Utilitiy Functions from Risk Theory to Finance.pdf

8/9/2019 Utilitiy Functions from Risk Theory to Finance.pdf

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74

UTILITY FUNCTIONS:FROMRISK THEORY TOFINANCE

Hans U. Gerber* and Gerard Pafumi

ABSTRACT

This article is a self-contained survey of utility functions and some of their applications. Through-out the paper the theory is illustrated by three examples: exponential utility functions, powerutility functions of the first kind (such as quadratic utility functions), and power utility functionsof the second kind (such as the logarithmic utility function). The postulate of equivalent expectedutility can be used to replace a random gain by a fixed amount and to determine a fair premium

for claims to be insured, even if the insurers wealth without the new contract is a random variableitself. Then n companies (or economic agents) with random wealth are considered. They areinterested in exchanging wealth to improve their expected utility. The family of Pareto optimal

risk exchanges is characterized by the theorem of Borch. Two specific solutions are proposed.The first, believed to be new, is based on the synergy potential; this is the largest amount thatcan be withdrawn from the system without hurting any company in terms of expected utility.The second is the economic equilibrium originally proposed by Borch. As by-products, the option-pricing formula of Black-Scholes can be derived and the Esscher method of option pricing canbe explained.

1. INTRODUCTIONThe notion of utility goes back to Daniel Bernoulli(1738). Because the value of money does not solve the

paradox of St. Petersburg, he proposed the moralvalueof money as a standard of judgment. Accordingto Borch (1974, p. 26),

Several mathematicians, for example Laplace, dis-

cussed the Bernoulli principle in the following cen-tury, and its relevance to insurance systems seemsto have been generally recognized. In 1832 Barroispresented a fairly complete theory of fire insurance,based on Laplaces work on the Bernoulli principle.For reasons that are difficult to explain, the principle

was almost completely forgotten, by actuaries andeconomists alike, during the next hundred years.

*Hans U. Gerber, A.S.A., Ph.D., is Professor of Actuarial Science at

the Ecole des HEC (Business School), University of Lausanne, CH-

1015 Lausanne, Switzerland, e-mail, [email protected].

Gerard Pafumi is a doctoral student at the Ecole des HEC (Business

School), University of Lausanne, CH-1015 Lausanne, Switzerland,

email, [email protected].

This is confirmed by Seal (1969, Ch. 6) and the ref-erences cited therein.

Utility theory came to life again in the middle of

this century. This was above all the merit of von Neu-mann and Morgenstern (1947), who argued that theexistence of a utility function could be derived from aset of axioms governing a preference ordering. Borchshowed how utility theory could be used to formulateand solve some problems that are relevant to insur-ance. Due to him, risk theory has grown beyond ruintheory. Most of the original papers of Borch have beenreprinted and published in book form (1974, 1990).

Economic ideas have greatly stimulated the devel-opment of utility theory. But this also means that sub-stantial parts of the literature have been written in astyle that does not appeal to actuaries.

The purpose of this paper is to give a concise butself-contained survey of utility functions and their ap-plications that might be of interest to actuaries. InSections 2 and 3 the notion of a utility function withits associated risk aversion function is introduced.Throughout the paper, the theory is illustrated bymeans of examples, in which exponential utility func-tions and power utility functions of the first and


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UTILITY FUNCTIONS: FROMRISKTHEORY TO FINANCE 75

second kind are considered, which include quadraticand logarithmic utility functions.

In Section 4, we order random gains by means oftheir expected utilities. In particular, a random gaincan be replaced by a fixed amount, the certainty equiv-

alent. This notion can be used by the consumer whowants to determine the maximal premium he or sheis willing to pay to obtain full coverage.

The insurers situation is considered in Section 5. Apremium that is fair in terms of expected utility typ-ically contains a loading that depends on the insurersrisk aversion and on the joint distribution of theclaims,S, and the random wealth, W, without the newcontract; certain rules of thumb in terms of Var[S]and Cov(S,W) are obtained. Section 6 presents a clas-sical result that can be found as Theorem 1.5.1 inBowers et al. (1997).

In Section 8, we consider n companies with randomwealth. Can they gain simultaneously by trading risks?The class of Pareto optimal exchanges is discussed andcharacterized by the theorem of Borch. Two more spe-cific solutions are proposed. The first idea is to with-draw the synergy potential, which is the largestamount that can be withdrawn from the system of the

n companies without hurting any of them. Then thisamount is reallocated to the companies in an unam-biguous fashion. The second idea, as presented byBuhlmann (1980, 1984), is to consider a competitiveequilibrium, in which random payments can be boughtin a market. Here the equilibrium price density plays

a crucial role. In Section 11 it is shown how optionscan be priced by means of the equilibrium price den-sity. This approach differs from chapter 4 of Panjer etal. (1998), which considers the utility of consumptionand assumes the existence of a representative agent.

2. UTILITY FUNCTIONSOften it is not appropriate to measure the usefulnessof money on the monetary scale. To explain certainphenomena, the usefulness of money must be mea-sured on a new scale. Thus, the usefulness of $x is

u(x), the utility(or moral value) of $x. Typically, xis the wealth or a gain of a decision-maker.

We suppose that a utility functionu(x) has the fol-lowing two basic properties:(1) u(x) is an increasing function ofx(2) u(x) is a concave function ofx.

Usually we assume that the function u(x) is twicedifferentiable; then (1) and (2) state that u(x) 0and u(x) 0.

The first property amounts to the evident require-ment that more is better. Several reasons are given

for the second property. One way to justify it is torequire that the marginal utilityu(x) be a decreasingfunction of wealth x, or equivalently, that the gain ofutility resulting from a monetary gain of $g, u(x g) u(x), be a decreasing function of wealth x.

Example 1

Exponential utility function (parameter a 0)

1axu(x) (1 e ), x . (1)

a

We note that for x , the utility is bounded and

tends to the finite value 1/a.

Example 2

Power utility function of the first kind (parameterss 0, c 0)

c1 c1s (s x)u(x) , x s. (2)

c(c 1)s

Obviously this expression cannot serve as a model be-yond x s. The only way to extend the definition be-yond this point so that u(x) is a nondecreasing andconcave function is to set u(x) s/(c 1) for x

s. In this sense s can be interpreted as a level of sat-uration: the maximal utility is already attained for thefinite wealth s. The special case c 1 is of particularinterest. Then

2xu(x) x , x s (3)

2s

is a quadratic utility function.

Example 3

Power utility function of the second kind (parameterc 0). For c 1 we set

1cx 1u(x) , x 0. (4)

1 c

For c 1 we define

u(x) ln x, x 0. (5)

Note that (5) is the limit of (4) as c 1.


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76 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME2, NUMBER 3

Remark 1

A utility function u(x) can be replaced by an equiva-lent utility function of the form

u(x) Au(x) B (6)

(with A 0 and B arbitrary). Hence it is possible tostandardize a utility function, for example, by requir-ing that

u() 0, u() 1 (7)

for a particular point . In Examples 1 and 2 this hasbeen done for 0; in Example 3 it has been donefor 1.

Remark 2

If we take the limit a 0 in Example 1, or s inExample 2, we obtain u(x) x, a linear utility func-

tion, which is not a utility function in the propersense. Similarly, the limit c 0 in Example 3 is

u(x) x 1.

Remark 3

In the following we tacitly assume that x s if u(x)is a power utility function of the first kind, and that

x 0 ifu(x) is a power utility function of the secondkind. The analogous assumptions are made when weconsider the utility of a random variable.

3. RISK AVERSIONFUNCTIONSTo a given utility functionu(x) we associate a function

u(x) dr(x) ln u(x), (8)

u(x) dx

called the risk aversion function. We note that prop-erties (1) and (2) imply that r(x) 0. Let us revisitthe three examples of Section 2.

For the exponential utility function (parameter a 0), we find that

r(x) a, x . (9)

Thus the exponential utility function yields a constantrisk aversion.For the power utility function of the first kind (par-

ameterss 0, c 0), we find that

cr(x) , x s. (10)

s x

Here the risk aversion increases with wealth and be-comes infinite for x s; this has the following inter-pretation: if the wealth is close to the level of satu-rations, very little utility can be gained by a monetarygain; hence there is no point in taking any risk.

For thepower utility function of the second kind(pa-rameterc 0), we obtain

cr(x) , x 0. (11)

x

Here the risk aversion is a decreasing function ofwealth, which may be typical for some investors.

If u(x) is replaced by an equivalent utility functionas in (6), the associated risk aversion function is thesame. In the opposite direction, if we are given therisk aversion functionr(x) and want to find an under-lying utility function, we look for a function u(x) that

satisfies the equationu(x) r(x)u(x) 0. (12)

Such a differential equation has a two-parameter fam-ily of solutions. To get a unique answer, we may stan-dardize according to (7) for some. Then the solutionis

x z

u(x) exp r(y)dy dz. (13)

Now suppose that r1(x) and r2(x) are two risk aver-sion functions with

r(x) r(x) for all x. (14)1 2

Let u1(x) and u2(x) be two underlying utility func-tions. Because of their ambiguity, they cannot be com-pared without making any further assumptions. If weassume however, that u1(x) and u2(x) are standard-ized at the same point , that is,

u () 0, u() 1, i 1, 2, (15)i i

then it follows that

u (x) u (x) for all x. (16)1 2

For the proof we observe that

x z

u (x) exp r(y)dy dz, ifx , i i

u (x) exp r(y)dy dz, ifx , i ix z

and use the assumption (14).


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4. PREFERENCEORDERING OF RANDOMGAINS

Consider a decision-maker with initial wealth w whohas the choice between a certain number of randomgains. By using a utility function, two random gainscan be directly compared: he or she prefers G1to G2,if

E[u(w G )] E[u(w G )], (17)1 2

that is, if the expected utility from G1 exceeds theexpected utility from G2. If the expected utilities areequal, he will be indifferent between G1and G2. Thusa complete preference ordering is defined on the setof random gains.

If we multiply (17) by a positive constantA and adda constant B on both sides, an equivalent inequalityin terms of the function is obtained. Henceu(x)u(x)

and define the same ordering and are consideredu(x)to be equivalent.

Example 4

Suppose that the decision-maker uses the exponentialutility function with parameter a and has the choicebetween two normal random variables,G1andG2, withE[Gi] i, Var[Gi] i 1, 2. Since

2 ,i

1aG 2 2iE[e ] exp a a , i i2

it follows that

E[u(w G )]i

1 12 2

1 exp aw a a . i ia 2Hence G1 is preferred to G2, if (17) is satisfied, thatis, if

1 12 2 a a . (19)1 1 2 22 2

Jensens inequality tells us that for any random vari-able G,

u(w E[G]) E[u(w G)]. (20)

Hence, if a decision-maker can choose between a ran-dom gain G and a fixed amount equal to its expecta-tion, he will prefer the latter. This brings us to thefollowing definition: The certainty equivalent, , as-sociated to G is defined by the condition that the de-cision-maker is indifferent between receivingG or thefixed amount . Mathematically, this is the conditionthat

u(w ) E[u(w G)]. (21)

From (20) we see that E[G]. Let us considertwo examples in which explicit expressions for canbe obtained:

For an exponential utility function, the certainty

equivalent is

1aG ln E[e ]. (22)

a

Note that it does not depend on w. By expanding thisexpression in powers of a, we obtain the simple ap-proximation

a E[G] Var[G], (23)

2

valid for sufficiently small values ofa.For aquadratic utility function,condition (21) leads

to a quadratic equation for . Its solution can be writ-ten as follows:

E[G] (s w E[G])

with

Var[G] 1 1. (24)

2 (s w E[G])For large values of s, we can expand the square rootand find the approximation

1 Var[G] E[G] . (25)

2 (s w E[G])In view of (10) we can write this formula as

1 E[G] r(w E[G]) Var[G], (26)

2

which is similar to formula (23).For a general utility function, it follows from (21)

that

1 u (E[u(w G)]) w. (27)

IfG is a gain with a small risk, the following moreexplicit approximation is available:

1 E[G] r(w E[G]) Var[G]. (28)

2

To give a precise meaning to this statement, we set

G zV, z 0 (29)z

where is a constant and Va random variable withE[V] 0 and Var[V] E[V2] 2. Hence E[Gz] and Var[Gz] z

2 2. Let(z) be the certainty equiv-alent ofGz, defined by the equation


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u(w (z)) E[u(w G)]. (30)z

The idea is to expand the function (z) in powers ofz :

2(z) a bz cz . . . (31)

If we set z 0 in (30), we obtain

u(w a) u(w ), or a . (32)

If we differentiate (30), we get the equation

(z)u(w (z)) E[Vu(w G)]. (33)z

Settingz 0 yields

bu(w ) E[V] u(w ) 0,

or b 0. (34)

Finally we differentiate (33) to obtain

2(z)u(w (z)) (z) u(w (z))

2 E[V u(w G)]. (35)z

Settingz 0 we obtain

22c u(w ) E[V ] u(w ), (36)

or

1 u(w ) 12 2c r(w ) . (37)

2 u(w ) 2

Substitution in (31) yields the approximation

1 2 2(z) r(w ) z 2

1 E[G] r(w E[G])Var[G], (38)z z z2

which explains (28).Let us now consider two utility functions u1(x) and

u2(x) so that

r(x) r(x) for all x, (39)1 2

and let 1 and 2 denote their respective certaintyequivalents. Then we expect that

. (40)1 2

To verify this result, we assume that the underlyingutility functions are standardized at the same point w 1. Then

u (x) u (x) for all x. (41)1 2

From this and the definitions of1 and 2, it followsthat

u (w ) 02 1

u (w )1 1

E[u (w G )]1

E[u (w G )]2

u (w ). (42)2 2

Since u2 is an increasing function, it follows indeedthat 1 2.

5. PREMIUM CALCULATIONWe consider a company with initial wealth w. Thecompany is to insure a risk and has to pay the totalclaimsS (a random variable) at the end of the period.

What should be the appropriate premium,P, for this

contract? An answer is obtained by assuming a utilityfunction,u(x), and by postulating fairness in terms ofutility. This means that the expected utility of wealth

with the contract should be equal to the utilitywith-out the contract:

E[u(w P S)] u(w). (43)

This is called the principle of equivalent utility. Equa-tion (43) determines P uniquely, but has no explicitsolution in general. Notable exceptions are the casesin which u(x) is exponential, where

1aSP ln E[e ], (44)

a

or quadratic, where we find that

Var[S]P E[S] (s w) 1 1 . (45) 2 (s w)IfS is a small risk, (43) can be solved approximatelyas follows:

1P E[S] r(w)Var[S ] (46)

2

(to see this, set S zV, with E[V] 0, and

expand P in powers ofz).In many cases a more realistic assumption is thatthe wealth without the new contract is a random vari-able itself, say W. Then P is obtained from the equa-tion

E[u(W P S)] E[u(W)]. (47)

Note that nowP depends on the joint distribution ofS and W.

Let us revisit the examples in which P can be cal-culated explicitly.


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Example 5

Ifu(x) (1 eax)/a, we find that

a(SW)1 E[e ]P ln . (48)

aWa E[e ]

Ifa is small, we can expand this expression in powersofa and obtain the approximation

a aP E[S] Var[S W] Var[W] (49)

2 2

a E[S] Var[S] a Cov(S,W). (50)

2

We note that (48) reduces to (44) in the case in whichS and Ware independent random variables. Also, weremark that (49) is exact in the case where S and Ware bivariate normal.

Example 6

Ifu(x) x x 2/2s, we find that

P E[S] (s E[W])

with

Var[S] 2 Cov(S,W) 1 1 . (51)

2 (s E[W])Note that this expression reduces to (45) with w re-placed by E[W], in the case in which S and W areuncorrelated random variables. For large values of s,

(51) leads to the approximation1 Var[S] 2 Cov(S,W)

P E[S] 2 s E[W]

1 E[S] r(E[W]){Var[S] 2 Cov(S,W)}.

2

(52)

6. OPTIMALITY OF A STOP-LOSSCONTRACT

We consider a company that has to pay the total

amount S (a random variable) to its policyholders atthe end of the year. We compare two reinsuranceagreements:(1) A stop-loss contract with deductible d. Here the

reinsurer will pay

S d ifS d(S d) (53) 0 ifS d

at the end of the year.

(2) A general reinsurance contract, given by a func-tion h(x), where the reinsurer pays h(S ) at theend of the year. The only restriction on the func-tion h(x) is that

0 h(x) x. (54)

We assume that the two contracts are comparable,in the sense that the expected payments of the rein-surer are the same, that is, that

E[(S d) ] E[h(S )]. (55)

Furthermore, we make the convenient (but perhapsnot realistic) assumption that the two reinsurancepremiums are the same. Then, in terms of utility, thestop-loss contract is preferable:

E[u(w S h(S))] E[u(w S (S d) )].

(56)

In this context, w represents the wealth after receiptof the premiums and payment of the reinsurance pre-miums.

The proof of (56) starts with the observation that aconcave curve is below its tangents, that is, that

u(y) u(x) u(x)(y x) for all x and y.

(57)

Using this for y w S h(S), x w S (S d )

, we get

u(wSh(S)) u(wS(Sd) )

u(wS(Sd) )(h(S )(Sd ) )

u(wS(Sd ) ) u(wd)(h(S)(Sd) ).

(58)

To verify the second inequality, distinguish the casesS d, in which equality holds, and S d, where

u(w S (S d) )(h(S) (S d) )

u(w S)h(S)

u(w d)h(S)

u(w d)(h(S) (S d) ).

Now we take expectations in (58) and use (55) to ob-tain (56).

7. OPTIMAL DEGREE OF REINSURANCEAgain we consider a company that has to pay the totalamount S (a random variable) at the end of the year.


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A proportional reinsurance coverage can be pur-chased. If P is the reinsurance premium for full cov-erage (of course P E[S ]), we assume that for a pre-mium of P the fraction S is covered and will bereimbursed at the end of the year (0 1.) Then

the optimal value of, is the value that maximizes,E[u(w P (1 )S)], (59)

where u(x) is an appropriate utility function andwhere the initial surplus, w, includes the premiumsreceived. In the particular case in which u(x) is theexponential utility function with parameter a, and Shas a normal distribution with mean and variance 2, the calculations can be done explicitly. The ex-pected utility is now

1(1 E{exp[aw aP a(1 )S]})

a

1 12 2 2

exp awaPa(1) a (1) . a 2It is maximal for

P 1 . (60)

2a

This result has an appealing interpretation. The opti-mal fraction that is retained is proportional to theloading contained in the reinsurance premium for fullcoverage, and inversely proportional to the companysrisk aversion and the variance of the total claims.

In finance, a formula similar to (60) is known as theMerton ratio, see Panjer et al. (1998, Ch. 4). The dif-ference is that for Mertons formula, the utility func-tion is a power utility function and S is lognormal,

while here the utility function is exponential and S isnormal.

8. PARETO OPTIMAL RISK EXCHANGESWe consider n companies (or economic agents). Weassume that company i has a wealth Wi at the end ofthe year and bases its decisions on a utility function

ui(x). Here W1, . . . , Wn are random variables with aknown joint distribution. Let W W1 . . . Wndenote the total wealth of the companies. A riskexchange provides a redistribution of total wealth.Thus after a risk exchange, the wealth of company i

will beXi; hereX1, . . . ,Xncan be any random variablesprovided that

X . . . X W, (61)1 n

that is, the total wealth remains the same. The valuefor companyi of such an exchange is measured by

E[u (X)].i i

A risk exchange . . . , is said to be Pareto (X , X)1 n

optimal, if it is not possible to improve the situationof one company without worsening the situation of atleast one other company. In other words, there is noother exchange (X1, . . . , Xn) with

E[u (X)] E[u (X)], for i 1, . . . , ni i i i

whereby at least one of these inequalities is strict. Ifthe companies are willing to cooperate, they shouldchoose a risk exchange that is Pareto optimal.

The Pareto optimal risk exchanges constitute a fam-ily with n 1 parameters. They can be obtained bythe following method: Choose k1 0, . . . , kn 0 andthen maximize the expression

n

k E[u (X)], (62) i i ii1

where the maximum is taken over all risk exchanges(X1, . . . , Xn). This problem has a relatively explicitsolution:

Theorem 1 (Borch)

A risk exchange . . . , maximizes (62) if and (X , X)1 nonly if the random variables are the same fork u(X)i i i

i 1, . . . , n.

Proof

(a) Suppose that . . . , maximizes (62). Let (X , X)1 nj h and let Vbe an arbitrary random variable. Wedefine

X X, for i j, h,i i

X X tV,j j

X X tV,h h

wheret is a parameter. Let

n

f(t) k E[u (X)]. (63)

i i i

i1

According to our assumption, the function f(t) has amaximum at t 0. Hence f(t) 0, or

k E[Vu(X)] k E[Vu(X)] 0. (64)j j j h h h

It is useful to rewrite this equation as

E[V{k u(X) k u(X)}] 0. (65)j j j h h h


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Since this holds for an arbitraryV, we conclude that

k u(X) k u(X) 0. (66)j j j h h h

This shows indeed that is independent ofi.k u(X)i i i

(b) Conversely, let . . . , be a risk exchange (X , X)1 n

so thatk u(X) (67)i i i

is the same random variable for all i. Let (X1, . . . , Xn)be any other risk exchange. From (57) it follows that

u (X) u (X) u(X)(X X). (68)i i i i i i i i

If we multiply this inequality byki, sum overi and use(67), we get

n n n

k u (X) k u (X) (X X) i i i i i i i ii1 i1 i1

n k u (X). i i ii1

Hence

n n

k E[u (X)] k E[u (X)]. i i i i i ii1 i1

This shows that expression (63) is indeed maximal for. . . , (X , X).1 n

Example 7

Suppose that all companies use an exponential utility

function,

1u (x) [1 exp(a x)],j jaj

where aj is the constant risk aversion of company j,j 1, . . . , n. From (67), we get

k exp(a X) (69)j j j

or

ln kln jX . (70)j a aj j

Summing over j, we obtain an equation that deter-mines:

n n ln k1 jW ln . (71)

a aj1 j1j j

Let us introduce a, which is defined by the equation

1 1 1 . . . . (72)

a a a1 n

Then it follows from (71) that

n ln kjln aW a . (73)

aj1 j

Substitution in (70) yields

n ln ka ln k a jiX W (74)i a a a aj1i i i jfor i 1, . . . , n. Thus company i will assume thefraction (or quota) qi a/ai of total wealth Wplus apossibly negative side payment

n ln kln k a jid . (75)i a a aj1i i jIt is easily verified that

q . . . q 1 (76)1 n

and

d . . . d 0. (77)1 n

We note that theqis are inversely proportional to therisk aversions and that they are the same for all Paretooptimal risk exchanges. Pareto optimal risk exchangesdiffer only by their side payments.

Example 8

Suppose now that all companies use a power utilityfunction of the first kind, such that

c1 c1

s

(s

x)j ju (x) , j 1, . . . , n, (78)j c(c 1)sj

where sj is the level of saturation of companyj. From(67) we get

cXjk 1 (79) j sj

or

sj 1/cX s . (80)j j1/ckj

Summing over j

, we obtain an equation which deter-mines:

n nsj 1/cW s . (81) j1/ckj1 j1jLet

s s . . . s (82)1 n


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denote the combined level of saturation. Then it fol-lows from (81) that

s W1/c . (83)n sj 1/ckj1

j

Substitution in (80) yields

s si i1/c 1/ck ki iX W s s (84)i n i ns sj j 1/c 1/ck kj1 j1j j

for i 1, . . . , n. Hence again is of the formXi

X q W d . (85)i i i

But note that now both the quotas and the side pay-ments vary, such that

d s q s. (86)i i i

If we write this result in the form

s X q (s W), (87)i i i

it has the following interpretation: The expressionsi is the amount that is missing for maximal satisfac-Xi

tion. It is a fixed percentage of s W, which is thetotal amount missing for all companies combined.

Example 9

Considern investors with identical power utility func-

tions of the second kind1cx 1

u (x) , j 1, . . . , n.j 1 c

From (67), we see that

ck X (88)j j

or

1/c 1/cX k . (89)j j

Summing over j, we get

n

1/c 1/cW k , (90)

jj1or

W1/c . (91)n

1/ck jj1

If we substitute this in (89) we see that

X q W (92)i i

with

1/ckiq . (93)ni

1/ck jj1Hence each investor assumes a fixed quota of total

wealth. As in the case of power utility functions of thefirst kind, the quotas vary, but now there are no sidepayments.

Example 10

Let n 2. Suppose that u1(x) x and u2(x) u(x),a utility function in the proper sense with u(x) 0.Then condition (67) tells us that

k k u(X).1 2 2

But this means that is a constant, say d. HenceX2 W d. This result is not really surprising: sinceX1

company 1 is not risk averse, it will assume all therisk!

We have presented selected examples in which thePareto optimal risk exchanges are of an attractivelysimple form. In general, this is not the case. The fol-lowing example illustrates the point.

Example 11

Let n 2. Suppose that u1(x) and u2(x) are powerutility functions of the second kind with parametersc1 1 and c2 2, that is, that

1u (x) ln x, u (x) 1 for x 0.1 2 x

From (67) we obtain the condition that

1 12 X (X) . (94)1 2k k1 2

Together with the condition that W, this X X1 2results in a quadratic equation. Its solution is

12X W (a 4aW a), (95)1 2

12X (a 4aW a), (96)2 2

witha k2/k1. Here and are obviously not linear X X1 2functions ofW.


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Example 7 helps us to understand the Pareto opti-mal risk exchanges in the general case. Let u1(x),. . . , un(x) be arbitrary utility functions, and let (X ,1. . . , be a Pareto optimal risk exchange. For givenX)nW w, . . . , maximizes expression (62). (X , X)1 n

Hence

is a function of the total wealth w.

X X(w)i iLet j h. According to Theorem 1

k u(X(w)) k u(X(w)). (97)j j j h h h

Differentiation with respect to w yields

d X d Xj h k u(X(w)) k u(X(w)) . (98)j j j h h hdw dw

Dividing (98) by (97) we see that

d X dXj h r(X(w)) r(X(w)) . (99)j j h hdw dw

From this and the observation that dX . . . dX dw, (100)1 n

it follows that

1r(X)j jdX dw, j 1, . . . , n. (101)j n 1 r(X)h1 h h

Thus the family of Pareto optimal risk exchanges canbe obtained as follows. For a particular value ofw, say

w0, we can choose . . . , Then X(w ), X(w ).1 0 n 0. . . , are determined as the solution of X(w), X(w)1 n

(101), subject to the boundary condition at w w0.As an application of (101), we revisit Examples 8

and 9. For a unified treatment, we suppose that

1 x , j 1, . . . , n. (102)jr(x)j

We want to verify that a Pareto optimal risk exchangeis of the form

X q W d , (103)j j j

or equivalently,

d X q dw (104)j j

for a set of quotas q1, . . . , qn and side payments d1,. . . , dn. From (101) and (102) we obtain

X X j j j jd X dw dw (105)njW

(X ) h hh1

with 1 . . . n. Hence, by (103)

q W d j j jdX dw (106)jW

To see when this ratio is equal to qj, we distinguishtwo cases:(1) If 0, it suffices to set

d j jq , j 1, . . . , n.j

(2) If 0, q1, . . . , qnare arbitrary quotas, and theside payments are fixed:

jd , j 1, . . . , n.j

Theorem 1 tells us that for a Pareto optimal riskexchange . . . , there is a random variable (X, X),1 nsuch that

k u(X), for i 1, . . . , n. (107)i i i

Since is a function of total wealth w, it X X(w)i ifollows that (w) is a function ofw. Differenti-ating (107), we get

k u(X)X. (108)i i i i

Dividing this equation by (107) and using (101), weobtain

1 . (109)n

1 r(X)h1 h h

This shows that is a decreasing function of totalwealth.

9. THE SYNERGY POTENTIALWe consider the n companies introduced in the pre-ceding section and assume that ( W1, . . . , Wn), theallocation of their total wealth W, is not Pareto opti-mal. How much can the companies gain through co-operation?

An answer is provided by the synergy potential .This is the largest amount x that can be extractedfrom the system without hurting any of the compa-nies, that is, such that there is a risk exchange (X1,. . . , Xn) with

X . . . X W x (110)1 n

and

E[u (X)] E[u (W)], for i 1, . . . , n. (111)i i i i


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It is clear that for x we must have equality in(111) and . . . , must be a Pareto optimal risk (X, X)1 nexchange ofW .

Example 12 (continued from Example 7)

Suppose that all utility functions are exponential.Since . . . , is a Pareto optimal risk exchange (X , X)1 nofW , it follows that

aX (W ) d , for i 1, . . . , n. (112)i iai

Then we use the condition that

E[u (X)] E[u (W)] (113)i i i i

to see that

aW aa d a Wi i i iE[e ]e E[e ], (114)

ora 1 1

a W aWi i d ln E[e ] ln E[e ]. (115)ia a ai i i

Summation overi yields an explicit expression for thesynergy potential:

n 1 1a W aWi i ln E[e ] ln E[e ]

a ai1 i

n

a W 1/ai i iE[e ]i1

ln . (116)aW 1/aE[e ]


We assume that the companies use power utility func-tions of the first kind. According to (87)

s X q (s W ). (117)i i i

From

E[u (X)] E[u (W)]i i i i

it follows that

c1 c1 c1q E[(s W ) ] E[(s W) ]. (118)i i i

Taking the (c 1)-th root and summing over i, weget

n

c1 1/(c1) c1 1/(c1)E[(s W ) ] E[(s W) ] . i ii1

(119)

This is an implicit equation for the synergy potential.


Suppose that each of n investors has a power utilityfunction of the second kind. Hence

X q (W ). (120)i i

FromE[u (X)] E[u (W)]i i i i

we get

1c 1c 1cq E[(W ) ] E[W ] ifc 1, (121)i i

and

ln q E[ln(W )] E[ln W] ifc 1. (122)i i

Taking the (1 c)-th root in (121) and summing overi, we obtain the equation

n

1c 1/(1c) 1c 1/(1c)E[(W ) ] E[W ] , (123)

i

i1

which determines ifc 1. By exponentiating (122)and summing over i we obtain the equation

n

E[ln( W)] E[ln W]ie e , (124)i1

which determines ifc 1.

Example 15

In the situation of Example 10, equality of the ex-

pected utilities implies thatX W E[W] (125)1 2

and d, whereX2

u(d) E[u(W)]. (126)2

Thus d , the certainty equivalent ofW2. It followsthat

W (X )1

E[W] . (127)2

We can use the synergy potential to construct a par-

ticular Pareto optimal risk exchange. The idea is tofirst extract from the companies and then to dis-tribute to the companies according to (101). Theresulting Pareto optimal risk exchange ( . . . , X, X)1 nofW is characterized by the condition that

E[u (X(W ))] E[u (W)], for i 1, . . . , n.i i i i

(128)


12/27



In the case of exponential utility functions we have

aX W d .i iai

To determine the side payments, we substitute thisexpression in (128) to see that

aW aa d a Wi i i iE[e ] e E[e ].

From this it follows that

a 1 1aW a Wi id ln E[e ] ln E[e ]. (129)i a a ai i i

Substituting for , we obtain finally the result that

na 1 1a W a Wj j i id ln E[e ] ln E[e ],i a a aj1i j i

for i 1, . . . , n. (130)

Example 17

For power utility functions of the first kind, we foundthat a Pareto optimal risk exchange . . . , is (X , X)1 nsuch that

s X q (s W).i i i

From this and (128), we obtain the condition that

c1 c1 c1q E[(s W ) ] E[(s W) ].i i i

Thus

1/(c1)c1E[(s W) ]i iq . (131) i c1E[(s W ) ]Finally, we use (119) to get an explicit formula for theresulting quota:

c1 1/(c1)E[(s W) ]i iq , for i 1, . . . , n.nic1 1/(c1)E[(s W) ] j j

j1

(132)

Example 18

For power utility functions of the second kind, wefound that qiW. From condition (128), we seeXithat

1/(1c)1cE[W ]iq , ifc 1, (133) i 1cE[(W ) ]and

E[ln W]ieq , ifc 1. (134)i E[ln( W)]e

From (123) and (124), it follows that

1c 1/(1c)E[W ]iq , ifc 1, (135)ni1c 1/(1c)E[W ] jj1

and

E[ln W]ieq , ifc 1. (136)ni

E[ln W]jej1

In the Appendix we derive Holders inequality andMinkowskis inequality as a by-product of Examples 12and 14.

10. MARKET AND EQUILIBRIUMAgain we consider the n companies that were intro-duced in Section 8. We concluded that the companiesshould settle on a Pareto optimal risk exchange. Be-cause this is a rich family, more definite answers aredesirable. In the last section we proposed a particularPareto optimal risk exchange. In this section an alter-native proposal, due to Borch and Buhlmann, is dis-cussed, which is based on economic ideas.

We suppose that random payments are traded in amarket, whereby the price H(Y) for any payment Y(arandom variable) is calculated as

H(Y) E[Y]. (137)

Here is a positive random variable. We assume thatH(Y) represents the price as of the end of the year.Hence the price of a constant payment must be iden-tical to this constant. Therefore we must have E[] 1. By writing the right-hand side of (137) as E[Y] E[Y] E[]E[Y], we see that the price of Y canalso be written in the form

H(Y) E[Y] Cov(Y, ), (138)

that is, the price of a payment is its expectation mod-

ified by an adjustment that takes into account themarket conditions. Alternatively, we can interpret theprice of a payment as its expectation with respect toa modified probability measure, Q, that is defined bythe relation

E [Y] E[Y] for all Y. (139)Q

In other words, is the Radon-Nikodym derivative oftheQ-measure with respect to the original probability


13/27


measure. For this reason Buhlmann (1980, 1984)callsa price density.

Company i will want to buy a payment Yi in orderto

maximize E[u (W Y H(Y))]. (140)i i i i

A payment solves this problem if and only if theYicondition

u(W Y H(Y)) E[u(W Y H(Y))]i i i i i i i i

(141)

is satisfied.To see the necessity of this condition, suppose thatis a solution of (140). Let Vbe an arbitrary randomYi

variable; we consider the family

Y Y tV.i i

According to our assumption, the function

f(t) E[u (W Y H(Y))]i i i i

is maximal for t 0. Hence

f(0) E[u(W Y H(Y))(V E[V])] 0.i i i i

(142)

We rewrite this equation as

E[V{u(W Y H(Y))i i i i

E[u(W Y H(Y))]}] 0. (143)i i i i

Since it is valid for all V, the random variable insidethe braces must be zero, and condition (141) follows.

To see that condition (141) is sufficient, consider apayment that satisfies (141) and any other paymentYY. From (57) it follows that

u (WYH(Y))i i i i

u (WYH(Y))i i i i

u(WYH(Y))(YH(Y)YH(Y))i i i i i i i i

u (WYH(Y))i i i i

E[u(WYH(Y))](YH(Y)YH(Y)).i i i i i i i i

(144)

Taking expectations and using the definition ofH, wesee that

E[u (W Y H(Y))] E[u (W Y H(Y))],i i i i i i i i

(145)

which completes the proof.We note that the optimal is unique apart from anYi

additive constant; hence is unique. It can Y H(Y)i ibe interpreted as the optimal payment that has azero price, and we refer to it as the net demand of

company i.Given, the random variable

n

[Y H(Y)] (146) i ii1

is the excess demand. The companies can maximizesimultaneously their expected utilities only if the ex-cess demand vanishes (this is the market clearing con-dition). This leads us to the following definition.

A price density and the payments . . . , Y , Y1 nconstitute an equilibrium, if (146) vanishes and if(141) is satisfied for i 1, . . . , n.

Note that an equilibrium induces a risk exchange. . . , with (X , X),1 n

X W Y H(Y), for i 1, . . . , n. (147)i i i i

Then condition (141) states that

u(X) E[u(X)], for i 1, . . . , n. (148)i i i i

From this and Theorem 1 it follows that the riskexchange implied by an equilibrium is Pareto optimal.Furthermore, (109) is satisfied with . In partic-ular, this shows that is a decreasing function of to-tal wealth.

The converse is true in the following sense. Supposethat (W1, . . . ,Wn) is already Pareto optimal; then W1,. . . , Wn and constitute an equilibrium, if we set

u(W)i i . (149)

E[u(W)]i i

Moreover,

Y H(Y) 0 for i 1, . . . , n.i i

This can be seen from (67) (with replaced by Wi )Xiand (141).


Assuming that all companies use exponential utilityfunctions, we gather from (141) that

1Y W ln , (150)i i iai

whereiis a constant. Hence the net demand of com-pany i is


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1 Y H(Y) W ln E[W]i i i iai

1 E[ln ]. (151)

ai

In the equilibrium the sum over i must vanish. Hence

10 W ln , (152)

a

where is a constant. Since E[] 1, it follows thatthe equilibrium price density is

aWe . (153)

aWE[e ]

Finally, a little calculation shows that

X W Y H(Y)i i i i

a a W E[W] E[W]ia ai i

a a W H(W) H(W) (154)ia ai i

in the equilibrium.


We assume that the companies use power utility func-tions of the first kind. Hence

c(s x)iu(x) , i 1, . . . , n,i csi

and

s X q (s W), i 1, . . . , n;i i i

see (87). Then according to (148) the equilibriumprice density is

cu(X) (s W)i i . (155)

cE[u(X)] E[(s W) ]i i

The equilibrium quotas are best determined from thecondition that H(Wi ) orH(X),i

H(W) H(s q (s W))i i i

s q s q H(W). (156)i i i

Hence

s H(W) E[(s W)]i i i iq ,i s H(W) E[(s W)]

for i 1, . . . , n. (157)


If all companies use the same power utility functionof the second kind,

cu(x) x , i 1, . . . , n,i

we know thatX q W, i 1, . . . , n;i i

see (92). Hence the equilibrium price density is

cu(X) Wi i . (158)

cE[u(X)] E[W ]i i

Again, the equilibrium quotas are best obtained fromthe condition that H(Wi ) qiH(W). ThusH(X)i

cH(W) E[W] E[W W]i i iq ,i c1H(W) E[W] E[W ]

for i 1, . . . , n. (159)

Remark 4

From (138) it follows that for any random variable Y

H(Y) E[Y] (H(W) E( W)) (160)

with

Cov(Y, ) , (161)

Cov(W, )

whereis now the equilibrium price density. Formula(160) is close to a central result in the capital-asset-

pricing model (CAPM). As an illustration, we revisitour three preceding examples. Thus

aWCov(Y, e ) (162)

aWCov(W, e )

in Example 19,

cCov(Y, (s W) ) (163)

cCov(W, (s W) )

in Example 20, and

cCov(Y, W ) (164)

c

Cov(W, W )in Example 21. Note that for c 1 (quadratic utilityfunctions), (163) reduces to the classical CAPM for-mula

Cov(Y, W) . (165)

Var[W]


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11. PRICING OF DERIVATIVE SECURITIESIn the equilibrium the price of a payment Y is H(Y),given by formulas (137), (138) or (139), where isthe equilibrium price density. Typically, the random

variable Y is the value of an asset or a derivative se-

curity at the end of a period. Under certain assump-tions, the price of a derivative security can be ex-pressed in terms of the price of the underlying asset.

First, we assume that the random variable has alognormal distribution, that is,

Z e , (166)

where Zhas a normal distribution, say with variance 2. Since

12E[] exp E[Z] (167) 2

must be 1, it follows that E[Z] (1/2) 2. Accordingto Formulas (153), (155), and (158), the assumptionof lognormality for means that W is normal in Ex-ample 19, that s Wis lognormal in Example 20, orthat Wis lognormal in Example 21.

Let us consider a particular asset. We denote itsvalue at the end of the period by S and assume thatthe random variable S has a lognormal distribution.Then we can write

RS s e , (168)0

where s0 is the observed price of the asset at the be-

ginning of the period, andRhas a normal distribution,say, with mean and variance 2. We assume that the

joint distribution of (Z, R) is bivariate normal withcoefficient of correlation. Then we obtain the follow-ing expression for the moment-generating function of

R with respect to the Q-measure:

tR tR Z tRE [e ] E[e ] E[e ]Q

12 2

exp t( ) t . (169) 2This shows that in the Q-measure the distribution of

Ris still normal, with unchanged variance 2 and newmean

. (170)Q

Luckily, there is a more practical expression for Q.Since s0 is the price of the asset at the beginning ofthe period,we have

s e H(S) e E [S], (171)0 Q

where is the risk-free force of interest. Hence weobtain the equation

1 R 2s e s E [e ] e s exp , 0 0 Q 0 Q 2

(172)

which yields

12 . (173)Q 2

Now let us consider a derivative security, whosevalue at the end of the period is f(S), a function ofthe underlying asset. Its price at the beginning of theperiod is

Re H(f(S)) e E [f(s e )], (174)Q 0

whereR is normal with mean given by (173) and var-iance 2. For example, for a European call option withstrike price K, f(S) (S K)

. Then (174) can be

calculated explicitly, which leads to the Black-Scholesformula.

Remark 5

The method can be generalized to price derivative se-curities that depend on several, say,m assets. Let

RiS s e , (175)i i0

denote the value at the end of the period of asset i,wheresi0 is the observed price of asset i at the begin-ning of the period, i 1, . . . , m. The assumption isnow that (Z, R1, . . . , Rm) has a multivariate normal

distribution. Then in the Q-measure (R1, . . . ,Rm) hasstill a multivariate normal distribution, with un-changed covariance matrix, but modified mean vector,such that

1E [R ] Var[R ], for i 1, . . . , m. (176)Q i i2

In the framework of Examples 1921, practical re-sults can also be obtained for derivative securities onassets for which S is a linear function ofW.

In Example 19 suppose that S qW. Then

aW SE[Se ] E[Se ]

E [S]

(177)Q aW SE[e ] E[e ]

with a/q. According to (171), the value of isdetermined from the condition that

SE[Se ]

e s . (178)0SE[e ]

Then the price of a derivative security with payofff(S)is given by the expression


16/27


SE[f(S)e ] e E [ f(S)] e . (179)Q SE[e ]

This is the Esscher method in the sense of Buhlmann.We note that it also works for assets whereS and W S are independent random variables: here

aS a( WS)E[Se e ]E [S] Q aS a( WS)E[e e ]

aS a( WS) aSE[Se ]E[e ] E[Se ] . (180)

aS a( WS) aSE[e ]E[e ] E[e ]

Hence a is determined from (178) with replacedbya.

In Example 20 we suppose that S q(s W). Then

c cE[S(sW) ] E[SS]E [S] . (181)Q c cE[(s W) ] E[S]

The value ofc is determined from the condition that1cE[S ]

e s , (182)0cE[S]

and the price of a derivative security with payofff(S)is given by the expression

cE[f(S)S] e E [ f(S)] e . (183)Q cE[S ]

In Example 21 we suppose again S qW. Then

c cE[SW ] E[SS ]E [S] . (184)Q c c

E[W ] E[S ]The value of c is now determined from the conditionthat

1cE[S ]

e s , (185)0cE[S ]

and the price of a derivative security with payofff(S)is given by the expression

cE[f(S)S ] e E [ f(S)] e . (186)Q cE[S ]

Formula (183) is also acceptable, ifc, the solution of

(182) is negative. In this case the solution of (185) ispositive, which leads to (186). But this is again (183),

with a negativec. Formulas (182) and (183) summa-rize the Esscher method that was proposed by Gerberand Shiu (1994a, 1994b).

Remark 6

Using (178), we can rewrite (179) as

SE[f(S)e ]e E [ f(S)] s . (187)Q 0 SE[Se ]

Similarly, (183) can be rewritten ascE[f(S)S]

e E [ f(S)] s . (188)Q 0 1cE[S ]

It may be surprising that does not appear in theseexpressions for the prices, but of course the values of and c are functions of.

Remark 7

The Esscher method summarized by Formulas (182)and (183) has some attractive features. For example,if S has a lognormal distribution, it has also a log-

normal distribution in theQ-measure. In particular, itreproduces the formula of Black-Scholes.

12. BIBLIOGRAPHICAL NOTESA broad, less self-contained review has been given byAase (1993). The article by Taylor (1992a) is highlyrecommended.

In Section 5 the premiums are determined by theprinciple of equivalent utility. If this principle isadopted in a dynamic model, there is an intrinsic re-lationship between the underlying utility function and

the resulting probability of ruin; see Gerber (1975).The optimality of a stop-loss contract of Section 6seems to have been discovered by Arrow (1963). Itsminimal variance property has been discussed by oth-ers, for example, by Kahn (1961).

The theorem of Borch in Section 8 can be found inthe books of Buhlmann (1970) and Gerber (1979). Insome of the literature, the family of utility functionssatisfying (102) is called the HARA family (hyperbolicabsolute risk aversion).

In Sections 9 and 10 we discussed Pareto optimalrisk exchanges of a specific form. Other proposalshave been discussed by Buhlmann and Jewell (1979)and by Baton and Lemaire (1981). Solutions that arenot Pareto optimal have been proposed by Chan andGerber (1985), Gerber (1984), and Taylor (1992b).


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ACKNOWLEDGMENTThe authors thank two anonymous reviewers for their

valuable comments.

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APPENDIX

ECONOMIC

PROOFS OF

TWO

FAMOUS

INEQUALITIESIn Section 9 we introduced the synergy potential. Byobserving that this quantity is non-negative, we canderive two mathematical inequalities in a nonconven-tional way. In Example 12, 0 implies that

n

aW 1/a a W 1/ai i iE[e ] E[e ] ; (189)i1

see (116). With the substitutions

a Wi iZ e , r a /a,i i i

Inequality (189) can be written asn n

r 1/ri iE Z E[Z ] . (190) i ii1 i1

Because the substitutions can be reversed, this ine-quality is valid for arbitrary random variablesZ1 0,. . . , Zn 0 and numbers r1 0, . . . , rn 0 with1/r1 . . . 1/rn 1. In the mathematical literature,Inequality (190) is known as Holders inequality.


18/27


The other inequality is Minkowskis inequality. Itstates that for p 1 and random variables Z1 0,. . . , Zn 0 the following inequality holds:

p 1/pn n

p 1/pE Z E[Z ] . (191)

i i

i1 i1

The proof starts with 0 in (119). Then it sufficesto set

Z s W, p c 1,i i i

and to observe that substitutions can be reversed. Ifp 1, the inequality sign in (191) should be reversed.This follows from Example 14, with the substitution

Z W, p 1 c.i i

In the limit p 0, we obtain

n n

exp E ln Z exp (E[lnZ]);

i i

i1 i1

this can be seen from (124). Note that (191) alsoholds for p 1, in which case it is known as the tri-angle inequality.

DISCUSSIONS

HANGSUCK LEE*Dr. Gerber and Mr. Pafumi have written a very inter-esting paper. My comments concern Section 7, on the

optimal fraction of reinsurance.In the particular case in which u(x) is an exponen-

tial utility function with parameter a and S has agamma distribution with parameters and , the cal-culation can be also done explicitly. IfS gamma(,), then E(S) / and Var(S) /2. For a(1 ) , the expected utility is

1awaPa(1)S(1 E[e ])

a

1

awaP 1 e ,

a a(1 )

which is maximal for

11 .

P a

*Mr. Lee is a graduate student in actuarial science, Department of

Statistics and Actuarial Science, at the University of Iowa, 241

Schaeffer Hall, Iowa City, Iowa 52242-1409.

To compare this result with the one in normal case,we rewrite it as

P E(S) E(S)1 .

a Var(S) P

If we assume, and P tend to infinity such that P E(S) and Var(S) remain constant, then

P E(S)1 .

a Var(S)

In another particular case in which u(x) is an ex-ponential utility function with parameter a, and S hasan inverse Gaussian distribution with parameters and (Bowers et al. 1997, Ex. 2.3.5), the calculationcan be again done explicitly. If S Inverse Gaus-sian(, ), then E(S) / and Var(S) /2. Fora(1 ) /2, the expected utility is

1 1awaPa(1)S awaP(1 E[e ]) 1 ea a

1/22a(1 )

exp 1 1 .

It is maximal for

2 2P (/) 1 .

2P 2a

To compare this result with that in normal case, werewrite it as

E(S)1

PP E(S) E(S)1 .a Var(S) 2 P

If we assume, and P tend to infinity such that P E(S) and Var(S) are constant, then

P E(S)1 .

a Var(S)

REFERENCE

BOWERS, N.L., JR., GERBER, H.U., HICKMAN, J.C., JONES, D.A., AND

NESBITT, C.J. 1997. Actuarial Mathematics, 2nd ed.Schaumburg, Ill.: Society of Actuaries.


19/27


ALASTAIR G. LONGLEY-COOK*As Charles Trowbridge (1989, p. 11) points out inFundamental Concepts of Actuarial Science, utilitytheory forms the philosophical basis of actuarial sci-ence, and yet there is barely a mention of the subject

in the actuarial literature beyond Chapter 1 ofActu-arial Mathematics (Bowers et al. 1997). Dr. Gerberand Mr. Pafumi do the profession a great service bypublishing this excellent summary of utility functionsand their applications.

The authors employ three general forms of utilityfunctions, the exponential and the power of the firstand second kind, in their examples. The reasonable-ness of these functions should be evaluated in anypractical application. In particular, the upper andlower bounds on the two kinds of power functions mayrender them unreasonable assumptions in situations

in which results can vary widely from expected.It is also generally agreed in finance theory that for

a utility function to be realistic with regard to eco-nomic behavior, its absolute risk aversion with regardto wealth W, defined as A(W) U( W)/U(W),should be a decreasing function ofW; and while thereis some debate over the slope of its relative risk aver-sion, defined asR( W) W[U(W)/U( W)], empir-ical evidence suggests that it should be constant overW(Rubinstein 1976). If the variable x in the papersfirst example [Formula (1)] is defined as change in

wealth Wwith respect to initial wealth W0, then the

negative exponential utility function satisfies the de-creasing-absolute and constant-relative risk aversioncriteria with respect to initial wealth W0. As the au-thors point out, the absolute risk aversion of thepower utility function of the first kind [Formula (10)]

increases with wealth, a condition that may prove un-realistic.

Note that, despite the differences in utility func-tions, the general form of a risk adjustment (whichreduces the expected value to its certainty equivalent)is dependent on the variance of gain, rather than thestandard deviation or some other measure or risk [seeFormulas (23), (26), and (28)]. This is consistent withmean/variance risk analysis and challenges the use ofother risk measures.

What is then surprising is that the classical valuefor given in Formula (165) is generated by the quad-ratic utility function, rather than the exponential. The

*Alastair G. Longley-Cook, F.S.A., is Vice President and Corporate

Actuary, Aetna, 151 Farmington Ave., RC2D, Hartford, Connecticut

06156, e-mail, [email protected].

expected return versus beta relationship of the cap-ital-asset-pricing model (CAPM)

E(r) r [E(r ) r]i f i M f

can be derived from a variance-proportionate risk pre-

mium (Bodie et al. 1996, p. 243)2E(r) r k .i f

So it is strange that the exponential utility functionassumption, with its variance-proportionate risk pre-mium [see Formula (23)] does not lead to the clas-sical value for . Further explanation of the utilityfunction assumptions inherent in CAPM would behelpful.

REFERENCES

BODIE, A., ET AL. 1996. Investments.Homewood, Ill.: Irwin.

BOWERS, N.L., JR., ET AL. 1986. Actuarial Mathematics. Itasca,Ill.: Society of Actuaries.

RUBENSTEIN, M. 1976. The Strong Case for the Generalized

Logarithmic Utility Model as the Premier Model of Finan-

cial Markets,Journal of Finance 31:55171.

TROWBRIDGE, C.L. 1989.Fundamental Concepts of Actuarial Sci-

ence. Itasca, Ill.: Actuarial Education and Research Fund.

HEINZ H. M LLER*UThe authors are to be congratulated for their very el-egant article that treats in a very concise and clearstyle the most important applications of utility theory

in insurance and finance. The article not only is anexcellent survey but also introduces the synergy po-tential of risk exchanges as a new concept. In the in-surance context this synergy potential measures in amost convincing way the welfare gain resulting fromreinsurance.

It may be of some interest to address the question,which types of utility functions lead to a decision-making consistent with empirical observations? Thefollowing result due to Arrow (1971) helps to answerthis question:

Assume that a risk-less and a risky asset are availableas investment opportunities. Then, under an in-crease of initial wealth investors with increasing (de-

creasing) risk aversion decrease (increase) the dollaramount invested in the risky asset.

*Heinz H. Muller, Ph.D., is Professor of Mathematics at the University

of St. Gallen, Bodanstrasse 4, CH-9000 St. Gallen, Switzerland,

e-mail, [email protected].


20/27


Therefore, Arrow postulates a decreasing risk aver-sion. According to this postulate power utility func-tions of the first kind and, in particular, the quadraticutility function may not be appropriate for practicalpurposes.

The results in the survey allow also for some com-ments on the properties of equilibrium in financialmarkets. As shown in Section 10, an equilibrium riskexchange . . . , with a price density is Pareto (X , X)1 noptimal, and there exist k1, . . . , kn such that (107)and (109) hold with . This provides the basisfor the construction of a fictitious investor represent-ing the market. Up to an additive constant the utilityfunctionu of this representative investor is given by

u(w) (w).

From (109) we conclude that u is strictly concave andfor the risk aversion of the representative investor weobtain

u(w) (w) 1r(w) .nu(w) (w) 1

r(X )h1 h h

Instead of the risk aversion

u(x)r(x) ,

u(x)

it is convenient in this context to use the risk toler-ance, which is defined by

1(x) .

r(x)

Hence the risk tolerance of the representative investoris given by

n

(w) (X ), h hh1

and the risk-sharing rule (101) can be written as

(X)j jdX dw, j 1, . . . , nj(w)

or

(X)j jX .j(w)

Taking logarithms and differentiating leads to

X (X) (w)j j j X j X (X) (w)j j j

1 [(X) (w)].j j

(w)

This implies in particular

sign(X) sign[(X) (w)],j j j

and we conclude

(X) (w) X 0 (*).j j j

() ()

In the context of financial markets,Wcorresponds tothe total market capitalization and Xj, j 1, . . . , n,denotes investorjs payoff as a function ofW. Accord-ing to Arrows postulate risk tolerances are increasingand (*) can be interpreted as follows:

Investors who are more (less) sensitive to wealthchanges than the market choose a convex (concave)payoff function.

As Leland (1980) pointed out in his article on risk-sharing in financial economics, convex payoff func-tions can be considered as generalized portfolio in-surance strategies. A discussion of the shape of payofffunctions in market equilibrium can be found, for ex-ample, in Chevallier and Mueller (1994).

The relationship (*) may be of some interest for theinvestment strategy of pension funds. Because of sol-

vency problems, such an investor may be more sensi-tive to wealth changes than the market for low valuesofw, that is, forw w0. For high values(X) (w)j jof w, funding problems disappear and (X) (w)j jfor w w0 may hold. This leads to a payoff function

Xj as depicted in Figure 1.A payoff function that is convex for loww and con-

cave for highw is, for example, obtained by investing

Figure 1

Payoff Function for a Pension Fund

Xj

ww

j'(Xj) > '(w) j'(Xj) < '(w)0


21/27


in the market portfolio, selling calls with high strikeprices and buying puts with low strike prices.

REFERENCES

ARROW, K.J. 1971. Essays in the Theory of Risk Bearing. Chi-

cago, Ill.: Markham.

CHEVALLIER, E., AND MUELLER, H.H. 1994. Risk Allocation in

Capital Markets: Portfolio Insurance, Tactical Asset Allo-

cation and Collar Strategies, ASTIN Bulletin 24, no. 1:

518.

LELAND, H.E. 1980. Who Should Buy Portfolio Insurance?

Journal of FinanceXXXV, no. 2:581596.

STANLEY R. PLISKA*This paper provides a survey of applications of utilitytheory to selected risk management and insuranceproblems. I would like to compliment the authors for

writing such a clear, interesting, and yet concise treat-ment of how utility functions can be used for decision-making in the actuarial context. Applications studiedrange from the fundamental problem faced by a com-pany of setting an insurance premium to esoteric is-sues such as synergy potentials.

I would also like to complement the authors by ex-tending their exposition in two directions. Both direc-tions have to do with the literature relating utility the-ory and financial decision-making. The first directionpertains to the utility functions themselves. An im-portant, classical treatment of the use of utility func-

tions for decision-making is by Fishburn (1970). Inrecent years, however, financial economists such asBergman (1985), recognizing the limited realism as-sociated with standard utility functions, have devel-oped some generalizations reflecting changing pref-erences across time. For example, Constantinides(1990) studied a utility function model that reflectshabit formation, and Duffie (1992) covered a relatednotion called recursive utility. It would be interestingto consider how in the actuarial context preferencesmight change with time.

Another direction the authors could have pursued

is the well-known application of utility functions forportfolio management. More than 25 years ago RobertMerton, the recent winner of a Nobel prize in econom-ics, wrote several papers (see his 1990 book) in whichfor continuous time stochastic process models of asset

*Stanley Pliska, Ph.D., is CBA Distinguished Professor of Finance, De-

partment of Finance, College of Business Administration, University

of Illinois at Chicago, 601 South Morgan Street, Chicago, Ill. 60607,


prices the objective is to maximize expected utility ofconsumption and/or terminal wealth. He employed adynamic programming approach, an approach that iselegant yet often impractical due to computationaldifficulties. More recently, martingale methods have

been employed to successfully solve these continuoustime optimal portfolio problems [for example, seePliska (1997) and the book by Boyle et al. (1998),

which the authors already reference]. Since maximiz-ing expected utility is the objective preferred by finan-cial economists for managing portfolios of assets andliabilities, the relevance for the actuarial and insur-ance industries is obvious.

REFERENCES

BERGMAN, Y.Z. 1985. Time Preference and Capital Asset Pricing

Models,Journal of Financial Economics 14:145159.

CONSTANTINIDES, G. 1990. Habit Formation: a Resolution of theEquity Premium Puzzle,Journal of Political Economy 98:

51943.

DUFFIE, D. 1992.Dynamic Asset Pricing Theory.Princeton, N.J.:

Princeton University Press.

FISHBURN, P. 1970.Utility Theory for Decision Making. New York:

John Wiley & Sons.

MERTON, R.C. 1990. Continuous Time Finance. Cambridge,

Mass.: Blackwell Publishers.

PLISKA, S.R. 1997. Introduction to Mathematical Finance: Dis-

crete Time Models. Cambridge, Mass.: Blackwell Publish-

ers.

ELIAS S.W. SHIU*This is a masterful paper on utility functions and manyof their applications in actuarial science and finance.I am particularly grateful for this paper because I once

wrote in TSA (Shiu 1993) a defense for the use ofutility theory.

We are indebted to the authors for giving a preciseexplanation of the approximation formula (28). Thisresult can be found in textbooks such as Bowers et al.(1997, Ex. 1.10.a) and Luenberger (1998, p. 256, Ex.8). The derivation of (28) outlined in these two bookscombines a second-order approximation with a first-order approximation. Formula (28) probably first ap-peared in Pratt (1964, Eq. 7). Pratt (1964, p. 125)

was rather careful in stating that he assumed the thirdabsolute central moment ofGzto be of smaller orderthan Var[Gz]; ordinarily, it is of order (Var[Gz])

3/2.

*Elias S.W. Shiu, A.S.A., Ph.D., is Principal Financial Group Founda-

tion Professor of Actuarial Science, Department of Statistics and Ac-

tuarial Science, University of Iowa, Iowa City, Iowa 52242-1409,



22/27


The formulas for the two kinds of power utility func-tions, as given by (2) and (4), are rather unsymmet-rical because there is an s in (2) but not in (4). Bymodifying (4) as

1c(x s) 1

u(x) , x s, (D.1)1 c

we can obtain a certain symmetry between the twokinds of utility functions. Then (11) becomes

cr(x) , (D.2)

x s

(89) becomes

1/c 1/cX k s , (D.3)j j j

and so on.My final comment is motivated by the reference to

the so-called Merton ratio in Section 7. The Mertonratio was also discussed in a paper in this journal(Boyle and Lin 1997). It means that an investor witha power utility function of the second kind will use aproportional asset investment strategy. The result wasderived by Merton (1969) using Bellmans equation.

An elegant proof using the insights from the martin-gale approach to the contingent-claims pricing theorycan be found in the review paper by Cox and Huang(1989, p. 283). In the context of discrete-time mod-els, the result was obtained by Mossin (1968); furtherdiscussions can be found in survey articles such asHakansson (1987) and Hakansson and Ziemba (1995)and in various papers reprinted in Ziemba and Vickson(1975). Merton (1969) also showed that an investor

with an exponential utility function would invest aconstant amount in the risky asset.

REFERENCES

BOWERS, N.L., JR., GERBER, H.U., HICKMAN, J.C., JONES, D.A., AND

NESBITT, C.J. 1997. Actuarial Mathematics, 2nd ed.

Schaumburg, Ill.: Society of Actuaries.

BOYLE, P.P., AND LIN, X. 1997. Optimal Portfolio Selection with

Transaction Costs, North American Actuarial Journal 1,

no. 2:2739.COX, J.C., AND HUANG, C.F. 1989. Option Pricing Theory and

Its Applications, in Theory of Valuation: Frontier of Mod-

ern Financial Theory, edited by S. Bhattacharya and G.M.

Constantinides, 27288. Totowa, N.J.: Rowman & Little-

field.

HAKANSSON, N.H. 1987. Portfolio Analysis, in The New Pal-

grave: A Dictionary of Economics, Vol. 3, edited by J.

Eatwell, M. Milgate and P. Newman, 91720. London: Mac-

millan.

HAKANSSON, N.H., AN DZIEMBA, W.T. 1995. Capital Growth The-

ory, in Handbooks in Operations Research and Manage-

ment Science, Vol. 9 Finance, edited by R.A. Jarrow, V.

Maksimovic, and W.T. Ziemba, 6586. Amsterdam: Else-

vier.

LUENBERGER, D.G. 1998. Investment Science. New York: Oxford

University Press.MERTON, R.C. 1969. Lifetime Portfolio Selection under Uncer-

tainty: The Continuous-Time Case, Review of Economics

and Statistics51:24757.

MOSSIN, J. 1968. Optimal Multiperiod Portfolio Policies,Jour-

nal of Business41:21529.

PRATT, J.W. 1964. Risk Aversion in the Small and in the

Large, Econometrica 32:12236. Reprinted with an ad-

dendum in Ziemba and Vickson (1975), pp. 11530.

SHIU, E.S.W. 1993. Discussion of Loading Gross Premiums for

Risk Without Using Utility Theory, Transactions of the

Society of ActuariesXLV:34648.

ZIEMBA, W.T., AND VICKSON, R.G., ed. 1975. Stochastic Optimi-

zation Models in Finance.New York: Academic Press.

VIRGINIA R. YOUNG*I congratulate the authors on writing an excellentsummary of applications of utility functions in evalu-ating risks, with special emphasis on insurance eco-nomics. Their paper will serve as a valuable referencefor actuaries, both practitioners and researchers.

In this discussion, I simply wish to point out thatArrows theorem on the optimality of stop-loss (or de-ductible) insurance is more generally true; see the au-thors Section 6. Specifically, suppose a decisionmaker orders risks according to stop-loss ordering;that is, a (non-negative) loss random variableXis con-sidered less risky under stop-loss ordering than a lossYif

t t S (x)dx S (x)dxX Y0 0

for all t 0, in which SXis the survival function ofX;namely,SX(x) Pr(X x). Then, for a fixed premium

P f(E[h(X)], in which f is a function such thatf(x) x and f(x) 1, the decision maker will preferstop-loss insurance with deductible d given implicitlyby

P f S (x)dx . Xd

*Virginia R. Young, F.S.A., Ph.D., is Assistant Professor of Business,

School of Business, 975 University Avenue, University of Wisconsin

Madison, Madison, Wisconsin 53706, e-mail, [email protected].


23/27


Van Heerwaarden, Kaas, and Goovaerts (1989) provethis result for f(x) (1 )x, while constraining theindemnity benefit to have a derivative between zeroand one. The result is true when one only constrainsthe indemnity benefit to lie between zero and the loss

amount, as in the authors Inequality (54); see Gollierand Schlesinger (1996).The optimality of stop-loss insurance is intuitive by

noting that the authors proof of Inequality (56) isindependent of the (increasing, concave) utility func-tion and by recalling that the common (partial) or-dering of random variables by risk-averse decisionmakers is stop-loss ordering (Wang and Young 1998).I encourage the authors and interested researchers toexplore whether or not one can generalize other re-sults from expected utility theory.

REFERENCESGOLLIER, C., AND SCHLESINGER, H. 1996. Arrows Theorem on

the Optimality of Deductibles: A Stochastic Dominance

Approach,Economic Theory7:35963.

VAN HEERWAARDEN, A.E., KAAS, R., AND GOOVAERTS, M.J. 1989.

Optimal Reinsurance in Relation to Ordering of Risks,

Insurance: Mathematics and Economics8:26167.

WANG, S.S., AND YOUNG, V.R. 1998. Ordering Risks: Expected

Utility Theory versus Yaaris Dual Theory of Risk, Insur-

ance: Mathematics and Economics, in press.

AUTHORS REPLY

HANS U. GERBER ANDGERARD PAFUMIMr. Lee shows how the optimal degree of proportionalreinsurance can be determined explicitly if S has agamma or an inverse Gaussian distribution. He alsoshows that Formula (60) is obtained in both cases asa limiting result. This raises the question, What re-sults can be obtained under the more general as-sumption that S has an infinitely divisible distribu-tion? Let

1 2 3(z) z z z . . .

2 6

denote the cumulant generating function of a randomvariable with infinitely divisible distribution, havingmean 1, variance 1, and third central moment . Nowsuppose that S has a distribution such that its mo-ment generating function is

zS (z/)M(z) E [e ] e (R.1)S

for some 0 (the shape parameter) and 0 (thescale parameter). Then

E [S] ,

Var[S] ,

2

3

E S . 3 This construction is illustrated by the following table.

Distribution (z)

Normal 1

2z z

2 0

Gamma ln(1 z) 2

Inverse Gaussian 1 1 2z 3

The expected utility (59) is

1 (1 exp(aw aP)M(a(1 ))).Sa

Using (R.1), we see that we must minimize the ex-pression

a(1 )aP .

Setting the derivative equal to 0, we gather that isobtained from the condition

a(1 )P 0. (R.2)

In general, there is no explicit formula for 1 How-.ever, it is possible to obtain an asymptotic formula.Let P , , , such that

P P E [S] constant,

Var[S] constant.

2

Substituting(z) 1 z (/2)z2 . . . in (R.2),we get

P E[S] a(1 )Var[S]

2a2

(1 ) Var[S] . . . 0.2

Finally, we develop 1 in powers of 1/and obtainthe formula

P E [S] P E [S]1 1 . . . .

a Var[S] 2 Var[S]

Again, Formula (60) results in the limit.


24/27


We appreciate the comments of Mr. Longley-Cook.He finds it strange that the assumption of exponentialutility functions does not lead to the classical expres-sion for . However, if Formula (162) is expanded inpowers of a, the classical Formula (165) can be ob-

tained as a first-order approximation. In this contextwe note that Formulas (23), (26), and (28) are alsofirst-order approximations. For a deeper discussion ofthe CAPM formula, refer to Section 8.2.4 of Panjer etal. (1998) and the references quoted therein.

Dr. Muller raises some very interesting points. Therisk tolerance function (the reciprocal of the risk aver-sion function) leads to simplification of some formulasand is a useful and appealing tool by its own.

Dr. Pliska and Dr. Shiu point out an important ap-plication of utility theory: the construction of an op-timal portfolio, that is, a portfolio that maximizes theexpected utility of an investor. This problem can in-deed be discussed in the framework of Sections 10 and11 of the paper. Consider an investor with wealth wat time 0 and utility function u(x) to assess the ter-minal wealth at time T. Let 0 denote the risklessforce of interest. In the market random payments canbe bought. Their price is given by a price density .Thus the price (due at time 0) for a payment ofY(dueat time T) is E [Y]. If the investor buys Y, theTeterminal wealth will be

TW we Y E [Y]. (R.3)T

The problem is to chooseYthat maximizesE [u(WT)].

In analogy to (141), the solution is characterized bythe condition

u(W) E [u(W )] (R.4)T T

with WT given by (R.3). For the utility functions ofExamples 1 to 3, explicit expressions for the optimalterminal wealth are obtained. For an exponential util-ity function,u(x) the optimal terminal wealthaxe ,is

1 1TW we ln E [ ln ]. (R.5)T a a

For a power utility function of the first kind, u(x) (1 x/s)c, x s, we obtain

Ts we1/cW s , (R.6)T 11/cE [ ]

and for a power utility function of the second kind,u(x) xc, the result is that

Twe1/cW . (R.7)T 11/cE[ ]

We note that WTis the solution of a static optimiza-tion problem. If we make appropriate additional as-sumptions about the market, we can determine theoptimal investment strategy, that is, the dynamicstrategy that replicates the optimal terminal wealth

WT. As in Section 10.6 of Panjer et al. (1998), assumethat two securities are traded continuously, the risk-less investment (which grows at a constant rate ),and a non-dividend-paying stock, with price S(t) attime t, 0 t T. We make the classical assumptionthat {S(t)} is a geometric Brownian motion, that is,

X(t)S(t) S(0)e

where {X(t)} is a Wiener process with parameters and 2 and parameters * (1/2)2 and 2 inthe risk-neutral measure. Then

h*S(T) *

T

e , with h*

, (R.8)

2S(0)

where is such thatE [] 1. Note thath* is definedas in Gerber and Shiu (1994) and Gerber and Shiu(1996), that is, as the value of the Esscher parameter

h, for which the discounted stock price process is amartingale under the transformed measure. As a prep-aration, we recall a result concerning the self-financ-ing portfolio that replicates the payoff of a European-type contingent claim. Consider a Europeancontingent claim with terminal dateTand payoff func-tion(z); that is, at time Tthe payoff(S(T)) is due.Let V(z, t) denote its price at time t, and (z, t) theamount invested in stocks in the replicating portfolioat time t, ifS(t) z. It is well known that

V(z, t)(z, t) z ; (R.9)

z

see Formula (10.6.6) in Panjer et al. (1998), page 95of Baxter and Rennie (1996), or Section 9.3 of Dothan(1990).

Let us revisit the three examples. From (R.5) and(R.8) we obtain

1 TTW we E[ln ] T

a a

h* h* ln S(T) ln S(0). (R.10)

a a

Consider a European contingent claim with terminaldateTand payoff function

1 T h*T(z) we E[ln ] ln z.

a a a

Its payoff differs from WT only by the constant


25/27


(h*/a) ln S(0). Since w is the initial price ofWT, itfollows that

h*TV(z, 0) w e ln z.

a

Hence, by (R.9), the initial amount invested in stocksmust be

h*T(z, 0) e .

a

Similarly, at time t, the amount invested in stocks inthe replicating portfolio is

h* *(Tt) (Tt)

e e , 0 t T. (R.11)2a a

Note that its discounted value is constant.For a power utility function of the first kind, we

gather from (R.6) and (R.8) that the optimal terminalwealth is

h*/cTs we S(T)T/cW s e . T 11/cE[ ] S(0)

Now consider a European contingent claim with ter-minal date Tand payoff function

Ts weh*/c(z) exp(T/c)z . 11/cE [ ]

Note that

h*/c(S(T)) (s W )S(0) . (R.12)

T

It follows that the initial price of the contingent claimis

T h*/cV(z, 0) (se w)z ,

withS(0) z. Then according to (R.9) we have

h*T h*/c(z, 0) (se w)z . (R.13)

c

To determine the replicating portfolio for WT, we re-write (R.12) as

h*/cW s (S(T))S(0) .

T

Hence the amount invested in stocks at time 0 mustbe

h*/c(S(0), 0) S(0)

which, by (R.13), simplifies to

h*T

(se w).c

Similarly, at time t the amount invested in stocks inthe replicating portfolio is

h*(Tt)

(se W)tc

* (Tt) (se W), (R.14)t2c

where Wt is the wealth at time t, a constant fractionof what is missing for total satisfaction.

For a power utility function of the second kind, theoptimal terminal wealth is

h*/cTwe S(T)T/cW e . T 11/cE [ ] S(0)

This time consider a European contingent claim withterminal date Tand payoff function

Twe h*/c(z) exp(T/c)z . 11/cE[ ]Its price at time 0 is

h*/cV(z, 0) z w.

Hence, by (R.9),

h*(z, 0) V(z, 0).

c

For the replicating portfolio of WT, the amount in-vested in stocks is the same constant fraction of totalwealth, that is,

h* w

c

at time 0, and

h* * W W (R.15)t t2c c

at time t.At first sight, expressions (R.11), (R.14), and (R.15)

are quite different. However, they can be written in acommon form: in all three cases the optimal trading

strategy is to invest the amount *

(Tt)e (R.16)2 (Tt) r(e W)t

at time t (0 t T) in stocks, where r is the riskaversion function. For a verification, simply use (9),(10), and (11) of the paper.

These results can be generalized to the case wheren 2 different types of stocks are traded. Let Sk(t)denote the price of stock k. We assume that


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{S1(t), . . . , Sn(t)} is an n-dimensional geometricBrownian motion with drift parameters 1, . . . , n

. . . , in the risk-neutral measure) and covar-(*, *1 niances ik. It is assumed that the covariance matrixhas an inverse (the precision matrix); its elements are

denoted by the symbol ik. Then we find the followinggeneralization of (R.8):*hn k nS (T)kT e , with h* (* ) k i i ikS(0)k1 i1k

defined as in Section 7 of Gerber and Shiu (1994),and again such that E [] 1. According to (R.5),(R.6), (R.7), the optimal terminal wealth is

1 TTW we E[ln ] T a a

n1 h* ln S (T)

k k

a k1

n1 h* ln S (0) k ka k1

for an exponential utility function,*h /cn kTs we S(T)kT/cW s e T 11/cE [ ] S(0)k1 k

for a power utility function of the first kind and*h /cn kTwe S(T)kT/cW e T 11/cE [ ] S(0)k1 k

for a power utility function of the second kind. In eachcase we can relate the optimal terminal wealth to thepayoff of an appropriately chosen European contin-gent claim. Such a payoff can be replicated by a dy-namic portfolio, whereby the amount kis invested instocks of type k at timet. Let V(z1, . . . , zn, t) denotethe price of the portfolio at time t (ifSk(t) zk, k 1, . . . , n). Then

V(z , . . . , z , t)1 n(z , . . . , z , t) zk 1 n k

zk

for k 1, . . . , n. See, for example, Formula (8.35)

in Gerber and Shiu (1996). The portfolio that repli-cates WT is the optimal investment strategy. For anexponential utility function, we find

Utilitiy Functions from Risk Theory to Finance.pdf

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