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UTILITY FUNCTIONS:
FROM RISK THEORY TO FINANCE
Hans U. Gerber* and Gérard 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, power
utility functions of the first kind (such as quadratic utility functions), and power utility functions
of the second kind (such as the logarithmic utility function). The postulate of equivalent expected
utility 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 insurer’s wealth without the new contract is a random variable
itself. Then n companies (or economic agents) with random wealth are considered. They are
interested 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 that
can be withdrawn from the system without hurting any company in terms of expected utility.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 can
be explained.

1. INTRODUCTION This is confirmed by Seal (1969, Ch. 6) and the ref-

The notion of utility goes back to Daniel Bernoulli erences cited therein.
(1738). Because the value of money does not solve the Utility theory came to life again in the middle of
paradox of St. Petersburg, he proposed the moral this century. This was above all the merit of von Neu-
value of money as a standard of judgment. According mann and Morgenstern (1947), who argued that the
to Borch (1974, p. 26), existence of a utility function could be derived from a
set of axioms governing a preference ordering. Borch
Several mathematicians, for example Laplace, dis- showed how utility theory could be used to formulate
cussed the Bernoulli principle in the following cen- and solve some problems that are relevant to insur-
tury, and its relevance to insurance systems seems ance. Due to him, risk theory has grown beyond ruin
to have been generally recognized. In 1832 Barrois theory. Most of the original papers of Borch have been
presented a fairly complete theory of fire insurance, reprinted and published in book form (1974, 1990).
based on Laplace’s work on the Bernoulli principle. Economic ideas have greatly stimulated the devel-
For reasons that are difficult to explain, the principle opment of utility theory. But this also means that sub-
was almost completely forgotten, by actuaries and stantial parts of the literature have been written in a
economists alike, during the next hundred years. style that does not appeal to actuaries.
The purpose of this paper is to give a concise but
self-contained survey of utility functions and their ap-
plications that might be of interest to actuaries. In
*Hans U. Gerber, A.S.A., Ph.D., is Professor of Actuarial Science at
Sections 2 and 3 the notion of a utility function with
the Ecole des HEC (Business School), University of Lausanne, CH-
1015 Lausanne, Switzerland, e-mail, hgerber@hec.unil.ch.
its associated risk aversion function is introduced.
†Gérard Pafumi is a doctoral student at the Ecole des HEC (Business Throughout the paper, the theory is illustrated by
School), University of Lausanne, CH-1015 Lausanne, Switzerland, means of examples, in which exponential utility func-
email, gpafumi@hec.unil.ch. tions and power utility functions of the first and

74
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UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 75

second kind are considered, which include quadratic Usually we assume that the function u(x) is twice
and logarithmic utility functions. differentiable; then (1) and (2) state that u9(x) . 0
In Section 4, we order random gains by means of and u0(x) , 0.
their expected utilities. In particular, a random gain The first property amounts to the evident require-
can be replaced by a fixed amount, the certainty equiv- ment that more is better. Several reasons are given
alent. This notion can be used by the consumer who for the second property. One way to justify it is to
wants to determine the maximal premium he or she require that the marginal utility u9(x) be a decreasing
is willing to pay to obtain full coverage. function of wealth x, or equivalently, that the gain of
The insurer’s situation is considered in Section 5. A utility resulting from a monetary gain of $g, u(x 1 g)
premium that is fair in terms of expected utility typ- 2 u(x), be a decreasing function of wealth x.
ically contains a loading that depends on the insurer’s
risk aversion and on the joint distribution of the Example 1
claims, S, and the random wealth, W, without the new Exponential utility function (parameter a . 0)
contract; certain rules of thumb in terms of Var[S]
and Cov(S, W ) are obtained. Section 6 presents a clas- 1
u(x) 5 (1 2 e2ax), 2` , x , `. (1)
sical result that can be found as Theorem 1.5.1 in a
Bowers et al. (1997). We note that for x → `, the utility is bounded and
In Section 8, we consider n companies with random tends to the finite value 1/a.
wealth. Can they gain simultaneously by trading risks?
The class of Pareto optimal exchanges is discussed and Example 2
characterized by the theorem of Borch. Two more spe- Power utility function of the first kind (parameters
cific solutions are proposed. The first idea is to with- s . 0, c . 0)
draw the synergy potential, which is the largest
amount that can be withdrawn from the system of the s c11 2 (s 2 x)c11
u(x) 5 , x , s. (2)
n companies without hurting any of them. Then this (c 1 1)s c
amount is reallocated to the companies in an unam-
Obviously this expression cannot serve as a model be-
biguous fashion. The second idea, as presented by
yond x 5 s. The only way to extend the definition be-
Bühlmann (1980, 1984), is to consider a competitive
yond this point so that u(x) is a nondecreasing and
equilibrium, in which random payments can be bought
concave function is to set u(x) 5 s/(c 1 1) for x $
in a market. Here the equilibrium price density plays
s. In this sense s can be interpreted as a level of sat-
a crucial role. In Section 11 it is shown how options
uration: the maximal utility is already attained for the
can be priced by means of the equilibrium price den-
finite wealth s. The special case c 5 1 is of particular
sity. This approach differs from chapter 4 of Panjer et
interest. Then
al. (1998), which considers the utility of consumption
and assumes the existence of a representative agent. x2
u(x) 5 x 2 , x,s (3)
2s
2. UTILITY FUNCTIONS is a quadratic utility function.
Often it is not appropriate to measure the usefulness
of money on the monetary scale. To explain certain Example 3
phenomena, the usefulness of money must be mea- Power utility function of the second kind (parameter
sured on a new scale. Thus, the usefulness of $x is c . 0). For c Þ 1 we set
u(x), the utility (or ‘‘moral value’’) of $x. Typically, x
x 12c 2 1
is the wealth or a gain of a decision-maker. u(x) 5 , x . 0. (4)
12c
We suppose that a utility function u(x) has the fol-
lowing two basic properties: For c 5 1 we define
(1) u(x) is an increasing function of x
(2) u(x) is a concave function of x. u(x) 5 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, VOLUME 2, NUMBER 3

Remark 1 Here the risk aversion increases with wealth and be-
A utility function u(x) can be replaced by an equiva- comes infinite for x → s; this has the following inter-
lent utility function of the form pretation: if the wealth is close to the level of satu-
ration s, very little utility can be gained by a monetary
ũ(x) 5 Au(x) 1 B (6) gain; hence there is no point in taking any risk.
(with A . 0 and B arbitrary). Hence it is possible to For the power utility function of the second kind (pa-
standardize a utility function, for example, by requir- rameter c . 0), we obtain
ing that c
r(x) 5 , x . 0. (11)
u(j) 5 0, u9(j) 5 1 (7) x

for a particular point j. In Examples 1 and 2 this has Here the risk aversion is a decreasing function of
been done for j 5 0; in Example 3 it has been done wealth, which may be typical for some investors.
for j 5 1. If u(x) is replaced by an equivalent utility function
as in (6), the associated risk aversion function is the
Remark 2 same. In the opposite direction, if we are given the
If we take the limit a → 0 in Example 1, or s → ` in risk aversion function r(x) and want to find an under-
Example 2, we obtain u(x) 5 x, a linear utility func- lying utility function, we look for a function u(x) that
tion, which is not a utility function in the proper satisfies the equation
sense. Similarly, the limit c → 0 in Example 3 is u0(x) 1 r(x)u9(x) 5 0. (12)
u(x) 5 x 2 1.
Such a differential equation has a two-parameter fam-
Remark 3 ily of solutions. To get a unique answer, we may stan-
In the following we tacitly assume that x , s if u(x) dardize according to (7) for some j. Then the solution
is a power utility function of the first kind, and that is
x . 0 if u(x) is a power utility function of the second
kind. The analogous assumptions are made when we u(x) 5 E exp F2E r( y)dyG dz.
x

j
z

j
(13)
consider the utility of a random variable.
Now suppose that r1(x) and r2(x) are two risk aver-
sion functions with
3. RISK AVERSION FUNCTIONS
r1(x) # r2(x) for all x. (14)
To a given utility function u(x) we associate a function
Let u1(x) and u2(x) be two underlying utility func-
2u0(x) d
r(x) 5 52 ln u9(x), (8) tions. Because of their ambiguity, they cannot be com-
u9(x) dx
pared without making any further assumptions. If we
called the risk aversion function. We note that prop- assume however, that u1(x) and u2(x) are standard-
erties (1) and (2) imply that r(x) . 0. Let us revisit ized at the same point j, that is,
the three examples of Section 2.
ui(j) 5 0, u9(j) 5 1, i 5 1, 2, (15)
For the exponential utility function (parameter a . i

0), we find that then it follows that

r(x) 5 a, 2` , x , `. (9) u1(x) $ u2(x) for all x. (16)
Thus the exponential utility function yields a constant For the proof we observe that

E exp F2E r ( y)dyG dz,

risk aversion. x z
For the power utility function of the first kind (par- ui(x) 5 i if x . j,
ameters s . 0, c . 0), we find that j j

u (x) 5 2E exp FE r ( y)dyG dz,

j j
c
r(x) 5 , x , s. (10) i i if x , j,
s2x x z

and use the assumption (14).

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UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 77

4. PREFERENCE ORDERING OF RANDOM u(w 1 p) 5 E[u(w 1 G)]. (21)

GAINS From (20) we see that p , E[G ]. Let us consider
Consider a decision-maker with initial wealth w who two examples in which explicit expressions for p can
has the choice between a certain number of random be obtained:
gains. By using a utility function, two random gains For an exponential utility function, the certainty
can be directly compared: he or she prefers G1 to G2, equivalent is
if 21
p5 ln E[e2aG]. (22)
E[u(w 1 G1)] . E[u(w 1 G2)], (17) a

that is, if the expected utility from G1 exceeds the Note that it does not depend on w. By expanding this
expected utility from G2. If the expected utilities are expression in powers of a, we obtain the simple ap-
equal, he will be indifferent between G1 and G2. Thus proximation
a complete preference ordering is defined on the set
a
of random gains. p < E[G] 2 Var[G], (23)
2
If we multiply (17) by a positive constant A and add
a constant B on both sides, an equivalent inequality valid for sufficiently small values of a.
in terms of the function ũ(x) is obtained. Hence u(x) For a quadratic utility function, condition (21) leads
and ũ(x) define the same ordering and are considered to a quadratic equation for p. Its solution can be writ-
to be equivalent. ten as follows:

Example 4 p 5 E[G] 2 (s 2 w 2 E[G])l

Suppose that the decision-maker uses the exponential with
utility function with parameter a and has the choice
between two normal random variables, G1 and G2, with
E[Gi] 5 mi, Var[Gi] 5 s 2i , i 5 1, 2. Since
l5 !1 1 (s 2 Var[G]
w 2 E[G])
2 1.2
(24)

E[e2aGi] 5 exp S
2ami 1
1 2 2
2
a si , D For large values of s, we can expand the square root
and find the approximation
1 Var[G]
it follows that p < E[G] 2 . (25)
2 (s 2 w 2 E[G])
E[u(w 1 Gi)]

F S DG
In view of (10) we can write this formula as
1 1
5 1 2 exp 2aw 2 ami 1 a 2s 2i . 1
a 2 p < E[G] 2 r(w 1 E[G]) Var[G], (26)
2
Hence G1 is preferred to G2, if (17) is satisfied, that
is, if which is similar to formula (23).
For a general utility function, it follows from (21)
1 1 that
m1 2 as 21 . m2 2 as 22. (19)
2 2
p 5 u21(E[u(w 1 G)]) 2 w. (27)
Jensen’s inequality tells us that for any random vari-
If G is a gain with a ‘‘small’’ risk, the following more
able G,
explicit approximation is available:
u(w 1 E[G]) . E[u(w 1 G)]. (20)
1
p < E[G] 2 r(w 1 E[G]) Var[G]. (28)
Hence, if a decision-maker can choose between a ran- 2
dom gain G and a fixed amount equal to its expecta-
tion, he will prefer the latter. This brings us to the To give a precise meaning to this statement, we set
following definition: The certainty equivalent, p, as- Gz 5 m 1 zV, z.0 (29)
sociated to G is defined by the condition that the de-
cision-maker is indifferent between receiving G or the where m is a constant and V a random variable with
fixed amount p. Mathematically, this is the condition E[V ] 5 0 and Var[V] 5 E[V 2] 5 s 2. Hence E[Gz] 5
that m and Var[Gz] 5 z 2s 2. Let p(z) be the certainty equiv-
alent of Gz, defined by the equation
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78 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

u(w 1 p(z)) 5 E[u(w 1 Gz)]. (30) u2(w 1 p1) 5 0

The idea is to expand the function p(z) in powers of 5 u1(w 1p1)
z:
5 E[u1(w 1 G)]
p(z) 5 a 1 bz 1 cz 2 1 . . . (31)
$ E[u2(w 1 G)]
If we set z 5 0 in (30), we obtain
u(w 1 a) 5 u(w 1 m), or a 5 m. (32) 5 u2(w 1 p2). (42)

If we differentiate (30), we get the equation Since u2 is an increasing function, it follows indeed
that p1 $ p2.
p9(z)u9(w 1 p(z)) 5 E[Vu9(w 1 Gz)]. (33)
Setting z 5 0 yields 5. PREMIUM CALCULATION
bu9(w 1 m) 5 E[V] u9(w 1 m) 5 0, We consider a company with initial wealth w. The
company is to insure a risk and has to pay the total
or b 5 0. (34)
claims S (a random variable) at the end of the period.
Finally we differentiate (33) to obtain What should be the appropriate premium, P, for this
contract? An answer is obtained by assuming a utility
p0(z)u9(w 1 p(z)) 1 p9(z)2u0(w 1 p(z))
function, u(x), and by postulating fairness in terms of
5 E[V 2u0(w 1 Gz)]. (35) utility. This means that the expected utility of wealth
with the contract should be equal to the utility with-
Setting z 5 0 we obtain
out the contract:
2c u9(w 1 m) 5 E[V 2] u0(w 1 m), (36)
E[u(w 1 P 2 S)] 5 u(w). (43)
or
This is called the principle of equivalent utility. Equa-
1 u0(w 1 m) 2 1 tion (43) determines P uniquely, but has no explicit
c5 s 5 2 r(w 1 m)s 2. (37)
2 u9(w 1 m) 2 solution in general. Notable exceptions are the cases
in which u(x) is exponential, where
Substitution in (31) yields the approximation
1
1 P5 ln E[eaS], (44)
p(z) < m 2 r(w 1 m) z 2s 2 a
2
or quadratic, where we find that

H 2 w) J
1
5 E[Gz] 2 r(w 1 E[Gz])Var[Gz],
!1 2 (sVar[S]
(38)
2 P 5 E[S] 1 (s 2 w) 12 . (45)
2

which explains (28).

Let us now consider two utility functions u1(x) and If S is a ‘‘small’’ risk, (43) can be solved approximately
u2(x) so that as follows:

r1(x) # r2(x) for all x, (39) P < E[S] 1

1
r(w)Var[S] (46)
2
and let p1 and p2 denote their respective certainty
equivalents. Then we expect that (to see this, set S 5 m 1 zV, with E[V] 5 0, and
expand P in powers of z).
p1 $ p2. (40)
In many cases a more realistic assumption is that
To verify this result, we assume that the underlying the wealth without the new contract is a random vari-
utility functions are standardized at the same point able itself, say W. Then P is obtained from the equa-
j 5 w 1 p1. Then tion

u1(x) $ u2(x) for all x. (41) E[u(W 1 P 2 S)] 5 E[u(W )]. (47)

From this and the definitions of p1 and p2, it follows Note that now P depends on the joint distribution of
that S and W.
Let us revisit the examples in which P can be cal-
culated explicitly.
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UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 79

Example 5 (2) A general reinsurance contract, given by a func-

If u(x) 5 (1 2 e 2ax
)/a, we find that tion h(x), where the reinsurer pays h(S ) at the
end of the year. The only restriction on the func-
1 E[ea(S2W )] tion h(x) is that
P5 ln . (48)
a E[e2aW]
0 # h(x) # x. (54)
If a is small, we can expand this expression in powers
of a and obtain the approximation We assume that the two contracts are comparable,
in the sense that the expected payments of the rein-
a a surer are the same, that is, that
P < E[S] 1 Var[S 2 W] 2 Var[W] (49)
2 2
E[(S 2 d)1] 5 E[h(S)]. (55)
a
5 E[S] 1 Var[S] 2 a Cov(S,W ). (50) Furthermore, we make the convenient (but perhaps
2 not realistic) assumption that the two reinsurance
We note that (48) reduces to (44) in the case in which premiums are the same. Then, in terms of utility, the
S and W are independent random variables. Also, we stop-loss contract is preferable:
remark that (49) is exact in the case where S and W E[u(w 2 S 1 h(S))] # E[u(w 2 S 1 (S 2 d)1)].
are bivariate normal.
(56)
Example 6
In this context, w represents the wealth after receipt
If u(x) 5 x 2 x 2/2s, we find that of the premiums and payment of the reinsurance pre-
P 5 E[S] 1 (s 2 E[W])l miums.
The proof of (56) starts with the observation that a
with concave curve is below its tangents, that is, that

!1 2 Var[S](s 22 E[W])
2 Cov(S,W ) u( y) # u(x) 1 u9(x)( y 2 x) for all x and y.
l512 .
2
(51)
(57)
Note that this expression reduces to (45) with w re-
Using this for y 5 w 2 S 1 h(S ), x 5 w 2 S 1 (S 2
placed by E[W ], in the case in which S and W are
d)1, we get
uncorrelated random variables. For large values of s,
(51) leads to the approximation u(w2S1h(S)) # u(w2S1(S2d)1)
1 Var[S] 2 2 Cov(S,W ) 1 u9(w2S1(S2d)1)(h(S)2(S2d)1)
P < E[S] 1
2 s 2 E[W]
# u(w2S1(S2d)1) 1 u9(w2d)(h(S)2(S2d)1).
1 (58)
5 E[S] 1 r(E[W]){Var[S] 2 2 Cov(S,W )}.
2
To verify the second inequality, distinguish the cases
(52) S . d, in which equality holds, and S # d, where
u9(w 2 S 1 (S 2 d)1)(h(S) 2 (S 2 d)1)
6. OPTIMALITY OF A STOP-LOSS
5 u9(w 2 S)h(S)
CONTRACT
We consider a company that has to pay the total # u9(w 2 d)h(S)
amount S (a random variable) to its policyholders at 5 u9(w 2 d)(h(S) 2 (S 2 d)1).
the end of the year. We compare two reinsurance
agreements: Now we take expectations in (58) and use (55) to ob-
(1) A stop-loss contract with deductible d. Here the tain (56).
reinsurer will pay

(S 2 d)1 5 H
S 2 d if S . d
0 if S # d
(53)
7. OPTIMAL DEGREE OF REINSURANCE
Again we consider a company that has to pay the total
at the end of the year. amount S (a random variable) at the end of the year.
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80 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

A proportional reinsurance coverage can be pur- that is, the total wealth remains the same. The value
chased. If P is the reinsurance premium for full cov- for company i of such an exchange is measured by
erage (of course P . E[S ]), we assume that for a pre-
E[ui(Xi )].
mium of wP the fraction wS is covered and will be
reimbursed at the end of the year (0 # w # 1.) Then A risk exchange (X˜ 1, . . . , X˜ n) is said to be Pareto
w̃, the optimal value of w, is the value that maximizes optimal, if it is not possible to improve the situation
of one company without worsening the situation of at
E[u(w 2 wP 2 (1 2 w)S)], (59)
least one other company. In other words, there is no
where u(x) is an appropriate utility function and other exchange (X1, . . . , Xn) with
where the initial surplus, w, includes the premiums
E[ui(Xi)] $ E[ui(X˜ i )], for i 5 1, . . . , n
received. In the particular case in which u(x) is the
exponential utility function with parameter a, and S whereby at least one of these inequalities is strict. If
has a normal distribution with mean m and variance the companies are willing to cooperate, they should
s 2, the calculations can be done explicitly. The ex- choose a risk exchange that is Pareto optimal.
pected utility is now The Pareto optimal risk exchanges constitute a fam-
ily with n 2 1 parameters. They can be obtained by
1
(1 2 E{exp[2aw 1 awP 1 a(1 2 w)S]}) the following method: Choose k1 . 0, . . . , kn . 0 and
a then maximize the expression

5
1
a F 1
exp 2aw1awP1a(12w)m 1 a2(12w)2s 2 .
2 G O k E[u (X )],
n

i51
i i i (62)

It is maximal for
where the maximum is taken over all risk exchanges
P2m (X1, . . . , Xn). This problem has a relatively explicit
1 2 w̃ 5 . (60)
as 2 solution:
This result has an appealing interpretation. The opti- Theorem 1 (Borch)
mal fraction that is retained is proportional to the
A risk exchange (X˜ 1, . . . , X˜ n) maximizes (62) if and
loading contained in the reinsurance premium for full ˜ i ) are the same for
only if the random variables kiu9(X i
coverage, and inversely proportional to the company’s
i 5 1, . . . , n.
risk aversion and the variance of the total claims.
In finance, a formula similar to (60) is known as the
Proof
Merton ratio, see Panjer et al. (1998, Ch. 4). The dif-
ference is that for Merton’s formula, the utility func- (a) Suppose that (X˜ 1, . . . , X˜ n) maximizes (62). Let
tion is a power utility function and S is lognormal, j Þ h and let V be an arbitrary random variable. We
while here the utility function is exponential and S is define
normal. Xi 5 X˜ i, for i Þ j, h,

Xj 5 X˜ j 1 tV,
8. PARETO OPTIMAL RISK EXCHANGES
We consider n companies (or economic agents). We Xh 5 X˜ h 2 tV,
assume that company i has a wealth Wi at the end of where t is a parameter. Let
the year and bases its decisions on a utility function
O k E[u (X )].
n
ui(x). Here W1, . . . , Wn are random variables with a f(t) 5 (63)
i i i
known joint distribution. Let W 5 W1 1 . . . 1 Wn i51
denote the total wealth of the companies. A risk
According to our assumption, the function f (t) has a
exchange provides a redistribution of total wealth.
maximum at t 5 0. Hence f 9(t) 5 0, or
Thus after a risk exchange, the wealth of company i
will be Xi; here X1, . . . , Xn can be any random variables ˜ j)] 2 khE[Vu9(X
kjE[Vu9(X
j h
˜ h)] 5 0. (64)
provided that
It is useful to rewrite this equation as
X1 1 . . . 1 Xn 5 W, (61)
j
˜ j) 2 khu9(X
E[V{kju9(X h
˜ h)}] 5 0. (65)
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UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 81

Since this holds for an arbitrary V, we conclude that Then it follows from (71) that
˜ j) 2 khu9(X
˜ h) 5 0.
O lna k .
n
kju9(X
j h (66) j
2ln L 5 aW 2 a (73)
This shows indeed that kiu9(X i
˜ i ) is independent of i. j51 j

(b) Conversely, let (X˜ 1, . . . , X˜ n) be a risk exchange Substitution in (70) yields

O lna k
so that n
a ln ki a j
˜ i) 5 L X̃i 5 W1 2 (74)
kiu9(X
i (67) ai ai ai j51 j

is the same random variable for all i. Let (X1, . . . , Xn) for i 5 1, . . . , n. Thus company i will assume the
be any other risk exchange. From (57) it follows that fraction (or quota) qi 5 a/ai of total wealth W plus a
ui(Xi ) # ui(X˜ i ) 1 u9(X
˜ i )(Xi 2 X˜ i ). (68) possibly negative side payment
i

O lna k .
n
If we multiply this inequality by ki, sum over i and use ln ki a j
di 5 2 (75)
(67), we get ai ai j51 j

O O O (X 2 X˜ ) It is easily verified that

n n n
kiui(Xi ) # kiui(X˜ i ) 1 L i i
i51 i51 i51 q1 1 . . . 1 qn 5 1 (76)

5 O k u (X˜ ).
n
and
i i i
i51
d1 1 . . . 1 dn 5 0. (77)
Hence We note that the qi’s are inversely proportional to the

O k E[u (X )] # O k E[u (X˜ )]. risk aversions and that they are the same for all Pareto
n n

i51
i i i
i51
i i i optimal risk exchanges. Pareto optimal risk exchanges
differ only by their side payments.
This shows that expression (63) is indeed maximal for
(X˜ 1, . . . , X˜ n). M Example 8
Suppose now that all companies use a power utility
Example 7 function of the first kind, such that
Suppose that all companies use an exponential utility
function, s c11
j 2 (sj 2 x)c11
uj(x) 5 , j 5 1, . . . , n, (78)
(c 1 1)s cj
1
uj(x) 5 [1 2 exp(2aj x)], where sj is the level of saturation of company j. From
aj
(67) we get

S D
where aj is the constant risk aversion of company j, c
j 5 1, . . . , n. From (67), we get X̃j
kj 12 5L (79)
sj
kj exp(2aj X˜ j) 5 L (69)
or
or
sj
ln L ln kj X̃j 5 2 1/c L1/c 1 sj. (80)
X̃j 5 2 1 . (70) kj
aj aj
Summing over j, we obtain an equation which deter-
Summing over j, we obtain an equation that deter-
mines L:
mines L:

O ks O s.
n n

O O
n n j
1 ln kj W52 L1/c 1 (81)
W52 ln L 1 . (71) j51
1/c
j j51
j
j51 aj j51 aj
Let
Let us introduce a, which is defined by the equation
s 5 s1 1 . . . 1 sn (82)
1 1 1
5 1...1 . (72)
a a1 an
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82 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

denote the combined level of saturation. Then it fol- If we substitute this in (89) we see that
lows from (81) that
X̃i 5 qiW (92)
s2W
L1/c 5 n
O
. (83) with
sj
1/c
j51 kj
k1/c
qi 5 i

O
n . (93)
Substitution in (80) yields k1/c
j
j51
si si
Hence each investor assumes a fixed quota of total
k1/c k1/c
X̃i 5 n i W 1 si 2 i
wealth. As in the case of power utility functions of the
O O
n s (84)
sj sj
first kind, the quotas vary, but now there are no side
1/c 1/c
j51 kj j51 kj payments.
for i 5 1, . . . , n. Hence again X̃i is of the form
Example 10
X̃i 5 qiW 1 di. (85) Let n 5 2. Suppose that u1(x) 5 x and u2(x) 5 u(x),
But note that now both the quotas and the side pay- a utility function in the proper sense with u0(x) , 0.
ments vary, such that Then condition (67) tells us that

di 5 si 2 qi s. (86) k1 5 k2u9(X˜ 2).

If we write this result in the form But this means that X̃2 is a constant, say d. Hence
X̃1 5 W 2 d. This result is not really surprising: since
si 2 X˜ i 5 qi(s 2 W ), (87) company 1 is not risk averse, it will assume all the
it has the following interpretation: The expression si 2 risk!
X̃i is the amount that is missing for maximal satisfac- We have presented selected examples in which the
tion. It is a fixed percentage of s 2 W, which is the Pareto optimal risk exchanges are of an attractively
total amount missing for all companies combined. simple form. In general, this is not the case. The fol-
lowing example illustrates the point.
Example 9
Example 11
Consider n investors with identical power utility func-
tions of the second kind Let n 5 2. Suppose that u1(x) and u2(x) are power
utility functions of the second kind with parameters
x12c 2 1 c1 5 1 and c2 5 2, that is, that
uj(x) 5 , j 5 1, . . . , n.
12c
1
From (67), we see that u1(x) 5 ln x, u2(x) 5 1 2 for x . 0.
x
kj X˜ j2c 5 L (88) From (67) we obtain the condition that
or 1 ˜ 1 ˜ 2
X 5 (X ) . (94)
X̃j 5 k 1/c
j L 21/c
. (89) k1 1 k2 2

Summing over j, we get Together with the condition that X˜ 1 1 X˜ 2 5 W, this

results in a quadratic equation. Its solution is
Ok
n
W5 1/c
j L21/c, (90) 1
j51 X̃1 5 W 2 (Ïa2 1 4aW 2 a), (95)
2
or
1
W X̃2 5 (Ïa2 1 4aW 2 a), (96)
L21/c 5 2
Ok
n . (91)
1/c
j with a 5 k2/k1. Here X˜ 1 and X˜ 2 are obviously not linear
j51
functions of W.
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UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 83

Example 7 helps us to understand the Pareto opti- aqjW 1 adj 1 bj

mal risk exchanges in the general case. Let u1(x), dX˜ j 5 dw (106)
aW 1 b
. . . , un(x) be arbitrary utility functions, and let (X˜ 1,
. . . , X̃n) be a Pareto optimal risk exchange. For given To see when this ratio is equal to qj, we distinguish
W 5 w, (X˜ 1, . . . , X˜ n) maximizes expression (62). two cases:
Hence X˜ i 5 X˜ i(w) is a function of the total wealth w. (1) If b Þ 0, it suffices to set
Let j Þ h. According to Theorem 1 adj 1 bj
qj 5 , j 5 1, . . . , n.
˜ j(w)) 5 khu9(X
kju9(X
j h
˜ h(w)). (97) b

Differentiation with respect to w yields (2) If b 5 0, q1, . . . , qn are arbitrary quotas, and the
side payments are fixed:
dX˜ j ˜
˜ j(w))
kju0(X ˜ h(w)) dXh.
5 khu0(X (98)
j
dw h
dw bj
dj 5 2 , j 5 1, . . . , n.
a
Dividing (98) by (97) we see that
Theorem 1 tells us that for a Pareto optimal risk
dX˜ j dX˜ h
rj(X˜ j(w)) 5 rh(X˜ h(w)) . (99) exchange (X˜ 1, . . . , X˜ n), there is a random variable L
dw dw such that
From this and the observation that L 5 kiu9(X
i
˜ i ), for i 5 1, . . . , n. (107)
dX˜ 1 1 . . . 1 dX˜ n 5 dw, (100) Since X˜ i 5 X˜ i(w) is a function of total wealth w, it
it follows that follows that L 5 L(w) is a function of w. Differenti-
ating (107), we get
1
L9 5 kiu0(X
˜ i )X9.
˜i (108)
rj(X˜ j) i
dX˜ j 5 n j 5 1, . . . , n.
O
dw, (101)
1 Dividing this equation by (107) and using (101), we
h51 rh(Xh)
˜ obtain

Thus the family of Pareto optimal risk exchanges can L9 1

52
O
. (109)
be obtained as follows. For a particular value of w, say L n
1
w0, we can choose X˜ 1(w0), . . . , X˜ n(w0). Then h51 rh(X˜ h)
X˜ 1(w), . . . , X˜ n(w) are determined as the solution of
This shows that L is a decreasing function of total
(101), subject to the boundary condition at w 5 w0.
wealth.
As an application of (101), we revisit Examples 8
and 9. For a unified treatment, we suppose that
1 9. THE SYNERGY POTENTIAL
5 ax 1 bj, j 5 1, . . . , n. (102) We consider the n companies introduced in the pre-
rj(x)
ceding section and assume that (W1, . . . , Wn), the
We want to verify that a Pareto optimal risk exchange allocation of their total wealth W, is not Pareto opti-
is of the form mal. How much can the companies gain through co-
X̃j 5 qjW 1 dj, (103) operation?
An answer is provided by the synergy potential h.
or equivalently, This is the largest amount x that can be extracted
dX˜ j 5 qj dw (104) from the system without hurting any of the compa-
nies, that is, such that there is a risk exchange (X1,
for a set of quotas q1, . . . , qn and side payments d1, . . . , Xn) with
. . . , dn. From (101) and (102) we obtain
X1 1 . . . 1 Xn 5 W 2 x (110)
aX˜ j 1 bj aX˜ j 1 bj
dX˜ j 5 dw 5
O
n dw (105) and
aW 1 b
(aX˜ h 1 bh)
h51 E[ui(Xi )] $ E[ui(Wi )], for i 5 1, . . . , n. (111)

with b 5 b1 1 . . . 1 bn. Hence, by (103)

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84 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

It is clear that for x 5 h we must have equality in Example 14 (continued from Example 9)
(111) and (X˜ 1, . . . , X˜ n) must be a Pareto optimal risk Suppose that each of n investors has a power utility
exchange of W 2 h. function of the second kind. Hence

Example 12 (continued from Example 7) X̃i 5 qi(W 2 h). (120)

Suppose that all utility functions are exponential. From
Since (X˜ 1, . . . , X˜ n) is a Pareto optimal risk exchange
of W 2 h, it follows that E[ui(X˜ i )] 5 E[ui(Wi )]

a we get
X̃i 5 (W 2 h) 1 di, for i 5 1, . . . , n. (112)
ai q12c
i E[(W 2 h)12c] 5 E[W 12c
i ] if c Þ 1, (121)
Then we use the condition that and
E[ui(X˜ i )] 5 E[ui(Wi )] (113) ln qi 1 E[ln(W 2 h)] 5 E[ln Wi] if c 5 1. (122)
to see that Taking the (1 2 c)-th root in (121) and summing over
E[e 2aW
]e ah2aidi
5 E[e 2aiWi
], (114) i, we obtain the equation

O E[W
n
or E[(W 2 h)12c]1/(12c) 5 12c 1/(12c)
] , (123)
i
i51
a 1 1
h 2 di 5 ln E[e2aiWi] 2 ln E[e2aW]. (115)
ai ai ai which determines h if c Þ 1. By exponentiating (122)
and summing over i we obtain the equation
Summation over i yields an explicit expression for the

Oe
n
synergy potential:
eE[ln( W2h)] 5 E[ln Wi]
, (124)
O
n i51
1 1
h5 ln E[e2aiWi] 2 ln E[e2aW]
i51 ai a which determines h if c 5 1.

P E[e
n
2aiWi 1/ai
] Example 15
5 ln i51
. (116) In the situation of Example 10, equality of the ex-
E[e2aW]1/a pected utilities implies that

Example 13 (continued from Example 8) X̃1 5 W 2 E[W2] (125)

We assume that the companies use power utility func- and X̃2 5 d, where
tions of the first kind. According to (87)
u(d) 5 E[u(W2)]. (126)
si 2 X˜ i 5 qi(s 2 W 1 h). (117)
Thus d 5 p, the certainty equivalent of W2. It follows
From that
E[ui(X˜ i)] 5 E[ui(Wi )] h 5 W 2 (X˜ 1 1 p)
it follows that 5 E[W2] 2 p. (127)
q c11
i E[(s 2 W 1 h)c11] 5 E[(si 2 Wi )c11]. (118) We can use the synergy potential to construct a par-
Taking the (c 1 1)-th root and summing over i, we ticular Pareto optimal risk exchange. The idea is to
get first extract h from the companies and then to dis-
tribute h to the companies according to (101). The
O E[(s 2 W )
n
E[(s 2 W 1 h)c11]1/(c11) 5 c11 1/(c11)
] . resulting Pareto optimal risk exchange (X˜ 1, . . . , X˜ n)
i i
i51 of W is characterized by the condition that
(119) E[ui(X˜ i(W 2 h))] 5 E[ui(Wi )], for i 5 1, . . . , n.
This is an implicit equation for the synergy potential (128)
h.
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UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 85

Example 16 (continued from Example 12) eE[ln Wi]

qi 5 E[ln( W2h)]
, if c 5 1. (134)
In the case of exponential utility functions we have e

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

X̃i 5 W 1 di.
ai E[W 12c]1/(12c)
qi 5 i
if c Þ 1,
O E[W
n , (135)
To determine the side payments, we substitute this 12c 1/(12c)
j ]
expression in (128) to see that j51

E[e2aW] eah2aidi 5 E[e2aiWi]. and

From this it follows that eE[ln Wi]
qi 5 if c 5 1.
Oe
n , (136)
a 1 1 E[ln Wj]
di 5 h 1 ln E[e2aW] 2 ln E[e2aiWi]. (129) j51
ai ai ai
Substituting for h, we obtain finally the result that In the Appendix we derive Hölder’s inequality and
Minkowski’s inequality as a by-product of Examples 12
O a1 ln E[e
n
a 2ajWj
1 and 14.
di 5 ]2 ln E[e2aiWi],
ai j51 j ai

for i 5 1, . . . , n. (130) 10. MARKET AND EQUILIBRIUM

Again we consider the n companies that were intro-
Example 17 duced in Section 8. We concluded that the companies
For power utility functions of the first kind, we found should settle on a Pareto optimal risk exchange. Be-
that a Pareto optimal risk exchange (X˜ 1, . . . , X˜ n) is cause this is a rich family, more definite answers are
such that desirable. In the last section we proposed a particular
si 2 X˜ i 5 qi(s 2 W ). Pareto optimal risk exchange. In this section an alter-
native proposal, due to Borch and Bühlmann, is dis-
From this and (128), we obtain the condition that cussed, which is based on economic ideas.
We suppose that random payments are traded in a
q c11
i E[(s 2 W 1 h)c11] 5 E[(si 2 Wi)c11].
market, whereby the price H(Y) for any payment Y (a
Thus random variable) is calculated as

qi 5 S E[(si 2 Wi )c11]
E[(s 2 W 1 h)c11] D1/(c11)

. (131)
H(Y) 5 E[CY].
Here C is a positive random variable. We assume that
(137)

Finally, we use (119) to get an explicit formula for the H(Y ) represents the price as of the end of the year.
resulting quota: Hence the price of a constant payment must be iden-
tical to this constant. Therefore we must have E[C] 5
E[(si 2 Wi )c11]1/(c11)
qi 5 for i 5 1, . . . , n. 1. By writing the right-hand side of (137) as E[Y ] 1
O E[(s 2 W )
n ,
j j
c11 1/(c11)
] E[CY] 2 E[C]E[Y], we see that the price of Y can
j51 also be written in the form
(132) H(Y) 5 E[Y] 1 Cov(Y, C), (138)

Example 18 that is, the price of a payment is its expectation mod-

ified by an adjustment that takes into account the
For power utility functions of the second kind, we
market conditions. Alternatively, we can interpret the
found that X̃i 5 qiW. From condition (128), we see
price of a payment as its expectation with respect to
that

S D
a modified probability measure, Q, that is defined by
1/(12c)
E[W 12c] the relation
qi 5 i
, if c Þ 1, (133)
E[(W 2 h)12c]
EQ[Y] 5 E[CY] for all Y. (139)
and
In other words, C is the Radon-Nikodym derivative of
the Q-measure with respect to the original probability
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86 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

measure. For this reason Bühlmann (1980, 1984) which completes the proof.
calls C a price density. We note that the optimal Ỹi is unique apart from an
Company i will want to buy a payment Yi in order additive constant; hence Y˜ i 2 H(Y˜ i) is unique. It can
to be interpreted as the optimal payment that has a
zero price, and we refer to it as the net demand of
maximize E[ui(Wi 1 Yi 2 H(Yi ))]. (140)
company i.
A payment Ỹi solves this problem if and only if the Given C, the random variable

O [Y˜ 2 H(Y˜ )]
condition n
(146)
i 1 Yi 2 H(Yi )) 5 CE[u9(Wi 1 Yi 2 H(Yi ))]
u9(W
i
˜ ˜ i
˜ ˜ i i
i51

(141) is the excess demand. The companies can maximize

is satisfied. simultaneously their expected utilities only if the ex-
To see the necessity of this condition, suppose that cess demand vanishes (this is the market clearing con-
Ỹi is a solution of (140). Let V be an arbitrary random dition). This leads us to the following definition.
variable; we consider the family A price density C and the payments Y˜ 1, . . . , Y˜ n
constitute an equilibrium, if (146) vanishes and if
Yi 5 Y˜ i 1 tV. (141) is satisfied for i 5 1, . . . , n.
According to our assumption, the function Note that an equilibrium induces a risk exchange
(X˜ 1, . . . , X˜ n), with
f(t) 5 E[ui(Wi 1 Yi 2 H(Yi ))]
X˜ i 5 Wi 1 Y˜ i 2 H(Y˜ i ), for i 5 1, . . . , n. (147)
is maximal for t 5 0. Hence
Then condition (141) states that
f9(0) 5 E[u9(W
i i 1 Yi 2 H(Yi ))(V 2 E[CV])] 5 0.
˜ ˜
˜ i) 5 CE[u9(X
u9(X
i i
˜ i)], for i 5 1, . . . , n. (148)
(142)
From this and Theorem 1 it follows that the risk
We rewrite this equation as exchange implied by an equilibrium is Pareto optimal.
Furthermore, (109) is satisfied with L 5 C. In partic-
i 1 Yi 2 H(Yi ))
E[V{u9(W
i
˜ ˜
ular, this shows that C is a decreasing function of to-
2 CE[u9(W
i i 1 Yi 2 H(Yi ))]}] 5 0.
˜ ˜ (143) tal wealth.
Since it is valid for all V, the random variable inside The converse is true in the following sense. Suppose
the braces must be zero, and condition (141) follows. that (W1, . . . , Wn) is already Pareto optimal; then W1,
To see that condition (141) is sufficient, consider a . . . , Wn and C constitute an equilibrium, if we set
payment Ỹ that satisfies (141) and any other payment u9(W i)
Y. From (57) it follows that C5 i
. (149)
E[u9(W
i i )]
ui(Wi1Yi2H(Yi )) Moreover,
# ui(Wi1Y˜ i2H(Y˜ i )) Y˜ i 2 H(Y˜ i ) 5 0 for i 5 1, . . . , n.
1 u9(W
i i1Yi2H(Yi ))(Yi2H(Yi )2Yi1H(Yi ))
˜ ˜ ˜ ˜ This can be seen from (67) (with X̃i replaced by Wi )
and (141).
5 ui(Wi1Y˜ i2H(Y˜ i ))
Example 19 (continued from Example 7)
1 CE[u9(W
i i1Yi2H(Yi ))](Yi2H(Yi )2Yi1H(Yi )).
˜ ˜ ˜ ˜
Assuming that all companies use exponential utility
(144) functions, we gather from (141) that
Taking expectations and using the definition of H, we 1
see that Ỹi 5 2Wi 2 ln C 1 k i, (150)
ai
E[ui(Wi 1 Yi 2 H(Yi ))] # E[ui(Wi 1 Y˜ i 2 H(Y˜ i ))], where k i is a constant. Hence the net demand of com-
(145) pany i is
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UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 87

1 Example 21 (continued from Example 9)

Y˜ i 2 H(Y˜ i) 5 2Wi 2 ln C 1 E[CWi]
ai If all companies use the same power utility function
of the second kind,
1
1 E[C ln C]. (151) u9(x) 5 x2c, i 5 1, . . . , n,
ai i

In the equilibrium the sum over i must vanish. Hence we know that

1 X̃i 5 qiW, i 5 1, . . . , n;
0 5 2W 2 ln C 1 k, (152)
a see (92). Hence the equilibrium price density is
where k is a constant. Since E[C] 5 1, it follows that u9(X ˜ i) W2c
the equilibrium price density is C5 i
5 . (158)
E[u9(X
i
˜ i )] E[W2c]
e2aW Again, the equilibrium quotas are best obtained from
C5 . (153)
E[e2aW] the condition that H(Wi ) 5 H(X˜ i ) 5 qi H( W ). Thus
Finally, a little calculation shows that H(Wi ) E[CWi] E[W2cWi]
qi 5 5 5 ,
X˜ i 5 Wi 1 Y˜ i 2 H(Y˜ i ) H(W ) E[CW] E[W2c11]

a a for i 5 1, . . . , n. (159)
5 W 1 E[CWi] 2 E[CW]
ai ai Remark 4
a a From (138) it follows that for any random variable Y
5 W 1 H(Wi ) 2 H(W ) (154)
ai ai H(Y) 2 E[Y] 5 b(H(W ) 2 E(W )) (160)
in the equilibrium. with
Example 20 (continued from Example 8) Cov(Y, C)
b5 , (161)
We assume that the companies use power utility func- Cov(W, C)
tions of the first kind. Hence where C is now the equilibrium price density. Formula
(si 2 x) c (160) is close to a central result in the capital-asset-
u9(x)
i 5 , i 5 1, . . . , n, pricing model (CAPM). As an illustration, we revisit
sic
our three preceding examples. Thus
and
Cov(Y, e2aW)
si 2 X˜ i 5 qi(s 2 W ), i 5 1, . . . , n; b5 (162)
Cov(W, e2aW)
see (87). Then according to (148) the equilibrium in Example 19,
price density is
Cov(Y, (s 2 W )c)
u9(X ˜ i) (s 2 W )c b5 (163)
C5 i
5 . (155) Cov(W, (s 2 W )c)
E[u9(X
i
˜ i )] E[(s 2 W )c]
in Example 20, and
The equilibrium quotas are best determined from the
condition that H(Wi ) 5 H(X˜ i ), or Cov(Y, W2c)
b5 (164)
Cov(W, W2c)
H(Wi ) 5 H(si 2 qi(s 2 W ))
in Example 21. Note that for c 5 1 (quadratic utility
5 si 2 qi s 1 qi H(W ). (156) functions), (163) reduces to the classical CAPM for-
Hence mula

si 2 H(Wi ) E[C(si 2 Wi )] Cov(Y, W )

qi 5 5 , b5 . (165)
s 2 H(W ) E[C(s 2 W )] Var[W]

for i 5 1, . . . , n. (157)
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88 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

11. PRICING OF DERIVATIVE SECURITIES

In the equilibrium the price of a payment Y is H(Y),
s0 5 e2d s0 EQ[eR] 5 e2d s0 exp SmQ 1
1 2
2
s ,D
given by formulas (137), (138) or (139), where C is (172)
the equilibrium price density. Typically, the random
which yields
variable Y is the value of an asset or a derivative se-
curity at the end of a period. Under certain assump- 1 2
tions, the price of a derivative security can be ex- mQ 5 d 2 s . (173)
2
pressed in terms of the price of the underlying asset.
First, we assume that the random variable C has a Now let us consider a derivative security, whose
lognormal distribution, that is, value at the end of the period is f (S), a function of
the underlying asset. Its price at the beginning of the
C 5 e Z, (166) period is
where Z has a normal distribution, say with variance e2d H( f(S)) 5 e2d EQ[ f(s0 eR)], (174)
n 2. Since

S D
where R is normal with mean given by (173) and var-
1 iance s 2. For example, for a European call option with
E[C] 5 exp E[Z] 1 n 2 (167)
2 strike price K, f(S) 5 (S 2 K )1. Then (174) can be
calculated explicitly, which leads to the Black-Scholes
must be 1, it follows that E[Z ] 5 2(1/2)n 2. According
formula.
to Formulas (153), (155), and (158), the assumption
of lognormality for C means that W is normal in Ex-
Remark 5
ample 19, that s 2 W is lognormal in Example 20, or
that W is lognormal in Example 21. The method can be generalized to price derivative se-
Let us consider a particular asset. We denote its curities that depend on several, say, m assets. Let
value at the end of the period by S and assume that Si 5 si0 eRi, (175)
the random variable S has a lognormal distribution.
Then we can write denote the value at the end of the period of asset i,
where si0 is the observed price of asset i at the begin-
S 5 s0 eR, (168) ning of the period, i 5 1, . . . , m. The assumption is
where s0 is the observed price of the asset at the be- now that (Z, R1, . . . , Rm) has a multivariate normal
ginning of the period, and R has a normal distribution, distribution. Then in the Q-measure (R1, . . . , Rm) has
say, with mean m and variance s 2. We assume that the still a multivariate normal distribution, with un-
joint distribution of (Z, R) is bivariate normal with changed covariance matrix, but modified mean vector,
coefficient of correlation r. Then we obtain the follow- such that
ing expression for the moment-generating function of 1
R with respect to the Q-measure: EQ[Ri] 5 d 2 Var[Ri], for i 5 1, . . . , m. (176)
2
EQ[etR] 5 E[CetR] 5 E[e Z1tR] In the framework of Examples 19–21, practical re-

5 exp F t(m 1 rns) 1

1 2 2
2 G
t s . (169)
sults can also be obtained for derivative securities on
assets for which S is a linear function of W.
In Example 19 suppose that S 5 qW. Then
This shows that in the Q-measure the distribution of
E[Se2aW] E[Se2aS]
R is still normal, with unchanged variance s 2 and new EQ[S] 5 2aW
5 (177)
mean E[e ] E[e2aS]

mQ 5 m 1 rns. (170) with a 5 a/q. According to (171), the value of a is

determined from the condition that
Luckily, there is a more practical expression for mQ.
Since s0 is the price of the asset at the beginning of E[Se2aS]
5 ed s0. (178)
the period, we have E[e2aS]

s0 5 e2d H(S) 5 e2d EQ[S], (171) Then the price of a derivative security with payoff f (S)
is given by the expression
where d is the risk-free force of interest. Hence we
obtain the equation
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 89 # 16

UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 89

E[ f(S)e2aS] Remark 6
e2d EQ[ f(S)] 5 e2d . (179)
E[e2aS] Using (178), we can rewrite (179) as
This is the Esscher method in the sense of Bühlmann. E[ f(S)e2aS]
We note that it also works for assets where S and W 2 e2d EQ[ f(S)] 5 s0 . (187)
E[Se2aS]
S are independent random variables: here
Similarly, (183) can be rewritten as
E[Se2aS e2a( W2S)]
EQ[S] 5 E[ f(S)Sc]
E[e2aS e2a( W2S)] e2d EQ[ f(S)] 5 s0 . (188)
E[S11c]
2aS 2a( W2S) 2aS
E[Se ]E[e ] E[Se ]
5 2aS 2a( W2S)
5 . (180) It may be surprising that d does not appear in these
E[e ]E[e ] E[e2aS]
expressions for the prices, but of course the values of
Hence a is determined from (178) with a replaced a and c are functions of d.
by a.
In Example 20 we suppose that S 5 q(s 2 W ). Then Remark 7
E[S(s2W )c] E[SSc] The Esscher method summarized by Formulas (182)
EQ[S] 5 5 . (181) and (183) has some attractive features. For example,
E[(s 2 W ) ]
c
E[Sc]
if S has a lognormal distribution, it has also a log-
The value of c is determined from the condition that normal distribution in the Q-measure. In particular, it
reproduces the formula of Black-Scholes.
E[S11c]
5 ed s0, (182)
E[Sc]
and the price of a derivative security with payoff f(S)
12. BIBLIOGRAPHICAL NOTES
is given by the expression A broad, less self-contained review has been given by
Aase (1993). The article by Taylor (1992a) is highly
E[ f(S)Sc] recommended.
e2d EQ[ f(S)] 5 e2d . (183)
E[Sc] In Section 5 the premiums are determined by the
In Example 21 we suppose again S 5 qW. Then principle of equivalent utility. If this principle is
adopted in a dynamic model, there is an intrinsic re-
E[SW2c] E[SS2c] lationship between the underlying utility function and
EQ[S] 5 2c
5 . (184)
E[W ] E[S2c] the resulting probability of ruin; see Gerber (1975).
The optimality of a stop-loss contract of Section 6
The value of c is now determined from the condition
seems to have been discovered by Arrow (1963). Its
that
minimal variance property has been discussed by oth-
E[S12c] ers, for example, by Kahn (1961).
5 ed s0, (185)
E[S2c] The theorem of Borch in Section 8 can be found in
the books of Bühlmann (1970) and Gerber (1979). In
and the price of a derivative security with payoff f(S) some of the literature, the family of utility functions
is given by the expression satisfying (102) is called the HARA family (hyperbolic
E[ f(S)S2c] absolute risk aversion).
e2d EQ[ f(S)] 5 e2d . (186) In Sections 9 and 10 we discussed Pareto optimal
E[S2c]
risk exchanges of a specific form. Other proposals
Formula (183) is also acceptable, if c, the solution of have been discussed by Bühlmann and Jewell (1979)
(182) is negative. In this case the solution of (185) is and by Baton and Lemaire (1981). Solutions that are
positive, which leads to (186). But this is again (183), not Pareto optimal have been proposed by Chan and
with a negative c. Formulas (182) and (183) summa- Gerber (1985), Gerber (1984), and Taylor (1992b).
rize the Esscher method that was proposed by Gerber
and Shiu (1994a, 1994b).
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 90 # 17

90 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

GERBER, H.U., AND SHIU, E.S.W. 1994a. ‘‘Martingale Approach

ACKNOWLEDGMENT
to Pricing Perpetual American Options,’’ ASTIN Bulletin
The authors thank two anonymous reviewers for their 24:195–220.
valuable comments. GERBER, H.U., AND SHIU, E.S.W. 1994b. ‘‘Option Pricing by
Esscher Transforms,’’ Transactions of the Society of Actu-
aries XLVI:99–140.
REFERENCES GERBER, H.U., AND SHIU, E.S.W. 1996. ‘‘Actuarial Bridges to Dy-
AASE, K.K. 1993. ‘‘Equilibrium in a Reinsurance Syndicate; Ex- namic Hedging and Option Pricing,’’ Insurance: Mathe-
istence, Uniqueness and Characterisation,’’ ASTIN Bulletin matics and Economics 18:183–218.
23:185–211. KAHN, P. 1961. ‘‘Some Remarks on a Recent Paper by Borch,’’
ARROW, K.J. 1963. ‘‘Uncertainty and the Welfare Economics of ASTIN Bulletin 1:263–72.
Medical Care,’’ The American Economic Review 53:941– LIENHARD, M. 1986. ‘‘Calculation of Price Equilibria for Utility
73. Functions of the HARA Class,’’ ASTIN Bulletin 16C:S91–
BATON, B., AND LEMAIRE, J. 1981. ‘‘The Core of a Reinsurance S97.
Market,’’ ASTIN Bulletin 12:57–71. PANJER, H., (editor), BOYLE, P., COX, S., DUFRESNE, D., GERBER,
BERNOULLI, D. 1954. ‘‘Exposition of a New Theory on the Mea- H., MÜLLER, H., PEDERSEN, H., PLISKA, S., SHERRIS, M., SHIU,
surement of Risk’’ (translation of ‘‘Specimen Theoriae No- E., AND TAN, K. 1998. Financial Economics: With Applica-
vae de Mensura Sortis,’’ St. Petersburg, 1738), Econome- tions to Investments, Insurance and Pensions. Schaum-
trica 22:23–36. burg, Ill.: The Actuarial Foundation.
BLACK, F., AND SCHOLES, M. 1973. ‘‘The Pricing of Options and SEAL, H.L. 1969. Stochastic Theory of a Risk Business. New York:
Corporate Liabilities,’’ Journal of Political Economy 81: Wiley.
637–55. TAYLOR, G.C. 1992a. ‘‘Risk Exchange I: A Unification of Some
BORCH, K. 1960. ‘‘The Safety Loading of Reinsurance Premi- Existing Results,’’ Scandinavian Actuarial Journal, 15–39.
ums,’’ Skandinavisk Aktuarietidskrift 43:163–84. TAYLOR, G.C. 1992b. ‘‘Risk Exchange II: Optimal Reinsurance
BORCH, K. 1962. ‘‘Equilibrium in a Reinsurance Market,’’ Econ- Contracts,’’ Scandinavian Actuarial Journal, 40–59.
ometrica 30:424–444. VON NEUMANN, J., AND MORGENSTERN, O. 1947. The Theory of
BORCH, K. 1974. The Mathematical Theory of Insurance. Lexing- Games and Economic Behavior, 2nd ed. Princeton, N.J.:
ton, Mass.: Lexington Books. Princeton University Press.
BORCH, K. 1990. Economics of Insurance, edited by K.K. Aase
and A. Sandmo. Amsterdam: North-Holland.
BOWERS, N., GERBER, H.U., HICKMAN, J., JONES, D., AND NESBITT, APPENDIX
C. 1997. Actuarial Mathematics, 2nd ed. Schaumburg, Ill.:
Society of Actuaries. ECONOMIC PROOFS OF TWO FAMOUS
BÜHLMANN, H. 1970. Mathematical Methods in Risk Theory. New INEQUALITIES
York: Springer-Verlag.
BÜHLMANN, H. 1980. ‘‘An Economic Premium Principle,’’ ASTIN In Section 9 we introduced the synergy potential. By
Bulletin 11:52–60. observing that this quantity is non-negative, we can
BÜHLMANN, H. 1984. ‘‘The General Economic Premium Princi- derive two mathematical inequalities in a nonconven-
ple,’’ ASTIN Bulletin 14:13–21. tional way. In Example 12, h $ 0 implies that

P E[e
BÜHLMANN, H., AND JEWELL, W.S. 1979. ‘‘Optimal Risk Ex- n
changes,’’ ASTIN Bulletin 10:243–62. E[e2aW]1/a # 2aiWi 1/ai
] ; (189)
CHAN, F.-Y., AND GERBER, H.U. 1985. ‘‘The Reinsurer’s Monopoly i51
and the Bowley Solution,’’ ASTIN Bulletin 15:141–48.
DU MOUCHEL, W. 1968. ‘‘The Pareto-Optimality of a n-Company see (116). With the substitutions
Reinsurance Treaty,’’ Skandinavisk Aktuarietidskrift 51: Zi 5 e2aiWi, ri 5 ai/a,
165–70.
GERBER, H.U. 1975. ‘‘The Surplus Process as a Fair Game— Inequality (189) can be written as

FP G
Utilitywise,’’ ASTIN Bulletin 8:307–22.

P E[Z ]
n n
GERBER, H.U. 1979. ‘‘An Introduction to Mathematical Risk
E Zi # ri 1/ri
i . (190)
Theory,’’ S.S. Huebner Foundation monograph. Philadel- i51 i51
phia.
GERBER, H.U. 1984. ‘‘Chains of Reinsurance,’’ Insurance: Math- Because the substitutions can be reversed, this ine-
ematics and Economics 3:43–8. quality is valid for arbitrary random variables Z1 . 0,
GERBER, H.U. 1987. ‘‘Actuarial Applications of Utility Func- . . . , Zn . 0 and numbers r1 . 0, . . . , rn . 0 with
tions,’’ in Actuarial Science, number 6 in Advances in the 1/r1 1 . . . 1 1/rn 5 1. In the mathematical literature,
Statistical Sciences, edited by I. MacNeill and G. Umphrey. Inequality (190) is known as Hölder’s inequality.
Dordrecht, Holland: Reidel, pp. 53–61.
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 91 # 18

UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 91

The other inequality is Minkowski’s inequality. It To compare this result with the one in normal case,
states that for p . 1 and random variables Z1 . 0, we rewrite it as
. . . , Zn . 0 the following inequality holds:
P 2 E(S) E(S)

FSO D G
1 2 w̃ 5
O E[Z ]
n p 1/p n .
a Var(S) P
E Zi # p 1/p
i . (191)
i51 i51
If we assume a, b and P tend to infinity such that P 2
The proof starts with h $ 0 in (119). Then it suffices E(S) and Var(S) remain constant, then
to set P 2 E(S)
1 2 w̃ —→ .
Zi 5 si 2 Wi, p 5 c 1 1, a Var(S)

and to observe that substitutions can be reversed. If In another particular case in which u(x) is an ex-
p , 1, the inequality sign in (191) should be reversed. ponential utility function with parameter a, and S has
This follows from Example 14, with the substitution an inverse Gaussian distribution with parameters a
and b (Bowers et al. 1997, Ex. 2.3.5), the calculation
Zi 5 Wi, p 5 1 2 c. can be again done explicitly. If S , Inverse Gaus-
In the limit p → 0, we obtain sian(a, b), then E(S) 5 a/b and Var(S) 5 a/b2. For
a(1 2 w) , b/2, the expected utility is
exp H F SO DGJ
E ln
n

i51
Zi $ O exp (E[ln Z ]);
n

i51
i 1
a
(1 2 E[e2aw1awP1a(12w)S]) 5
1
a S1 2 e2aw1awP

HF S D GJD
this can be seen from (124). Note that (191) also 1/2
2a(1 2 w)
holds for p 5 1, in which case it is known as the tri- 3 exp a 12 12 .
angle inequality. b
It is maximal for
DISCUSSIONS P2 2 (a/b)2 b
1 2 w̃ 5 3 .
P2 2a
HANGSUCK LEE*
To compare this result with that in normal case, we
Dr. Gerber and Mr. Pafumi have written a very inter-
rewrite it as
esting paper. My comments concern Section 7, on the
optimal fraction of reinsurance.
In the particular case in which u(x) is an exponen-
P 2 E(S)
S11
E(S)
P D E(S)
tial utility function with parameter a and S has a 1 2 w̃ 5 .
gamma distribution with parameters a and b, the cal- a Var(S) 2 P
culation can be also done explicitly. If S , gamma(a, If we assume a, b and P tend to infinity such that P 2
b), then E(S) 5 a/b and Var(S) 5 a/b2. For a(1 2 E(S) and Var(S) are constant, then
w) , b, the expected utility is
P 2 E(S)
1 1 2 w̃ —→ .
(1 2 E[e2aw1awP1a(12w)S]) a Var(S)
a

5
1
a S1 2 e2aw1awP S b
b 2 a(1 2 w) DD
a

, REFERENCE
BOWERS, N.L., JR., GERBER, H.U., HICKMAN, J.C., JONES, D.A., AND
NESBITT, C.J. 1997. Actuarial Mathematics, 2nd ed.
which is maximal for

S D
Schaumburg, Ill.: Society of Actuaries.
a 1
1 2 w̃ 5 b2 3 .
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.
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92 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

ALASTAIR G. LONGLEY-COOK* ‘‘expected return versus beta’’ relationship of the cap-

ital-asset-pricing model (CAPM)
As Charles Trowbridge (1989, p. 11) points out in
Fundamental Concepts of Actuarial Science, utility E(ri) 5 rf 1 bi[E(rM) 2 rf]
theory forms the philosophical basis of actuarial sci-
can be derived from a variance-proportionate risk pre-
ence, and yet there is barely a mention of the subject
mium (Bodie et al. 1996, p. 243)
in the actuarial literature beyond Chapter 1 of Actu-
arial Mathematics (Bowers et al. 1997). Dr. Gerber E(ri) 2 rf 5 ks 2.
and Mr. Pafumi do the profession a great service by
So it is strange that the exponential utility function
publishing this excellent summary of utility functions
assumption, with its variance-proportionate risk pre-
and their applications.
mium [see Formula (23)] does not lead to the clas-
The authors employ three general forms of utility
sical value for b. Further explanation of the utility
functions, the exponential and the power of the first
function assumptions inherent in CAPM would be
and second kind, in their examples. The reasonable-
helpful.
ness of these functions should be evaluated in any
practical application. In particular, the upper and
lower bounds on the two kinds of power functions may REFERENCES
render them unreasonable assumptions in situations BODIE, A., ET AL. 1996. Investments. Homewood, Ill.: Irwin.
in which results can vary widely from expected. BOWERS, N.L., JR., ET AL. 1986. Actuarial Mathematics. Itasca,
It is also generally agreed in finance theory that for Ill.: Society of Actuaries.
a utility function to be realistic with regard to eco- RUBENSTEIN, M. 1976. ‘‘The Strong Case for the Generalized
nomic behavior, its absolute risk aversion with regard Logarithmic Utility Model as the Premier Model of Finan-
to wealth W, defined as A( W ) 5 2U0( W )/U9( W ), cial Markets,’’ Journal of Finance 31:551–71.
should be a decreasing function of W; and while there TROWBRIDGE, C.L. 1989. Fundamental Concepts of Actuarial Sci-
is some debate over the slope of its relative risk aver- ence. Itasca, Ill.: Actuarial Education and Research Fund.
sion, defined as R( W ) 5 2W[U0( W )/U9( W )], empir-
ical evidence suggests that it should be constant over HEINZ H. MÜLLER*
W (Rubinstein 1976). If the variable x in the paper’s
first example [Formula (1)] is defined as change in The authors are to be congratulated for their very el-
wealth W with respect to initial wealth W0, then the egant article that treats in a very concise and clear
negative exponential utility function satisfies the de- style the most important applications of utility theory
creasing-absolute and constant-relative risk aversion in insurance and finance. The article not only is an
criteria with respect to initial wealth W0. As the au- excellent survey but also introduces the synergy po-
thors point out, the absolute risk aversion of the tential of risk exchanges as a new concept. In the in-
power utility function of the first kind [Formula (10)] surance context this synergy potential measures in a
increases with wealth, a condition that may prove un- most convincing way the welfare gain resulting from
realistic. reinsurance.
Note that, despite the differences in utility func- It may be of some interest to address the question,
tions, the general form of a risk adjustment (which which types of utility functions lead to a decision-
reduces the expected value to its certainty equivalent) making consistent with empirical observations? The
is dependent on the variance of gain, rather than the following result due to Arrow (1971) helps to answer
standard deviation or some other measure or risk [see this question:
Formulas (23), (26), and (28)]. This is consistent with
Assume that a risk-less and a risky asset are available
mean/variance risk analysis and challenges the use of
as investment opportunities. Then, under an in-
other risk measures.
crease of initial wealth investors with increasing (de-
What is then surprising is that the classical value
creasing) risk aversion decrease (increase) the dollar
for b given in Formula (165) is generated by the quad-
amount invested in the risky asset.
ratic utility function, rather than the exponential. The

*Alastair G. Longley-Cook, F.S.A., is Vice President and Corporate *Heinz H. Müller, Ph.D., is Professor of Mathematics at the University
Actuary, Aetna, 151 Farmington Ave., RC2D, Hartford, Connecticut of St. Gallen, Bodanstrasse 4, CH-9000 St. Gallen, Switzerland,
06156, e-mail, longleycookag@aetna.com. e-mail, Heinz.Mueller@unisg.ch.
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 93 # 20

UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 93

Therefore, Arrow postulates a decreasing risk aver- ˜j

X0 t9(X
j
˜ j) t̂9(w)
sion. According to this postulate power utility func- 5 ˜j 2
X9
˜X9j tj(Xj)
˜ t̂(w)
tions of the first kind and, in particular, the quadratic
utility function may not be appropriate for practical 1
5 [t9(X˜ ) 2 t̂9(w)].
purposes. t̂(w) j j
The results in the survey allow also for some com-
This implies in particular
ments on the properties of equilibrium in financial
markets. As shown in Section 10, an equilibrium risk ˜ j 5 sign[t9(X
sign(X0) j
˜ j) 2 t̂9(w)],
exchange (X˜ 1, . . . , X˜ n) with a price density C is Pareto
and we conclude
optimal, and there exist k1, . . . , kn such that (107)
and (109) hold with L 5 C. This provides the basis t9(X
j
˜ j) . t̂9(w) —→ X0
˜j . 0 (*).
for the construction of a fictitious investor represent- (,) (,)

ing the market. Up to an additive constant the utility In the context of financial markets, W corresponds to
function û of this representative investor is given by the total market capitalization and X̃j, j 5 1, . . . , n,
û9(w) 5 C(w). denotes investor j’s payoff as a function of W. Accord-
ing to Arrow’s postulate risk tolerances are increasing
From (109) we conclude that û is strictly concave and and (*) can be interpreted as follows:
for the risk aversion of the representative investor we
obtain Investors who are more (less) sensitive to wealth
û0(w) C9(w) 1 changes than the market choose a convex (concave)
r̂(w) 5 2 52 5
O
. payoff function.
û9(w) C(w) n
1
h51 rh(X˜ h ) As Leland (1980) pointed out in his article on risk-
Instead of the risk aversion sharing in financial economics, convex payoff func-
tions can be considered as generalized portfolio in-
u0(x) surance strategies. A discussion of the shape of payoff
r(x) 5 2 ,
u9(x) functions in market equilibrium can be found, for ex-
ample, in Chevallier and Mueller (1994).
it is convenient in this context to use the risk toler-
The relationship (*) may be of some interest for the
ance, which is defined by
investment strategy of pension funds. Because of sol-
1 vency problems, such an investor may be more sensi-
t(x) 5 .
r(x) tive to wealth changes than the market for low values
of w, that is, t9(X
j
˜ j) . t̂(w) for w , w0. For high values
Hence the risk tolerance of the representative investor
of w, funding problems disappear and t9(X j
˜ j) , t̂9(w)
is given by
for w . w0 may hold. This leads to a payoff function

O t (X˜ ), X̃j as depicted in Figure 1.

n
t̂(w) 5 h h A payoff function that is convex for low w and con-
h51
cave for high w is, for example, obtained by investing
and the risk-sharing rule (101) can be written as
tj(X˜ j ) Figure 1
dX˜ j 5 dw, j 5 1, . . . , n
t̂(w) Payoff Function for a Pension Fund
or
tj (X˜ j )
X̃9j 5 . ;
Xj
t̂(w)
•
Taking logarithms and differentiating leads to

w0 w
; ) > t'(w)
tj'(X ; ) < t'(w)
tj'(X
j j
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 94 # 21

94 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

in the market portfolio, selling calls with high strike prices the objective is to maximize expected utility of
prices and buying puts with low strike prices. consumption and/or terminal wealth. He employed a
dynamic programming approach, an approach that is
elegant yet often impractical due to computational
REFERENCES difficulties. More recently, martingale methods have
ARROW, K.J. 1971. Essays in the Theory of Risk Bearing. Chi- been employed to successfully solve these continuous
cago, Ill.: Markham. time optimal portfolio problems [for example, see
CHEVALLIER, E., AND MUELLER, H.H. 1994. ‘‘Risk Allocation in
Pliska (1997) and the book by Boyle et al. (1998),
Capital Markets: Portfolio Insurance, Tactical Asset Allo-
which the authors already reference]. Since maximiz-
cation and Collar Strategies,’’ ASTIN Bulletin 24, no. 1:
5–18.
ing expected utility is the objective preferred by finan-
LELAND, H.E. 1980. ‘‘Who Should Buy Portfolio Insurance?’’ cial economists for managing portfolios of assets and
Journal of Finance XXXV, no. 2:581–596. liabilities, the relevance for the actuarial and insur-
ance industries is obvious.

STANLEY R. PLISKA*
REFERENCES
This paper provides a survey of applications of utility
theory to selected risk management and insurance BERGMAN, Y.Z. 1985. ‘‘Time Preference and Capital Asset Pricing
Models,’’ Journal of Financial Economics 14:145–159.
problems. I would like to compliment the authors for
CONSTANTINIDES, G. 1990. ‘‘Habit Formation: a Resolution of the
writing such a clear, interesting, and yet concise treat-
Equity Premium Puzzle,’’ Journal of Political Economy 98:
ment of how utility functions can be used for decision- 519–43.
making in the actuarial context. Applications studied DUFFIE, D. 1992. Dynamic Asset Pricing Theory. Princeton, N.J.:
range from the fundamental problem faced by a com- Princeton University Press.
pany of setting an insurance premium to esoteric is- FISHBURN, P. 1970. Utility Theory for Decision Making. New York:
sues such as synergy potentials. John Wiley & Sons.
I would also like to complement the authors by ex- MERTON, R.C. 1990. Continuous Time Finance. Cambridge,
tending their exposition in two directions. Both direc- Mass.: Blackwell Publishers.
tions have to do with the literature relating utility the- PLISKA, S.R. 1997. Introduction to Mathematical Finance: Dis-
ory and financial decision-making. The first direction crete Time Models. Cambridge, Mass.: Blackwell Publish-
ers.
pertains to the utility functions themselves. An im-
portant, classical treatment of the use of utility func-
tions for decision-making is by Fishburn (1970). In ELIAS S.W. SHIU*
recent years, however, financial economists such as
This is a masterful paper on utility functions and many
Bergman (1985), recognizing the limited realism as-
of their applications in actuarial science and finance.
sociated with standard utility functions, have devel-
I am particularly grateful for this paper because I once
oped some generalizations reflecting changing pref-
wrote in TSA (Shiu 1993) a defense for the use of
erences across time. For example, Constantinides
utility theory.
(1990) studied a utility function model that reflects
We are indebted to the authors for giving a precise
habit formation, and Duffie (1992) covered a related
explanation of the approximation formula (28). This
notion called recursive utility. It would be interesting
result can be found in textbooks such as Bowers et al.
to consider how in the actuarial context preferences
(1997, Ex. 1.10.a) and Luenberger (1998, p. 256, Ex.
might change with time.
8). The derivation of (28) outlined in these two books
Another direction the authors could have pursued
combines a second-order approximation with a first-
is the well-known application of utility functions for
order approximation. Formula (28) probably first ap-
portfolio management. More than 25 years ago Robert
peared in Pratt (1964, Eq. 7). Pratt (1964, p. 125)
Merton, the recent winner of a Nobel prize in econom-
was rather careful in stating that he assumed the third
ics, wrote several papers (see his 1990 book) in which
absolute central moment of Gz to be of smaller order
for continuous time stochastic process models of asset
than Var[Gz]; ordinarily, it is of order (Var[Gz])3/2.

*Stanley Pliska, Ph.D., is CBA Distinguished Professor of Finance, De- *Elias S.W. Shiu, A.S.A., Ph.D., is Principal Financial Group Founda-
partment of Finance, College of Business Administration, University tion Professor of Actuarial Science, Department of Statistics and Ac-
of Illinois at Chicago, 601 South Morgan Street, Chicago, Ill. 60607, tuarial Science, University of Iowa, Iowa City, Iowa 52242-1409,
e-mail, srpliska@uic.edu. e-mail, eshiu@stat.uiowa.edu.
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 95 # 22

UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 95

The formulas for the two kinds of power utility func- HAKANSSON, N.H., AND ZIEMBA, W.T. 1995. ‘‘Capital Growth The-
tions, as given by (2) and (4), are rather unsymmet- ory,’’ in Handbooks in Operations Research and Manage-
rical because there is an s in (2) but not in (4). By ment Science, Vol. 9 Finance, edited by R.A. Jarrow, V.
modifying (4) as Maksimovic, and W.T. Ziemba, 65–86. Amsterdam: Else-
vier.
(x 2 s)12c 2 1 LUENBERGER, D.G. 1998. Investment Science. New York: Oxford
u(x) 5 , x . s, (D.1)
12c University Press.
MERTON, R.C. 1969. ‘‘Lifetime Portfolio Selection under Uncer-
we can obtain a certain symmetry between the two tainty: The Continuous-Time Case,’’ Review of Economics
kinds of utility functions. Then (11) becomes and Statistics 51:247–57.
MOSSIN, J. 1968. ‘‘Optimal Multiperiod Portfolio Policies,’’ Jour-
c
r(x) 5 , (D.2) nal of Business 41:215–29.
x2s PRATT, J.W. 1964. ‘‘Risk Aversion in the Small and in the
Large,’’ Econometrica 32:122–36. Reprinted with an ad-
(89) becomes
dendum in Ziemba and Vickson (1975), pp. 115–30.
X̃j 5 k1/c
j L21/c 1 sj, (D.3) SHIU, E.S.W. 1993. Discussion of ‘‘Loading Gross Premiums for
Risk Without Using Utility Theory,’’ Transactions of the
and so on. Society of Actuaries XLV:346–48.
My final comment is motivated by the reference to ZIEMBA, W.T., AND VICKSON, R.G., ed. 1975. Stochastic Optimi-
the so-called Merton ratio in Section 7. The Merton zation Models in Finance. New York: Academic Press.
ratio was also discussed in a paper in this journal
(Boyle and Lin 1997). It means that an investor with
a power utility function of the second kind will use a VIRGINIA R. YOUNG*
proportional asset investment strategy. The result was I congratulate the authors on writing an excellent
derived by Merton (1969) using Bellman’s equation. summary of applications of utility functions in evalu-
An elegant proof using the insights from the martin- ating risks, with special emphasis on insurance eco-
gale approach to the contingent-claims pricing theory nomics. Their paper will serve as a valuable reference
can be found in the review paper by Cox and Huang for actuaries, both practitioners and researchers.
(1989, p. 283). In the context of discrete-time mod- In this discussion, I simply wish to point out that
els, the result was obtained by Mossin (1968); further Arrow’s theorem on the optimality of stop-loss (or de-
discussions can be found in survey articles such as ductible) insurance is more generally true; see the au-
Hakansson (1987) and Hakansson and Ziemba (1995) thors’ Section 6. Specifically, suppose a decision
and in various papers reprinted in Ziemba and Vickson maker orders risks according to stop-loss ordering;
(1975). Merton (1969) also showed that an investor that is, a (non-negative) loss random variable X is con-
with an exponential utility function would invest a sidered less risky under stop-loss ordering than a loss
constant amount in the risky asset. Y if

REFERENCES
E S (x)dx # E S (x)dx
0
t

X
0
t

BOWERS, N.L., JR., GERBER, H.U., HICKMAN, J.C., JONES, D.A., AND for all t . 0, in which SX is the survival function of X;
NESBITT, C.J. 1997. Actuarial Mathematics, 2nd ed.
namely, SX(x) 5 Pr(X . x). Then, for a fixed premium
Schaumburg, Ill.: Society of Actuaries.
P 5 f(E[h(X)], in which f is a function such that
BOYLE, P.P., AND LIN, X. 1997. ‘‘Optimal Portfolio Selection with
Transaction Costs,’’ North American Actuarial Journal 1,
f(x) $ x and f9(x) $ 1, the decision maker will prefer
no. 2:27–39. stop-loss insurance with deductible d given implicitly
COX, J.C., AND HUANG, C.F. 1989. ‘‘Option Pricing Theory and by

SE D
Its Applications,’’ in Theory of Valuation: Frontier of Mod- `
ern Financial Theory, edited by S. Bhattacharya and G.M. P5f SX (x)dx .
Constantinides, 272–88. Totowa, N.J.: Rowman & Little- d

field.
HAKANSSON, N.H. 1987. ‘‘Portfolio Analysis,’’ in The New Pal-
grave: A Dictionary of Economics, Vol. 3, edited by J. *Virginia R. Young, F.S.A., Ph.D., is Assistant Professor of Business,
Eatwell, M. Milgate and P. Newman, 917–20. London: Mac- School of Business, 975 University Avenue, University of Wisconsin–
millan. Madison, Madison, Wisconsin 53706, e-mail, vyoung@bus.wisc.edu.
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 96 # 23

96 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

Van Heerwaarden, Kaas, and Goovaerts (1989) prove a

E [S] 5 ,
this result for f(x) 5 (1 1 l)x, while constraining the b
indemnity benefit to have a derivative between zero
and one. The result is true when one only constrains a
Var[S] 5 ,
the indemnity benefit to lie between zero and the loss b2
amount, as in the authors’ Inequality (54); see Gollier
and Schlesinger (1996).
The optimality of stop-loss insurance is intuitive by
E FS D G
S2
a
b
3

5g
a
b3
.

noting that the authors’ proof of Inequality (56) is This construction is illustrated by the following table.
independent of the (increasing, concave) utility func-
tion and by recalling that the common (partial) or- Distribution u(z) g
dering of random variables by risk-averse decision
1 2
makers is stop-loss ordering (Wang and Young 1998). Normal z1 z 0
2
I encourage the authors and interested researchers to Gamma 2ln(1 2 z) 2
explore whether or not one can generalize other re- Inverse Gaussian 1 2 Ï1 2 2z 3
sults from expected utility theory.

The expected utility (59) is

REFERENCES
1
GOLLIER, C., AND SCHLESINGER, H. 1996. ‘‘Arrow’s Theorem on 5 (1 2 exp(2aw 1 awP)MS(a(1 2 w))).
a
the Optimality of Deductibles: A Stochastic Dominance
Approach,’’ Economic Theory 7:359–63. Using (R.1), we see that we must minimize the ex-
VAN HEERWAARDEN, A.E., KAAS, R., AND GOOVAERTS, M.J. 1989. pression

S D
‘‘Optimal Reinsurance in Relation to Ordering of Risks,’’
Insurance: Mathematics and Economics 8:261–67. a(1 2 w)
awP 1 au .
WANG, S.S., AND YOUNG, V.R. 1998. ‘‘Ordering Risks: Expected b
Utility Theory versus Yaari’s Dual Theory of Risk,’’ Insur-
ance: Mathematics and Economics, in press. Setting the derivative equal to 0, we gather that w̃ is
obtained from the condition

AUTHORS’ REPLY P2
a
b
u9 S
a(1 2 w̃)
b D 5 0. (R.2)

HANS U. GERBER AND GÉRARD PAFUMI In general, there is no explicit formula for 1 2 w̃. How-
Mr. Lee shows how the optimal degree of proportional ever, it is possible to obtain an asymptotic formula.
reinsurance can be determined explicitly if S has a Let P → `, a → `, b → `, such that
gamma or an inverse Gaussian distribution. He also
shows that Formula (60) is obtained in both cases as a
P2 5 P 2 E[S] 5 constant,
a limiting result. This raises the question, What re- b
sults can be obtained under the more general as- a
sumption that S has an infinitely divisible distribu- 5 Var[S] 5 constant.
b2
tion? Let
Substituting u9(z) 5 1 1 z 1 (g/2)z2 1 . . . in (R.2),
1 g
u(z) 5 z 1 z 2 1 z 3 1 . . . we get
2 6
P 2 E[S] 2 a(1 2 w̃)Var[S]
denote the cumulant generating function of a random
variable with infinitely divisible distribution, having ga2
2 (1 2 w̃)2 Var[S] 1 . . . 5 0.
mean 1, variance 1, and third central moment g. Now 2b
suppose that S has a distribution such that its mo-
Finally, we develop 1 2 w̃ in powers of 1/b and obtain
ment generating function is
the formula

H J
MS(z) 5 E [e zS] 5 eau(z/b) (R.1)
P 2 E[S] g P 2 E [S]
1 2 w̃ 5 12 1... .
for some a . 0 (the shape parameter) and b . 0 (the a Var[S] 2b Var[S]
scale parameter). Then
Again, Formula (60) results in the limit.
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 97 # 24

UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 97

We appreciate the comments of Mr. Longley-Cook. We note that WT is the solution of a static optimiza-
He finds it strange that the assumption of exponential tion problem. If we make appropriate additional as-
utility functions does not lead to the classical expres- sumptions about the market, we can determine the
sion for b. However, if Formula (162) is expanded in optimal investment strategy, that is, the dynamic
powers of a, the classical Formula (165) can be ob- strategy that replicates the optimal terminal wealth
tained as a first-order approximation. In this context WT. As in Section 10.6 of Panjer et al. (1998), assume
we note that Formulas (23), (26), and (28) are also that two securities are traded continuously, the risk-
first-order approximations. For a deeper discussion of less investment (which grows at a constant rate d),
the CAPM formula, refer to Section 8.2.4 of Panjer et and a non-dividend-paying stock, with price S(t) at
al. (1998) and the references quoted therein. time t, 0 # t # T. We make the classical assumption
Dr. Müller raises some very interesting points. The that {S(t)} is a geometric Brownian motion, that is,
risk tolerance function (the reciprocal of the risk aver-
S(t) 5 S(0)eX(t)
sion function) leads to simplification of some formulas
and is a useful and appealing tool by its own. where {X(t)} is a Wiener process with parameters m
Dr. Pliska and Dr. Shiu point out an important ap- and s2 and parameters m* 5 d 2 (1/2)s2 and s2 in
plication of utility theory: the construction of an op- the risk-neutral measure. Then

S D
timal portfolio, that is, a portfolio that maximizes the h*
S(T) m* 2 m
expected utility of an investor. This problem can in- C 5 eaT , with h* 5 , (R.8)
deed be discussed in the framework of Sections 10 and S(0) s2
11 of the paper. Consider an investor with wealth w where a is such that E[C] 5 1. Note that h* is defined
at time 0 and utility function u(x) to assess the ter- as in Gerber and Shiu (1994) and Gerber and Shiu
minal wealth at time T. Let d . 0 denote the riskless (1996), that is, as the value of the Esscher parameter
force of interest. In the market random payments can h, for which the discounted stock price process is a
be bought. Their price is given by a price density C. martingale under the transformed measure. As a prep-
Thus the price (due at time 0) for a payment of Y (due aration, we recall a result concerning the self-financ-
at time T) is e2dT E [CY ]. If the investor buys Y, the ing portfolio that replicates the payoff of a European-
terminal wealth will be type contingent claim. Consider a European
WT 5 wedT 1 Y 2 E[CY]. (R.3) contingent claim with terminal date T and payoff func-
tion P(z); that is, at time T the payoff P(S(T)) is due.
The problem is to choose Y that maximizes E[u(WT)]. Let V(z, t) denote its price at time t, and h(z, t) the
In analogy to (141), the solution is characterized by amount invested in stocks in the replicating portfolio
the condition at time t, if S(t) 5 z. It is well known that
u9(WT) 5 CE[u9(WT)] (R.4) ­V(z, t)
h(z, t) 5 z ; (R.9)
with WT given by (R.3). For the utility functions of ­z
Examples 1 to 3, explicit expressions for the optimal see Formula (10.6.6) in Panjer et al. (1998), page 95
terminal wealth are obtained. For an exponential util- of Baxter and Rennie (1996), or Section 9.3 of Dothan
ity function, u9(x) 5 e2ax, the optimal terminal wealth (1990).
is Let us revisit the three examples. From (R.5) and
1 1 (R.8) we obtain
WT 5 wedT 2 ln C 1 E[C ln C]. (R.5)
a a 1 aT
WT 5 wedT 1 E [C ln C] 2
For a power utility function of the first kind, u9(x) 5 a a
(1 2 x/s)c, x , s, we obtain h* h*
dT
2 ln S(T) 1 ln S(0). (R.10)
s 2 we a a
WT 5 s 2 C1/c, (R.6)
E[C111/c] Consider a European contingent claim with terminal
and for a power utility function of the second kind, date T and payoff function
u9(x) 5 x2c, the result is that
we dT
P(z) 5 we dT
1
1
a
E [C ln C] 2
aT
a
2
h*
a
ln z.
WT 5 C21/c. (R.7)
E[C121/c] Its payoff differs from WT only by the constant
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98 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

2(h*/a) ln S(0). Since w is the initial price of WT, it Similarly, at time t the amount invested in stocks in
follows that the replicating portfolio is
h* 2dT h*
V(z, 0) 5 w 2 e ln z. 2 (se2d(T2t) 2 Wt )
a c
Hence, by (R.9), the initial amount invested in stocks m 2 m*
5 (se2d(T2t) 2 Wt ), (R.14)
must be cs2
h* where Wt is the wealth at time t, a constant fraction
h(z, 0) 5 2 e2dT.
a of what is missing for total satisfaction.
For a power utility function of the second kind, the
Similarly, at time t, the amount invested in stocks in
optimal terminal wealth is
the replicating portfolio is

2
h* 2d(T2t)
a
e 5
m 2 m* 2d(T2t)
as2
e , 0 # t # T. (R.11)
WT 5
wedT
E[C121/c]
e2aT/c S D
S(T)
S(0)
2h*/c

This time consider a European contingent claim with

Note that its discounted value is constant.
terminal date T and payoff function
For a power utility function of the first kind, we
gather from (R.6) and (R.8) that the optimal terminal
wealth is P(z) 5 E [Cwe dT

121/c
]
exp(2aT/c)z 2h*/c.

WT 5 s 2
s 2 wedT aT/c
E[C111/c]
e S D
S(T)
S(0)
h*/c

.
Its price at time 0 is
V(z, 0) 5 z2h*/c w.
Now consider a European contingent claim with ter-
Hence, by (R.9),
minal date T and payoff function

P
h*
s 2 wedT h(z, 0) 5 2 V(z, 0).
(z) 5 exp(aT/c)z h*/c. c
E [C111/c]
For the replicating portfolio of WT , the amount in-
Note that

P(S(T)) 5 (s 2 W )S(0)
vested in stocks is the same constant fraction of total
h*/c
T . (R.12) wealth, that is,
It follows that the initial price of the contingent claim h*
2 w
is c
V(z, 0) 5 (se2dT 2 w)z h*/c, at time 0, and
with S(0) 5 z. Then according to (R.9) we have h* m 2 m*
2 Wt 5 Wt (R.15)
h* c cs2
h(z, 0) 5 (se2dT 2 w)z h*/c. (R.13)
c at time t.
At first sight, expressions (R.11), (R.14), and (R.15)
To determine the replicating portfolio for WT, we re-
are quite different. However, they can be written in a
write (R.12) as

P(S(T))S(0)
common form: in all three cases the optimal trading
2h*/c
WT 5 s 2 . strategy is to invest the amount
Hence the amount invested in stocks at time 0 must m 2 m*
be e2d(T2t) (R.16)
s2r(ed(T2t)Wt )
2h(S(0), 0) S(0)2h*/c at time t (0 # t , T ) in stocks, where r is the risk
which, by (R.13), simplifies to aversion function. For a verification, simply use (9),
(10), and (11) of the paper.
h* These results can be generalized to the case where
2 (se2dT 2 w).
c n $ 2 different types of stocks are traded. Let Sk(t)
denote the price of stock k. We assume that
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 99 # 26

UTILITY FUNCTIONS: FROM RISK THEORY TO FINANCE 99

{S1(t), . . . , Sn(t)} is an n-dimensional geometric h*

2 k
(se2d(T2t) 2 Wt ), (R.18)
Brownian motion with drift parameters m1, . . . , mn c
1 . . . , m*
(m*, n in the risk-neutral measure) and covar-
iances sik. It is assumed that the covariance matrix and for a power utility function of the second kind,
has an inverse (the precision matrix); its elements are the amount invested in stocks of type k at time t is
denoted by the symbol tik. Then we find the following h*k
generalization of (R.8): 2 W. (R.19)
c t

P SSS (T)
(0)D
O (m* 2 m )t
n hk* n
C 5 eaT k
, with h*k 5 Again, (R.17), (R.18), and (R.19) can be written in
i i ik
k51 k i51 a common form. Now the optimal trading strategy
defined as in Section 7 of Gerber and Shiu (1994), consists of investing the amount
and again a such that E [C] 5 1. According to (R.5),
O (m 2 m*)t
n

(R.6), (R.7), the optimal terminal wealth is h*k i i ik

2d(T2t) i51
2 e 5 e2d(T2t)
dT
1 aT r(ed(T2t) Wt ) r(ed(T2t) Wt )
WT 5 we 1 E[C ln C] 2
a a (R.20)

O h* ln S (T)
n
1 of stock of type k at time t. It follows that the total
2 k k
a k51 amount invested in stocks at time t is

1 O h* ln S (0) O O (m 2 m*)t
n
1 n n
k k
a k51 k51 i51
i i ik
e2d(T2t). (R.21)
for an exponential utility function, r(ed(T2t) Wt )

P SSS (T)
(0)D
hk*/c
s 2 wedT aT/c n Hence at any time the amount invested in stock of
WT 5 s 2 e k
type k must be the constant fraction
E[C111/c] k51 k

O (m 2 m*)t
n
for a power utility function of the first kind and

PS D
i i ik
2hk*/c i51
wedT n

O O (m 2 m*)t
Sk(T) n n (R.22)
WT 5 e2aT/c
E[C121/c] k51 Sk(0) i i ik
k51 i51
for a power utility function of the second kind. In each
of the total amount invested in stocks. Note that this
case we can relate the optimal terminal wealth to the
fraction does not depend on the utility function.
payoff of an appropriately chosen European contin-
If we divide expressions (R.16), (R.20), or (R.21) by
gent claim. Such a payoff can be replicated by a dy-
Wt, we obtain the Merton ratios. For more results and
namic portfolio, whereby the amount hk is invested in
further background, refer to Chapter 8 of Duffie
stocks of type k at time t. Let V(z1, . . . , zn, t) denote
(1992) and the annotated references.
the price of the portfolio at time t (if Sk(t) 5 zk, k 5
Needless to say, we share Dr. Shiu’s enthusiasm for
1, . . . , n). Then
utility functions. We were pleased to see that utility-
­V(z1, . . . , zn, t) related papers by Longley-Cook (1998) and Frees
hk(z1, . . . , zn, t) 5 zk
­zk (1998) have been published by the NAAJ. Dr. Shiu
proposes a more symmetric treatment of power utility
for k 5 1, . . . , n. See, for example, Formula (8.35)
functions. The utility function in his Formula (D.1) is
in Gerber and Shiu (1996). The portfolio that repli-
standardized at the point j 5 1 1 s. If s , 0, it may
cates WT is the optimal investment strategy. For an
be natural to standardize it at j 5 0, which yields the
exponential utility function, we find that the amount
formula
h*k 2d(T2t)
2 e (R.17) (x 2 s)12c 2 (2s)12c
a u(x) 5 , x . s.
(1 2 c)(2s)2c
must be invested in stocks of type k at time t. For a
Then, in the limit s → 2`, we obtain u(x) 5 x.
power utility function of the first kind, the corre-
sponding amount is
Name /7956/02 08/13/98 09:38AM Plate # 0 NAAJ (SOA) pg 100 # 27

100 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 2, NUMBER 3

Dr. Young points out a generalization of Arrow’s re- REFERENCES

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way as in Example 19, that is, by first determining the GERBER, H.U., AND SHIU, E.S.W. 1996. ‘‘Actuarial Bridges to Dy-
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paper. print, University of Wisconsin–Madison, School of Busi-
We are most grateful to the six discussants for their ness.
valuable and stimulating comments.
Additional discussions on this paper can be submitted
until January 1, 1999. The authors reserve the right to
reply to any discussion. See the ‘‘Submission Guide-
lines for Authors’’ for detailed instructions on the sub-
mission of discussions.