 1997 107.6 1.61% 4.08% 5.80% 13,868.54 14.98% 1998 108.6 0.93% 5.03% 5.08% 13,648.84 -1.58% 1999 110.5 1.75% 5.14% 6.25% 17,977.46 31.71% 2000 113.5 2.71% 5.87% 5.59% 19,309.35 7.41% 2001 116.4 2.56% 3.81% 5.75% 16,881.75 -12.57% 2002 119.0 2.23% 3.06% 5.37% 14,782.01 -12.44% 2003 122.3 2.77% 3.01% 5.14% 18,732.48 26.72% 2004 124.6 1.88% 2.55% 4.86% 21,444.89 14.48%
Appendix B Economie Variables: Model The model used to generate the économie scénarios is described in Chapter 5. Ail the
numerical values for the parameters of this model are given in Table B.l.
Table B.l Parameters of the multivariate model
Variable a M a <Plas, 9c <PB <pY
Inflation 0.095 -2.123 0.089 0.680 Wage growth 0.027 -3.672 0.299 -0.249 -0.784 T-bills 0.011 -3.323 0.314 0.894 -0.343 YTM on bonds 0.131 -1.745 0.054 -0.178 -0.327 0.177 Common stock 6.787 1.927 0.020 -0.299 -0.027 -0.004 0.042
Appendix C Working Historiés The working historiés that were used in this thesis are the same that Cooper (1997) used in
her paper (see Table Al, p. 44). The names given to those historiés are also the same. We
reproduce them hère in Table C l . A value of 1 indicates full-time and any lesser value
indicates what fraction of full-time is worked. A value greater than 1 indicates that the
history includes overtime, or additional pay treated as salary for the calculation of the
pension benefit.
Table C l Employment pattern of différent categ ories of employées
History Age range
History 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64
Full-Time 1 1 1 1 1 Break and Mixed 0 0.4 0 0.75 0 Late Break 0 1 1 0 0 Break and Part-Time 0 0.4 0.4 0.75 0.75 0.75 Early Break and Full-Time 0 0 1 1 Late Break and Full-Time 0 0 1 Short Break and Full-Time 0 1 1 Long Break and Full-Time 0 0 0 Late Start 0 1 0 1 Full-Time and Bonus 1.2 1.2 1.2 1.2 1.2 Cyclical 0 1 0 1 0 0
Appendix D Salary Scale We used the salary scale found in the course notes by Adam (2006), reproduced in Table
D.l. The "average" value is the arithmetic average of the "maie" and "female" values.
Table D.l Salary scale
Age Maie Female Average Age Maie Female Average 41 1.220 1.120 1.170 42 1.240 1.140 1.190 43 1.260 1.160 1.210 44 1.280 1.180 1.230
20 1.000 1.000 1.000 45 1.300 1.200 1.250 21 1.000 1.000 1.000 46 1.320 1.220 1.270 22 1.000 1.000 1.000 47 1.340 1.240 1.290 23 1.000 1.000 1.000 48 1.360 1.260 1.310 24 1.000 1.000 1.000 49 1.380 1.280 1.330 25 1.000 1.000 1.000 50 1.400 1.300 1.350 26 1.010 1.000 1.005 51 1.410 1.310 1.360 27 1.020 1.000 1.010 52 1.420 1.320 1.370 28 1.030 1.000 1.015 53 1.430 1.330 1.380 29 1.040 1.000 1.020 54 1.440 1.340 1.390 30 1.050 1.000 1.025 55 1.450 1.350 1.400 31 1.060 1.010 1.035 56 1.460 1.360 1.410 32 1.070 1.020 1.045 57 1.470 1.370 1.420 33 1.080 1.030 1.055 58 1.480 1.380 1.430 34 1.090 1.040 1.065 59 1.490 1.390 1.440 35 1.100 1.050 1.075 60 1.500 1.400 1.450 36 1.120 1.060 1.090 61 1.500 1.400 1.450 37 1.140 1.070 1.105 62 1.500 1.400 1.450 38 1.160 1.080 1.120 63 1.500 1.400 1.450 39 1.180 1.090 1.135 64 1.500 1.400 1.450 40 1.200 1.100 1.150 65 1.500 1.400 1.450
Appendix E Mortality For the probabilities of dying at différent âges, we hâve used the UP-94 mortality table,
together with the AA scale which we used to project the UP-94 rates from 1994 to 2015
(Society of Actuaries UP-94 Task Force 1995). Both the UP-94 mortality table and the AA
scale vary by âge and gender. The values provided by the AA scale are rates of decrease of
the mortality that are applied geometrically. For instance, according to the UP-94 table, the
probability of dying next year for a 70-year-old maie is 0.025516 and the applicable rate of
decrease using the AA scale is 0.015. We obtain the probability that would apply in 2015
as follows:
0.025516 xO-O.OlS)2015"'994 =0.018577. (E.l)
Ail the results of those computations hâve been rounded to six décimais. We hâve then
taken the arithmetic average of the maie and female probabilities to dérive the probabilities
applicable to the employée we hâve assumed to be "genderless". Ail the probabilities
projected to 2015 are found in Table E.l, on the next two pages. (Note that that table does
not hâve any value for âge 0.)
Table E.l Mortality table: UP-94 projected to 2015 using AA scale
Age Qx
Age Qx
Age Maie Female Average Age Maie Female Average 1 0.000417 0.000374 0.000396 31 0.000795 0.000339 0.000567 2 0.000281 0.000243 0.000262 32 0.000812 0.000361 0.000587 3 0.000234 0.000182 0.000208 33 0.000821 0.000375 0.000598 4 0.000182 0.000136 0.000159 34 0.000822 0.000390 0.000606 5 0.000167 0.000123 0.000145 35 0.000824 0.000407 0.000616 6 0.000160 0.000115 0.000138 36 0.000834 0.000427 0.000631 7 0.000153 0.000108 0.000131 37 0.000862 0.000451 0.000657 8 0.000141 0.000096 0.000119 38 0.000890 0.000478 0.000684 9 0.000137 0.000092 0.000115 39 0.000928 0.000510 0.000719 10 0.000139 0.000092 0.000116 40 0.000974 0.000556 0.000765 11 0.000146 0.000097 0.000122 41 0.001028 0.000601 0.000815 12 0.000159 0.000104 0.000132 42 0.001090 0.000647 0.000869 13 0.000180 0.000116 0.000148 43 0.001153 0.000687 0.000920 14 0.000214 0.000139 0.000177 44 0.001217 0.000722 0.000970 15 0.000248 0.000166 0.000207 45 0.001289 0.000745 0.001017 16 0.000281 0.000190 0.000236 46 0.001377 0.000775 0.001076 17 0.000309 0.000209 0.000259 47 0.001487 0.000817 0.001152 18 0.000331 0.000218 0.000275 48 0.001611 0.000886 0.001249 19 0.000348 0.000219 0.000284 49 0.001745 0.000961 0.001353 20 0.000364 0.000217 0.000291 50 0.001894 0.001072 0.001483 21 0.000389 0.000215 0.000302 51 0.002064 0.001202 0.001633 22 0.000417 0.000217 0.000317 52 0.002260 0.001386 0.001823 23 0.000461 0.000223 0.000342 53 0.002522 0.001592 0.002057 24 0.000510 0.000228 0.000369 54 0.002799 0.001815 0.002307 25 0.000576 0.000233 0.000405 55 0.003180 0.002083 0.002632 26 0.000660 0.000245 0.000453 56 0.003634 0.002428 0.003031 27 0.000704 0.000251 0.000478 57 0.004186 0.002825 0.003506 28 0.000730 0.000262 0.000496 58 0.004828 0.003251 0.004040 29 0.000754 0.000276 0.000515 59 0.005433 0.003739 0.004586 30 0.000776 0.000305 0.000541 60 0.006112 0.004296 0.005204
Table E.l Mortahty table: UP-94 projected to 2015 using AA scale (contmued)
Age qx
Age <lx
Age Maie Female Average Age Maie Female Average 61 0.007035 0.004929 0.005982 91 0.165331 0.129977 0.147654 62 0.007944 0.005644 0.006794 92 0.184016 0.143326 0.163671 63 0.009174 0.006462 0.007818 93 0.200281 0.160765 0.180523 64 0.010348 0.007375 0.008862 94 0.217754 0.175968 0.196861 65 0.011624 0.008358 0.009991 95 0.240847 0.191985 0.216416 66 0.013266 0.009382 0.011324 96 0.259307 0.208817 0.234062 67 0.014732 0.010418 0.012575 97 0.277148 0.231277 0.254213 68 0.015882 0.011384 0.013633 98 0.300372 0.250291 0.275332 69 0.017377 0.012300 0.014839 99 0.317240 0.270296 0.293768 70 0.018577 0.013288 0.015933 100 0.334024 0.291053 0.312539 71 0.020316 0.014170 0.017243 101 0.358560 0.318956 0.338758 72 0.022297 0.015641 0.018969 102 0.376699 0.340960 0.358830 73 0.024425 0.017019 0.020722 103 0.396884 0.364586 0.380735 74 0.026657 0.018909 0.022783 104 0.418855 0.389996 0.404426 75 0.029758 0.020607 0.025183 105 0.440585 0.415180 0.427883 76 0.032674 0.023004 0.027839 106 0.460043 0.438126 0.449085 77 0.036900 0.026318 0.031609 107 0.475200 0.456824 0.466012 78 0.041900 0.029436 0.035668 108 0.485670 0.471493 0.478582 79 0.047616 0.032809 0.040213 109 0.492807 0.483473 0.488140 80 0.054006 0.036551 0.045279 110 0.497189 0.492436 0.494813 81 0.061022 0.040778 0.050900 111 0.499394 0.498054 0.498724 82 0.068611 0.045604 0.057108 112 0.500000 0.500000 0.500000 83 0.074950 0.050896 0.062923 113 0.500000 0.500000 0.500000 84 0.083142 0.056576 0.069859 114 0.500000 0.500000 0.500000 85 0.090219 0.064189 0.077204 115 0.500000 0.500000 0.500000 86 0.098153 0.072923 0.085538 116 0.500000 0.500000 0.500000 87 0.109611 0.083055 0.096333 117 0.500000 0.500000 0.500000 88 0.122895 0.092738 0.107817 118 0.500000 0.500000 0.500000 89 0.134967 0.105590 0.120279 119 0.500000 0.500000 0.500000 90 0.151168 0.117372 0.134270 120 1.000000 1.000000 1.000000
