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
-