251700.pdf

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1/15

Some Fundamental Theorems of Risk ManagementAuthor(s): N. A. Doherty

Source:The Journal of Risk and Insurance,

Vol. 42, No. 3 (Sep., 1975), pp. 447-460Published by: American Risk and Insurance AssociationStable URL: http://www.jstor.org/stable/251700 .

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2/15

Some

Fundamental Theorems

of

Risk Management

N.

A.

DOHERTY

ABSTRACT

This paper

explores the

role of premium

loadings

in risk management

decisions. It

shows

how the expected

utility

hypothesis

might be

applied

to risk management situations and how rational insurance decisions

depend

on

the nature of

the

premium loading.

Finally

the

paper

examines

the

'moral hazard'

argument

that insurance

discourages ex-

penditure

on

loss prevention.

The relationship

between

insurance and

loss

prevention is

shown

to depend on

the size

and nature

of the

premium

loading.

The

contribution which

risk management

has made

to the

management

sciences

has been its stress

on the interdependence

between

the alternative

methods

of

handling

risk. It

therefore

becomes

inappropriate

for

a

firm

to formulate an optimal insurance program without reference to its pro-

gram

for loss prevention.

Conversely,

the

viability of

any given

loss pre-

vention

program

depends

on

the

level

of

insurance

and

the conditions

under

which it is

arranged.

The

purpose

of

this

paper

is to

show how

the

nature

and

extent of this interdependence

rest upon

the premium

structure

and,

in

particular,

the

size

and structure

of the

premium loadings.

The importance

of

the

premium

loadings

for

optimal

insurance and

risk

management

decisions

has

been

recognized

by

several economists and,

collectively, their works form a theoretical basis for risk management.

Some

of

the

more important

conclusions

from

this literature are brought

together

here

and this

is used as

a

basis

for

further

exploration

of the

relationship

between risk

management

decisions

and

premium

loadings.

Risk

Management

and

Expected

Utility Hypothesis

An individual

or

firm starts

with

an

initial

level

of

wealth, A,

which is

exposed

to

the

prospect

of destruction

or

damage by

some

specified

perils.

The

analysis

of this

paper

is

appropriate

in

the case where

the

wealth is

in the form of a proprietorial interest in certain productive assets or con-

sumer

goods.

N. A. Doherty,

B.Phil.,

F.C.I.I.,

is Stewart

Wrightson

Research Fellow in

the

University

f Nottingham,

United

Kingdom.

This

paper

was submitted

n

April,

1974.

This

paper

is

part

of a wider

study

into the

economicsof insuranceand

loss pre-

vention

sponsored

y

Stewart

Wrightson,

Ltd.

The authorwishes to

recordhis thanks

to them

and

to an

anonymous

eferee

of this

journal

who has

made many

helpful

suggestions.

(447)


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3/15

448

The

Journal of

Risk

and Insurance

The potential

losses

to

the

individual can

be

described

in

the

form of

a

probability

distribution such as that shown in

Figure

la. Here the

value

0

on

the

x

axis

represents

a zero loss and the value

ON is

the maximum

Loss

)

Expected

income

distribution

distribution

1~~a lb

FIGURE

a

FIGURE b

possible

loss

the

individual

can sustain.

It

is

useful to consider

this

loss

distribution

in

a

slightly different form, where the horizontal axis records

not the loss

itself but how much wealth or income remains after

the loss

and/or any insurance

premium has been

deducted.

In

Figure lb, OA

is

the

initial

wealth

and the outcome x

= 0

represents a total loss. Figure

lb

is

simply

a

mirror image of la. This

distribution is referred

to as

the

expected income distribution. Clearly

the individual can change the

shape

of his

expected

income distribution

by

purchasing

insurance

and

in

the

limiting case full

insurance would be

represented by

a

distribution which

showed

an

outcome

of

A

-

R with an

associated probability of

1. The

term R

denotes the

insurance premium.

Decisions

facing

the firm about

whether

it

purchases

insurance,

or

how much it purchases, might be graphically represented as a choice

between various

probability

distributions

each

of

which

corresponds

to

a different level

of

insurance.

One device for

ranking competing

distribu-

tions

is

the

expected utility hypothesis' by

which each

distribution

can be

associated with

a real

number

and

the

ordering

of

these real

numbers

provides

a

preference

ordering

over

the various

probability distributions.

Figure

2

combines

four

separate

diagrams,

the

first

of which, 2(a),

shows

the

probability

distribution

f(x) appearing

in

lb which

shows

the

income prospect

if no

insurance

is

purchased.

Directly underneath

2a

is

a utility of income function U1(x). This satisfies the requirement of dimin-

ishing marginal

utility

of

income,2

which

implies

aversion

to

risk. The

horizontal

axis

of

2b

has

the

same variable. as

the horizontal axis

of

2a and

furthermore they

are

drawn

on

the

same

scale.

Part

2c is

merely

a

45

'See for example

K.

H.

Borcb, The

Economics

of Uncertainty, Princeton Studies

in Mathematical

Economics No.

2,

Ch. 3.

2

For

our application,

the

utility

function

must

also satisfy the well-known Von

Neumann/Morgenstern

axiom and be determined

up

to linear

transformation.



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Some

Fundamental

Theorems

of Risk

Management 449

*%

4

U

j

jNO

I

I

J

-V9 ____________._ __ _____ ____ __ _ ___/&

FIGURE

2.

The Bernoulli

Theorem

o l

~ ~ t

O

U~~~~~

0

X~~~~~~~

FIGURE

3.

Risk Aversion

and the Demand

for Insurance

/dgree

line which

renders the

horizontal

axis

of 2c

directly comparable

-with

the

vertical

axis

of 2b. The

final

part

of

Figure

2

can now be used

to

-construct

what

might

be

called

the

utilityequivalents

of the

probability

Zlistributions;

t is

from

these

that a

preference ordering

can

be obtained.



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450 The

Journal of

Risk and

Insurance

The

utility equivalent of the probability

distribution can

be constructed

as follows. For any possible outcome, e.g. (x

=

x:), there is

an associated

probability,

p(xj),

shown on the vertical axis of 2(a). If

u,

is individual

I's

utility function, then xj can be given

a

utility value

or index

U(xj).

The co-ordinates

U(xj)

and

P(xj)

can now be plotted in 2(d) and

are

shown

with an

X. By repetition the whole distribution g(U1(x) )

can

be

constructed.3 If one wishes to establish a preference ordering

over

several

probability distributions drawn in 2a then all that needs to be done

from

the

expected utility hypothesis is to compare the

means of

the

corresponding distributions appearing in 2d.

The

standard propositions about insurance purchasing under actuarially

fair

premiums can now be clearly illustrated. The actuarial premium

is

OA-OD where

1

is the mean of f(x). With full insurance the individual's

income will be

'1

with certainty. This prospect is shown as

0

on the U (x)

axis. However, if he does not insure, the expected utility is the mean of

g(U,(x)

) which is

(?

on the U(x) axis. The ordering is such that

the

full insurance

prospect

is

preferred. The result depends on the concavity

of the utility function to the

x

axis. If it had been linear,

(0

and

?

would

coincide

indicating

that the

individual

is

indifferent between

full

insur-

ance and

self

insurance. If the

utility function were convex

(0

and

0

would have the reverse

ordering thereby indicating

a

preference for

self

insurance

even

at actuarial

premiums.

Turning

back to

U1(x),

if

the

premium quoted by

the insurers is OA-

OX1

then

a situation of

indifference between full insurance and self insur-

ance

results since

the

expected

utilities

from

these alternatives are equal.

The difference

between OID

and OA1

measures

the

risk

premium or, in other

words

the maximum sum

the

individual

is

willing to pay

in

order to off-

load

the

risk.

The risk

premium

is

dependent upon the intensity of aversion

to risk.4

Figure

3 is

drawn

along

similar

lines

to Figure

2

but a second

utility function U2(x) has been added.

U2(x)

is

constructed

to

display

a

lower

intensity of aversion to risk than

U1

(

x)

and

the

effect

is

that the

risk

premium is reduced to O-D-OX2.

In

the limiting

case where

the utility function is linear, the risk premium

will

be zero. These

propositions

are

of

an

'all

or

nothing' nature, whereas

risk management

has

underlined

the

possibility that the optimal insurance

strategy may

well

involve

some measure

of risk

sharing between the

insurer and

the

insured.

The

next

section

considers the

optimality

condi-

tions for

various

forms

of

risk

sharing.

Optimality

Conditions

for Insurance and Risk

Retention

Risk

sharing

is

commonly practiced

in

many

insurance

markets

but

it

appears

in various

forms. Coinsurance occurs where the insurers

limit their

liability

to

a

given percentage

of

any

loss. A

deductible

is a

fixed

sum

3

Formally g(U1(x) )

=

f

(x)

for

all

x.

4

Intensity

of

risk aversion

is

measured

by

the coefficient

of

risk

aversion. The

absolute coefficient

of

risk aversion

is

given by U /U'

for each

level

of

x.

See J. W.

Pratt

Risk Aversion

in the

Large

and in the

Small,

Econometrica

32, (Jan.-April

1964) and K. J. Arrow Aspects of the Theory of Risk Bearing, Helsinki 1965.



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Some Fundamental

Theorems

of

Risk

Management

451

deduction

from

each claim

so

that the

insured

himself

bears

losses

below

this sum. Where

the loss exceeds

this

sum

the insurers

pay the

difference.

A franchise

is similar

to a deductible

but,

unlike

it,

the insurers

are

fully

liable

for losses

over the

deductible

amount.

A fourth

form

of

risk sharing

is the

first loss

arrangement.

Here,

the insurers

will pay

the

amount

of

the loss or a

given sum,

whichever

is the lower.

The text concentrates

on

the optimality

conditions

for

coinsurance

and deductibles,

although

an

attempt

is made later

to order

these and other

forms

of risk

sharing.

If

the income

prospect

is

represented

in

the form

of

a discrete

prob-

ability distribution

the

expected

utility is

given

by

n

I

pJU(A

-

L;)

where

pi is

the

probability

that

a

loss

of

size

Li

will

occur

A

is initial

wealth

U is

the

appropriate

utility

function

n

is

the

number

of

possible

outcomes

(including

the

outcome

in

which

no

loss

occurs)

Fortunately

the

presentation

can be simplified

very considerably

by

con-

sidering a binomial probability distribution in which only two outcomes

are

possible.

This

simplification

does

not affect

the

main

conclusions

of

the

paper,

as

can

be

demonstrated

easily

by

appropriate

differentiation

with

a

multi

outcome

distributions

There

is a

probability

p

that a loss

will occur

and

it will assume

a

single

value L. This

state

is denoted

by

subscript

0.

There

is

a

residual

probability

(1-p)

that no loss

will

occur;

that

state

being

denoted

by

subscript

1.

If no

insurance

is

purchased,

the

expected

utility

of the income prospect

is given by:

6

U (1 -

p)U(A)

+

pU(A- L)

Coinsurance

If the

individual

or

firm

coinsures,

the

expected

utility

is given

by:

III(i) U

(1

-

p)U(A

-

=w

L)

+

pU(A-

=w L

-

(1-

)L)

where

a

is the

proportion

of insurance

7r

is the

price of

insurance.7

5Many of

the writers cited

here

take

advantage

of

this

simplification because

it can

be used to produce fairly general results.

6The presentation here is similar to

that

of

Ehrlich and

Becker. However, a differ-

ence

does arise in the definition of the

price

of insurance.

Here,

it is defined as

the

rate

at which

certainty income

can

be transformed into

income

in

state

0

only. The

reason

for using

this definition

is that

insurers

rarely make the

payment

of

premium

contingent

on whether

a

loss

occurs.

Thus

the

E

& B

symbols

are

equivalent

to

my

aL

-

a7rL

or,

in other

words, my

net

insurance

payment

in state 0 is

functionally

related to r.

See

I

Ehrlich

and

G. S.

Becker,

Market

Insurance,

Self

Insurance

and

Self Protection

J.P.E. Vol. 80, 1972.

7

Incomes

in

each state

of the world are

defined net of

tax. If

premiums and

uninsured

losses are both

tax

deductible,

then

the

real

price

of

insurance

is

the rate at which

net of tax premiums can be used to make good contingent net of tax losses.



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Some Fundamental

Theorems

of Risk Management

453

Deductibles

The optimality

conditionsfor deductibles turn out to be similarto those

for coinsurance.

The

expected

utility

from

the income

prospect

is

given by:

III(iv)

U

=

(1

-

p)U(A-

(L

-))

+

pU(A

-

wr(L-)-)

where A

is the deductible

The

maximisingconditions

are

given

by

III

(v)

dd (1

-

p)Ulr +

pU(

r-1)

=

0

which implies

full insurance if the premium

is

actuarially

fair

(7

=

p);

and

III

(v) d2U

(_-

U Tr2 +

2

2

(1- )Ul

1

pUO(

X

-1)

<

0

which

holds if U

<

0. With a proportionate loading

the first order

con-

dition

is:

III(vii)

(1

-

p)(l

+

m)pU{ +

p((1

+

m)p

-

M)UI

0

which if m > 0 and p > 0 implies that

U01

> U I which in turn implies

less

than

full insurance.

If

the

premium

includes

a

lump

sum

loading,

the result

is the same as

that considered

under

coinsurance.

If insurance

is purchased

at

all,

then full insurance

will

be optimal.

However,

the lump

sum tax

itself might

be sufficient

to dissuade

firmsfrom

buying insurance.

Summarising,

he Mossin

and Smiththeorems

show that partial

insurance

will

always

be optimal where

a

proportionate

oading

is imposed.12

At

first

sight,

it seems that

most property

insurances

are calculated

on

the

basis of proportionate oadings. A common practice of property insurers

is

to use target

loss ratios

as the basis

of premium

calculations.'3

The

premiums

for

each

class of insureds

are

calculated at

a set ratio of

the

expected

claims costs.'4

Where coinsurance

is

undertakenat

some

given

percentage

the premium

s usually,

though

not

always,

the

same

percentage

of

the

full

insurance

premium

and the

assumptions

of Mossin

and

Smith

seem

appropriate.

However,

when risk

sharing takes

the form of

deductibles,

franchises

or

first

loss

policies,

then

the premium

loadings

are not

likely to be

of a

simple proportionate nature if only because this would involve a detailed

12

Sometimes the

insured is only offered a limited number of choices. For example,

he may be offered

either full cover at a stated premium or a given deductible at

an

alternative

premium. In

these cases

it is

impossible

to

anticipate the

ordering

of these

alternatives unless the

insured's utility function is exactly specified.

13

See Report on the

Supply of Fire Insurance, The Monopolies Commission, London

HMSO, 2nd August

1972, paras

168-178.

14Each

class

contains a group

of

insureds

who

are considered to be relatively

homogeneous in terms

of loss expectancies.



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454

The

Journal of Risk

and

Insurance

knowledge

of

the loss

density

function.

Premium

structures

often contain

crude

or

arbitrary premium

reductions for

deductibles

and, in the

case

of

larger

insureds, the

premium reduction

is often

determined

by

bargain-

ing between the insurer and the insured and/or his brokers. It is there-

fore

important

to state the

conditions under

which risk

sharing may

be

optimal in

terms of the

premium

saving and in

a way which

applies

to

each of

the forms

of risk

sharing.

In

Figure

4

the

distribution

f(x)

represents the

individual's

expected

income if he does

not

insure (i.e. it is

comparable with

f (x)

in Figures

lb,

2a and 3a). If

he fully

insures, then

his

expected income

will be

the

initial wealth OA

minus

the

required premium R1.

In

other words

his

expected income is OB with a

probability

of one.

R,

(OA

-

OC) is the

premium for

coinsurance at

the

rate

of

fifty percent

of all

losses and

the

appropriate distribution for

50

percent

coinsurance

is

g(x). The

diagram

is

constructed so that

the mathematical

value of

g(x)

is

equal to OB

or,

in

other

words, the actuarial

values

of the

coinsurance

option

and

the

full

insurance option

are

equal.

If the

utility equivalent distribution

had been

drawn,

as in

Figures

2

and

3, and the

utility

function were

linear,

then

clearly

the

firm

would

be

indifferent between full insurance

and

coinsurance.

If,

however,

the utility

function were concave, the expected utility theorem, as illustrated in

Figures

2

and

3

suggests

that full

insurance

would

be

preferred.

Further-

more,

concavity

of

the

utility

function

suggests

that the

coinsurance

option

would

never

be

chosen unless its

actuarial

value

exceeded

that

of the

full insurance

option.

arc

RI

FIGURE

4.

The

Preconditions

for

Risk

Retention

This

conclusion

can

be

presented

in

another manner. The

actuarial

value of the

coinsurance

option

can

only

exceed that

of the

full

insurance

option

if

the

premium

reduction

for

coinsurance

is

greater

than

the actuarial

value

of the

uninsured

risk.

In

order to

induce

a risk

averter

to

coinsure,



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Some Fundamental Theorenm

of Risk Management 455

it is a necessary, but not sufficient, condition

that

the

reduction in

premium

exceeds the actuarial value of the uninsured risk. The term risk sharing

can be substituted for coinsurance in

the last sentence since it can be

shown that it also applies to other forms of partial insurance.

This coinsurance condition is compatible

with the theorems of Mossin

and Smith since they assume the premium loading to be a percentage

markup on the actuarial value.'5

On the other hand, if the premium

loading

is

independent of the actuarial

value, i.e. a lump sum tax,

then

the conditions for risk sharing are not

met,

and as seen above, the firm

will

either fully insure or escape the

tax altogether by self insuring. In

principle the application of the condition to more complex premium struc-

tures is simple, though of course in practice knowledge of the probability

distribution

is

required.

Arrow ' has shown that, given a choice between two policies of equiva-

lent

actuarial value, a

risk

averter would prefer one with a deductible

to

one

subject to coinsurance. This ranking

can be extended to produce

a

preference ordering over the various forms of risk sharing.

The effects of the various forms of risk sharing on the individual's

ex-

pected income distribution are shown in Figure 5. In all cases the

maximum

attainable

wealth

is the initial

wealth

OA minus the premium

OA-GB.

The effect of coinsurance is to compress the possible outcomes over a

smaller

range of values of

x

and

in figure 5(a). Insuring with a deductible

has

the effect of chopping off the tail

of

the

original density function

and

replacing this with a probability mass

at the wealth value relating to

the

deductible

(i.e. c

in

figure 5(b) ).

A franchise has a similar effect to

the

deductible but

the

probability

mass is

replaced

at a wealth level

relating

to

the initial

wealth minus the insurance premium (B

in

fig. 5(d) ).

The

first loss policy effectively extracts the probability mass relating to losses

between

zero

and the upper

limit

and replaces

this

with

a

probability

mass at the initial wealth minus the insurance premium (see fig. 5(c) ).

The

model

developed

earlier can now

be

used

to yield a preference

ordering

over the various forms of

risk

sharing,

on the

assumption

that

the

actuarial

values of

all

the

alternative policies

are

identical. This

is done

by pairing

off

the

distributions and

comparing the

means

of

their

utility

equivalents.

To

illustrate,

the

deductible

policy

and the

first

loss

policy

are

compared

in

fig.

6. In

part (a)

the

probability

distributions

relating

to the two

options

are

constructed;

the

deductibles option denoted by

diagonal shading and the first loss option is not shaded. The actuarial

values

of

the two

options

are identical at

4a,

but in

part (b)

which shows

the

utility equivalent

distributions

it

is

seen

clearly

that the

deductible

option

has a

higher utility

value

(

(2)

on the

U(x) axis)

than

the

first

loss

option ( (1)

on

the

U(x) axis.)

This outcome

is a

necessary

conse-

15

If the insurers charge a premium of (1 + m) times the expected claims cost then

the premium reduction

will

be (1 + m) times

the

expected value

of the

uninsured

risk.

6K. J. Arrow, Uncertainty and the Welfare Economics

of

Medical Care, A.E.R.

Vol. LIII No. 5, (December 1963).



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456 The Journal

of

Risk and Insurance

Thy)

i,

T t~~~~D~vdb/

zQB-OC

X

x

piu6

~FrJ

E

Loss

frnchis

4

x~~~~ a

FIGURE

5.

Alternative Methods

of Risk Sharing

a)educyibcK

f

ow

(6)

cle

fte

Pva

4

>

|;e

dedmlcfibk_

)

< |

s/4

k

0

0p

1

2

FIGURE 6.

Comparison of

Deductible and First Loss Policies

quence

of

the

concavity

of the

utility

function and similar comparisons

show

the

deductiblespolicy

to be

preferred

o

each

of

the

other alternatives

and

both

the

franchise

and

coinsurance

policies

preferred

to the first

loss

policy.



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12/15

Some

Fundamental Theorems

of

Risk

Management

457

Interdependence

Between Insurance and

Loss

Prevention

Loss

prevention provides an

alternative strategy towards risk.

The

firm

may

be able to reduce the

probability

that a certain undesirable

event will

occur, or

reduce the financial

impact

if

it

does

occur, by

means of

safety

devices. These may include

sprinkler

systems, burglar alarms, safer

work-

ing procedures,

etc. The optimal size

for a loss

prevention program

is

defined

where the

costs

and

benefits are

equated

at

the

margin.

However,

these

costs and

benefits depend on the

level of

insurance

and

the conditions

under which it is

transacted.

The effect of risk transfer is to

relieve

the

insured

of the direct financial

consequences of certain events

and

this

in

turn

implies

a zero

sum

gain

from

the

installation of

safety

devices.

This

may lead to a change in loss probabilities, which is described as moral

hazard.

Counteracting this, the insurers

might devise

premium structures

which grant premium reduction for

loss prevention

systems.

For example,

under

an

actuarially

fair

premium

structure, the expected

return from a

loss

prevention

program

for a firm

which

is

fully insured is the

reduction

in

the expected

value

of

the

loss; this is, of

course, the

same expected

return as

in

the

case where

no

insurance is

purchased.

The

interdependence

between

insurance

and loss

prevention is shown

by considering

exogenous

changes

in

the

level

of

insurance on

the optimal-

ity conditions for loss prevention. This will reveal whether insurance and

loss

prevention

are

complements

or

substitutes.

The

utility of

the income

prospect

is:

IV(i)

U

=

(1

-

p(r))U(A

-

=

L

w(r)

-

r

+

p(r)U(A-

=Lr

(r)

-(1 -)L

-

r)

where r

is

the

level

of

expenditure

on

loss

prevention,

and the

other

symbols

are

as

expressed

earlier.

The optimal level of loss prevention for any given level of insurance is

given

by

the first

order condition:

IV(ii)

au

-pb(r)

U

(A

-

as ''L

-

r)

-

U(A

-

(r)L

-

(1

-L

-

r

-(1

-

p(r))U;(

-

'(r)L

+

1)

-

p(r)UO(s:

v(r)L

+

1)

-

0

and the

appropriate

second order

condition. In

order

to show

the effect of

exogenous changes

in

the level of

insurance

on

self

protection

expenditure,

take the derivative of

the

first order

condition;

this

gives dr/da

=-N/m

where

M

is the second

order

condition,

which

will

be

negative,

so that

the

sign

of

dr/da

will

be

the same as

the

sign

of

N.

N

is

given by:

IV(iii)

N

-

p' (r){UL

i(r)

-

UoL(

ir(r)

-

1)}

-(1

-

p(r]UlL

:'(r)

+

(1

-

p(r))U (

-L

:'(r)

+

1)L

x(r)

-p(r)UOL

I'

(r)

+

p(r)U (

-L

r

I(r)

+

1) (L(

ir) -

1))

In

the

actuarial

case, 7r(r)

=

p(r)

and N

simplifies

to:

IV(iv) Lp'

(r)(2p(r)

-

1)(U;

-

U1)

+ L(

Lp(r)

+

1)p(r)(1

-

p(r))(

-

Ulf

1

p 0)(U

~



8/9/2019 251700.pdf

13/15

458

The

Journalof

Risk

and

Insurance

Ehrlich

and

Becker

point out

that in

this

case

market

insurance and

loss

prevention

may

turn

out

to

be

complementary if

p is

not

very

small

and

U

is

concave.17,

18

However,

a

similar

result

may

occur

where

the

premium includes a loading factor. For example, if the premium includes

a

lump

sum

loading

(premium=

aLp (r)

+ k)

then

N

would be

the

same as in

the

actuarial

case.

Since

M

also

would

be

unchanged

then

the

rate of

substitution

would

be

identical.

A

more

interesting

case is

where

the

loading is

related to the

sum

insured.

By

substituting a

premium of

the

form

ap (r)

L

+

akL into

IV (i)

and

appropriate

differentiation,

N

emerges as

follows:

IV(v)

pL(r)L(2p(r)

+

k

-

-

UO)

-L(

Lp'(r)

+

1)f(p(r)

-

P

-

kp(r))(U' - U )

+kUl}

As

with the

actuarial

case,

this

may

result

in

complementarity

between

insurance and

loss

prevention if

p is

not

very

small

and

U is

concave.

For

example,

if

U is

quadratic

then

insurance

and

loss

prevention

will be

complements

if

(2p(r)

+

k

-

1) > 0.

(This is a

sufficient,

but

not

neces-

sary,

condition.)

The

widespread

use

of target loss

ratios

implies

a

proportionate

loading

factor. By substituting

7r

r)

=

(1 + m )p

(

r) where m>1 and appropriate

differentiation,

then

N

becomes:

IV(vi)

p'

(r)L{(2p(r)(1

+

m)

-

-

-

mUjj-

.

+

L

(

L(1

+

m)p'(r)

+

1){P(r)(1

-

p(r)

-

mp(r))

(U

-

U )

+

mp(r)U';}

The conditions for

a

positive

sign

for this

expression

are

more

complex.

For

example,

if

U is

quadratic

then

N

will

certainly

be

positive

if

(a) U2p

r)(1

+

m)

-

1) > 0 and

(b)

Ulf

PI

W

.Tr

p(r)(

L(1

+7;

mp

W

-+1)

and

may

be

positive

if

only

one

of these

conditions

is

met.

Furthermore,

complementarity

is

likely

to

occur

at lower

levels of

p

under

a

proportion-

ate

loading

than

in

the

actuarial or

lump

sum

cases.

It

may

also

be

noted

that the

lower

the coefficient

of

risk

aversion

the

more

likely

it

is that

insurance and loss prevention will turn out to be complementary. In the

limit,

dr/da

will

certainly

be

positive

if U

is

linear

since

the

firm

will

maximize

its

expected

return.

It

is

interesting

to

compare

the

expected

payback

to

loss

prevention

expenditure

under

different

assumptions

about

the

premium

loading.

In

Figure

7,

the

expected

payback

to a loss

prevention

program

is

defined

I

See Ehrlich

and

Becker, op.

cit., p. 642.

8

Note hat

(aLp'(r) + 1)

is

positive by

the first order

condition.



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8/9/2019 251700.pdf

15/15

460

The Journal

of

Risk

and

Insurance

defining

and

identifying premium

loadings. In

practice, if

the premium

loading is

to be

related

mathematically

to the

actuarial value

of the

policy

then

the

expression must

contain

some

random

error

because

the

actuarial

value itself is not known with certainty. Perhaps one of the strongest points

of

the

rational

model

is

that it

draws attention

to

the

type of

information

which is

required for

better risk

management.

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