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u n d a m e n t a l s o f

P e n s i o n u n d i n g

Charles L Trowbridge

I . In t r o d u c t i o nAmong the too l s o f t he pens ion ac tua ry a re a va r i e ty

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

he re be ca l l ed f u n d i n g m e t h o d s . B y f u n d i n g m e t h o d i s

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

which the bene f i t s a re to be f inanced . The choice of

funding me thod in no way a f fec t s t rue ove r -a l l cos t s ,

which a re a func t ion of t he bene f i t s t o be prov ided and

certa in other factors such as ra tes of morta l i ty, interest ,

a n d e m p l o y e e w i t h d ra w a l . T h e f u n d i n g m e t h o d i s , h o w -

ever , t he cont ro l li ng fac tor i n de te rmin ing how m uch o f

the eventua l cos t i s t o be pa id a t any pa r t i cu la r po in t o f

t ime . Funding me thod , a s employed in th i s pape r ,sh o u l d n o t b e c o n f u se d w i t h f u n d i n g medium i .e . the

vehic l e ( such a s Dep osi t Adm ini s t ra tion of Se l f -Adm in-

i s t e red Trus t ) by means of which the funding a r range-

ment s a re ca r r ied ou t .

T h e f u n d i n g m e t h o d s c o m m o n l y u se d i n th e p e n s i o n

f i e ld a re pe rhaps fa i r ly we l l unde rs too d b y the ac tua r i e s

wh o use them, b u t t he ac tua r i a l l i te ra ture on th is su b jec t

i s ex t remely spa rse . The c l a ss i c Br i t i sh pape rs on pen-

s ions devote themse lves l a rge ly to the t echniques of

va lu ing compl i ca t ed bene f i t s . They pu t l i t t l e o r no

emphas i s on the poss ib l e va r i a t ions in funding me thod ,

re ly ing a lmo st en t i re ly on wh a t i s e ssen t i a l ly ind iv idua l

l eve l p remium funding . Pe rhaps the bes t desc r ip t ion of

the va r ious funding m e thods w i l l be found in the Bul -

l e tin on Sec t ion 23(p) pu t ou t by the U .S . Treasury

Depar tment . Even th i s i s on ly a ve ry ske tchy and supe r -

f ic ia l t reatment , and the beginner in the pension f ie ld

pre t ty much has to d ig the ideas ou t fo r h imse l f . Thi s

pape r a t t empt s , i n som e measure , t o g e t a t l eas t the fun-

damen ta l s o f pens ion fund ing in to ac tua r i a l l i te ra ture .

Pa r t I I fo l lowing in t roduces ce r t a in fundamenta lc o n c e p t s , a m o n g th e m th e m a t u r e p o p u l a t i o n a n d

m a t u r e f u n d c o n c e p t s . B y m e a n s o f t h e E q u a t i o n o f

Matur i ty a log ica l c l a ss i fi ca t ion sys t em for t he va r ious

funding me thods i s dev i sed . Assumpt ions and no ta t ion

necessa ry for ac tua r ia l ana lys i s a re se t fo r th .

Pa r t HI desc r ibes and c l a ss i f i e s va r ious me thods

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

use among ac tua r i e s ac t ive in the pens ion f i e ld . The

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

funding (under the r ig id condi t ions of an in i t i al ly s t a -

t i ona ry popula t ion) a s a sor t o f t heore t i ca l base on

which to bu i ld a more prac t i ca l unders t anding .Pa r t IV looks in to the cha rac t e r i s t i c s o f t hese me th-

ods under l e ss i dea l i s t i c condi t ions . Cer t a in seeming

incons i s t enc ie s w hich a r i se in p rac t i ce a re expla ined .

Pa r t V in t roduces the ra the r t r eache rous sub jec t o f

ad jus tme nt fo r ga ins and losses , and desc r ibes va r ious

m e t h o d s o f m a k i n g su c h a d j u s t m e n t .

II. F u n d a m e n t a l C o n c e p t s

A s s u m p t i o n s N o t a t i o n

M atu r e Popu la t ion oncep t

Al l ac tua r i e s a re fami l i a r w i th the se rv i ce t ab le

de r ived f rom es t ima tes o f ra t e s o f dea th , w i thdrawa l ,

and new h i rings . The Ix co lum n of th i s tab l e represen t s

approx ima te ly the age d is t r ibu tion of the emp loye e

g r o u p af t e r the group reaches wha t we ca l l a s t a t ion-

a ry condi t ion .

II. Funda mentals of Pens ion Funding 101



Most employee groups today are immature; i .e. , they

contain more younger members and fewer pensioners

than the l, s column o f the underly ing service table

would indicate. Yet most of us accept the idea that anyemployee group of sufficient size can be assumed (for

want of better information) to approach a mature or sta-

tionary condit ion eventually. It seems logical, therefore,

to employ this mature population concept in the classi-

fication of funding methods.

quation of MaturityIt is apparent that a pension fund, like any other

fund, grows or shrinks as income exceeds outgo, or vice

versa. Contributions and interest make up income. Ben-

efits paid are outgo. Thus if benefits (B) and contribu-

tions (C) are both assumed payable at the beginning of

a year, and if the fund (F) is measured at the beginning

of the year (prior to either contributions or benefits then

due), the following relationship holds.

v A F = C d F - B (1)

where AF is the change in F over the year and d is the

rate of discount.

It is the essence of the mature population concept

that benefits (B) eventually become stationary. More-

over, it is characteristic of all of the funding methods

described in this paper that at or after the time when the

employee population becomes stationary, the contribu-

tion (C) and the fund (F) reach (or approach) a constant.

AF therefore becomes zero and equation (1) becomes

C d F = B (2)

where C, F, and B are all constants. Equation (2) can be

thought of as an Equation of Maturity.

Note that this equation does not necessarily hold as

soon as the populat ion reaches maturity. Sufficient time

must have elapsed so that C and F have reached their

ultimate levels as well. In point of time the concept of a

m a t u r e f u n d may therefore be one step beyond the idea

of a m a t u r e p o p u l a t i o n .

Classi f ication of F un ding M ethods

In the Equation of Maturity, B and d are entirely

independent of the funding method. Therefore, in the

ultimate situation, the various funding methods differ

only as to the relative sizes of F and C. At one extreme

F = O a nd C = B ; at the other C= 0 an dF =B /d .

Between these two extremes lie the funding methodscommonly employed.

It is logical to classify these funding methods in

ascending order of F (or descending order of C, which

is the same thing). This scheme of classification will be

used throughout this paper.

ssumptions

The actuarial analysis of the ultimate situation to

which a given funding method leads is materially sim-

plified if a mature population is assumed, not after

many years, but right from the inauguration of the plan.The concept of an i n i t i a l l y mature population (both as

to active and retired lives) is therefore employed as a

starting point and as a base on which to build. The unre-

ality of the assumption that the employee population is

stationary f r o m t h e b e g i n n i n g is nonetheless recog-

nized, and observations as to the more realistic situation

follow in Parts IV and V.

Moreover, since this paper concerns itself only with

fundamentals, complications arising from benefit

increases, death benefits, etc., are avoided by assuming

the simplest benefits possible. Unless otherwise indi-

cated, the algebraic statements and demonstrations

found in this paper are based on the following assump-

tions.

Assume a population, stationary from the moment

the pension plan is established, such that the number

attaining age x in a given year is 1~. It is immaterial to

this discussion whether the table is of the single or mul-

tiple decrement type, so long as Ix÷~ epresents the survi-

vors one year hence of the group l~. It is likewise

immaterial whether lx represents numbers of lives, or

whether it be thought of as dollars of salary; i .e. , the 1~

used in this paper can be thought of as meaning s f l~ in

cases where a salary scale (a function of age only) is

introduced.Further assume a single retirement age r, and that the

pension benefit for each life (or each 1 of salary)

reaching retirement age is 1.00 payable annually in

advance. Assume that the plan provides no death or

withdrawal benefits of any description.

102 S o c i e ty o f A c tu a r i e s 5 0 t h A n n i v e r s a r y M o n o g r a p h



o t a t i o n

Let a be youn ges t age in the se rv ice tab le, so tha t the

s ta t iona ry popula t ion i s suppor ted by lo new ent ran tsyear ly.

Le t ¢o be l imi t ing age of se rv ice tab le .

Le t C , r epresent the t th annua l cont r ibu t ion to the

pens ion p lan , payab le annua l ly in advance . Supersc r ip ts

to the le f t ind ica te the funding m e thod un der cons ide r -

a t ion . Fo r exam ple EA~c represents the f i r s t cont r ibu-

tion u nde r entry age norm al, an d AC** represen ts the

ul t ima te cont r ibu t ion f f aggrega te funding i s used .

Le t F , r epresent the fund (or r e se rve ) bu i l t up a f te r

t yea r s (be fore cont r ibu t ion or bene f i t s then due ) . Aga in

supersc r ip ts ind ica te funding me thod.

III D escription and Classif icat ion o f

Funding Methods

lass Funding

Under the scheme of c la ss i f ica t ion prev ious ly

desc r ibed , C lass I i s log ica l ly a ss igned to wha t i s com -

mo n ly k n o w n a s p a y as y o u g o f u n d in g . N o c o n l ri b u -

t i o n s a r e ma d e t o t h e p l a n b e y o n d th o se imme d ia t e ly

necessa ry to m ee t bene f i t paym ents f a l l ing due . Contr i -

butions (PCt) are exac tly equal to ben efits for a l l valuesof t , and PF is zero fo r a l l values o f t.

S ince the in i t ia l ly ma ture popula t ion prev ious ly

desc r ibed produces cons tan t bene f i t paym ents , pay-as-

you -go funding for such a group produces leve l cont r i -

butions equal to ,o

]~ lx .r

lass H Funding

f f n o f u n d in g w h a t so e v e r i s c o n t e mp la t e d f o r a c ti v e

l ives, bu t i f the presen t va lue o f fu ture pens ion bene f i ts

i s cont r ibu ted for each l i f e a s i t r eaches re t i r ement , weh a v e w h a t h a s c o m e to b e k n o w n as t e r min a l f u n d in g .

S ince th is me thod produces h ighe r eventua l cont r ibu-

t ions and low er eventua l r e se rves than any of the o the r

c o mmo n me th o d s e x c e p t C la s s I , t e r min a l f u n d in g i s

assign ed to Class IT.

When te rmina l funding i s appl ied to an in i t ia l ly

matu re popula t ion , a l l cont r ibu t ions except the f i rs t a re

equa l and can be quant i ta t ive ly expressed a s l~ r . The

pr inc ip le of fu l l funding for a l l r e t i r ed l ives r equi re s ,

however , tha t the f i r s t cont r ibu t ion be cons ide rab ly

grea te r to fund the bene f i t s o f those a l ready beyond

re t i r emen t age a t the t im e the p lan i s inaugura ted . Theinit ia l contr ibution is in fact

0

r c i = ]~ lx a~ and exceedsr

the ult im ate level contr ibu tion TC** = l~ , by ]~ Ix t~.

r + l

This ex t ra cont r ibu t ion in the f i r s t yea r a r i se s

b e c a u se t h e p la n w a s n o t a lw a y s i n e x i s t e n c e b ut c a me

in to be ing a f te r ce r ta in ind iv idua ls had a l read y re t i red .

Here we f ind the f i rs t sugges t ion of norm al cos t and

acc rued l iab i l ity , two concepts f r equent ly em ploy ed

in the pen s ion bus iness .N o r m a l C o s t i s c o mm o n ly u n d e r s to o d t o me a n th e

l e v e l o f c o n t r i b u t i o n w h ic h a f u n d in g me th o d w o u ld

cur ren t ly produce , were i t no t for a la te s ta r t in paying

for benefits . A c c r u e d L i a b i l i t y me a su r e d a t a n y t ime ,

represents the d i f fe rence be tw een the then presen t va lue

of fu ture bene f i t s and the present va lue of fu ture norm al

cos ts . The por t ion o f the acc rued l iab i li ty no t of f se t by

assets is called the u n f u n d e d a c c r u e d l i a b i l i t y . T h e

acc rued l iab il i ty , when m easured a t the e s tab l i shment o f

the p lan , is com mo nly re fe r red to a s the i n i t ia l a c c r u e d

l iabi l i ty .

Un der C lass I I o r te rmina l funding appl ied to an in i -t ia l ly ma ture group we have seen tha t normal cos t i s

represented by l ,~ , , and the in i t ia l acc rued l iab i l i ty by

co

]~ Ix t~ . The acc rued l iab il i ty does not cha nge wi thr + l

t h e p a s sa g e o f t ime i f t h e g r o u p i s ma tu r e f r o m th e

beginning . On ce the acc rued l iab i l i ty has been pa id of f ,

lrar

r + l

a n d th e f u n d a me n ta l E q u a t io n o f Ma tu r i t y c a n b e

c h e c k e d o u t b y t h e i d e n t it y

l r~ir d Z lxax - ~ lx •

r+l r

Note tha t TF ,, the u l t im a te re se rve bui l t up , and the

acc rued l iab i l ity a re , a s w e m ight expec t , a lgebra ica l lyidentical .

H . F u n d a m e n t a l s o f P e n s i o n F u n d i n g 103



l ass I I I F un d i n g

T h e s o - c al l ed u n i t c r e d i t o r s i n g l e p r e m i u m

m e t h o d o f f u n d i n g i s t h e f i rs t m e t h o d h e r e c o n s i d e r e dt h a t f u n d s i n a n y r e s p e c t f o r e m p l o y e e s n o t y e t r e t ir e d .

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

o d s y e t t o b e c o n s i d e r e d , i t i s h e r e c l a s s i fi e d a s C l a s s H I .

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

p e n s i o n t o b e p r o v i d e d a t r e t i re m e n t a g e w i l l b e d i v i d e d

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

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

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

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

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

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

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

t h e a c c r u e d l i a b i l i t y i s o f t e n r e f e r r e d t o a s t h e p a s t s e r-v ice l i ab i l i ty .

T o t h e e x t e n t p r a c t i c a b l e t h e u n i t s a s s i g n e d t o v a r i -

o u s y e a r s a r e e q u a l i n a m o u n t . F o r a n y i n d i v i d u a l ,

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

v a l u e o f a d e f e rr e d a n n u i t y c o m m e n c i n g a t a g e r i s an

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

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

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

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

r - I

1 y l . . . { ? i xr a

a

T h e a c c r u e d l i a b i l i t y is

r I to

r ± a ~ X - a ) t . . . . ? i x + ~ t x ? i x •a r

U n d e r t h i s m e t h o d o f f u n d i n g t h e i ni ti al c c r u e d l i a

b il it y c a n b e p a i d o f f i n a v a r i e t y o f w a y s . A c o m m o n

m e t h o d i s t o a m o r t i z e t h e l ia b i l it y b y m e a n s o f a n a n n u -

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

p a y m e n t b e c o m i n g k % o f t h e i n i t i a l a c c r u e d l i a b i l i t y ,

w h e r e k = 1 0 0 / ? i ~ . A r e q u i r e m e n t in s o m e p l a n s

u s i n g u n i t c r e d i t fu n d i n g i s t h a t t h e a c c r u e d l i a b i l i t y a s

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

v i d u a l r e t i re s . I n a n y c a s e , o n c e t h e a c c r u e d l i a b i l it y i sf u l l y fu n d e d

UC= 1 ~-~r _ a E l . . . I?ix

a

a n d

U F . = 1r~ l to

r _ a E X - a ) l . . . . ] ? i X E l x a x •

a r

O n c e a g a i n t h e u l t i m a t e f u n d a n d t h e a c c r u e d l i a b i l -

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

T h e a l g e b r a i c i d e n t i t yr - |

1 lr _ a ~ . . . . [ ? i x

a

r - I t o t o

d F l 7 . . . . [? ix+~lx? ix] - -Y f fXr

i s, o f c o u r se , a n e x p r e s s i o n o f th e E q u a t i o n o f M a t u r i t y

a p p l i e d t o C l a s s I I I fu n d i n g . N o t e t h a t i t is a l s o a n a l g e -

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

o f f , b u t i n s t e a d i s a m o r t i z e d i n p e r p e t u i t y b y p a y i n g

i n t e r e s t a l o n e , u n i t c r e d i t f u n d i n g f o r a n i n i t i a l ly m a t u r ep o p u l a t i o n d e g e n e r a t e s i n t o p a y a s y o u g o .

l a ss I V F un d i ng

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

c la s sed toge the r , because we wi l l s ee tha t once the u l t i -

m a t e c o n d i t i o n h a s b e e n r e a c h e d t h e s e m e t h o d s p r o d u c e

iden t ica l con t r ibu t ions and bu i ld up iden t ica l r e ser ves .

1 Entry Age Normal Method

T h i s m e t h o d , a s i t s t i t l e i m p l i e s , v i s u a l i z e s t h e n o r -

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

f i t o v e r t h e w o r k i n g l i f e t i m e o f s u c h e m p l o y e e • T h e n o r -

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

a g e a i s t h e r e f o r e

r-~l?io

aa:~_--:-

T h e a c c r u e d l i a b i l i t y a s to a n y i n d i v i d u a l a g e x ( x < r ) i s

r-~l?ia ..r- x ]?ix ?ia.~--:-~ x:r--:~

I f w e l o o k a t th e g r o u p i n s t e a d o f t h e i n d i v i d u a l , w e

f ind the acc r ued l i ab i l i ty i s

r-I ca . . r-I

Z I . . . . ]? i, +Z l x i ~ x ~ - a l a a z l x ? i x ~ _ - : - ~ .a a

a r a

W h e n t h i s la s t e x p r e s si o n i s w r i t t e n i n t h e f o r m

to - ' ( , - o l ? i o . . ~~_flx?ix + ~.~ l . . ., I?ix ..-- - - ax:~---~

aa:r_--: ~ /r a

104 S o c i e ty o f A c t u a r i e s 5 0 t h A n n i v e r s a r y M o n o g r a p h



i t is a p p a r e n t t h a t t h e i n i t ia l a c c r u e d l i a b i l i t y is s i m p l y

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

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

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

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

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

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

c o m p u t e d ) a t a g e a . T h e n o r m a l c o s t ,

r -~ iao

a a : r - ' : '~

f o r e a c h a c t i v e l i fe , i s o f c o u r s e

r-oltia ~-, .- z _ ~ i x

aa:r_- '= a

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

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

l e v el p a y m e n t s f o r a fi x e d n u m b e r o f y e a rs . T h e r e m a y

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

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

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

c o m p l e t e l y l i q u i d a t e d , E ~ C = i s t h e n o r m a l c o s t

I~ r I. . ' 5 7 - ' -aa:r__.: ~ ~ x

°

a n d

0 3 r - I

- ~ . d l x ( i x + I . . . . l a x . ' = - - a x : r - - -: ~ I •

r 7 \ ao:r-- :~

Once again the u l timate fund under the r ig id c ondi-t ions imposed becomes ident ica l wi th the unchanging

accrued l iab i l i ty. Once a gain an a lgebraic iden t i ty

• r i

r -~ la°z txaa:r_-: ~ a

i o r , : °a ~, tx i ix E l l x ..--------axr_-~ll----21x

r a a : ~ , / 1 r

p r o v e s o u t t h e E q u a t i o n o f M a t u r i t y , a n d a t t h e s a m e

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

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

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

2 . I n d i v id u a l L e v e l P r e m i u m F u n d i n g

A s e c o n d C l a s s I V m e t h o d f u n d s t h e b e n e f it s a s to

a n y i n d i v i d u a l f r o m d a t e o f e n t r y ( o r d a t e p l a n i s e st a b -

f i s h e d , i f l a t e r) t o r e t i r e m e n t d a t e a s a l e v e l a m o u n t ( o r

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

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

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

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

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

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

h i s f u t u r e w o r k i n g l i f e t i m e , o r i n e x a c t l y t h e s a m e m a n -

n e r a s t h e n o r m a l c o s t .

F o r a n i n d i v i d u a l a g e x w h e n t h e p l a n i s i n a u g u r a t e d ,

i n d i v i d u a l l e v el p r e m i u m f u n d i n g r e q u ir e s a p a y m e n t o f

r -x[ax

ax.r--:~

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

v i c e . B u t n o t e t h a t s i n c e

r-x[a~

a x : r - -: -~

c a n b e e x p r e s s e d a s

r -olao r - x l a . . . l a o )_ _ a ,. .7(ia:r_--:-~ \ ~ix:r_- ' :~ a a ~ '

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

v i e w e d a s t h e n o r m a l c o s t ( i . e . , t h e c o s t f o r n e w

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

r x l a . . . . i a o

a x . r _ - - a a : r _ - - :~

E x t e n d i n g t h i s c o n c e p t t o t h e e n t i r e p o p u l a t i o n , w e

s e e t h a t t h e i n i t ia l c o n t r i b u t i o n t o t h e p l a n i s s i m p l y

r - I

~ c , = ~ t x T x l a x + y : : ~a x : r _ - : ~°

r I a co

r - ° l a ° ~ / x 7 . / . .- - - -- | t~ 2 . l : x~o~_-~ o 7 ~ ao r---~: r

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

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

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

W e f i n d t h e s i t u a t i o n t y e a r s a f t e r t h e i n a u g u r a t i o n o ft h e p l a n t o b e a s f o l l o w s :

' c , . , = l r - ' l a x - 'x - - -- -- - o l a o

a + t + l a x - t : r - x + t l a a a r -- : ~

l a , - i r - , : ~ r _ o l a o- a I a t ~ . ~ I r - x + t l ~ x - t

= .. 2 . ~ G + 2 ~ I . . - - ~ Ia a ' r - - : ~ a a + t + l \ a x - t r - x + t l a a . r - - ': ~ /

I I. F u n d a m e n t a l s o f P e n s i o n F u n d i n g 105



T h e n o r m a l c o s t r e m a i n s l e v e l b u t t h e a c c r u e d l i a b i l-

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

a f t e r r - a year s ac crued l i ab i l i t y i s a l l pa i d o f f .

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

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

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

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

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

i n it i a l p e n s i o n e r s . I t c a n o f c o u r s e b e d e m o n s t r a t e d t h a t

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

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

i n i t ia l ac crued l i ab i li t y .

3 . A g g r e g a t e F u n d i n g

T h e p r i n c ip l e b e h i n d t h e a g g r e g a t e m e t h o d i s t h a t o f

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

p r e s e n t v a l u e o f f u t u r e c o n t r i b u t io n s , w h e r e t h e c o n t r i -

bu t i on per ac t i ve l i f e (o r per do l l a r o f sa l a ry ) per ye ar i s

a s s u m e d c o n s t a n t . I t m a y s e e m a t f ir s t t h o u g h t t h a t t h e

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

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

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

l eve l l y over fu t u re l i f e year s .

T h i s s u p p o s i t i o n r e g a r d i n g t h e a g g r e g a t e m e t h o d i s

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

t a k e n i n t o a c c o u n t , b o t h i n v a l u i n g p r e s e n t v a l u e o f

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

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

g a t e c o n t r i b u t i o n u n d e r o u r r i g i d c o n d i t i o n s i s e x a c t l y

0}

~ l ~

r

w h i c h w e r e c o g n i z e a s t h e p a y - a s - y o u - g o p a y m e n t .

S i n c e t h e c o n t r i b u t io n j u s t e q u a l s t h e b e n e f i ts , n o

r e s e r v e s b u i l d u p a n d c o n t r i b u t i o n s c o n t i n u e t o d u p l i -

ca t e t he l eve l C l ass I con t r i bu t i on .

T h e c o m m o n u s e o f t h e a g g r e g a te m e t h o d , h o w e ve r ,

i g n o r e s n e w e n t r a n t s . T h e e f f e c t , o f c o u r s e , i s t o s u b -

t r ac t v / d l . . . . l a , f r o m t h e n u m e r a t o r a n d v /dloi iof r o m t h e d e n o m i n a t o r o f e q u a t io n ( 1 ) o f D e m o n s t r a t i o n I.

S i nce , w here A , B , C , and D are pos i t i ve cons t an t s , i f

A C~>~

t h e nA A C

> - -

B B + D

i t f o l l ow s t ha t AC1 (new en t r an t s d i s r ega rded ) i s g rea t e r

t h a n t h e l e v e l p a y - a s - y o u - g o p a y m e n t i f

r I CO

E l . . . . l a ~ Z / ~ a ~a r ~

r - I

~ lxiix:r_--:-~a

~_o[iio

i a : r - - ~

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

p r i nc i p l e .

T h e i g n o r i n g o f n e w e n t r a n t s t h e r e f o r e p r o d u c e s , i n

t he f i r s t year , a con t r i bu t i on i n excess o f benef i t s , and

s t a rt s t h e a c c u m u l a t i o n o f a r e s e rv e .

I n a n y y e a r t h e r e a f t e r

r I ¢o A

Z l . . . [t~x+ Z l , ti~ - F , - Ir I

AC, = o r ~ , Ix .r l

~ lxiix.r__:~ °a

A s F t i ncreases , AC, decreases . I t can be show n t ha t as

AC, decreases , t he i nc rem en t t o AF,,

d e c r e a s e s . T h e f u n d c o n t i n u e s t o i n c r e a s e , b u t a t a

s l ower and s l owe r r a t e , so l ong as AAF, i s pos i t ive , i.e.,

s o l o n g a s

ol

A C , > ~ l x - - d A F , .

r

I t i s s h o w n i n D e m o n s t r a t i o n I I t h a t u n d e r t h i s p r o-

cess AC, app roa ches asym pt o t i ca l l y it s l i mi t

AC . = r-°la°. , -~l~a a ~

a

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

C l a s s I V m e t h o d s . S i m i l a r ly A F a p p r o a c h e s , b u t n e v e r

r eaches , a l i m i t i den t i ca l t o E~ 'NF. and neF® The agg re-

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

accrue d l i ab i l i t y is pa i d o f f r a t her r ap i d l y a t t he beg i n -

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

accru ed l i ab i l i t y is com pl e t e l y pa i d o f f on l y a t in f i n i t y .

I f f o r i n s ta n c e t h e a v e r a g e t e m p o r a r y a n n u i t y y ( s e e

De mo ns t r a t i on H) i s 100 / k , t he f i rs t pay m en t t oward t he

accrue d l iab i l i t y i s k o f t he accru ed li ab i li t y . La t e r

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

106 Society o f Actuaries 50th Anniversal~y Mo nograph



acc rued l i ab i l i t y . Compare the foregoing wi th k% fund-

ing of the acc rued l i ab i li t y under en t ry a ge normal ,

wh ere the k% appl i e s to the fu l l acc rued l i ab i l i ty ra the r

than to the unfunde d por t ion o n ly . t

I t can be demonstra ted that the ini t ia l contr ibut ion

under the aggregate method is general ly lower than that

under individual level premium funding. A temporary

annuity ax.,_--:-~ which decreases with advancing age is a

suff icient, but not necessary, condi t ion fo r AC < ~ C I . I f ,

due to heavy wi thdrawa l a s sumpt ions a t young ages ,

//~ ~ increa ses throug h a significant portion of its range,

there m ay b e rare ex cept ions to the general re la t ionship.

4 A t ta i n e d A g e N o r m a l

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

a t t a ined age norm al me tho d , a r is ing f rom ce r ta in

Clas s HI cha rac te r i s ti c s in w ha t i s e s sen t i a l ly a C las s IV

method .

Tota l bene f i t s a re d iv ided in to pas t s e rv ice and fu ture

se rv ice bene f i t s exac t ly a s under un i t c red i t funding ,

and as under Clas s HI funding the re i s comple te f ree -

dom as to the manner in which the pas t s e rv ice l i ab i l i t y

sha l l be pa id of f . For fu ture s e rv ice bene f i t s , however ,

the aggrega te me thod i s adopted .

The f i r s t yea r cont r ibu t ion toward fu ture s e rv ice

b e c o m e s

1 r I

r _ a ~ r - x ) l . . . Ia x - I~ , i x .

r t

X lxiix:~_.mm oa

Since th i s amount i s somewha t h ighe r than the Clas s HI

n o r m a l c o s t

1 r I

r _ a Z l . . . . l a x

a

(which i s l eve l under our in i t i a l ly ma ture popula t ion

as sumpt ions ) , i t is appa ren t tha t fu ture s e rv ice cont r ibu-

t ions under a t t a ined age normal a re o f a dec reas ing

nature .Future s e rv ice cos t s a f t e r the f i r st yea r a re com mo nly

ca lcu la t ed in the form

r I

I l ax + ~ l ~ t i ~ - Unfunded pas t NF

servic e liability - ,- t_ i

r I

a

We perhaps ge t a be t t e r i dea of the e s sen t i a l cha rac -

t e r is t i c s o f a t t a ined ag e normal , howev er , i f we expres s

the t th fu ture s e rv ice cont r ibu t ion in the iden t i ca l fo rm

1 r I

r - _ a ~ r - x ) I . . . l a ~ - f - i

r I

X lxiix:~_-r-:ma

r 1

a

wh ere f , i s t ha t por t ion of AANF bui l t up by the a ccum u-

la t ed exces s (wi th in t e res t ) o f the a t t a ined age normal

fu ture s e rv ice cont r ibu t ion o ve r the un i t c red i t one .

As f , g rows the a t t a ined age norm al fu ture cont r ibu-

t ion dec reases . I t can be show n tha t f , approache s a s a

l imi t t he amount by which Clas s IV acc rued l i ab i l i t yexceeds the Clas s HI acc rued l i ab i l i t y , and tha t i f t he

in i t i a l pas t s e rv ice l i ab i l i t y i s comple te ly l iqu ida ted

A ~ C t and AAr~F have a s l imi t s ~ C ~ and EANF. re spec -

tively.

At t a ined age normal i s t he re fore a t rue Clas s IV

method . I t s acc rued l i ab i l i t y is ac tua l ly a s g rea t a s under

the o the r Clas s IV methods , bu t a t t a ined age normal

looks a t t he acc rued l i ab i l i t y in two pa r t s . The me thod

impo ses no re s tr i c t ions a s to how the pas t s e rv ice

pa r t , equa l in magni tude to the Clas s HI acc rued l i ab i l -

i ty , sha ll be funded . T he s econd p or t ion i s l iqu ida ted b y

the dec reas ing acc ru ed l i ab i li t y payment s , w hich a re the

exces s of the fu ture s e rv ice cont r ibu t ion ove r the u l t i -

ma te fu ture s e rv ice cont r ibu t ion . S imi la r i ty w i th the

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

the aggrega te me thod a l l acc rued l i ab i l i t y i s l i qu ida ted

by r ig id dec reas ing payment s , under a t t a ined age nor -

ma l on ly a por t ion of the acc rued l i ab i li t y is so funded

and the funding as to the remain ing acc rued l i ab i l i t y i s

unspec i f i ed .

l a ss V F un d i n g

B e y o n d t h e v a r i o u s v a r i a t i o n s o f C l a s s I V f u n d i n g

prev ious ly d i s cus sed , t he re i s no th ing of a p rac t i ca ln a t u r e , b u t f u n d i n g m e t h o d s w h i c h p r o d u c e h i g h e r

eventua l re se rves and lower eventua l cont r ibu t ions than

any of the me tho ds so fa r d i s cus sed a re , o f course , t heo-

re t i ca l ly pos s ib le . Pe rha ps the s im ples t o f these i s in i t ia lf u n d i n g , w h e r e a n e m p l o y e e ' s b e n e f it s a r e f u ll y f u n d e d

as soon as he i s h i red .

H Fundam enta l s o f Pens ion Funding 107



H e r e n o r m a l c o s t is 1 . . . l a a , a c c r u e d l i a b i l i t y a n d

t F . . a r e b o t h

r - I

Z l . . . . l a x d¢ E l x a . x '

a l r

a n d E q u a t i o n o f M a t u r i t y i s ex p r e s s e d b y

r-~ o, ] ~

I . . . l a o d [ ~ l . . . . [ a ~ + r ~ l x a x - Ix

~ a + 1 r

l a ss V I F u n d i n g

E v e n l e ss p r a c t ic a l t h a n C l a s s V , b u t i n c l u d e d h e r e

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

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

a n o t h e r a n a c c r u e d l i a b i l i ty o f

co

1 / d ~ l x i s fu l l y p a i d o f f , i n t e r e s t o n t h e f u n d s b u i l t

r~0

u p w i l l e x a c t l y m e e t t h e b e n e f it p a y m e n t s ~ _ l ~ .

r

I l l u s tr a t i o n o f n i t i a l l y M a t u r e S i tu a t i o n

I t m a y b e e n l i g h t e n i n g t o i l l u s tr a te t h e f o r e g o i n g d i s -

c u s s i o n o f t h e o p e r a t io n o f th e v a r i o u s f u n d i n g m e t h o d s

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

b y m e a n s o f a n u m e r i c a l e x a m p l e . T a b l e I sh o w s t h e 1

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

u p o f e x a c t l y 1 , 0 0 0 a c t iv e a n d 1 5 0 r e t ir e d l i v e s, m a i n -

t a i n e d b y 1 0 0 n e w e n t r a n t s e a c h y e a r al l a g e 3 0 . E a c h

y e a r 1 o f t h e a c t i v e l i v e s r e t i re a n d 9 d i e o r w i t h -

d r aw . T h e c o m b i n e d r a te o f d e a th a n d w i t h d r a w a l i s

1 6 a t a g e 3 0 , a n d a p p r o x i m a t e l y e q u i v a l e n t t o t h e

S t a n d a r d A n n u i t y T a b l e q ~ a t r e t i r ed a g e s .

T A B L E I

x x x x x x0 . . . . . . . . . . . .

3 1 . . . . . . . . . . . .

3 2 . . . . . . . . . . . .

3 3 . . . . . . . . . . . .

3 4 . . . . . . . . . . . .

3 5 . . . . . . . . . . . .

3 6 . . . . . . . . . . . .

3 7 . . . . . . . . . . . .

3 8 . . . . . . . . . . . .

3 9 . . . . . . . . . . . .

4 0 . . . . . . . . . . . .

4 1 . . . . . . . . . . . .

4 2 . . . . . . . . . . . .

4 3 . . . . . . . . . . . .

4 . . . . . . . . . . . .

4 5 . . . . . . . . . . . .

4 6 . . . . . . . . . . . .

4 7 . . . . . . . . . . . .

4 8 . . . . . . . . . . . .

4 9 . . . . . . . . . . . .

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

5 1 . . . . . . . . . . . .

5 2 . . . . . . . . . . . .

5 3 . . . . . . . . . . . .

5 4 . . . . . . . . . . . .

1 0 0

8 4

7 1

6 0

5 1

4 4

4 0

3 6

3 4

3 2

3 0

2 8

2 7

2 6

2 5

2 4

2 3

2 2

2 1

2 0

1 9

1 8

1 7

1 6

1 5

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

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

5 7 . . . . . . . . . . . .

5 8 . . . . . . . . . . . .

5 9 . . . . . . . . . . . .

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

6 1 . . . . . . . . . . . .

6 2 . . . . . . . . . . . .

6 3 . . . . . . . . . . . .

4 . . . . . . . . . . . .

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

6 6 . . . . . . . . . . . .

6 7 . . . . . . . . . . . .

6 8 . . . . . . . . . . . .

6 9 . . . . . . . . . . . .

7 0 . . . . . . . . . . . .

7 1 . . . . . . . . . . . .

7 2 . . . . . . . . . . . .

7 3 . . . . . . . . . . . .

7 4 . . . . . . . . . . . .

7 5 . . . . . . . . . . . .

7 6 . . . . . . . . . . . .

7 7 . . . . . . . . . . . .

i 7 8 . . . . . . . . . . . .

i 7 9 . . . . . . . . . . . .

1 4

1 3

1 3

1 2

1 2

11

11

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1 0

1 0

1 0

1 0

9

9

9

8

8

8

7

7

7

6

6

6

5

0 . . . . . . . . . . . .

8 1 . . . . . . . . . . . .

8 2 . . . . . . . . . . . .

8 3 . . . . . . . . . . . .

8 4 . . . . . . . . . . . .

8 5 . . . . . . . . . . . .

8 6 . . . . . . . . . . . .

8 7 . . . . . . . . . . . .

8 8 . . . . . . . . . . . .

8 9 . . . . . . . . . . . .

9 0 . . . . . . . . . . . .

9 1 . . . . . . . . . . . .

9 2 . . . . . . . . . . . .

9 3 . . . . . . . . . . . .

9 4 . . . . . . . . . . . .

9 5 . . . . . . . . . . . .

1 0 8 Socie ty of Actuaries 50th Anniversary Monogra ph



Table I I i l l us t ra t es the yea r ly cont r ibut ion and the

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

metho ds , a s sum ing 2V2% in te res t , a benef i t o f $420

annua l ly , and the s t a t ionary pop ula t ion o f Table I .T w e n t y y e a r s h a s b e e n c h o s e n a s t h e p e r i o d o f a m o r ti -

za t ion of the in i t i a l acc rued l i ab i l i ty for those funding

methods pe rmi t t ing such t rea tment .

I V M o d i f i c a t i o n s f o r I n i ti a l ly

I m m a t u re F u n d

Let us a t th i s poin t abandon one o f the r ig id as sum p-

t io n s p r e v i o u s l y i m p o s e d a n d l o o k i n to t h e c o m m o n s it -

ua t ion where the group i s n o t ini t ia l ly m ature, but is to a

grea te r or l e s s ex tent immature . For the present w e wi l lcont inue to as sume tha t a l l ac tua r i a l a s sumpt ions a re

rea l i zed , l eaving the ques t ion of ac tua r i a l ga ins and

los ses to Par t V . As w e aband on the as sum pt ion tha t the

group i s in i t i a l ly mature ( though we re t a in the concept

tha t the popula t ion wi l l eventua l ly approach a s t a t ion-

a ry condi t ion) , we rep lace the lx of the s t a t ionary popu -

la t ion by the l o f the immature pop ula t ion . As we

should expec t , t he ident i t i e s expres s ing the Equa t ion of

Matur i ty do not hold a f t e r th i s subs t i tu t ion unt i l such

t ime as the popu la t ion has becom e mature and the l~' s

approach the l~ ' s. Mo reover , w e f ind tha t the conc lu-

s ions previous ly reached for the in i t i a l ly mature fund

mu s t be mod i f i ed in s evera l o the r re spec t s .

N o rm a l C o st s N o L o n g e r L ev el

I f t he in it i a l group i s immature i t fo l lows tha t C las s I

funding wi l l p roduce cont r ibut ions which a re in i t i a l ly

very low, but which inc rease ra the r rap id ly , eventua l ly

leve l ing of f when m atur i ty o f the group i s a t t a ined .

C las s I I funding requ i res cont r ibut ions which t end to

f luc tua te ra the r wide ly as number of re t i rement s va ry

f rom year to yea r . Moreover , benea th th i s e r ra t i c i sm of

cont r ibut ions i s an under ly ing t endency for cos t s t o

i nc rease , s ince as the group matures the n um ber re t i ringeach y ea r t ends to grow, even i f t he s i ze of the s t a f f a s a

wh ole remains s t a tionary .

The normal cos t for C las s HI or uni t c redi t funding

( for a g iven s t a f f and benef i t s ) remains cons tant i f ac tu-

axial assumptions are real ized, and i f t h e a v e r a g e a g e o f

the ac t ive s t a f f doe s not change . The average age h e re

meant i s not the s imple a r i thmet i c mean, but the age

c o r r e s p o n d in g t o t h e w e i g h t e d a v e r a g e s i n g le p r e m i u m

defe r red annui ty , where the s ingle premium a t each age

i s we ighted by uni t s be ing funded a t such age . I f t he

group i s in i t ia l ly immature , however , i t i s ax iomat i c tha t

th i s average age wi l l s lowly inc rease and normal cos t s

wi l l s lowly r i s e be for e eventua l ly l eve l ing of f . Thi s r i sem a y b e p r o n o u n c e d i f t h e g r o u p i s u n u s u a l l y y o u n g a t

the es t ab l i shment of the p lan .

The poss ib i l i ty of inc reas ing n ormal cos t s , even i f a ll

ac tua r i a l a s sumpt ions a re rea l i zed , i s not e l imina ted

under C las s IV funding . The expec ted inc rease in aver -

age age of the ac t ive l i fe group wi l l no t , i n i t s e l f , p ro-

duce inc reas ing normal cos t s . Leve l normal cos t s do ,

h o w e v e r , d e p e n d u p o n t h e a v e r a g e a g e o f n e w e n t ra n t s

i n t o t he p lan . I f t h is average ent ry age remains cons tant

and o the r ac tua r i a l a s sumpt ions a re rea l i zed , normal

cos t s wi l l remain cons tant (a s suming s t a f f and benef i t

l eve l s do not change) . Aga in th i s average ent ry age i snot a s imple a r ithmet i c mean, bu t the age cor responding

t o t he we ighted average l eve l premium where the l eve l

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

those en te r ing a t such age .

A c c r u e d L i a b i l i ty N o L o n g e r C o n st a n t

W e h a v e p r e v i o u s l y s e e n t h at u n d e r t h e a s s u m p t i o n

of an in i t i a l ly mature popula t ion the acc rued l i ab i l i ty

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

n o t c h a n g e w i t h t h e p a s s a g e o f t im e . t t akes no mathe-

mat i ca l demons t ra t ion to convince us tha t , i f t he popu-

lat ion is ini t ia l ly immature, the accrued l iabi l i ty wil l

r i s e a s the popula t ion grow s o lde r.

As a coro l l a ry we f ind tha t the funds wi l l g row

bey ond the in i t ia l acc rue d l i ab i l i ty up to the l eve l of the

ul t ima te acc rued l i ab i l i ty (as suming in i t i a l acc rued l i a -

b i l i ty i s comple te ly funded) . The excess of the u l t ima te

over the in i t i a l acc rued l i ab i li ty i s bui l t up by the ea r ly

year exces s of normal cos t s p lus in t e res t on the in i t i a l

acc rued l i ab i l i ty ov er benef i t paym ents .

N o rm a l Cost P l u s n t e r est on A c c r u ed

L i a b i l i t y N o L onge r den t i c a l t o Pay A sYou Go

We found ea r l i e r tha t for an in i t i a l ly mature group a

cont r ibut ion equa l to normal cos t p lus in t e res t on the

in i ti a l acc rued l i ab i li ty was ex ac t ly equa l to benef i t pay -

ment s ; accordingly no funds were bui l t up and C las s I

funding resu l t ed . Thi s was t rue regard les s of whe ther

normal cos t s and acc rued l i ab i l i ty were those of C las sII , I / I , IV, V, orV I.

H F u n d a m e n t a l s o f P e n s i o n F u n d i n g 109



T A B L E I I

C lass I C l ass I I C l ass HI C lass IV Class V Class VI

Pay A s

You GoTerminalFunding

Unit Credit20 Yr.

Entry AgeNormal 20

Yr.

IndividualLevel Prem.

Attained Age Initial Fund-Aggregate Normal 20 ing

Yr. 20 Yr.

In i ti a l Accru ed L i ab i l i t y

CompleteFunding 20

Yr

No n e ]$ 5 0 2 ,1 0 4 $ 1 ,2 0 6 ,9 2 4 I $ 1 ,4 7 1 ,8 7 3 $ 1 ,4 7 1 ,8 7 3 $ 1 ,4 7 1 ,8 7 3 $ 1 ,4 7 1 ,8 7 3 ] $ 1 ,7 0 6 ,1 7 3 ] $ 2 ,5 8 3 ,0 0 0

N o r m a l C o s t

B e g . o fY e a r

$ 6 3 ,0 0 0 J$ 5 0 ,7 5 3 $ 3 3 ,5 6 3 I 5 2 7 ,1 0 1 $ 2 7 ,1 0 1 $ 2 7 ,1 0 1 , $ 2 7 ,1 0 1 j $ 2 1 ,3 8 6 j N o n e

Co n t r i b u t i o n s

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I f t he o r i g in a l g r o u p i s i m m a t u r e t h e p a y m e n t o f n o r -

r e a l c o s t p l u s i n t e r e s t o n l y o n t h e i n i t i a l a c c r u e d l i a b il -

i t y d i f f e r s f r o m p a y - a s - y o u - g o i n t w o r e s p e c t s : ( 1 ) t h e

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

i n c r e a s i n g , a n d ( 2 ) a f u n d i s b u il t u p , a t a n y t i m e t b e i n g

e q u a l i n a m o u n t t o t h e e x c e s s o f t h e a c c r u e d l i a b i l it y a t

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

d i f f e re n c e s t h e a u t h o r p r e f e r s t o c o n s i d e r t h e s e m e t h o d s

c o n t e m p l a t i n g n o a m o r t i z a t i o n o f t h e in i t ia l a c c r u e d l i a -

b i l i ty a s C l a s s I m e t h o d s .

110 Society of Actuaries 5 th Anniversary Monograph



Gene ra l Rel a t i onsh i p B etween N o rm a l

Cost and c cr u ed L i a bi l i t y M ay N o t H o l d

I n P a r t 111 w e f o u n d t h a t , i f t h e p o p u l a t i o n w a s i n i -t i a l l y m a t u r e , t h e f u n d i n g m e t h o d p r o d u c i n g t h e h i g h e r

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

v i c e v e r s a . I n t h e i n i t i a l l y i m m a t u r e c a s e t h i s g e n e r a l

r e l a t i o n s h i p m a y n o t h o l d i m m e d i a t e l y . F o r e x a m p l e ,

t h e C l a s s I p a y - a s - y o u - g o p a y m e n t m a y b e i n i t ia l ly

l o w e r t h a n f ir st y e a r t e rm i n a l f u n d i n g n o r m a l c o s t , e v e n

t h o u g h C l a s s I I f u n d i n g p r o d u c e s a n a c c r u e d l ia b i li t y

a n d C l a s s I h a s n o n e . T h e e x p l a n a t i o n i s, o f c o u r s e , t h a t

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

o t q u i t e s o o b v i o u s i s t h e s i t u a t io n w e f i n d i f t h e

u n i t c re d i t m e t h o d o f f u n d in g , a p p l i e d t o a g i v e n g r o u p ,

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

n o r m a l ( a r e s u lt o n e w o u l d e x p e c t ) a n d y e t t u r n s u p a

l o w e r i n i t i a l n o r m a l c o s t a s w e l l . S u c h a r e s u l t i s d u e t o

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

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

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

t e m p o r a r y f e a t u r e o n l y , a n d t h e p r e s e n t v a l u e o f a ll n o r -

m a l c o s t s i s h i g h e r u n d e r C l a s s 111 f u n d i n g , e v e n t h o u g h

t h e n o r m a l c o s t in e a r l y y e a r s m a y b e l o w e r .

I l l u st r a ti o n o f n i t i aUy mm a t u r e

S i tua t i on

B y c h a n g i n g t h e p r e c e d i n g i l l u s t r a t i o n s o m e w h a t w ec a n m a k e a g o o d n u m e r i c a l r e p r e s e n t a t i o n o f th e c o u r s e

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

i m m a t u r e s i t u a t i o n . T a b l e I l l r e p r e s e n t s a n i m m a t u r e

p o p u l a t i o n o f 1 , 0 0 0 a c t i v e l i v e s , w i t h n o r e t i r e d l i v e s

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

e x a c t l y i n a c c o r d a n c e w i t h t h e s e r v i c e t a b l e i l lu s t r a t e d

i n T a b l e I , a n d i f su f f i c i e n t n e w e n t r a n t s c o m e i n a t a g e

3 0 e a c h y e a r t o k e e p a c t i v e s t a f f u p t o 1 , 0 0 0 , t h e i n i-

t ia l ly i m m a t u r e g r o u p w i l l s lo w l y a p p r o a c h t h e s t a t io n -

a r y p o p u l a t i o n s h o w n i n T a b l e I.

T a b l e I V i l l u s t r a t e s t h e e f f e c t o f t h e s e v e r a l f u n d i n g

m e t h o d s u n d e r t h e s e c o n d i t io n s . T w o C l a s s I m e t h o d s

b e s i d e s p a y - a s - y o u - g o a r e h e r e il lu s t ra t e d , b o t h o f

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

r e s e r v e s t h a n C l a s s I T. B e c a u s e C l a s s V a n d V I a r e p r a c -

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

f r o m t h e i l l u s t ra t i o n .

T A B L E I H

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6 3 . . . . . . . . . . . .

4

65 and up . . .

l

H. Fundam enta l s o f Pens ion Funding 111



V Ad justmen t for Actuarial G ains

and Lo sses

In Part HI the operation of the various funding meth-

ods was described under conditions of (1) an initially

mature population, and (2) experience strictly in accor-

dance with the actuarial assumptions. The first of these

ideal conditions was abandoned in the discussion of the

initially immature population in Part IV. To complete

the transition from the ideal to the realistic, we now

abandon the second of the rigorous idea l conditions

and look into the practical situation where the actuarial

assumptions are never exact ly realized.

O r i g i n o f c tu a r i a l G a i n s a n d L o sses

The calculation of the contribution for any given

year under any funding method is always based on a set

of assumptions or estimates. As the actual experience

unfolds it is found that each of these estimates is in

error to a greater or less extent, and that these errors

give rise to what have come to be known as actuarial

gains or losses. The reader can undoubtedly enumerate

/~1o~ of the sources of gains or losses, the net effect ofwhich is the total actuarial gain or loss for any

particular period. Some of these sources may be over-

looked in thinking through pension valuation problems,

however. As an aid to clear thinking, a partial list of

sources of actuarial gains is therefore here included. Ineach case the converse represents a source of actuarial

loss. Under certain plan provisions or particular funding

media any of the following may have no effect (or even

the opposite effect). In general, however, an element of

actuarial gain arises if:

1. Rates of employee mortal ity are higher than

assumed.

2. Rates of employee withdrawal (especially nonvested

withdrawal) are higher than assumed.

3. Rate of interest earned is higher than assumed.

4. Benefits which cannot be determined exactly are

overestimated. This could arise, for example, byassuming too steep a salary scale for benefits based

on salary, or underestimating Social Security benefits

under a $100 less Social Security plan.

5. Retirements occur at a higher age than assumed.

6. The value of the pension fund assets appreciates.

7. Errors of various types, overstating the liabilities, are

corrected.

8. Provision for expenses of administration is overly

adequate.

De term i n a t i on o f mou n t o f c tu a r i a l

Ga i n o r L o ss

It is seldom practical to determine the actuarial gainor loss for a given period by summing the various com-

ponents. It is ordinarily not too difficult, however, to

obtain the total gain or loss directly. The most conve-

nient procedure for doing so depends somewhat on the

method of funding.

An approach to the computation of gain or loss

which has wide application is the comparison between

(1) funds actually on hand at the end of the period, and

(2) funds expected in accordance with the assump-

tions made. The latter is invariably the accrued liability

at the end of the period, less the expected unamor-

tized initial accrued liability, i .e. unamortized initial

liability at the beginning of the period, with interest to

end of period, less payments within the period toward

the initial accrued liability, with interest to end of

period.

Under either unit credit (Class III) or entry age nor-

mal (Class IV) the desired result is obtained without

difficulty by this general procedure.

Under Class 11 funding, where initial accrued liabil-

ity is ordinarily paid off immediately, the gain or loss is

measured by the excess of actual funds over present

value of all benefits for reti red lives.

Gains or losses under individual level premium fund-

ing c a n be obtained in exactly the generalized mannerpreviously set forth. But since under this method pay-

ments toward the initial accrued liability are not imme-

diately evident, and since expected funds are equal to

the sum of the individual level premium reserves, it i~

more convenient to compare the actual funds with the

level premium reserve. The calculation of the reserve

item is somewhat arduous, and accordingly the adjust-

ment for gains and losses is difficult under this method

of funding, except under insured plans where no losses

occur and dividends declared represent the gains.

The dollar a m o u n t of actuarial gains and losses

involves some difficulty under the aggregate and

attained age normal methods as well. The generalized

procedure previously suggested is theoretically accu-

rate, but is practically difficult because the calculations

of the payments toward the initial accured liability con-

sist of a year-by-year comparison of contributions made

with the Class IV normal cost. The redeeming feature

of both these methods, from a gain and loss viewpoint,

is that adjustment for gain and loss can be easily made

without previous determination of the absolute amount

of such gain or loss.

112 Soc ie ty o f Ac tuar ie s 50 th Anniver sary Mon ograph



T A B L E I V

Class I Class II Class HI Class IV

AttainedEntry Age Term ina l U ni tCredi t Ent ryAge Individual AgePay As Unit Cred it No rmal nt. Normal Aggregate

You Go Int. Only O n ly Fu nd ing 20 Yr. 20 Yr. Le ve l rem. Normal20 Yr.

Initial Accrued Liability

N o n e $ 4 3 1 , 9 2 4 1 5 6 6 1 , 3 1 5 N o n e $ 4 3 1 , 9 2 4 i $ 6 6 1 , 3 1 5 ~ $ 6 6 1 , 3 1 5 $ 6 6 1 , 3 1 5 $ 6 6 1 , 3 1 5

Ult imate Accrued Liabi l i ty

No ne $1,206,924 ]51,471,873 $ 502 ,104 $1,20 6,924 ~$1,471,873 151,471,873 $1,471,873 $1,471,873

Initial Normal Cost

No ne $ 26,371 ] $ 27,101 No ne $ 26,371 $ 27,101 ] $ 27,101 $ 27,101 $ 27,101

Ult imate Normal Cost

$ 63,000 $ 33,563 ] $ 27,101 $ 50,753 $ 33 ,5 63 $ 27,1011 $ 27,101 $ 27,101 $ 27,101

Beg . of ContributionsYear

....... $ ......... $ 95,5 91

10 . . . . .15 . . . . . .20 . . . . .21 . . . . . .25 . . . . . .

30 . . . . .35 . . . . . .40 . . . . . .50 . . . . . .Limit...

$ 8402,1003,5435,326

17,27030 0064 0 5 8 24 2 3 5 64 8 1 5 8

5 4 4 4 36 2 9 9 96 5 5 5 964.24963.000

36,90637,90238,77139,56240,23442,32443,43744,36744,54345,229

45,46844,01443,61243,92344,098

$ 43 ,230vt

It

ii

tt

iI

It

tt

II

tt

iv

tv

iv

Iv

43,230

10,15115,22618,45623,07039,04142,29544,13444,40945,316

55,82963,44250,36949,22750,753

$ 53,40254,39855,26756,05856,73158,82159,93360,86334,00834,694

34,93433,48033,07733,38833,563

E n d o fYear Funds

10 . . . . .

15 . . . . . .20 . . . . . .,21 . . . . . .125 .. . . . .3 0 . . .. . .35 . . . . . .

0

50 . . . . . .Limit...

N o n eit

II

II

TI

II

Iv

Iv

It

tt

tt

It

II

N o n e

68,488vv

vv

wv

tt

iv

tl

27,101it

wt

tt

tt

M

27,101

$ 1 2 6 , 4 8 8112,387101,472

92,77885,06157,23542,03234,06033,00229,971

27,90027,101

vt

it

27, 101

89 86784 68579 99575 72859 23343 33137 73036 85834 015

31 56829 94928 93027 86727 101

$ 77,88975,90374,10672,47970,99965,27761,48458,94731,52130,233

29,12528,39127,92927,44827,101

$ 37,82976 762

116 269156 094195 77738O 564

528 079638 136656 331719 556781 305803 511793 576770 124775 000

$ 44,31188,869

133,249177,260220,543417,303

570,864682,428700,385760,785815,616834,594826,569806,286810,558

N o n e$ 9,54323,23639,10358,267

178,161

288,992364,714375,937410,149454,999528,172536,121501,0025 0 2 , 1 0 4

$ 54,737111,002168,273226,307284,655569,997

831,2831,070,0601,088,2551,151,4801,213,2301,235,4351,225,5001,202,0481,206,924

70,200141,293212,873284,763356,622707,342

1,035,0961,343,7431,361,7001,422,1001,476,9311,495,9101,487,8841,467,6011,471,873

$ 129,651247,228355,265455,613548,731918,561

1,160,8171,315,8681,339,1771,413,8221,475,9051,495,910 ~1,487,8841,467,6011,471,873

$ 97,981191,683281,125366,515447,840794,067

1,090,1391,251,6911,277,3471,362,7291,438,7991,471,3781,472,0261,460,9551,471,873

$ 79,836158,772236,547313,120388,263737,424

1,039,1071,302,0391,323,4841,395,2011,459,6551,484,7961,480,6991,464,5881,471,873

H Fundamentals of Pension Funding 113



Techn ique o f Ga in o r L oss A d ju stmen t

T h e f u n d i n g m e t h o d s d i s c u s s e d in t h is p a p e r e m p l o y

o n e o r e i t h e r o f t w o t e c h n iq u e s i n d e t e r m i n in g h o wmuch cont r ibut ions wi l l be ad jus ted to recognize previ -

ous ac tua r ia l ga ins or los ses .

I mmedia te M ethod

T h e i m m e d i a t e m e t h o d m a k e s u p a n y l o s s o r o f f -

se t s any ga in , a s soon as such ga in or los s i s ev ident , by

addi t ion to or deduc t ion f rom the n ext cont r ibut ion .

Pay- as -yo u-go funding invar iab ly and autom at i ca l ly

adjus t s immedia te ly for ga in or los s . Thi s i s ev ident i f

we th ink of the cont r ibut ion (ac tua l benef i t payments )

as the sum of expec ted benef i t payments p lus ad jus t -

ment for ga in or los s . C las s I I or t e rmina l funding a l so

employs the immedia te ad jus tment t echnique .

Ful ly insured p lans ( those employing convent iona l

group annui t i e s , g roup pe rmanent , o r ind iv idua l insur -

ance pol i c i es ) in mos t cases apply any d iv idend aga ins t

the next cont r ibut ion . To the ex tent tha t the d iv idend

imm edia te ly re f l ec t s ac tua l exper i ence , ac tua r ia l ga in i s

r e c o g n i z e d a t o n c e . T h e i n s u r an c e c o m p a n y g u a r a n t e e s

e l imina te the poss ib i l i ty of ac tua r i al l os s .

Immedia te ad jus tment i s a l so theore t i ca l ly poss ib le

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

commonly used in s evera l of them. Due to the d i f f i cu l -

t i e s of ca lcu la t ion i t is s e ldom em ploy ed wi th the aggre -

ga te or a t t a ined age normal funding methods .

Spread M ethod

The spread metho d mak es the ga in or los s ad jus t -

ment in easy s t ages , by spreading the ad jus tment in to

the fu ture , such tha t the present v a lue of fu ture ad jus t -

men t s i s equa l to the dol l a r amou nt of the cur rent ga in

or los s . Ordina r i ly the method of spreading fo l lows the

n o r m a l c o s t o f th e p a r ti c u la r f u n d i n g m e t h o d e m p l o y e d ,

so tha t the ad jus tment for ga in (or los s ) becomes a

dedu c t ion f rom (or addi t ion to) fu ture normal cos t s .T h e s p r e a d a d j u s tm e n t m e t h o d i s t h e o n l y c o n v e n i e n t

scheme under aggrega te or a t t a ined age normal fund-

ing , has of t en been used wi th en t ry age normal , and i s

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

W hen ga ins and los ses a re spread und er e i the r of these

las t two method s the t e rm f rozen in it i al li ab i l ity i s

f requent ly emp loyed, to d i st ingui sh f rom the immed ia te

a d j u s tm e n t f o r m s o f th e s e s a m e t w o m e t h o d s ?

M echan i cs o f Sp read A d jus tmen t U nderC lass V M ethods

T h e e n t r y a g e n o r m a l - - f r o z e n i n i t i a l l i a b i l i t y - -

method re l i e s on the equa t ion

N o r m a l c o s t =

Present Va lue Al l B enef i t s

U n a m o r t i z e d In i ti a l Accru ed Liabi l i ty - F und

W e i g h t e d A v e r a g e T e m p o r a r y A n n u i t y

which i s an ident i ty so long as Fun d represent s the

funds which would have been bui l t up i f a l l ac tua r i a l

as sumpt ions were rea l i zed . By replac ing expec ted fund

by ac tua l fund in the r ight s ide of th i s equa t ion , we

automat i ca l ly compute normal cos t ad jus ted for ac tua r -

i a l ga in or los s . The adjus tment becomes the dol l a r

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

average t empora ry annui ty . The same proces s , repea ted

in fu ture va lua tions , re spreads the unrecognized por t ion

of the ga in or los s over fu ture l i fe yea rs of the then

ac t ive group (a t the s ame t ime spreading new ga ins or

los ses in the s am e fash ion) .

The amor t i za t ion of the ga in or los s by th i s me thod

i s ident i ca l to the amor t i za t ion of the acc rued l i ab i l i tyu n d e r t h e a g g r e g a te m e t h o d o f f u n d i n g . A s m i g h t b e

expec ted the ad jus tment for the ga in or los s of any

per iod i s never comple ted , but approaches ze ro as tha t

period fal ls far ther and farther into the pas t .

Und er aggrega te and a t t a ined age normal , t he subs t i-

tu t ion of ac tua l for expec ted fund has exac t ly the same

ef fec t a s under en t ry age normal . Ga ins or los ses a re

aga in spread in the dec reas ing asymptot i c manner

desc r ibed above , and the ad jus tment i s au tomat i c ,

en ta il ing no mo re w ork than i f ga in or los s d id not ex i s t .

Illustration. We can i l lus t ra t e ad jus tment for ga ins

and los ses as fa r a s i t is d i scussed in th i s paper b y going

back to the i l l us tra t ion in Table IV . f f a t t he end o f the

four th yea r , for example , the fund suf fe rs a los s of

$10,00 0 through d eprec ia t ion of s ecur i ti e s , t he immedi -

a t e ad jus tment me thod ca l l s for an ex t ra cont r ibut ion of

$10,000 a t the beginning of the 5 th yea r . Table V fo l -

lowing shows the amor t i za t ion of th i s los s in fu ture

years when spread adjus t ing i s employed.

114 Society of Actuaries 50th Anniversao Monograph



VI Conclusion

I n t h e B u r e a u o f I n t e r n a l R e v e n u e ' s " B u l l e t i n o n

S e c t i o n 2 3 ( p ) " a n d R e g . 1 1 1, S e c . 2 9 .2 3 ( p ) m a y b ef o u n d t h e T r e a s u r y r u l e s a s t o t h e m a x i m u m c o n t r i b u -

t i o n f o r w h i c h f u l l t a x d e d u c t i o n c a n b e c l a i m e d . A b r i e f

s t a t e m e n t o f t h e m a x i m u m c o n t r i b u t i o n u n d e r t h e v a r i -

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

d i scu ss io n .

C l a s s I a n d C l a s s I I f u n d i n g a r e n o t s p e c i f ic a l l y r e c -

o g n i z e d . P r e s u m a b l y c o n t r i b u t i o n s w o u l d b e f u l l y

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

f o r o n e o f t h e r e c o g n i z e d f u n d i n g m e t h o d s .

U n i t c r e d i t a n d e n t r y a g e n o r m a l f u n d i n g a r e l u m p e d

t o g e t h e r a s " C l a u s e ( i i i ) " m e t h o d s . P r o v i d e d t h e a c t u a r -

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

b o t h a r e f u r y d e d u c t i b l e u p to n o r m a l c o s t p l u s 1 0 % o f

th e in i t i a l acc r u ed l i ab i l i ty .

T A B L E V

Y e a r E x ~ a C o n t r i b u t i o n A d d e d

t o N o r m a l C o s t

5 . . . . . . . . . . . . . . . . . $ 1, 03 0

6 . . . . . . . . . . . . . . . . . . . . 9 48

7 . . . . . . . . . . . . . . . . . . . . 8 72

8 . . . . . . . . . . . . . . . . . . . . 8 03

9 . . . . . . . . . . . . . . . . . . . . 7 39

10 . . . . . . . . . . . . . . . . . . . . 6 812 0 . . . . . . . . . . . . . . . . . . . . 2 94

3 0 . . . . . . . . . . . . . . . . . . . . 1 24

4 0 . . . . . . . . . . . . . . . . . . . . . 51

5 0 . . . . . . . . . . . . . . . . . . . . . 21

A g g r e g a t e a n d i n d i v i d u a l l e v e l p r e m i u m f u n d i n g

m a k e u p t h e " C l a u s e ( i i ) " m e t h o d s . C o n t r i b u t i o n s u n d e r

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

t e m p o r a r y a n n u i t y d o e s n o t d r o p b e l o w 5 . A s t o i n d i v i d -

u a l l e v e l p r e m i u m f u n d i n g t h e T r e a s u r y s e t s fo r t h v a r i-

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

o b t a i n f u l l d e d u c t i o n . T h e e f f e c t o f t h e s e t e s t s i s t o l i m i tm a x i m u m d e d u c t i o n to t h a t u n d e r t he " n o r m a l c o s t p lu s

1 0 % " r u l e e s t a b l i s h e d f o r e n t r y a g e n o r m a l .

A t t a i n e d a g e n o r m a l i s d e s c r i b e d a s a " s p e c i a l "

m e t h o d w i t h b o t h C l a u s e ( i i) a n d C l a u s e ( i i i) c h a r a c t e r-

i s ti c s . C o n t r i b u t i o n s a r e f u l l y d e d u c t i b l e i f t h e p a s t s e r-

v i c e l i a b i l it y p a y m e n t i s n o g r e a t e r t h a n 1 0 % .

T h e T r e a s u r y s p e c i f i c a l ly r e q u ir e s p e r i o d i c a d j u s t -

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

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

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

T r e a s u r y p o s i t i o n o n s p r e a d a d j u s t m e n t i s n o t t o o c l ea r ,

t h e B u l l e t i n d e s c r i p t i o n o f a g g r e g a t e , a t t a i n e d a g e n o r -m a l , a n d t h e f r o z e n i n i t i a l l i a b i l i t y f o r m o f e n t r y a g e

n o r m a l s e e m s t o i m p l y a p p r o v a l o f s p re a d a d j u s t m e n t .

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

t h a n 1 0 % p e r y e a r , s i n c e fo r t a x p u r p o s e s t h e y a r e c o n -

s id e r ed ad d i t io n s to th e in i t i a l acc r u ed l i ab i l i ty . Sp r ead

f u n d i n g a s p r e v i o u s l y d e s c r i b e d w i l l o r d i n a r i l y k e e p

e x t r a c o n t r i b u t i o n s f o r a c t u a r i a l l o s s W i t h i n t h e 1 0 %

m a x i m u m , a n d a p p e a r s t o b e a n a c c e p t a b l e t e c h n i q u e

f o r l o s s e s a s w e l l a s g a i n s .

Appendix

D e m o n s t r a t i o n I

Pr esen t v a lu e b en ef i t sAC~ = Presen t value x Curren t act ive l ives

f u tu r e ac t iv e l i f e y ea r s

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

r - I t o~ . . . I ,i ~ + ~ t ~ a ~ . I - I . . . . I I ~ a ( V } v 2 l . . . . ) r - I

a r - I r ~ . d l x o 1 )

~ l~ii~:r_--:~+ l, i i,:,_-:-~ v + v z + .... )

a

u t

r - I

~ , l . . . . l i i ~ - v / d l r i i , - v / d l . . . . l a . (2)a

o 0 )

E l , i i ~ - 1 / d E l ~ - v / d l , i i,

r r

r - I r - I

3 )

~ l~i i, :,_---~ - 1 / d ~ l~ - v / d l , iio.,_-=~ (4)

v + v 2 + . . .. - -- v / d . (5)

o

Substi tu t ing (2) , (3) , (4) , and (5) in (1) AC1 = ~ l ~ .

r

D e m o n s t r a t i o n I I

L e t

r - I O ~ co

b = ~ . . . . I ~ i ~ + ~ l ~ a . - 1 / d E l ~ - v / d l . . . . I?i.

a r r

1 1. F u n d a m e n t a l s o f P e n s i o n F u n d i n g 115



r - I r - I

E i J i , : ~ _ - ~ 1 / d ~ l x - v / d l j i ~ : ~ _ - - ~

r - I r - I

E l E la a

p = E l , .

N o w

b - AF t _ IA C t - -

a n d

= A FA F t , - 1 + A C , - P ) 1 + i )

E 1 ) ]^ F ,_ , 1 - y + b / y - p ) l + i )

A F 0 = 0

A F 1 = ( b / y - p ) ( 1 + i )

A F 2 = b / y - p ) l + i ) I I + l + i ) 1 - ~ ) ]

A F , = ( b / y - p ) ( 1 + i )[ 1 + s + S 2 + . . . 1 S t - l ]

• 1 - s

= ( b / y - p ) ( 1 + OT .~ -~ _s

w h e r e

s = (1 + i ) ( 1 - 1 / y ) .

N o w

O < s < l

a s

l < y < l / d ;

(1)

2 )

3 )

b u t

r - I

Z lxax:,--~a

y = > 1 , si n ce //,.~_-:-~> 1r - I

a

a n d

r - I

1 / d ~ 1 , - v/d l. ii .:~_-=-~1

y = <r - I d '

a

s ince v / d l j i o ~ > 1 .

a n d

. . L s = 0t -...) ~

) (1 + i ~ b - p y

^ F . = ( b / y - p ~ -l- L- 's -s ) = 1 - d r(4)

F r o m ( 1 )

AC. b - AF ._ _ p - b d ( 5 )

y 1 - d y

S u b s t i t u t i n g t h e r i g h t h a n d f o r m s o f t h e d e f i n i t i o n s o f ya n d b i n ( 5 ) w e o b t a i n

• r - I

A C . = ~ . - ° ] a a ~ l , . (6)aa:r-- :~ a

F r o m ( 4 )

A F . = b - p y = b ( 1 - d y ) - y ( p - b d )

1 - d y 1 - d y

. p - b d = b _ A C .= o - Y l _ d y Y

r - I

= Z t . . . . l a ~ Z t , a ,

• r - I

r - o la . E l / i ~_ - ~ . (7)aa.r_ :

12




D i s c u s s i o n s

C e c i l J N e s b i t t

In this paper Mr. Trowbridge has done an excellent

job of classifying and illustrating the various methods

of pension funding. From time to time we have dis-

cussed similar ideas at Michigan but have never orga-

nized a complete analysis such as the author presents.

We knew, for instance, that the contributions and fund

under the aggregate method for a mature population

would approach limits but we did not realize what is

now fairly obvious, that those limits would be the con-

tribution and fund for the entry age normal cost method.

Throughout the paper the author uses discrete func-

tions which, of course, are convenient for the calcula-tion of illustrations. For purposes of exploring the

theory, continuous methods have some advantages.

With a few changes in assumptions and notation it is

easy to obtain continuous function formulas parallel to

the discrete function formulas of the paper. For exam-

ple, if we assume that the retirement benefit is $1 per

year payable momently from age r, and let AC, equal the

annual rate of contribution at time t under the aggregate

method, and A F , t h e fund at time t, then corresponding

to formulas of Demonstration II, we have

b = ~ l ~ . r _ x l a ~ d x 1 j~l.~(tx dx =-Tr-~-~l..1 r_ ~l ~ .

y =~ l~.r_-~dx _ ~[(T, - T r ) - l a a a . r _ - - ~ ]

~l~dx T, - T,

b - A F tA C t

d A F , ) = A C t d t + A F t ~ ) d t - p d t

o r

whence

b - p y - [ ( i / y ) -S l tA F = _ l _ - - ~ y e +

b - p y .

1 - 8 y

from which it may be shown that

A F® = b - p y

1 - ~ yDemonstration I is related to the general average

premium concept discussed by Feraud in A c t u a r i a l

T e c h n i q u e a n d F i n a n c i a l O r g a n i z a t i o n o f S o c i a l I n s u r -

a n c e , page 28. By general average premium is meant

that contribution which if paid in respect to all present

and future participants would be sufficient to provide

benefits for all present and future participants, ff r~~

denotes such a premium for a mature population whose

members are to receive the momently benefit indicated

above, and i f r~~ s payable momently, then

o r

1~ C A T a - T r ) = ~Tr

c ~ r

T ~ - T , (1)

The premium n c is independent of the interest rate

and the method amounts to pay-as-you-go funding.

If the population is immature to the extent that it con-

tains individuals up to age r only but is otherwise distrib-

uted according to the service table, and if the general

average premium for this case is denoted by n ~, then

o r

n m ~ T . - T , ) = _g l r a r )

~, lrar (2)= T~- Tr

The justification for formula (2) is that at each

moment d t in the future, benefits of value L f i r d t will be

incurred and so the total present value of benefits for

present and future participants will be 1/8(l ra~). This is

a terminal (or maturity) funding method but differs fromthe Class II funding illustration in Table IV in that the

contribution remains level from year to year by reason

of the assumed service table distribution below age r.

ff the population is just commencing to be built up

from Ia new entrants each year at age a, and n~ denotesthe general average premium for this ease, then

.1 ~(1. Ifi.)'~ ' ~ ( l a a a : r - - '7 ~ ) = r - a

o r

~. = r-a l a aa~.,--~ ' (3)

that is, ~ is the entry age normal cost.

H . F u n d a m e n t a l s o f P e n s io n F u n d i n g 117



For the cur rent cos t funding s i tua t ion wi th genera l

average p rem ium n ~, the acc rued l i ab il i ty , de f ined as the

present v a lue of benef i t s for a ll p resent and fu ture pa r -

t i c ipant s l e s s the present va lue of a l l fu ture premiumsfrom such pa r t i c ipant s , remains cons tant a t 0 . The

accrued l i ab i li ty in regard to jus t presen t pa r t i c ipant s i s

l c

w h i c h m a y b e r e d u c e d t o

. ¢ n .

~ t n - ~ ) l~a~ ~_-~. (4)

The acc rued l i ab i l i ty for fu ture new ent rant s , w i th

va lue of benef i t s expres sed in t e rms of normal co s t ,

m a y b e w r i t t en a s

1 n 1 c

~ Tt l~a~:~_-=~)-~ T t l,a~.~_-=~)

which i s the nega t ive of (4) . Thus the acc rued l i ab i l i ty

for present pa r t i c ipant s i s ba lanced by an ant i c ipa ted

ga in in regard to fu ture new ent rant s to g ive a to t a l

accrued l iabi l i ty of O.

For the matur i ty funding method wi th genera l ave r -

age p rem ium ~:m, the ac crued l iabi l i ty for re t i red part ic i -

p a n t s u l ti m a t e ly b e c o m e s

. . . . 1 T o - T ~ )t T r - n ) 3 (5a )

The acc rued l i ab i l i ty for ac t ive pa r t i c ipant s f rom

ages a to r i s

m

w h i c h r e d u c e s t o

. m t l ~

~ t n - n l oa a :~_ -~ . 5 b )

The ga in in respec t to fu ture new ent rant s i s a l so

(5b) ; henc e the to t a l acc rued l i ab i li ty for re t ired , ac t ive

and fu ture pa r t i c ipant s bec om es jus t (5a ) , t he l i ab i l i ty

for the re t i red group.

For the en t ry age normal cos t funding s i tua t ion wi th

g e n e r al a v e r ag e p r e m i u m ~ , b e n e f it s a n d p r e m i u m s f o r

fu ture new ent rant s exac t ly ba lance and the acc rued l i a -

b i l i ty for th i s group i s a lways 0 , and the to t a l acc rued

l iabi l i ty is the l iabi l i ty for presen t part ic ipants (act ive or

ret i red). I t i s convenient , however, in computing the

total l iabi l i ty to calculate benefi t values for both present

and fu ture pa r t i c ipants and proc eed s imi la rly for the ca l -cu la t ion of premium va lues . Thus , b y the t ime the ac t ive

group has grown, according to the s e rv ice t ab le , t o

inc lude pe rsons up to age r the to t a l acc rued l i ab i li ty i s

n m~ [ l r l r - n ( T a - T r ) ] = ~ (lt - n n ) ( T a - % ) . (6)

At t ime t years la ter the total accrued l iabi l i ty is

1 [ l . + , a . + , + T . - T . + , ) - 7 r ~ T a - T ~ ) ]~5

and the ul t imate total accrued l iabi l i ty is

_I [Tr rc (T. Tr )]

r

(7)

n

(xc _ rr )(Ta - Tr ) . (8)

In the foregoing d i scuss ion , ga ins in respec t to fu ture

new ent rant s have been t aken in to account . I t should not

be infe r red , however , t ha t I favor the d i scount ing of

such gains in regard to actual pension plans ; in fact , I

usua l ly t ake the op pos i t e a t t it ude . Wh atever our a t t i t ude

be toward tha t ques t ion , we should be wi l l ing to exam-

ine and unders t and the poss ib i li t ie s . The pa per of Mi lesM . D a w s o n , T h e A c t u ar i al B a s i s o f C o m p u l s o r y I n su r -

ance , was a good s t ep in tha t d i rec tion .

For some whi l e Michigan s tudent s have been pre -

sented problems a long the l ines indica ted in the above

di scuss ion . M y thanks a re due to the au thor for g iv ing a

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

W R u l o n W i ll ia m s o n

Mr. Trowbr idge ' s Tables I I and IV sugges t tha t

present management i s na tura l ly cons ide ra te of fu ture

m a n a g e m e n t ' s s o l v e n cy .T w o F e d e r a l p r o g r a m s o f p e n s i o n s h a v e h a d p r e s s

a t t en t ion l a t e ly , one because the concern seems too

la rge ly mis s ing , the o the r b ecause o f an e l ec tion .

1 . The F edera l C iv i l Se rv ice R e t i rement S ys tem has

ac tua r i a l repor t s based on Mr . Trowbr idge ' s C las sIV-1 f inanc ing method. Las t month Cha i rman Ram-

speck had two l inked a r t i c l es in the Wa shington Post .He sa id tha t the re was $4 b i l l ion in the fund, ag a ins t

118 Soc ie ty o f Ac tuaries 50th Anniversary Monograph



accrued l i ab il i ti e s o f $9 b i l l ion . T he $4 b i l l ion as se t s

fe l l shor t of the tw o requi rement s , l i ab i li t ie s on exi s t -

ing pens ioners and the guaran teed re turn of cont r ibu-

t ions wi th in t e res t to employees and the i r surv ivors .Thi s l e f t no th ing toward the employer ' s l i ab i l i ty to

a c t i v e w o r k i n g e m p l o y e e s . C o n g r e s s m a n M u r r a y

quo ted the ra tio of 12 to 1 for benef i t paymen ts and

employee cont r ibut ion for ex i s t ing pens ioners .

2 . OA SI is be ing presented as the b igges t pens ion

plan , and the b igges t l i fe insurance sys tem.

On October 7 in Mis s i s s ippi (pres s re l ease of Oc to-

ber 3) the Federa l Secur i ty Adm ini s t ra tor s a id the

FSA se rv iced 157 mi l l ion Am er icans for heal th , we l -

fa re and educa t ion . He sa id tha t 100 mi l l ion of them

h a d O A S I w a g e r e c o r d s i n B a l ti m o r e .

A l s o o n O c t o b e r 3 , G o v e r n o r S t e v e n s o n s a i d i n

C o l u m b u s , O h i o : T o d a y 65 m i ll i o n p e o p l e h a v e

bui l t up subs tant i a l equi t i e s in the Soc ia l Secur i ty

s y s t e m . W h e n y o u a n d y o u r w i f e r ea c h t h e a g e o f 6 5 ,

your sha re in the re t i rement fund wi l l amount to the

equiva lent of a $15,00 0 annui ty . The prod uc t of 65

mi l l ion and $15,000 i s about a t ri l li on dol l a rs ; of 100

mi l l ion and $15,000, $1 ½ t r i ll i on , o f 157 mi l l ion and

$15,000, $2¼ t r i l l i on . F rom age 18 ( the end of

depen dent ch i ldhood ) to 65 ( the age for e l ig ib i li ty

t o O A S r ' ) i s 4 7 y e a r s . T o p p e n s i o n n o w i s $ 8 5 .

Abjec t pover ty i s s a id to begin be low $2,000 a yea r .

T h e m o n t h l y p e n s i o n c o r r e s p o n d i n g t o $ 2 , 0 0 0 i s$65. Us ing tha t a s the average pens ion expe c ted a t

6 5 , t h e y e a r l y u n i t w o u l d b e $ 1 . 4 0 p e r y e a r o f p r e -

sumpt ive work . Us ing only the t r i l l i on f igure , and

Mr. Trowbr idge ' s uni t me thod, C las s I I I , a no- in te r -

es t base wou ld show about a ha l f - tr i ll i on acc rued l i a -

b i l i ty . But us ing 2% in te res t , U .S . L i fe Table Whi te

Males , 1939-1941, pure annui t ie s , and som e ra the r

anc ient age d i s t r ibut ions , might cu t the acc rued l i a -

b i l i ty to $150 b i l l ion . The present t rus t fund i s about

10% of tha t . 2% in te res t on the acc rued l i ab i l i ty

would t ake $3 b i l l ion . $280 b i l l ion of l i fe insurance

a t the annua l dea th ra t e of 6 pe r 1 ,000 wou ld ca l l for

provi s ion of $1½ bi l l ions . The cur rent l i ab i l i ty forone uni t of de fe r red annui ty cou ld run $6 ½ bi l l ion .

The annua l loa d wou ld reach $11 b i l l ion . Current t ax

col l ec t ion i s abou t $4 b i l l ion .

The Miss i s s ippi speech a l so sa id : W e a re condu c t -

ing the bu s ines s o f Soc ia l Secur i ty so e f f i c i en t ly tha t

we have been able to expand the benef i t s . Thi s

month , wi th few except ions , each check was l a rge r

than the previous one by a t l eas t $5 or 12%, which-

ever was m ore . W e had a l i t tl e t rouble ge t t ing the b i l l

p a s s e d b y C o n g r e s s . B u t it w e n t t h r o u g h T h e p r o s -

pec t tha t b i ennia l ly a t each Congres s iona l e l ec t ion$20 b i l l ion addi t iona l acc rued l i ab i l i ty i s to be

accepted i s an in t r iguing one .

The p ic ture of my worr i ed countenance in the f i r s t

n u m b e r o f t h e n e w Li fe Magazine , b e n e a t h w h i c h w a s

the c l a im tha t I would f igure the lowes t ra t es on Soc ia l

Secur i ty , s eem s to have been proph e t i c .

Clark T Foster

Mr. Trow br idge has been g ui l ty of an unders t a t emen t

in the in t roduc t ion to h i s va luable paper in desc r ib ing

the need for a t ex t on pens ion funding methods . Hepoin t s out tha t the beginner in the pens ion f i e ld , i n h i s

a t t empts to educa te h imse l f , mus t re ly on the Bul le t in

on Section 23 p) , put out by the Uni t ed S ta t es Treasury

D e p a r t m e n t . R e m e m b e r i n g t h e h o p e l e s s f e e l i n g I h a d

when f i r s t s tudying tha t complex document , I have the

fee l ing tha t anyone wi th no o the r means of l ea rn ing

about pens ion funding would remain a beginner a l l h i s

fife.

There a re two poin t s I would l ike to ra i s e in connec-

t ion with this paper; f i rs t , to int roduce two addi t ional

funding c l as ses , which might be re fe r red to as 1½ and

2k'2 , and secon d, to com me nt on severa l me thods o f

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

desc r ibed .

C l a s s 1 ½ b e l o n g s s o m e w h e r e b e t w e e n C l a s s 1 , p a y -

as -yo u-go metho d, and C las s 2 , t e rmina l funding . I t has

b e e n u s e d i n a n u m b e r o f c a s e s, p a r ti c u la r ly i n s o m e o f

the negot i a t ed s t ee l p l ans , a s a means of l eve l ing the

cos t in the f i r s t few years when the t e rmina l funding

c o s t i s o ft e n q u i t e h ig h b e c a u s e o f t h e l a r g e n u m b e r o f

employees immedia te ly e l ig ib le to re t i re . The present

va lue of fu ture pens ion benef i t s i s pa id in to the fund in

ins ta llment s over a pe r iod of up to f ive yea rs a f t e r each

em ploy ee reaches re t i remen t ra the r than in a lum p sum

at the t ime of re t i rement .C las s 2½ l i e s be tw een the te rmina l funding of C las s

2 and the fu l l funding o f C las s 3 or 4 . In t e rmina l fund-

ing , no cont r ibut ions a re made unt i l an employee

re t ire s . Und er C las s 3 or C las s 4 funding , a cont r ibut ion

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

u n d e r t h e p l a n. U n d e r C l a s s 2 ½ f u n d i n g, c o n t r i b u t io n s

a r e m a d e o n l y f o r e m p l o y e e s w h o h a v e r e a c h e d a c e r -

t a in age or have comple ted a ce r t a in pe r iod of s e rv ice ,

IL Fundamentals o f Pension Funding 119



despite the fact that they are considered as being cov-

ered under the plan before satisfying such age or ser-

vice requirements. The advantage of this method lies in

the elimination of administrative records and actuarialcalculations on the young or short-service employees

who are most likely to terminate employment. It is par-

ticularly convenient to fund benefits only for employees

age 35 or over if the plan provides benefits at age 65

based only on years of service up to a maximum o f 30

years. Similarly, it is convenient to fund only for

employees over age 40 if the maximum benefits are

granted after 25 years o f service.

In some cases, it is preferable to fund only for

employees who have completed, say, two or three years

of service. Upon an employee s completion of such a

period of service, records are established, and his total

estimated benefit is funded over his remaining years of

employment.

Under any such program of funding, the annual cost

for each employee for wh om benefits are being funded

is greater than if funding had started at employment, but

since a large group of employees is not having any ben-

efits funded, the reserves at any time are less than they

would otherwise be. The pattern of contributions from

year to year depends on the maturity of the group and

its age distribution. In a new organization with a rela-

tively low average age, the cost is likely to increase

sharply as more and more employees pass the age at

which funding commences.This Class 2½ funding becomes identical with

Class 2, or terminal funding, if the age at which benefits

are funded is the retirement age. Similarly, this Class

2½ funding becomes identical with the Class 3 or Class

4 full funding methods if benefits are funded for

employees as soon as they are eligible for coverage

under the plan.

There are a number of ways by which the various

funding methods are frequently combined. It is com-

mon, for instance, in a plan providing normal benefits

in accordance with a percentage formula, but in which

benefits are subject to a certain minimum, to fund thepercentage accruals on a Class 3 unit credit method and

to fund any additional benefit requited by the minimum

on a Class 2 terminal funding basis at retirement. Simi-

larly, in a plan allowing retirement at any time after age

65 with additional benefits accruing as a result of ser-

vice after 65, it is convenient to assume that each

employee will retire at 65 and fund such benefits on a

Class 3 or a Class 4 program, funding any additional

benefits resulting from service after age 65 on a termi-

nal funding arrangement at the end of each year of ser-

vice after age 65. The cost o f such additional benefits is

normally offset by the savings resulting from paymentsthat would otherwise have been made to the employee

during his period of postponement.

Another frequently used combination of methods is

to establish a past service liability on the Class 3 unit

credit method and to fund the future service benefits on

a Class 4 individual level premium method. Occasion-

ally, this combination might be further complicated by

the use of Class 2 terminal funding for the purchase of

disability benefits.

Just as the funding methods themselves may be com-

bined, the various methods of handling actuarial gains

and losses may also be combined. Frequently, futureservice gains are immediately used as a credit against a

plan s normal cost, whereas past service gains are tem-

porarily ignored, serving to shorten the period over

which the past service liability is funded. This arrange-

ment is possible as long as a corporation s total contri-

butions for any year fall within the Internal Revenue

Bureau s specified maximum. Occasionally, certain

types of gains from either past or future service are

taken immediately while others are spread over a period

of years. For example, a loss resulting from salary

increases in a plan involving an assumed salary scale

may be spread over the period to an employee s retire-

ment, while all other gains or losses are immediatelyrecognized. Alternatively, the loss from salary changes

might be allocated between past and future service,

with the future service loss recognized at once and the

past service loss spread over the past service funding

period.

R o b e r t L i n k

One can visualize the population of a group as an

organism which passes through a period of growth, a

period of maturity, and finally senescence. This is a

rather idealist ic description since the characteristics ofgrowth, maturity and senescence are usually obscured

by such extraneous factors as ups and downs of the eco-

nomic cycle, changes in the characteristics of the partic-

ular industry, etc. The theory of most pension funding

methods is most easily examined on the assumption that

one has a stable group which can be expected to remain

stable for a number of future years. However, this idea is

realized so infrequently in practice that the examination

120 Society of Actuaries 5 th Anniversaly Mo nograph



and comparison of funding methods on the basis of a

stable population may create or promote misconceptions

rather than otherwise. Thus, any mathematical theory of

pension funding must take as its laboratory a group pop-ulation which is assumed to be subject to change in its

composition as to ages, salaries and so forth.

Mr. Trowbridge is to be congratulated for a paper

which (within the limited range of my own reading)

appears to be the first attempt to state the definitions,

axioms, and theorems of a true science of pension fund-

ing methods. His concept of the immature group gets

the science of pension funding off immediately on the

right foot. His descriptions of various funding methods

and their operation in the context of a simple group

population should probably be required reading for stu-

dents of pension funding; they might well be adopted asthe foundation for any future developments along these

lines.

I am sure that Mr. Trowbridge will not be offended if

I suggest that his paper has barely scratched the surface

of a great body of potential scientific knowledge of var-

ious funding methods. Further points to be examined

(and which have been intentionally avoided by Mr.

Trowbridge) are such matters as:

a) The effect of differences between the actuarial

assumptions and the true experience of the particular

group;

b) Extension of his theories to cover the more realistic

situation of multiple entry ages;

c) Examination of various methods o f estimating pen-

sion costs with respect to their appropriateness in

predicting future costs;

d) Miscellaneous matters such as the choice of correct

and meaningful turnover rates, salary scales, etc.;

e) Special problems arising from the introduction of

unusual benefits or employee contributions (in partic-

ular those arising from superimposition of an alter-

nate benefit formula on an existing scale of benefits).

In describing the trend of normal costs under various

classes of funding, Mr. Trowbridge has tended to give

further documentation to what I believe to be an over-worked thesis. This thesis is that rising costs under

plans which are funded by the unit credit cost method

are due mainly to the increase in the average premium

age of the group. Occasionally an employer has

requested that we explain the reason for rising costs

under a deferred annuity plan and predict the trend of

these costs for the future. We have found that of the

total rise in cost only a small part was usually attribut-

able to increasing average premium ages. The rest was

due to such factors as:

a) The tendency, as a group progresses toward maturity,

for a higher percentage of the total lives in the groupto find themselves in the group of eligible employees;

b) Generally rising salaries (which have an intensified

effect under an approximately integrated unit plan);

c) Amendments of the group annuity contract with

respect to the rate basis of purchase.

The attempt to analyze trends of cost in terms of an

initially immature group population leads to a rather

interesting result. One tends to think of the asymptotic

approach f rom the l ~ distribution of Mr. Trowbridge s

immature population to the l~ distribution of his station-

ary population as a smooth progression of uniform

direction. Actually this asymptotic approach looks morelike a decreasing sine wave. This is somewhat evident

from Mr. Trowbridge s illustrations; the terminal fund-

ing amounts shown in Table 4 rise until the 35th year,

drop again to the 50th year, and reach an ultimate level

higher than that of the 50th year. It can easily be seen

that if lives leave a group only by retirement (there

being no deaths or withdrawals at all) the population

would tend to repeat itself on a cycle of r - a years. The

decrements have a damping effect on this tendency.

This wave motion makes it just a little trickier to draw

conclusions from numerical illustrations.

The Equation of Maturity can also be written in such

a way as to exclude the liability for retired lives (this is

equivalent to paying the benefit in a lump sum at retire-

ment age). In this alternate form, the equation looks like

this:

C d F = v R i i r

where R is the total annual income for new retirements.

For certain purposes, there seems to be some merit in

extending Mr. Trowbridge s notation to embrace two

variables, entry age and attained age. This leads to a set

of select functions which can be identified by the sub-

script x, y (x representing attained age and y represent-

ing entry age). ff one assumes a constant percentage

distribution of entry ages for each generation of newentrants, one should ultimately come up with a station-

ary population expressed by a distribution consisting of

the values of Ix.~ . M r . Trowbridge s algebraic identities

based on the Equation of Equilibrium will still apply for

the unit credit cost method and the entry age normal

cost method, since these equations can be expressed for

the double variable case in terms of the sums of various

items for each entry age.

H . F u n d a m e n t a l s o f P e n s i o n F u n d i n g 121



A l i t t l e cau t ion i s needed , however , when approach-

ing the prob lem of u l t ima te co s t under an aggrega te

funding m e thod . The E qua t ion o f Matur i ty can be wr i t -

t en w i th the u l t ima te cos t pe r l i f e expressed a s anunkn own (us ing the al t e rna te fo rm o f the Equa t ion of

Matur i ty ) , a s fo l lows:

V E l r , ynyC i r .~ . A C P L . ~ I x yy x y

x,y A p x.y

whe re C PL i s t he u l t ima te no rmal co s t pe r l if e and By i s

the annua l ra t e o f re t ir ement i ncom e for an en t ran t a t

age y . I f we s o lve th is equa t ion , t he normal cos t pe r l i f e

tu rns ou t t o be

Z Iy,,By _y[ay,

ACPL® = ;.'

Y

and the normal cos t t u rns ou t to be

AC~ = ACPL~EIx.y.X y

Put in words , t he u l t ima te normal cos t under any

aggrega te funding me thod i s a cons t an t amount pa id

each yea r fo r each ac t ive l i f e , such cons t an t amount t o

be the sam e as the amount wh ich , i f pa id ove r t he fu ture

l i fe t ime of each of t he en t ran t s o f one ca l endar yea r ,would prov ide the bene f i t s fo r t hese en t ran t s . Thi s

resul t , in re t rospect , i s not part icular ly surprising. This

normal cos t i s no t p rec i se ly the sam e as tha t under ind i-

v idua l en t ry age funding , s ince the l a t t e r would be

expressed a s fo l lows:

E ~ c V I B -YIgiY'Y~ Z _ ~ x , y y ~

x,y Oy, y r-- '~

T h e d i f f e re n c e b e t w e e n A C . a n d e ~ C i s t h at A C .

soc ia l i zes the cos t a s be tw een en tran t s who, in E~C,

have d i f fe ren t l eve l p remium cos t s . The re i s an ac tua l

numer i ca l d i f fe rence be tween the two, which i s p roba -b ly un impor t an t i n most cases . The d i f fe rence ought t o

be recognized , however , i n any theore t i ca l d i scuss ion .

The two a re iden t i ca l when B y . r _ y l i i y y + a y , y:r-- =~ is

cons t an t fo r a l l va lues o f y.

I t m a y b e f e l t b y so m e t h a t t h i s t y p e o f a n a l y s i s o f

funding me th ods i s o f li t tl e p rac t ica l va lue . As a yo ung

ac tua ry s t ruggl ing w i th prac t ica l ques t ions o f funding , I

be l i eve tha t many prac t i ca l ques t ions which have

caused me grea t d i f f i cu l ty in the pas t can be answered

by th i s pape r and i ts seque lae .

Hilary L Seal

T h e a u t h o r h a s d i v id e d m e t h o d s o f f u n d in g p e n s i o n

plans in to s ix c l a sses , one o f t hese be ing fur the r subdi -

v ided in to four d i f fe ren t me thods . H e has thus spec i f i ed

n ine d i f fe ren t funding me thods . However , by in t roduc-

ing the conce pt o f a l t e rna t ive imm edia t e o r spread

adjus tment s on account o f t he ga ins o r l osses tha t can

occu r in seven o f t hese me thods , h e has e f fec t ive ly pro-

v i d e d u s w i th sixteen di f fe ren t ways of funding a pen-

s i o n p l a n . H o w m a n y o f t h e se w o u l d b e a c c e p t a b l e t o

t h e T r e a su ry f o r t a x d e d u c t io n p u r p o se s?

Judging f rom the op in ions exp ressed in the i r Bulletin

o f Ju n e 1 9 4 5 t h e sp r e a d m e t h o d o f a d j u s ti n g f o r gains

w o u l d n o t b e a c c e p t a b l e i f th e f u n d i n g w a s b a se d o n ( i)

un i t c red i t o r ( i i) i nd iv idua l l eve l p remium methods . On

the o the r hand , losses c o u l d b e m a d e su b j e c t t o so m e

d e g r e e o f sp r e a d b y u s i n g t h e T r e a su r y ' s S p e c i a l

10% base in conjunc t ion wi th the un i t c red i t me thod .

Fur the r, t he aggrega te m e thod as d esc r ibed in the Bulle-

tin automat i ca l ly uses the spread me tho d of ad jus t -

ment , though i t i s l i ke ly tha t t he imm edia t e type of

a d j u s t m e n t c o u l d b e a d o p t e d f o r gains. Natural ly, the

m e t h o d s c l a ss i fi e d b y t h e a u t h o r a s V a n d V I w o u l d n o t

be accep tab le for t ax deduc t ion .

The ne t r e su l t o f these t ax cons ide ra t ions i s t o reducethe au thor ' s s ix t een funding me thods to the n ine

employed in prac t i ce , namely :

( 1 ) P a y - a s - y o u - g o

(2) Terminal

(3) Uni t c red i t w i th immedia t e ga ins ad jus tment

(4) Ent ry age normal w i th imm edia t e ga ins ad jus tmen t

(5) Ent ry age normal w i th spread ga ins ad jus tment

(frozen ini t ia l l iabi l i ty)

(6) Ind iv idua l l eve l p remium wi th immedia t e ga ins

ad jus tment

(7) Agg rega te w i th spread ga ins ad jus tmen t

(8) A t t a ined age wi th imm edia t e ga ins ad jus tment(9) A t t a ined age wi th spread ga ins ad jus tment

A c o m p a r i so n o f t h e la s t s e v en o f t h e se f r o m t h e

i n c o m e t a x v i e w p o i n t w a s m a d e b y t h e sp e a k e r i n t h e

recen t ly publ i shed Proceedings of the Conference of

Actuaries in Public Practice, Vol. i i , 1952. It will be

obse rved f rom tha t pape r t ha t , con t ra ry to Mr . Trow-

br idge ' s op in ion , t he Treasury ' s publ i shed v i ews on

122 Socie~ of Actuaries 50th Anniversary Monograph



spreading gains are clear and, with one exception, rea-

sonably consistent.

I mention, in conclusion, that methods (3), (8) and

(9) above suffer from a serious limitation: they can beapplied only where the benefits for each employee may

be regarded as accruing each year on a level, or gradu-

ally changing, scale. The methods are difficult to apply,

for example, to plans with a lump-sum death benefit or

where the pensionable earnings base is the average of

the five years' earnings preceding retirement.

C h a l m e r s L W e a v e r

This discussion is offered to supplement this excel-

lent paper with the answer to a question that occurs on

reading the paper. The author demonstrates that if thetabular assumptions as to mortality and interest prevail

in the mature state the entry age normal and aggregate

methods lead to identical ultimate funds and annual

contributions. The question is how these two methods

compare at maturity when the population mortality and

interest earnings are not tabular. It is to be demonstrated

that if the tabular assumptions as to mortality and inter-

est are conservative the aggregate method produces a

larger fund and smaller annual contribution at maturity

than does the entry age normal method. The converse is

true if the tabular assumptions as to mortality and inter-

est are liberal.

The equations at maturity in this more general stateare symbolical ly the same as those in the paper. The dif-

ference is that now the annuities in all the formulas

given involve tabular assumptions as to mortality and

interest that may differ from the mortality indicated by

the l's of the population and the actual interest rate

earned. The latter are combined with these annuities in

the formulas. These more general condit ions hold in the

symbolic defini tions of b, y, and p given in the paper,

and in

T = r°[~ r~x

ao:~_--~ a

We have, as in the paper,

E A N F . = b - T y

EANC EAN. p - a r . p - d b + d T y

A F . = b - A C . y = b - p y1 - d y

A C . _ b - A F . _ p - d b = p - d A F -

y 1 - d y

= p - d b + d a c . y .

From these we obtain

^ F . = E A r ~ F . ( T - A C . ) y

^ C . = E A N c .- - d ( T A C . ) y .

Note that if tabular assumptions are realized

~ A N c . = T a n d t h e n E A N c . = A C ® = T .

I f t a b u l a r a s s u m p t i o n s a r e c on s e r v a t i v e i t i s o b v i o u s

t h a t u n d e r t h e e n t r y a g e n o r m a l m e t h o d

e A Nc . = T - v G = T - I - d ) G ,

w h e r e G i s t h e g a i n t h at w o u l d t u r n u p a t t h e e n d o f t h e

y e a r i f T w e r e t h e c o n t r ib u t i o n.

v G = T ( 1 - d y ) + d b - p .

Then

AC. = T - ( 1 -d )G -d Ty + d AC®y

1 - dAC® = T - I _ - ' ~ y G

AF . = Ear~ + 1 - d. l _ - ~ y G y .

Since in practice d y < 1 we have

i f G > O

AF > E N F .

AC < EANc.,

and if G < 0

F. < E A ~ F ®

AC > E A N c

The aggregate method has a fundamental weakness.

If the tabular assumptions are modestly on the conser-

vative side the fund will grow to a materially higher

H. Funda menta l s o f Pens ion F t , nd tng 123



l im i t than t he funds t ha t w ou ld be he ld by t he en t ry age

norma l me thod . The conve rse i s a l so t rue . The me thod

i s ve ry sens i t i ve t o t he t abu l a r a ssumpt ions and shou ld

no t be used un l e ss t he re i s f r equen t r e ad jus tmen t o fthe se a ssumpt ions t o r ea l i s t i c va lue s . Mos t a c tua r i e s

w o u l d r a t h er u s e m o d e s t l y c o n s e r v a ti v e t a b u la r a s s u m p -

t i ons , check t he se l e ss f r equen t ly , and use a l e ss sens i -

t i ve me th od i n t he mean t ime .

C o n c l u s i o n . - - I f t abu l a r a ssumpt ions a re conse rva -

t i ve , t hen unde r any fund ing me thod t he immedia t e

ad jus tmen t o f ga ins wou ld s t il l le ave a fund t ha t wou ld

b e a d e q u a t e . T h e s p r e a d m e t h o d o f a d j u s t m e n t w o u l d

bu i ld up add i t i ona l funds . I f t abu l a r a ssumpt ions a re l ib -

e ra l , t hen unde r any fund ing me thod t he immedia t e

ad jus tme n t o f l osse s wou ld s t il l l e ave a fund t ha t wou ld

b e i n a d e q u a te . T h e s p r e a d m e t h o d o f a d j u s tm e n t w o u l ddraw do wn the fund t o a l ower l eve l .

A num er i ca l i l lu s t r a ti on ha s been p repa red . The fund

and annua l con t r i bu t i on a t ma tu r i t y have been com-

p u t e d f o r act ive l i ve s fo r a popu la t i on w i th en t ry age 35

and re t i r emen t age 65 . The popu la t i on ha s CSO mor t a l -

i t y , and t u rnove r o f 5% a t ages unde r 50 g raded t o no

tu rnove r a t age s 6 0 and ove r . The ea rned ra t e o f i n te re s t

i s se t a t t h ree va lue s , 2%, 2½% and 3%. The t abu l a r

a ssumpt ions a re CSO mor t a l i t y , no t u rnove r , and 2½%

in te re s t . The f i gure s fo r the un i t c red i t me thod a re a l so

inc luded .

2% Interest FC

t

2½ % Interest I FIc

3% Interest FC

Uni tCredi t

En t ry AgeNorma l

Aggregate

$932 ,000 ,$1 ,114 ,000 $1 ,607 ,00049,400 i 45,800 , 36,100

$932,000 $1,114,000 $1,768,00044 ,600 40 ,200 24 ,200

$932,000 $1,114,000 $1,961,00039,800 34,600 9,900

George E Immerwahr

Mr. T rowbr idge ' s pape r answers a l ong-s t and ingneed i n ac tua ri a l l i t e ratu re fo r a de sc r ip t i on an d an a lys i s

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

Uni t ed S t a t e s .

In d i scuss ing t he T rea sury ru l e s r e l a t i ng t o l im i t a -

t i ons app ly ing t o deduc t ions fo r pens ion con t r i bu t i ons ,

Mr . T rowbr idge s t a t e s t ha t t he T rea sury pos i t i on on

spread ad jus tmen t fo r ga ins i s no t t oo c l ea r , bu t t ha t

approva l o f sp read ad jus tmen t i s impl i ed by i t s de sc r ip -

t i on o f t he aggrega t e , t he a t t a ined age norma l , and t he

f rozen i n i ti a l l i ab i li t y me thods i n t he June 1945 Bul le t in

on Sect ion 23 p) of t he In t e rna l Revenue Code . The

Treasury r egu l a t i ons on t he ma t t e r , a s r ev i sed i n

No vem ber 1948, s t a te t ha t i n de t e rmin ing t he cos t sand l im i t a ti ons an ad jus tm en t sha l l be mad e on accoun t

o f any expe r i ence mo re f avorab l e than t ha t a ssumed in

the ba s i s o f l im i t a ti ons fo r p r i o r yea r s , and , un l e ss su ch

ad jus tmen t s a re cons i s t en t l y made eve ry yea r by r educ -

ing t he l im i t a t i ons o the rwi se de t e rmined by any

dec rea se i n l i ab i l i t y o r cos t a r i s i ng f rom expe r i ence i n

the nex t p reced ing t axab l e yea r more f avorab l e t han t he

a ssumed expe r i ence on which t he cos t s and l im i t a t i ons

w e r e b a s e d , t h e a d j u s t m e n t s h a ll b e m a d e b y s o m e o t h e r

me tho d approved by t he Com miss ione r . 3, These r egu-

l a t i ons wou ld imply t he accep t ab i l i t y o f ad jus tmen t s

made by t he me thods se t fo r th i n t he 1945 Bullet in ,s ince t ha t bu l l e t i n ha s no t been revoked , o r by va r ious

o the r me thods sa t i s f ac to ry t o t he Commiss ione r . W hi l e

no ampl i f i c a t i on o f t he 1945 Bullet in h a s b e e n p u b -

f i shed , i t wou ld appea r , f rom the t ypes o f ad jus tmen t

regu l a r ly emplo yed by a num ber o f consu l t i ng ac tua r ie s

and i nsu rance compan ie s , t ha t t he fo l l owing p rac t i c e s

wo uld be found sa t is f ac to ry .

1 . For p l ans wh e re sp read ad jus tmen t i s impl i c i t i n

the fund ing me thod , e.g., i n t he aggrega t e me thod o r

frozen ini t ia l l iabi l i ty method as descr ibed in Mr.

T rowbr idge ' s pape r , fu l l con t r i bu t i ons de t e rmined i n

a c c o r d a n c e w i t h t h e m e t h o d w o u l d b e d e d u c t i b l e

p rov ided t he fo l l owing cond i t i ons (de s igned t o p re -

ven t i n i ti a l ove r fund ing ) a re m e t :

a ) adequa t e a l l ow ance i s made fo r wi thdrawa l s and

mor t a l i t y , and a l l o the r a ssumpt ions a re r ea sonab l e ;

and

b) no subs t an t ia l p ropor t i on o f t he con t r i bu t ions i s pa id

o n b e h a l f o f e m p l o y e e s w h o a r e u n l ik e l y t o r e c e iv e

any benef i t s .

Cond i t i on (b ) c an pe rhaps be s t be sa t i s f i ed fo r t he

typ i ca l p l an by e l im ina t i ng f rom cove rage ( a t l e a s t fo r

t h e p u r p o s e o f c o m p u t i n g c o n t r i b u ti o n s ) t h o s e e m p l o y -ee s w ho fa l l be low a spec i f i ed age , such a s 30 , o r t hose

wi th l e ss t han a g iven numb er o f yea r s o f se rv i ce , such

as 3 fo r sa l a r i ed employees o r 5 fo r hour ly -pa id

e m p l o y e e s , o r t h o s e w h o f a i l to m e e t s o m e a p p r o p r ia t e

combina t i on r equ i rement o f age and yea r s o f se rv i ce .

A l l o w a n c e f o r w i t h d r aw a l s m a y s o m e t i m e s b e o m i t t ed

f rom cond i t i on ( a ) whe re no sa l a ry sca l e i s a ssumed

and whe re b road enough e l im ina t i on f rom cove rage i s

made unde r cond i t i on (b ) .

124 Society of Actuaries 50th Anniver saly Monograph



2 . F o r p l a n s f u n d e d b y t h e e n t r y a g e n o r m a l ( n o n f r o -

z e n ) m e t h o d o r t h e u n it c r e d i t m e t h o d , a m e t h o d m a y

b e u s e d u n d e r w h i c h ( r a t h e r t h a n r e q u i r i n g t h e f u l l

i m m e d i a t e a d j u s t m e n t e a c h y e a r d e s c r i b e d i n P a r tV I I I o f t h e 1 9 4 5 Bulletin the a m o u n t o f d e d u c t i b l e

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

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

d e t e r m i n e d l e s s a n y e x c e s s o f ( a ) a m o u n t s o f c o n t ri -

b u t i o n s a c t u a l l y t a k e n a s d e d u c t i o n s i n p a s t y e a r s ,

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

d e d u c t e d b a s e d o n s u c h c u r r e n t ly r e d e t e r m i n e d c o s t .

A n i l l u s tr a t i o n o f t h e a p p l i c a t i o n o f t h i s m e t h o d t o a

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

p l a n f u n d e d b y t h e e n t r y a g e n o r m a l m e t h o d w o u l d

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

s a m e p r i n c i p l e .

I L L U S T R A T I O N O F M A X I M U M D E D U C T I B L E C O N T R I B U T I O N S U N D E R

C O N V E N T I O N A L N O N C O N T R I B U T O R Y G R O U P A N N U I T Y P L A N

( M i n o r i n t e r e s t a d j u s t m e n t s i g n o r e d )

1. Ini t ia l pa s t service c os t as de term ined a t ince pt ion of pla n . .. .. .. .. .. .. .2 . R educ t ions in in i t i a l pa s t s e rv ice cos t due to w i thd rawa l s in yea r

(o the r than dea ths o r i l l-hea l th t e rmina t ions ) , whe the r pa s t s e rv ice

annu i t i e s had been p urchas ed fo r w i thd rawing mem bers o r no t . .. .. ..3 . Ini t ia l pas t serv ice cos t as r edet erm ined a t be ginn ing of ye ar .. .. .. .. .. .4. G ro ss cu rr en t ser vi ce co sts in ye ar .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

5 . C os t c red i t s a l lowed aga ins t cu r ren t s e rv ice con t r ibu t ions in yea r ,a r i s ing f rom

a) C ancel la t ion o f pas t servic e annui t ies a l ready p urch ased . .. .. .. .. .. ..

b) Cancel la t io n of curren t service annui t ies a l ready purcha sed . .. .. .. . i

c) To tal .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .6. To tal co nt rib uti on s pa id in ye ar .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .7 . Po r t ion o f con t r ibu t ions deduc t ib l e (and deduc ted ) unde r m e thod

re fe r red to in th i s d i s cus s ion

a ) Gros s c u r ren t s e rv ice cos t s l e s s cu r ren t s e rv ice cos t c red i t s(4 less 5b ) ... . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . .

b) 10% of red eterm ined ini t ia l p as t se rvice cos t (1 0% of 3) . .. .. .. .. .. .. .

c ) C umu la t ive deduc t ions fo r p rev ious yea rs(su m o f 7 g fo r a ll p re vi ou s ye ar s) .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .

d) Su m of 7a fo r all pr ev iou s ye ars . . . .. . . .. . . . .. . . . .. . . .. . . . .. . . .. . . . .. . . .. . . . .. . . .. .

e ) C ur ren t y ea r ' s 7 b mul t ip l i ed by num ber o f p rev iou s yea rs . .. .. .. .. ..f ) Ad jus tmen t = s um o f p rev ious yea rs ' a c tua l deduc t ions l e s s

deduc t ion on red e te rmined ba s i s (7c l e s s s um of 7d and 7e ) . .. .. ...

g ) Dedu c t ib l e con t r ibu t ion fo r cu r ren t yea r

(7 a plu s 7b less 73') .. . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . .

Y E A R O F P L A N

F k s t S e c o n d T h i r d

100 , 000

6 , 000100 , 000

15 , 000

0

0

03 0 , 0 0 0

15,000

10,000

25 , 000

7 , 000

94 , 00016,000

2 , 000

5 0 0

2 , 50030 , 000

15 , 500

9 , 400

25 , 00015 , 000

9 , 400

6 0 0

24 , 300

2 , 00087 , 000

18 , 000

5 0 09 0 0

1,40030 , 000

17 , 100

8 , 700

4 9 , 3 0 0

30 , 50017 , 400

1,400

2 4 , 4 0 0

H. Fundamentals of Pension Funding 125



W i ll i am M R a e

It is somewhat difficult to visualize the relationship

between the various, apparently unrelated, fundingmethods. Mr. Trowbridge has very ably demonstrated

their relationship by the algebra pertaining to a mature

population. I have also found the following general rea-

soning approach to be quite helpful. It is a prospective

approach.

Every funding method calls for determining the

group to be valued. This is usually all pensioners and

present employees, or all except those not yet meeting

certain minimum age and service requirements.

Having determined the group to be valued, every

funding method can be viewed as calling for the calcu-

lation of the present value of all future benefits for thevaluation group a s a c l o s e d g r o u p . This present value,

less funds on hand, is then split into two parts, (a)

unfunded accrued liability and (b) present value of

future normal costs. The split is dictated by the funding

method chosen. Each part is then amortized over a

period of years in the manner dictated by the particular

funding method. Under some methods e .g . , aggregate

method, individual level premium method) the amorti-

zation scheme is the same for both parts. The different

amortization schemes of the various funding methods

cause the different incidence of annual cost between the

methods.

The amortization scheme, in dollars, for (b) above

can be level a s t o a n i n d i v i d u a l or increasing as to an

individual e .g . , entry age normal with salary scale

method, unit credit method), but will decrease in the

aggregate as the closed valuation group is assumed to

retire, die or withdraw.

The valuation process in subsequent years can be

viewed in exactly the same fashion, subject to whatever

adjustment for gains and losses is called for by the par-

ticular funding method. In subsequent years we will

again be valuing a closed group, but the composition of

the closed group will be different from that of the pre-

ceding year. New lives will have been added. These,

broadly speaking, counterbalance the exits of the previ-

ous year. As a consequence the total normal cost will

not actually decrease from year to year as might be

inferred from the preceding paragraph.

It is theoretically possible to value an open group

rather than a closed group, making assumptions as to

new entrants in future years. Mr. Trowbridge does this

in his Demonstration I. In practice it is rarely, if ever,

done for private pension plans.

F r a n k L . G r i f f i n J r.

The author is to be complimented on a clear exposi-

tion of the nature o f various methods of budgeting pen-

sion costs. While the paper deals with matters largely

theoretical, and therefore does not lend itself to a dis-

cussion from the standpoint of the strictly practical

problems faced by consulting actuaries, nonetheless his

general approach, omitting the mathematical symbol-

ism, is sometimes found useful by consultants in dispel-

ling for their clients the technical mysteries of

different methods. Furthermore, an extension of the

principles set forth in the paper makes possible an

appraisal of the results obtained by using various meth-

ods in an actuarial valuation of costs, considering both

the nature of the employee group and the purpose to be

served by the particular valuation.

For his classification of funding methods, the author

has made use of the Equation of Maturity, C + d F = B ,

in which only the size of the ultimate contribution (C)

and fund (F) will vary according to the method. Using

the so-called mature population concept, he determines

the ultimate C and F, by means of which the funding

methods are classified in a logical order--namely, in

ascending order of F (descending order of C). Omitting

Classes V and VI, which were included in the paper fortheoretical reasons only, the remaining classes are: (I)

pay as you go, (1I) terminal funding, and two classes (Ill

and W) of funding in advance of retirement. The separa-

tion of funding in advance of retirement into Classes III

and IV was necessitated on the basis of the ultimate con-

tribution required, a point on which further comment

will be made later.

One or two comments relative to the mature popula-

tion concept which forms the basis of the author's pre-

sentation may be in order. A mature age distribution

and a stationary population are not one and the same

for purposes of a pension forecast, since the size of anemployee group may remain stationary indefinitely

without its having reached a mature age distribution.

The difference, of course, can be brought about by a

varying number of new hires each year or by hirings at

many different ages, rather than a uniform number of

hires each year at the youngest age of the service table

which is the unrealistic assumption inherent in the con-

ventional maturity concept.




In the case of a well established organization, the

assumption of a constant work force moving toward

maturity in its age distribution is probably as defensible

as any other approach. However, the mature age distri-bution which the group might be considered to reach

would not be of the form usually assumed, namely, pro-

portionate at all ages to the g column of the service

table. The latter would be true, as indicated in the pre-

ceding paragraph, only if all new entrants came into

service at the youngest age of the service table. If, for

example, a constant number of annual new entrants

were distributed in fixed ratios at each age from 20 to

40, the ultimate mature age distribution would be in

proportion to the 1~ column only at ages 40 and over.

Below age 40, the distribution would be in proportion

toy X

: Z ( H y + l y )y m

where H r is the percentage of total hi.rings at age y.

In practice, none of the ideal conditions of a

mature population (either initially or in the ultimate)

will ever be found. Notwithstanding this fact, the con-

cept may serve a useful purpose as a limiting value in a

pension projection. For example, if the actuary wishes

to compare the results of a valuation by any particular

cost method, with a projection of pension payouts con-

sidering future new entrants, the reasonableness of his

results for a going concern or the relative trend ofcosts by different methods may be made apparent.

Since the actuary is confronted, not with a mature

group, but with a group of unknown future age distribu-

tion and size, practical considerations usually dictate

that any valuation he makes (Class l]I or IV) be limited

to the group of employees existing on the date of valua-

tion, without allowance on any empirical basis for any

new entrants of the future. Depending on the actuarial

cost method, the resulting costs may or may not reason-

ably approximate the long range requirements, even in a

case where it is thought that the work force will remain

constant in the future; and a projection o f payouts (Class

I) or terminal funding requirements (Class II), taking

into account future new entrants on a reasonable basisfor maintaining the work force, may help to establish

the relative merits or deficiencies of different Class Iii

or IV valuation methods for a going concern.

Obviously, if we were in a position to predict the

new entrants of the future with any accuracy, the pro-

jected requirements by Class I or II would be exactly

equivalent financially to the contributions developed, in

turn, by the initial and successive future valuations of

the plan, by any valuation method selected. Therefore,

the result obtained by a particular method in a single

valuation, measured against a long range projection of

disbursements, affords an indication of the reliability ofsuch result in relation to future requirements, or, what is

the same thing, the relative trends which contributions

determined by different valuation methods will follow

in future years.

The accompanying chart, prepared for a large orga-

nization, sets out the projected payouts and terminal

funding requirements against the indicated annual con-

tributions determined from an initial valuation by the

entry age normal method. In this chart, the discounted

value of payouts into perpetuity, considering new

entrants, is practically identical to the discounted value

of contributions into perpetuity, i f su c h c o n t r i b u t i o n sd e t e r m i n e d i n r e s p e c t o f t h e p r e s e n t e m p l o y e e g r o u p

o n l y w e r e t o r e m a i n a t t h e i r o r i g i n a l l y d e t e r m i n e d l e ve l .

The propriety of the valuation method for a continuing

plan and a going concern is thus reasonably well

established. In contrast, if the " u n i t purchase method

had been employed in this case, the indicated level of

contribution (initially determined amount) would have

been much less, leading to the conclusion that contribu-

tions by such a method would increase in the future if

the group were to maintain its size.

I I. F u n d a m e n t a l s o f P e n si o n F u n d i n g 127



CHART 1

PROJECTION OF PENSION CONTRIBUTIONS, PAYOUTS, AND TERMINAL FUNDING REQUIREMENTS

A s s u m i n g : (1) Constant Work Force Supported by New Hires with Identical Entry Ages as the Original Group(2) Mortality, Disability, Withdrawal, and Interest as Assumed in Valuation and

(3) Indefinite Continuation of the Plan without Change

ollars 0 0 0 o m i t t e d )

3,2OO

2,800

2,4OO

2,009

1,600

1,200

I

3 0 Y e a r f u n d i n g o f a c c r u e d l i a b il i ty p l u s n o r m a l c o s t

Pension Pay-outs ]

Pay- o u t s u l t im at e leve l

Te m in al u n d n.9. u. l t. i m .at e ev .eJ ~

N o r m a l c o s t

I 1 1 2 1 3 I I s 1 6 1 7 1 8 1 9 1 1 0 1 I 1 6 1 1 2 o l 1 2 s l 1 3 01 1 3 s l 1 4 o l I ~ 1 I s 0 1Years rom Inceptionof Plan

Other actuarial assumptions being the same, the

entry age normal method always produces a higher

accrued or past service liability than the unit purchase

method. The relative size of the normal (or current ser-

vice) costs, however, will depend on the existing age

distribution on date of valuation. Examples of compara-

tive figures by the two methods, derived from other

cases, are as follows:

Past Service Normal Current

Liabili ty Service) ostCase A: Entry age normal.. $37,908,000 $2,461,000

Unit purchase .. .. . 21,895,000 2,401,000

Case B: Entry age normal.. $ 820,000 $ 76,000Unit purchase ..... 601,000 71,000

The wide difference in the results of initial valuation

by the two methods, when it is a certainty that all meth-

ods must produce the same capitalized value of contri-

butions, points up the absurdity of trying to compare

such results without recognizing the different purposes

they are intended to serve. The difference in purpose,

implicit in the author's separate treatment of Class 11I

and IV methods, may be stated briefly as follows. From

one viewpoint (that of a going concern and a continuing

plan), the funding requirements developed by the valua-

tion should take into account not only the past but also

the future requirements on a basis which will tend to

equalize long range trends in the age distribution. The

entry age normal method does this to the maximum

extent possible for a group assumed to be stationary insize. From another viewpoint (that of establishing liqui-

dation values under a terminat ing plan, i .e. accrued lia-

bilities without regard to the future), the requirements

developed by the valuation will take into account only

the past. The unit purchase method is the only one

which provides this particular answer, and it will do no

more.

The structure of Class IV methods, adapting them to

the requirements of a going conce rn is therefore such

128 S o c i e ty o f A c t u a r ie s 5 0 t h A n n i v e r sa r y M o n o g r a p h



Class Normal Cost ] Accrued Liability (Ultimate Fund)

Algebraically

1½ (or II minus)

2½ (or III minus) .......

21/2 (or IV minus) .......

r + 4

ar nr

r Ir _ a _ 5 Z l x r - x [ ( ix

a + 5

I ~ r - I

r-~-51 ~+5%-~1

aa +5: r-a-sIa + 5

r 4

E l x a x ~ r ~ l l X ~ . . . . . . ,r + l

1 r - I o

r _ a _ 5a~ +5 (x -a- 5 ) lx . r_~ l iix + ~_flx ii~r

r - I 0

Z l~ , -~ l i ix + Z lJ i~ r- -zla~+'..~-g 1~~.r---~a + 5 • aa 5 r-a-5~ a + 5

Numerically

1V2 (or II minus) ....... ~ $53,206 $ 401,542

2½ (or HI minus) ....... $35,410 $1,131,2082½ (or IV minus) ....... $30,978 $1,312,888

Mr. Link is also quite correct that the initial ly imma-

ture group approaches the ultimate mature state asymp-

totically from both sides. Under the conditions stated

for Table IV, the group passes from badly immature to

somewhat overmature, then back to slightly immature,

etc. This came as somewhat of a surprise to the author,

as it did to Mr. Link.

Mr. Foster's Classes 1Vg. and 2V2 can be presented

algebraically and numerica lly in the same fashion as theother methods have been analyzed in the paper. The fol-

lowing table presents the formulas for normal cost and

ultimate fund for the initially stationary population,

assuming that in Class 1V2 the funding period beyond

retirement is uniformly five years, and assuming in

Class 2V9. a waiting period of five years.

Class 21/9. naturally breaks into two subclasses,

depending on whether Class III or Class IV funding

becomes subject to the waiting period. Perhaps Mr. Fos-

ter would permit me to call these HI minus and IV

minus. The numerical illuslxation can be considered an

addition to Table II, and to the limiting situation inTable IV.

For practical work involving Class 2V2 it is not

uncommon to offset the cheapening of the funding

method due to exclusion of certain lives from the fund-

ing by overconservative assumptions with respect to the

included lives. This tends to remove the minus and

bring the funding closer to Classes III or IV.

Mr. Rae analyzes the various funding methods by

means of the concept of an ever changing closed

group. This analysis is particularly appealing because

it follows exactly the kind of group actually employed

in practical valuations. The open group approach to

which he refers is of considerable theoretical interest,

even i.hough the necessity for assumptions as to future

new entrants eliminates it for most practical work.

Dr. Nesbit t's (and Mr. Feraud's) general average

premium is of course a result of the open group

approach. The general average premium rff in the sta-tionary population assumed is equivalent to the pay-as-

you-go contribution, which is in turn equivalent to n ,

or what the paper refers to as Class IV normal cost,

plus interest on the Class IV accrued liability. If we

now shift our frame of reference and think o f nc instead

of re as the normal cost, the corresponding accrued

liability becomes 0. It is under these latter definitions

that the anticipated gains from future new entrants off-

set the shortage of funding in respect to the initial

group. Dr. Nesbitt states that in general he does not

advocate the discounting of such gains; I assume this

means that he ordinarily recommends that the accruedliability (in the sense used in the paper) should eventu-

ally be funded.

Mr. Griffin views Class III funding as essential ly ret-

rospective, looking back at benefits accrued. On the

other hand he thinks of the Class IV methods as funda-

mentally prospective, and points out that under certain

conditions the init ial normal cost is representative of the

ultimate cost. These conditions involve among others an

unchanging average entry age.

130 Soc ie ty o f Ac tuar ie s 50 th A nniversa ly Monograph



Suppose , however , t ha t p resen t h i r ing po l i c i e s i nd i -

ca t e a d i f fe ren t en t ry age for fu ture new en t ran t s t han

the ave rage for i n i t i a l pa r t i c ipan t s . In such cases the

above cha rac t e r i s t i c o f Class IV funding can be pre -se rved on ly by a ssuming for t he in i t i a l g roup the same

pa t t e rn o f h i ring ages a s i s i nd ica t ed for t he fu ture . I f i t

i s impor t an t t ha t fu ture normal cos t s rema in re l a t ive ly

cons tan t , t h i s modi f i ca t ion of t he usua l exac t en t ry age

m e t h o d w o u l d s e e m t o b e a p p r o pr i at e .

Dr . Sea l a tt empt s to pu t a p rac ti ca l empha s i s on w ha t

i s e ssen t i a l ly a t heore t i ca l pape r . Af t e r examining the

theore t i ca l poss ib i l i t i e s t o de t e rmine which a re accep t -

ab le to the Treasury , he produces a l i s t o f n ine prac t i -

ca l me tho ds , t h ree of which he l imi t s to ce rt a in types

of bene f i t fo rmulas .

Dr . Sea l ' s i nc lus ion of the imm edia t e ad jus tmen t

form of a t t a ined ag e normal in a l i s t o f p rac t ica l me tho ds

i s ra the r surpr i s ing , s ince immedia t e ad jus tment fo r

ga ins or l osses i s a s d i f f i cu lt to ma ke in th is m e thod as i t

i s i n t he aggrega te me thod . Pe rhaps Dr . Sea l i s t h ink ing

of immedia t e ad jus tment i n re spec t t o the pas t se rv i ce

por t ion ( th is appea rs t o be feas ib le ) , bu t w i th a spread of

ga ins or l osses a r i s ing f rom the fu ture se rv i ce por t ion .

Evident ly Dr . Sea l f i nds someth ing no t appa ren t t o

the au thor in the Bulle tin on 23 p) , l ead ing h im to the

conc lus ion tha t spread ad jus tment fo r ga ins i s no t

accep tab le under un i t c red i t funding . True , t he Bul le t in

does no t spec i f i ca l ly pe rmi t t he prac t i ce in ques t ion ; nor

does i t ru l e aga ins t i t . The same s i tua t ion ex i s t s i nrega rd to the t echnique desc r ibed by Mr . Immerwahr ,

w h i c h h e h a s f o u n d t o b e a c c e p t a b l e d e sp i t e i t s n o n -

inc lus ion in the Bulle tin.

Mr. Imme rwah r ' s r emarks cen te r a round the Trea -

sury regula t ions w i th re spec t t o ad jus tmen t fo r ac tua r ia l

ga ins . H i s two condi t ions under which the spread

adjus tment t echnique i s accep tab le appea r t o be e ssen-

t i a l ly the same . I f I unde rs t and h im cor rec t ly he s t a t e s

tha t spread ad jus tment i s accep tab le p rov ided tu rnover

i s adequa te ly recognized- -e i the r by a rea l i s t i c w i th-

drawa i a ssum pt ion , o r by suf f i c i en t e l imina t ion o f shor t

s e r v i c e e m p l o y e e s f r o m t h e f u n d i n g .I t is i n t e re s ting to no te tha t he has found accep tab le a

mod i f i ca t ion of imm edia t e ad jus tment . Thi s modi f i ca -

t ion appea rs t o amo unt t o the spread ing o f ga ins a r is ing

wi th in the in i t i a l acc rued l i ab i l i t y ove r t he min imum

funding pe r iod for such l i ab i l i t y , even though grea t e r

ga ins m ay oc cur in a pa r t icu la r yea r .

Mr . Weaver reaches the conc lus ion tha t i f ga ins p re -

d o m i n a t e t h e sp r e a d a d j u s t m e n t f o r m o f a n y f u n d i n g

m e t h o d p r o d u c e s a h i g h e r f u n d t h a n t h e c o r re sp o n d i n g

i m m e d i a t e a d j u s t m e n t f o r m ; b u t c o n v e r se l y a l o w e r

f u n d i s p r o d u c e d b y sp r e a d a d j u s t m e n t i f a s su m p t i o n s

a re unconse rva t ive and losses p reva i l . The va l id i ty o fM r . W e a v e r ' s c o n c l u s i o n c a n b e d e m o n s t r a t e d r a t h e r

eas i ly by s imple gene ra l r easoning . I f the re i s no change

in a ssumpt ions (and under these condi t ions a change

w o u l d s e e m t o b e a p p r o p r ia t e ) sp r e a d a d j u s t m e n t t e n d s

to exaggera t e the ove r funding a r i s ing f rom assumpt ions

tha t p rove to be too conse rva t ive , and a l so t ends to

accentua te any under funding a r i s ing f rom too l i be ra l

a ssumpt ions .

T h e r e a d e r o f M r . W e a v e r ' s d i s c u s s i o n sh o u l d r e a li z e

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

aggrega te me thod ( spread ad jus tment t echnique ) and

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

The f rozen in i ti a l l i ab i l it y fo rm of en t ry age norm al

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

cont r ibu t ion a s aggrega te , even i f t abu la r a ssumpt ions

a re no t rea l i zed ( sub jec t t o Mr . L ink ' s except ion a s to

mul t ip l e en t ry ages) .

The au thor does no t fee l pa r t i cu la r ly qua l i f i ed to

c o m m e n t o n M r . W i U i a m so n ' s o b se r v a t i o n s r e g a r d i n g

the funding of t he Fede ra l Civ i l Se rv ice Re t i rement

S y s t e m a n d O A S I . M r. W i l l ia m so n ' s c o m m e n t s b r i n g t o

mind , how ever , an ear l i e r s tudy of pens ion fu nding

which might w e l l be broug ht t o the a tt en t ion of t hose

interested in this subject . I refer to Actuar ia l S tudy

No. 10 of the Off i ce of t he Ac tua ry , Soc ia l Secur i tyBoard , en t i t led Var ious M ethod s of F inanc ing Old-

A g e P e n s i o n P l a n s M r . W i l l ia m so n , M r . R . J. M y e r s ,

and Mr . E . A . Rasor were the au thors o f t h is pamph le t ,

which i s an exce l l en t p r imer on funding me thod , wr i t -

t en in 1938 a t t he t ime of t he cont rove rsy ove r re se rve

f inanc ing o f OA SI .

S ince the publ i ca t ion of t he pape r t he Treasury pos i -

t i o n w i t h r e sp e c t t o m a x i m u m d e d u c t i o n s u n d e r i n d i -

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

t h e C o m m i ss i o n e r ' s a c q u i e sc e n c e i n th e Saalfie ld dec i -

s ion . I t now appea rs t ha t t he cont r ibu t ions ca l l ed for by

ind iv idua l l eve l p remium funding a re fu l ly deduc t ib l e ,e v e n i f i n e x c e s s o f t h e n o r m a l c o s t p l u s 1 0 % m a x i -

m u m f o r e n t ry a g e n o r m a l .

T h e a u t h o r w a n t s t o t h a n k t h e s e v e r a l p e r so n s w h o

par t i c ipa t ed in the d i scuss ion o f the pape r , each of

w h o m h a v e i n o n e w a y o r a n o t h e r a d d e d t o p u b l i sh e d

k n o w l e d g e r e g ar d i ng m e t h o d s o f p e n s i o n f u n d in g . E v e n

so the au thor wou ld l ike to echo Mr . L ink ' s s t a t ement t o

the e f fec t t ha t t he re i s a l ong w ay ye t t o go .

H. Fundamentals of Pension Funding 131



n d N o t e s

1. A p ecul i a r i ty o f the aggrega te me tho d i s t ha t t he

as sumpt ion of heav ie r dea th or w i thdrawa l ra t e ssom et imes l eads to a h ighe r in i ti a l con t r ibu t ion . T he

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

annui ty , t he reby inc reas ing the pe rcen tage k . The

inc rease in k may b e enou gh to of f s e t t he dec rease in

normal cos t and a cc rued l i ab i li t y .

2. ff i t seem s to the reader that froz en ini tia l l iabi l i ty

i s s o m e th i n g o f a m i s n o m e r f o r a m e t h o d u n d e r

which funding of the acc rued l i ab i li t y is con tem-

p l a te d , h e m a y p r e f e r t h e t e r m i n o l o g y s u g g e s t e d b yMr. Rae in TS A 1, 274. Fro zen ini tia l l iabi l i ty

might be be t t e r app l i ed to the Clas s I me thods

desc r ibed on page 33 .

3 . Sec t ion 29 . 23(p) 4 of Regu la t ions 111, as rev i s ed

by T . D . 5666 .

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