2.3Expected utility
In this section, we focus on individual financial assets under the assumption that their payoff distributions at a fixed time are known, and without any regard to hedging opportunities in the context of a financial market model. Such asset distributions may be viewed as lotteries with monetary outcomes in some interval on the real line. Thus, we take M as a fixed set of Borel probability measures on a fixed interval S ⊂ ℝ. In this setting, we discuss the paradigm of expected utility in its standard form, where the function u appearing in the von Neumann–Morgenstern representation has additional properties suggested by the monetary interpretation. We introduce risk aversion and certainty equivalents, and illustrate these notions with a number of examples.
Throughout this section, we assume that M is convex and contains all point masses δx for x ∈ S. We assume also that each μ ∈ M has a well-defined expectation
Remark 2.31. For an asset whose (discounted) random payoff has a known distribution μ, the expected value m(μ) is often called the fair price of the asset. For an insurance contract where μ is the distribution of payments to be received by the insured party in dependence of some random damage within a given period, the expected value m(μ) is also called the fair premium. Typically, actual asset prices and actual insurance premiums will be different from these values. In many situations, such differences can be explained within the conceptual framework of expected utility, and in particular in terms of risk aversion.
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Definition 2.32. A preference relation 
on M is called monotone if
The preference relation is called risk averse if for μ ∈ M
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It is easy to characterize these properties within the class of preference relations which admit a von Neumann–Morgenstern representation.
Proposition 2.33. Suppose the preference relation has a von Neumann–Morgenstern representation
Then:
(a) is monotone if and only if u is strictly increasing.
(b) is risk averse if and only if u is strictly concave.
Proof. (a): Monotonicity is equivalent to
holds for all distinct x, y ∈ S and α ∈ (0, 1). Hence,
i.e., u is strictly concave. Conversely, if u is strictly concave, then Jensen’s inequality implies risk aversion:
with equality if and only if μ = δm(μ).
Remark 2.34. In view of the monetary interpretation of the state space S, it is natural to assume that the preference relation is monotone. The assumption of risk aversion is more debatable, at least from a descriptive point of view. In fact, there is considerable empirical evidence that agents tend to switch between risk aversion and risk seeking behavior, depending on the context. In particular, they may be risk averse after prior gains, and they may become risk seeking if they see an opportunity to compensate prior losses. Tversky and Kahneman [272] propose to describe such a behavioral pattern by a function u of the form
where c is a given benchmark level, and their experiments suggest parameter values λ around 2 and γ slightly less than 1. Nevertheless, one can insist on risk aversion from a normative point of view, and in the sequel we explore some of its consequences.
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Definition 2.35. A function u : S → ℝ is called a utility function if it is strictly concave, strictly increasing, and continuous on S. A von Neumann-Morgenstern representation
in terms of a utility function u is called an expected utility representation.
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Any increasing concave function u : S → ℝ is necessarily continuous on every interval (a, b] ⊂ S; see Proposition A.7. Hence, the condition of continuity in the preceding definition is only relevant if S contains its lower boundary point. Note that any utility function u(x) decreases at least linearly as x ↓ inf S. Therefore, u cannot be bounded from below unless inf S > −∞.
From now on, we will consider a fixed preference relation on M which admits a von Neumann–Morgenstern representation
in terms of a strictly increasing continuous function u : S → ℝ. The intermediate value theorem applied to the function u yields for any μ ∈ M a unique real number c(μ) for which
It follows that
i.e., there is indifference between the lottery μ and the sure amount of money c(μ).
Definition 2.36. The certainty equivalent of the lottery μ ∈ M with respect to u is defined as the number c(μ) of (2.11), and
is called the risk premium of μ.
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If u is a utility function, risk aversion implies that
for every lottery μ with μ ≠ δm(μ). Hence, monotonicity yields that
In particular, the risk premium ϱ(μ) associated with a utility function is always nonnegative, and it is strictly positive as soon as the distribution μ carries any risk.
Remark 2.37. The certainty equivalent c(μ) can be viewed as an upper bound for any price of μ which would be acceptable to an economic agent with utility function u. Thus, the fair price m(μ) must be reduced at least by the risk premium ϱ(μ) if one wants the agent to buy the asset distribution μ. Alternatively, suppose that the agent holds an asset with distribution μ. Then the risk premium may be viewed as the amount that the agent would be ready to pay for replacing the asset by its expected value m(μ).
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Example 2.38 (St. Petersburg Paradox). Consider the lottery
which may be viewed as the payoff distribution of the following game. A fair coin is tossed until a head appears. If the head appears on the nth toss, the payoff will be 2n−1€. Up to the early 18th century, it was commonly accepted that the price of a lottery should be computed as the fair price, i.e., as the expected value m(μ). In the present example, the fair price is given by m(μ) = ∞,but it is hard to find someone who is ready to pay even 20€. In view of this “paradox”, posed by Nicholas Bernoulli in 1713, Gabriel Cramer and Daniel Bernoulli [26] independently introduced the idea of determining an acceptable price as the certainty equivalent with respect to some utility function. For the two utility functions
proposed, respectively, by G. Cramer and by D. Bernoulli, these certainty equivalents are given by
and this is within the range of prices people are usually ready to pay. Note, however, that for any utility function which is unbounded from above we could modify the payoff in such a way that the paradox reappears. For example, we could replace the payoff 2n by u−1(2n) for n ≥ 1000, so that 
The choice of a utility function that is bounded from above would remove this difficulty, but would create others; see the discussion on pp. 85–89.
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Exercise 2.3.1. Prove (2.12).
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Given the preference order on M, we can now try to determine those distributions in M which are maximal with respect to . As a first illustration, consider the following simple optimization problem. Let X be an integrable random variable on some probability space (Ω,F, P) with nondegenerate distribution μ ∈ M. We assume that X is bounded from below by some number a in the interior of S. What is the best mix
of the risky payoff X and the certain amount c, which also belongs to the interior of S? If we evaluate X λ by its expected utility E [ u(Xλ) ] and denote by μλ the distribution of X λ under P, then we are looking for a maximum of the function f on [0, 1] defined by
If u is a utility function, f is strictly concave and attains its maximum in a unique point λ∗ ∈ [0, 1].
Proposition 2.39. Let u be a utility function.
(a) We have λ∗ = 1 if E[ X ]≤ c, and λ∗ > 0 if c ≥ c(μ).
(b) If u is differentiable, then
and
Proof. (a): Jensen’s inequality yields that
with equality if and only if λ = 1. It follows that λ∗ = 1 if the right-hand side is increasing in λ, i.e., if E[ X ]≤ c.
Strict concavity of u implies
with equality if and only if λ ∈ {0, 1}. The right-hand side is increasing in λ if c ≥ c(μ), and this implies λ∗ > 0.
(b): Clearly, we have λ∗ = 0 if and only if the right-hand derivative 
of f satisfies 
see Appendix A.1 for the definition of 
and 
Note that the difference quotients
and that they converge to
as λ ↓ 0. By Lebesgue’s theorem, this implies
If u is differentiable, or if the countable set {x | uʹ+(x) ≠ uʹ−(x)} has μ-measure 0, then we can conclude
i.e., 
if and only if
In the same way, we obtain
If u is differentiable at c, then we can conclude
This implies fʹ−(1) < 0, and hence λ∗ < 1, if and only if E[ X ] > c.
Exercise 2.3.2. As above, let X be a random variable with a nondegenerate distribution μ ∈ M. Show that for a differentiable utility function u we have
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Example 2.40 (Demand for a risky asset). Let S = S1 be a risky asset with price π = π1. Given an initial wealth w, an agent with utility function u ∈ C1 can invest a fraction (1 − λ)w into the asset and the remaining part λw into a risk-free bond with interest rate r. The resulting payoff is
The preceding proposition implies that there will be no investment into the risky asset if and only if
In other words, the price of the risky asset must be below its expected discounted payoff in order to attract any risk averse investor, and in that case it will indeed be optimal for the investor to invest at least some amount. Instead of the simple linear profiles X λ, the investor may wish to consider alternative forms of investment. For example, this may involve derivatives such as max{S, K} = K + (S − K)+ for some threshold K. In order to discuss such nonlinear payoff profiles, we need an extended formulation of the optimization problem; see Section 3.3 below.
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Example 2.41 (Demand for insurance). Suppose an agent with utility function u ∈ C1 considers taking at least some partial insurance against a random loss Y, with 0 ≤ Y ≤ w and P[ Y ≠ E[ Y ] ] > 0, where w is a given initial wealth. If insurance of λY is available at the insurance premium λπ, the resulting final payoff is given by
By Proposition 2.39, full insurance is optimal if and only if π ≤ E [ Y ]. In reality, however, the insurance premium π will exceed the “fair premium” E[ Y ]. In this case, it will be optimal to insure only a fraction λ∗Y of the loss, with λ∗ ∈ [0, 1). This fraction will be strictly positive as long as
Since the right-hand side is strictly larger than E[ Y ] due to (2.13), risk aversion may create a demand for insurance even if the insurance premium π lies above the “fair” price E[ Y ]. As in the previous example, the agent may wish to consider alternative forms of insurance such as a stop-loss contract, whose payoff has the nonlinear structure (Y − K )+ of a call option.
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Let us take another look at the risk premium ϱ(μ) of a lottery μ. For an approximate calculation, we consider the Taylor expansion of a sufficiently smooth and strictly increasing function u(x) at x = c(μ) around m := m(μ), and we assume that μ has finite variance var(μ). On the one hand,
On the other hand,
where r(x) denotes the remainder term in the Taylor expansion of u. It follows that
Thus, α(m(μ)) is the factor by which an economic agent with von Neumann-Morgenstern preferences described by u weighs the risk, measured by 
in order to determine the risk premium he or she is ready to pay.
Definition 2.42. Suppose that u is a twice continuously differentiable and strictly increasing function on S. Then
is called the Arrow–Pratt coefficient of absolute risk aversion of u at level x.
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Example 2.43. The following classes of utility functions u and their corresponding coefficients of risk aversion are standard examples.
(a) Constant absolute risk aversion (CARA): α(x) equals some constant α > 0. Since α(x) = −(log uʹ)ʹ(x), it follows that u(x) = a − b · e−αx. Using an affine transformation, u can be normalized to
(b) Hyperbolic absolute risk aversion (HARA): α(x) = (1 − γ)/x on S = (0, ∞) for some γ < 1. Up to affine transformations, we have
Sometimes, these functions are also called CRRA utility functions, because their “relative risk aversion” xα(x) is constant. Of course, these utility functions can be shifted to any interval S = (a,∞). The “risk-neutral” limiting case γ = 1 would correspond to an affine function u.
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Exercise 2.3.3. Compute the coefficient of risk aversion for the S-shaped utility function in (2.9). Sketch the graphs of u and its risk aversion for λ = 2 and γ = 0.9.
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Proposition 2.44. Suppose that u and 
are two strictly increasing functions on S which are twice continuously differentiable, and that α and 
are the corresponding Arrow–Pratt coefficients of absolute risk aversion. Then the following conditions are equivalent.
(a) α(x) ≥ (x) for all x ∈ S.
(b) u = F ◦ for a strictly increasing concave function F.
(c) The respective risk premiums ϱ and 
associated with u and satisfy ϱ(μ) ≥ (μ) for all μ ∈ M.
Proof. (a)⇒(b): Since is strictly increasing, we may define its inverse function, w.
Then F (t) := u w(t)is strictly increasing, twice differentiable, and satisfies u = F ◦ .
For showing that F is concave we calculate the first two derivatives of w:
Now we can calculate the first two derivatives of F:
and
This proves that F is concave.
(b)⇒(c): Jensen’s inequality implies that the respective certainty equivalents c(μ) and 
(μ) satisfy
Hence, ϱ(μ) = m(μ) − c(μ) ≥ m(μ) − (μ) = (μ).
(c)⇒(a): If condition (a) is false, there exists an open interval O ⊂ S such that (x) > α(x) for all x ∈ O. Let 
:= (O), and denote again by w the inverse of . Then the function F (t) = u (w(t)) will be strictly convex in the open interval by (2.15). Thus, if μ is a measure with support in O, the inequality in (2.16) is reversed and is even strict – unless μ is concentrated at a single point. It follows that ϱ(μ) < (μ), which contradicts condition (c).
As an application of the preceding proposition, we will now investigate the structure of those continuous and strictly increasing functions u on ℝ whose associated certainty equivalents have the following translation property:
where the translation μt of μ ∈ M by t ∈ ℝ is defined by
Here we also assume that M is closed under translation, i.e., μt ∈ M for all μ ∈ M and t ∈ ℝ.
Lemma 2.45. Suppose the certainty equivalent associated with a continuous and strictly increasing function u : ℝ → ℝ satisfies the translation property (2.17). Then u belongs to C∞(ℝ).
Proof. Let λ denote the Lebesgue measure on [0, 1]. Then
and this implies f ∈ C1(ℝ) with
Thus, u(x) = f (x − c(λ)) is in C1(ℝ), which implies that fʹ ∈ C1(ℝ) by (2.19), hence f ∈ C2(ℝ). Iterating the argument we get u ∈ C∞(ℝ).
The following proposition implies in particular that a utility function u that satisfies the translation property (2.17) is necessarily a CARA utility function of exponential type as in part (a) of Example 2.43.
Proposition 2.46. Suppose the certainty equivalent associated with a continuous and strictly increasing function u : ℝ → ℝ satisfies the translation property (2.17). Then u has constant absolute risk aversion and is hence either linear or an exponential function. More precisely, there are constants a ∈ ℝ and b, α > 0 such that u(x) equals one of the following three functions
Proof. For t ∈ ℝ let ut(x) := u(x + t), and denote by ct(μ) the corresponding certainty equivalent. For μ ∈ M we have
It follows that ct(μ) = c(μ) for all t ∈ ℝ and μ ∈ M. Therefore,
for all t ∈ ℝ. Since u is smooth by Lemma 2.45, we may apply Proposition 2.44 to conclude that the respective Arrow-Pratt coefficients αt(x) = −utʹʹ (x)/utʹ (x) and α(x) = −uʹʹ(x)/uʹ(x) are equal for all t and x. But αt(x) = α(x + t), and so α(x) does not depend on x. If α > 0, we see as in Example 2.43 that u is of the form u(x) = a − be−αx. If α = 0, u must be linear. And if α < 0, we must have u(x) = a + beαx.
Now we focus on the case in which u is a utility function and preferences have the expected utility representation (2.10). In view of the underlying axioms, the paradigm of expected utility has a certain plausibility on a normative level, i.e., as a guideline of rational behavior in the face of risk. But this guideline should be applied with care: If pushed too far, it may lead to unplausible conclusions. In the remaining part of this section we discuss some of these issues. From now on, we assume that S is unbounded from above, so that w + x ∈ S for any x ∈ S and w ≥ 0. So far, we have implicitly assumed that the preference relation on lotteries reflects the views of an economic agent in a given set of conditions, including a fixed level w ≥ 0 of the agent’s initial wealth. In particular, the utility function may vary as the level of wealth changes, and so it should really be indexed by w. Usually one assumes that uw is obtained by simply shifting a fixed utility function u to the level w, i.e., uw(x) := u(w + x). Thus, a lottery μ is declined at a given level of wealth w if and only if
Let us now return to the situation of Proposition 2.39 where μ is the distribution of an integrable random variable X on (Ω,F, P), which is bounded from below by some number a in the interior of S. We view X as the net payoff of some financial bet, and we assume that the bet is favorable in the sense that
Remark 2.47. Even though the favorable bet X might be declined at a given level w due to risk aversion, it follows from Proposition 2.39 that it would be optimal to accept the bet at some smaller scale, i.e., there is some γ∗ > 0 such that
On the other hand, it follows from Proposition 2.49 below that the given bet X becomes acceptable at a sufficiently high level of wealth whenever the utility function is unbounded from above.
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Sometimes it is assumed that some favorable bet is declined at every level of wealth. The assumption that such a bet exists is not as innocent as it may look. In fact it has rather drastic consequences. In particular, we are going to see that it rules out all utility functions in Example 2.43 except for the class of exponential utilities.
Example 2.48. For any exponential utility function u(x) = 1− e −αx with constant risk aversion α > 0, the induced preference order on lotteries does not depend at all on the initial wealth w. To see this, note that
is equivalent to
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Let us now show that the rejection of some favorable bet μ at every wealth level w leads to a not quite plausible conclusion: At high levels of wealth, the agent would reject a bet ν with huge potential gain even though the potential loss is just a negligible fraction of the initial wealth.
Proposition 2.49. If the favorable bet μ is rejected at any level of wealth, then the utility function u is bounded from above, and there exists A > 0 such that the bet
is rejected at any level of wealth.
Proof. We have assumed that X is bounded from below, i.e., μ is concentrated on [a,∞) for some a < 0, where a is in the interior of S. Moreover, we can choose b > 0 such that
is still favorable. Since u is increasing, we have
for any w ≥ 0, i.e., also the lottery 
is rejected at any level of wealth. It follows that
Let us assume for simplicity that u is differentiable; the general case requires only minor modifications. Then the previous inequality implies
where
due to the fact that is favorable. Thus,
for any x in the interior of S. This exponential decay of the derivative implies u(∞) := limx↑∞ u(x) < ∞. More precisely, if A := n(|a| + b) for some n, then
Take n such that γn ≤ 1/2. Then we obtain
i.e.,
for all x such that x − A ∈ S.
Example 2.50. For an exponential utility function u(x) = 1 −e−αxν , the bet defined in the preceding lemma is rejected at any level of wealth as soon as 
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Suppose now that the lottery μ ∈ M is played not only once but n times in a row. For instance, one can think of an insurance company selling identical policies to a large number of individual customers. More precisely, let (Ω,F, P) be a probability space supporting a sequence X1, X2, . . . of independent random variables with common distribution μ. The value of X i will be interpreted as the outcome of the ith drawing of the lottery μ. The accumulated payoff of n successive independent repetitions of the financial bet X1 is given by
and we assume that this accumulated payoff takes values in S; this is the case if, e.g., S = [0, ∞).
Remark 2.51. It may happen that an agent refuses the single favorable bet X at any level of wealth but feels tempted by a sufficiently large series X1, . . . , Xn of independent repetitions of the same bet. It is true that, by the weak law of large numbers, the probability
(for ε := m(μ)) of incurring a cumulative loss at the end of the series converges to 0 as n ↑ ∞. Nevertheless, the decision of accepting n repetitions is not consistent with the decision to reject the single bet at any wealth level w. In fact, for Wk := w + Z k we obtain
i.e., the bet described by Z n should be rejected as well.
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Let us denote by μn the distribution of the accumulated payoff Zn. The lottery μn has the mean m(μn) = n · m(μ), the certainty equivalent c(μn), and the associated risk premium ϱ(μn) = n · m(μ) − c(μn). We are interested in the asymptotic behavior of these quantities for large n. Kolmogorov’s law of large numbers states that the average outcome 
converges P-a.s. to the constant m(μ). Therefore, one might guess that a similar averaging effect occurs on the level of the relative certainty equivalents
and of the relative risk premiums
Does cn converge to m(μ), and is there a successive reduction of the relative risk premiums ϱn as n grows to infinity? Applying our heuristic (2.14) to the present situation yields
Thus, one should expect that ϱn tends to zero only if the Arrow–Pratt coefficient α(x) becomes arbitrarily small as x becomes large, i.e., if the utility function is decreasingly risk averse. This guess is confirmed by the following two examples.
Example 2.52. Suppose that u(x) = 1 − e−αx is a CARA utility function with constant risk aversion α > 0 and assume that μ is such that 
Then, with the notation introduced above,
Hence, the certainty equivalent of μn is given by
It follows that cn and ϱn are independent of n. In particular, the relative risk premiums are not reduced if the lottery is drawn more than once.
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The second example displays a different behavior. It shows that for HARA utility functions the relative risk premiums will indeed decrease to 0. In particular, the lottery μn will become attractive for large enough n as soon as the price of the single lottery μ is less than m(μ).
Example 2.53. Suppose that μ is a nondegenerate lottery concentrated on (0, ∞), and that u is a HARA utility function of index γ ∈ [0, 1). If γ > 0 then 
and 
hence
If γ = 0 then u(x) = log x, and the relative certainty equivalent satisfies
Thus, we have
for any γ ∈ [0, 1). By symmetry,
see, e.g., part II of §20 in [20]. It follows that
Since u is strictly concave and since μ is nondegenerate, we get
i.e., the relative certainty equivalents are strictly increasing and the relative risk premiums ϱn are strictly decreasing. By Kolmogorov’s law of large numbers,
Thus, by Fatou’s lemma (we assume for simplicity that μ is concentrated on [ε,∞) for some ε > 0 if γ = 0),
hence
Suppose that the price of μ is given by π ∈ c(μ), m(μ). At initial wealth w = 0, the agent would decline a single bet. But, in contrast to the situation in Remark 2.51, a series of n repetitions of the same bet would now become attractive for large enough n, since c(μn) = ncn > nπ for
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Remark 2.54. The identity (2.21) can also be written as
where An+1 = σ(Zn+1, Zn+2, . . . ). This means that the stochastic process 
n = 1,2. . . , is a backwards martingale, sometimes also called reversed martingale. In particular, Kolmogorov’s law of large numbers (2.22) can be regarded as a special case of the convergence theorem for backwards martingales; see part II of §20 in [20].
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Exercise 2.3.4. Investigate the asymptotics of cn in (2.20) for a HARA utility function 
with γ < 0.
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