Download 1 (Robust) Expected Utility

1
(Robust) Expected Utility
Thus far, we established representation results for preference orderings on a set of lotteries.
In this section we discuss the paradigm of expected utility maximization in its standard
form where the function u appearing in the vNM representation has additional properties.
Subsequently, we discuss an extension, called the Savage representation, and robust expected
utilities.
1.1
Expected Utility
Throughout we assume that S ⊂ R and consider a convex set of probability measures M that
contains all point masses δx for x ∈ S. We also assume that all µ ∈ M have a well defined
expectation
∫
m(µ) :=
xµ(dx).
Definition 1.1. A preference ordering ≤ on M is called monotone if x > y implies δx > δy .
It is called risk averse if for all µ ∈ M, µ ̸= δm(µ) one has
δm(µ) > µ.
It is easy to characterize the preceding properties within the class of preferences that have
a vNM representation.
Proposition 1.2. Let ≤ have a vNM representation with function u. Then ≤ is monotone
if and only if u is strictly increasing. Moreover, it is risk averse if and only if u is strictly
concave.
Proof: The characterization of monotonicity is straightforward. Regarding the second
assertion notice that risk aversion implies
δαx+(1−α)y > αδx + (1 − α)δy
for all x, y ∈ S, α ∈ (0, 1) (chose µαδx + (1 − α)δy ). Hence
u(αx + (1 − α)y) > αu(x) + (1 − α)u(y),
i.e., u is strictly concave. Conversely, if u is strictly concave, then Jensen’s inequality yields
∫
∫
U (δm(µ) ) = u( xµ(dx)) ≥ u(x)µ(dx) = U (µ)
2
with equality if and only if µ = δm(µ) .
Definition 1.3. We call a strictly increasing and strictly concave function u : S → R a
utility function.
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For the rest of this section, we assume that the preference ordering admits a vNM representation with a utility function u. If µ ̸= δm(µ) then there exists x, y ∈ S such that
u(x) < U (µ) < u(y). Hence an application of the intermediate-value theorem to the strictly
increasing function u : S → R yields a unique real number c(µ) such that
u(c(µ)) = U (µ).
If µ = δm(µ) , then obviously c(µ) = δm(µ) .
Definition 1.4. The number c(µ) is called the certainty equivalent of the lottery µ. The
number ρ(µ) := m(µ) − c(µ) is called the risk premium.
Risk aversion implies via Jensen’s inequality that c(µ) ≤ m(µ) and
c(µ) < m(µ) ⇔ µ ̸= δm(µ) .
In particular, the risk premium is strictly positive whenever µ carries some risk. As a first
application of expected-utility theory, we then consider the following optimization problem.
Let X be an integrable random variable on (Ω, F, P) bounded from below with law µ. What
is the best mix
Xλ = (1 − λ)X + λc
of X and the deterministic payoff c? To answer this question we evaluate Xλ by its expected
utility E[u(Xλ ] and denote its law by µλ . Then we are looking for a maximum of the function
f on [0, 1] given by
∫
f (λ) := U (µλ ) =
u(x)dµλ (dx).
Strict concavity of u yields
∫
u ((1 − λ)x + λc) dµ
∫
≥ (1 − λ) u(x)µ(dx) + λu(c)
f (λ) =
(1)
= (1 − λ)u(c(µ)) + λu(c)
= (1 − λ)f (1) + λf (0)
with equality if and only if λ ∈ 0, 1. So, f is strictly concave and hence it attains its maximum
in a unique point λ∗ .
Lemma 1.5.
a) We have λ∗ = 1 if E[X] ≤ c and λ∗ > 0 if c ≥ c(µ).
b) If u ∈ C 1 then λ = 1 if and only if E[X] ≤ c and λ = 0 if and only if c ≤
E[Xu′ (X)]
E[u′ (X)] .
Proof: In order to establish part a) notice that it follows from Jensen’s inequality that
f (λ) ≤ u (E[Xλ ]) = u ((1 − λ)E[X] + λc)
with equality if and only if λ = 1. Thus, λ∗ = 1 if the right-hand side is increasing in λ.
Moreover, the right hand side in (1) is increasing in λ if c ≥ c(µ) so λ∗ > 0.
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As for part b) let us denote the derivatives of a function g from the left and the right by
′ and g ′ , respectively. Then λ∗ = 0 if and only if f ′ (0) ≤ 0. Our goal is then to identify
g−
+
+
f+′ (0) in terms of u. To this end, notice first that since X is bounded from below and u is
strictly concave, the difference quotients
u(Xλ ) − u(X)
u(Xλ ) − u(X)
=
(c − X)
λ
Xλ − X
are almost surely bounded by
u′+ (a)|c − X|
if a ≤ min{c, X},
an integrable random variable, and converge to
u′+ (X)(c − X)+ − u′− (X)(c − X)−
(we differentiate u at X in the direction (c − X)). By Lebesgue’s theorem this implies
f+′ (0) = E[u′+ (X)(c − X)+ ] − E[u′− (X)(c − X)− ].
Since u ∈ C 1 we deduce that
f+′ (0) = E[u′ (X)(c − X)].
So, f+′ (0) ≤ 0 if and only if
c≤
E[Xu′ (X)]
.
E[u′ (X)]
The second assertion follows in the same way.
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Example 1.6. Consider an investor with utility function u ∈ C 1 and initial wealth w > 0
who may partially insure herself against a random loss 0 ≤ Y ≤ w with P[Y ̸= E[Y ]] > 0.
We assume that insurance of λY is available at a cost λπ. The resulting final payoff is then
given by
Xλ := (1 − λ)(w − Y ) + λ(w − π).
In view of the preceding lemma, full insurance is optimal if π ≤ E[Y ]. In reality, π will exceed
the fair premium E[Y ] in which case the optimal demand is positive as long as
π<
E[Y u′ (w − Y )]
,
E[u′ (w − Y )]
that is, risk aversion generates a demand for insurance even if the premium lies above the
“fair value”.
We are now going to motivate the definition of the so-called Arrow-Pratt coefficient of
(absolute) risk aversion. To this end, assume that u is smooth enough and consider the Taylor
approximation at c(µ) around m = m(µ):
u(c(µ)) ≈ u(m) + u′ (m)(c(µ) − m) = u(m) − u′ (m)ρ(µ).
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On the other hand
∫
u(c(µ)) =
u(x)µ(dx)
)
∫ (
1 ′′
′
2
=
u(m) + u (m)(x − m) + u (m)(x − m) + r(x) µ(dx)
2
1 ′′
≈ u(m) + u (m)V ar(µ)
2
and so
1
ρ(µ) ≈ α(m)V ar(µ) where
2
α(m) := −
u′′ (m)
.
u′ (m)
Definition 1.7. Let u be twice continuously differentiable. The coefficient
α(x) := −
u′′ (m)
u′ (m)
is called the Arrow-Pratt coefficient of absolute risk aversion.
Two classes of utility functions are given special attention in the literature. The first is
the class of CARA (constant absolute risk aversion) utilities that satisfy α(x) = α. Since
α(x) = −(ln u′ )′
it follows that u(x) = a−be−αx . Thus, up to an affine transformation, CARA utility functions
are of the form u(x) = 1 − e−αx .
Remark 1.8. Since monotone transformations of preference functionals describe the same
preferences, the following preference functionals defined over random variables are equivalent:
U (X) = E[1 − e−αX ],
Ũ (X) =
1
ln E[−e−αX ].
α
Since Ũ satisfies the translation property
Ũ (X + m) = Ũ (X) + m
for all m ∈ R. In particular, whatever an agent’s investment opportunities, her optimal
choice will always be independent of her wealth (represented by m).
The second class is the class of HARA (hyperbolic absolute risk aversion) utilities for
which α(x) = 1−γ
x on S = (0, ∞) for some γ ∈ [0, 1). Up to an affine transformation, one has
u(x) = log x for γ = 0
1 γ
u(x) =
x for 0 < γ < 1.
γ
We close this section with a brief comment on the implications of monotonicity properties
of the coefficients of risk aversion. Let us therefore consider a utility function u and a strictly
increasing smooth concave transformation F . Then
−
(F ◦ u)′′
u′′ F ′′ ◦ u ′
=
−
− ′
u.
(F ◦ u)′
u′
F ◦u
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Thus the coefficients of risk aversion α and α̃ associated with u and ũ := F ◦ u satisfy
α(x) ≤ α̃(x).
More generally, one has the following result.
Proposition 1.9. Suppose that u and ũ are two utility functions on S of class C 2 with
corresponding Arrow-Pratt coefficient of absolute risk aversion α and α̃. Then the following
conditions are equivalent:
a) α̃(x) ≥ α(x) for all x ∈ S.
b) ũ = F ◦ u for a strictly increasing concave transformation F .
c) The associated risk premiums ρ and ρ̃ satisfy ρ̃(µ) ≥ ρ(µ).
1.2
Robust expected utility
The need for robust version of the expected utility paradigm can best be motivated by means
of the Elberg paradox. Assume you are faced with a choice between two urns, each containing
100 balls which are either red or black. In the first urn, the proportion p of read balls is known,
say p = 0.51. In the second urn, the proportion p̃ is unknown. Suppose you collect 1.000
Euro if you draw a red ball and zero if you draw a black ball. Which urn would you chose?
Typically, people chose the first urn; they also chose the first urn if they receive 1.000 Euro
when drawing a black ball and zero if drawing a red ball. But this behavior is not compatible
with the expected-utility theory. For any subjective belief p̃ choosing urn 1 in the first case
means p ≥ p̃ while choosing urn 1 in the second case means p ≤ p̃. So, unless p = p̃ for which
there is no reason, most people violate the expected utility paradigm. Obviously, it makes a
difference for most people if they know the distribution of balls, or not.
1.2.1
The Savage representation
Instead of assuming that the distribution of asset payoffs are known and that preferences
are defined on a set of probability measures, we now take as our basic objects the assets
themselves. More precisely, we take as our commodity space X a st of bounded measurable
functions X (“asset payoffs”) defined on a measurable space (Ω, F) which does not necessarily
carry a reference measure.
Definition 1.10. let ≤ be a preference relation on X . A representation U of the form
∫
U (X) = u(x)Q(dx)
where Q is a probability measure on (Ω, F) is called a Savage representation.
Let µQ,X denote the distribution of X under the subjective probability measure Q. Then
a reference order on X of the preceding from induces a preference order on the set
MQ := {µQ,X : X ∈ X }
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with vNM representation
∫
UQ (µQ,X ) =
u(x)µQ,X (dx).
On this level the previous section specifies conditions on UQ that guarantee that u is a utility
function. While the savage representation provides a possibly remedy to the Allais paradox,
it does not provide a remedy to the Elseberg paradox. This can only be achieved by looking
at “robust expected utilities” (coherent risk measures).
1.3
Robust expected values
Our aim is now to characterize those preference relations on X which admit a representation
of the form
(2)
U (X) = inf EQ [u(X)].
Q∈Q
In fact, we consider a slightly more general model than the one just described. Let
Mb := {µ ∈ M1 R : µ has compact support}
be the set of all boundedly supported probability measures on R, fix a measurable space
(Ω, F) and define the commodity space as
X̃ := conv{X̃(ω; dy) : there exists c ≥ 0 s.t. X̃(ω; [−c, c] = 1) for all ω ∈ Ω}.
A preference order on X then extends to X̃ by
∫ ∫
u(y)X̃(ω; dy)Q(dω) = inf EQ [ũ(X̃)]
Ũ (X̃) inf
Q∈Q
Q∈Q
where ũ is an affine function on Mb (R) defined by ũ =
6
∫
udµ.