With this choice W does not depend on a and the resulting risk measure in this case,
it is a E[x − r
2
] exhibits the homogeneity property
8
in a. If, for example, t is rather small, then the risk is small and the use of quadratic utility
is well justified. Consequently, using quadratic utility 3 in the investor’s objective function 10 we arrive to the following maximisation problem
E[U
∗
W ] = max
a
a E[x − r] − 1
2 γ
a
2
E[x − r
2
]. The first-order condition of optimality of a gives
a = 1
γ E[x − r]
E[x − r
2
] ,
11 which is the famous Arrow’s solution see Arrow 1971, p. 102. Finally, using the
expression for the optimal value of a, we obtain that the maximum expected utility of the investor is given by
E[U
∗
W ] = 1
2γ E[x − r]
2
E[x − r
2
] .
Note that for any investor the higher the value of
E[x−r]
2
E[x−r
2
]
, the higher the maximum expected utility irrespective of the value of γ . Thus, the value of
S R = E[x − r]
E[x − r
2
] ,
12 which is nothing else than the absolute value of the Sharpe ratio
9
see Sharpe 1966, can be used as the ranking statistics in the performance measurement of risky assets.
Observe that the Sharpe ratio is usually presented based on the assumption that either E[x − r] 0 or short sales are restricted. Here we also allow for profitable short
selling strategies. Therefore we compute the absolute value of the Sharpe ratio in order to properly measure the performance.
3. Assumptions, Definitions, and Notation
The purpose of this paper is to generalise the mean-variance analysis and some of its important results presented in the preceding section. Before proceeding to the analysis,
in this section we would like to present the assumptions, definitions, and notation.
Assumption 1. We suppose that the decision maker’s utility function is continuous and increasing in wealth.
Assumption 2. We suppose that the decision maker’s utility function generally has a kink at the reference point W
and U W =
U
+
W if W ≥ W ,
U
−
W if W W .
8
In the landmark paper of Artzner et al. 1999, the authors argue that a sensible risk measure should satisfy several properties. One of these properties is the homogeneity property. This
says that if one invests the amount a in the risky asset, the measure that assesses the investment risk should be a homogeneous function in a. In other words, ‘twice the risk is twice as risky’.
9
Observe that E[x − r
2
] = Var[x] + E[x] − r
2
≈ Var[x] for a small t.
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This means that above the reference point the utility function is given by U
+
, whereas below the reference point the utility function is given by U
−
. The continuity assumption gives
U
−
W = U
+
W = U W
. 13
Assumption 3. The decision maker regards the outcomes below the reference points as losses, while the outcomes above the reference point as gains. Consequently, we will
refer to U
−
as the utility function for losses, and to U
+
as the utility function for gains.
Assumption 4. We suppose that the left and right derivatives of U at the reference point exist and finite. We denote the first-order left-sided derivative
U
1 −
W = lim
h→0−
U
−
W − U
−
W + h
h .
Similarly, we denote the first-order right-sided derivative
10
U
1 +
W = lim
h→0+
U
+
W + h − U
+
W h
. The higher-order one-sided derivatives of U at the reference point are denoted in the
similar manner by U
2 −
W , U
2 +
W , etc.
Assumption 5. We suppose that the decision maker generally distorts the objective probability distribution. More formally, suppose that the cumulative objective probability
distribution of a gamble X is given by F
X
. Then the expected payoff of the gamble in the objective world is
E[X ] =
∞ −∞
xdF
X
x. Observe that the integral above is a Lebesgue-Stieltjes integral which is defined for
both discrete and continuous distributions. We denote by Q
X
the cumulative distorted probability distribution function of X . Under the distortion of probabilities, the expected
payoff of the gamble for the decision maker is given by E [X ] =
∞ −∞
xdQ
X
x. Note that E[·] is generally not an expectation since there are some types of distortions
for which
∞ −∞
d Q
X
= 1. Since the probability distortion may depend either on the cumulative objective
distribution function or on whether the outcomes of X are interpreted as losses or gains, we need to distinguish between the probability distortion of X and −X . For this
purpose we denote by Q
¯ X
the cumulative distorted probability distribution function of the complementary
11
gamble ¯ X = −X. Observe that generally
E [X ] = E[X],
10
To be more precise, we need to denote the left derivative as U
1 −
W − and the right
derivative as U
1 +
W +, but this would enlarge the notation.
11
By a complementary gamble ¯ X = −X we mean a gamble which is obtained by changing
the signs of the outcomes of gamble X .
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This says that the expected value of X under distortion of probabilities is generally different from the expected value of X under objective probabilities. This holds true for
all moments of distribution of X . Moreover,
E [−X] = −E[X]. This says that under the same rule of probability distortion the expected value of −X is
generally not equal to the expected value of X with the opposite sign. Illustrations are provided in Appendix A.
Assumption 6. If W = W
+ X where X is some gamble, then we suppose that
∞ −∞
U W dQ
X
= U W .
14 This is true when either the sum of all distorted probabilities is equal to 1 that is,
∞ −∞
d Q
X
= 1 or the utility function is zero at the reference point that is, as in PTCPT U W
= 0.
Definition 1 Small risk distribution. As formalised by Samuelson 1970, a prob- ability distribution belongs to a family of ‘compact’ or ‘small risk’ distributions if as
some specified parameter goes to zero, the distribution converges to a sure outcome. To demonstrate the construction of a small risk distribution, suppose that the decision
maker’s wealth W = W
+ x and x = tε where ε is some random variable. As t goes to zero, the probability distribution of W converges to the sure amount W
.
Definition 2 The measure of loss aversion. The measure of loss aversion is given by
λ =
U
1 −
W U
1 +
W .
15 This mesasure of loss aversion was proposed by Bernartzi and Thaler 1995 and
formalised by K¨obberling and Wakker 2005. Observe that if the decision maker does not exhibit loss aversion, then λ = 1. Loss aversion implies λ 1. Conversely, loss
seeking behaviour implies λ 1. Finally note that since the first-order derivatives of U are positive follows from Assumption 1, the value of λ is also positive, that is,
λ
0.
Definition 3 Two measures of risk aversion. Recall the measure of risk aversion 2 that was introduced by Arrow and Pratt. Since the utility function of the decision maker
generally has different functions for losses and gains, we introduce: the measure of risk aversion in the domain of gains
γ
+
= − U
2 +
W U
1 +
W ,
and the measure of risk aversion in the domain of losses γ
−
= − U
2 −
W U
1 −
W .
If, for example, γ
+
0, then the utility function for gains is concave which means that the decision maker is risk averse in the domain of gains. By contrast, if γ
+
0, then the utility function for gains is convex which means that the decision maker is risk seeking
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in the domain of gains. Finally, if γ
+
= 0, then the decision maker is risk neutral in the domain of gains.
Definition 4 Lower and Upper Partial Moments. If the decision maker’s random wealth is W and the reference point is W
, a lower partial moment of integer order n under the distortion of probability is given by
LP M
n
W , W = −1
n W
−∞
w − W
n
dQ
W
w . The coefficient −1
n
is chosen to bring our definition of a lower partial moment in correspondence with the definition of Fishburn 1977. An upper partial moment of
order n under the distortion of probability is given by U P M
n
W , W =
∞ W
w − W
n
dQ
W
w . The lower and upper partial moments of order n computed in the objective world are
denoted by LPM
n
W , W and UPM
n
W , W respectively. Note here that if X is some
gamble, then without the distortion of probabilities LPM
n
X , 0 = U PM
n
−X, 0. This is generally not the case with probability distortion. That is, generally
LP M
n
X , 0 = UPM
n
−X, 0, because in the computation of LPM
n
X , 0 we employ Q
X
, whereas in the computation of U PM
n
−X, 0 we use Q
¯ X
. The main reason for the introduction of a new definition of a lower partial moment is the necessity to distinguish between the computation of
moments of distribution for X and −X . See Appendix A that illustrates the computation of partial moments with and without probability distortion.
Definition 5 Optimistic probability distortion. We say that a decision maker is optimistic if he overweights the probabilities of favourable outcomes. For an optimistic
decision maker E[X ] 0 if X is a fair gamble in the objective world, that is, E[X ] = 0. Moreover, observe that for an optimistic decision maker E[−X] 0 as well. Illustrations
are provided in Appendix A.
Definition 6 Pessimistic probability distortion. We say that a decision maker is pessimistic if he overweights the probabilities of unfavourable outcomes. For a pes-
simistic decision maker E[X ] 0 if X is a fair gamble in the objective world. Moreover, observe that for a pessimistic decision maker E[−X] 0 as well. For illustrations, see
Appendix A.
Definition 7 Performance Measure. By a performance measure we mean a score numbervalue attached to each financial asset. A performance measure is related to
the level of expected utility provided by the asset. That is, the higher the performance measure of an asset, the higher level of expected utility the asset provides.
4. Mean-Partial Moments Utility and Piecewise-Quadratic Utility Function
