Uncertainty Aversion, Risk Aversion, and the Optimal Choice of Portfolio
Author(s): James Dow and Sérgio Ribeiro da Costa Werlang
Source: Econometrica , Jan., 1992, Vol. 60, No. 1 (Jan., 1992), pp. 197-204
Published by: The Econometric Society
Stable URL: https://www.jstor.org/stable/2951685
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Econometrica, Vol. 60, No. 1 (January, 1992), 197-204
UNCERTAINTY AVERSION, RISK AVERSION, AND THE OPTIMAL
CHOICE OF PORTFOLIO
BY JAMES Dow AND SE1RGIO RIBEIRO DA COSTA WERLANG
1. INTRODUCTION
IN THIS PAPER we describe some implications for economic analysis of a model of
decision making under uncertainty which generalizes the expected-utility model accepted
by most economists as a representation of rational behavior. The model we use is the
model of expected utility under a nonadditive probability measure, which seeks to
distinguish between quantifiable "risks" and unknown "uncertainties." An axiomatic
treatment of the model may be found in Schmeidler (1982, 1989), Gilboa (1987), and
Gilboa and Schmeidler (1989).
The focus of this paper is the problem of optimal investment decisions. Under the
standard theory of expected utility, an agent who must allocate his or her wealth between
a safe and a risky asset will buy some of the asset if the price is less than the expected
(present) value. Conversely the agent will sell the asset short when the price is greater
than the expected value. Our main theorem is a generalization of this result to the case
of uncertainty. We also provide a definition of an increase in perceived uncertainty, and
analyze the effects of an increase on the investment decision.
The problem of making decisions under uncertainty has been of central importance to
economics and statistics throughout the development of these disciplines. The expectedutility theory, which owes its axiomatic development to von Neumann and Morgenstern
(1947), initiates from the work of Bernoulli (1730). Savage (1954) has made a persuasive
case that rational behavior necessarily entails actions represented by such a utility
function and by a prior subjective probability distribution over possible events. For
example, an agent gambling on the toss of a coin about which he knows nothing may
behave qualitatively differently from when he knows whether the coin is biased and if so
by how much. According to Savage, this distinction would be unreasonable: in the first
case the agent should behave exactly as if he knew that the bias was equal to some value
(of course, this value need not be the "true" value since the agent does not have
sufficient information).
Nevertheless, for both theoretical and empirical reasons economists have developed
models which generalize the expected-utility model. One group of these models is based
on a distinction between risk and uncertainty: the idea was proposed by Knight (1921)
and has been further explored by Ellsberg (1961) and Bewley (1986) among others. In the
series of papers referred to above, Schmeidler and Gilboa have given an axiomatic
development of a model which incorporates this distinction. Based on a weakening of the
independence axiom, the model entails maximizing expected utility with a nonadditive
probability measure. With a nonadditive probability measure, the "probability" that
either of two mutually exclusive events will occur is not necessarily equal to the sum of
their two "probabilities." If it is less than the sum, then expected-utility calculations
using this probability measure will reflect uncertainty aversion as well as (possibly) risk
aversion. The reader may be disturbed by "probabilities" that do not sum to one. It
should be stressed that the probabilities, together with the utility function, provide a
representation of behavior. They are not objective probabilities.
Although the expected-utility model has been questioned, there is one factor which is
strongly in its favor. While the theory of consumer behavior under certainty has only the
most pedestrian empirical implications (homogeneity of degree zero and continuity of
the demand function, and symmetry and negative semi-definiteness of the Slutsky matrix
This research was initiated while the authors were at the University of Pennsylvania.
197
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198 JAMES DOW AND SERGIO RIBEIRO DA COSTA WERLANG
where demand is differentiable), the theory of expected utility yields some strong
predictions, in particular the results on local risk neutrality and on complete insurance
with actuarially fair policies. A generalization of the theory which eliminated the
independence axiom completely would also lead to the loss of these useful predictions.
The purpose of this paper is to show that the model of expected-utility maximization
with nonadditive probabilities reflecting uncertainty aversion preserves strong results
which are analogous to these. We focus on the local risk-neutrality theorem (Arrow
(1965)).
According to this result, an agent who starts from a position of certainty will invest in
an asset if, and only if, the expected value of the asset exceeds the price. The amount of
the asset that is bought depends on the agent's attitude to risk. This result holds in the
absence of transactions costs whenever it is possible to buy small quantities of an asset.
Conversely, if the expected value is lower than the price of the asset the agent will wish
to sell the asset short. Consequently an agent's demand for an asset should be positive
below a certain price, negative above that price, and zero at exactly that price. In case
there are many risky assets, this price will not necessarily be the expected value.
With a nonadditive subjective probability distribution over returns on the asset, we
show that this result has a straightforward analog which is intuitively plausible and is
compatible with observed investment behavior. There is an interval of prices within
which the agent neither buys nor sells short the asset. At prices below the lower limit of
this interval, the agent is willing to buy this asset. At prices above the upper end of this
interval, the agent is willing to sell the asset short. The highest price at which the agent
will buy the asset is the expected value of the asset under the nonadditive probability
measure. The lowest price at which the agent sells the asset is the expected value of
selling the asset short. This reservation price is larger than the other one if the beliefs
reflect uncertainty aversion: with a nonadditive probability measure, the expectation of a
random variable is less than the negative of the expectation of the negative of the
random variable. The computation of expected values with nonadditive probability
measures is explained below.
These two reservation prices, then, depend only on the beliefs and aversion to
uncertainty incorporated in the agent's prior, and not on attitudes to risk. This result is
the nonadditive analog of the local risk-neutrality result.
The local risk-neutrality result has a counterpart in the analysis of insurance. An agent
who can buy actuarially fair insurance in any amount will choose to be fully insured. It
follows from the results presented here that there will be a range of premium costs at
which the agent buys full insurance (the model, like Savage's model, has no objective
probabilities and hence actuarial fairness is not defined).
We suggest that a reasonable person may not act according to Savage's model.
Maximizing utility with a nonadditive prior may be a reasonable model of rational
behavior in some circumstances. However, we do not argue that this model is the only
way, nor necessarily the best way, to represent genuine uncertainty. What we show here
is that it provides a tractable framework for economic analysis of the types of problems
for which expected-utility theory itself is useful.
In terms of empirical implications of the Schmeidler-Gilboa model, broadly similar
types of behavior could be caused by transactions costs or asymmetric information, or
the preferences in Bewley's (1986) model. The main difference is that in each of those
three cases there is a tendency not to trade, whereas in Schmeidler-Gilboa there is a
tendency not to hold a position. In other words the agent's frame of reference here is the
safe allocation, rather than the status quo.
In this paper we have set out the simplest investment decision to analyze, namely
where there is only one uncertain asset. In case there are several assets the analysis
becomes more complex because one must consider statistical dependence of the risks
and uncertainties of the different asset returns. We hope to pursue this issue in the
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UNCERTAINTY
AVERSION
199
future. We have also refrained here from describing equilibrium interaction among many
uncertainty averse traders. Dow, Madrigal, and Werlang (1989) discuss this in relation to
the no-trade theorem.
The organization of the rest of the paper is as follows. In Section 2 we present a
simple example which illustrates the basic features of the model. In Section 3 we present
a definition of an increase in uncertainty aversion and results on expectation of a random
variable with a nonadditive distribution. In Section 4 we give our main theorem on asset
choice under uncertainty. The Appendix contains mathematical results for reference.
Several of the proofs are omitted for brevity, and are available on request from the
authors.
2. AN EXAMPLE
In this section we present an example which illustrates the portfolio decisions of an
agent whose preferences are represented by a nonadditive probability measure. The
example is based on a risk-neutral agent and an asset which can take only two possible
values. The agent has wealth W and the (present) value of the asset will be either high,
H, or low, L. The probabilities of these two outcomes are vr and rr' respectively. If
vT + IT' < 1 the agent's decisions reflect uncertainty aversion. We stress that the nonadditive prior represents both the presence of uncertainty and the agent's aversion to it. For
instance, in this example we could have vT = T' = 1/2, which does not necessarily mean
that the agent "knows" the risk with certainty. It could be that the agent thinks both
outcomes are equally likely and is not averse to uncertainty.
Consider the expected return from buying one unit of the asset at price p. The value
will be at worst (L - p) net of the price, but with probability IT it will be (H - p), that is,
an improvement of (H-L) over the worst outcome. The assessment of this possible
improvement reflects its uncertainty: the expected payoff from buying one unit of the
asset is [L + 7r(H - L)] - p. If the price p is less than [L + 7r(H - L)], a risk neutral
investor will buy the asset.
Now consider the return from selling one unit of the asset short. The payoff will be
(p - H) if the asset is worth H, which is the worst outcome. With probability IT' it wi
Payoff
Expected gain Expected gain
from buying from short sale
L +1r (H -L) -p p -H +,-'(H -L)
L
H
Price
Long position y '. Short position
FIGURE 1.-Expected gains from buying and selling short one u
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200 JAMES DOW AND SERGIO RIBEIRO DA COSTA WERLANG
increase to (p - L). The expected payoff is therefore p - H+ irr'(H - L). Thus if p
exceeds H - rr'(H - L), the -investor will sell the asset short. Because vT + 7T' < 1,
H - rr'(H - L) > L + rr(H - L). At prices in between these two numbers the investor
will not hold the asset. Figure 1 shows the expected payoff from buying and selling the
asset as a function of p.
This example illustrates how the expected value is computed under a nonadditive
distribution. In this case, E(X) = L + rr(H - L) (the details are given in the Appendix).
It should be clear from the discussion that adding a constant to a random variable or
multiplying it by a positive constant has the same effect on its expectation. On the other
hand, -this property does not hold for negative constants: -E( -X) = H + w'(H - L), so
that -E(-X) > E(X). It is this inequality which gives rise to the interval of prices with
no asset holdings.
A closely related representation of decisions is to suppose that the agent evaluates
expected utility for a set of prior (additive) probability distributions and acts to maximize
the minimum of expected utility over these priors (see Gilboa and Schmeidler (1989)). At
one extreme, the agent considers only one prior-a "known" distribution-and acts
according to the standard theory of expected utility. At the other extreme, if all prior
distributions over outcomes are considered, the agent considers only the worst possible
outcome. In the above example we would consider a set of additive priors where the
chance of a high return lies between vT (at least) and 1 - IT' (at most). The payoff from
buying a unit of the asset is then
Min{L +A(H-L)-ptA E [7r,1-IT']) =L +rr(H-L) -p,
and from selling it short,
Min{p-H+A(H-L)IA E [7r',1 -IT]} =p-H+IT'(H-L).
3. UNCERTAINTY AVERSION
We define a measure of uncertainty aversion, following an idea of Schmeidler (1989)
for the case of two states of nature. The reader should refer as necessary to the
Appendix for the notation, the definition of nonadditive probabilities, and a summary of
their mathematical properties.
3.1. DEFINITION: Let P be a probability and A c 2 an event. The uncertainty aversion
of P at A is defined by
c(P, A) = 1 -P(A) -P(Ac).
This number measures the amount of probability "lost" by the presence of uncertainty
aversion. It gives the deviation of P from additivity at A. Notice that c(P, A) = c(P, Ac),
which is natural.
3.2. LEMMA: c(P, A) = 0 for all'events A c (2 if, and only if, P is additive.
The proof is omitted.
3.3. EXAMPLE: Constant Uncertainty Aversion. Let (2 be finite with n elements and let
the event space be the power set of (2, 2Q. For all GO E (2, set P({wl) = (1 - c)/n, where
c E [0, 1]. For A c (, A # (, define P(A) = ,, E AP({(w). It is easy to verify that
c(P, A) = c, VA (2, 0. In other words this is a distribution with constant uncertainty
aversion. In general a nonadditive probability need not be so simple.
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UNCERTAINTY AVERSION 201
3.4. EXAMPLE: Maximin Behavior. A person wit
completely uninformed maximizes the payoff of the worst possible outcome. Suppose
c(P, A) = 1 for all events A 12, 0. Then P(A) = 0 for all A Q2. Let u: R -* R be
the utility function of the agent. Then:
Eu =fudp = P(u > a)da.
Let u = infx e u(x). Then P(u > u) = 1 and P(u > u + ?) = O VE > 0. Therefore
u
Eu=frlda=u= inf u(x).
0
XER
This "maximin" rule was proposed by Wald (1950) for situations of complete uncertainty,
and Ellsberg (1961) and Rawls (1971) also suggest that this rule should be considered in
such circumstances. Simonsen (1986) is a recent application to the theory of inflationary
inertia.
We now proceed to extend this "local" measure of uncertainty aversion to the whole
range of two nonadditive probabilities.
3.5. DEFINITION: Given two nonadditive probabilities P and Q defined on the same
space of events, we say that P is at least as uncertainty averse as Q if for all events A c Q2,
c(P, A) > c(Q, A).
The terminology is clumsy, but shorter than alternatives such as "P reflects at l
much perceived uncertainty as Q," etc. This definition allows us to formalize the
statement that the gap between buying and selling prices increases as the uncertainty
aversion increases.
3.6. THEOREM: The following statements are equivalent:
(i) P is at least as uncertainty averse as Q.
(ii) For all random variables X for which the integrals are finite,
-Ep( -X) -EpX >-EQ( -X) -EQX.
PROOF: (i) =* (ii): Let A(a) = {w E Q2 IX(o) > a}. Then
EpX= f0[P(A(a))
- 1] da + fP(A(a)) da.
_00o
Notice that { e Q I - X ) > a} = A( -a)c. Thus
Ep(-X) = [P(A( a)c) -1] da + f P(A(-)c) da
_00o
=f [P(A(a)c) -iJda + J_P(A(a)) da.
Hence
-Ep( -X) - Ep(X) = [1 - P(A(a)) - P(A(a)c)] da.
_00
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202 JAMES DOW AND SERGIO RIBEIRO DA COSTA WERLANG
By the same argument,
-EQ(-X) -EQ(X) = f_[1 - Q(A(a)) - Q(A(a)c)] da.
Since P is at least as uncertainty averse as Q, the result follows immediately.
(ii) = (i): For all events A E X, define the random variable X = 1A (the characteristic
function of the set A). Then EPX =P(A), Ep(-X) = P(AC) - 1, EQX = Q(A), and
EQ(-X) = Q(AC) - 1. Applying (ii) to X, we get (i). Q.E.D.
The next example illustrates the effect of uncertainty aversion on the difference
between -E(-X) and E(X).
3.7. EXAMPLE: Let X be a random variable with X =inf, E-X
sup, EQ Xo) < oo Let P be an additive probability, and fix c e [0
nonadditive probability which is obtained by uniformly increasing the uncertainty aver-
sion from P: let Pj(n2) = 1, and PJ(A) = (1 - c)P(A) for A # 2. It is easy to verify that
c(Pc, A) = c for all A # 12,0, and that
EPX= cX+ (1- c)EPX and
-EP (-X) = cX + (1-c)EPX.
Thus -E.( -X) - EPX = c(X - X), which is increasing in the uncertainty aversion c in
accordance with Theorem 3.6. Here we have taken an additive distribution and squeezed
it uniformly. A risk-neutral agent whose behavior is represented by this distribution will
maximize a weighted average of the worst possible outcome and the expectation of the
additive distribution. Ellsberg (1961) suggested this as an ad hoc decision rule; this
example provides some rationale for the rule.
4. PORTFOLIO CHOICE
In this section we derive our main result, namely that there will be
from E(X) to -E(-X), at which the investor has no position in the asset. At prices
below these, the investor holds a positive amount of the asset, and at higher prices he
holds a short position. Notice that this range of prices depends only on the beliefs and
attitudes to uncertainty incorporated in the agent's prior, and not on the attitudes
towards risk captured by the utility function.
Let W> 0 be the investors' initial wealth, u > 0 the utility function, and X a random
variable with nonadditive distribution P. We assume that u is C2, u' > 0, and u" < 0.
4.1. LEMMA: Suppose EX< oo and -E(-X2)< oo. For A E Ra define f(A)= Eu(W+
AX). Then: (i) f is right-differentiable at A = 0; (ii) f4(0) = u'(W)EX.
The proof is omitted. We now proceed to the main result, namely the behavior of the
risk-averse or risk-neutral agent under uncertainty aversion. Suppose the investor is
faced with the problem of choosing the sum of money S he will invest in an asset. The
present value of one unit of the asset next period is a random amount X with
nonadditive probability distribution P. We characterize the demand for the asset as a
function of the price.
4.2. THEOREM: A risk averse or risk neutral investor with certain wealth W, who is faced
with an asset which yields X per unit, whose price is p > 0 per unit, will buy the asset if
p < EX and only if p < EX. He will sell the asset if p > -E( -X) and only if p > -E( -X).
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UNCERTAINTY AVERSION 203
PROOF: By Jensen's inequality (see the Appendix):
Eu(W- S + (S/p)X) < u(E[W- S + (S/p)X]).
If EXAC E X (here Ac means the set of elements of Q2 not
in A). n is the set of states of nature and the elements of X are called events. A function P:
X [0, 1] is a nonadditive probability if (i) P(0) = 0, (ii) P(W2) = 1, and (iii) P(A) 6 P(B) if A c B.
We impose an additional restriction (see Gilboa and Schmeidler (1989), Schmeidler (1986), and
Shafer (1976)): (iv) VA, B E X, P(A U B) + P(A n B) > P(A) + P(B). In Section 3 of the paper we
show that this corresponds to uncertainty aversion.
A real valued function X: 12 -* DR is said to be a random variable if for all open sets 0 of 1R,
X-1(0) ELX. The expected value of a random variable X is defined as:
EX =f Xdp =fJ (P(X > a)- 1) da + P(X > a) da,
whenever these integrals exist (in the improper Riemann sense) and are finite. Notice that since
P(X> a) = P(X> a) a.e., the expressiop for the expected value may also be written with strict
inequalities. When it is necessary to distinguish between P and other distributions, we write Ep
The following properties of the integral are either proved in the papers referred to above, or else
can be proved immediately:
(i) X > Y =EX > EY;
(ii) E(X + Y) > EX + EY;
(iii) - E( -X) > EX;
(iv) Va >0 and be R, E(aX+b)=aEX+b;
(v) For all concave functions u: R1 -* GR, Eu(X) < u(EX)
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204 JAMES DOW AND SERGIO RIBEIRO DA COSTA WERLANG
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