ACTUARIAL RESEARCH CLEARING HOUSE 1 9 8 8 VOL. 3

PRICING INSURANCE PRODUCTS USING CAPITAL BUDGETING

by M. Michel Rochette

ABSTRACT

The purpose o f t h i s paper i s f i r s t t o acqua in t a c t u a r i e s

w i th a p rocedure w i d e l y used i n t h e f i n a n c e area~ C a P i t a l

Budqet inq . Th is t echn ique i s u s e f u l f ~ t h e process o f d e c i s i o n

making when a f i r m - m a n u f a c t u r i n g o r f i n a n c i a l - dec ides t o commit

l o n g - t e r m funds t o a p r o j e c t .

Second, t he a p p l i c a b i l i t y o f t h i s p rocess t o t he p r i c i n g o f

an i nsu rance p roduc t w i l l be demonst ra ted a long w i t h an example,

F i n a l l y , a comparison o f t h e main f e a t u r e s o f c a p i t a l

budget ing t o an e x i s t i n g a c t u a r i a l t echn ique - Anderson 's method -

w i l l be made.

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CAPITAL BUDGETING

Cap i ta l budgeting i s a procedure used more and more o f t e n by

manufactur ing f i r m s in o r d ~ to m a k e long- term investment

dec is ions wh i le maximizing the value of the f i r m . Some examples

of c a p i t a l budgeting dec i s ions are the f i n a n c i a l consequences of

buying a p iece of equipment, b u i l d i n g a new p l a n t , or en te r i ng a

new market.

The dec is ion as t o whether or not a company should undertake

a long- term p r o j e c t i s based on the net presen t valqe c r i t e r i o n

(NPV) or the p r o j e c t ' s i n t e r n a l r a t e o f r e t u r n . Both of these

c r i t e r i a w i l l lead to the same conclus ion in most circumstances

but the former i s p r e f e r r e d s ince i t w i l l maximize d o l l a r s t o the

f i r m and not r e l a t e d percentages.

A b r i e f d e s c r i p t i o n of the NPV method f o l l o w s . I f the

present value of the p r o j e c t ' s f u t u r e cash f l ows i s g rea te r than

i t s i n i t i a l cash investment - f o r example, cost of buying a p iece

of equipment or cost of b u i l d i n g a new p lan t - , then the p r o j e c t

w i l l be p r o f i t a b l e f o r the f i r m t h a t undertakes i t . A lso, i f the

f i r m can choose between many p r o j e c t s , i t w i l l i nves t in those

t h a t have the l a r g e s t net present values up t o the l i m i t of the

company's investment budget. By doing sop the company w i l l

maximize the value of the shareholders" c la im in the long run.

In order t o apply t h i s procedure, t h ree elements are

necessary: an ~st imate of the p e r i o d i c n~t cash f l ows generated

throughout the l i f e of a p r o j e c t , a d iscount r a t ~ in order t o take

t h e i r present va lue, and a c r i t e r i o n in order t o make the f i n a l

dec i s i on .

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CASH FLOWS

The cash f l o w s can be d i v i d e d i n two groups: t hose t h a t a re

r e l a t e d t o t h e o p e r a t i o n ~ o f • p r o j e c t and those genera ted f rom

f i s c a l p o l i c i e s .

The fo rmer i n c l u d e s the d i r e c t revenues genera ted f rom t h e

p r o j e c t - s a l e s volume - minus t h e d i r e c t expenses i n c u r r e d i n

o r d e r t o r e a l i z e those s a l e s . For example, t h e c o s t s i n c u r r e d i n

o rde r t o produce the goods which a re s o l d a re the o n l y expenses

which should be cons i de red . In f a c t , i n d i r e c t or overhead

e×penses should never be taken i n t o account a t t h i s s tage because

t h e y can d i s t o r t t he d e c i s i o n process depending on how t h e y a re

a l l o c a t e d . A l so , i t i s i m p o r t a n t t o base t h e c a l c u l a t i o n s on

a f t e r - t a x cash f l o w s because i t i s necessary t o repay t h e i n i t i a l

i nves tment which i s made w i t h a f t e r - t a x money.

The second p a r t o f t h e cash f l o w s a s s o c i a t e d w i t h a p r o j e c t

comes f rom the d e d u c t i b i l i t y f o r t ax purposes o f t he d e p r e c i a t i o n

expenses o f the d e p r e c i a b l e goods used i n t h e p r o d u c t i o n p rocess .

In f a c t , t h i s deduc t i on t imes t h e marg ina l t a x r a t e o f t h e company

a l l o w s i t t o reduce the t a x e s t h a t i t w i l l have t o pay t o t h e t a x

a u t h o r i t y g iven i t s revenues and o t h e r expenses; c o n s e q u e n t l y ,

t h i s t ax s h i e l d i s an a d d i t i o n a l cash i n f l o w t h a t i s " r e c e i v e d " o r

" n o t p a i d " by t he company when the p r o j e c t i s i n f c~ce .

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DISCOUNT RATE

Once the cash f l ows f o r each per iod - month, qua r t e r , year -

throughout the l i f e of the p r o j e c t have been determined, i t i s

necessary t o d iscount them t o the present . The d iscount r a t e i s a

weighted average of the cos ts o f the d i f f e r e n t sources of

f i n a n c i n g a v a i l a b l e t o the f i r m inc lud ing bonds,

preferred and common s tock , and i n t e r n a l earn ings. The value

c a l c u l a t e d by the ch ie f f i n a n c i a l o f f i c e r of the company i s equal

t o the r i s k - f r e e r a t e p lus a r i s k premium determined by the market

when i t lends funds t o the company. I t i s based on an eva lua t i on

of the r i s k i n e s s of f u t u r e cash f l ows . This i s the d iscount r a t e

t h a t should be used unless the cash f l ows of the p r o j e c t inc lude

some r i s k margins in which case the r i s k - f r e e r a t e should be the

d iscount r a t e .

DECISION CRITERION

F i n a l l y , the p r o j e c t i s a p r o f i t a b l e investment dec is ion i f

the present value of the net cash f l ows at the aforementioned

d iscount r a t e i s g r e a t e r than the i n i t i a l investment requ i red t o

undertake the p r o j e c t . (NPV > O)

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PRICING AN INSURANCE CONTRACT USING CAPITAL BUDSETING

I t becomes ex t reme ly easy t o app ly t he p reced ing p r i n c i p l e s

i n o rde r t o p r i c e an i nsu rance p r o d u c t . The t h r e e e lements t h a t

a re necessary t o app ly t h i s p rocedure can be r e s t a t e d as f o l l ~ s s

CASH FLI]~;S

In the case o f a l i f e i nsu rance p o l i c y , t he main ~ash i n f ~ o ~

t h a t i s r e c e i v e d year a f t e r year i s the premium. The investment

income i s o f course a cash i n f l o N but i t i s taken i n t o account

th rough the d i scoun t r a t e . The main o p e r a t i n g y e a r l y ~ s h

o u t f l o w s a re the d i r e c t a d m i n i s t r a t i v e expenses, t he premium t a x ,

the death and t he su r rende r c la ims , t he pa id d i v i d e n d s , and t he

t axes .

A lso, the insurance company can deduct , on i t s income tax

r e t u r n , t he i n c r e a s e in t he p o l i c y §~@cal r e s e r v e . I t p l a y s t h e

same r o l e as the d e p r e c i a t i o n e~pense ~or a manu fac tu r ing c ~ a n y .

Both reduce t he t a x a b l e p r o f i t and t he t a x e s t h a t have t o be p a i d .

In mathematical terms, the net cash f l ~ s a f t e r t a x e s o f t he

i nsu rance p roduc t i n year t a re :

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C F L O W S ~ = (CF I I ~ - CF O U T ~ ) I ( 1 - T ) + T $ F I S C A L RESERVlE~

where

CF INt= GROSS PFIEMIUMS~

CF OUT~= DIRECT ADMINISTRATIVE EXPENSES~

+ CLAIMS~: DEATHS~ + ~ S ~ + PAID DIVIDENDS~

T= MARGINAL TAX RATE Of: THE COMPANY

DISCOUNT RATE

The r i s k - f r e e r a t e shou ld be used t o d i s c o u n t t he expected

cash f l o w s over the t o t a l d u r a t i o n of t he i nsu rance p roduc t . In

f a c t , i f the a c t u a r y has a l r e a d y taken i n t o account an a p p r o p r i a t e

p r o v i s i o n f o r adverse d e v i a t i o n s in h i s / h e r e s t i m a t i o n o f t he

m o r t a l i t y and lapse r a t e s , i t i s thus unnecessary t o add so re

conserva t i sm t o the premium by d i s c o u n t i n g the cash ~lows a t a

r a t e h i g h e r than the r i s k - f r e e r a t e .

DECISION CRITERION

F i n a l l y , t he a c t u a r y w i l l de te rmine t h e g ross premium such

t h a t the p resen t v a l u e o f t he f u t u r e ne t cash f l o w s s i n u s t he

a c q u i s i t i o n expenses - inves tment by an i nsu rance company i n t o

p o l i c y - i s p o s i t i v e . A lso , he /she can se t an e x p l i c i t p r o f i t

o b j e c t i v e f o r the va lue o f t he NPV. He/she should no t rely and

a n t i c i p a t e t he i m p l i c i t p r o f i t t h a t i s b u i l t i n t o the p roduc t

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th rough c o n s e r v a t i v e assumpt ions. I f t h e ~ were se t t o zero~

the insu rance c o n t r a c t i n t e r n a l r a t e o f r e t u r n would be ec~JaI t o

the d i s c o u n t r a t e . Th is i s an a c c e p t a b l e s i t u a t i o n but i t does

not add v a l u e t o t he i nsu rance cod~pany i n t h e long run.

In o r d e r t o i l l u s t r a t e t h e p reced ing i deas , the h y p o t h e t i c a l

l i f e i nsu rance p lan t h a t f o l l o w s w i l l be u s e f u l . I t s main

c h a r a c t e r i s t i c s a re :

- O r d i n a r y l i f e i nsu rance p o l i c y issued t o a man, aged 30.

- B e n e f i t s :

-Death b e n e f i t : $10 000, l e v e l t h roughou t the l i f e o f t he

p o l i c y and payab le a t t he end o f t he year

of death .

-Cash va lues are equal t o t he r e s e r v e and are payab le a t t he

end o( the year o f w i t h d r a w a l .

- D i v i d e n d s are pa id a t t he end o f each p o l i c y yea r .

-Assumpt ions:

- M o r t a l i t y r a t e s based on the CIA 69-75, Se lec t & U l t i m a t e .

-Lapse r a t e s a re equal t o 10%,87.,6%, and 5% t h e r e a f t e r .

-Expenses: - A c q u i s i t i o n s : $100 + 60% o f t he g ross premium

- A d m i n i s t r a t i v e : 5% o f t h e g ross premium in a l l

p o l i c y yea rs .

- C a s h - f l o w o b j e c t i v e : $5 per 1000 o f i nsu rance f o r c e .

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L e t ' s a l s o assume t h a t t h e r i s k - f r e e r a t e i s 8Z i n t h e

economy. At t h e same t i m e , t h e company w i l l i n v e s t t he proceeds

f rom the p o l i c y in bonds which w i l l y i e l d 11Z. Through t h e i n s i d e

b u i l d - u p o f t h e cash va l ue - which I assumed i s nan t a x a b l e - and

t h e d i v i d e n d s d e c l a r e d , t h e p o l i c y h o l d e r s w i l l r e c e i v e 9"£ on t h e i r

i n ves tmen t s .

For a $10 000 l i f e i n s u r a n c e p o l i c y , t h e g ross premium t h a t

w i l l make the NPV equal t o t h e cash-- f low o b j e c t i v e i s $121.

I f an a c t u a r y were t o p r i c e the i nsu rance p roduc t o f t h e

p reced ing example us ing Anderson 's method - see d i f f e r e n c e s be low-

t he gross premium ob ta i ned would be $124. Anderson 's method leads

t o a more c o n s e r v a t i v e va l ue f o r the g ross premium than the one

determined w i t h t he c a p i t a l budget ing t e c h n i q u e . The d i f f e r e n c e

i s due p a r t l y t o t he f a c t t h a t Anderson 's method d i s c o u n t s

cons tan t p o s i t i v e ea rn ings w h i l e c a p i t a l budget ing d i s c o u n t s

i n c r e a s i n g l y n e g a t i v e net cash f l o w s . The f o l l o w i n g t a b l e shows

the f i r s t t w e n t y - s i x va lues o f t he ea rn i ngs and the co r respond ing

cash f l o w s a s s o c i a t e d w i t h t he same p r o d u c t .

AGE E A R N I N G S NET CASH FLOWS /

30 ( 1 3 8 . 8 9 ) ( 4 7 . 5 7 ) 31 3 3 . 9 0 8 6 . 8 4 :32 3 3 . 9 0 8 0 . 9 4 33 3 3 . 9 0 7 5 . 0 5 34 3 3 . 9 0 67 . 04 35 :3:3.90 5 8 . 6 6 36 3 3 . 9 0 5 0 . O0 37 3 3 . 9 0 4 0 . 9 4 38 3 3 . 9 0 3 1 . 2 7 39 3 3 . 9 0 21 . 20 40 :33 .90 10. 72 41 3 3 . 9 0 ( 0 . 2 7 ) 42 33.90 ( 11 . 80)

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2) Al~ays c o n s i d e r s a f t e r - t a x

cash f l o ~ s .

2) T a x a t i o n i s o f t e n i i n o r e d .

3) The d i s c o u n t r a t e i s based on

the marg ina l cos t o f c a p i t a l o r i s

the r i s k - f r e e r a t e .

3) The d i l c o u n t r a t s i s l ~ f t l n

de te rmined • s th@ r a t e o~ r e t u r n on

the s u r p l u s ( e q u i t y ) o f t he company.

There are a few reasons ~hy c a p i t a l budge t i ng s i g h t be • mQre

a p p r o p r i a t e method t o p r i c e an insu rance p r o d u c t .

F i r s t p e a r n i n g s as usL~d i n AfldL~rson's l u l t h o d cannot be splN1t

l i k e cash and they can be man ipu la ted more e a s i l y than t h e i r

cc~rr espondi ng cash f l o ~ s th rough t he cho ice o f d i f f e r e n t

assumpt ions i n the c a l c u l a t i o n o f t he r e s e r v e o f a p o l i c y .

Second, an i nsu rance company does not i n v e s t s u r p l u s ( e q u i t y )

when z t xssues a c o n t r a c t as i s assumed i n Anderson 's method. In

r e a l i t y , i t I n i t i a l l y i n v e s t s cash mhhen i t u n d e r w r i t e s a

p ropos l t i on and issues a p o l i c y , and then pays i n cash a

commission t o the agent or t o the b r o k e r . So, i ns tead o f p r i c i n g

• p roduc t us ing ea rn i ngs , a c t u a r i e s should de te rm ine t he premium

such t h a t t he i n i t i a l ne t cash inves tment (i~hich i s equal t o the

a c q u i s i t i o n expenses minus t he f i r s t premium) w i l l be r ~ a i d by

the r e a l f u t u r e a f t e r - t a x cash f l o ~ s over a l l p o l i c y years Nhlch

~ i l l a f f e c t t he i nsu rance company ~hen i t i s sues • p o l i c y o f t h i s

t ype .

T h i r d , i n one o f t he S o c i e t y c~ i~c tuar l ss " s tudy no tes on

p r i c i n g us ing Anderson 's method, i t i s ment ioned t h a t t he d i s c o u n t

r a t e shou ld " r e p r e s e n t t he i n t e r e s t expected by the company o~

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s u r p l u s i n v e s t e d in new b u s i n e s s . " Th is approach i s u n c l e a r and

can lead t o d i f f e r e n t r a t e s . F i r s t , t he d i s c o u n t r a t e can be

determined f rom the va l ues shown in t he f i n a n c i a l s ta temen ts .

However, i t would r e ~ l e c t pas t r e s u l t s and i t would be

i n a p p r o p r i a t e t o use i t t o d i s c o ~ n t f u t u r e v a l u e s . Second, t he

r a t e o f r e t u r n on e q u i t y cou ld be used as t he d i s c o u n t r a t e .

However, i t s va lue can v a r y whether i t i s determined by management

or by t he s h a r e h o l d e r s th rough t h e p r i c e o f t he coi4pany's sha res

on the markets i f they a re p u b l i c l y t r a d e d . A lso , t h i s assessment

a p p l i e s t o the company as a whole and t h i s r a t e i n c l u d e s r i s k

premiums which a re unnecessary i f t he a c t u a r y has a l r e a d y taken

c o n s e r v a t i v e m o r t a l i t y , i n t e r e s t , and expense assumpt ions f o r t h e

i nsu rance p lan .

CONCLUSION

F i n a l l y , p r i c i n g an insu rance p lan us ing the t r a d i t i o n a l

Anderson method OF a m o d i f i c a t i o n o f i t i s l e s s a p p r o p r i a t e

because i t i s concerned w i t h the e a r n i n g s a s s o c i a t e d w i th a p o l i c y

r a t h e r than the cash f l o w s a f f e c t i n g t he company. Earn ings can be

c a l c u l a t e d but they do no t e x i s t ! They a re mere ly an accoun t ing

i n v e n t i o n t h a t smooths revenues and expenses. They do no t r e f l e c t

t h e d a i l y o p e r a t i o n s o f a company; consequen t l y , t he cash f l o w s o f

a p roduc t shou ld be the a c t u a r y ' s p r imary concern. He/she shou ld

use them t o p r i c e a p roduc t a p p l y i n g t h e p r i n c i p l e s o f c a p i t a l

budget ing . Then, the a c t u a r y can de te rm ine t he impact o f t h e

i n t r o d u c t i o n o f a p roduc t on the ea rn ings o f t he company. Not t h e

o t h e r w a y a r o u n d !

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BIBLIOGRAPHY

-Par t 7 Study Note: Gross Premiums fo r Ind iv idua l L i f e In iurance,

Society o~ Actuar ies, Chicago, I l l i n o i s .

-Essen t ia l s o# Financia l Manaq~ment by George E. Pinches, Second

Ed i t i on , Harper & Row Publ ishers, New York, 1986.

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