PRINCING INSURANCE PRODUCTS USING CAPITAL BUDGETING

A C T U A R I A L RESEARCH CLEARING HOUSE 1 9 8 8 VOL. 3 PRICING INSURANCE PRODUCTS USING CAPITAL BUDGETING

by M. Michel Rochette

ABSTRACT

The with

a

purpose of procedure

Budqetinq.

this

paper

is

first

used

in

the

finance

area~

CaPital

f~

the process of

decision

widely

This technique is

making when a f i r m

useful

to acquaint

-manufacturing or financial

-

actuaries

decides to

commit

long-term funds to a project.

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

a

will

this

process to

the

pricing

of

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

comparison

b u d g e t i n g t o an e x i s t i n g

of

of

the

actuarial

1 7 9

features

technique -

be made.

-

main

-

of

capital

A n d e r s o n ' s method -

CAPITAL BUDGETING

Capital

budgeting i s

manufacturing

firms

in

a p r o c e d u r e used more and more o f t e n by ord~

make

to

d e c i s i o n s w h i l e maximizing t h e value o f of

capital

long-term

the firm.

S o m e examples

budgeting d e c i s i o n s are the f i n a n c i a l

buying a p i e c e o f

investment

consequences

equipment, b u i l d i n g a new p l a n t ,

or

entering

of a

new m a r k e t .

The a

d e c i s i o n as t o whether o r n o t a company should u n d e r t a k e

long-term project is

(NPV)

based

or t h e p r o j e c t ' s

criteria

will

on t h e n e t presen t

internal

lead t o t h e

the former i s

firm

and n o t r e l a t e d p e r c e n t a g e s . brief

its of

initial

p r e f e r r e d since i t

description

present value of

cash i n v e s t m e n t -

be p r o f i t a b l e

firm

can choose between

that

have

the

for

investment

order

necessary:

to

maximize d o l l a r s

method

follows.

f u t u r e cash f l o w s i s for

example, c o s t o f then

the f i r m t h a t undertakes i t . many p r o j e c t s ,

budget.

it

will

apply

of

present value,

By

doing

sop

the shareholders" claim in

an ~ s t i m a t e o f

throughout the l i f e their

NPV

this the

procedure,

If

and a

decision.

-

1 8 0

-

in

than

the

project

Also,

if

the

those of

the

company w i l l

the long run.

three

a discount rat~ in

criterion

the

buying a p i e c e

invest in

the

to the

greater

elements

p e r i o d i c n~t cash f l o w s

a project,

these

circumstances

l a r g e s t n e t p r e s e n t v a l u e s up t o t h e l i m i t

maximize t h e v a l u e o f In

the

will

Both o f

most

b u i l d i n g a new p l a n t - ,

will

company's

of

the project's

equipment o r c o s t o f

return.

same conclusion in

but

A

rate of

valqe c r i t e r i o n

order t o

are

generated

order to take make the f i n a l

CASH FLOWS

The

cash f l o w s can be d i v i d e d

related fiscal

to

the operation~ of

two groups:

project

those that

and t h o s e

are

generated from

policies.

The

former includes the

project

-

s a l e s volume

-

to

realize

order

to

p r o d u c e t h e goods

which

should

eﾃ用enses they

those sales.

be

after-tax

it

is

is

important

made w i t h

The second p a r t

of

expenses

of

In fact,

this it

authority

to

fact,

is

to

base

this

or

for

s t a g e because they are

repay the initial

a

the production process. of

reduce t h e t a x e s t h a t

pay t o

its is

r e v e n u e s and

an a d d i t i o n a l

will

other

-

181

have t o

-

t h e company the tax

expenses; c o n s e q u e n t l y ,

cash i n f l o w

by t h e company when t h e p r o j e c t

project

the depreciation

deduction times the marginal tax rate

given

on

money.

tax purposes of

it

in

overhead

the calculations

necessary to

after-tax

in

t h e o n l y expenses

indirect

account at

t h e d e p r e c i a b l e goods used i n

tax shield

"not paid"

expenses i n c u r r e d

t h e cash f l o w s a s s o c i a t e d w i t h

from the deductibility

from the

p r o c e s s d e p e n d i n g on how

cash f l o w s b e c a u s e i t

i n v e s t m e n t which

allows

In

the decision

Also,

direct

which a r e s o l d a r e

s h o u l d n e v e r be t a k e n i n t o

allocated.

revenues generated

For example, t h e c o s t s i n c u r r e d

considered.

can d i s t o r t

comes

direct

minus t h e

order

this

窶｢

in

is

that in

is

"received"

fc~ce.

or

DISCOUNT RATE

Once t h e cash f l o w s f o r throughout

the life

of

each p e r i o d -

the

month,

q u a r t e r , year

p r o j e c t have been d e t e r m i n e d ,

it

n e c e s s a r y t o d i s c o u n t them t o t h e p r e s e n t .

The d i s c o u n t r a t e

weighted

different

average

financing

and

calculated

of that

costs to

by t h e c h i e f

of

firm

and

financial

r a t e plus a r i s k

officer

earnings.

margins i n

a of

bonds, The v a l u e

o f t h e company i s

equal

premium d e t e r m i n e d by t h e market It

f u t u r e cash f l o w s .

is

based on an e v a l u a t i o n

This i s

should be used u n l e s s t h e cash f l o w s o f

some r i s k

is

is

sources

including

internal

l e n d s f u n d s t o t h e company.

the r i s k i n e s s of

the

the

common s t o c k ,

to the risk-free it

the

available

preferred

when

of

-

which case t h e r i s k - f r e e

the discount rate

the

project

r a t e should

include be

the

discount r a t e .

DECISION CRITERION

Finally, the

the project

present value of

discount rate

is

is

the

a profitable n e t cash

flows at the

g r e a t e r than t h e i n i t i a l

undertake the p r o j e c t .

(NPV > O)

-

182-

investment

decision

if

aforementioned

investment

required

to

PRICING AN INSURANCE CONTRACT USING CAPITAL BUDSETING

It

becomes e x t r e m e l y e a s y t o

in order to price are

apply the

an i n s u r a n c e p r o d u c t .

necessary to apply this

preceding

The t h r e e

principles

elements

that

p r o c e d u r e can be r e s t a t e d a s f o l l ~ s s

CASH FLI]~;S

In that

is

income

t h e case o f

received year after is

through

of

the

outflows the

a life

insurance policy, year is

t h e premium.

course a

cash i n f l o N b u t

discount

rate.

are the direct

The

it

main

administrative

d e a t h and t h e s u r r e n d e r

t h e main ~ash i n f ~ o ~

claims,

is

The

investment

taken into

operating

account

yearly

~sh

e x p e n s e s , t h e premium t a x ,

the paid dividends,

and t h e

taxes.

Also, return,

t h e i n s u r a n c e company

the increase in

same r o l e

the policy

ยง~@cal r e s e r v e .

its

income

tax

It

plays

the

as t h e d e p r e c i a t i o n e~pense ~ o r a m a n u f a c t u r i n g c ~ a n y .

Both r e d u c e t h e t a x a b l e p r o f i t

In

can d e d u c t , on

mathematical terms,

insurance product in year t

and t h e t a x e s t h a t

t h e n e t cash f l ~ s are:

-

183

-

have t o

after

be p a i d .

taxes of the

CFLOWS~

=

(CF

II~

-

CF

OUT~)I(1-T)

+

T$

FISCAL

RESERVlE~

where

CF I N t = GROSS PFIEMIUMS~ CF OUT~= DIRECT ADMINISTRATIVE EXPENSES~ CLAIMS~:

+

DEATHS~ + ~

S

~

+ PAID DIVIDENDS~

T= MARGINAL TAX RATE Of: THE COMPANY

DISCOUNT RATE

The

risk-free

rate

s h o u l d be

cash f l o w s o v e r t h e t o t a l fact,

if

used t o d i s c o u n t t h e

d u r a t i o n of

the insurance

t h e a c t u a r y has a l r e a d y t a k e n i n t o

provision

for

adverse deviations

in

mortality

and

lapse rates,

thus

conservatism rate

to

it

t h e premium

is

product.

In

a c c o u n t an a p p r o p r i a t e

his/her

estimation

unnecessary to

by d i s c o u n t i n g t h e

h i g h e r than the r i s k - f r e e

expected

of

add

the sore

cash ~ l o w s a t

a

rate.

DECISION CRITERION

Finally, that

the

the actuary will

present value

acquisition

expenses -

policy

positive.

-

objective anticipate

is

for

of

the future

investment Also,

the value of

the implicit

determine the gross

h e / s h e can

profit

-

n e t cash f l o w s

by an i n s u r a n c e

t h e NPV.

1 8 4

that

-

premium such sinus the

company

s e t an e x p l i c i t

into

profit

He/she should

n o t r e l y and

is

the

built

into

product

through

conservative assumptions.

the insurance contract the

discount rate.

internal

This is

If

rate

the

~

were s e t t o

of return

would be

an a c c e p t a b l e s i t u a t i o n

zero~

ec~JaI

but

it

to

does

n o t add v a l u e t o t h e i n s u r a n c e cod~pany i n t h e l o n g r u n .

In life

order to

insurance

characteristics

-Ordinary life

illustrate plan

the preceding ideas, the hypothetical

that

follows

will

be

useful.

Its

main

are:

insurance policy

i s s u e d t o a man, aged 30.

-Benefits: -Death benefit:

$10 000, policy of

level

throughout the life

and p a y a b l e a t

t h e end o f

of

the

the year

death.

-Cash v a l u e s a r e e q u a l t o t h e r e s e r v e and a r e p a y a b l e a t end o ( t h e y e a r o f

the

withdrawal.

-Dividends are paid at

t h e end o f each p o l i c y

year.

-Assumptions: -Mortality

r a t e s based on t h e CIA

-Lapse r a t e s a r e equal t o -Expenses:

69-75, S e l e c t & U l t i m a t e .

10%,87.,6%, and 5% t h e r e a f t e r .

-Acquisitions:

$100 + 60% o f

-Administrative:

5% o f t h e g r o s s premium i n policy

-Cash-flow objective:

$5 p e r 1000 o f

-

1 8 5

t h e g r o s s premium

-

years. insurance force.

all

Let's economy.

also At

of

that

t h e same t i m e ,

from the p o l i c y build-up

assume

in

the

rate

t h e company w i l l

bonds w h i c h w i l l

t h e cash v a l u e -

the dividends declared,

risk-free

yield

which I

is

invest

11Z.

the

8Z

in

proceeds

Through t h e

assumed i s

inside

nan t a x a b l e -

the policyholders

will

F o r a $10 000 l i f e

insurance policy,

the gross

make t h e NPV e q u a l

to

the

and

r e c e i v e 9"ÂŁ on t h e i r

investments.

will

If

an

a c t u a r y were

t h e cash--flow o b j e c t i v e to

price

the

t h e g r o s s premium o b t a i n e d w o u l d be $124. a more c o n s e r v a t i v e v a l u e

determined is

due

with

partly

constant

first

cash f l o w s

to

positive

increasingly the

the capital the

fact

negative net

while

cash f l o w s .

values of

associated with

AGE

that

technique.

Anderson's

below-

capital

t h a n t h e one The d i f f e r e n c e

method

discounts

budgeting

discounts

The f o l l o w i n g

table

shows

t h e e a r n i n g s and t h e c o r r e s p o n d i n g

t h e same p r o d u c t .

E A R N I N G SNET CASH FLOWS (138.89) 33.90 33.90 33.90 33.90 :3:3.90 33.90 33.90 33.90 33.90 :33.90 33.90 33.90

-

the

A n d e r s o n ' s method l e a d s

/

30 31 :32 33 34 35 36 37 38 39 40 41 42

that

$121.

see d i f f e r e n c e s

t h e g r o s s premium

budgeting

earnings

twenty-six

for

is

insurance product of

p r e c e d i n g e x a m p l e u s i n g A n d e r s o n ' s method -

to

premium

186-

(47.57) 86.84 80.94 75.05 67 . 04 58.66 5 0 . O0 40.94 31.27 21 . 2 0 10. 72 (0.27) ( 11 . 80)

43 44 45 46 47 48 49 50 51 52 53 54 55

33.90 33.90 33.90 33.90 33.90 33.90 33.90 33.90 33.90 33.90 33.90 33.90 33.90

(23.85) (36.75) (51.53) (65.90) (80.97) (gb. b2) (I12.7&) (129.40) (147.07) (166.15) (IE~.83) (208.72) (231.27)

In o r d e r t o see t h e u s e f u l n e s s o f t h e two methods, a brief

comparaison o f

Anderson's price

insurance

principles

of

make

them.

method,

as

traditionally

products,

capital

let's

is

budgeting.

a

used

direct

by

actuaries to

application

HoNever, i t

is

of

dif;erent

the

i n some

respects:

CAPITAL BUDGETING

1) the

Determines initial

the cash

ANDERSON" S METHOD

NPV b y

subtracttn

investment

9

from

1)

Dtt~mines

the

Initial

{r~

the

t h e ;a~h f l o w s

stu-Dlus

over t h e d u r a t i o n o f

the product or

value

CAPITAL i~Ji)~TIN6 VS ~UIDEI~'S METHOD

tmmty

-

187-

equivalent

MPV by ~ u b t r a c t t n g

the present value of

sef-vice.

the

investment

of

the

years

larninGs or

less.

usually

of

the

prel~ent over

2)

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

2)

Taxation is

often

3)

The d i l c o u n t

rats

iinored.

cash f l o ~ s .

3)

The d i s c o u n t r a t e

the marginal cost the risk-free

is

based on

of capital

or

is

d e t e r m i n e d • s th@ r a t e

rate.

the surplus

T h e r e a r e a f e w r e a s o n s ~hy c a p i t a l appropriate

method t o

Firstp like

cash and

t h e y can cash

assumptions in Second, zt

reality,

•

flo~s

and to

the

acquisition the real ~ill

as i s

issues

a

t h e agent or

initial

be • mQre

the

cash

policy, to

c a n n o t be splN1t

more e a s i l y

assumed

choice

than

of

their

different

invest

surplus

mhhen and

it

then So,

n e t cash i n v e s t m e n t

pays

t h e i n s u r a n c e company ~hen i t

in

a

cash

a

instead of pricing the

(i~hich i s e q u a l

premium) w i l l over all

In

underwrites

should determine

cash f l o ~ s

(equity)

i n A n d e r s o n ' s method.

the broker.

actuaries

after-tax

on

t h e company.

the reserve of a policy.

e x p e n s e s minus t h e f i r s t

future

affect

of

invests

product using earnings,

such t h a t

through

the calculation

Initially

p r o p o s l t i on commission

AfldL~rson's l u l t h o d

an i n s u r a n c e company does n o t

it

o~ r e t u r n

of

budgeting sight

be m a n i p u l a t e d

xssues a c o n t r a c t

(equity)

l~ftln

an i n s u r a n c e p r o d u c t .

e a r n i n g s as usL~d i n

cc~rr e s p o n d i ng

when

price

is

policy

premium to

the

be r ~ a i d

by

years

issues • policy

Nhlch of

this

type.

Third, pricing rate

in

one o f

t h e S o c i e t y c~

u s i n g A n d e r s o n ' s method,

should

"represent the

-

it

interest

1 8 8

-

is

i~ctuarlss"

study notes

mentioned t h a t e x p e c t e d by

on

the discount

t h e company

o~

s u r p l u s i n v e s t e d i n new b u s i n e s s . " can

lead

to different

determined

from

However,

it

rates.

the values

would

rate

of

return

However, i t s

First,

shown

re~lect

i n a p p r o p r i a t e t o use i t

This approach i s the

c o u l d be

is

o r by t h e s h a r e h o l d e r s t h r o u g h t h e p r i c e

applies

to

premiums

they are publicly

t h e company

as a w h o l e

conservative

mortality,

interest,

and

can

be

statements.

it

values.

would

be

Second,

the

the discount

rate.

d e t e r m i n e d by management of

traded.

which a r e u n n e c e s s a r y i f

and

used as

v a l u e can v a r y w h e t h e r i t

on t h e m a r k e t s i f

financial

results

to disco~nt future

on e q u i t y

discount rate

in the

past

unclear

the

coi4pany's

Also,

and t h i s

this

rate

shares

assessment

includes risk

t h e a c t u a r y has a l r e a d y

taken

and e x p e n s e a s s u m p t i o n s f o r

the

insurance plan.

CONCLUSION

Finally, Anderson because i t rather

pricing

method is

OF

an

insurance plan

a modification

concerned w i t h

c a l c u l a t e d b u t t h e y do n o t e x i s t !

the daily a

operations of

them t o p r i c e

Then, t h e

introduction

of

other

less

They a r e m e r e l y

appropriate

E a r n i n g s can be an

accounting

They do n o t r e f l e c t

a company; c o n s e q u e n t l y , t h e cash f l o w s o f primary concern.

a product applying

budgeting.

is

t h e company.

smooths r e v e n u e s and e x p e n s e s .

p r o d u c t s h o u l d be t h e a c t u a r y ' s

use

it

the traditional

the earnings associated with a policy

t h a n t h e cash f l o w s a f f e c t i n g

invention that

of

using

a c t u a r y can

He/she s h o u l d

the principles

determine the

of

impact o f

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

way around!

-

1 8 9

-

capital the

Not t h e

BIBLIOGRAPHY

- P a r t 7 Study Note:

Gross Premiums f o r I n d i v i d u a l L i f e I n i u r a n c e , S o c i e t y o~ A c t u a r i e s , Chicago, I l l i n o i s .

- E s s e n t i a l s o# F i n a n c i a l Manaq~ment by George E. Pinches, Second E d i t i o n , Harper & Row P u b l i s h e r s , New York, 1986.

-

1 9 0

-