full 100 marks insurance project (blackbook) for bachelor of banking & insurance and bms. i do not...

8/10/2019 Full 100 Marks Insurance Project (blackbook) for Bachelor Of Banking & Insurance and BMS. I Do not Own It.

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

1 9 8 1 V O L 3 3

INSURANCE COMPANY GROWTH

D A L E S . H A G S T R O M

A B S T R A C T

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

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

d e v e l o p e q u i v a l e n t p r o f it s t a n d a r d s f o r a l l o p p o r t u n i t i e s th a t u s e s u r p l u s ,

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

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

p r of it s t a n d a r d s . M i c r o e c o n o m i c t h e o r y p r o v i d e s a t e c h n i c a l s e t ti n g fo r th e

d e f in i ti o n o f p ro f i t e q u i v a l e n c e , u s i n g t h e c o n c e p t s o f m a r g i n a l p r o f i t a b i l i ty

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

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

t h a t a r e e q u a l in p r o p o r t i o n t o t h e p r o d u c t s ' m a r g i n a l u s e o f s u r p l u s . A f t e r

c o n s i d e r i n g t e c h n i c a l a n d p r a c t i c a l p r o b l e m s in th e a p p l i c a t i o n o f t h i s p ri n -

c i p l e , t h e p a p e r t o u c h e s o n q u e s t i o n s o f e q u i ty . A n e w e q u i t y p r i n c i p l e i s

p r o p o s e d f o r d e b a t e .

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

OTH s t o c k a n d m u t u a l i n s u r a n c e c o m p a n i e s s e e k t o g r o w . D i f f e re n t

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

f o r m e a s u r e m e n t m a y i n c l u d e p r e m i u m i n c o m e in fo r c e , l if e i n su r -

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

e a r n in g s p e r s h a r e a c c o r d i n g to g e n e r a l l y a c c e p t e d a c c o u n t i n g p r i n c i p le s

( G A A P ) . F o r m u t u a l i n s u r a n c e c o m p a n i e s , g ro w t h i n t h e s e m e a s u r e s m a y

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

s u r a n c e c o m p a n y s e e k t o g ro w f a s t e r t h a n t h e c o m p a n y ' s e x p e n s e s i n fl at e,

t o k e e p d o w n u n i t e x p e n s e s . ' S o m e c o m p a n i e s f e e l t h a t t h e y n e e d t o g r o w

e n o u g h t o m a i n t a in t h e i r m a r k e t s h a r e . W h a t e v e r t h e g r o w t h t a r g e t, h o w -

e v e r , t h e i n s u r a n c e c o m p a n y f a c e s a d if fi c ul t r e s o u r c e - a l l o c a t i o n p r o b l e m

w h e n i t s e e k s t o o p t i m i z e g r o w t h . T h e s o l u t i o n m a y l ie i n a n a p p l i c a t i o n o f

m i c r o e c o n o m i c t h e o r y t o a c t u a r i a l t h e o r y .

RSA,

V, 36, 1360; VI, 307--8, 310, 644.

251


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252 INSURANCE COMPANY GROWTH

A company's growth is limited by its key resources: agents, employees,

managemen t, space, compute rs, cash, and surplus. 2 For any one company,

a single resource may be the limiting factor.

For example, in some companies the size and productivity of the agency

force may be the limiting factor; increased sales of one insurance product

(or other financial service) may come at the cost of reduced sales of another

product. Even for a comp any with an increasing

v o l u m e

of sales, the

n u m b e r

of new sales may be constant. For some companies, certain insurance prod-

ucts act as door- ope ners that lead to additional sales. Some products

may be ad d- on s whose sales can be achieved with no additional sales

effort. In these situations, productivity of the sales force is not the limiting

factor. Before a company can plan a strategy to optimize growth, it must

know its own operations well enough to know not only which are its limited

resources but also what are the dynamic relationships among its key re-

sources.

Most resources, including productive agents, can be acquired, in time,

at the cost of surplus. This makes surplus a kind of common denominator

among resources. Furthe rmore, gross premium price competition has forced

in su ra nc e companies to rely more on surplus than on premium margins to

absorb e.xperience fluct uati ons) Surplus levels have declined, however, rel-

ative to liabi lities .' A low level of surplus may limit a company 's ability to

underwrite additional risks. Thus, surplus is not only a kind of common

denomin ato r for scarce resources but also a potentially scarce resource

itself.

Although in the past surplus usually has not been a scarce resource for

mutual insurance companies, capital and surplus have been scarce resources

for stock insurance companies. Therefore, the pricing methods developed

for stock insurance companies have highlighted the returns on capital in-

vested in underwriting new business. As mutual insurance companies find

that current earnings cannot easily increase surplus to match both real and

inflation-related growth, they may start to use pricing methods similar to

those of stock insurance companies. ~ Such a pricing method could be as

follows:

In mutua l in surance companies with limited surplus, all gains after divi-

dends to policyholders are used to support additional growth. Thus, such

mutual companies can price a block of new insurance contracts assuming

2 R S A ,

I, 928--29;I1. 224-29.

3 R S A , I, 890-91; IV, 181; VI, 657.

4 R S A . Ill. 33-35, 956; Vi, 647--48.

R S A .

I11,915: VI, 101,309.


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INSURANCE COMPANY GROWTH 253

that the block requires a commit ment of surplus from the company at issue

and is stripped of statutory gains (or other releases of surplus commitments)

as soon as they emerge. Such gains support the sale of additional new

business. In these mutual companies, growth is limited ultimately by the

return on surplus from the business written.

Compare this with the pricing method of stock insurance companies. In

stock insurance companies, the profits not retained to support growth gen-

erally are paid out as dividends to stockholders. Thus, stock companies can

price a block of new insurance contracts assuming that the block requires

a commitment of capital and surplus from stockholders at issue and pays

out all statutory gains (and other releases of surplus and capital commit-

ments) as re turns of capital or as dividends to stockholders as soon as the

gains emerge. Stock insurance compan ies hold surplus in addition to capital

to absorb experience fluctuations, jus t as mutual insurance companies do.

Relying on surplus is less disruptive than reflecting such fluctuations in

stockholder dividends or going unexpectedly to the capital markets. The

growth of stock insurance companies is limited in the long run by their

ability to write business whose return on surplus and capital is at least equal

to the return on capital sought by stockholders. 6

Surplus and capital are resources that must be used or relied upon to

achieve growth of business (or to change the mix of the comp any' s business).

This paper assumes that surplus (taken together with capital) is the p r i m a r y

constraint on an insurance comp any 's growth.

If the net worth of an insurance company is defined as the sum of surplus

plus the value of the existing business in force plus the productive capacity

of its agency force, 7 then optimum growth occurs when the insurance com-

pany maximizes its annual additions to net worth, subject to the constr aint

of limited surplus. This is optimum growth for a stock insurance company

because in most instances it maximizes the value of the company to its

stockholders. This is optimum growth for a mutual insurance company

because in most instances it means that the company's service capacity,

now and in the future, is as great as possible. That is, if surplus is the

primary constraint on service capacity, then growth in net worth, which

over time can emerge as surplus, means growth in future service capacity.

If surplus is the primary constraint on an insurance comp any 's growth

in net worth, then surplus must be budgeted correctly if growth is to be

optimized.

6 RSA,

V, 857; VI, 99. However, see also

RSA,

IV, 812.

7 RSA,

III, 52; V, 970.


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Section 1I of this paper discusses four technical problems that arise in

the efficient allocation of the surplus budget. 8Section III present s the paper's

central theme: an insurance company can optimize growth only if surplus

is budgeted in such a way that all actions are expected to yield equivalent

profits in proportion to the amount of surplus used as a result of those

actions. Subsequent sections give applications of the methods developed

in this paper and identify certain practical problems. The final section dis-

cusses questions of equity and encourages debate.

The methods of this paper can be applied even if a company currently

is not limited by its surplus but expects to be so limited after a large sales

effort.

This paper uses statutory surplus (defined to include capital in a stock

insurance company) plus the mandatory securities valuation reserve (to

simplify the consideration of investments) as the definition of surplus be-

cause this is the quanti ty that is most often found to be a limiting resource. 9

l I . F O U R T E C H N I C A L P R O B L E M S IN T H E A L L O C A T I O N

O F T H E S U R P L U S B U D G E T

Profit is the contribution to surplus? A risk venture is some op-

portunity that an insurance company can pursue in the hope of making a

profit. For example, the com pan y may develop and issue certain insurance

contracts rather than others; it may make certain investments rather than

others; it may choose to develop a larger or more professional agency force

in various ways; or it may establish computer networks or effect other

administrative reorganizations. All of these are risk ventures.

A. Use of Surplus

The first technical problem in the allocation of the surplus budget is that

different risk ventures use surplus in different ways.

Consider first the expend iture of a specific amount of surplus to undertake

a risk venture. For example, in the case of an insurance risk venture , some

money actually is spent for commissions, marketing expenses, and issue

expenses. These outlays, plus the amount needed to meet statutory reserve

8 T h e a l l o c a t io n o f s u r p l u s h e r e d o e s n o t m e a n t h e a l l o c a ti o n o f f u n d s t h a t d e t e r m i n e s t h e c l a s s

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

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

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

9 R a t h e r t h a n s t a t u t o r y s u r p l u s , o n e c o u l d c o n s i d e r s u r p l u s p l u s l o n g -t e r m s u b o r d i n a t e d d e b t .

F o r o n e c o m m e r c i a l b a n k ' s v i e w s o n i n c l ud i n g l o n g - te r m d e b t in it s ca p i ta l b a s e s e e H o w a r d [ 7 ] .

T h e q u a n t i f ic a t i o n o f a n y s u r p l u s a m o u n t i s o p e n t o q u e s t io n ; s e e RSA, V I , 6 4 6 .

io RS A, V, 29.


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sort all hazards into four groups, identified by the parameter in proportion

to which the reliance upon surplus is measured. We label the parameter

Pt, where, in year t,

P[ = Amount of assets;

P~ = Amount of liabilities;

P~ = Amou nt of premiums in force; and

P~ = Net amoun t at risk.

We definef~ to be the ratio of surplus committed to buffer against all hazards

in hazard group k in year t, to the parameter Pi. Following Trowbridge ([13],

p. 216), we measure the amount of surplus used as a buffer in year t as

B S , = ~ f'~ P~ .

k = l

Yearly renewable term reinsurance ceded will produce a reduction in the

amount of surplus commi tmen t needed to buffer against the mortality risk.

Both

BS,

and P~ are negative for reinsurance ceded.

Buffer surplus corresponds to Donald Cody's corporate solidity surplus

( R SA , III, 29-32) or to Robert Lin k's minimum target strategic surplus

( R SA , III, 957).

Table I subjec tive ly classifies some risk ventures according to their prin-

cipal use of surplus. The table differentiates among the various parameters

that measure the use of surplus as a buffer, on the assumption that one

parameter suffices to measure each risk venture's reliance on surplus. Of

course, specific insurance product designs, reserve requ irements, and mar-

kets can produce risk ventures with these generic descriptions that should

be classified differently.

For risk ventures that rely upon surplus as a buffer, we seek to define a

buf fer book profi t that will be comparable to the book profit defined for

risk ventures that require an expenditure of surplus. We treat the change

in surplus commitment as a charge against buffer book profits at the start

of the year. We assu me that any policyholder dividend calculation provides

only for those federal income tdx amounts in excess of the federal income

tax on investmen t earnings on surplus. Thus, we use the after-tax investment

earnings on surplus to calculate a rate, i, earned on surplus. Since the book

profit for a risk venture that requires surplus to be expended is defined as


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I N S U R A N C E C O M P A N Y G R O W T H

2 5 7

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

b o o k p r o f i t t o i n c l u d e t h e a f t e r - t a x i n v e s t m e n t y i e ld , i , o n t h e a s s e t s u n -

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

a r d s . T h u s , w e d e f i n e t h e " b u f f e r b o o k p r o f i t " i n y e a r t a s

B B P , = d ( B S , ) - ( B S , - B S , _ ~ ) ,

w h e r e

d = i / ( I +

i ) , a n d

B S o = 0 : 3

' I ~ B L E I

TYPO LO G Y O F RISK V ENTURES

U S E O F S U R P L U S

E x p e n d i t u r e

B u f f e r ( r e l a t e t o a s -

s e t s )

B u f f e r ( r e l a t e t o l i a -

b i l i t i e s )

B u f f e r ( r e l a t e t o

p r e m i u m s i n

f o r c e )

B u f f e r ( r e l a te t o n e t

a m o u n t a t r i s k )

T I M E O F P R O F I T

EMERGENCE

I n a f e w y e a r s

O v e r m a n y

y e a r s


O v e r m a n y

y e a r s


O v e r m a n y

y e a r s


O v e r m a n y

y e a r s


O v e r m a n y

y e a r s

P R O F I T V O L A T I L IT Y

Predictable Risks

C o m p u t e r i z a t i o n

o f a d m i n i s t r a -

t i v e t a s k s

I n d i v i d u a l i m -

m e d i a t e a n n u -

i t ie s w i t h

s t r a i n

C o m m e r c i a l p a -

p e r

M o r t g a g e s ;

b o n d s

A u t o m o b i l e l i a -

b i l i t y i n s u r -

a n c e

N o n e

H o m e o w n e r

p r o p e r t y i n -

s u r a n c e

N o n e

I n d i v i d u a l t e r m

l i f e i n s u r a n c e

N o n e

Volatile Risk s

M a s s m a r k e t i n g o f

g r o u p t e r m i n s u r -

a n c e

I n d i v i d u a l d i s a b i l i t y i n -

c o m e i n s u r an c e ( n o n

c a n c e l a b l e )

C o m m o n s t o c k s

R e a l e s t a t e

G r o u p l o n g - t e r m d i s a -

b i l i t y i n s u r a n c e ;

m e d i c a l m a l p r a c t i c e

i n s u r a n c e

P h a r m a c e u t i c a l p r o d -

u c t l i a b i l i t y i n s u r -

a n c e

G r o u p t e r m h e a l t h i n -

s u r a n c e ; a v i a t i o n

r e i n s u r a n c e ; c o m -

m e r c i a l p r o p e r t y i n -

s u r a n c e

N o n e

G r o u p a c c i d e n t a l d e a t h

i n s u r a n c e

I n d i v i d u a l r e v e r t i b l e

t e r m l if e i n s u r a n c e

1 3 R S A ,

IV, 815, 817; VI, 96-97. We might want to redefine d as

i/(I + j'}

o n c e j h a s b e e n

defined below.


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cussed in Section III. We propose the internal rate of return on the margin

as the appropriate time value of surplus because it will be the most appro-

priate discriminator between those risk ventures at the margin among which

the company must choose.

To give some feel for the resul ts of using an internal rate of return as the

time value of surplus, we digress to discuss the properties of an average

internal rate of return, ra ther than a marginal internal rate of return. The

average internal rate of retu rn may be greater than the marginal internal rate

of return because of market saturation, or it may be less than the marginal

internal rate o f return because of fixed expenses not charged on the margin.

The main property of an average internal rate of return is that a company

can grow at that rate indefinitely. Take a new line of business in which each

unit of new business generates a sequence of augmented book profits whose

net present value, discounted at the rate j, is zero. The sales of that product

can grow at rate j indefinitely on a self-supporting basis after the line of

business reaches surplus equ ilibrium at the end of the lifetime of the con-

tracts issued in the first year. That is, after surplus commitments are made

for all contracts issued during the lifetime of the first year's issues , then the

surplus commitment required by any contracts issued later is provided by

the release of surplus from the in-force business. The surplus committed

to the line of business in the lifetime of the first year 's issues is not released

by the line of business (unless the growth slows below rate./3, but the line

of business requires no additional surplus commitment from the company.

This finite surplus commi tment to a growing business is achieved because

the internal rate of return equals the growth rate of surplus needed to support

the business . A co mpany can grow at ra te j indefinitely if the average internal

rate of return implicit in its in-force and new business is j.

A company somet imes sets its desired rate of growth (and surplus growth)

in relation to the growth rate of the national economy or of the insurance

industry. However, a compan y should not use such a desired rate of surplus

growth as the time value of surplus if its desired rate of growth has been

set without regard to the internal rates of return available in risk ventures

in the marketplace. If an unreal istically high desired rate of growth is used

as the time value of surplus, the re in vest ment rate implicit in comparisons

of risk ventures with different times of profit emergence will not be mean-

ingful, and the profitability comparisons will be misleading. We conclude

that available internal rates of return set the limits on the choice of a mean-

ingful time value of surplus to be used to compare profitability.

How do we compare profitability? The most surplus-efficient risk ventures

for a company to assume are those with the highest ratio of net profits to


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INS UR A NC E C OMP ANY GR OW TH 2 61

surplus comm itm ent used. We propose the following ratio to compare the

attractiveness of alternative risk ventures:

~ A B P , ( I + J~ l -'

t = l

e "~- M

- A B P , ( I + j ) l-~

t = l

where we define M to be the earliest duration at which the denominator

(if positive) reaches a maximum, or at which the d enominato r (if negative)

reaches a mi nimum. That is, in the positive case,

2

~ -AB P, ( I + j ) l -~ / > - A B P , ( 1 + j ) l - , for all t 4: M .

s = l $ = 1

The denominator is the initial surplus commitment, TSj, plus the present

value of any increases in TS,. The numerator is just the present value of

augmented book profits. Thus, a risk venture with a higher ratio R is more

profitable in relation to its use of the limited resource, surplus, than is a

second risk venture with a lower ratio R.

If all acceptable risk ventures have a co mmon internal rate of return, j,

then they are all equally profitable, and the present value of augmented

book profits will be zero. A compa ny could use the internal rate of return,

rather than

R ,

as the basis for profitability compar isons in most cases. There

are cases, however, where profit comparisons cannot be based on the in-

ternal rate of return, j6 This paper will use ratio R for profit compari sons ,

although the methods presented here can be adapted to the use of the

internal rate of return as a profit standard.

C. Profit Volatility

The third technical problem in the allocation of the surplus budget is the.

different profit volatilities inherent in the different risk ventures. Table 1

subjec tively classifies risk ventures as to position within the volatili ty spec-

trum. The profitability demanded of a volatile risk ven ture should be greater

~6 In general, there are problems with using internal rates o f return to set stan dards. There can

be mutually exclusive risk ventures. Because there can be m ore than one solution to a polynomial

equation, there can be more than one internal rate of return that arises from a stre am of augmented

book profits. This can often occur when a new product will be reinsured and the two ventures

are considered as one. Internal rates of return (especially on marginal augmented book profits)

can be very high for small risk ventures that cannot be replaced at the same rate. (See [t4], pp.

272, 275, and Appendix A.)


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than the profitability of a more predictable risk venture. There are several

different ways to define the greater profitability demanded of a more volatile

risk venture.

When the surplus invest ment

IS,

is negligible, the simplest way to define

the greater profitability demanded of a more volatile risk venture is to use

the ratio R defined in the preceding section, with the term

BS,

reflecting the

level of volatility. One way to measure the surplus relied on is to determine,

on the basis of ruin theory, how seriously the company's surplus can be

impaired by the worst loss expected in a very long period (such as a century).

Risk ventures using more buffer surplus will tend to have a lower ratio R

and be less attractive.

A second way to define the greater profitability demanded of a more

volatile risk venture is most useful if there will be an initial expenditure of

surplus and there is no chance that more expendi tures will be required. The

buffer surplus

BS,

is negligible. I fq is the probabili ty that the profit sequence

A B P ,

is interrupted at any duration, and if we define

A j = (1 + j ) q / ( l - q ) ,

then th~ expected present value of augmented book profits is

(1 -

q ) ' - ' A B P , ( I + j ) ' - ' = ~ A B P , ( I + j + A j ) ~ - , .

t = l t = l

In this case, it is convenient if the rate used to discount

A B P ,

in the nu-

merator of the ratio R is j + Aj, where the increment Aj recognizes the risk

of not recovering the surplus already spent. The lower ra te j is still used in

the denominator to discount the sequence of changes in surplus commit-

ments, -A BP ,. Because of the compounding of the discount rate, this

method discounts more heavily those profits that lie farther in the future

and seem, therefore, less certain. The increment Aj equals zero for any risk

venture whose future profits seem certain.

A third way to define the greater profitability demanded of a more volatile

risk venture is a combination of the first two ways. The buffer surplus term

BS,

reflects the risk that additional surplus may be required, and the incre-

ment Aj to j in the nume rator reflects the risk that the surplus investment

ISt

may not be fully recovered and that the buffer surplus will not be released

as soon (or may never be released if an experience fluctuation wipes it out).

A fourth way to define the grea ter profitability demanded of a more volatile

risk venture involves the calculation of the numerator of the ratio R. It is

useful whether or not additional surplus may be at risk. A probabilistic


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I N S U R A N C E C O M P A N Y G R O W T H 2 63

m o d e l ( u s in g d e t e r m i n i s t ic o r M o n t e C a r l o m e t h o d s ) is u s e d . .7 M u l t ip l e

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

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

I n e i t h e r t h e d e t e r m i n i s t ic m o d e l o r th e M o n t e C a r l o m o d e l , e a c h p r o f i t

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

w e i g h t s w i ll d i f f e r f r o m u n i t y i f u t i l it y d o e s n o t v a r y l i n e a r l y w i t h d o l l a r s .

F o r e x a m p l e , a c o n s e r v a t i v e c o m p a n y ( o r a s m a l l c o m p a n y w i t h a l o w

s u r p lu s ) a s s i g n s p o o r o u t c o m e s g r e a t e r r e l a t i v e w e i g h t s , s i n c e p o o r o u t -

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

p o t e n t i a l l o ss a s w e l l a s t h e f r e q u e n c y o f lo s s e s . I t i s p a r t i c u l a r ly v a l u a b l e

w h e n a n e w i n s u r a n c e p r o d u c t w i l l r e q u i r e l a rg e a d m i n i s t r a t iv e e x p e n s e s

in t h e fu t u r e w h e t h e r o r n o t t h e n e w p r o d u c t s e l l s w e l l , su c h a s , f o r e x a m p l e ,

i n d iv i d u a l v a r i a b l e a n n u i t i e s w h e r e a c o m p a n y s e ts u p it s o w n s e p a r a t e

a c c o u n t r e q u i r in g a m a i n t e n a n c e e x p e n s e f o r a n i n de f in i te p e r i o d .

A f if th w a y t o d e f in e t h e g r e a t e r p r o f i t a b i l i t y d e m a n d e d i s a v a r i a t i o n o n

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

g r e a t e r t h a n a g i v e n v a l u e Rh~ h w i t h a c e r t a i n f r e q u e n c y ( in t h e u n i v e r s e o f

s i m u l a t e d o u t c o m e s ) . F u r t h e r , a r i s k v e n t u r e is n o t a s s u m e d i f t h e r a t io R

i s le s s t h a n a g i v e n v a l u e R *ow w i t h a c e r t a i n f r e q u e n c y .

D . M ar g i na l P r o fi tab i li ty an d U s e o f Sur p l us

T h e f o u r t h p r o b l e m in th e a l l o c a t i o n o f t h e s u r p l u s b u d g e t i s t o r e c o g n i z e

m ar g i na l

p r o f i ta b i l i ty a n d u s e o f s u r p l u s a s d i s t in c t f r o m

a v e r a g e

p r o f i t -

a b i l i ty a n d u s e o f s u r p lu s .

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

o f a c t i o n o r o n th e b e s t p r i c e a t w h i c h t o o f f e r i n s u r a n c e . A n i n s u r a n c e

c o m p a n y m a y f a c e at d i f f e re n t t i m e s a s m a n y a s f o u r d i s t in c t t y p e s o f

d e c i s i o n s t o m a k e c o n c e r n i n g r i s k v e n t u r e s :

. Buy~introduce.

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

a ) B u y a c e r t a i n in v e s t m e n t r is k v e n t u r e o r b u y s o me o t h e r in v e s t m e n t r is k v e n t u r e r

wi th l e ss r i sk ;

b ) G o a h e a d w i t h a c o m p u t e r , a d mi n i s t r a t i v e , o r a g e n c y - d e v e l o p m e n t r is k v e n t u r e

or f i nd an a l t e rna t ive t ha t cos t s l e ss ;

~ c) A s s u m e a b l o c k o f r e i n s u r a n c e b u s i n e s s ;

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

o f b u s i n e s s ;

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

surp lus .

2. Pricing new insurance sales. T h e i n s u r a n c e c o m p a n y m u s t d ec i d e w h a t p r ic e l ev e l

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

17 RSA, I I I , 222-2 3 ,231 ; V, 34-- -41. See a l so Be ekm an a nd Fuel l ing [31.


14/50

2 6 4 I N S U R A N C E C O M P A N Y G R O W T H

B y

price

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

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

i n a d v a n c e .

3. Repricing in-force insurance. I n c e r t a i n s i t u a t i o n s , t h e i n s u r a n c e c o m p a n y c a n

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

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

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

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

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

i n f o r c e i n s o m e m a r k e t s r e s p o n d s m o r e s l ow l y t o a c h a n g e i n p ri c e l ev e l t h a n

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

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

b y a c h a n g e i n p r ic e l e v e l f o r n e w b u s i n e s s .

4. Sell~cede~discontinue.

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

a ) S e ll a n i n v e s t m e n t r i s k v e n t u r e ;

b ) C a n c e l s o m e c o m p u t e r , a d m i n i s t r a t i v e , o r a g e n c y - d e v e l o p m e n t r i sk v e n t u r e s ;

c ) C e d e a n i n - f o r c e b l o c k o f i n s u r a n c e b u s i n e s s t o a r e i n s u r e r ;

d ) D i s c o n t i n u e o f f e r in g a p a r t ic u l a r i n s u r a n c e p r o d u c t , l e a v e a p a r t ic u l a r in s u r a n c e

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

c a n c e l e d ;

e ) R a i s e a d d i t io n a l c a p i ta l a n d s u r p l u s b y a n o f f e r in g o f s t o c k a n d s u b o r d i n a t e d

d e b e n t u r e s .

A l l o f th e r i s k v e n t u r e s i n t h e " s e l l / c e d e / d i s c o n t i n u e " d e c i s io n g r o u p m a y r e l e a s e

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

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

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

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

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

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

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

s t o o d t o b e c a l c u l a t e d u s i n g m a r g i n a l b o o k p r o f it s , B ,, a n d m a r g i n a l b u f f e r

s u r p l u s , BS,.

H o w d o w e c a l c u l a t e m a r g i n a l b o o k p r o f i t s ? F o r a b u y / i n t r o d u c e d e c i s i o n ,

w e ' c a l c u l a t e m a r g i n a l b o o k p r o f i t s a s th e d i f fe r e n c e b e t w e e n t h e b o o k p r o f i t s

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

d e c i d e s n o , n o t t o go a h e a d , w i t h t h e r i sk v e n t u r e t h a t r e q u i r e s a s u r p l u s

c o m m i t m e n t . I n c e r t a i n s i tu a t i o n s , t h e " n o " d e c i s i o n m e a n s t h a t t h e c o m -

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

s u r p l u s c o m m i t m e n t .

F o r e x a m p l e , a n i n v e s t m e n t i n r ea l e s t a t e c a n b e c o n s i d e r e d i n c o m p a r i s o n

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

b e t w e e n i n i ti a l i n v e s t m e n t s , t h e y e a r - b y - y e a r e x c e s s o f n e t r e n t a l i n c o m e


15/50


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

o f p r i n c ip a l u n d e r t h e m o r t g a g e , a l l c r e a t e t h e s e q u e n c e o f m a r g i n al b o o k

p r o f i ts f o r a n i n v e s t m e n t i n r e a l e s t a t e .

T h e m a r g in a l b o o k p ro f it s o f a n e w c o m p u t e r s y s t e m a r e m a d e u p o f t h e

i ni ti al d e v e l o p m e n t a l e x p e n s e ( a l o ss ) a n d t h e n t h e e x c e s s o f t h e e x p e n s e

o f p r o c e s s in g u n d e r t h e o ld s y s t e m o v e r t h e e x p e n s e o f p r o c e s s in g u n d e r

t h e n e w s y s t e m ( a s e r ie s o f g a i n s ).

A n a g e n c y d e v e l o p m e n t r is k v e n t u re i s b e s t c o m p a r e d w i th a n a l t e rn a t iv e

s a l e s - d e v e l o p m e n t s tr a t e g y t h at r e q u i r e s l e ss s u r p l u s c o m m i t m e n t . I f d if -

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

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

w i th t h e s u r p l u s c o m m i t m e n t r e q u i r e d , t h e n t h e m a r g i n a l b o o k p r o f it s o f a n

a g e n c y d e v e l o p m e n t r i s k v e n t u r e a r e b e s t c a l c u l a t e d a s th e d i f f e re n c e in

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

d i ff e re n c e b e t w e e n t h e e x p e n s e s o f a g e n c y d e v e l o p m e n t a n d t h e e x p e n s e s

o f a lt e r n a t i v e s a le s d e v e l o p m e n t .

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

b y c o m p a r i n g t h e i n it ia l s u r p l u s e x p e n d i t u r e ( a lo s s ) a n d t h e l a t e r b o o k

p r o f it s ( g a in s ) w i t h t h e e a r n i n g s s i t u a t i o n a s s u m i n g n o r e i n s u r a n c e . T h e t a x

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

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

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

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

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

p l a n n e d p r i c e l e v e l , c l a i m c o s t s , i n t e r e s t r a t e s , a n d p e r s i s t e n c y . T h e a s -

s u m p t i o n i n th e c a l c u l a t io n o f m a r g in a l b o o k p r o f it s th a t d i f f e rs th e m o s t

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

is t he e x p e n s e a s s u m p t i o n . C e r t a in e x p e n s e s v a r y w i th v o l u m e : c o m m i s -

s io n s a n d p r e m i u m t a x e s w i t h p r e m i u m i n c o m e , a n d i ss u e e x p e n s e s w i t h

n u m b e r o f c o n t r a c t s o r w i th s i z e o f r i sk u n d e r w r i t te n . T h e s e a r e variable

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

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

direct-periodic

e x -

p e n s e s . C e r t a in o t h e r e x p e n s e s d i r e c t l y r e l a t e d to t h e i n t r o d u c t i o n o f t h e

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

direct-f ixed

e x p e n s e s . V a r i a b l e, d i r e c t -

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

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

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

p r o d u c t l i ne i s c h a r g e d f o r a n y d i r e c t - p e r i o d i c o r d i re c t - f ix e d e x p e n s e s ( a s

w e l l a s v a r i a b l e e x p e n s e s ) i t w i l l c a u s e , b u t a d d i t i o n a l s a l e s i n e x i s t i n g

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

c o m p a n y i s a l r e a d y in c u r r in g th o s e e x p e n s e s o n e x i s ti n g s a le s . T h e r e i s ,


16/50

2 66 I N S U R A N C E C O M PA N Y G R O W T H

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

b e a c h i e v e d . I s t h e p r i c e fo r t h e e x i s ti n g p r o d u c t l i n e r e d u c e d , o r d o w e

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

s a l es in t h e n e w p r o d u c t l in e b e c a u s e t h e a g e n c y f o r c e c a n s el l o n l y s o

m u c h b u s i n e s s in a y e a r ? F i n a l l y , c e r t a i n o t h e r e x p e n s e s a r e n o t d i r e c t ly

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

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

i n d i r e c t - p e r i o d i c

e x p e n s e s . ~ 8

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

p e r i o d i c a n d w h i c h a r e v a r i a b l e . A c r o s s - s e c t i o n a l s tu d y t o d a y w o u l d s ug -

g e s t t h a t m o s t e x p e n s e s a r e v a r i a b l e o v e r t h e l o n g r u n ; l ar g e c o m p a n i e s

h a v e l a rg e e x p e n s e s , s o m e w h a t in p r o p o r t i o n to t h e i r b u s i n e s s i n f o r c e ;

s m a ll c o m p a n i e s h a v e s m a l l e r e x p e n s e s . H o w e v e r , s o m e c o m p a n i e s , f or t he

i n t e r m e d i a t e t e r m , c a n n o t a f f o rd t o v i e w t h e i r e x p e n s e s a s v a r i a b l e ; e ve n"

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

a l a r g e b r a n c h m a n a g e r a n d c a r e e r a g e n c y s y s t e m . ~ 9

T h e m a r g i n a l b o o k p r o f i ts f o r a n e w p r o d u c t l i n e a r e i n a s e n s e e q u a l t o

t h e d if f e r e n c e b e t w e e n t h e tw o

a v e r a g e

b o o k p r o f it f ig u r e s t h a t a r e e x p e c t e d

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

p a r t i c u l a r p r i c e l e v e l . O n c e t h e d e c i s i o n t o i n t r o d u c e t h e p r o d u c t l in e i s

m a d e , t h e r e i s a s e c o n d m a r g i n a l b o o k p r of it c a l c u l a t io n m a d e t o d e t e r m i n e

t h e o p t i m u m p r i c e l e v e l . W e d i s c u s s b e l o w t h e m a r g in a l b o o k p r o fi t c a l-

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

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

i n it ia l c o s t ( a l o s s ) a n d t h e f u t u r e d i v i d e n d p a y m e n t s s a v e d ( a s e r i e s o f

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

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

i t d o e s f e e l r e a s o n a b l y c o n s t r a i n e d b y s u r p l u s . -'

O n c e a n i n s u r a n c e c o m p a n y h a s m a d e t h e d e c is i o n t o i n t r o d u c e a n i n-

s u r a n c e r i s k v e n t u r e , a n d u n t i l it h a s m a d e t h e d e c i s i o n t o d i s c o n t i n u e s e l li n g

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

T h e c o n s i d e r a t i o n s i n s e tt in g a c o m m i s s i o n s c a l e a r e s i m i l a r t o , a n d i n te r -

r e l a t e d w i t h , t h e c o n s i d e r a t i o n s i n se t t i n g a p r i c e l e v e l . W e s h a l l s p e a k o n l y

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

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

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

t h a t w i ll b e s o l d i n a y e a r a t th a t p r i c e l e v e l. I f t h e c o m p a n y d e c r e a s e s t h e

p r i c e o n t h e i n s u r a n c e r i s k v e n t u r e , t h e n , u s u a l l y , t h e s a l e s v o l u m e i n -

c r e a s e s . A l l u n i t s s o l d d u r i n g t h e y e a r , n o t j u s t t h e a d d i t i o n a l u n i t s , g e t t h e

la RS A, Ill, 32, 38; IV, 206-8; VI, 98-100, and 645. See also LOMA [10l.

t9 RS A, IV, 208.

20 RS A,

IV, 825.


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I N S U R A N C E C O M P A N Y G R O W T H 2 6 7

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

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

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

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

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

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

s o l d a t t h e o r i g in a l p r i c e a n d o n a n y a f f e c t e d u n i t s in f o r c e . 2~ T h e c h a n g e

i n b o o k p r o f it c o r r e s p o n d i n g t o a p r i c e r e d u c t i o n o n u n i ts t h a t w o u l d h a v e

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

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

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

m a r k e t d i f f e r e n t ia t i o n s o th a t i t c a n c h a n g e p r i c e s o n l y i n t h e s p e c i f ic s e g -

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

p r o fi ts f o r a g i v e n a m o u n t o f a d d i t i o n a l s u r p lu s c o m m i t m e n t . S u c h n a t u r a l

m a r k e t d i f fe r e n t ia t io n m a y b e b a s e d o n a g e , s e x , o c c u p a t i o n , i n c o m e , g e o -

g r a p h i c a r e a , m a r k e t i n g m e t h o d , t a x q u a l if ic a t io n s ta t u s , s i z e o f c a s e , o r

c l i e n t s o p h i s t i c a t i o n .

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

u n d e r w r i t in g c h a n g e s r a t h e r t h a n b y p r i c e r e d u c t io n s . T h i s a v o i d s t h e l o s s

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

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

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

e x p e r i e n c e o n t h e a d d i t i o n a l sa l e s i f t h e u n d e r w r i t i n g s t a n d a r d s a r e r e l a x e d .

F o r a d e c i s i o n o n r e p r i c i n g in - f o r c e b u s i n e s s w h e n t h i s is d i s t i n c t fr o m

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

a r e e q u a l t o t h e c h a n g e s i n p r e m i u m i n c o m e ( a l o s s ) l e s s t h e c h a n g e s i n

p r e m i u m - r e l a t e d v a r i a b l e e x p e n s e s c a u s e d b y a s l i gh t re d u c t i o n in p r ic e

l e v e l, p lu s th e m a r g i n a l b o o k p r o f i ts on c o n t r a c t s t h a t w o u l d h a v e l a p s e d

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

F o r a s e l l / c e d e / d i sc o n t i n u e d e c i s i o n th e c o m p a n y c a l c u l a t e s m a r g in a l b o o k

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

f o r a b u y / i n t r o d u c e d e c i s i o n . T h e d e c i s i o n is b a s e d o n t h e f a c t s a n d o p t i o n s

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

r e l a te s to t h e d e c i s i o n w h e t h e r t o d i s c o n t i n u e s a l e s in a n i n s u r a n c e p r o d u c t

l in e . I f t h e i n s u r a n c e c o m p a n y i s n o t a b l e t o c a n c e l e x i s t i n g b u s i n e s s i n

f o r c e , d i re c t - p e r i o d i c e x p e n s e s s h o u l d n o t b e c h a r g e d a g a i n s t th e m a r g i n a l

b o o k p r of it s c a l c u l a t e d f o r th e c h o i c e o f c o n ti n u i n g t o w r i te n e w b u s i n e s s ,

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

w r i ti n g n e w b u s i n e s s . T h e s ig n c o n v e n t i o n a s s u m e d in th i s p a p e r f o r m a r -

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

21 B r a g g [ 4 ] g i v e s s o m e e s t i m a t e s f o r l if e i n s u r a n c e .


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IN S U R A N C E C O M P A N Y G R O W T H 2 69

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

S u p p o s e a n i n s u r a n c e c o m p a n y h a s a l a rg e a m o u n t o f g r o u p l i fe i n s u r a n c e .

S i n c e t h e c l a im c o s t s o n a n i n c r e m e n t a l p i e c e o f g r o u p l if e b u s i n e s s a r e

l a r g e ly i n d e p e n d e n t ( b a s e d o n p r o b a b i l i ty t h e o r y ) o f th e a v e r a g e g r o u p l i fe

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

t o t h e e x i s t i n g g r o u p l if e b u s i n e s s , t h e

m a r g i n a l

b u f f e r s u r p l u s p e r d o l l a r

o f n e w g r o u p l if e p r e m i u m m a y b e l e ss t h a n t h e

a v e r a g e

b u f f e r s u r p l u s p e r

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

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

a r e e x p o s e d ( c e n t ra l li m it t h e o r e m ) . O n t h e o t h e r h a n d , s u p p o s e t h e c o m -

p a n y h a s a la r g e a m o u n t o f g r o u p h e a l th i n s u r a n c e . T h e v a r i a t i o n in c la i m

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

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

h e a l t h b u s i n e s s . T h e m ar g i na l b u f f e r s u r p l u s f a c t o r f o r g r o u p h e a l th m a y

b e n o l e s s t h a n t h e

a v e r a g e

b u f f e r s u r p l u s f a c t o r f o r g r o u p h e a l t h . T h e

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

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

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

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

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

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

i n p r ic e l e v e l ( o r f r o m a n i n c r e a s e i n c o m m i s s i o n s , i f t h a t i s t h e b a s i s f o r

c h a n g e a s s u m e d in th e c a l c u l a t i o n o f m a r g i n a l b o o k p r o f it s) . T h e m a r g i n a l

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

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

w i t h r e s p e c t t o a ll th e u n i t s t h a t w o u l d h a v e b e e n s o l d a t t h e o r i g in a l p r i c e ,

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

i n - f o r c e b u s i n e s s m o v e s w i th t h e p r i c e l e v e l o f n e w b u s i n e s s ) .

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

p r i c i n g n e w b u s i n e s s , t h e m a r g i n a l b u f f e r s u r p l u s a t a p a r t i c u l a r p r ic e l e v e l '

i s t h e c h a n g e in b u f fe r s u r p lu s n e e d e d b y t h e i n - f o rc e b u s i n e s s b e c a u s e o f

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

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

i n p r i c e l e v e l .

E v e n i f a r is k v e n t u r e ' s u s e o f b u f f e r s u r p lu s i s m e a s u r e d in p r o p o r t i o n

t o p r e m i u m i n c o m e i n f o r c e , w e m u s t r e c o g n i z e t h a t b u ff e r s u r p lu s i n c r e a s e s

i f t h e r e i s a r e d u c t i o n in p r e m i u m ( p r i c e l e v e l ) . M a r g i n a l f ~ is u n r e l a t e d t o

a v e r a g e f ~.

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

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


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270

INSURANCE COMPANY GROWTH

ginal buffer surplus commit ted is that when a decision is made to sell, cede,

or discontinue, the marginal buffer surplus committed is negative, because

surplus is released.

II|. STRATEGY FOR OPTIMUM GROWTH GIVEN A SURPLUS CONSTRAINT

A. Basic Microeconomic Principle

A basic principle of economics is that a company maximizes the produc-

tion of a good, subjec t to a single limited resource, when marginal product ion

divided by marginal use of the limited resource is equal across the various

ways of producing the good. No reallocation of the limited resource from

one way to another way of producing the good will increase the total output

of the good. :4

B. Strategy of Equal Marginal Results

We have defined optimum growth as a maximization of annual additions

to net worth, subject to the constr aint of limited surplus. To obtain optimum

growth, a company with limited surplus dan adopt the following strategy.

First, the company estimates the surplus that will become available from

augmented book profits on business in force, year by year, over the next

several years. Since the com pan y is looking for just the net amount of

surplus becoming available, these amounts are reduced by overhead ex-

penses and other expenses that are charged ultimately against none of the

marginal book profits on risk ventures that the company will assume over

the next several years. Next, the compa ny estimates the surplus that would

be committed, year by year, for each of the risk ventures expected to be

available in the next several years. For insurance risk ventures, the surplus

commitment will depend on the price level. The company then examines,

for each risk venture, the marginal ratio R, defined above to be the ratio

of the present value of marginal augmented book profits to the marginal

surplus commitment required for the risk venture. Finally, the company

goes through a three-step iterative process.

To start the iterative process, the company tentatively plans to assume

only those most profitable risk ventur es (and to cede or discontinue those

least profitable risk ventures) such that, in aggregate, exactly the amount

of surplus becoming available in the year (or over several years) is com-

mitted. Those risk ventures with ratio R greater than or equal to a threshold

value R t are ten tat ive ly accepted, and those with ratio R less than R ~ are

discontinued or tenta tively ceded. We define R t to be the value of R such

that the summat ion of the first-year surplus commi tment s over all risk yen-

24 Hendersonand Quandt ([6], pp. 64-65).


21/50


tures (other than reinsurance ceded and other risk ventures in the sell/cede/

discontinue decision group) with marginal ratios R greater than or equal to

R ~, less the summat ion of the first-year surplus relief over all reinsurance-

ceded risk ventures (or surplus released on other risk ventures in the sell/

cede/discontinue decision group) with R less than or equal to R ~, is equal

to the surplus that will become available to support growth in the next one,

or several, years. (If many risk ventures require increasing surplus com-

mitments in later years, the company must check that the surplus that will

become available in the next several years can cover these additional de-

mands from the risk ven tures assumed in the first year.) George Dinney has

called a process of selecting the most profitable risk ventures first a de-

cision-tree pro cess .

The second step in the iterative process is to reprice all

insurance

risk

ventures so that, on the margin, the ir ratios R will equal R ~. We will discuss

what this means in the various types of decision situations.

For an insurance risk venture requiring a buy /introduce decision there are

two cases; either the initial ratio R is equal to or greater than R ', or the

initial ratio R is less than R ~. If the risk vent ur e' s ratio R (calculated us ing

a yes /no comparison) is equal to or greater than R I, then the comp any

calculates the risk venture 's marginal ratio R using a change-of-price com-

parison. The company changes the price level so that the marginal ratio R

(calculated using a change-of-price comparison) equals R ~. The ratio R (using

a yes/no comparison) will still exceed R t, either because additiona l units

were added at a marginal profit greater than R t if the price level was reduced,

or because unit s were removed at a loss of marginal profit less than R ~ if

the price level had to be increased. It is not efficient to reduce the ratio R

calculated using a yes/no comparison to the R' level, because the additional

units sold do not produce enough profit to justify the additional surplus

commitment. The ratio R calculated on a yes/no basis is really the com-

parison of two

average

profits, rather than a true

marginal

ratio. If R is le ss

than R t, the company must examine different strategies for the risk venture

to find a strategy (perhaps involving price level, commissions, marketing

focus, product design, etc.) that produces a ratio R (calculated on a yes/no

basis) greater than or equal to R ~. If such a strategy is found, then this

second case has been reduced to the first case. The company alters the

price level so that the marginal ratio R (calculated using a change-of-price

basis) equals R t.

For an insurance risk venture requiring a pricing-new-insurance-sales

decision, the price level generally should be reduced if the initial ratio R

RSA,

I I , 22 7. S t a l e y [ 12 ] s u g g e s t s a s i m i l a r m a x i m i z a t i o n w i t h o u t t h e s u r p l u s c o n s t r a i n t .


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2 72 I N S U R A N C E C O M P A N Y G R O W T H

( c a l c u l a t e d u s i n g a c h a n g e - o f - p r i c e c o m p a r i s o n ) i s g r e a t e r t h a n R ' . A l t e r -

n a t i v e l y , i f t h e i n i t i a l r a t i o R is le s s t h a n R ' , t h e p r i c e l e v e l s h o u l d b e

i n c r e a s e d s o t h a t t h e r e a r e f e w e r s a le s a n d m o r e p r o f i t o n e a c h s a l e . W h e n

R r e a c h e s R ' , t h e n e x t i n c r e a s e i n p r i c e w o u l d h a v e e l im i n a t e d a n a c c e p t a b l e

p r o fi t i n c o m p a r i s o n w i t h t h e s u r p l u s u s e d , s o t h e p r i c e is f ix e d a t t h a t l e v e l .

F o r a n i n s u r a n c e r i s k v e n t u r e r e q u i r i n g a r e p r i c i n g - in - f o r c e - i n s u r a n c e d e -

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

p r i c e l e v e l t h e l a p s e s e r o d e t h e t o t a l p r o f i t ab i l i t y , a n d a t a l o w e r p r i c e l e v e l

t h e r e d u c e d p r e m i u m m a r g i n e r o d e s t h e to t a l p r o fi t a b i li ty . T h i s p r i c e l e v e l

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

t h e m a r g i n a l s u r p l u s c o m m i t m e n t i s n e g li g ib l e , p e r h a p s o n l y e q u a l to t h e

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

l a p s e i f t h e r e i s a s l ig h t p r ic e r e d u c t i o n . B e c a u s e t h e m a r g i n a l p r o f i t a b i l i ty

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

t h a n R ' . A s t h e p r i c e l e v e l d e c l i n e s , t h e m a r g i n a l p r o f i t a b i l i ty d r o p s t h r o u g h

z e r o a t th e p o i n t o f m a x i m u m t o t a l p r o f i ta b i li ty . I n t h e m e a n t i m e , a s th e

p r ic e l e v e l d r o p s , t h e m a r g i n a l s u r p l u s c o m m i t m e n t i n c r e a s e s . A t s o m e p r ic e

l e v e l , t h e r a t i o R w i l l e q u a l R ' , a n d t h i s is t h e p r i c e l e v e l to u s e .

F o r a n i n s u r a n c e r i s k v e n t u r e r e q u i r i n g a d i s c o n t i n u e d e c i s i o n , t h e r e a r e

t w o c a s e s ; e i t h e r t h e i n i t i a l r a t i o i s l e s s t h a n R ~, o r t h e i n i ti a l r a t i o is g r e a t e r

t h a n R ' . I f t h e i n i t i a l r a t i o R ( u s i n g a y e s / n o c o m p a r i s o n ) i s l e s s t h a n R t ,

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

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

m a r k e t s t r a t e g y , p r ic e l e v e l , c o m m i s s i o n s c a le , a n d s o f o r th . T h e a v e r a g e

p r o fi t lo s t i s n o t v e r y g r e a t i n c o m p a r i s o n w i t h t h e s u r p l u s t h a t i s r e l e a s e d . '6

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

R ( c a l c u l a t e d u s i n g a y e s / n o c o m p a r i s o n ) g r e a t e r t h a n R ' - - t h i s i s t h e s e c o n d

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

B e c a u s e t h e p r i c e s o f i n s u r a n c e r i s k v e n t u r e s a r e s e t i n t h e s e c o n d s t e p

o f t h e i te r a t iv e p r o c e s s u s i n g , a m o n g o t h e r t h i n g s , m a r g i n a l e x p e n s e s , w e

d i g r e s s h e r e to d i s c u s s p r i c i n g o n a m a r g i n a l b a s i s . E x p e n s e c h a r g e s o n a

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

26 The com pany does not sim ply cut com missions or increase prem iums on a relatively un-

profitable product w ithout an analysis of wh ethe r this improves the ratio R.

If an insurance risk venture sells easily but u ses too m uch surplus, it is not efficient simply to

limit insurance sales by setting a quota. To impose a surplus b udget, it is mo re efficient to raise

the prem ium charged for the insurance. This w ill reduce u se of surplus, increase p rofits, and

reduce sales. For the sam e nu m be r of sales allow ed under a quota (or mo re), a higher profit will

be earned. See also

RSA,

V, 44; VI, 315.

The marketing strategy dictates a price level in relation to the competition. See RSA, IV,

206-7. The marketing strategy may be based on current market share and desired market share.

which in turn is related to ratio R and to growth prospects for that insurance market.


23/50

I N S U R A N C E C O M P A N Y G R O W T H 2 7 3

t h e m a r g i n a l b o o k p r o f it s a r e s u f f i c ie n t , in t o ta l , t o c o v e r e x p e n s e s t h a t a r e

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

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

w o r t h i n a y e a r o n a m a r g i n a l b a s is a f t e r w e h a v e f i n i s h e d o p t i m i z i n g g r o w t h ,

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

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

( o r o t h e r ) e x p e n s e s .

O n c e a l l i n s u r a n c e p r o d u c t s a r e r e p r i c e d o r r e d e s g n e d s o t h a t th e i r r a t i o s

R ( c a l c u l a t e d o n a p r i c e - c h a n g e b a s is ) e q u a l R t, t h e s u r p l u s s o u g h t b y a l l

r is k v e n t u r e s w i t h R g r e a t e r t h a n o r e q u a l t o R ~ w i l l b e g r e a t e r t h a n t h e

s u r p lu s a v a i l a b l e ( i n c lu d i n g t h e s u r p l u s r e l e a s e d b y r i sk v e n t u r e s i n t h e s e l l /

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

R I ) . T h i s l e a d s u s t o t h e t h i r d s t e p i n t h e i t e r a t i v e p r o c e s s .

T h e t h i rd s t e p i s t o i n c r e a s e t h e p r i c e l e v e l s o n a l l i n s u r a n c e r i sk v e n t u r e s

u n ti l th e f o l l o w i n g t w o c o n d i t i o n s a r e a c h i e v e d :

1. All i n s u r a n c e r i s k v en t u re s h av e o n t h e m arg i n t h e i r r a t i o s R ( ca l cu l a t ed u s i n g a

ch an g e -o f -p r i ce co m p ar i s o n ) eq u a l t o a n ew t h re s h o l d v a l u e , R 2. h i g h e r t h an R I .

2 . T h e s u m m a t io n o f t h e f i rs t -y e a r su r p lu s c o m m i t m e n t o v e r

a l l

r i s k v en t u re s ( ex -

c l u d i n g t h o s e f ro m t h e s e l l / c ed e / d i s co n t i n u e d ec i s i o n g ro u p ) wh o s e r a t i o s R ( ca l -

cu l a t ed u s i n g a y es / n o co m p ar i s o n ) a r e g re a t e r t h an o r eq u a l t o t h e n ew R 2, l e s s

t h e s u m m at i o n o f t h e f i r s t -y ea r s u rp l u s r e l ea s ed o v e r -a l l r i s k v en t u re s f ro m t h e

s e l l / c ed e / d i s co n t i n u e d ec i s i o n g ro u p wh o s e r a t i o s R ( ca l cu l a t ed u s i n g a y es / n o

co m p ar i s o n ) a r e l e s s t h an o r eq u a l t o R : , i s eq u a l t o t h e s u rp l u s t h a t w i l l b e co m e

av a i l ab l e . T h e co m p an y d o es a s s u m e t h o s e r i s k v en t u re s t h a t en t e r t h e f i r s t s u m -

m a t i o n an d s e l l s , c ed es , o r d i s co n t i n u es t h o s e r i s k v en t u re s t h a t en t e r t h e s eco n d

s u m m at i o n .

A t th is s ta g e , t h e c o m p a n y h a s c o m p l e t e d t h e it e r a t i v e p r o c e s s . N o r e a l-

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

t h e y e a r ' s b u s i n e ss . T h e c o m p a n y d o e s n o t a s s u m e n o n i n s u r a n c e r is k v e n -

t u r e s th a t a r e le s s p r o f i t a b le , in c o m p a r i s o n w i t h th e r e q u i r e d s u r p l u s c o m -

m i t m e n t , t h a n t h e t h r e s h o l d v a l u e R- '. T h e c o m p a n y d o e s n o t s e ll p r o d u c t

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

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

g 2

F i g u r e I i s a s c h e m a t i c r e p r e s e n t a t i o n o f t h e n e w r is k v e n t u r e s a v a i l a b le

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

t h e p r i c e o n n e w i n s u r a n c e r is k v e n t u r e s e f f ic i e n t ly a n d h a s c h o s e n n e w r i s k

v e n t u r e s e f f i c ie n t ly , t h e r i s k v e n t u r e s a s s u m e d w i l l a l l h a v e a r a ti o R e q u a l

t o o r g r e a t e r t h a n R 2. T h e i n c r e m e n t a l r is k v e n t u r e s o n w h i c h t h e c o m p a n y

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

o r ig i n a n d h a s a s l o p e e q u a l t o th e c o m m o n r a t io R 2. I f r i s k v e n t u r e A h a s


24/50

2 7 4 I N S U R A N C E C O M P A N Y G R O W T H

a m o r e a t t r a c t i v e r a t i o R , i t w i l l l ie t o t h e l e f t o f , a n d a b o v e , t h e l i n e i n

F i g u r e 1. T h e c o m p a n y s h o u l d a s s u m e r i sk v e n t u r e A a n d b i d a g g r e s s iv e l y

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

o t h e r r i s k v e n t u r e s s i m i la r t o A w i t h o u t a s s u m i n g A a t a lo w e r p r o f i t, t h e n

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

r i s k v e n t u r e s u n t i l , o n t h e m a r g i n , R = R 2.

F i g u r e 1 i l lu s t r a t e s w h y t h e i n s u r a n c e c o m p a n y s e e k s a l o w r a t i o R o n

r e i n s u r a n c e c e d e d ( a n d o t h e r r is k v e n t u r e s f r o m t h e s e l l / c e d e / d i s c o n t in u e

d e c i s i o n g r o u p ) b u t a h i g h r a t io R o n a l l r i s k v e n t u r e s a s s u m e d . B o t h r u l e s

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

I f r i s k v e n t u r e B h a s a l e s s a t t r a c t i v e r a t i o R . i t w i ll li e t o t h e r i g h t o f ,

a n d b e l o w , t h e l i ne in F i g u r e 1. T h e c o m p a n y s h o u l d d e c l i n e r i s k v e n t u r e

B ( u n l e s s it i s a n i n s u r a n c e r i sk v e n t u r e t h a t t h e c o m p a n y c o u l d r e d e s i g n

o r r e p r i c e ) . I t i s e a s i e r t o d e c l i n e a r is k v e n t u r e t h a t i s a p o s s i b l e i n v e s t m e n t ,

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

d i ff i cu l t t o d e c i d e t o l e a v e a n i n s u r a n c e p r o d u c t l in e if t h i s d e c i s i o n l e a v e s

a g a p i n t h e i n s u r a n c e c o m p a n y ' s p r o d u c t p o r t fo l io . I f i t i s i m p o s s i b l e t o

Net Present

Valae of

Marginal

Augmented

Book Profits

+1

- I 0

Issuance of comm on

(stock company) or

Issuance of subordinated

debenture (mu tual

company)

Risk venture A (to be pursued

aggressively)

M ore attractive real estate

already bought

A g e n c y ~ ' ~ " ~ N e w individual

d e ~ iit'e in'suranceseries

f f G roup.long.- R isk v en ture B

" ~ I term d isab ility (to b e re vtsed

+ 1 ins ur an ce or avoided)

ceded

Present Value of

stock Sequence of Marginal

Surplus Commitment

Increments

F IG . l . - - A n a l y s i s o f p o t e n t i a l r i sk v e n t u r e s , a r b i t r a r y s c al e . R e l a t iv e p l a c e m e n t

o n l i n e d e p e n d s o n s i z e o f i n c r e m e n t a l c h a n g e o f su r p l u s c o m m i t m e n t s a n d o f i n c re -

m e n t a l c h a n g e o f a u g m e n t e d b o o k p r o fi ts . S l o p e o f l in e e q u a l s l o w e s t a c c e p t a b l e

r a t i o R : . A l l r i s k v e n t u r e s a b o v e a n d t o t h e l e f t o f t h e l i n e s h o u l d b e a s s u m e d i n

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


25/50

INSURANCE COMPANY GROWTH 275

get risk venture B's ratio up to R 2, and if failure to keep selling or failure

to introduce the product line is of concern to the insurance company, then

the company can quantify what value product line B has for the other lines

of business. This value may be added to the marginal augmented book

profits of product line B and deducted from the marginal augmented book

profits of the lines helped by the maintenance or introduction of product

line B. 27 The marginal profits here are those calculated on the basis of a yes/

no comparison.

For some insurance companies, the marketing force is as much the limiting

resource as is the surplus. One goal, therefore, o f a career agency compan y

is to help new agents survive their first few years. Certain simple insurance

products may help because they are more easily sold by new agents than

are insurance products directed at sophisticated markets. If these simple

products help new agents survive, then the calculation of profitability for

the simple products should include a recognition that these products con-

tribute more than near-te rm statutory profits---they help develop the agency

force. 28 Exp ense alloca tions that allocate the heavy expenses incurred on

financing new agents to the products that new agents sell should be recon-

sidered in this regard.

C. Miscel laneous Comments on Strategy

Theoretically, the company need only undercut the competition slightly

in extremely price-sensitive markets to capture the whole market. At that

point in pricing, the marginal ratio R is discon tinuous. The next reduct ion

in price produces a reduction in profit but no additional sales. The ratio R

goes from a value above R 2 to a value below R 2. In such a price-sensitive

market, the company should anticipate that the competition will respond

by cutting its price to recapture the business. The company may consider

undercu tting the competition immediately to the ultimate level to which it

would be willing to go in a price war. This ult imate level is the lowest price

level at which the competition's prices could be set and at which the in-

surance company 's marginal ratio R would then be equal to R 2. If the

competition is willing to go below that price, then the insu rance company

drops out of the competition without having wasted a lot of money, time,

27 These additionsand deductionsare for strategic planning,not for statutory accounting.

OthSr theoretical benefitsof a particularcorporate policyor strategy may be difficult o quantify

in practice. Thus, the growth optimizationmay be done subject to management-specifiedcon-

straints. For example, managementmay believe the company should always maintain a large

presence in the ordinary life insurancebusiness. However, to optimize growth rationally, it is

best to quantifywherever possible. See

RSA,

Ill, 223.

28 RSA,

Ill, 45.


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27/50

I N S U R A N C E C O M PA N Y G R O W T H 2 77

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

r e s e r v e s d i d n o t c r e a t e t h e n e e d f o r a d d i t io n a l b u f fe r s u r p lu s . T h e n t h e o n l y

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

r e s e r v e s i n t h e U n i t e d S t a t e s .

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

i s g i v e n t a x d e d u c t i o n s o n t h e p o l i c y h o l d e r ' s sh a r e o f in v e s t m e n t i n c o m e .

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

a n d t h a t t h e c o n s e r v a t i v e r e s e r v e i s a li fe i n s u r a n c e r e s e r v e . W e w i ll s h o w

t h a t th e n e t p r e s e n t v a l u e o f a u g m e n t e d b o o k p r o f i ts f r om h o l d in g a c o n -

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

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

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

e n d o f th e p o l i c y y e ar )~ T h e s u r p l u s c o m m i t m e n t , h o w e v e r , i s p o s i t iv e .

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

a v o i d e d . T h a t t h e n e t p r e s e n t, v a l u e o f a u g m e n t e d b o o k p r o f i ts is n e g a t iv e

c a n b e d e m o n s t r a t e d a s f o l l o w s . B u f f e r s u r p l u s i s z e r o , s o b o o k p r o f i t s e q u a l

a u g m e n t e d b o o k p r o f i

full 100 marks insurance project (blackbook) for bachelor of banking & insurance and bms. i do not...

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