Lecture 05

UNIT ONE

CHAPTER THREE

THE TIME VALUE

OF MONEY

Lesson 5

Chapter 3

The time value of money

Unit 1

Core concepts in financial management

After reading this lesson you will be able to: -

Understand what is meant by "the time value of money." Describe how the interest rate can be used to adjust the value of cash flows to a

single point in time.

Calculate the future value of an amount invested today. Calculate the present value of a single future cash flow. Understand the relationship between present and future values. Understand in what period of time money doubles Understand shorter compounding periods Calculate & understand the relationship between effective & nominal interest

rate.

Use the interest factor tables and understand how they provide a short cut to calculating present and future values.

You all instinctively know that money loses its value with time. Why does this happen?

What does a Financial Manager have to do to accommodate this loss in the value of

money with time? In this section, we will take a look at this very interesting issue.

Why should financial managers be familiar with the time value of money?

The time value of money shows mathematically how the timing of cash flows, combined

with the opportunity costs of capital, affect financial asset values. A thorough

understanding of these concepts gives a financial manager powerful tool to maximize

wealth.

What is the time value of money?

The time value of money serves as the foundation for all other notions in finance. It

impacts business finance, consumer finance and government finance. Time value of

money results from the concept of interest.

This overview covers an introduction to simple interest and compound interest, illustrates

the use of time value of money tables, shows a approach to solving time value of money

problems and introduces the concepts of intra year compounding, annuities due, and

perpetuities. A simple introduction to working time value of money problems on a

financial calculator is included as well as additional resources to help understand time

value of money.

Time value of money

The universal preference for a rupee today over a rupee at some future time is because of

the following reasons: -

Alternative uses/ Opportunity cost Inflation Uncertainty

The manner in which these three determinants combine to determine the rate of interest

can be represented symbolically as

Nominal or market rate of interest rate = Real rate of interest + Expected rate of

Inflation + Risk of premiums to

compensate uncertainty

Basics

Evaluating financial transactions requires valuing uncertain future cash flows. Translating

a value to the present is referred to as discounting. Translating a value to the future is

referred to as compounding

The principal is the amount borrowed. Interest is the compensation for the opportunity

cost of funds and the uncertainty of repayment of the amount borrowed; that is, it

represents both the price of time and the price of risk. The price of time is compensation

for the opportunity cost of funds and the price of risk is compensation for bearing risk.

Interest is compound interest if interest is paid on both the principal and any accumulated

interest. Most financial transactions involve compound interest, though there are a few

consumer transactions that use simple interest (that is, interest paid only on the principal

or amount borrowed).

Under the method of compounding, we find the future values (FV) of all the cash

flows at the end of the time horizon at a particular rate of interest. Therefore, in this

case we will be comparing the future value of the initial outflow of Rs. 1,000 as at the

end of year 4 with the sum of the future values of the yearly cash inflows at the end of

year 4. This process can be schematically represented as follows:

PROCESS OF DISCOUNTING

Under the method of discounting, we reckon the time value of money now, i.e. at

time 0 on the time line. So, we will be comparing the initial outflow with the sum of the

present values (PV) of the future inflows at a given rate of interest.

Translating a value back in time -- referred to as discounting -- requires determining

what a future amount or cash flow is worth today. Discounting is used in valuation

because we often want to determine the value today of future value or cash flows.

The equation for the present value is:

Present value = PV = FV / (1 + i) n

Where:

PV = present value (today's value),

FV = future value (a value or cash flow sometime in the future),

i = interest rate per period, and

n = number of compounding periods

And [(1 + i) n] is the compound factor.

We can also represent the equation a number of different, yet equivalent ways:

Where PVIFi,n is the present value interest factor, or discount factor.

In other words future value is the sum of the present value and interest:

Future value = Present value + interest

From the formula for the present value you can see that as the number of discount

periods, n, becomes larger, the discount factor becomes smaller and the present value

becomes less, and as the interest rate per period, i, becomes larger, the discount factor

becomes smaller and the present value becomes less.

Therefore, the present value is influenced by both the interest rate (i.e., the discount rate)

and the numbers of discount periods.

Example

Suppose you invest 1,000 in an account that pays 6% interest, compounded annually.

How much will you have in the account at the end of 5 years if you make no

withdrawals? After 10 years?

Solution

FV5 = Rs 1,000 (1 + 0.06) 5 = Rs 1,000 (1.3382) = Rs 1,338.23

FV10 = Rs 1,000 (1 + 0.06) 10 = Rs 1,000 (1.7908) = Rs 1,790.85

What if interest was not compounded interest? Then we would have a lower balance in

the account:

FV5 = Rs 1,000 + [Rs 1,000(0.06) (5)] = Rs 1,300

FV10 = Rs 1,000 + [Rs 1,000 (0.06)(10)] = Rs 1,600

Simple interest is the product of the principal, the time in years, and the annual interest

rate.

In compound interest the principal is more than once during the time of the investment.

Compound interest is another matter. It's good to receive compound interest, but not so

good to pay compound interest. With compound interest, interest is calculated not only

on the beginning interest, but also on any interest accumulated in the meantime.

I hope you have understood the concept of simple interest and compound interest. It is

explained with the help of a graph, which is self-explanatory.

NNooww lleett uuss ssoollvvee aa pprroobblleemm ffoorr Compound Interest vs. Simple Interest

Example

Suppose you are faced with a choice between two accounts, Account A and Account B.

Account A provides 5% interest, compounded annually and Account B provides 5.25%

simple interest. Consider a deposit of Rs 10,000 today. Which account provides the

highest balance at the end of 4 years?

Solution

Account A: FV4 = Rs 10,000 (1 + 0.05) 4 = Rs 12,155.06

Account B: FV4 = Rs 10,000 + (Rs 10,000 (0.0525)(4)] = Rs 12,100.00

Account A provides the greater future value.

Present value is simply the reciprocal of compound interest. Another way to think of

present value is to adopt a stance out on the time line in the future and look back toward

time 0 to see what was the beginning amount.

Present Value = P0 = Fn / (1+I) n

Table A-3 shows present value factors: Note that they are all less than one.

Therefore, when multiplying a future value by these factors, the future value is

discounted down to present value. The table is used in much the same way as the other

time value of money tables. To find the present value of a future amount, locate the

appropriate number of years and the appropriate interest rate, take the resulting factor and

multiply it times the future value.

How much would you have to deposit now to have Rs 15,000 in 8 years if interest is 7%?

= 15000 X .582 = 8730 Rs

Consider a case in which you want to determine the value today of $ 1,000 to be received five years from now. If the interest rate (i.e., discount rate) is 4%,

Problem

Suppose that you wish to have Rs 20,000 saved by the end of five years. And suppose

you deposit funds today in account that pays 4% interest, compounded annually. How

much must you deposit today to meet your goal?

Solution

Given: FV = Rs 20,000; n = 5; i = 4%

PV = Rs 20,000/(1 + 0.04) 5 = Rs 20,000/1.21665

PV = Rs 16,438.54

Q. If you want to have Rs 10,000 in 3 years and you can earn 8%, how much would you

have to deposit today?

Rs 7938.00 Rs 25,771 Rs 12,597

Using Tables to Solve Future Value Problems

A-1 for future value at the end of n yrs

A-3 for present value at the beginning of the year

Compound Interest tables have been calculated by figuring out the (1+I) n values for

various time periods and interest rates. Look at Time Value of Money Future Value

Factors.

This table summarizes the factors for various interest rates for various years. To use the

table, simply go down the left-hand column to locate the appropriate number of years.

Then go out along the top row until the appropriate interest rate is located.

For instance, to find the future value of Rs100 at 5% compound interest, look up five

years on the table, and then go out to 5% interest. At the intersection of these two values,

a factor of 1.2763 appears. Multiplying this factor times the beginning value of Rs100.00

results in Rs127.63, exactly what was calculated using the Compound Interest Formula.

Note, however, that there may be slight differences between using the formula and tables

due to rounding errors.

An example shows how simple it is to use the tables to calculate future amounts.

You deposit Rs2000 today at 6% interest. How much will you have in 5 years?

=2000*1.338=2676

The following exercise should aid in using tables to solve future value problems. Please

answer the questions below by using tables

1. You invest Rs 5,000 today. You will earn 8% interest. How much will you have in 4

years? (Pick the closest answer)

Rs 6,802.50 Rs 6,843.00 Rs 3,675

2.You have Rs 450,000 to invest. If you think you can earn 7%, how much could you

accumulate in 10 years? ? (Pick the closest answer)

Rs 25,415 Rs 722,610 Rs 722,610

3.If a commodity costs Rs500 now and inflation is expected to go up at the rate of 10%

per year, how much will the commodity cost in 5 years?

Rs 805.25 Rs 3,052.55 Cannot tell from this information

Now we will talk about the cases when the interest is given semi annually, quarterly,

monthly.

The interest rate per compounding period is found by taking the annual rate and dividing

it by the number of times per year the cash flows are compounded. The total number of

compounding periods is found by multiplying the number of years by the number of

times per year cash flows is compounded.

The formula for this shorter compounding period is

FFVVnn = PV0 (1+i/m)n*m

Consider the following example. You deposited Rs 1000 for 5 yrs in a bank that offers

10% interest p.a. compounded semiannually, what will be the future value.

=1000 (1+. 10/2) 5*2

For instance, suppose someone were to invest Rs 5,000 at 8% interest, compounded

semiannually, and hold it for five years.

The interest rate per compounding period would be 4%, (8% / 2)

The number of compounding periods would be 10 (5 x 2)

To solve, find the future value of a single sum looking up 4% and 10 periods in the

Future Value table.

FV = PV (FVIF)

FV = Rs 5,000(1.480)

FV = Rs 7,400

Now let us solve a problem for Frequency of Compounding

Problem

Suppose you invest Rs 20,000 in an account that pays 12% interest, compounded

monthly. How much do you have in the account at the end of 5 years?

Solution

FV = Rs 20,000 (1 + 0.01) 60 = Rs 20,000 (1.8167) = Rs 36,333.93

In what period of time money will be doubled?

Investor most of the times wants to know that in what period of time his money will be

doubled. For this the rule of 72 is used.

Suppose the rate of interest is 12%, the doubling period will be 72/12=6 yrs.

Apart from this rule we do use another rule, which gives better results, is the rule of 69

= .35 + 69

int rate

= .35 + 69

12

= .35 + 5.75 = 6.1 yrs

Practice Problems

What is the balance in an account at the end of 10 years if Rs 2,500 is deposited today

and the account earns 4% interest, compounded annually? Quarterly?

If you deposit Rs10 in an account that pays 5% interest, compounded annually, how

much will you have at the end of 10 years? 50 years? 100 years?

How much will be in an account at the end of five years the amount deposited today is Rs

10,000 and interest is 8% per year, compounded semi-annually?

Answers

1.Annual compounding: FV = Rs 2,500 (1 + 0.04) 10 = Rs 2,500 (1.4802) = Rs 3,700.61

Quarterly compounding: FV = Rs 2,500 (1 + 0.01) 40 = Rs 2,500 (1.4889) = Rs3,722.16

2.

10 years:

FV = Rs10 (1+0.05) 10 = Rs10 (1.6289) = Rs16.29

50 years:

FV = Rs10 (1 + 0.05) 50 = Rs10 (11.4674) = Rs114.67

100 years:

FV = Rs10 (1 + 0.05) 100 = Rs10 (131.50) = Rs 1,315.01

3. FV = Rs 10,000 (1+0.04) 10 = Rs10,000 (1.4802) = Rs14,802.44

For example, assume you deposit Rs. 10,000 in a bank, which offers 10% interest per

annum compounded semi-annually which means that interest is paid every six months.

Now, amount in the beginning = Rs. 10,000

Rs.

Interest @ 10% p.a. for first six = 500

Months 10000 x 21.0 =10500

Interest for second

6 months = 10500 x 21.0 = 525

Amount at the end of the year = 11,025

Instead, if the compounding is done annually, the amount at the end of the year will be

10,000 (1 + 0.1) = Rs, 11000. This difference of Rs. 25 is because under semi-annual

compounding, the interest for first 6 moths earns interest in the second 6 months.

The generalized formula for these shorter compounding periods is

FVn = PV mxn

MK

+1

Where

FVn = future value after n years

PV = cash flow today

K = Nominal Interest rate per annum

M = Number of times compounding is done during a year

N = Number of years for which compounding is done.

Example

Under the Vijaya Cash Certificate scheme of Vijaya Bank, deposits can be made for

periods ranging from 6 months to 10 years. Every quarter, interest will be added on to the

principal. The rate of interest applied is 9% p.a. for periods form 12 to 13 months and

10% p.a. for periods form 24 to 120 months.

An amount of Rs. 1,000 invested for 2 years will grow to

Fn = PV mn

MK

+1

Where m = frequency of compounding during a year

= 1000 8

410.1

0 +

= 1000 (1.025)8

= 1000 x 1.2184 = Rs. 1218

Effective vs. Nominal Rate of interest

We have seen above that the accumulation under the semi-annual compounding scheme

exceeds the accumulation under the annual compounding scheme compounding scheme,

the nominal rate of interest is 10% per annum, under the scheme where compounding is

done semi annually, the principal amount grows at the rate of 10.25 percent per annum.

This 1025 percent is called the effective rate of interest which is the rate of interest per

annum under annual compounding that produces the same effect as that produced by an

interest rate of 10 percent under semi annual compounding.

The general relationship between the effective an nominal rates of interest is as follows:

= 11

m

m + k

where r = effective rate of interest

k = nominal rate of interest

m = frequency of compounding per year.

Example

Find out the effective rate of interest, if the nominal rate of interest is 12% and is

quarterly compounded?

Effective rate of interest

= (1 + mk )m 1

= (+ 412.0 )4 1

= (1 + 0.03)4 -1 = 1.126 -1

= 0.126 = 12.6% p.a. compounded quarterly

By now you should have clear understanding of

Compounding Discounting Doubling period (Rule of 72) Doubling period (Rule of 69) Shorter compounding periods Effective vs. Nominal Rate of interest

By now you should be an expert in using the following two tables:

A-1 The Compound Sum of one rupee FVIF A-3 The Present Value of one rupee PVIF

IMPORTANT The inverse of FVIF is PVIF i.e. inverse of FVIF is PVIF.

IMPORTANT Slide 1

3-1

Chapter 3Chapter 3

Time Value of Money

Time Value of Time Value of MoneyMoney

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3-2

The Time Value of MoneyThe Time Value of MoneyThe Time Value of Money

The Interest Rate Simple Interest Compound Interest Amortizing a Loan

The Interest Rate Simple Interest Compound Interest Amortizing a Loan

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3-3

Obviously, $10,000 today$10,000 today.

You already recognize that there is TIME VALUE TO MONEYTIME VALUE TO MONEY!!

The Interest RateThe Interest RateThe Interest Rate

Which would you prefer -- $10,000 $10,000 today today or $10,000 in 5 years$10,000 in 5 years?

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3-4

TIMETIME allows you the opportunity to postpone consumption and earn

INTERESTINTEREST.

Why TIME?Why TIME?Why TIME?

Why is TIMETIME such an important element in your decision?

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3-5

Types of InterestTypes of InterestTypes of Interest

Compound InterestCompound InterestInterest paid (earned) on any previous

interest earned, as well as on the principal borrowed (lent).

Simple InterestSimple InterestInterest paid (earned) on only the original

amount, or principal borrowed (lent).

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3-6

Simple Interest FormulaSimple Interest FormulaSimple Interest Formula

FormulaFormula SI = P0(i)(n)SI: Simple InterestP0: Deposit today (t=0)i: Interest Rate per Periodn: Number of Time Periods

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3-7

SI = P0(i)(n)= $1,000(.07)(2)= $140$140

Simple Interest ExampleSimple Interest ExampleSimple Interest Example

Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

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3-8

FVFV = P0 + SI = $1,000 + $140= $1,140$1,140

Future ValueFuture Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (FV)Simple Interest (FV)Simple Interest (FV)

What is the Future Value Future Value (FVFV) of the deposit?

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3-9

The Present Value is simply the $1,000 you originally deposited. That is the value today!

Present ValuePresent Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (PV)Simple Interest (PV)Simple Interest (PV)

What is the Present Value Present Value (PVPV) of the previous problem?

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3-10

0

5000

10000

15000

20000

1st Year 10thYear

20thYear

30thYear

Future Value of a Single $1,000 Deposit

10% SimpleInterest7% CompoundInterest10% CompoundInterest

Why Compound Interest?Why Compound Interest?Why Compound Interest?

Futu

re V

alue

(U.S

. Dol

lars

)

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3-11

Assume that you deposit $1,000$1,000 at a compound interest rate of 7% for

2 years2 years.

Future ValueSingle Deposit (Graphic)Future ValueFuture ValueSingle Deposit (Graphic)Single Deposit (Graphic)

0 1 22

$1,000$1,000FVFV22

7%

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3-12

FVFV11 = PP00 (1+i)1 = $1,000$1,000 (1.07)= $1,070$1,070

Compound InterestYou earned $70 interest on your $1,000

deposit over the first year.This is the same amount of interest you

would earn under simple interest.

Future ValueSingle Deposit (Formula)Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)

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3-13

FVFV11 = PP00 (1+i)1 = $1,000$1,000 (1.07)= $1,070$1,070

FVFV22 = FV1 (1+i)1= PP0 0 (1+i)(1+i) = $1,000$1,000(1.07)(1.07)= PP00 (1+i)2 = $1,000$1,000(1.07)2

= $1,144.90$1,144.90You earned an EXTRA $4.90$4.90 in Year 2 with

compound over simple interest.

Future ValueSingle Deposit (Formula)Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)

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3-14

FVFV11 = P0(1+i)1

FVFV22 = P0(1+i)2

General Future Value Future Value Formula:FVFVnn = P0 (1+i)n

or FVFVnn = P0 (FVIFFVIFi,n) -- See Table ISee Table I

General Future Value FormulaGeneral Future General Future Value FormulaValue Formula

etc.

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3-15

FVIFFVIFi,n is found on Table I at the end of the book or on the card insert.

Valuation Using Table IValuation Using Table IValuation Using Table I

Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.1663 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

1.145

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3-16

FVFV22 = $1,000 (FVIFFVIF7%,2)= $1,000 (1.145)= $1,145$1,145 [Due to Rounding]

Using Future Value TablesUsing Future Value TablesUsing Future Value Tables

Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.1663 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

1.145

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3-17

Julie Miller wants to know how large her deposit of $10,000$10,000 today will become at a compound annual interest rate of 10% for 5 years5 years.

Story Problem ExampleStory Problem ExampleStory Problem Example

0 1 2 3 4 55

$10,000$10,000

FVFV55

10%

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3-18

Calculation based on Table I:FVFV55 = $10,000 (FVIFFVIF10%, 5)

= $10,000 (1.611)= $16,110$16,110 [Due to Rounding]

Story Problem SolutionStory Problem SolutionStory Problem Solution

Calculation based on general formula:FVFVnn = P0 (1+i)nFVFV55 = $10,000 (1+ 0.10)5

= $16,105.10$16,105.10

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3-19

We will use the RuleRule--ofof--7272..

Double Your Money!!!Double Your Money!!!Double Your Money!!!

Quick! How long does it take to double $5,000 at a compound rate

of 12% per year (approx.)?

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3-20

Approx. Years to Double = 7272 / i%

7272 / 12% = 6 Years6 Years[Actual Time is 6.12 Years]

The Rule-of-72The RuleThe Rule--ofof--7272

Quick! How long does it take to double $5,000 at a compound rate

of 12% per year (approx.)?

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3-21

Assume that you need $1,000$1,000 in 2 years.2 years.Lets examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually.

0 1 22

$1,000$1,0007%

PV1PVPV00

Present ValueSingle Deposit (Graphic)Present ValuePresent ValueSingle Deposit (Graphic)Single Deposit (Graphic)

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3-22

PVPV00 = FVFV22 / (1+i)2 = $1,000$1,000 / (1.07)2= FVFV22 / (1+i)2 = $873.44$873.44

Present Value Single Deposit (Formula)Present Value Present Value Single Deposit (Formula)Single Deposit (Formula)

0 1 22

$1,000$1,0007%

PVPV00

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3-23

PVPV00 = FVFV11 / (1+i)1

PVPV00 = FVFV22 / (1+i)2

General Present Value Present Value Formula:PVPV00 = FVFVnn / (1+i)n

or PVPV00 = FVFVnn (PVIFPVIFi,n) -- See Table IISee Table II

General Present Value FormulaGeneral Present General Present Value FormulaValue Formula

etc.

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3-24

PVIFPVIFi,n is found on Table II at the end of the book or on the card insert.

Valuation Using Table IIValuation Using Table IIValuation Using Table II

Period 6% 7% 8% 1 .943 .935 .926 2 .890 .857 3 .840 .816 .794 4 .792 .763 .735 5 .747 .713 .681

.873

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3-25

PVPV22 = $1,000$1,000 (PVIF7%,2)= $1,000$1,000 (.873)= $873$873 [Due to Rounding]

Using Present Value TablesUsing Present Value TablesUsing Present Value Tables

Period 6% 7% 8%1 .943 .935 .9262 .890 .8573 .840 .816 .7944 .792 .763 .7355 .747 .713 .681

.873

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3-26

Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000$10,000 in 5 years5 years at a discount rate of 10%.

Story Problem ExampleStory Problem ExampleStory Problem Example

0 1 2 3 4 55

$10,000$10,000PVPV00

10%

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3-27

Calculation based on general formula:PVPV00 = FVFVnn / (1+i)nPVPV00 = $10,000$10,000 / (1+ 0.10)5

= $6,209.21$6,209.21

Calculation based on Table I:PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5)

= $10,000$10,000 (.621)= $6,210.00$6,210.00 [Due to Rounding]

Story Problem SolutionStory Problem SolutionStory Problem Solution

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3-28

Types of AnnuitiesTypes of AnnuitiesTypes of Annuities

Ordinary AnnuityOrdinary Annuity: Payments or receipts occur at the end of each period.

Annuity DueAnnuity Due: Payments or receipts occur at the beginning of each period.

An AnnuityAn Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

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3-29

Examples of AnnuitiesExamples of Annuities

Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings

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3-30

Parts of an AnnuityParts of an AnnuityParts of an Annuity

0 1 2 3

$100 $100 $100

(Ordinary Annuity)EndEnd of

Period 1EndEnd of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

EndEnd ofPeriod 3

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3-31

Parts of an AnnuityParts of an AnnuityParts of an Annuity

0 1 2 3

$100 $100 $100

(Annuity Due)BeginningBeginning of

Period 1BeginningBeginning of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

BeginningBeginning ofPeriod 3

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3-32

FVAFVAnn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 + R(1+i)0

Overview of an Ordinary Annuity -- FVAOverview of an Overview of an Ordinary Annuity Ordinary Annuity ---- FVAFVA

R R R

0 1 2 n n n+1

FVAFVAnn

R = Periodic Cash Flow

Cash flows occur at the end of the period

i% . . .

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3-33

FVAFVA33 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0

= $1,145 + $1,070 + $1,000= $3,215$3,215

Example of anOrdinary Annuity -- FVAExample of anExample of anOrdinary Annuity Ordinary Annuity ---- FVAFVA

$1,000 $1,000 $1,000

0 1 2 3 3 4

$3,215 = FVA$3,215 = FVA33

7%

$1,070

$1,145


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3-34

Hint on Annuity ValuationHint on Annuity Valuation

The future value of an ordinary annuity can be viewed as

occurring at the endend of the last cash flow period, whereas the future value of an annuity

due can be viewed as occurring at the beginningbeginning of

the last cash flow period.

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3-35

FVAFVAnn = R (FVIFAi%,n)FVAFVA33 = $1,000 (FVIFA7%,3)

= $1,000 (3.215) = $3,215$3,215

Valuation Using Table IIIValuation Using Table IIIValuation Using Table III

Period 6% 7% 8%1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.2464 4.375 4.440 4.5065 5.637 5.751 5.867

3.215

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3-36

FVADFVADnn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 + R(1+i)1

= FVAFVAnn (1+i)

Overview View of anAnnuity Due -- FVADOverview View of anOverview View of anAnnuity Due Annuity Due ---- FVADFVAD

R R R R R

0 1 2 3 nn--11 n

FVADFVADnn

i% . . .

Cash flows occur at the beginning of the period

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3-37

FVADFVAD33 = $1,000(1.07)3 + $1,000(1.07)2 + $1,000(1.07)1

= $1,225 + $1,145 + $1,070= $3,440$3,440

Example of anAnnuity Due -- FVADExample of anExample of anAnnuity Due Annuity Due ---- FVADFVAD

$1,000 $1,000 $1,000 $1,070

0 1 2 3 3 4

$3,440 = FVAD$3,440 = FVAD33

7%

$1,225$1,145


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3-38

FVADFVADnn = R (FVIFAi%,n)(1+i)FVADFVAD33 = $1,000 (FVIFA7%,3)(1.07)

= $1,000 (3.215)(1.07) = $3,440$3,440

Valuation Using Table IIIValuation Using Table IIIValuation Using Table III

Period 6% 7% 8%1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.2464 4.375 4.440 4.5065 5.637 5.751 5.867

3.215

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3-39

PVAPVAnn = R/(1+i)1 + R/(1+i)2

+ ... + R/(1+i)n

Overview of anOrdinary Annuity -- PVAOverview of anOverview of anOrdinary Annuity Ordinary Annuity ---- PVAPVA

R R R

0 1 2 n n n+1

PVAPVAnn

R = Periodic Cash Flow

i% . . .


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3-40

PVAPVA33 = $1,000/(1.07)1 + $1,000/(1.07)2 + $1,000/(1.07)3

= $934.58 + $873.44 + $816.30 = $2,624.32$2,624.32

Example of anOrdinary Annuity -- PVAExample of anExample of anOrdinary Annuity Ordinary Annuity ---- PVAPVA

$1,000 $1,000 $1,000

0 1 2 3 3 4

$2,624.32 = PVA$2,624.32 = PVA33

7%

$ 934.58$ 873.44 $ 816.30


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3-41

Hint on Annuity ValuationHint on Annuity Valuation

The present value of an ordinary annuity can be viewed as

occurring at the beginningbeginning of the first cash flow period,

whereas the present value of an annuity due can be viewed as

occurring at the endend of the first cash flow period.

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3-42

PVAPVAnn = R (PVIFAi%,n)PVAPVA33 = $1,000 (PVIFA7%,3)

= $1,000 (2.624) = $2,624$2,624

Valuation Using Table IVValuation Using Table IVValuation Using Table IV

Period 6% 7% 8%1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.5774 3.465 3.387 3.3125 4.212 4.100 3.993

2.624

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3-43

PVADPVADnn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1= PVAPVAnn (1+i)

Overview of anAnnuity Due -- PVADOverview of anOverview of anAnnuity Due Annuity Due ---- PVADPVAD

R R R R

0 1 2 nn--11 n

PVADPVADnnR: Periodic Cash Flow

i% . . .


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3-44

PVADPVADnn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02$2,808.02

Example of anAnnuity Due -- PVADExample of anExample of anAnnuity Due Annuity Due ---- PVADPVAD

$1,000.00 $1,000 $1,000

0 1 2 33 4

$2,808.02 $2,808.02 = PVADPVADnn

7%

$ 934.58$ 873.44


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3-45

PVADPVADnn = R (PVIFAi%,n)(1+i)PVADPVAD33 = $1,000 (PVIFA7%,3)(1.07)

= $1,000 (2.624)(1.07) = $2,808$2,808

Valuation Using Table IVValuation Using Table IVValuation Using Table IV

Period 6% 7% 8%1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.5774 3.465 3.387 3.3125 4.212 4.100 3.993

2.624

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3-46

1. Read problem thoroughly2. Determine if it is a PV or FV problem3. Create a time line4. Put cash flows and arrows on time line5. Determine if solution involves a single

CF, annuity stream(s), or mixed flow6. Solve the problem7. Check with financial calculator (optional)

Steps to Solve Time Value of Money ProblemsSteps to Solve Time Value Steps to Solve Time Value of Money Problemsof Money Problems

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3-47

Julie Miller will receive the set of cash flows below. What is the Present Value Present Value at a discount rate of 10%10%?

Mixed Flows ExampleMixed Flows ExampleMixed Flows Example

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $100PVPV00

10%10%

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3-48

1. Solve a piecepiece--atat--aa--timetime by discounting each piecepiece back to t=0.

2. Solve a groupgroup--atat--aa--timetime by firstbreaking problem into groups ofannuity streams and any singlecash flow group. Then discount each groupgroup back to t=0.

How to Solve?How to Solve?How to Solve?

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3-49

Piece-At-A-TimePiecePiece--AtAt--AA--TimeTime

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $10010%

$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$ 62.09$ 62.09

$1677.15 $1677.15 = = PVPV00 of the Mixed Flowof the Mixed Flow

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3-50

Group-At-A-Time (#1)GroupGroup--AtAt--AA--Time (#1)Time (#1)

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $10010%

$1,041.60$1,041.60$ 573.57$ 573.57$ 62.10$ 62.10

$1,677.27$1,677.27 = = PVPV00 of Mixed Flow of Mixed Flow [Using Tables][Using Tables]

$600(PVIFA10%,2) = $600(1.736) = $1,041.60$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57

$100 (PVIF10%,5) = $100 (0.621) = $62.10

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3-51

Group-At-A-Time (#2)GroupGroup--AtAt--AA--Time (#2)Time (#2)

0 1 2 3 4

$400 $400 $400 $400$400 $400 $400 $400

PVPV00 equals$1677.30.$1677.30.

0 1 2

$200 $200$200 $200

0 1 2 3 4 5$100$100

$1,268.00$1,268.00

$347.20$347.20

$62.10$62.10

PlusPlus

PlusPlus

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3-52

General Formula:FVn = PVPV00(1 + [i/m])mn

n: Number of Yearsm: Compounding Periods per Yeari: Annual Interest RateFVn,m: FV at the end of Year nPVPV00: PV of the Cash Flow today

Frequency of CompoundingFrequency of Frequency of CompoundingCompounding

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3-53

Julie Miller has $1,000$1,000 to invest for 2 years at an annual interest rate of

12%.Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)

= 1,254.401,254.40Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2)

= 1,262.481,262.48

Impact of FrequencyImpact of FrequencyImpact of Frequency

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3-54

Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2) = 1,266.771,266.77

Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2) = 1,269.731,269.73

Daily FV2 = 1,0001,000(1+[.12/365])(365)(2) = 1,271.201,271.20

Impact of FrequencyImpact of FrequencyImpact of Frequency

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3-55

Effective Annual Interest RateThe actual rate of interest earned (paid) after adjusting the nominal

rate for factors such as the number of compounding periods per year.

(1 + [ i / m ] )m - 1

Effective Annual Interest RateEffective Annual Effective Annual Interest RateInterest Rate

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Documents

Lecture 05