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Risk Management and the Theory of the FirmAuthor(s): J. David Cummins

Source:The Journal of Risk and Insurance,

Vol. 43, No. 4 (Dec., 1976), pp. 587-609Published by: American Risk and Insurance AssociationStable URL: http://www.jstor.org/stable/252028 .

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Risk

Management

and

the

Theory

of the

Firm

J.

DAVID CUMMINS

ABSTRACT

This

article integrates risk

management

decision variables into the

theory

of the

firm under

risk

and

develops

risk

management

decision

rules consistent with

the firm's overall

objectives.

The

theoretical con-

struct chosen for extension to the risk management problem is the

capital asset

pricing model.

Utilizing

this

model,

decision

rules

are

developed

for

optimal

proportional

retention, selection

of

aggregate

deductibles and

choosing

reserving policies.

The results

differ

signifi-

cantly from

the

expected

value

decision rules

developed by previous re-

searchers.

The

article

concludes by

examining

the

impact

on

the decision

rules of

relaxing

some of

the key assumptions

of

the model.

Recent

years

have

witnessed

significant

progress

in the

application of

quantitative

decision

tools to the

solution

of risk

management

problems.

These developments have ranged from purely theoretical exercises to highly

particularized practical

applications and

have

encompassed

the

entire

spectrum of

quantitative

methodology. For

example, a model

for

the

selection of

optimal

deductibles has been

developed and

applied

by Allen

and

Duvall

[1,

4].

Optimal

insurance and

loss

prevention

decisions

have

been

the

subject

of

a

paper

by

Shpilberg and

de

Neufville

[16]

which

employed a

decision

theoretic

framework. A

study which

involved the

testing of

alternative decision

tools,

such as

utility

theory and

the

worry

factor method, has been conducted by Neter and Williams [13] while

a

computer simulation

model of self

insurance

of

workers'

compensation

losses

was

developed by Mortimer

[11].

The

fitting of

loss

distributions

to

facilitate

the

more

precise estimation

of

future claim

costs

has

been em-

phasized

by Hartman

and

Siskin

[9].

Finally, Head

[8]

has

illustrated

how

capital

budgeting

can be

applied in

the context

of

risk management

decision

making.

These

and

numerous

other studies

have

provided the basis

for

significant

improvements in

risk

management

analysis.

However, as

is perhaps

in-

evitable in a developing field, most of the articles have concentrated on

J.

David

Cummins

is

Assistant

Professor

of

Insurance

in the Wharton

School, Univer-

sity

of

Pennsylvania.

Professor

Cummins is

Research Director

of the

S. S.

Huebner

Foundation and

is

a member

of

the

Editorial

Board

of the

Journalof Risk

and

Insurance.

He

is

the author

of An

Econometric

Model

of

the

Life

Insurance Sector of

the U.S.

Economy and

co-authorof

Consumer

Attitudes

Toward

Auto and

Homeowners

Insurance.

This

paper

was

submitted for

publication

June,

1975.

It was

presented to

the

1975

Risk

Theory

Seminar.

(

587)


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588 The

Journal

of Risk and Insurance

local rather

than

global optimization.

That is, the overall objectives of

the

firm generally

have been

recognized

only implicitly or

peripherally

in

the formulation

of the

risk

management

decision

rules.' This approach

is

potentially misleading,since rules which appear appropriatestrictly from

a

risk

management

point of

view may be

suboptimal

in the context

of

the

firm's broader goals. An

appropriatecomplement

to the

existing risk

management literature

thus

would seem to be

a general

theory which

integrates

risk management

nto the overall

business setting.

The purpose

of

this article is to develop

such a theoretical

framework.

As microeconomictheory

is a highly

developed discipline,

this

paper

does not attempt

to derive a totally

new theoretical

model. Rather,

the

goal

is

to adapt existing models

to incorporate

risk management

variables

and parameters.Because of the nature of the risk managementproblem,

a theoretical model is required

which

views the firm in

the context

of

uncertainty.

After

consideringa number

of alternatives, he

model chosen

for scrutiny in this paper

is the capital asset

pricing model

developed

by

finance theorists.2 The applicability of this

model was

suggested

to

the author by a recent

paper by Schramm

and Sherman [15].

That article

exploredthe idea

that firms

can managethe variability

of their

net income

streams

by varying

their advertising

and research and development

ex-

penditures. The

present paper

focuses on the

effect of the

selection of

alternativerisk managementdevices on that variability.In this regard,the

capital asset pricing

model

is first utilized to

derive some

fundamental

optimization

rules.

These rules

are

then expressed

in

terms

of familiar

risk

management

variables. Because

the

model rests

on

a

number of

simplifying

assump-

tions,

the paper

explores

the

realism

of

some

of

these

assumptions

n the

risk management

context and the impact

of their

relaxationon

the decision

rules. The paper

concludes

with a discussion

of the

potential

for the em-

pirical applicationof the model. The emphasisof this article is on theory

rather

than

on

practical applications

and the

analysis

is

conducted at a

high

level

of

abstraction.

Hopefully,

the theoretical

results

presented

eventually

will

provide guidelines

for

the

development

and testing

of

practical

risk management

decision

rules.

However,

much research

remains

before

direct

practical application

is

possible.

The

Capital

Asset

Pricing

Model

The

capital

asset

pricing

model as

developed

in the

theory

of

finance

is

a two period model in which the participantsare consumersand business

firms.

At

the

beginning

of Period

1,

consumersare assumed

to come to

the

market with

"quantities

of

resources-labor,

which

will

be sold to

some

firm,

and portfolioassets,

that

is,

shares

of firms

...

that must

be allocated

to

current

consumption

cl

.

.

. and

a

portfolio

investment

whose market

1

This

problem

has been recognized

in an

earlier

paper

by

Mehr and

Forbes

[10].

2This

theory

is

discussed

in

numerous

books

and articles including Fama

and

Miller

[51

and

Mossin

[11].



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Risk

Management

589

value

at

the

beginning of Period

2

determines

the

level of

consumptions c2"

for that

period.

Consumers

are assumed to be risk averse

expected

utility

maximizers

and

to

behave as

if

the

probability

distributions

of return

on

their portfolios can be completely summarized in terms of two para-

meters,

the

expected

value

and some measure

of

the

dispersion

of the

returns.

The business

firms

in

the market organize

production

activities which

are

financed

by

selling

shares in their Period 2

output

values to

consumers

The

firms

are assumed

to operate

according to the decision rule: maximize

Period I

market value.4

The solution of the

model involves

the

simultaneous

optimal

allocation of funds by

consumers

between consumption

cl and

investment and the

maximization by

firms of their Period 1

market values.

A number of assumptions are inherent in the capital asset pricing model:

1.

Perfect

capital markets. This

assumption

is similar to that found

in the

conventional

economic theory

of perfect competition.

Among its

major

implications for the instant

case are that

no transactions costs

or taxes

exist,

and that

"no

firm is

large enough to affect

the

opportunity set

facing consumers."

2.

Consumerrisk aversion.

3.

Symmetric

stable

distributions. Consumers

are

assumed to behave as if

distributions

of returns on all

portfolios

are symmetric stable with

the

same value

of

the

characteristicexponent. This

assumption permits

the

summarization

of returndistributions

by two

parameters-the mean

and

a

measure of

dispersion.

For

convenience, this analysismakes

the added

assumption

that

the distributions are

normal;

i.e.,

that

the

measure

of

dispersion

s

the

variance.

4.

Riskless

borrowing

and

lending.

The

assumption

is

that consumers can

borrow and lend

as

much

as

they

wish at a

risk

free rate of

interest.

5.

Homogeneous

expectations.

This

assumption

means

that

"all

consumers

have

the same set

of

portfolioopportunities

. .

and all view

the

prob-

ability

distributions of

returns

associated with the various available

portfolios n the same

way."

6

All

five

major

assumptions

are

crucial

to

the

model,

but the last

four

make the

capital

asset

pricing

model

unique.

The

third

assumption

im-

plies that consumers

can summarize

all

distributions of

return in terms

of

the mean

and

a measure

of the

dispersion

of those

distributions. Com-

bined with

the

second

assumption,

the

use of

two

parameter

distributions

permits

the consumer

to consider

only

those

distributions

which are

mean,

variance

efficient. Assumptions

4

and

5 further

simplify

the

problem by

permitting

all

consumers

first to

view

the

efficient set

in

the

same

way

(assumption 5) and then (through assumption 4) to focus attention on

one

common

point

on that

frontier,

the

market

portfolio.

Thus, the

assump-

3

Fama and Miller

[5],

p.

277.

4For a

more complete

discussion

of

the firm's

objectives

see Fama

and Miller [5],

pp.

299-301. Schramm

and Sherman

[15]

have

shown

that when risk is taken

into

account,

the value maximization

goal may

give

rise

to

business behavior

which

manifests

itself

as either

growth

or

sales

maximization.

5

Fama

and

Miller

[51,

p.

277.

i

7i2A

D.

9287



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.590

The Journal

of

Risk

and

Insurance

tions

lead

to

market

equilibrium

conditions

in

which

individual

utility

functions are

not

explicitly

present.

Although capital

market equilibrium

in

the foregoing

model

implies

a number of interesting relationships, the one selected for the analysis

of

risk

management

decisions

is

the

pricing equation

for the

Period 1

value of

a firm:

E(l)Rf

(1)

where

P

-

the

equilibrium

market

value of

the

jth

firm

at the

beginning

of

Pqriod

1

(in

equilibrium,

this

quantity

is

maximized);

- the market value of the firm at the beginning of Period 2;

E(f) -

P i

+

Rf)

S

M

. -f

m

(

-'the market value

at the

beginning

of Period 2 of the

market

m

portfolio

("market

wealth");

PM

market

value

at the

beginning

of

Period

1

of

the

market

portfolio;

Rf

a

the risk free borrowing-lending rate; and

p

a

the correlation

coefficient

between

the return on the

jth

firm

and that on the market

portfolio.

The

placement of a

tilde over

a symbol

indicates that it

represents a

random variable.

According

to the

pricing

equation,

the Period

1

marketvalue

of the

firm

is the

present

value,

computedat the

risk

free rate, of the

firm's

expected

value at Period 2 less the present value of a risk charge. The risk charge

is

equal

to the

product of

the market

price of

a unit of

standard

deviation

(Sm),

the

correlation

coefficient

of the

firm'sreturn

with

that of the

market

and

the

standard

deviation

of the firm's

Period 2 market

value. Of

interest

is

that the

risk

penalty of the

jth finn

varies

directly

with its

correlation

coefficient

with market

return.

That

relationship exists

because a stock

with

a

relatively

low correlation

with

market

return is

more valuable

to

investors in terms

of

diversification.

Optimal Risk ManagementDecisions

General

Optimization

Rules

In

most

developments

involving the

capital

asset pricing

model,

the

parameters of

the

distribution

of

returns of the

individual

firms

remain

invariant. In

order

to apply the

model

in the risk

managementcase,

how-

ever, that

assumption

must be relaxed

to

permit the firm

to

vary within

7

Ibid., p.

296.



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Risk

Management

591

limits

its mean

and

variance

of

return.

In particular,

he assumption

that

the

objective

of

the

firm

is

to vary

E(V)

and

a(V1)

to maximize

ts

Period

1 value,

P5.

This

procedure

is an

accurateabstraction

of

the

risk

manage-

ment process as most decisions involving the use of deductibles and self

insurance

hinge

on the

tradeoff

between

savings

in the expected

cost

of

pure

risk

and

the

increased

risk

faced

by

the

firm

as a

result

of

reductions

in

its

insurance

coverage.

For

example,

when

a

firm

elects

a larger

deductible

the

usual

motivation

is

that

its insurance

premium

is reduced

by

a

larger

amount

than

the

resultant

increase

in

expected

retained

claims

and

settlement

costs.

In

return

for

this

savings,

however,

the

firm

must

be

willing

to

accept

a

higher

degree

of

variability

in

its

income

stream

as

an

unexpectedly

large

numberof losses could result in total loss costs substantially n excess of

the

original

premium.

Intuitively,

the anticipation

is

that

the

firm

would

increase

its

deductible

to

the

point

at which

the

value

of

the

premium

savings

is offset

precisely

by the

cost

of

the

increased

variability

in

the

income

stream.

A

noteworthy

assumption

in this analysis

is

that

the

firm

can

obtain

a

net

savings

by

varying

its degree

of

pure

risk retention.

Such a

savings

is

not

necessarily

available

in the world

of

perfect

markets

and

the

assump-

tion is equivalent to the introductionof a degree of imperfectionin the

insurance

market.

Investigations

of the

precise

role

of

insurance

n

economic

equilibrium

and the

conditions

under

which

the

hypothesized

savings

would

be

available

are

beyond

the

scope

of

this

study

but

constitute

in-

triguing

avenues

for

future

research.

This

risk

management

process,

under

the

assumption

that

savings

through

risk

retention

are

available,

can

be

expressed

in a

more

precise

manner

through

the use

of the capital

asset pricing

model.

Thus,

accepting

a larger

deductible

or

adopting

a

self

insurance

program

results

in

an

increase

in both

E(V' and

O);

I-e.,,

;D

>o

and

3D.L

>

0

where

D

stands

for

the

retention

of

the insured

The

optimization

rule thus

becomes:

Maximize:

Equation

(1)

with

respect

to D.

In

general

terms,

the

first-

and second-order

conditions

for a maximum

are

the following:

First-order

Condition

3D 3D

mjm

a3D

no

UD

)/(

aD

is

P

(2t



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592

The Journal

of

Risk and

Insurance

Second-order

Condition

a2p

921( )

32a('

)

BD2 3D2

D

3D2

When written as

(2)',

the

first

order

condition for a

maximum

can

be

given

an

interesting

interpretation.

Implied

is

that the

firm

should

in-

crease its

risk

retention to the

point

at

which

the

marginal

rate

of

sub-

stitution

between return

and risk is

equal

to

the

price placed

by the

market

on a unit

of

standard deviation

multiplied by

the

correlation

co-

efficient between

the firm

and the

market.

Beyond

that

point,

the firm

will not

receive

an

increase in

expected

return

adequate

to

compensate

for

the additional

risk.

Those familiarwith the capital asset pricing model will recognize that

the

derivation of

expression (2)

involves

additional

simplifying

assump-

tions.

In

particular

the

assumption

that the risk

management

decisions

of

the

individualfirm have

no

impact either

on

the

marketprice

of

risk,

S.,

or the

correlation

between the

jth

firm's

return

and

that

of

the

market;

i.e.,

the

assumption

s

that

aSm/JD

and

apjm/aDare

equal

to zero.

Those

assumptions

are not

supportable

in

a strict

mathematical sense as

the

parameters

of

the

firm are

components of the market

parameters.

How-

ever,

the

former

condition can

be justified

by

noting that the

model

is

couched in terms of perfect competition. Thus, even though the firm's

risk-return

characteristicsare

components of

Sm,

the number of firms

in

the market is

so large

that any

change on

the part of

one

particular

firm

has no

measurable

effect

on the market

set.

The

latter

condition, ap

/1D

=

0,

is

considerably

more

restrictive, but

its

use in an

approximate

ense can be

justified

heuristically

by recognizing

that

most of the

covariancebetween

the

returns of the

various

firms

in

the

market

arises

from the

nature

of their

responseto

economic

fluctuations.

A change in retention, on the other hand, involves the assumption of

nsks

generally

uncorrelatedwith

the business

cycle

and are

approximately

independent across

the

set of all firms.8

Thus,

additional

pure risk as-

sumption should

affect the

correlation

with

marketreturns

only

minimally,

if

at

all.

A

version

of

the

first-order

ondition which

permits

pjm

o vary

with

D,

while

retaining

the

assumption that

aD

0

,

is

presented as

equation

(4):

aP

aE(

)

acov(VI,

3D aD a a

3D

-

(V

a

OD

M

aCov(tJ

e

k)

When air " 0,

j # k,

equation (4)

reduces

to:

"A

number

of

well-known

real

world phenomena

are contrary to

this

assumption.

For

example, disability

income

claim filings are

negatively

correlated

with the business

cycle,

while

auto

collision claims

exhibit a

positive

correlation.

However, in

general,

a

reasonable

hypothesis

is

that pure

risk

occurrences

demonstrate a lower

degree of

interfirm

correlation

than

returns

from

productive

activities.



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Risk Management

593

UP

aE(V)

S aa2tV()

CO?)

DD

DD

a0?m

D

j

)

Readers concerned about the assumption that =0 may wish to

9D

reformulate the

subsequent analysis

in

terms of

equation

(4)

or

(5).

An

importantobservation s

that in a

theoreticalsense the risk

manage-

ment

decision affects the

production

characteristicsof the firm;

i.e., it

alters the

probability distribution

of the

firm'stotal earnings at

Period

2.

Since

the

standard capital

asset pricing model

assumptionsare in

effect,

once the

risk

management

decision has been made,

how the

resulting

operations are

financed is

irrelevant; i.e., the mix

of debt and equity

financing has no impact on the marketvalue of the firm.9Thus, whether

the insurance

premium

and/or the retained losses

are paid for out of

equity or

debt-generated

financial resources is a

matter of

indifference

to the

firm's

shareholders.

Proportional

Retention

In

order to

explore further

the implications of

expression (2)

for risk

management

decisions, to define

E(V

)

and

Cov(VjVm)

in terms of

more familiar risk management variables seems appropriate.Thus, one

can

state that

v

=t

-

C

(6)

:1

'i 21

where

v'

W

net income of the

firm

considering

all

revenues and

all

expenses

other

than

those associated with

pure risk; and

C

=

the

pure risk costs of the

firm.

By

focusing on the

componentsof

Cj,

it is

then

possible to establish some

more precise decision rules.

In

this

section,

the

assumption

s

that

the firm can

base its retention on

a

quota

share

arrangement

on

"Original

erms."

n

that

case,

,-(1

.-c)G2

a(-

+

EJ)

(

where

G

-

the insurance

premium

which the

firm

would be charged for

complete,

first

dollar

coverage;

L - the

firm's

losses for

the

year;

Ej

=

the

cost of

administeringhe

pure

risk

of the

firm

f

all

losses

are

retained;10

nd

a

=

the

proportion

f the

risk

retained

by

the firm.

9

This

is

the familiar

Modigliani-Miller

Proposition

I.

For a

proof

of this

result

see

Fama

and

Miller

[5], pp.

160-164.

"'In

a

more general

formulation

Ej

would

have

fixed

and

variable

components with

the

latter

being a

function of

Li.

The

simpler

formulation is

utilized

here to

reduce the

computational

burden.



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594

The Journal

of

Risk and

Imurance

Based

on

this

arrangement,

E&H

) - E(01)

-

l

-

a)G

+

aE(L;

)

+

]l

(8)

02(;t^-

U2(Bt)

+ 32cU2C)

- 2aCO"

.i;

(9)

:1

1

-

i'i

Note

that the insurance

premium

contributes

nothing

to the variance

of

the finnds

ncome

stream;

iLe,

by

purchasing

complete

insurance

coverage

the

firm can transfer

its entire

pure

risk

to the

insurer.

The

presence

of

the

covariance

term reflects

the fact that

losses

due

to

pure

risks

give

rise to indirect

as well as direct

costs. For

example,

a

business

interruption due to

a fire could result in

a decline

in

future

revenuesif some customers ail to returnwhen the firmresumesoperations.

The

subsequent

analysis

assumes

that all costs

of

purerisk

can be

subsumed

under

Lj

and,

therefore,

that

Cov(tLJ

0.

Based

on

that

assumption

and

equations

(2),

(6)

and

(7), the

first

order

condition for

a

maximum

in the

proportional

retention

case

is

the

following:

Oa2(j )

(G~j E(L;j)

-

EJ)

-

szpjm=_

Li

(10)

a

j

Solving

for

a,

one

obtains:

(G1a

Ei~

1

)s

~

(11)

a(j

)Smm

Verbally, tyis

directly

related to

the

marginal

reduction in

expected loss

costs

arising

from

an

additional

dollar

of

retention

(i.e.,

to

Gj-

E(tj)

-

En). (Note

that

retention

is not

feasible if this

term is

e

0.

Further-

more,

if

a

exceeds

1

the

analysis

suggests

that the

firm-

hould

become an

insurer.)

The value

of

a

also

bears a

direct

relationship o

the

ratio of

the

standard

deviation of

the

firm's

ncome

stream,

?(Vj),

and the

varianceof

its loss

cost

distribution.Thus,

the more

significant

the

pure risk

in

terms

of the

total risk

of

the

firm,

the

less of it

the

firm

should

retain.

Finally,

a

is inverselyrelated to the product of the marketprice of risk and the cor-

relation

coefficient;

.e.,

for

a

given

Sma

firm

whose

returns

are more

highly

correlated

with those

of the

market

should have a

lower

retention.

Aggregate

Deductibles

and

Self Insurance

Reserves

The

foregoing

model

should

be

relevant for

a

number

of

risk

manage-

ment

situations. For

example,

if

a

is

close to

1,

the

model

implies

that

complete

self

insurance

is

in

order.

Furthermore,

applying

the loss

unit



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Risk

Managenwnt

595

concept and

making

the assumptionthat loss units

are independent,

the

decision

rule

could

be

applied separately

to the various pure risks facing

the firm.

For

instance,

the computation

may give

one answer for the

workers' compensation

risk and another for

that

of products

liability.

Nevertheless,many risk managementsituations exist to which (11) does

not apply.

If

a

is substantially less

than

1

but

greater than zero, for

example,

the results have

little practical meaning

as quota share

arrange-

ments

are

rarely

effected

between

non-insurance irms and insurers.The

more

common

arrangement,

of

course,

is the

use of

a

deductible;

and,

consequently,

the model

now will

be adapted

to

apply to the

deductible

selection

decision.

Aggregate

Deductible

Selection.The first case considered

s that

in which

the firmis faced with the choice of an optimal aggregate deductible;i.e.,

a deductible under

a

contract

in

which the insurer

agrees to pay all

losses

in

excess of a

total

annual

amount,

D. The retention

of the firm

under such

an arrangement

s:

L

-

,t

if

D;

and

Ji

i

*D

if

D

CL2)

AD

~ 1

,where

a5 the losses retained by

the

insured.

The

relevantparameters

can be

developed as follows:

H(t-)

J

dP(i)

+

DfdF(j)

(13)

D

N2R)

-

J

dFti

)

+

D2JdP(%)

E2(R)

(14)

ID

Differentiationwith

respect

to D

then yields (after

some simplification):

ID

w

-

F(D)

5

3D

2[

1

-

F

(D)]1[D

-

y~gj

Substituting these expressions

nto

equation (2), the first-order

condition

for

a

maximum,

one

obtains:

3G

[1

-

F(D)][D

-

Oi)+

Sp

}

(17)



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596 The

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Risk and

Insurance

The left side of equation

(17) is derived by recalling equation (6) and

noting that

Cj

in this case

is

equal to the insurancepremium

plus

the losses

and

administrative

xpenses

retained under the aggregate deductible plan.

Although theoretically he optimaldeductiblecan be obtainedby solving

expression (17) for D, a

generalized explicit solution

to

the

equation

can-

not

easily be

derived.

Nevertheless,

this

version

of

the

optimization

condi-

tion suggests

two

important

observations about the use

of

aggregate

deductibles. First,

an

increase in the

aggregate

deductible

always

increases

the variance

of retained

losses; i.e.,

aa2(i)/laD

>

o.

This

relationship

can

i D

be verified,

under the

assumption

that

>

o

and

that

i .

>

o,

by

noting that

D

P

U(R

(t

dh)

+

D

UPt

D

<

H(t

)

+

DfdF(?)

- D.

0

D

Thus ,

D

-

E(?~R)

>

0

and, from

(16),

aa2CR)/aD

>

0.

Second,

it should be

evident

that a

necessary (although

not

sufficient)

condition

for

the

adoption

of an

aggregate

deductible

is

that

the left side

of

equation (17)

must

be

positive

for some

value

of

D. That

term,

of

course,

represents

the net

marginal

expected

savings

in loss costs

arising

from

the

use

of a

deductible; i.e.,

the

marginalpremium

reduction

aGJ/aD

less the

marginal

increase

in

the

expected retention,

1

-

F(D).

If the

expected increase in retained losses always is greater than the premium

reduction,

the firm has no incentive to adopt a deductible. Likewise, if

the term

is

zero; i.e.,

if

the

premium

is

actuarially

air

at the margin, the

firm

should not

adopt

the

deductible because

it

would be accepting a larger

variance with

no

offsetting

increase

in

expected return. The condition is

not

sufficient because a positive expected reduction in loss costs must be

large enough to offset the

resulting increase in the variance, with the re-

quisite relationshipbetween

the two

effects

determinedby the parameters

of the


model.

The Optimal Level of Reserves. Another interesting case involves the

choice

of an

optimal

buffer or

reserve

fund to

accompanya retention pro-

gram.

In order to

focus

exclusively

on

this

decision,

the

analysis is initially

conducted

under the

assumption

that

an

optimal aggregate deductible

already

has been

selected

and

that, given

that

deductible, the firm wishes

to

establish

an

optimal

size

reserve fund. The

case in

which the optimal

deductible

and

reserve fund

are established

simultaneously s considered

subsequently.



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Risk Management

597

To

solve the

buffer

fund

problem

it is useful to redefine

Vj

as follows:

t e

Gj

(1

+

Rj) Bj

AdR

RC)

-

A

-Bj)

Rj

t

Bj

<

O

D

Two

additional variables have

been introduced:

Rj

=

the

rate of

return

on

shares of firm

j;

and

Bj

=

the

buffer or reserve fund for loss costs associated with pure

risks.

Equation (18) is equivalent to that utilized to derive the optimaldeductible

OV

#-%

(LJ-Bj)Rj, Bj

<

L

<

D

with the

exceptionof 2

terms,

Bj

(Rj-Rf

) and

{#

(D-Bj)Rj,

LiND

which are designed to represent the effect of reserves on the net

revenues

of the

firm. The

former term

appears

in the

equation

with a

negative

sign and represents

the

opportunity

cost

of holding reserve funds;

i.e.,

if

the firm maintainsreserves of

Bj*,

it

sacrificesearnings

of

RjBj*,

which

could be earned

if

those funds

were

invested

in its

production

processes,

but earns instead the lower amountRfBj*, which representsthe earnings

on

the reserve fund invested

in

relatively liquid,

risk free

assets.

The

latter

term, on the other

hand,

represents

the

firm's

sacrifice

in

earnings

if

losses

exceed

the

buffer

fund.

In

that

instance,

the

equation implies

that

the

firm loses its

earnings

for the

entire

year

on funds

paid

out

as losses.1'

That

aspect

of the

equation

constitutes a built-in

penalty

for

failure

to

maintain adequate reserves. That

is

the case because

the

equation

suggests that losses paid from reserve funds

are

paid at

the

end

of

the

year while

loss

payments

in

excess

of

reserve funds

are made

at

the

begin-

ning of the year. To understandthis point, note that for losses greater

than the reserves, lost interest

is

equal

to

Ri

times

the

excess

while

the

buffer

fund

is

assumed

to earn

interest

at

the

risk

free

rate for

the

full

year.

If

losses less than

the

buffer

fund

were

paid prior

to the

end of

the

year,

the

firm

could

not

be

credited with

the entire

amount,

BjRf.

A more

nearly

realistic version

of

the model

might

utilize

the

following

cost

ele-

ment

for

the buffer fund case:

-

-(Bj -

Lj) (Rf/

2) ,

L'j D

-

L

"The loss

paymentshemselvesrereflectedn L1.



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598

The

Journal

of

Risk and

Insurance

This

expression

makes

the

alternative

assumption

hat losses are

paid, on

the average, midway

through

the year. Although

such an expression

is

more

realistic than the one

included

in equation (18),

the latter

formula-

tion has been utilized in the analysis. That decision was based on two

principal

considerations.

First, the desire to

compare

the results of the

present

model

with those obtained by

Duvall and

Allen

[4],

whose

formula-

tion is

consistentwith equation

(18) and second,

the difficulty

of fully re-

flecting the

consequencesto the

firm

of uninsured

ossesdifferent rom

those

expected,

due

to the abstract nature

of the

two period capital

market

model.

The

assumption

inherent in equation (18) helps

to remove this

limitation

as it

exacts a more significant

penalty

for losses

'in excess of

reserves

than the

alternative expression.1

For the reader

reluctant

to

accept this assumption, he optimalityconditionscan easily be derived by

performing

the

appropriatecomputations

under

the alternative

formula-

tion. These results

differmidetail but

not in substance

fromthose

presented

below.

Before

proceeding

with the computations,

it seems appropriate

to

re-

iterate

that the optimization

is carried out under

the assumption

that

Rj

and D are fixed; i.e., the optimal

buffer

fund will be derived

conditional

on

those

two variables.Taking expected

values

and

differentiating

equa-

tion

(18)

with

respect

to

Bj,

one

obtains

(after

some

simplification):

3B

0

)

- (j-

Rf)

+

RjEl

-

(Bj)J

(19)

This

result

is

equivalent

to that

obtained by

Duvall and

Allen

[4,

p. 501]

if one assumes

no taxes.

As

explained

in

their

article,

(Rj

-Rf)

represents

the

opportunity

cost to

the firm of increasingreserves by $1,

while

the

latter

term

is

the

expected

savings

of

the

firm

as

a result

of the

additional

dollar of reserves.

To obtain the

variance contribution

of the

buffer

fund, note that

when

Rj

and

D

are fixed,

the first three terms

of equation (18)

do not contribute

to the

variance

of Va. Differentiation

with respect

to

Bj

of the

variance

contribution

of

the last two

terms

yields 392B

Although

these

com-

putations

are

straightforward,

they are cumbersome;

and,

accordingly,

are presented in an Appendix,

which

is

available

from

the author. In

the

Appendix

it

also is demonstrated

hat

aac2(Vj)/aBj

<

0

for

0

< B,

D

that,

for a

fixed

deductible,

the firm can reduce its variance through

the

use of a

reserve fund.

An intuitive rationale

for

this

variance

reduction

is

that

earnings

be-

come

more

predictable

as a

result

of

the

earmarking

of

funds

for

the

12

The

penalty robably

s

realistic

ue

o

the

iquidity

roblems

hich

n

unexpected

lss

couldpresent.



8/9/2019 252028.pdf

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Risk

Management 599

payment

of

unexpectedly large

retained losses. This function

of

the

reserve

fund

becomes

readily

apparent

when

one

thinks

of a

real-world

case in which

a firm can cushion the

impact

on

reported

earnings

of

an

uninsuredloss by drawing down its reserve fund. This increasedstability

in earnings might be expected

to

give

rise

to a

corresponding

tabilization

in the

price

of

the firm's

stock.

As

one might anticipate,

however,

the

use

of

a

reserve fund

generally

involves some

sacrifice of

expected

return

by

the firm. This

anticipation

is

the

case as

(19), although positive

for

small

Bj,

is

a

monotonically

de-

creasing

function of

that variable

and

can be

expected

to be

negative

for

a buffer fund of

any

substantial size. The

implications

of this result

for

the

selection

of

an

optimal

buffer fund

can

be

given

more

precise

form

by

setting up the maximizationcondition [from (2) and (19)]:

aa2(v

)fas

Rf

-

RF(B)

=

SmPjm

(20)

j 3 mjin

2a(Vj)

Since

au2(Vj)/aBj

<

0,

it is clear that if

[Rf

-

RjF(B,)]

is

positive fox

all

Bj

-?

D,

the buffer

fund

should be set

equal

to the

deductible.

In that

case the

use of

the

buffer fund would

both increase

expected

returnand

decrease the variance over the entire range of possible values of B,. In

view of the fact that

[Rt

-

RjF(Bj)]

may

become

negative for

B,

< D,

however,

it

seems

appropriate

o

examine

the most

likely

outcomes

under

that condition.

These

are

illustrated in

Figures la and lb.

Rf

E

(Vj)

]

Rf

E'(VY)

]

iB

optimum

8

ptimum

I

~~~~~~~~f

cr21t'1

~~~I

I

Figure

'a

Figure

lb

SELECTION

OF

AN OPTIMUM

RESERVE

FUND

FOR A PREDETERMINED DEDUCTIBLE

*V

In the

figures,

the curve

E'(Vj),

which

represents

[Rf

-

RjF(Bj)],

is

equal

to

Rf

when

Bj

=0, and

then gradually

declines,

crossing the B,

axis

for

B,

< D. The other

curve

f[Cr2'(V,)

,

which represents

aa2(0

)/aB

SMP

2a



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600

The

Journal

of

Risk and Insurance

is below the

Bj

axis for all

Bj

>0 and

:

D and is drawn with a minimum

for some

Bj

< D.

As

demonstrated n the Appendix available from the

author,

the

curve is unambiguouslynegative, but the U-shape portrayed

by

the

figures is only

one

possible outcome. It is possible for the curve

to be monotonically decreasing; but if so, outcomes similar to those

illustrated n Figures la and lb still would representthe only alternatives.

In

Figure la, the value of the variance reduction resulting from

the

use of the buffer fund always is greater in absolute value than

E'(Vj).

In that case, the optimal solution would be at

Bj

=

D. Figure lb illustrates

the more likely outcome; i.e., the two curves intersect yielding an optimal

reserve fund less than the deductible. In both cases,

however,

the optimal

buffer fund occurs when

E'(MV)

< 0. This finding indicates that expected

value

decision

making

would result in a buffer fund of suboptimal size

as

in that case

the

fund is chosen where

E(VJ,)

=

.ys

Simultaneous Selection

of

Deductibles

and Reserves.

The

preceding

analysis has developed optimization

conditions

for an

aggregate

deductible

in the absence of

a

bufferfund

and for

a buffer

fund

given

a

predetermined

deductible.

While those decision

rules

were

useful

in

focusing

attention

on the individual

variables,

theoretical

precision requires

that a

more

general

result

be obtained

which

will

permit

the simultaneous

selection of

optimal

values for

the

decision

variables,

D

and

B,.

To

facilitate

the anal-

ysis, it is helpful to expressthe problemas follows:

Maximize:

Equation (1),

with

Vj

given by (18)

Subject

to:

Bj

D

Forming

the

Lagrangian,

Z

=

P,

-

k(Bj

-

D),

and

taking partial

deriva-

tives,

the

necessary

conditions for a maximumare obtained:

az i

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Jaa2c

)fa

aZ

DE (V )

a

S

/aD

Q.

i~)

a

0

(21)

aD 1 + Rf 'D mjm 2a( )

9z

1

3E(R

)

aa2(~

)/aB

3Bj ' (B Smf) 2M

i

)

-

X

*

0

(22)

j f j

~~~~2a(~

X

>

0; B.

-

I)

<

0; A(B1

-

D)

-

0

(23)

Because

the constraint

s

an

inequality,

the

conditions

are

obtained

accord-

ing

to

the Kuhn-Tucker ules.

The

term

aE(V)

is

given by equation (19)

and

aa2(V)/9Bj

is

pre-

sented

in the

Appendix

available from

the

author.

However,

the

cor-

responding partial

derivatives with

respect

to

D

are different

from those

implied by equations (15)

and

(16)

because

those

expressions

did not

explicitly recognize

Rj

and

because

they

were

developed

under

the as-

2

Se

Duval

and

Allen

(41

p. 501.



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Risk

Managenent

601

sumption that

no

reserve

fund would be used. With the

introduction of

those

factors,

it

can

be

shown

that:

DFE(v

G

an

a

_

a~:

(1 + R - - F(D)I(I + R) (24

and

-Z(9

-

2(1

+

Rj)t1

-

F(D)]{jL

dF(L)

+ jL(1

+

Ri)dF(i)

0 B;

+

BjRF (j

) - (1

+

R; )DF-(D) (25)

Under the assumptionthat

JdF(Ly

*

o and that the probability mass of

0

the

distributionover the

range

0

"I

<

D

is

not all concentratedat

Lj

=

D,

it

is

easy

to show that

aa2()/9D

>

0.

Furthernore, when BA=O, expres-

sion (25)

is

equivalent to (16) with the

explicit

recognition

of Ra. With

the introduction of a reserve fund,

aa2(t

)/aD

declines, and this decline

is

uninterrupted

or

O

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602 The Journal of

Risk and

Insurance

Impact of Selected Assumptions

The preceding analysis of risk management decisions in the context

of

the capital asset pricing model has been conducted subject to a number

of simplifying assumptions. Many of these are no more restrictive in the

risk

management case than for other types of microeconomic analysis and

have not been analyzed further in

this

article on the ground that such

a discussion more properly belongs in an article on the capital asset pricing

model itself. However, two of the assumptions are particularly limiting in

the

risk management case, and the purpose of this section is to determine

the

effect of their relaxation on the risk management decision rules. The

two assumptions are: (1) that there are no taxes and (2) that the return

distributions are symmetric stable.

Taxes and Risk Management Decisions

The assumption about taxation is restrictive in the risk management

case because of the rules regarding taxation of casualty losses,

insurance

premiums, and contributions to self insurance reserves. Although

the tax

law in this area is

complex,

for theoretical purposes it is generally correct

to assume that

the

first

two items are deductible while

the

third

is not.

In

other words, the tax law contains

an

apparent bias

in

favor

of

insurance

and against self insurance.

The tax rules can be incorporated into the model by dividing Bj in

equation (18) by (1

-

t) where t is the corporate income tax

rate. This

operation

indicates

that,

when taxes

are present,

Bj/(1

-

t) dollars (before

taxes)

rather

than

Bj

dollars are needed to set

up

a buffer

fund of

size

Be. Carrying

out

the

optimization

with that modification

for

the most

general case discussed above yields a decision

rule

equivalent to equation

(26)

with terms

analogous

to

equations

(24), (25), 19, and

V2(Jv)

.

As

one

would expect, it can be shown that the introduction of taxes results both

in

a

smaller

buffer fund and a lower deductible than

in

the no-tax case;

i.e., the tax law does appear to be biased in favor of insurance.

Decision Rules with Non-SymmetricDistributions

The

significance

of the

symmetry assumption derives from the fact that

many

loss

distributions

are known to be positively skewed.14 However,

this

does not

necessarily invalidate the results obtained above utilizing

the capital market model. That is the case as the model is based on

distributions of total

return,

which have generally been shown to be

symmetrical.15 Apparently, the risk management aspects of the typical

firm's operations combine with its other activities in such a way that the

total

distribution retains the symmetry property. This observation,

in

fact,

is

what the central

limit theorem would lead one to expect for a

14

See, for

example, the

results

obtained by

Hartman

and

Siskin

[9].

15

Fama and

Miller

[51,

p. 260.



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Risk Management

603

large

firm

with many (approximately)

independent activities. Another

factor which

may

limit the

impact

of the skewness

problem

is that

the

risk inherent in

the large loss tails of

loss distributionsusually is trans-

ferredby the firmto an insurancecompany.Thus, althoughthe distribution

of

retained losses

is

not

precisely

symmetrical,most of

the skewness effect

is

no longer present. As a consequence

of these factors, the foregoing

decision

rules

probably constitute a satisfactoryapproximationo the more

precise

theoretical results which would

be obtained if higher moments

of the loss

distributions

were

considered.

In

spite of

the foregoing considerations,optimizationrules are

developed

below which

recognize

the fact

that loss distributions

are

not

symmetrical.

These

rules are

presented for two majorreasons. First, it is possible that

for some firms the insurance capacity crisis, the general trend toward self

insurance,

and

the changing economic and legal environment will cause

risk

management decisions to play an

increasingly important role in the

determination

of

net

income.

Thus,

distributionsof returns for

those

firms

may

in

the

future acquire a more pronounceddegree of (negative)

skew-

ness. An

industry in which such a trend may be present is

the

drug

in-

dustry,

which

is

beset by burgeoning products liability problems.

The

second reason

for the development of a more general set of decision rules

is

to provide an indication of the nature of

the precise optimumconditions

in

the risk managementcase. This

precision is desired on the ground that

one

should fall

back

on

the central limit theorem

only as

a

last resort and

that even

then it would be helpful to

know the extent to which

mean-

variance

analysis

leads to a departure

from the exact result.

When the

symmetry assumption

is

removed,

the effect on

the

capital

asset

pricing

model

is

significant. If one

is

no

longer willing

to

assume

symmetry,one can no longer summarize

distributionsof

return

in terms

of their

means

and

variances

(assuming, of course, that utility

functions

depend on more than the first two moments of the return distributions).

Thus,

mean-variance

fficiency

ceases to

be

a useful

concept,

the

advantage

derived from the

assumption

of

risk

free

borrowing

and

lending

is

lost,

and

utility

functions

must

be

introduced

explicitly

into the

optimization

problem.

Unfortunately,

the introduction

of

utility

functions

gives

rise

to

other

fundamental

issues.

For

example,

not

immediately apparent is whose

utility function should

be introduced.

Consistency with the capital asset

pricing model

requires that the function

be that of an investor, but more

than one investor generally holds sharesin a firm and interpersonalutility

comparisonsusually are not acceptable.

Recent

developments

in

the

theory of finance

may offer a possible solu-

tion

to

this

dilemma.

Friend

and

Blume

[7]

have shown

that

the

relative

risk

aversion coefficient

of an individual investor can be

approximatedby

the following function:

16

16

See Friend and

Blume

[7], p.

10. Risk

aversion

coefficients are

discussed

in

Arrow

(2]

and

Pratt

[141.



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604

The

Journal of

Risk and Insurance

11+1

Ck

where C - the harmonic mean of the

Ck,

and

ek

-

a disturbance term with zero mean.

In other

words,

a

reasonableassumption

s

that

in

terms of

expected

values

a

constant relative risk aversion coefficient for the

entire

securities

mar-

ket can be developed. Thus, to

utilize

a utility

function for the

entire

market in lieu

of one or

a series

of individual

investors'

utility

functions

may be acceptable.'7 This procedure

is

equivalent to assuming

that the

market, through

the

actions

of

a large

number of individual

investors,

arrives at a ranking of probability distributionsof return on risky assets

which possesses the properties required by the axioms

underlying

utility

theory.

The use of

a

marketutility function would retain

a

key advantage of

the

more restrictiveform of the capital asset pricing model; i.e., decision rules

could be developed and applied without constructing individual or cor-

porate utility functions.

As the use of the market ndex involves

a

number

of complex

problems,

however, that development has been consigned

to a future research project. This paper follows instead the more con-

ventional approach of adopting the utility function of the firm'smanagers

as

the

appropriate

index.

When

the

model

is couched

in terms of

utility analysis,

the

basic optimiza-

tion

rule becomes:

Maximize:

EMU(O)]

w

fU(j)dF(j) (28)

where

t-

he Period 2 value of the firm;

U(-)

-

the

utility function

of Period

2

value

at

the

beginning

of

Period 1;

and

F(')

-

the distribution

function

of Period 2 value of the

firm.

Expression (28)

indicates that the

objective

of

the firm is

to maximize

the

expected

Period

1

utility

of its Period 2

marketvalue. This objective

is

analogous

to the

capital

market model

objective of maximization of the

risk adjusted expected

Period

1

value

of the firm. One should recognize

that the discountingprocessis now implicit in the utility function although

that operation generally

will

be ignored

in the

examples presented below.

To

utilize

expression (28)

to

derive risk management decision

rules,

the

first

step

is

to

substitute

equation (6)

or

equation (18) for Va. Then

the

first order conditions

for a

maximum

are

obtained by setting the appro-

priate

derivatives

equal

to

zero. As the risk

management

maximization

s

"I

The

foregoing

onclusion

s that

of the

author

and

not

of

Friend

and

Blume.



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Risk

Management 605

conditional

on

the other

stochastic

variables

which affect the firm and

because

those

variables

have

been assumed to be

independent

of

total

losses,

the expected value operation

can

be

carried

out with

respect

to

the

marginaldistributionof total losses ratherthan the complete distribution

of

Vs. For the

simplest case,

proportional

etention,

the general

rule would

be:

..L...

f

U'(V

)(G-

-L

-

E

)d(L

)

(29)

act

j j ij iJ

where

Vj

is defined

by equations (6)

and

(7). By

solving

that

equation

for

a

one would obtain the optimal level of proportional etention.

For

the aggregate

deductible-buffer

und

version

of the

model,

the

opti-

mization

problem

is:

Maximize: EE

U(J)

(30)

Subject

to:

BJ

<

D

By

~

~D

to

where

E[U(Y:)]

-

RJ(Yi)dH(?

+

fU(V2j)dF(j)

+

JU(V3J)dF(tJ);

o

B

I)

1

- G

(1

+ R;) -

BJ(RJ

-

Rf)

-

J;

V2:m

V1J

(i

- B

)Rj;

and

V3

-

,

+

Ad

-

D

-

(D

-

B

J)RJ.

The

maximization

would be carried out

according

to the Kuhn-Tucker

conditions with

the solution of

the

resulting equations

giving optimal

values for

Bj

and D.

In

order to provide an

illustration

of

the

nature

of the decision rules

generated by

the

utility

model,

the

maximization

problem is solved below

for

the

proportional retention

case under the

assumptions that losses

follow a gamma distributionand that the utility function of the firm is

exponential. The

use of an

exponentialutility

function

implies that the

firm's

absolute

risk

aversion coefficient

is

constant

and that its relative

risk

aversion

is

increasing.

These

properties mean that the

firm would

have the

same

tendency

to

insure

a

potential

source

of

loss of

a

given

size

regardless

of

its

wealth

and

that

if

the

magnitude

of the

insurable

loss

and

the

firm'swealth

increased

proportionally

t

would

be more

likely

to

insure

after

than

before

the increase.

While

one

may quarrel with the



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606

The

Journal f

Risk

and

Insurance

reasonablenessof these

properties, it

is importantto recall

that risk man-

agement

decisions have only a

marginal mpact

on the wealth of the firm.

Thus,

for a

given firmat a particular

ime,

constant

absoluterisk aversion

probablydoes not constitute a serious departure rom realitywhile relative

risk aversion properties

are not

especially relevant. If one is

unwilling to

accept this

reasoning,

however, other

functions are

availablewhich possess

more

acceptable

properties.'8

With

gamma distributedlosses

and

exponentialutility,

the

proportional

retention

problem becomes:

Maximize:

E[U()]

y

(C

-

Cej)

r

(r?)r.eL

i

e

d

(31)

0

where

r--

the

absolute

risk

aversion

parameter,

and

A,

r

= parameters

of

the

gamma

distributionof

losses.

The

integrationeasily

is

accomplished

by recognizing

that

the

first

term

inside the

integral

integrates

to C while

the

second

can

be

written

as

-Ce--v

multiplied by

the

moment

generating

function of the

gamma

dis-

tribution, where V

=

V -

(l

-

)G1 -

cEj

and

the

parameter

of

the

mo-

ment generatingfunction is

ya.

Thus, one can write:

E[U(6'

)J

C -

Ce

Y(V)(l

-

Xa)-r (32)

j

A)

Differentiationwith

respect

to

a

yields:

BE1U(

)]

|-

.-Cey(v.

)

(-Y)

G

E)

(3:3)

-Bej(V)(.r(l

-

Y;frl(

The optimal solutionfor

a

then

becomes:

C1=

r

a

Y

(;

-

+

T

(34)

This

condition

indicates that the

level of retention

varies inversely

with the

degree

of

risk

aversion

displayed by

the

firm,

a result which is

both

reasonable and

consistent

with

intuitive

expectations.

Not so intui-

tively apparent is the reason that

a

varies inversely with (Gj

-

Ej). The

rationale for

this

relationship becomes clearer

when one

recalls that

retention

is feasible

only when

Gj

> E (

Lj)

+

Ej.

Thus, the

relationship

between

Gj

and

Ej

is

reflective of

the relativesignificanceof

the stochastic

and

non-stochastic

portions of the

total cost of the

retention program.

18

For

example,

the log

utility

function

exhibits

constant

relative

and

decreasing

absolute

risk

aversion.

See Pratt

[14].



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Risk Management 607

Consequently, f

Ej

increases relativeto the total, one can

say in an intui-

tive sense that the riskiness of the

retention program declines

and that

a

can be set at a

higher level.

Empirical

Comsiderations

The

preceding

analysis has integrated risk management

variables

into

the theory of capital

market equilibrium. The resulting

decision rules

indicate that the

firm must consider more than expected

values when

developing its program

for dealing

with pure risk. As a theoretical con-

struct, the model

should be useful in focusing attention

on the

relation-

ships between expected

costs, risk,

and

other

parameters nvolved in

risk

retentionprograms. However, there appearsto be reasonto doubt its prac-

tical applicability.

One

problem

is the abstract nature of the model and

the number of

simplifying assumptionsemployed.

Thus, the decision

rules may be too

simplistic and may ignore too much

relevant informationfor real-world

application.A second

problem is that,

due to the magnitudeof the market

parameters and other variables appearing

in the model,

most of the

decision rules may

reduce for practical

purposes to expected value criteria.

An

interesting possibility

for future

research would be to test this conten-

tion through the use of real-world data. However, even if practical appli-

cation

is

not feasible,many of the foregoing

relationships till can be useful

for

organizing

and

analyzing risk management

data. Furthermore,he con-

struction

and use of utility functions appears to offer significant

potential

for

practical applications.

Summary and Conclusions

In

this article

the


model

of

the

theory

of

finance

has been adapted and applied to the development of risk management

decision rules. The

fundamental

concept underlying the analysis

is that

when

considering

a retention program,

a firm must recognize

not

only

the

savings

in

expected loss costs but also

the

increase

in

risk

accompany-

ing such a program.

The model reveals

that the firm should increase

its

retention to the point at which the marginal rate of substitution

between

expected

return

and risk is

equal

to the market price of risk multiplied

by

the

correlation coefficient between

the firm's

returns

and those of

the

market.

This

rule

was applied

more

specifically

for

the cases

of

propor-

tional retention and of aggregatedeductible selection. The latter problem

was solved both

with

and without the

use

of

a reserve

fund.

The

optimization

rules reveal that

the

degree

of

proportional

retention

is

inversely

related to

the ratio of

the variance

of

the

firm's oss

distribution

and

its

standard deviation

of total

return.

When

an

aggregate

deductible

is

adopted,

the

use of

a

reserve

fund

results

in a decreasein the

variance

attributable

to

the deductible.

The

recognition

of

this

negative

variance

contribution

mplies

that the

optimal

reserve fund should be chosen

where



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608

The

Journal

of Risk and

Insurance

<

0,

suggesting

that

expected

value

decision

making

leads

to

2,j

a reserve fund of suboptimalsize.

The decision

rules based

on the

capital asset

pricing

model can be

rendered

more

realistic by introducing

taxation and

by recognizing

that

loss distributions

are rarely

symmetrical.

When

taxes

are introduced,

it

can

be demonstrated

that the

optimal solution

implies

lower

values

for

both the aggregate

deductible

and

the

reserve fund.

The

relaxation

of

the

symmetry

assumption

necessitates

the adoption

of utility

functions.

Although the capital

asset pricing model decision

rules probably

have

limited

empirical

applicability,

utility analysis represents

a

potentially

fruitful alternativefor those who wish to avoid exclusive reliance on ex-

pected

values in

practical situations.

REFERENCES

1. Allen,

Tom C.

and

Duvall,

Richard

M. A Theoretical

and Practical

Approach

to Risk

Management.

New

York: American

Society of

Insurance

Manage-

ment,

1971.

2. Arrow, Kenneth

J. Essays

On the

Theory

of Risk Bearing.

Chicago:

Mark-

ham Publishing

Company,

1971.

3.

Borch, Karl. The Mathematical Theory of Insurance. Lexington, Mass.:

Lexington

Books,

D. C. Heath

and

Company, 1974.

4.

Duvall,

Richard

M. and Allen

Tom C. "Least

Cost Deductible

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Journal

of Risk and

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40, no. 4 (December

1973):

497-507.

5.

Fama,

Eugene

F. and

Miller, Merton

H. The

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Finance.

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Ill.: Dryden

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6.

Feller,

William.

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Introduction

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Probability

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Its

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2nd ed. New

York:John

Wiley & Sons,

1971.

7.

Friend,

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and Blume,

Marshall

E.

"The Demand for Risky

Assets."

Working Paper No. 1-74a. Philadelphia: Rodney L.

White Center

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University

of Pennsylvania,

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Head, George

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