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T R A N S A C T I O N S O F S O C I E T Y O F A C T U A R I E S

1 9 6 2 V O L . 1 4 P T . 1 N O . 3 9 A B

A N N U I T Y V A L U E S D I R E C T L Y F R O M

T H E M A I ( E I - I A M C O N S T A N T S

J O H N A . M E R E U

I N T R O D U C T I O N

w R Y a c t u a r ia l s t u d e n t i s f a m i l ia r w i t h M a k e h a m ' s f a m o u s F i r s t

L a w o f M o r t a l i t y u n d e r w h i c h t h e fo r ce of m o r t a l i t y a t a n y a g e

is g i v e n b y t h e fo ll o w in g e q u a t i o n :

~ . =

A q - B a * ( 1 )

Se v e ra l mo r t a l i t y t a b l e s w h i c h h a v e b e e n o r a r e p r e s e n t l y i n u s e a r e

M a k e h a m i z e d . Fo r s u c h t a b le s t h e l a b o r i n v o l v e d i n c a l c u l a ti n g j o i n t l if e

va lues i s reduced because i t i s poss ib le to use equ iva len t equa l ages to

the ages o f the l ives un der cons ide ra t ion .

A p a r t f r o m t h e a p p l i ca t io n t o j o i n t l if e p r o b le m s t h e f a c t t h a t M a k e -

h a m ' s L a w h o l d s i s t a k e n i n t o a c c o u n t o n l y i n t h e o r i g i n a l c o n s t ru c t i o n

o f t h e mo r t a l i t y t a b l e . O n c e t h e t a b l e i s c o n s t ru c t e d c o mmu t a t i o n fu n c -

t io n s a n d o t h e r v a l u e s a r e o b t a i n e d i n t h e s a me m a n n e r a s fo r m o r t a l i t y

tables in genera l .

T h e m o r t a l i t y t a b l e a n d c o m m u t a t i o n f u n c t i o n s a r e t o o l s w h i c h t h e

a c t u a ry e mp l o y s i n c a l c u l a t i n g a n n u i t y , i n s u ra n c e a n d o t h e r l i f e c o n -

t i n g e n c y fu n c ti o n s . H o w e v e r , w h e re t h e g o v e rn i n g l a w o f m o r t a l i t y is

r e p re s e n te d b y a m a t h e m a t i c a l e q u a t i o n i t s h o u l d b e p o s s ib le t o e x p res s

l if e c o n t i n g e n c y fu n c t i o n s i n t e rms o f t h e p a ra m e t e r s o f th e m o r t a l i t y la w

d i r e c tl y w i t h o u t d e ve lo p in g th e s t a n d a r d m o r t a l i t y t a b l e a n d c o m m u t a -

t ion fun c t ion too ls .

Whe the r i t i s wor th whi le to ob ta in expres s ions fo r l i f e con t ingency

fu n c t i o n s d i r e c t l y f ro m t h e g o v e rn i n g l aw o f m o r t a l i t y i t se l f w ill d e p e n d

o n a n u m b e r o f f ac t o r s. A n i m p o r t a n t c o n s i d e ra t i o n w i ll b e t h e e a se i n

p ra c t i c e o f u s i n g t h e fo rmu l a d e v e l o p e d fo r d e t e rmi n i n g t h e v a l u e o f a

p a r t i c u l a r f u n c t i o n . A s e c o n d c o n s id e ra t io n w o u l d b e t h e n u m b e r o f ca l-

c u l a t io n s t o b e m a d e . I f o n l y a f e w c a lc u l a ti o n s a r e t o b e m a d e i t m i g h t

b e w o r t h w h il e t o a v o i d co n s t ru c t in g t h e m o r t a l i t y t a b le a n d c o m m u t a -

t i o n fu n c t i o n s . A t h i rd c o n s i d e ra t i o n i s t h e c a l c u l a t i n g e q u i p me n t

ava i l ab le .

In t h i s p a p e r t h e r e is d e v e l o p ed a fo rm u l a fo r t h e c o n t i n u o u s a n n u i t y

a , : ~ i n t e rm s o f t h e M a k e h a m p a r a m e t e r s A , B , a n d c , a n d t h e f or ce o f

269

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270 A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E I I A M C O N S T A N T S

interest. s m a n y life ontingency functions an be expressed in ter ms of

a~:~, the se functions also will enefit ro m a useful orm ula for a~:~.

At t h is o i n t it s h o u l d b e r e c og n i ze d t h a t Mr. Em o r y McCl i n t o c k , i n

a pa pe r presented to the Institute f Actuaries in Ju ly 1874 entitled On

C o m p u t a t i o n o f An nu i ti es o n M r . M a k e h a m ' s H y po th e si s, d e ve l op e d a

f o r m u l a f or t h e co n t in u o us a n n u i t y d~ i n t e r m s o f t h e t h re e Ma k e h a m c o n -

s t an t s a n d t h e f o r ce o f i nt er es t. s Mr. McC l i n t o c k ' s p a p e r c a m e t o m y

a t t e n t i o n o n l y a f t e r th i s p a p e r h a d b e e n d r a f te d a n d a s t h e m e t h o d s a l-

t h o u g h s i m i l a r a r e n o t i d e n t i c a l , I h a v e f o u n d i t m o r e c o n v e n i e n t t o

p r o c e e d a l o n g t h e l in e s o ri g i n a ll y p l a n n e d a n d t h e n t o c o m p a r e t h e f i na l

r e s u lt s w i t h M r . M c C l i n t o c k ' s . B e c a u s e o f t h e p a s s ag e o f t i m e s in c e M r .

M c C l i n t o c k ' s p a p e r w a s p u b l i sh e d , i t is l ik e l y t h a t i t is n o t n o w w e ll

k n o w n a n d I t h e r e f o r e c o n s i d e r i t w o r t h w h i le t o h a v e i t r e - e x a m i n e d in

t h e l i g h t o f p r e s e n t - d a y a c t u a r i a l t h in k i n g .

T I ~ C O N T I N U O U S A N N U I T Y ax:n--]

Con s ide r the c on t inuo us an nu i ty a .:.--1. I t can be r epres en t ed i n i n t egra l

fo rm as fo l lows :

f o

: - q =

v * . t p , d t .

( 2 )

E x p r e s s e d a s a f u n c t i o n o f t h e f o r c e s o f i n t e r e s t a n d m o r t a l i t y ,

e - f ° " x + h a h d t . ( 3 )

f . e _ ~ t t

a x : n - l . o

F o r a M a k e h a m T a b l e t h e e x p r e s s io n f o r a ~:~ , a f t e r s u b s t i t u t io n f o r th e

f o rc e o f m o r t a l i ty , b e c o m e s :

in

: - q = e - ( a + ~ ) t ,

e - ( B : + t A ~ ' c ) . e ( B : A n c ) d t .

( 4 )

Th i s c om pl i ca t ed express ion fo r a.:.--] can be s impl i fi ed b y m aking t he

fo ll owing subs t i t u t i on s :

B c x

K,- (5)

l n c

H - A + ~

( 6 )

In c

B c x + '

- K ~ ' c ' . (7)

Y - - l n c

t I n i s u s e d t o d e n o t e t h e n a t u r a l l o g a r i t h m o r l o g a r i t h m t o b a s e e .

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ANNUITY VALU ES D/RECTLY FROM MAKEHAM CONSTANTS 271

It fol lows that

and

t - I n y / K ~ ) ( 8 )

In c

d t - d y

( 9 )

y l n c "

i,, K x t n

d,v

-- eK* fK~ x+"~ - ~ ) - g l n

" - -- U d Y

( l o )

Defining

and

it follows tha t

and

8 K x H f K x q ' n e - -u

= I n c K . J K . f + z

d y .

(11)

e - - Y

/ (y ) - (12 )

y~+H

F ( K . ) = £ ? f ( y ) d y ,

( 1 3 )

F ( K , ) - - F ( K ~ + . )

( 1 4 )

a : ~ =

K , . f ( K . ) . l n c

F ( K . )

a ~ = K . . f ( K ~ )

. l n c " ( 1 5 )

Defining

F ( K . )

~ ( K . ) ( 1 6 )

f ( K , )

b y ~ ( K . ) , i t fo llo w s t h at ~ - K , ' l n ~ "

T H E N A T U R E O r T H E r l Y NC TI ON S F ( K) , f K ) , ¢ K )

I t i s n o t p o s s i b l e t o e v a l u a t e e x a c t l y t h e d e f i n i t e i n t e g r a l r e p r e s e n t e d

b y F K ) . M e t h o d s f o r a p p r o x i m a t i n g F K ) a r e t h e r e f o r e r e q u i r e d . T h e

o p t i m u m m e t h o d i s t h e o n e w h i c h p r o d u c e s v a l u e s o f F K ) t o a d e s i r e d

d e g r e e o f a c c u r a c y w i t h t h e m i n i m u m a m o u n t o f c a l c u l a t i o n . A s at is f a c-

t o r y m e t h o d i s o n e w h i c h p r o d u c e s s uf f i ci e n tl y a c c u r a t e v a l u e s w i t h a

tolerable a m ou nt of calculat ion;

Graphical ly

F(K)

can be represented by the shaded a rea ( in F ig . 1)

subten ded b y the curve f (y) to the r ight of the ord ina te K . I f the d ot t ed

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2 72 A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E H A M CO N S T A N T S

curve r ep resen ts the g raph fo r e - ' /A u+H, the a rea to the r igh t of K under

the do t ted cu rve is g iven by

f K ~

'

e--v e--K

K I+ H d y = K l+ H = I ( K ) .

Th e ra t io of the smaller area

F ( K )

to the larger area

f ( K )

is ~(K).

As K increases, both

f ( K )

an d

F ( K )

become very smal l and ~(K) ap-

proaches uni ty .

~'IRST

M E T H O D F O R A P P R O X T M A T I N G

F K )

Using the E uler-M acL aurin form ula the valu e of a definite integral can

be approx imated f rom the sum o f even ly spaced o rd inates th roughou t the

area and values of the der ivat ives a t the l imits of in tegrat ion . Because

l~Gum~ 1

the f-cu rve s and all their der iva tives beco m e zero at infinity, only the

values of the der ivat ives a t the lower bound ary af fect F ( K ) . Accordingly

F ( K ) = L " f ( y ) d y

( 1 7 )

- - ~ . ~ f ( K q - t ) - - ½ f ( K ) + ~ - ~ I ' ( K ) - - ~z-6¢ " ' ¢ K )

$=0

an d

( . _ L _ K ' ÷ "

\ K + 1 2

t - o ( 1 8 )

(K~ I+R

]

+

g--2

\ 'U - + - ~ ] + " " " J "

= f ( K ) [ 1 ' 1 + / / 1

f ' ( K )

- - t - - - K - j . ( 1 9 )

f ' " ( K )

=

- - f ( K )

[1- -b

1

+__1t3

K

( 2 0 )

+ 3 ( I + H ) ( 2+H ) K 4 I+H) 2+//) 3+//)]K . .

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A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E H A M CO N S T A NT S 2 73

C o l l e c t i n g t e r m s , w e h a v e :

1 . - ,

\ K + I ]

(

K

I + H .

+e-~k-g-~] + . . . ] - - . 0 8 1 9 - - . 0 7 9 2 - - - - g - - - t . . . ( 2 1 )

+ . 0 0 4 2 ( 1 -4- H ) K (2 + H ) + . 0 0 1 4 ( 1 + H ) ( 2 +K H ) ( 3 + H ) t "

T A B L E

l - - F ( 1 0 )

Ratios tof(10) of

d (1 0) . . . . . . . . . .

3 (11) . . . . . . . . . . i i i i i i ~ i i i i i i i i i i i i i

](12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

]'(13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

/ ( 1 4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

/(15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

f ( 1 6 )

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

/(17) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

f(18 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~ / ( 1 0 4 - 0 +½ f(10 ) . . . . . . . . . . . . . . . . .

t = l

- .0 8 19 /(1 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- . 0 7 9 2 / ( 1 0 ) (1 + H ) /l O . . . . . . . . . . . . . . . . . . .

- -F 0 0 4 2 f( I 0 ) 1 + / / ) ( 2 + / / ) / 1 0 0 . . . . . . . . . . .

+ .0014]'(10) (1 +1/) (2 + H ) (3 +H )/100 0 . . . . .

~ ( i 0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~(10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

q ,( lO )/lO = a In c . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 0 e F (1 0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H=O

1 ( I 0 ) =.0000045

.5000

.3344

. 1 1 2 8

.0383

.0131

.0045

.0015

.0005

.0002

1 . 0 0 5 3

- - .0819

- - .0079

+ .0001

+ . 0 0 0 0

.9156

.0000041

.09156

.000004

1 t = 1

f ( 1 0 ) = . 0 0 0 0 0 0 4 5

.5000

.3040

. 0 9 ~

.0295

.0093

.0030

.0010

.0003

.0001

. 9 4 1 2

- - .0819

-- .0158

+ . 0 0 0 2

+ . 0 0 0 0

.8437

.00000038

.08437

.000004

I t i s e v i d e n t t h a t t h e a b o v e f o r m u l a s u f f er s f r o m t w o s e r i ou s d i s a d v a n -

t a g e s . A g r e a t d e a l o f c a l c u l a ti o n i s i n v o l v e d a n d t h e f o r m u l a c a n b e u s e d

o n l y if t h e v a l u e s o f h i g h e r d e r i v a t iv e s n o t t a b u l a t e d c a n b e ig n o re d . I t

i s o n l y f o r v e r y l ar g e K t h a t t h e a b o v e f o r m u l a i s p r a c t ic a l , s o t h a t i t is

p r o b a b l y l i m i t e d t o o b t a i n i n g t h e v a l u e o f g , f o r c e n t e n a r i a n s a n d o l de r .

A s i t w i ll b e u s e f u l f o r f o r m u l a s t o b e d e v e l o p e d l a t e r , w e s h a l l u s e f o r -

m u l a 2 1 t o c a l c u l a t e i n T a b l e 1 F ( 1 0 ) a n d 4 ( 1 0 ) f o r H = 0 a n d H = 1,

t w o e x t r e m e v a l u e s o f l i t .

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274

A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E H A M C O N S T A N T S

S E C O N D M E T H O D O F A P P R O X I M A T I N G

F ( K )

A s

F ( K ) = f Z f ( y ) d y ,

i t s d e r i v a t i v e is g i v e n b y

g - - K

F ' ( K ) = - - J ( K ) - K I+ H

K S K

= _ K - I - U + K - H - - - -

K 1 - H

Kn-l-tt

- F . . . - F ( - 1 ) " + 1 - F

I n t e g r a t i n g , w e h a v e

K - U K I - H

F ( K ) + C o ns ta nt of In te g ra t io n = -- - i f - n - 1 --------H

o r

K2-1t Kn-tl

( 2 H ) I 2 F . . . + ( - - 1 ) " + 1 -

- _ ( n - B ) l n _

] - . . . , 0 < H < I

( 2 2 )

( 2 3 )

( 2 4 )

g 2

= - - l n K + . K - - - ~ - _ +

. . .

. . . . ( 2 5 )

-I--(---

i ) '~ +i g + . H = O .

D e f i n i n g t h e s e r i e s o n t h e r i g h t - h a n d s i d e o f t h e e q u a t i o n a s G ( K ) , w e

h a v e

F ( K )

"1- C o n s t a n t o f I n t e g r a t i o n = G ( K )

( 2 6 )

F ( ~" ) + C o n s t a n t o f I n t e g r a t i o n = G (o o ) .

B u t F ( ~ o ) = 0 . . '. C o n s t a n t o f I n t e g r a t i o n = G ( o o ) . ( 2 7 )

. ' . F ( K ) - - G ( K ) - . G ( o o ) . ( 2 8 )

E V A L U A T I O N O F T H E C O N T I N U O U S A N N U I T Y

U S I N G S E C O ND A P P R O X I M A T I O N

U s in g t h e s e c on d a p p r o x i m a t i o n a b o v e , w e o b t a i n b y s u b s t i t u t i o n f r om

( 1 4 ) t h a t

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ANNUITY VALUES DIRECTLY FROM MAK.EttAM CONSTAN TS 275

G ( K , ) - - G ( K , + , ) ( 2 9 )

a = : " - ] = K . ' f ( K = ) ' l n c

a n d

G ( K=) - -G ( ¢o )

( 3 0 )

a * = K , . / ( ~ - ~ , ) : ~ c

To eva lu a t e d , fo r some g iven age x r equ i re s t he fo l lowing s t eps :

S t e p 1 . D e t e r m i n e K = f r o m e q u a t i o n 5 .

S te p 2. D e t e r m i n e / / f r o m e q u a t io n 6 .

• K 2 K *

Step 3 . D eterm ine the success ive te rm s, 1, K , /2 ' [3 ' e tc . , un t i l an in-

s ign if i can t t e rm is reached t ak ing i n to acc oun t t he n um be r o f dec i-

m al p laces se lec ted , and l i s t resul t s .

S t ep 4 . D iv ide success ive t e rm s b y - H , 1 - H , 2 - - H , e t c . , an d l is t

resul ts .

S t ep 5 . A l t e rn a t e ly sub t r ac t an d add the serie s o f te rm s de t e rm ined

in s tep 4 .

S t ep 6 . De te rm ine K 8 us ing l oga r i t hm t ab les .

S t ep 7 . D iv ide t he r e su l t o f s t ep 5 by t he r e su l t o f s t ep 6 t o ob t a in

a ( g ) .

S t e p 8 . D e t e r m i n e e - K b y a l t e r n a t e l y a d d i n g a n d s u b t r a c t i n g t e r m s o f

s tep 3 , and check resul t , if poss ib le , f rom a se t of m ath em at ic a l t ables .

S t e p 9 . C o m p u t e t h e d e n o m i n a t o r f o r t h e a n n u i t y v a l u e K=. f ( K , ) .

In

c = (e-K/K~) . In

c , us ing s t e ps 6 and 8 .

S t e p 1 0. O b t a i n G ( ~ ) - 1 / t t b y i n t e r p o l a t i o n i n t h e s e c o n d c o l u m n

o f T a b l e 4, a n d h e n c e G ( ~ ) . A n a l t e r n a t e m e i h o d w o u ld b e t o u se

e q u a t i o n 35 a n d p u b l i sh e d t a b le s o f t h e G a m m a F u n c t i o n .

S t e p 11. D e t e r m i n e t h e n u m e r a t o r fo r th e a n n u i t y v a l u e b y s u b t r a c t -

i ng t he r e su l t o f st ep 10 f rom the r e su l t o f s t ep 7 .

S t e p 1 2. D e t e r m i n e t h e v a l u e o f t h e a n n u i t y , b y d i v is io n o f t h e r e s u l t

o f s tep 11 by t he r e su l t o f s t ep 9 .

I t w i l l b e a p p a r e n t t h a t s t e p s 2 a n d 1 0 n e e d b e p e r f o r m e d o n l y o n c e

f o r a n y m o r t a l i t y t a b l e a n d i n t e r e s t r a t e . A p a r t f r o m s t e p s 3 t o 5 t h e

a m o u n t o f c a lc u l a ti o n f o r e a c h st e p w ill b e a p p a r e n t . T h e l a b o r i n v o l v e d

in s t eps 3 t o 5 c l ea r ly depends on t he magni tude o f K= i t s e l f . Cons ide r

T a b l e 2 w h i c h s h o w s t h e n u m b e r o f t e r m s w h i c h m u s t b e t a k e n i n s t e p

3 t o ob t a in a v a lue cor r ec t t o f ive dec ima l s . Fo r t he va lues o f K= appea r -

i n g i n t h e t a b le t h e a p p l ic a b le a g e o n t h e a - 1 9 4 9 F e m a l e T a b l e i s s h o w n

also. "

T h e q u a n t i t y G ( ~ ) a p p e a ri n g in t h e n u m e r a t o r f o r d= is t h e l i m i t t o

which G( K) approaches as K be 'comes inf in i te . As Table 2 sugges t s , the

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2 7 6 A N N U I T Y V A L U E S D I R E C T L Y F R O M M A I L E H A M C O N S T A N T S

l a b o r i n v o l v e d i n c o m p u t i n g G K ) u s i n g s t e p 3 b e c o m e s t o o g r e a t f o r

l a r g e r v a l u e s o f K . H o w e v e r , f o r l a r g e v a l u e s o f

K , F ( K )

c a n b e r e a d i ly

d e t e r m i n e d u s i n g t h e f i r s t m e t h o d o f a p p r o x i m a t i o n o f t h i s p a p e r . S i n c e

G ( o~ ) = G ( K ) - - F ( K ) , i t c a n b e d e t e r m i n e d b e s t b y s e l ec t in g a v a l u e o f

K w h e r e b o t h F ( K ) a n d G ( K ) c a n b e p r a c t ic a l l y a n d i n d e p e n d e n t l y d e -

t e r m i n e d . T a k i n g K = 1 0 a s a s a t i s f a c t o r y v a l u e , w e h a v e

G ( co ) = G ( I O ) - F ( I O )

1

- 1 0 ~ [ 1 0 ~ ' G ( 1 0 ) - 1 0 H ' F ( 1 0 ) | ( 3 1 )

- 1 0 B [ 1 0 ~ . G ( 1 0 ) - - . 0 0 0 0 0 4 1

T A B L E 2

N U M B E R O F T E R M S T O B E C A L -

C U L A T E D U N D E R S T E P 3

[ see Tab le 1 ] .

.01 . . . . . . . . . . . . . .

.02 . . . . . . . . . . . . .

.0S . . . . . . . . . . . . . .

. 1 . . . . . . . . . . . . . . .

, 2 . . . . . . . . . . . . . . .

. 5 . . . . . . . . . . . . . . .

1 . . . . . . . . . . . . . . . . .

2 . . . . . . . . . . . . . . . . .

5 . . . . . . . . . . . . . . . . .

A g e

45

51

59

65

71

79

86

92

100

N u m b e r o f

T e r m s

2

2

3

3

4

6

9

12

21

F r o m T a b l e 1 i t i s a p p a r e n t t h e v a lu e o f 1 0 R . F (1 0 ) r e m a i n s r e la t i v e l y

s m a l l a n d c o n s t a n t f o r a ll H . T h e r e f o r e G ( m ) c a n b e d e t e r m i n e d f o ll ow -

i ng t h e s a m e p r o c e d u r e a s d e s c ri b e d a b o v e f o r G ( K ) w i t h K = 10 , a n d

w i t h t h e d e d u c t i o n o f . 0 0 0 0 0 4 i n s t e p 3 . I n T a b l e 4 a r e t a b u l a t e d v a l u e s

o f G ( o ~ ) f o r a w i d e r a n g e o f v a l u e s o f H . A l s o t a b u l a t e d a r e v a l u e s o f

G ( ~ ) -- 1 / H , w h i c h a p p e a r s t o b e b e t t e r s u i te d f o r i n t e rp o l a t io n t h a n is

G ( o ~ ) i ts el f. T a b l e 4 o r t h e G a m m a F u n c t i o n r e fe r re d to i n e q u a t i o n 3 5

c a n t h e r e f o r e b e c o n v e n i e n t l y u s e d f o r o b t a i n i n g t h e r e q u i r e d v a l u e o f

C(~).

a 'E SrtN G r u ~ m ~ r a o D O N 1 -u ~ a - 1 9 4 9

r E ~ E

r~i ~ wira 2½ ncm ~sr

T o i l lu s t r a t e t h e s e c o n d m e t h o d i n p r a c t i c e , t h e c a l c u l a ti o n s i n T a b l e 3

w e r e d o n e f o r t h e a - 1 9 4 9 F e m a l e T a b l e a n d 2 ½ % in t e re s t, f o r w h i c h t h e

M a k e h a m c o n s t a n t s a re g iv e n o n p a g e 3 85 o f

T S A I .

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ANNU ITY VALUES DIRECTLY FROM MAKEHAM CONSTANTS 277

Detailed Calculation for Age 65

T h e c a l c u l a t i o n s for t h e a n n u i ty a t 6 5 wil l n o w b e d e v e lo p e d in d e t a il .

S tep 1

logto c = .0 49 ( bv def ini t i on)

log10 c 65= 65 X. 04 9 = 3.1 85

• c 65= 153 1 .1

In c = .04 9 X 2 .3 025 851

= • 1 1 2 8 2 6 6 7 ( c o n v e r t i n g t o n a t u r a l l o g a r i t h m s )

TABLE 3

a-1949 FEMALE TABLE AND 2½% INTEREST

ffi .0246926 H--- . 22772

A ffi

.001

G( ~) -- 5.25592

Bffi .0000070848535* In c=~ . 11282667

c= 1.1194379

A g e

50 .. . . 281.84

55 . .. . 495.45

60 .. . . 870.96

65 . .. . 1,531•1

70 .. . . 2,691.5

75 .. . . 4,731.6

80 . . . . 8,317.6

85 . .. . 14,622

90 .. . . 25,704

95 .. . . 45,186

100 . . . . 79,433

• 0030

• 0045

.0072

.0118

.0201

.0345

•0599

.1046

•

1831

.3211

• 5638

K~

.017698

.031111

.054691

.096144

.16901

.29712

•52230

•91818

1•6141

2.8374

4.9879

F(K)

5. 80554

4.51040

3.39110

2. 43729

1.64344

1.00964

•53946

.23185

•07031

.01199

•

00079

20.900

18.712

16.379

13.952

II.506

9.136

6.952

5.048

3.491

2.300

1.480

Estimated

a f f i t

20.402

18.214

15.882

13.455

11.010

8.641

6.459

4.559

3.008

1.829

1.029

Published

az

20.404

18. 215

15. 882

13.455

11.010

8.642

6.459

4.560

3.012

1.838

1 •012

A = .001 ,

I t

follows that

t J z = A + B ' c = , w h e r e

* In defining the constants for the a-1949 Tables the authors used

CO1Ogo p. = A l-Be ~, w h e r e

B = .

0000075 ,

B , _ B ln c

c - - l

t az obtained from approximate relationship

- I 1

log~o c --- . 0 4 9 .

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278 A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E H A M C O N S T A N T S

B - . 0 0 0 0 0 7 0 8 4 8 5 3 5

B ¢65

K 6 5 - - - - . 0 9 6 1 4 4

,In c

( b y d d i n i t i o n )

S t e p 2

A = . 0 0 1 ( b y d e f i n i ti o n )

= . 0 2 4 6 9 2 6 ( i = . 0 2 5 )

H - A + ~ _ . 2 2 7 7 2

In c

S t e p 3 S t e p 4

T1 = 1 = 1 . 0 0 0 0 0 0 T -2 = 4 . 3 9 1 3 5 8

H

T 2 = K = . 0 9 6 1 4 4

T 2 - . 1 2 4 4 9 4

1 - - H

g 2 T$

T a = - ~ - . 0 0 4 6 2 2 2 - -- -- -- -H - "0 0 2 6 0 8

K a T 4

T 4 - ~ - - - . 0 0 0 1 4 8 3 - - / / - - . 0 0 0 5 3

K 4 T 5

T5 - - . 0 0 0 0 0 4 - - - . 0 0 0 0 0 1

[ 4 4 - - H

g 5 T6

T e = - ~ - = . 0 0 0 0 0 0 5 - - H . 0 0 0 0 0 0

S t e p 5

K g . G ( K )

= 4 . 5 1 3 2 9 6

S t e p 6

lo g K = 5 . 9 8 2 9 2 = - - 1 . 0 1 7 0 8

lo g K H = . 2 2 7 7 2 X ( - - 1 . 0 1 7 0 8 ) = - . 2 3 1 6 1 = ] - . 7 6 8 3 9

K H = . 5 8 6 6 6

S t e p 7

G(K) -

4 . 5 1 3 2 9 6

- 7 . 6 9 3 2 1

. 5 8 6 6 6

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A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K F . H A M C O N S T A N T S 279

S t e p 8

e - g = . 9 0 8 3 3 4

S t e p 9

K s . I n c =

. 9 0 8 3 4 4 X . 1 1 2 8 2 6 6 7

. 5 8 6 6 6

= . 1 7 4 6 9 1

S t e p 1 0

1

a ( ~ ) - - ~

.2 2 . . . . . . . . . 8 5 2 1 4

.2 3 . . . . . . . . . 8 6 8 2 3

. 2 2 7 7 2 . . . . . 8 6 4 5 6

1

- - = 4 . 3 9 1 3 6

H

G ( ~ ) = 5 . 2 5 5 9 2

S tep 11

G ( K ) - - G ( a , ) = 7 . 6 9 3 2 1 - - 5 . 2 5 5 9 2

= 2 . 4 3 7 2 9

S tep 12

2 . 4 3 7 2 9

a 6 5 - . 1 7 4 6 9 1 1 3 . 9 5 2

P R O G R A M FOR C A L C U L A T I N G ~ z In c

A p r o g r a m w a s c o d e d f o r t h e c a l c u la t io n o f ~ , I n c b y a n e l ec t ro n i c c o m -

p u t e r , u s i n g t h e f o r m u l a

~x ln c = eK . K s [ G ( K ) - - G ( ~ ) ] . ( 3 2 )

A r a n g e o f v a l u e s o f K f r o m .0 0 1 t o 1 0, a n d f o r H f r o m 0 t o .6 4 w a s s e l e c t e d .

T h e p r o g r a m u s e d f o rm u l a 3 1 t o c o m p u t e G ( ~ ) f o r e a c h H . T h e s er ie s o f

t e rm s m a k i n g u p K H . G ( K ) w a s c a l c u l a t e d b y f i r s t c o m p u t i n g t h e s e r i e s

K ~ K 8 K n

K , - ~ , - ~ , . . . , - - ~ , . . . l an d t h e n s u c c e ss i v el y d i v i d in g e a c h t e r m b y t h e

v a l u e ( n - fir), b e f o r e a l t e r n a t e l y a d d i n g a n d s u b t r a c t i n g t e r m s . T h e v a l u e

o f e ~r w a s o b t a i n e d b y a d d i n g t h e s a m e s e r ie s b e f o r e d i v i d i n g e a c h t e r m b y

( n - fir). F r o m a n a p p r o x i m a t e v a l u e o f K "°1 a v a l u e t o n i n e d e c i m a l s w a s

c o m p u t e d u s i n g N e w t o n ' s m e t h o d o f s u c c e s s i v e a p p r o x i m a t i o n s . B y r e -

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280 ANN UIT Y VALUES DIRECTLY FROM MAKEHAM CONSTANTS

p e a te d mu l t i p l i c a t io n a ll t h e r e q u i r e d v a lu e s o f K H we re d e t e rmin e d .

In K w a s c o mp u te d f ro m th e s e r ie s

f X 2 X ~ X n

'~

In K = 100 ~ ,x - - -~ -+- ~- - - . . . -a - ( - - 1 )n - -+ . . .

n J t

wh er e x = K '° l _ 1 •

In T a b l e s 4 t o 7 a r e p r e se n te d e x t r a c t s f ro m th e p ro g ra m re su l t s . I a m

TABLE 4

VALUES OF G( ~)

lit G(oo) G(w) - 1/// H G(co) G(oo) - 1//7

( H ~ 0 ) ( H ~ 0 )

D . . . .

,01.

,02.

. 0 3 .

, 0 4 .

.05.

.06.

.07.

.08.

. 0 9 .

• I 0 . .

.11 . .

.12. .

.13. .

.14. .

.15. .

.16. .

.17. .

.18. .

.19. .

.20.

.21.

.22.

.23.

.24.

.25 . . . . . .

.26 . . . . . .

.27 . . . . . .

.28 . . . . . .

.29 . . . . . .

.57722

100.58720

50.59737

33.94107

25.61830

20.62907

17.30672

14.93697

13.16270

11.78548

10.68629

9.78937

9.04423

8.41592

7.87946

7.41656

7.01348

6.65975

6•34719

6.06937

5.82115

5.59836

5.39760

5.21606

5.05140

4.90167

4.76521

4.64062

4.52669

4.42240

.57722

.58720

.59737

.60774

.61830

.62907

.64005

.65126

.66270

.67437

•68629

.69846

• 71090

.72361

.30 . . .

. 31 . . .

. 32 . . .

• 33. . .

. 34 . . .

. 35 . . .

.36• . .

.37 . . .

. 38 . . .

. 39 . . .

. 4 0 . . .

. 41 . . .

. 42 . . .

4.32685

4.23928

4.15901

4.08547

4.01813

3.95656

3.90036

3.84918

3.80273

3.76074

3.72298

3.68925

3.65936

.43 . . . . . . 3.63317

.99352

1.01347

1.03401

1.05517

1.07695

1.09942

1.12258

1.14648

1.17115

1.19664

1.22298

1.25023

1.27841

1.30759

.73660

.74989

.76348

.77740

.79163

.80621

. 8 2 1 1 5

.83646

.85214

.86823

.88473

.90167

.91906

.93692

.95526

.97412

. 4 4 . . . . . .

.45 . . . . . .

.46 . . . . . .

.47 . . . . . .

.48 . . . . . .

. 4 9 . . . . . .

.50.- . . . . .

.51 . . . . . .

.52 . . . . . .

.53 . . . . . . .

. 5 4 . . . . . .

• 55. . . . . .

.56 . . . . . .

.57 . . . . . .

.58 . . . . . .

.59 . . . . . .

. 6 0 . . . . . .

.61 . . . . . .

.62 . . . . . .

. 6 3 . . . . . .

. 6 4 . . . . . .

3.61055

3.59139

3.57569

3.56310

3.55384

3.54779

3.54491

3.54520

3.54867

3.55533

3.56523

3.57843

3.59499

3.61500

3.63857

3.66583

1.33782

1.36917

1.40168

1.43544

1.47051

1.50697

1. 54491

1. 58442

1. 62559

1. 66854

1. 71338

1. 76025

1. 80928

1. 86061

1.91443

1.97091

3.69693 2.03026

3.73205 2.09271

3.77138 2.15848

3.81516 2.22786

3.86365 2.30115

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TABLE

5

V~ uE s oF G(K)

K H = O H = . 2 [ 1 = . 4 I H = . 6

. O01 . . . . .

.002 . . . . .

.004 . . . . .

.008 . . . . .

. 0 1 . . .

.02 .

. 0 4 . . .

. 0 8 . . .

. 1 . .

. 2 . .

. 4 . .

. 8 . .

L .

L .

6.90876

6.21661

5.52546

4.83629

4.61514

3.93192

3.25848

2.60416

2.40014

1.79987

1.27960

.88781

.79660

.62611

.58099

.57726

.57722

19.91033

17.33728

15.10051

13.15886

12.59076

10.98805

9.61261

8.44896

8.11827

7.22888

6.55706

6.11565

6.02230

5.86135

5.82391

5.82117

5.82115

39.64876

30.06814

22.81768

17.33847

15.87890

12.11320

9.29954

7.22679

6.69067

5.37083

4.50193

4.00383

3.90829

3.75611

3.72501

3.72300

3.72298

o . . . . . . . .

TABLE 6

VALUES

OF F(K)

I H =.2 H • .4 H = . 6

105.31749

69.58748

46.04782

30.56112

26.81056

17.94865

12.18370

8.48572

7.61644

5.65470

4.52913

3.96620

3.86838

3.72429

3.69842

3.69694

3.69693

K H = O

.i

.001 .. . . . i 6.33154

• 002 . . . . . 5.63939

• 004 . . . . . 4.94825

• 008 . . . . . 4.25908

.01 . . . . . . 4.03793

.02 . . . . . . 3.35471

.04 . . . . . . 2.68126

• 08 . . . . . . 2.02694

. 1

. . . . . . . 1 .82293

. 2 . . . . . . . 1 .22265

• 4 . . . . . . . . 70238

. 8 . . . . . . . . 31059

I . . . . . . . . . . 21938

2 . . . . . . . . . . 04889

. . . . . . . . . i .00378

i i i i i i i i

o . 0 o 0 0 4

14.08918

11.51613

9.27936

7.33771

6.76961

5.16690

3.79146

2.62781

2.29712

1.40774

.73591

.29450

.20116

.04020

.00277

.00002

35.92578

26.34516

19.09470

13.61549

12.15592

8.39022

5.57656

3.50380

2.96769

1.64785

.77895

• 28085

• 18531

.03313

.00203

.00002

101.62055

65.89055

42.35089

26.86419

23.11363

14.25172

8.48676

4.78879

3.91951

1.95776

.83220

.26926

.17145

•02736

.00148

.00001

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282 A N N U I T Y V A L U E S D IR E C T L Y F R O M M A K E H A M C O N S TA N T S

T A B L E 7

VALUES OF a. In c

K, Hffi0 Hffi.2 H= . 4 Hffi.6

. 0 0 1 .

. 0 0 2 .

. 0 0 4 .

. 0 0 8 .

. O l . . . . . . . . .

.0 2 . . . . . . . . .

. 04 . . . . . . . . .

.0 8 . . . . . . . . .

. 1 .

.2 .

.4 .

.8 .

I . . .

2 . . .

4 . . .

8 . . .

6 . 3 3 7 9

5 . 6 5 0 7

4 . 9 6 8 1

4 . 2 9 3 3

4 . 0 7 8 5

3 . 4 2 2 5

2 . 7 9 0 7

2 . 1 9 5 8

2 . 0 1 4 6

1 . 4 9 3 4

1 . 0 4 7 8

.6912

.5963

.3613

.2062

. l l

3 . 5 4 2 6

3 . 3 2 9 5

3 . 0 8 7 9

2 .8161

2 .7221

2 . 4 1 0 6

2 . 0 7 3 0

1 .7177

1 . 6 0 1 8

1 .2462

.9140

.6268

.5468

.3412

.1992

. 1 0

2 . 2 6 9 0

2 . 1 9 7 8

2 . 1 0 6 1

1 . 9 8 9 5

1 . 9 4 5 9

1 . 7 9 0 1

1 . 6 0 1 6

1 . 3 8 2 0

1 . 3 0 5 7

1 . 0 5 7 3

. 8 0 5 5

.5717

. 5 0 3 7

. 3 2 3 0

.1925

. 1 0

1 . 6 1 2 2

1 . 5 8 6 0

1 . 5 4 8 2

1 . 4 9 4 5

1 . 4 7 3 0

1 . 3 9 0 5

1 . 2 8 0 4

1 . 1 3 9 8

1 . 0 8 8 1

. 9 1 0 4

.7164

.5242

. 4 6 6 0

. 3 0 6 4

.1861

. 1 0

indebted to Mr. Ronald Ryan (A.S.A.) of my company for the develop-

ment

o f t h e c o m p u t e r p r o g r a m .

J O IN T A N N U IT IE S

The method which has been described for evaluating single life annui-

ties

c a n a l so b e a p p l i e d t o v a l u e j o i n t a n n u i t i e s . C o n s i d e r t h e g e n e r a l c a s e

w i t h l i v e s o f a g e s

x ~, x ~ , . . . , x , .

f°

*l . . . . ~ = v t" t p . , ~ , . . . . f i t

= . e - - f , ( ~ . , + h + . . , + h + ' + . . + h ) d h d t~ e - - S t |

J O

= f o ~ e - S t e - - f ~ C ~ A + B , h ) c , x ' + , x ' + . . . + c ~ ') ahdt

= f o o e _ ( n A + g ) t e _ ( B c l / l n c )(c X l+ cX ~ + . . , .[ .e x,,)

J o

L e t

K=K.I-q-Kx, q- . . . - I -g z:

n A - t- ~

l ~ n - - - -

I n c

y = K C *

• e ( B / I n e ) ( ~ t + c x 2 + ' " " + ~ ' ) d l .

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ANNUITY VALUES DIRECTLY FROM MAKEItAM CONSTANTS

f K c :°

~ x l z , . z n ~ e - - H n l n ( y / K ) e _ • / * e K .

- - e K

co - - H n ' ~ d y

Vn-c y

e K " K U , , f ' ~ e - "

- -ln c J K -y-i-+T,,d y

2 8 3

d y

y-ln c

~ b ( K )

K l n c "

In conclusion, to evalu ate a joint l ife an nu ity w e eva luate a single l ife

annu i ty a t an age where the K-va lue is the sum o f the K-va lues fo r the

individual ages concerned and

n A + ~

In c

COMPARISON WITH THE MCCLINTOCK METHOD

Under the second method presented in th is paper

a ~ l n c = e x . K R [ G ( K ) - - G ( o o ) ] ( 3 2 )

- e K . K H . G ( ~ ) + ( I + K + . . . - K ~ + . . . )

K "

X ( I + I _ _ - ~ K H + . . . + ( - - 1 ) n + t

( n _ H ) l n ~- . . . ) .

Multiplying the two ser ies and collecting terms, i t can be shown that

a * l n c = - e K ' K S ' G ( ° ° ) - [ -

-~ H ( 1 - H )

( 3 3 )

K "

+ . . . - I H ( 1 - - H ) . . . ( n - - H ) ~- . . . .

The method used by M r . M cCl in tock diff ers f rom the second method

of th is pape r because of the fac t th at he in tegrated the in tegral express ion

for the annui ty ( formula 11) by par ts twice and obta ined

1 f K e X . K X ~ ~

a * l n C = - H -~ H ( H )

H ( 1 = H )

e - " Y l - U d Y "

Working with the integral . ,

f

~ e -~ . y t - H d y

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2 8 4 A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E H A M C O N S T A N TS

i n s t e a d o f

K ~ e - ~ . y - t - H d y

a n d u s i n g th e s a m e a p p r o a c h a s i n t h e se c o n d m e t h o d , M r . M c C l i n t o c k

o b t a i n e d a c o n s t a n t o f i n t e g r a t i o n w h i c h h e i d e n ti f ie d a s 1 "( 2 - -

1 t ) ,

fo r

w h i c h t h e n u m e r i c a l v a l u e c a n b e o b t a i n e d f r o m a p u b l i s h e d t a b u l a t i o n

o f t h e G a m m a F u n c ti o n .

H i s f i n a l r e s u l t p r e s e n t e d i n t h e n o t a t i o n o f t h i s p a p e r i s

e K ' K F ( 2 - - H ) + I I

~ = ln c - -

H ( 1 - - H ) + H ( - - H )

( 3 4 )

K -

+ . . . q b . . . .

~ / ( 1 - ~ / ) . . . ( n - H )

C o m p a r i n g (3 4 ) a n d ( 3 3 ), w e s ee t h a t

a ( o o ) - r ( 2 - H ) H # 0 ( 3 5 )

H ( 1 - - H )

F o r H = 0 it a p p e a r s f r o m T a b l e 4 t h a t

G ( co ) I"=°=u~0Ltim[a( ~ ) - ~ •

If so,

G( co )[ H =o = l i ra

H ~ 0

r ( 2 - H ) - l + H

//(i-H)

= l i r a - - 1 " ( 2 - - H ) + 1

H~0 1 - - 2 H

= I - r ' ( 2 ) .

B u t s i n c e

r ( n + l ) = n r ( n )

r ' ( n + 1 ) = n r ' ( n ) + r ( n ) .

• F o r n = 1 ,

( L ' H o s p i t a l ' s R u l e )

r ( 2 ) =

--- 1 - - ~, ( w h e r e 7 i s t h e E u l e r C o n s t a n t ) .

• G ( ~o ) [ H = 0 = 7 = . 5 7 7 2 1 5 . . . . ( 3 6 )

T h i s r e s u l t a g r e e s w i t h t h e c a l c u l a t i o n s .

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A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E H A M C O N S T A N T S

285

POSSIBLE APPLICATIONOF MAKEHAMTABLES

TO PROJECTED MORTALITY

I t s e e m s s u r p r i s i n g t h a t M r . M c C l i n t o c k ' s p a p e r s h o u l d a p p a r e n t l y

h a v e r e m a i n ed d o r m a n t o v e r t h e y e a r s . U s i ng his a p p r o a c h , t h e r e w o u ld

s e em t o b e e v e n m o r e v a l u e i n h a v i n g a t a b l e M a k e h a m i z e d t h a n f o r t h e

a d v a n t a g e s a r i si n g in t h e e v a l u a t i o n o f j o i n t l i fe a n n u i t i e s . E v e n w i t h

s in g le l if e f u n c t i o n s t h e a p p r o a c h m a k e s i t p o s s ib le t o s t u d y t h e e f fe c t s o f

i n t er e s t a n d m o r t a l i t y c h an g e s w i t h o u t p r o d u c i n g a c o m p l e t e ne w m o r -

t a l i t y t ab le .

I t is p o ss ib le t h a t M a k e h a m t a b le s m a y b e u s e d e v e n w h e r e a p r o g r es -

s iv e i m p r o v e m e n t in m o r t a l i t y is a ss u m e d . I t is c o m m o n u n d e r s u c h c i r -

c u m s t a n c e s t o h a v e a d i f fe r e n t m o r t a l i t y t a b l e a p p l i c a b le t o e a c h c a l e n d a r

y e a r o f b i r th . I n o r d e r t h a t e a c h su c h g e n e r a t i o n t a b l e fo ll ow M a k e h a m ' s

L a w i t i s n e c e s s a r y t h a t t h e a n t i c i p a t e d i m p r o v e m e n t i n m o r t a l i t y b e

r e fl e ct e d i n t h e c h a n g e in t h e M a k e h a m p a r a m e t e r s f r o m o n e g e n e r a ti o n

t a b l e t o t h e n e x t .

T h e a n t i c ip a t e d y e a r l y i m p r o v e m e n t in m o r t a l i t y a s in d i c a t e d b y p r o -

j e c t i o n sc a le s w h i ch h a v e b e e n p u b l i s h e d i n t h e Tra n sa c t i on s is a c o n s t a n t

p e r c e n t a g e o f t h e m o r t a l i t y f o r t h e p r e v i o u s y e a r , w i t h t h e p e r c e n t a g e

d e c re a s in g b y a g e so t h a t n o i m p r o v e m e n t is e x p e c t e d a t a d v a n c e d a g es .

A m e t h o d o f re f le c ti n g t h e a n t i c i p a t e d p a t t e r n o f i m p r o v e m e n t i n t h e

M a k e h a m c o n s t a n t s i s t o a s s u m e n o c h a n g e i n t h e c o n s t a n t A , t o g e t h e r

w i t h a p e r c e n t a g e in c r e a se i n t h e c o n s t a n t c a n d a p e r c e n t a g e d e c r e a se i n

t h e c o n s t a n t B s u ch t h a t n o i m p r o v e m e n t re s u lt s a t s o m e a d v a n c e d a g e ,

s a y 1 00 , a n d s o t h a t t h e p e r c e n t a g e i m p r o v e m e n t a t s o m e a ge , s a y 50 , is

i n a c c o r d a n c e w i t h a s e l e ct e d p r o j e c t i o n s ca le .

T o i l lu s t r a te , l e t t h e f o r c e o f m o r t a l i t y a p p l i c a b le f o r t h e y e a r o f b i r t h

y a n d a t t a i n e d a g e x b e g iv e n b y

w h e r e

- - ~ C y - - 1

~ x , v - 1 - - A 1 - - 1 + = 1 - - 1 0 - - - 0 '

w h e r e r~ is t h e p e r ce n t a g e a n n u a l i m p r o v e m e n t i n / ~ - - A .

I f noo = 0 a nd rso = 1 . 25% (P ro jec t ion Sca le B) , we can , by so lv ing

equ a t ions , de t e rm ine tha t b = 2 .55 an d s = .025. r~ i s t he n de te rm ine d

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286 ANNUITY VALUES DIRECTLY FROM M AKEHAM CONSTANTS

f o r ea c h a ge . T a b l e 8 c o m p a r e s r , w i t h t h e p e r c e n t a g e i m p r o v e m e n t g iv e n

b y J e n k i n s a n d L e w i n P r o j e c t i o n S c ale B .

T h e a s s u m p t io n s m a d e i n t h e a b o v e a n a l y si s a re f i r s tl y t h a t i m p r o v e -

m e n t s i n t h e r a t e a n d i n t h e f o r c e o f m o r t a l i t y w o u l d b e o f t h e s a m e m a g -

n i tu d e , s e c o n d l y t h a t t h e a s s u m p t i o n o f n o i m p r o v e m e n t i n t h e c o n s t a n t

A i s n o t s i g n i f i c a n t , a n d t h i r d l y t h a t p r o i e c t i o n s c a l e s c a n b y n a t u r e b e

f r e e l y a d j u s t e d t o f i t t h e m e t h o d o f a p p l i c a t i o n a s l o n g a s t h e o v e r - a ll

p a t t e r n o f e x p e c t e d i m p r o v e m e n t is n o t c h a n g e d . I t w ill b e n o t ic e d t h a t

t h e p r o j e c t i o n s c a le d e v e l o p e d f o r r: i s v i r tu a l ly l i nea r.

TABLE 8

C O M P A R I S O N O F rz

W I T H

P R O J E C T I O N S C A L E B

Percentage Im-

provement in q :

Age rz under Project ion

S c a l e B

5 0 . . . . . . . . . . .

6 0 . . . . . . . . . . .

6 5 . . . . . . . . . .

70 . . . . . . . . . .

75 . . . . . . . . . .

8 0 . . . . . . . . . .

85 . . . . . . . . . .90 . . . . . . . . . .

100 . . . . . . . . . .

1.2s%

1 . 0 0

.87

.75

.63

.50

.37.25

0

1.25%

1.20

1.10

.95

.75

.50

.250

0

I n c o n c l us i o n , t h e f o r m u l a f o r e v a l u a t i n g a n n u i t i e s o n a M a k e h a m t a b l e

a s d e v e l o p e d i n t h i s p a p e r s h o u l d b e sa t is f ac t o rY f o r m a k i n g c a l c u l a t io n s

r ec o g n iz in g c a l e n d a r y e a r o f b i r t h a s a f a c t o r , p r o v i d e d t h e i m p r o v e m e n t

i n m o r t a l i t y i s a n t i c i p a t e d i n s u c h a m a n n e r t h a t t h e g e n e r a t io n m o r t a l i t y

t a b l e f o r e a c h y e a r o f b i r t h f o l l o w s M a k e h a m ' s L a w . I t i s h o p e d t h a t

s o m e p r a c t i c a l a p p l i c a t i o n m i g h t b e m a d e w h e n s o m e e x p e r i m e n t a t i o n

w i t h a t a b l e is d e s ir a b le , b e f o r e p r o c e e d i n g w i t h t h e v o l u m i n o u s c a l c u la -

t i o n s n e c e s s a r y t o p r o d u c e a c o m p l e t e s e t o f t a b le s .

R E F E R E N C E S

"O n the Com puta t ion of Ann ui ties ," E m ory M cClin tock, J I A XVIII , 242-247.

A N ew M ortal i ty B asis for Annuit ies ," Jenkins & Lew, TSA I, 385.

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D I S C U S S I O N O F P R E C E D I N G P A P E R

A. M . N I E S S E N :

I w a s g r e a t l y i n t r i g u e d b y M r . M e r e u s p a p e r b e c a u s e i t i s o n e o f t h e

r a r e e x c u r s i o n s i n t o t h e a p p l i c a t i o n o f c a l c u l u s t o c e r t a i n p r a c t i c a l a c t u -

a r i a l p r o b l e m s . I m u s t c o n f e s s , h o w e v e r , t h a t I w a s s o m e w h a t d i s a p -

p o i n t e d i n t h e p r a c t i c a l r e s u l t s w h i c h t h e p a p e r o f f er s . I f I u n d e r s t a n d

M r . M e r e u s p r o c e d u r e c o r r e c t l y , t h e t a s k o f c a l c u l a t i n g a s i n g l e a n n u i t y

v a l u e b y h i s m e t h o d i s v e r y f o r m i d a b l e a n d m a y , i n f a c t, c o n s u m e a s m u c h

t i m e a s t h e p r e p a r a t i o r l f a c o m p l e t e m o r t a l i t y t a b l e b y t h e u s e o f m o d e m

o ff i ce m e t h o d s . I t is e n o u g h t o t a k e a l o o k a t t h e c a l c u l a t i o n s s h o w n i n

t h e p a p e r f o r ~ 5 t o s e e t h a t m y c o n t e n t i o n i s n o t w i t h o u t v a l i d i t y .

T h e p r e m i s e o f M r . M e r e u s p a p e r i s t h a t i s o l a t e d a n n u i t y v a l u e s b a s e d

o n c e r t a i n p a r a m e t e r s i n a M a k e h a m f o r m u l a w i l l b e r e l i e d u p o n e v e n

t h o u g h t h e m o r t a l i t y t a b l e a s s o c i a t e d w i t h t h e s e v a l u e s h a s n e v e r b e e n

s e e n . I c e r t a i n l y w o u l d n o t w a n t t o r e l y o n s u c h a n n u i t y v a l u e s i n a n y

w a y a n d I d o u b t w h e t h e r a n y o t h e r a c t u a r y , i n c l u d i n g t h e a u t h o r o f t h e

p a p e r h i m s e l f , w o u l d b e w i l l i n g t o d o s o. T h i s o b s e r v a t i o n d o e s n o t n e c e s -

s a r i l y d e t r a c t f r o m t h e s i g n i f i c a n c e o f M r . M e r e u s c o n t r i b u t i o n ; w h a t I

a m s a y i n g i s m e r e l y t h a t t h e a t t e m p t t o j u s t i f y t h e p a p e r o n p r a c t i c a l

g r o u n d s s e e m s t o b e r a t h e r f a r f e t c h e d . P e r h a p s t h e l a c k o f p r a c t i c a l

s ig n i fi ca n c e is t h e r e a s o n w h y M r . M c C l i n t o c k ' s p a p e r o n t h e s a m e s u b j e c t

h a s r e m a i n e d d o r m a n t f o r s o m e 90 y e a r s , a f a c t t h a t s e em s t o b e s u rp r is in g

t o M r . M e r e u .

W h i l e o n t h e s u b j e c t o f a e s t h e t i c v e r s u s p r a c t i c a l v a l u e s i n c e r t a i n

p h a s e s o f a c t u a r i a l w o r k , I w o u l d l ik e t o m a k e a f e w c o m m e n t s o n t h e

s o - ca l le d l aw s o f m o r t a l i t y in g e n e r a l, a n d o n M a k e h a m ' s l a w i n p a r t i c u -

l a r. I t is m y b e l ie f t h a t t h e s e la w s a r e a r e l i c o f 1 9 t h c e n t u r y t h i n k i n g

w h e n i t w a s f e l t t h a t a l l p h e n o m e n a a r e s u b j e c t t o n e a t a n d d e f i n i t e

m a t h e m a t i c a l f o r m u l at io n s . M a k e h a m ' s l a w h a s n e v e r b e e n a g r e a t s c ie n -

t if ic su c c es s a n d t o d a y i t n o l o n g e r e n j o y s t h e e s t e e m i n w h i c h i t w a s h e l d

i n y e a r s p a s t . A s f o r th e p r a c t i c a l a d v a n t a g e s o f e a s y c a l c u l a t i o n o f j o i n t

l i fe func t ions , t h i s a lso is no longer a g rea t a s se t becau se l en g th y ca l cu la -

t i o n s p r e s e n t n o p r o b l e m t o m o d e m e q u i p m e n t . I t i s , t h e r e f o r e , m y

o p i n i o n t h a t t h e M a k e h a m l a w a n d o t h e r s o -c a ll ed l aw s o f m o r t a l i t y

s h o u l d n o l o n g e r b e u s e d i n t h e c o n s t r u c t i o n o f m o r t a l i t y t a b l e s . T h e

s a m e g oe s a ls o f o r fa n c y g r a d u a t i o n p r o c e d u r e s s u c h a s t h e H e n d e r s o n A

287

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28 8 A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E H A M C O N S T A N T S

a n d B f o rm u la s . It is m o r e i m p o r t a n t t o c o n c e n t r a t e o n fit t h a n o n

s m o o t h n e s s . A t l e a s t , t h i s i s t h e a p p r o a c h w h i c h w e h a v e b e e n f o l l o w i n g

a t t h e R a i l r o a d R e t i r e m e n t B o a r d w h e n e v e r w e h a d o c c as io n t o c o n s t ru c t

a n e w m o r t a l i t y t a b l e .

MOIlAMED ~' . AMER:

M r . M e r e u m e n t i o n e d t h a t h i s m e t h o d w o u l d h a v e p r a c t ic a l v a l u e if

o n l y f e w c a l c u l a ti o n s a r e t o b e m a d e . T h e p u r p o s e o f t h i s d i sc u s si o n is

t o p r e s e n t t w o a l t e r n a t i v e a p p r o a c h e s t h a t m a y b e u s e d to r e d u c e th e s iz e

o f c a l c u l a t i o n s n e e d e d . T h e p r o o f s o f a l l t h e f o r m u l a s a r e g i v e n i n t h e

A p p e n d i x . A s f a r a s p o s s i b l e , t h e s a m e n o t a t i o n u s e d b y M r . M e r e u i s

u sed he re .

C o n s i d e r a b l e p a r t o f t h e c a l c u l a t io n i n M r . M e r e u ' s p a p e r w a s t o d e t e r -

m i n e

f K ~ e-vy-~ -R d y

X

d e n o t e d b y F(Kx). H o w e v e r , t h is F(K, ) can be exp re s sed i n ~ e rms o f t he

i n c o m p l e t e g a m m a f u n c t i o n f o r w h i c h t a b u l a t e d v a l u e s a r e a v a i l a b l e .

1 1

F ( K , ) = - ~ U ( K x ) - - ~ [ 1 - - I ( K , , - - H ) I I ' ( 1 - - H ) ? ( 1 )

w h e r e

e - - K x fo Kx e-~y-HdY

U ( K . ) = K . f ( K . ) = ~ a n d I ( K . , - H ) =

I ' ( 1 - - H ) "

K a r l P e a r s o n ' s T a b l e I I I ~ g ive s l og I ' ( u , p ) , w he re

I ' ( u , p) = I ( u , p)

~ + 1

a n d t h e a r g u m e n t u s e d i s n o t K , b u t

u - - K , / n / p

+ 1. T h e r e a s o n f o r

us ing l og I '(u, p) i s t h a t i n t e r p o l a t i o n i n t h e I(u, p) t a b l e w o u l d b e u n -

s a t i s fa c t o r y , f o r m o r e t e r m s w o u l d b e n e c e s s a r y t o g e t s a t i s f a c t o r y v a l u es .

G l o v e r ' s T a b l e s s g i v e l o g P ( n ) f o r n = 1 .5 to n = 1 .9 99 , b u t t h e v a l u e s

needed a r e fo r n f rom .5 t o .999 . So t he r e l a t i onsh ip

r ( 2 - H )

r (1 -H) =

1 - H

c a n b e u s e d t o g e t t h e r e q u i r e d v a l u e s .

1 See the Appen dix for p roof .

* Tables o f the Incomplete G am m a Function, Cam br idge U nive rs i ty P ress (1922) .

3 T ab le s o f A p p l i ed M a t h em a t i c s . . . e tc . , ed i ted by J . W. Glover .

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D I S C U S S I O N 2 8 9

Method A

In the Appendix the following formulas are proved:

D , = k U ( K , ) ( 2 )

k i s an a rb i t r a ry cons tan t

~q~= ( k + l n c ) F ( K = )

( 3 )

M = = X [ U K = ) - - i - ~ F ( K = ) ] . ( 4 )

S= and ~.~ are inconvenient to obta in d irect ly as th ey involve the evalua-

tion of the integral

fK = e -v In yd y

x yl+H

So all tha t is needed would be U(K=) and F(K=) for any par t icu lar age x .

Th e former can be evalua ted b y us ing tab les of logari thms, the la t ter can

be e valuated us ing equ at ion (1) .

Fo r a ny specif ic value of log

I ' (u,

p) , a b i -var ia te in terpola t ion form ula

such as the following can be used:

Z (U0q-~, P0"t-~) = Z (U 0 , P0)"-~-½ ~ [ g (~b/'l, P0 ) - -g (U-- l , P0) ]

"J l- ~ [ Z (UO, P l) - - Z (~/~0, P -- l) ] "Jl-~ ~'r] [ g (U l, Pl ) ( 5 )

- - Z ( U - l , P l ) - - Z ( U l , P - l ) "-}- Z ( U - l , P - l )

1 .

As a check of the appl icabi l ity and accu racy of th is m ethod, ca lcu la t ion

of a , for ages 65 and 75 are g iven below using the a-1 94 9 female tab le . F or

all ages ab ov e 50, p and n are con stants ,

p = - - H = - . 2 2 7 7 2 , ~ / ~ - - ] - = . 8 7 8 7 9 5 , n = . 2 7 7 2 ,

r ( p + t ) r ( 2 - . 2 2 7 7 2 ) . 9 2 4 3 1 8

= 1 - . 2 2 7 7 2 = . 7 7 2 2 ~ = 1 " 1 9 6 8 6 9 "

The rate of interest is 2½c7o.

Age 65

K 6 5 = . 0 9 6 1 4 3

K85 . 0 9 6 1 4 3

u = v ~ p + l . 8 7 8 7 9 5 . 1 0 9 40 3

= . 0 9 4 0 3

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290 ANNUITYVAL UES DIRECTLY FR OM MAKEHAM CONSTANTS

log I ' = 1 . 9 7 4 9 9 6 + . 0 4 7 0 1 5 ( 1 ".9 5 8 1 8 9 - 1 . 9 9 2 1 0 7 )

+ . 2 7 7 2 ( 1 . 9 7 3 8 2 9 - - 1 " . 9 7 6 0 5 4 )

+ . 0 1 3 0 3 2 6 ( 1 - . 9 5 8 1 4 2 + 1 . 9 9 4 2 9 0 - - 1 . 9 8 9 8 0 3

- - 1 " . 9 5 8 1 3 8 ) = 1 - .9 7 2 8 4 3

( p + l ) l o g u = . 7 7 2 2 8 ( 1 . 0 3 9 0 2 9 2 ) = 1 . 2 5 7 8 6 2

log I = l o g I ' + ( p + 1 ) l o g u = 1 . 2 3 0 7 0 4

I = . 1 7 0 1 0

e - - K , ~

U ( K e s )

= - - = 1 . 5 4 8 3 0 1

K ~ ~772

F(K65)

= [ 1 . 5 4 8 3 0 1 - - ( 1 - - . 1 7 0 1 0 ) ( 1 . 1 9 6 8 6 9 ) ] + . 2 2 7 7 2

- - 2 . 4 3 7 2 9

d65 = F ( K e s ) + [ U ( K 6 5 ) In c ]

2 . 4 3 7 2 9

1 . 5 4 8 3 0 1 X . 1 1 2 8 2 6 6 7

This agrees with Mr. Mereu 's value for aes.

Ag8 75

" K 7 5 ~ -

log I ' =

= 1 3 . 9 5 2 .

. 2 9 7 1 1 1

. 3 3 8 0 8 8

. 3 8 0 8 8

1 . 9 4 1 6 8 5 + . 1 9 0 4 4 ( 1 . 9 2 5 4 8 2 - 1 . 9 5 8 1 8 9 )

+ . 2 7 7 2 ( 1 . 9 4 2 7 42 - 1 . 9 4 0 5 4 1 )

+ . 0 5 2 7 9 ( 1 . 9 2 7 6 2 4 + 1 . 9 5 8 1 3 8 - - 1 . 9 5 8 1 4 2

- - 1 . 9 2 3 2 6 1 ) = 1 . 9 3 6 2 9 7

( p + 1 ) lo g u = . 7 7 2 2 8 ( 1 . 5 2 9 0 2 0 7 ) = 1 . 6 3 6 2 7 2

log I = l o g I ' + ( p + 1 )l og u = 1 . 5 7 2 5 6 9

I = . 3 7 3 7 4

. , /

e - - K 7 5

U (K 76) . . . . . 9 7 9 4 7 2

K ~ : 2772

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D I S C U S S I O N

F(K76) = [ . 9 7 9 4 2 - - ( 1 - - . 3 7 3 7 4 ) 1 . 1 9 6 8 6 9 ] + . 2 2 7 7

= 1 . 0 0 9 6 7

291

F( K75)

1 . 0 0 9 6 7

= 9 . 1 3 6 .

a~5= U(KTs)lnc . 9 7 9 4 7 2 X . 1 1 2 8 2 6 6 7

O f c o u r s e , v a l u e s o f c o m m u t a t i o n f u n c t i o n s c a n b e o b t a i n e d i f o t h e r

th an a~ i s need ed u s ing equ a t i ons (2 ) , ( 3 ) , and (4 ) .

T h e m e t h o d c a n s ti ll b e a p p l ie d f o r ag e s u n d e r 5 0 i f a p p r o p r i a t e v a l u e s

of p a re used .

Method B

B e c a u s e o f t h e t i m e e l e m e n t , I d i d n o t i n v e s t i g a t e t h i s m e t h o d t h o r -

o u g h l y b u t i t se e m s to b e e v e n f a s t e r t h a n m e t h o d A f o r a n y c a l c u l a ti o n

r e la t e d t o a n y M a k e h a m i z e d m o r t a l i t y ta b l e.

I f t a b l e s o f U(Kx)a n d F(K~) a r e c a l c u la t e d f o r v a l u e s o f u a n d p t h a t

m i g h t b e n e e d e d , t h e n i n t h e f u t u r e f o r a n y M a k e h a m i z e d t a b l e a l l t h a t

w i l l be needed i s t o de t e rmine

Bc •

K X ~ - -

In c

f o r o n l y t h e r e q u i r e d v a l u e s o f x , t o g e t h e r w i t h

A - F d

In c

T h e n b y i n t e r p o l a t i o n i n e a c h o f t h e t w o t a b l e s , t h e r e q u i r e d v a l u e s o f

U(K~) a n d .F(K~)c a n b e f o u n d a n d f o r m u l a s ( 2 ), ( 3 ) , a n d ( 4) c a n t h e n b e

u s e d t o e v a l u a t e t h e a c t u a r i a l f u n c t i o n s .

I f t a b l e s f o r t h e i n c o m p l e t e F - f u n c t i o n t a b u l a t e d f o r t h e a r g u m e n t K ~

i n s t e a d o f u c a n b e f o u n d , i t w ill b e m o r e c o n v e n i e n t .

T h i s m e t h o d , h o w e v e r , h a s t o b e t e s t e d t o s e e if i t w i ll g i v e s u ff ic i e n tl y

a c c u r a t e r e s u l t s .

S o m e r e c e n t t a b l e s w e r e n o t M a k e h a m i z e d . W h e t h e r t h i s w a s b e c a u s e

o f u n s a t i s f a c t o r y f i tt in g o f M a k e h a m f o r m u l a o r a s a r e s u l t o f t h e e x t e n -

s i v e u s e o f e l e c t r o n i c c o m p u t e r s , i t r e m a i n s t o b e s e e n i f M r . M e r e u ' s

m e t h o d o r a n y o f t h e m e t h o d s p r e s e n t e d i n i t s d i s c u s s i o n w i l l r e n e w t h e

i n te r e st in M a k e h a m f o rm u l a .

A P P E N D I X

F O R M U L A S A N D P R O O F S

1 . F ( p + I ) =

e-uyndy

( 6 )

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292

ANNUITY VALUES DIRECTLY FROM MAKEHAM CONSTANTS

f o r

Fu (p d- 1 ) = e-*y pdy ( 7 )

r~(p+ 1) = i (y , p). (8)

r ( p + l )

P e a r s o n ' s t a b l e s u s e u =

y / x / p

d - 1 a s a r g u m e n t f o r r e a so n s t h a t d o n o t

e x i s t f o r o u r p u r p o s e . I (u, p) is t h e s a m e a s I (K , , p ) e x c e p t t h a t t h e a r g u -

m e n t u s e d f o r t a b u l a t i o n i s u = K J x , / p -]- 1 a n d t h u s t h e s e t w o f u n c t io n s

a r e h e r e u s e d i n t e r c h a n g e a b l y . N o r m a l l y

- . 7 < p = - H < 0

0 < K ~ < 8 .

2 . P r o o f o f f o r m u l a ( 1)

f K ~ e -~' 1 e - r ~ 1 L ~ e - ~ d y

F ( Kx ) = y~-gg dy - H K R H yH

z x

l - - I ( y , p ) = [ f o ~ e - ~ y ' d y - - f o ~ e - ~ f ' d y ]

+ fo ~ e - ~ f , d y --

I K a e y

yh d y = r ( 1 - H I [ 1 - I ( K ~ , - H I ]

= r ( 1 - H ) [ 1 - I ( u , - H ) ]

1 1

F ( K . ) = - ~ U ( K . ) - ~ r ( 1 - H ) [ 1 - / ( u , - H ) ] .

f

~ e - ~ y p d y ( 9 )

r ( p + l )

( 1 0 )

( i i )

3 . P ro o f o f f o r m u la s (2 ) , ( 3 ) , an d (4 )

i ) D , - f u n c t i o n : F r o m M r . M e r e u ' s e q u a t i o n s ( 1 4 ) a n d ( 1 5 )

N ~ N ~ + , , F ( K ~ ) - - F ( K . + , , )

D . D ~ U ( K x ) l n c

N ~ F ( K , )

D ~ U ( K x ) l n c

a n d

T h u s

or

Nx+~ F ( K .+ . )

°

D,+,, U ( K ,-~ ) ln c

D . + ~ / D .

= U ( K ~ + n ) / U ( K . )

Dx¢ = U ( K x )

(12)

( 1 3 )

( 1 4 )

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i.e.,

DISCUSSION

293

( 1 5 )

( 1 6 )

D . = k U ( K z )

go = X U ( K o )

k = lo / U ( Ko )

i i ) N . - f u n c t i o n

N ~ = D z ~ = k U ( K ~ ) . F ( K ~ )

k

U ( K ~ ) l n c = l n c . F ( K ~ ) ( 1 7 )

i i i ) M~-funct ion

[ ]

~ - - D , - - 6 N z = X

U ( K ~ ) - - ~ - - ~ . F ( K , )

( 1 8 )

i v ) S ~, R ~ a r e i n c o n v e n i e n t t o e v a l u a t e d i r e c t l y i n t e r m s o f t h e M a k e -

h a m c o n s t a n t s f o r t h e y i n v o l v e

f K°~ e-v

In y

y1 n

d y

a s a n o t h e r f u n c t i o n t o b e e v a l u a t e d .

4 . I n t e r p o l a t i o n

A l t h o u g h i t i s n e i t h e r n e c e s s a r y n o r d e s i r a b le t o u s e m a n y t e r m s f o r t h e

i n t e r p o l a t i o n , f o r m u l a ( 5 ) i s e x t e n d e d t o 4 t h d i f f e r e n c e s i n p a g e x i i o f

P e a r s o n ' s t a b l e s o f t h e i n c o m p l e t e F - f u n c t i o n a l o n g w i t h t w o o t h e r f o r -

m u l a s f o r o t h e r s it u a t io n s . T h e f a c t t h a t , a l o n g w i t h t h e v a l u e s o f l o g

I ' (u , p) , t h e t a b l e h a s 62 a n d 64 o f t h i s f u n c t i o n m a k e s E v e r e t t ' s f o r m u l a

t h e o n e t o b e u s e d .

F o r m o r e d e t a i l s , s e e

On the Construction of Table.s and Interpolation,

P a r t I I , b y P e a r s o n . T r a c t s f o r C o m p u t e r s , N o . I I I . C a m b r i d g e U n i v e r -

s i ty P ress .

DONALD

B .

M_~IER AND FRA NK A. W ECK :

M r . M e r e u ' s i n t e re s t in g p a p e r g iv e s a c o m p l e t e l y d e v e l o p e d m e t h o d

o f o b t a i n i n g e x a c t a n n u i t y v a l u e s f o r a M a k e h a m i z e d t a b l e g i v e n o n l y

t h e M a k e h a m c o n s t a n ts , w h e r e o n e d o e s n o t w i s h . to g o t o t h e t r o u b l e

o f c a l c u l a t i n g c o m m u t a t i o n f u n c t i o n s .

W h e r e o n l y a p p r o x i m a t e a n n u i t y v a l u e s a r e d e s ir e d a n d t a bl e s o f a n -

n u i t y v a l u e s a r e a v a i l a b l e o n a n o t h e r t a b l e w h i c h f o l l o w s M a k e h a m ' s

l aw , a n a l t e r n a t i v e a p p r o a c h w h i c h m a y b e u s e d i s a s fo ll ow s .

T o a p p r o x i m a t e a n n u i t y v a l u e s o n a t a b l e w h e r e t h e f o rc e o f i n t e r e st

a n d f o r c e o f m o r t a l i t y a r e , r e s p e c t iv e l y ,

F o r c e o f i n t e r e s t = 6 '

F o r c e o f m o r t a l i t y = u'. =" A ' "4- B '( d ) ~,

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294

A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E I I A M C O N S T A N T S

a n d a s su m i n g t h a t w e h a v e a v a i la b le a n n u i t y v a l u e s o n a m o r t a l i t y ta b l e ,

r e f e r r e d t o h e r e i n a s t h e " r e f e r e n c e t a b l e , " w h e r e t h e f o r c e o f m o r t a l i t y i s

t~ , = A + Be *,

_

t h e n a d e s i re d a n n u i t y v a l u e a.:~-~ m a y b e o b t a i n e d b y e v a l u a t in g t h e r i g h t -

h a n d s id e o f t h e f o l lo w i n g e q u a t i o n :

a ' 1

. : . 7

=

- ~ ¢ I . : ~ , ( 1 )

w h e r e t h e a n n u i t y o n t h e r ig h t - h a n d s id e o f t h e e q u a t i o n is b a s e d o n t h e

m o r t a l i t y o f th e r e f e r e n c e t a b l e a n d a f o rc e o f i n t e r e s t e q u a l t o 7 a n d w h e r e

k = log c '

log c

~ = - ~ ( ~ ' + A ' ) - A

+[ "']

= k x +

l o g ~ •

S h o u l d t h e M a k e h a m c o n s t a n t s b e e x p r es se d in t h e c o lo g f o r m s o t h a t

B ( c - - 1 )

c o lo g , p . - - A + f l c *, w h e r e / ~ = lo g . c '

t h e n z b ec o m e s

z = k x +

t o g \ 8 c ' - - '

w h i le t h e e x p r es s io n s f o r k a n d ~, a r e u n c h a n g e d f r o m t h o s e s h o w n a b o v e .

T h e s e r e l a t i o n s h i p s f o l l o w f r o m t h e f a c t t h a t w h e r e t h e f o r c e o f m o r -

t a l i t y f o l l o w s M a k e h a m ' s l a w , a c h a n g e i n t h e v a l u e o f A i s e q u i v a l e n t

t o a c h a n g e i n i n t e r e s t r a t e , a c h a n g e i n B i s e q u i v a l e n t t o a c h a n g e i n

a g e , a n d a c h a n g e i n t h e c o n s t a n t c r e s u lt s i n a s t r e t c h i n g o r co n d e n s i n g

o f t h e e f f ec t o f a g e a n d m u s t b e c o m p e n s a t e d f o r b y c h a n g e s i n t h e i n t e r e s t

r a t e a n d r a t e o f p a y m e n t .

D erivation o f Eq uat io n (1)

T h e d e s i r ed a n n u i t y m a y b e e x p r es s ed i n te r m s o f t h e M a k e h a m c o n -

s t an t s and t he fo rce o f i n t e r e s t a s fo l l ows :

~' - f%-f/tS'+a'+B'(¢J~+nlah

dr . ( 2 )

z : n T ~ J O ~ , o . . , " . . .

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L e t

DISCUSSION 295

(c')~+h = ckC~+n).

B ' = B c ~°

6 ' + A ' =

k ( 7 + A ) ,

w h e r e k , 0 a n d 7 a r e c o n s t a n t s d e f i n e d b y t h e f o r e g o in g r e l a ti o n s h i p s.

W e c a n t h e n w r it e

g~ = f . - -r t lk ~ V + A ~ + M ' O ÷ k ( * + h ) l ~ h a ,

" :~ J o . . . . . . ( 3 )

I f w e n o w s u b s t i t u t e t h e v a r i a b l e m f o r k h . so ' tha t d h = 1 / k . d m , t h e

l im i t s o f i n t e g r a t i o n i n t h e e x p o n e n t b e c o m e 0 t o k t a n d w e h a v e

a~:~-' = /"e-fo kt t~ +A +~ l/~)e ~( ~+°)+me~ d r . ( 4 )

N ow i f we le t c" = (1/k)c~(~+°) an d replac e t b y I / k . r , s o t h a t dt =

1 / k . d r ,

a n d c h a n g e t h e l i m i t s o f i n t e g r a t i o n a c c o r d i n g l y , w e o b t a i n

r . k n ~ ' f l "

1 ] e -Jo [ v + A + n ~ ' * ' ~ d r ( 5 )

1

= ~ a~,:k.--i, ( 6 )

w h e r e 7 is th e f o r c e o f i n t e r e s t a n d t h e m o r t a l i t y t a b l e i s t h e r e f e r e n c e

tab le .

A p p l i c a t i o n t o C u r t a t e A n n u i t i e s

U s i n g t h e a p p r o x i m a t e r e l a t i o n s h i p

a n d t h e f a c t t h a t

a n d

' D"

D . ~ = ~+___~_ ~ , . . ~ p , ( 8 )

D ' D ~

6 ' + u ' + = k ( ~ + ~',+ k.), ( 9 )

t h e f o ll o w in g e x pr e ss io n s m a y b e d e r i v e d f o r u s e i n c a se s i n v o l v i n g c u r t a t e

a n n u i t i e s :

11 a ~ - - k - 1 k s - 1

a ' . : ~ = ~ .:k .l 2 ( 1 - - ~ k " ' k " P * ) - ~ 1 2

. t o )

X [(~,+ m) - ~k"'~,P,C'r+U,+~).1 t,

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296 ANNU ITY VALUES DIRECTLY FROM M AKEHAM CONSTANTS

_

a :k~-- - ½( 1 -- ~'k".k,~p ) + 1--2 ( 1 1 )

X [ ( ~ ' + m ) - g k , , . ~ p , ( ~ + ~ , + ~ ) 1 .

E q u a t i o n (1 0 ) w o u l d b e u s e d w h e r e c u r t a t e a n n u i t y v a l u e s a r e a v a il a b le

o n t h e r e fe r en c e t a b l e a n d e q u a t i o n (1 1) w h e r e a n n u i t y v a l u e s o n t h e

re fe rence t ab l e a r e con t inuous . For mos t purposes t he t h i rd t e rm of t he

r igh t -hand member i n equa t ions (10) and (11) cou ld be i gnored .

Exam ples o f the Me thod

E x a m p l e 1 : A p p r o x i m a t i o n o f a~s o n t h e a - 1 9 4 9 F e m a l e T a b l e w i t h 2 ~ %

in te res t us ing t he 1941 CSO as t h e r e f e rence tab l e .

T h e n e c e s s a r y v a l u e s o f t h e M a k e h a m c o n s t a n t s in t h e f o r m u l a s f o r

colog,pz fo r t he a -194 9 an d 1941 CSO Tab le (which is a M ak eh am t ab l e

above age 15) a r e :

a-194 9 1941 CSO

A ' = . O 0 1 A = . 0 0 1 6 0 4 9 6 6

f l ' = . 0 0 0 0 0 7 5 /~ = . 0 0 0 1 5 1 0 2 2 4

c ' = 1 . 1 1 9 4 3 7 9 c = 1 . 0 8 9 1 3 1 7 0

lo glo c ' = . 0 4 9 loglo c = . 0 3 7 0 8 0 4

6 ' = . 0 2 4 6 9 2 6

k log to c ' . 0 49

. . . . 1 . 3 2 1 4 5 3

log~o c . 0 3 7 0 8 0 4

"y = - ~ ( ~ ' + A ' ) - - A

1

1 . 3 2 1 4 5 3 ( ' 0 2 4 6 9 2 6 q - ' 0 0 1 ) - - . 0 0 1 6 0 4 9 6 6

= . 0 1 7 8 3 7 7

k x + l_ ._ L _ , r t ~ ' ( c - l )

z = L ° g l ° [ N - c r = 1 )

1 { r.o oo oo 7_ 55 x . 0 8 9 1 3 1 7 1 t

= 1 . 3 2 1 4 5 3 ( 6 5 ) - { . 0 3 7 0 8 0 4 l o g to L . O O O 1 5 1 0 2 X . 1 1 9 4 3 7 9

= 8 5 . 8 9 4 4 - - 3 8 . 5 9 4 1

= 4 7 . 3 0 0 .

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D I S C U S S I O N 297

T h e e f fe c t iv e a n n u a l i n t e r e s t r a t e c o r r e s p o n d i n g t o t h e f o r c e o f i n t e r e s t

= .0178377 i s 1 .800 %. g 'C /

L e v e r ' s m e t h o d

(JIA,

Vol . ~ 1920, p . 171) i s one of m a n y which

m a y b e u s e d t o e v a l u a t e ~ L S %4~.8 on the 1941 CSO T ab le . Fo l lowing th i s

m e t h o d w e m i g h t p r o c e e d as fo l lo w s :

a2% = 1 7 . 4 9 0 4 ( s t r a i g h t l in e i n t e r p o l a t i o n b e t w e e n

4 7 . 3

t a b u l a te d v a lu e s a t 2 % )

2%

~1-]--- 17 .0 1 1 2

a~z-%-]= 1 7 . 6 5 8 0

a ~% 1 7 . 4 9 0 4 ( s t r a i g h t l in e i n t e rp o l a t io n ) ~

- - -

a 0 %

2 2 . 9 2 ( s t r a i g h t li ne i n t e r p o l a ti o n )

4 7 . 3 e47.8 ~--"

o%

a~.--¢-~ = 2 2 . 9 2

. 2

a - -~ ,w h e re m = 2 1 . 7 4 0 9 - { - ~ - ( 2 2 . 9 2 - - 2 1 . 7 4 0 9 ) = 2 1 . 8 5 8 8 .

4 7 . 3 m J

E v a l u a t i n g b y l o g a r i t h m s

1 ( 1 y i .8 5 . 8

1.80"/0 - - ~ k ~ /

a ~ = . 0 1 8 = 1 7 . 9 3 9 9

, l r 1so , k - - l ]

1 [ 1 7 . 9 3 9 9 . 3 2 1 4 5 3 ]

= ' 1 . 3 2 1 4 5 3 -

= 1 3 . 4 5 4 .

A

mo re accurate value

o f h

in a~--~1 w here h is betwee n 0 and 1, may be obtained

if d esired by using the following:

h - g - - 2 K ( 1 - g ) ,

where

a + - - - ~ - a ~

K - -

a~+---i- - a ~ "

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300 ANNUITY VALUES DIRECTLY FROM MAKEHAM CONSTANTS

k 1 ]

, 1 f ~ ~s5 % 2

a 65 = - ~ L a ~ . o o ~

- - -

_ 1 [ 9 . 7 4 8 1 q . 1 1 0 0 4 2 ]

. 8 8 9 9 5 8 L 2 J

= 1 1 . 0 1 5 .

T he pub l i shed v a lue i s 11 .01345.

Application to Pro jected M ortality

I f m o r t a l i t y i m p r o v e m e n t is re fl ec te d b y c h a n g i n g t h e M a k e h a m c o n-

s t a n t s a c c o r d i n g t o t h e f o r m u l a g i v e n b y M r . M e r e u , n a m e l y t h a t t h e

fo rce of m o r t a l i t y fo r t h e y e a r o f b i r t h y + t a n d a t t a i n e d a g e x is g i v e n b y

Z

#~, ~ + t = A + B u + t c v + t ,

where

c.+ t = ( 1 + 1--~0) t c~

a n d w h e re Bu a n d c u a r e t h e M a k e h a m c o n s t a n t s f o r t h e y e a r o f b i r t h

y , t h e n a n a n n u i t y v a l u e f o r a t t a i n e d a g e x a n d y e a r o f b i rt h y + t m a y

b e a p p ro x i m a t e d b y t h e fo ll o w i n g fo rm u l a :

1

w h e re ~, is t h e fo rc e o f i n t e r e s t a n d m o r t a l i t y is t h a t f o r y e a r o f b i r t h y

a n d w h e re

tw , I - l O 0 + s-I

]

t

= 1 - t log c~ ' to g[ i -6-6

i t -

" ~ = k t ~ - t - A ( k t 1)1

z = k t X + L i o g ~ '

T. N. E. GREVILLE:

I t is c e r t a in l y w o r t h p o i n t i n g o u t t h a t l if e c o n t i n g e n c y fu n c t i o n s b a s e d

o n a M a k e h a m i z e d m o r t a l i t y t a b l e c a n b e c o m p u t e d d i r e c t l y f r o m t h e

M a k e h a m c o n s t a n ts , w i t h o u t r e so r ti n g to c o m m u t a t i o n c o lu m n s . I f o n l y

a f e w c a l c u l at io n s a r e t o b e m a d e , t h i s m a y w e l l b e w o r t h w h i le .

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DISCUSSION 301

I t is t h e p u r p o s e o f t h i s d i sc u s si o n t o d e s c r ib e a m e t h o d o f c a l c u l a t i o n

b a s ed o n p u b li sh e d t a b le s o f t h e i n c o m p l e te g a m m a f u n c t io n , w h i c h m a y

b e s im p l e r t h a n t h o se su g g e s te d b y M r . M e r e u a n d M r . M c C l i n to c k . A t

l e a s t it h a s t h e a d v a n t a g e o f n o t r e q u i r in g t h e s u m m a t i o n o f a s e ri es .

E q u a t i o n ( 1 5 ) o f t h e p a p e r c a n b e w r i t t e n i n t h e f o r m

~H Kz f co _H _le_ ~d ,

a x l n c = _ ~ x e

JK~ y Y

( 1 )

w h e r e

K = B c ~ _ ~ , - - A H =A+_____~

In c In c ' In c

T h e i n c o m p l e te g a m m a f u n c t i o n i s d e fi n ed a s

£

, ( p + 1 ) = e - ' t ~ d t . ( 2 )

T h u s r ~ ( / , + 1 ) - - r ( p + 1 ) is t h e (c o m p l e t e ) g ~ m m a f u n c t i o n . I t w o u l d

a p p e a r t h a t t h e in t e g r a l c o n t a i n e d i n e q u a t i o n ( 1 ) o f t h is d is c u s si o n c o u l d

t h e r e f o r e b e ex p re s se d a s r ( - . N ) - - I ' K ~ ( - :H ) . H o w e v e r , th i s d oe s n o t

w o r k , b e c a u s e t h e i n t e g r a l in ( 2) d o e s n o t c o n v e r g e f o r n e g a t i v e v a l u e s

o f t h e a r g u m e n t p + 1 .

T o g e t a r o u n d t h i s d i f f i c u l t y , w e i n t e g r a t e b y p a r t s i n ( 1 ) a n d o b t a i n ,

a f t e r s o m e s i m p l i fi c at io n ( d r o p p i n g t h e s u b s c r i p t o f K ) ,

a , (A + 6 ) = 1 - - KHeK[I'(1 - - H ) - - I ' x (1 - - H ) ] . ( 3 )

A s i t a p p e a r s t h a t t h e v a l u e o f H w i l l a l w a y s f a l l b e t w e e n 0 a n d 1 , s o

t h a t 1 - H w i ll a l w a y s b e a p o s i t i v e q u a n t i t y , w e n o w h a v e a s a t i s f a c t o r y

express ion.

Tables o f the Incomplete r-Fu nctio n t

d o e s n o t g i v e t h e v a l u e s o f t h e

i n c o m p l e t e g a m m a f u n c t i o n d i r e c tl y ( b e c a u se o f t h e e x t r e m e l y w i d e r a n g e

o f v a l u e s a s su m e d b y t h a t f u n c t io n ) b u t o f o t h e r r e l a te d f u n c t io n s f r o m

w h i c h th e i n c o m p l e te g a m m a f u n c t i o n c a n r e a d il y b e c o m p u t e d . T h e t a b le

t h a t g iv e s t h e m o s t a c c u r a t e r e su l ts f o r t h e p r e s e n t p u r p o s e is T a b l e I I I ,

w h i c h g i v e s v a l u e s o f t h e l o g a r i t h m s o f a f u n c t i o n

I ' (u , p )

d e f in e d b y

r , ( p + 1 ) = u ~ + , r ( p + 1 ) I ' (u , p ) ,

where u = x / ~ / p + 1 . Su bs t i t u t i o n o f t h i s express ion i n ( 3 ) g ives , a f t e r

s o m e a l g eb r a ic m a n i p u l a t i o n ,

a , ( a + ~) = 1 - e a r ( 1 - H ) [ K n -- K ( 1 - - H ) ~ - H ) / 2 I ' ( u , - - H ) ] .

t E dited by Karl P earson and published in 1922 by the Cam bridge University Press.

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3 0 2 A N N U I T Y V A L U E S D I R E C T L Y

FROM

M A K E H A M C O N S T A N T S

T h e c o r r e s p o n d i n g e x p r e s s i o n f o r a=:h- i s c o m p l i c a t e d , a n d i t i s p r e f e r a b l e

to work with whole-life annuity values, using

a=:,---i = a = - E=a=+,

and

D E = = e - n ( A + $ ) + K z - K z + n .

Taking as an illustration the one worked out in detail in the paper,

namely, a65 for the a-1949 female table with 2~°~o interest, we have

A+6 = .0256926

K= .096144

H= .22772

K n = .58666

e~ = 1.10092

F(1--H) = 1.19688

u = K / v ' I - - t l =

.109405

(1

- - t t ) - ~ l - m / 2 I ' ( u , - - t t )

= 1.03791.

The first five values above are taken from the paper, the value of F(1 --

H) was obtained from Biornetrika Tables fo r Statisticians, volume I, 2 and

the last value listed was computed with the help of tables of logarithms

after obtaining log I t (u , - - .H) by straight-line interpolation from Tables

of the Incomplete P-Function.

The result obtained for ~5 is 13.952, the

same as in the paper.

D O N A L D A. J O N E S A N D C E C I L J. N E S BI T T :

A l m o s t f i f t y y e a r s a f t e r M r . M c C l i n t o c k u s e d t a b l e d v a l u e s o f t h e

g a m m a f u r i c t i o n i n h i s c a l c u l a t i o n s o f a = , K a r l P e a r s o n i n 1 9 2 2 t a b u l a t e d

v a l u e s o f t h e i n c o m p l e t e g a m m a f u n c t i o n . I t o c c u r r e d t o u s t o e x p l o r e

t h e p o s s i b i l i t y o f u s i n g P e a r s o n s t a b l e s f o r t h e p u r p o s e o f M r . M e r e u s

p a p e r . T h e i n c o m p l e t e g a m m a f u n c t i o n i s

f o x

=( l+p ) = y P e - v d y , p > - - l , x > _ 0 ;

F=(1 + p) is the well-known gamma function and for this case the sub-

script will be omitted. Pearson, as a tabulating convenience, gave values of

r . , ~ ( l + p )

I ( u , p ) =

r ( l + p )

in the main part of his tables.

In order to express a= in terms of tabled values, we integrated once

by parts in the right member of

Kz~ H f oo --Y - - I - -H.

( ln c ) a = = e 6 , j l q e y a y ,

2 Edited by E. S. Pearson and H. O. Hartley and published in 1956 by the Cam-

bridge University Press.

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DISCUSSION 303

wi th 0 < H < 1 to avo id specia l cases . ( I t i s w or th no t ing th a t M cC l in -

t o c k r e ma rk e d t h a t h i s p o w e r s e r i e s ma y b e d e v e l o p e d b y r e p e a t e d i n -

t egra t ion b y pa r t s . ) Th i s p rocess , fo l lowed b y sp l i t ting o f the range o f

i n t e g ra ti o n a n d u s e o f t h e i n d i c a t e d n o t a t i o n s , y i e l d s

1 e ~ x v ( 1 - - H ) [ l _ i H

( l n c ) ~x = H H

W hile th is fo rm ula pu t s a~ in t e rm s o f t ab led func t ions , the in te rpo la -

t ion necessa ry to o b ta in th e des i red degree o f accu racy in

K ~ H )

I ( 7 1 - - H '

r e q u i r e d n o le ss c o m p u t a t i o n t h a n t h e s erie s u s e d b y M r . M e re u . I n f a c t ,

fo r .75 < H < 1 and K < 1 .5v /1 - - H , Pea rso n sugges t s the in tegra t ion

of the pow er series fo r e ~ y - H t o o b t a i n a d e q u a t e a c c u ra c y . T h e e x a mp l e s

we have s tud ied would t end to conf i rm a power se r i e s approach such a s

u t i l i z e d b y M r . M e re u , e x c e p t a t e x t r e me h i g h a g e s w h e re t h e Pe a r s o n

tables are general ly efficient .

T h e re is t h e o b v i o u s o b s e rv a t io n t h a t i f a m o r t a l i t y t a b l e w i th a M a k e -

h a m g ra d u a t i o n is t o b e u s e d e x t e n s iv e l y fo r a v a r i e t y o f a n n u i t y c a lc u l a-

t io n s , c o m m u t a t i o n c o lu mn s w o u l d b e a p r a c t i c a l n e c e ss i ty . Su ch c o l u mn s

a re read i ly ob ta ined by e lec t ron ic compute rs and p rov ide a ve ry f l ex ib le

ca lcu la t ion too l . Th us , fo r ins tance , the ca lcu la t ion o f a d i sc re te va ry ing

a n n u i t y w o u l d se em t o b e m o r e fe a si bl e b y m e a n s o f c o m m u t a t i o n f u n c -

t i o n s t h a n b y d i r e c t c o m p u t a t i o n f r o m t h e M a k e h a m c o n s t a n t s . A l s o ,

t h e p a r a m e t e r A o f t h e M a k e h a m l aw i s o ft e n r e p la c ed b y a p o l y n o m i a l

a t the le ss adv anc ed ages , and th i s c rea te s an o th e r d i f f i cu l ty fo r d i rec t

c a l c u l a ti o n o f a n n u i t y v a l u e s .

D e s p i t e t h e s e p r a c ti c a l l i m i t a ti o n s , t h e a u t h o r ' s m e t h o d s a r e o f i n t e r e s t

i n s h o w i n g h o w a n a l y s i s m a y o b v i a t e t h e a r i t h m e t i c r e q u i re d fo r c o m p u t -

i n g a n n u i t y v a lu e s b y t h e u s u a l c o m m u t a t i o n p ro ce ss. W h e re o n l y is o l a te d

v a l u e s a r e r e q u i r e d fo r e x p e r i me n t a l p u rp o s e s , h i s me t h o d s ma y p ro v e

t h e i r w o r t h .

E. WARD EMERY:

M r. M e re u s t a r t s w i t h a fu n c t i o n o f a g e ,

viz . ,

t h e M a k e h a m f o r m u l a

fo r t h e fo r c e o f mo r t a l i t y , a n d s e t s o u t t o i n t e g ra t e i t t o o b t a i n v a l u e s

o f g ~:~ . A l t h o u g h h e m i g h t h a v e i m p r o v e d u n d e r s t a n d i n g b y i d e n t i fy i n g

h i s i n t e g ra l t o k n o w n m a t h e m a t i c a l f u n c t i o n s , h e w o u l d s ti ll h a v e l a c k e d

a n a d e q u a t e s e t o f ta b l e s. U n l e ss h i s T a b l e 7 is ma t e r i a l l y e x p a n d e d o n e

w o u l d t u rn r a t h e r q u i c k l y t o e l e c t ro n i c c o mp u t e r s t o e v a l u a t e h i s f u n c -

t io n s . I f t h e n s u c h c o m p u t e r s a r e t o b e u s e d , h o w c a n t h e y b e s t b e u s e d ?

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3 0 4 A N N U I T Y V A L U E S D I R E C T L Y F R O M M A K E H A M C ON ST AN TS

T h e c o n c l u s io n I r e a c h e d is t h a t t h e w a y t o m a k e t h e i n t e g r a t i o n o f t h e

M a k e h a m f o r m u l a m o s t m e a n i n g f u l is t o c o n s t r u c t re g u la r c o m m u t a t i o n

f u n c t i o n s i n t h e m e m o r y o f t h e c o m p u t e r a n d t h e n w r it e a s m u c h o r a s

l i t t l e a s m a y b e d e s i r e d .

I n o r d e r t o e x a m i n e t h e c o m p u t e r d i ff ic u lt ie s I w r o t e a n d a p p l i ed s u c h

a p r o g r a m f o r a p u n c h e d c a r d c o m p u t e r i n o u r o ff ic e. A n i n f o r m a t i o n c a r d

s u p p l i e s

A , B , c , i , y

a n d t a b l e i d e n t i f i c a t i o n . T h e c o m p u t e r p u n c h e s q ~ ,

~ y a n d l~ o n t h i s i n f o r m a t i o n c a r d . T h e r e l a t i o n s h i p c o lo g ,p x = A + B c x

is a s s u m e d t o a p p l y f o r a g e s y a n d o v e r. 1 0 0 0 B a n d e a c h o f t h e o t h e r c o n -

s t a n t s is t a k e n t o e i g h t d e c i m a l p l a c e s , w i t h o n l y c a l lo w e d t o b e a s la r g e

a s u n i t y . A l t e r n a t i v e l y c m a y b e r e p l a c e d w i t h l o g l 0 c a n d a c o d e t o s o

i n d i c a t e .

T h e p r o g r a m is d e s c r i b e d b e l o w i n a s e ri es o f s t e p s . I t is t o b e u n d e r -

s t o o d t h a t t h e p r o g r a m s t a r t s a t s t e p 1 a n d p r o c e e d s i n s e q u e n c e t o th e

n e x t h i g h e r s t e p u n l e ss t h e r e i s a s p ec if ic s t a t e m e n t t o t h e c o n t r a r y .

1 . R e a d i n f o r m a t i o n c a r d .

2 . C o n v e r t lo gl0 c t o c i f t h a t is n e c e s s a r y . T h i s c o n v e r s i o n r e q u i r e s c o m -

p u t i n g u = l o g ic = lo g,1 0 .1 og ~0 c a n d e x p a n d i n g t h e e x p o n e n t i a l s e -

ries e~.

3 . S e t th e a g e x to ze r o a n d s e t B ~ a n d v~ t o t h e k n o w n v a l u e s f o r x = 0 .

4 . T e s t w h e t h e r x = ¢0 a c c o rd i n g t o a t e s t d i s c u s s e d b e l o w . I f x = ~0

t h e n p r o c e e d t o s t e p 6 ; o t h e r w i s e p r o c e e d t o s t e p 5 .

5 . A d v a n c e t h e a g e o n e y e a r a n d d e t e r m i n e n e w B c~ a n d v* b y m u l t i p l y -

i n g b y c a n d d i v i d i n g b y ( 1 + i ) r e s p e c t i v e l y a n d r e t u r n t o s t e p 4 .

6 . Se t l~ = 1 , p~ = 0 , an d N , , , + I = 0 .

7 . C o m p u t e q, , D , , N , a n d a , b y t h e f a m i l i a r r u le s , t h a t is , q , = 1 - p x,

D , = v ~ l, , N , = N , +~ + D , , ~ /, = N J D ~ . O f c o u r s e , x = w t h e f ir s t

t i m e t h r o u g h t h i s s t e p .

8 . T e s t w h e t h e r x = y . I f x = y th e n t e r m i n a t e t h e c o m p u t a t i o n a n d

p u n c h q ,, d~, a n d l~ o n t h e i n f o r m a t i o n c a r d a n d g o t o s t e p 1 f o r t h e

n e x t c a r d ; o t h e r w i s e p r o c e e d t o s t e p 9 .

9 . D e c r e a s e t h e a g e o n e y e a r a n d d e t e r m i n e n e w v* a n d B c* b y m u l t i p l y -

i n g b y ( 1 + i) a n d d i v i d i n g b y c r e s p e c t i v e l y .

1 0. C o m p u t e c o l o g ep ,

= A + B c ~ .

1 1 . C o m p u t e p , b y e x p a n d i n g t h e e x p o n e n t i a l s e r i e s e ~ , where u =

-7 c o l o g , p , .

12. C o m p u t e l , b y d i v i d in g l ,+ ~ b y p , a n d r e t u r n t o s te p 7 .

T h e q u a n t i t i e s

A + B c

a re s u m m e d f ro m x = 0 t o x = o~ - - . 1 , a n d

w is r e c o g n i z e d a s t h e a g e a t w h i c h t h i s s u m e x c e e d s 1 6 . S i n c e l~ = 1 a n d

e~6 is a p p r o x i m a t e l y t e n m i l li o n, i t f ol lo w s t h a t i f t h e M a k e h a m f o r m u l a

h e l d t h r o u g h o u t l if e t h e n l0 w o u l d b e a p p r o x i m a t e l y t e n m i ll io n . A t s t e p

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DISCUSSION 305

3 a t e s t a m o u n t i s s e t e q u a l t o z e r o . A t s t e p 4 i f t h e t e s t a m o u n t d o e s

n o t e x c e ed 1 6 t h e n x # ¢0 a n d A + B ~ is a d d e d t o t h e t e s t a m o u n t .

T h e e x p a n s i o n o f t h e e x p o n e n t i a l s e r i e s

r e q u i re d a c e r t a i n a m o u n t o f s p e c ia l a t t e n t i o n • E a c h t e r m i n th i s se r ie s

i s c o m p u t e d f r o m t h e p r e v i o u s o n e b y m u l t i p l y i n g b y u a n d d i v i d i n g b y

n,

w i t h t h e e x p a n s i o n t e r m i n a t i n g w h e n a z e r o t e r m t o t h e n u m b e r o f

d e c i m a l p l a c e s u s e d is r e a c h e d . I n t h e u s u a l c a s e u = - - c o l o g ,p ~ a n d i s

n e g a t i v e . H o w e v e r , i t is a ls o n e c e s s a r y t o b e a l~ le t o h a n d l e t h e c a s e w h e r e

u = l o g, c a n d is p o s i t iv e . I t w a s d e c id e d t h a t i t w o u l d b e u s e f u l f r o m a

c o m p u t e r s ta n d p o i n t if e v e r y t e r m u n / n w e r e to b e d e t e r m i n e d t o ei g h t

d e c i m a l s , w i t h u s o l i m i t e d t h a t t h e a b s o l u t e v a l u e o f e v e r y t e r m w o u l d

b e le ss t h a n 1 0. T h i s s iz e r e q u i r e m e n t a l o n e l i m i t e d t h e a b s o l u t e v a l u e

o f u t o b e l e s s t h a n t h e c u b e r o o t o f 6 0 o r s l i g h t l y l e s s t h a n 4 . H o w e v e r ,

i t i s a l so d e s i r a b le t o k e e p t h e p o s s ib l e e ff e c ts of d r o p p e d d e c i m a l s s m a l l ,

w h ic h is s o m e w h a t m o r e r e s tr i ct i v e . T h e m e t h o d o f d e t e r m i n i n g t h e t e r -

m i n a l a g e s h o u ld k e e p t h e a b s o l u t e v a lu e o f u n o t m u c h l a r g e r t h a n 2 a n d

k e e p t h e a c c u m u l a t i v e e f f e c t o f d r o p p e d d e c i m a l s s m a l l.

T h e v a l u e s o f v~ a n d eo lo g,p ~ a r e d e t e r m i n e d t o f o u r t e e n d e c i m a l p l a c e s.

H o w e v e r , w h e n u s e d t o d e t e r m i n e D , a n d p , t h e y a r e f ir s t r o u n d e d t o

e i g h t d e c i m a l p la c e s . T h i s h a s b e e n f o u n d t o b e a p r a c t i c a l w a y o f a l w a y s

u s i n g t h e s a m e v a l u e s o f v a s th o s e f o u n d i n p u b l i s h e d t a b l e s . P a r a l l e l

t r e a t m e n t o f ~ a n d B ~ p r o v e d a c o m p u t in g c o n v e ni en c e a n d c e r ta i n ly

d i d n o t d e c r e a s e t h e a c c u r a c y o f t h e d e t e r m i n a t i o n o f B c~ . H o w e v e r , i t

s h o u l d b e u n d e r s t o o d t h a t c w a s l im i t e d t o e i g h t d e c i m a l p l a ce s a n d h e n c e

e x a c t d u p l i c a ti o n o f l o g a r i t h m c a l c u l a ti o n s w o u l d n o t b e a t t a i n e d a t h i g h

ages .

A l t h o u g h t h i s c o m p u t e r w o u l d b e c l a s s i f i e d a s s m a l l t o i n t e r m e d i a t e ,

t h e t i m e r e q u i r e d t o c o m p u t e 065 f o r 1 5 ta b l e s w a s a b o u t s i x m i n u t e s , o r

2 4 s e co n d s p e r t a b l e .

A s l i g h t v a r i a t i o n o f t h e p r o g r a m p e r m i t s t h e i n f o r m a t i o n c a r d t o b e

t r e a t e d a s a m a s t e r c a r d w h i c h s u p p l i e s i n f o r m a t i o n b u t i s n o t f u r t h e r

p u n c h e d . B l a n k c a r d s w h i c h f o l l o w i t a r e p u n c h e d e a c h t i m e t h e t e s t a s

t o w h e t h e r x = y i s m a d e . T a b l e i d e n t i f i c a t i o n , a g e ,

qx, gx, l~, D,,

a n d

N ~ a r e p u n c h e d o n t h e s e c a r d s f r o m a g e ~0 d o w n t h r o u g h a g e y. W h e n x

b e c o m e s le ss t h a n y t h e b a s ic c o m p u t a t i o n c e as e s a n d t h e b a l a n c e o f t h e

b l a n k c a r d s a r e p a s s e d w i t h o u t p u n c h i n g . T h e t i m e r e q u i re d t o f e ed c a rd s

a p p r o x i m a t e l y t r i p l e s t h e c o m p u t i n g t i m e f o r a t a b l e .

T h e p r o g r a m h a s b e e n a p p l i e d t o re p r o d u c e a n u m b e r o f f a m i l ia r ta b le s .

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306 ANNUITY VALUES DIRECTLY FRO M M AKE HA~ CONSTANTS

T h e r e p r o d u c t i o n i s v e r y g o o d e x c e p t n e a r t h e t e r m i n a l a g e w h e r e t h e

l~ f u n c t i o n s a r e q u i t e s m a ll. F o r e x a m p l e , t h e e q u i v a l e n t o f th e a - 1 9 4 9

f e m a l e t a b l e f o r a g es 5 0 t h r o u g h 1 0 8 a g r e e d w i t h t h e p u b l i sh e d m o r t a l i t y

r a t e s exc ep t fo r a d i f f e r ence o f one i n t he l a s t p l ace a t ages 57, 96 , 103

a n d 1 0 5; t h e v a l u e s o f ax a t 2 ~ o i n t e r e s t a g r e e d u p t h r o u g h a g e 1 01 .

A l so s o m e t a b l e s o f a x , w e r e c h e c k e d q u i t e s a t is f a c t o r il y b y d e t e r m i n i n g

a~ us ing 2A and 2B in p l ace o f A and B .

T h e G a - 1 9 5 1 t a b le a n d t h e 1 95 8 CS O ta b l e a re t w o i m p o r t a n t r e c e n t

t ab l es w h i ch a r e n o t M a k e h a m g r a d u a t e d b e c a u se n o s a t is f a c to r y f it c o u l d

b e m a d e . T h e s e s a m e c o m p u t e r s w h i c h m a k e p o s s i b l e t h e p r o g r a m d e -

s c r i b e d h e r e a l s o m a k e p o s s i b l e t h e d i r e c t c o m p u t a t i o n o f m u l t i p l e l i f e

f u n c t i o n s a n d h e n c e h a v e c o n t r i b u t e d t o a t r e n d a w a y f r o m M a k e h a m

g r a d u a t e d t a b l e s .

WILLIAM H. BUI~LING:

F o r m a n y a c t u a r i a l p r o b l e m s w e r e a l l y w o u l d l ik e t o re v i e w th e a n -

s w e rs p r o d u c e d b y c o m b i n i n g t h e r e s u lt s f r o m a w i d e v a r i e t y o f m o r t a l i t y ,

t a x a n d " w i t h d r a w a l " a s s u m p t io n s w i t h d i f f e r e n t s e ts o f p o s si bl e p r o b a -

b i l i t i e s . T h e r e s u l t a n t a r r a y o f

"expectations"

w o u l d b e i m m e a s u r a b l y

s u p e r i o r f o r p r a c t i c a l d e c is io n - m a k i n g t h a n is t h e o n e " e x p e c t a t i o n " t h a t

h a s g e n e r a l l y b e e n p o s s ib l e w i t h t h e r e l a t iv e l y p r i m i t i v e c o m p u t a t i o n m e t h -

o d s o f t h e p a s t . I t w a s a p l e a s u r e t o s ee a n e x p l o r a t o r y p a p e r o f th i s

n a t u r e a n d I t h i n k w e s h o u l d s h o w t h e a u t h o r e n o u g h a p p r e c i a t i o n s o

t h a t m o r e y o u n g p e o p l e will b e e n c o u r a g e d t o s e a r c h f o r t h e s o p h i s t ic a t e d

m a t h e m a t i c s a n d m a c h i n e s w e n e e d.

I n c i d e n t a l l y , w h e n m y a c t u a r i a l s t u d i e s r e a c h e d l i f e c o n t i n g e n c i e s , I

r a n i n to t r o u b le s w i t h t h e o l d G e o r g e K i n g t e x t b o o k a n d I w e n t b a c k t o

m y f o r m e r h ig h s c h o o l t e a c h e r f o r h e lp . H e t o o k o n e lo o k a t t h e c o m m u -

t a t i o n t a b l e s a t t h e b a c k o f t h e b o o k a n d s a i d , " Y o u w ill p r o b a b l y f o r g e t

h o w t o u s e th e s e c o lu m n s , b u t y o u w ill a l w a y s r e m e m b e r w h i ch c o m e s a f t e r

w h i c h - - t h e y a r e D ( a m n ) N ( o n ) S ( e n s e ) . "

( A U TH O R ' S R EV I EW O F D I S C U S S I O N )

JOHN A. MZEREU:

I a m g r a t e f u l f o r t h e v a r i e t y o f i n t e r e s t i n g d i s c u s s i o n s t h a t w e r e

p r o m p t e d b y t h e p a pe r .

I n t h e i n t r o d u c t i o n t o m y p a p e r , I d i d s t a t e t h a t w h e t h e r i t i s w o r t h

w h i le t o o b t a i n e x p r es s io n s f o r l if e c o n t i n g e n c y f u n c t io n s d i r e c t l y f r o m t h e

g o v e r n i n g l a w o f m o r t a l i t y w i ll d e p e n d o n a n u m b e r o f f a c t o rs , o f w h ic h

I l i s te d th r e e . B o t h M r . N 'ie sse n a n d M r . E m e r y o b s e r v e t h a t m o d e r n

c o m p u t i n g e q u i p m e n t m a k e s i t p o ss ib le t o p r o d u c e a c o m p l e te m o r t a l i t y

t a b l e in a fe w m i n u t e s , a n d M r . E m e r y h a s d e s c ri b e d a p r o g r a m s u c c e ss -

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DISCUSSION 307

f u l l y u s e d f o r t h e p u r p o s e . W h i le t h e i r o b s e r v a t i o n s a r e a c c u r a t e i n t h i s

r e s p ec t , t h e re i s a la r g e co s t f a c t o r i n v o l v e d w h i c h m i g h t m a k e i t p r u d e n t

t o c a r r y o u t a f e w d e s k c a l c u l a t i o n s b e f o r e c o m m i t t i n g t o e x p e r i m e n t a l

c a l c u l a t i o n s t h e c o m p u t e r r e s o u r c e s .

M r . N i e ss e n f in d s m y m e t h o d f o r c o m p u t i n g a s in g le a n n u i t y f o r m i -

d a b le . T h i s is a n a t u r a l o b s e r v a t i o n a n d a s i m p l e r m e t h o d w o u l d c e r t a i n l y

b e a p p r e c i a t e d . P o s s i b l y t h e m e t h o d d e s c r i b e d b y M r . W e c k a n d M r .

M a i e r is th e a n s w e r . H o w e v e r , th e s t a n d a r d m e t h o d o f p r o d u c i n g a t a b l e

o f m o r t a l i t y a n d c o m m u t a t i o n f u n c t i o n s is a f o r m i d a b l e o n e t o o a n d c o u ld

n o t i n a f e w m i n u t e s p r o d u c e t h e h a n d f u l o f a n n u i t y v a l u e s w h ic h m i g h t

b e s u f f i c i e n t f o r t h e u s e t o b e m a d e o f t h e m . U n d e r t h e m e t h o d o f t h e

p a p e r , a fe w v a l u e s c a n b e p r o d u c e d f o r a c o m p l e t e l y n e w t a b l e i n a

r e a s o n a b l e p e r i o d o f t i m e a n d w i t h o u t c o m m i t t i n g e x p e n s i v e c o m p u t i n g

m a c h i n e r y p r e m a t u r e l y .

M r . N i e s s e n d o e s n o t s e e h o w o n e c o u l d r e l y o n a n n u i t y v a l u e s c o m -

p u t e d f r o m a t a b l e w h ic h h a s n e v e r b e e n se e n. T h e s a m e o b s e r v a t i o n c a n

b e m a d e a b o u t a t a b l e w h ic h h a s b e e n se en . I d o n o t s ee t h a t i t m a t t e r s

w h e t h e r a t a b l e h a s b e e n s e e n o r n o t s e e n i n t h e u s u a l s e n se , a s l o n g as t h e

t a b l e o r v a l u e s c o m p u t e d f r o m t h e t a b l e s a t i s f y t h e t e s ts o f r e p r o d u c i n g

e x p e c te d v a lu e s w h i c h t h e a c t u a r y w a n t s t o i m p o s e .

I n h i s c l o s i n g r e m a r k s , M r . N i e s s e n a p p e a r s t o q u e s t i o n c e r t a i n p r i n c i -

p le s o f g r a d u a t i o n a n d s t a t e s t h a t i t is m o r e i m p o r t a n t t o c o n c e n t r a t e o n

f i t t h a n o n s m o o t h n e s s . T h e m o n o g r a m o n g r a d u a t i o n p r e p a r e d b y M r .

M o r t o n D . M i l l e r e x p l a i n s h o w g r a d u a t i o n i s c h a r a c t e r i z e d b y t h e t w o

e s s e n t i a l q u a l i t i e s - - - s m o o t h n e s s a n d f i t . " T h e g r a d u a t e d s e r i e s s h o u l d b e

s m o o t h a s c o m p a r e d w i t h t h e u n g r a d u a t e d s er ie s, b u t i t sh o u l d b e c o n -

s i s te n t w i t h t h e i n d i ca t io n o f t h e u n g r a d u a t e d s e ri e s. " " I n t h e a p p l i c a t i o n

o f a m e t h o d t o a s pe ci fi c p r o b l e m , t h e c i r c u m s t a n c e s o f t h a t p r o b l e m

d i c t a t e t h e n a t u r e a n d e x t e n t o f th e c o m p r o m i s e to b e e ff e ct e d b e t w e e n

f it a n d s m o o t h n e s s . " T h e r e is j u s t a s m u c h n e e d t o g r a d u a t e o b s e r v e d

r e s u l t s a s t h e r e e v e r w a s a n d a c h i e v e a p r o p e r b a l a n c e b e t w e e n s m o o t h -

n e ss a n d f i t. M a k e h a m i z a t i o n h a p p e n s t o b e o n e of th e m e t h o d s a v a i l a b le .

I t is n o t n e c e s s a r y t h a t o n e b e li e ve i n M a k e h a m ' s L a w a s a la w o f m o r -

t a l i t y b e f o r e u t il i zi n g i t a s a p r a c t i c a l g r a d u a t i o n t o o l.

M r . E m e r y s u g g e s t s t h e p o s s i b i l i t y o f m a t e r i a l l y e x p a n d i n g T a b l e 7 ,

I n c, a s a l a b o r - s a v i n g d e v i c e fo r c a lc u l a t i n g a n n u i t i e s . S u c h a n e x p a n s i o n

c o u ld b e v e r y u s e fu l p r o v i d e d t h a t t h e a r g u m e n t s o f t h e f u n c t io n c o u l d b e

s p a c e d s u f f ic i e n tl y c lo s e t o g e t h e r s o t h a t l i n e a r i n t e r p o l a t i o n i n t h e t a b l e

w o u l d b e a c c u r a te , a n d p r o v i d e d f u r t h e r t h a t t h e re s u l t in g t a b l e w a s n o t

t o o v o lu m i n o u s . I n s t e a d o f t a b u l a t i n g t h e r e s u l ts a t e q u i s p ac e d i n t e rv a l s

o f K , e q u i s p a ce d i n t e rv a l s o f s o m e f u n c t io n o f K m i g h t b e m o r e s u it a bl e .

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308 A N N U IT Y V A L U E S D IR E C TL Y F R O M MA K E H A M C O N ST A N T S

D r . N e s b i t t a n d D r . J o n e s , D r . G r e v il le , a n d M r . A m e r a ll s h ow h o w

t h e i n t e g ra l f o r th e c o n t i n u o u s a n n u i t y c a n b e e v a l u a t e d u s in g p u b l is h e d

v a l u e s o f t h e i n c o m p l e t e g a m m a f u n c t i o n . H o w e v e r , f o r t h e r a n g e o f K

v a l u e s l ik e l y t o b e re q u i r e d D r . N e s b i t t a n d D r . J o n e s d o n o t r e c o m m e n d

u s e o f t h e s e t a b l e s b e c a u s e o f t h e l a b o r i n v o l v e d i n i n t e r p o l a t i o n .

M r . M a i e r a n d M r . W e c k h a v e u s ed a c o m p l e t e ly n ew a p p r o a c h t o th e

p r o b l e m a n d d e r i v e a f o r m u l a r e l a t i n g t h e a n n u i t y o n t h e n e w t a b l e t o

a n a n n u i t y o n a r e f e r e n c e t a b l e . I n e f f e c t , t h e y g i v e u s a u s e f u l p a s s p o r t

f r o m o n e M a k e h a m t a b l e t o a n o t h e r . T h e i r a p p l i c a t i o n t o p r o j e c t e d m o r -

t a l it y p ro v i d es a d j u s t m e n t s t h a t m a y b e m a d e t o a b as e t ab l e t o p ro v i d e

f o r m o r t a l i t y i m p r o v e m e n t .

I a m t h a n k f u l t o M r . B u r l i n g f o r h i s w o r d s o f e n c o u r a g e m e n t .