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
The purpose of this section is to generalise mean-variance utility 1 and quadratic utility 3. We consider a decision maker with random wealth W and reference point W
. The
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decision maker generally distorts the objective probability distribution of W such that the cumulative distorted probability distribution function is Q
W
. The decision maker’s expected generalised behavioural utility is, therefore, given by
E [U W ] =
W −∞
U
−
w dQ
W
w +
∞ W
U
+
w dQ
W
w . We apply Taylor series expansions for U
−
w and U
+
w around W which yields
E [U W ] =
W −∞
∞ n=0
1 n
U
n −
W w − W
n
dQ
W
w +
∞ W
∞ n=0
1 n
U
n +
W w − W
n
dQ
W
w =
∞ −∞
U W dQ
W
w +
∞ n=1
1 n
U
n −
W
W −∞
w − W
n
dQ
W
w +
∞ n=1
1 n
U
n +
W
∞ W
w − W
n
dQ
W
w = U W
+
∞ n=1
1 n
U
n −
W −1
n
LP M
n
W , W +
∞ n=1
1 n
U
n +
W U PM
n
W , W ,
16 supposing that the Taylor series converge and the integrals exist also recall
Assumption 6.
Theorem 1. If either utility functions U
−
and U
+
are at most quadratic in wealth or the probability distribution of W belongs to the family of small risk distributions, then
the equivalent expected utility can be written as the following mean-partial moments utility
E [ U W ] = E[W − W
] − λ − 1LPM
1
W , W −
1 2
λγ
−
LP M
2
W , W + γ
+
U P M
2
W , W ,
17 which means that we can assume the following piecewise-quadratic form for the utility
function U U W =
W − W −
1 2
γ
+
W − W
2
if W ≥ W ,
λ W − W
−
1 2
γ
−
W − W
2
if W W .
18
Proof. If utility functions U
−
and U
+
are at most quadratic, then U
n −
W = 0 and
U
n +
W = 0 for n ≥ 3. If the probability distribution belongs to the family of small risk
distributions,
12
then we can assume that all the terms in 16 with LPM
n
W , W and
12
If the decision maker distort the objective probabilities, we suppose that the probability distribution of W belongs to the family of small risk distributions under distortion of
probabilities.
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U P M
n
W , W , n ≥ 3, are of smaller order than the second partial moments. Hence,
if we keep only the first and the second partial moments of distribution, the expected utility is
E [U W ] = U W +
2 n=1
1 n
U
n −
W −1
n
LP M
n
W , W +
2 n=1
1 n
U
n +
W U PM
n
W , W .
Since the utility function is unique up to a positive linear transformation and U
1 +
W 0, an equivalent expected utility can be given by
E [ U W ] = E
U W − U W U
1 +
W = −
U
1 −
W U
1 +
W LP M
1
W , W + UPM
1
W , W +
1 2
U
2 −
W U
1 −
W U
1 −
W U
1 +
W LP M
2
W , W +
U
2 +
W U
1 +
W U P M
2
W , W .
To arrive at 17 recall Definitions 2 and 3 and note that U PM
1
W , W −
LP M
1
W , W = E[W − W
].
Corollary 2. In the EUT framework, mean-partial moments utility 17 reduces to mean-variance utility 1, and piecewise-quadratic utility function 18 reduces to
quadratic utility function 3.
Proof. For a von Neumann-Morgenstern utility function the left and right derivatives of U at point W
are equal. Hence, for a von Neumann-Morgenstern utility function λ
= 1 and γ
−
= γ
+
= γ . Finally note that LPM
2
W , W + UPM
2
W , W = E[W −
W
2
].
Corollary 3. If λ = 1, γ
+
= 0, and there is no probability distortion, then mean-partial moments utility 17 reduces to mean-semivariance utility of Markowitz
E[ U W ] = E[W − W
] − 1
2 γ
−
LPM
2
W , W .
Remark 1. Observe that mean-semivariance utility is a particular case of mean-partial moments utility when the decision maker exhibits no loss aversion, risk aversion in the
domain of losses, and risk neutrality in the domain of gains.
Remark 2. Note that the result of Theorem 1 does not encompass the piecewise-power utility function of PTCPT. This utility is defined by U
+
W = W − W
α
and U
−
W = −λW − W
β
, with 0 α 1 and 0 β 1. Observe that one-sided derivatives of this utility function at the reference point do not exist. However, the study
of the financial decision making in the PTCPT framework is rather straightforward since the value function of PTCPT can be written in terms of partial moments without
any approximation
E [U W ] = UPM
α
W , W − λLPM
β
W , W ,
19 where LPM
β
W , W =
W −∞
W − w
β
dQ
W
w . In contrast to mean-partial moment utility 17 where the decision maker’s attitudes towards risk above and below the
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reference point are determined by the values of γ
+
and γ
−
, the decision maker’s attitudes towards risk in the PTCPT utility are determined by the values of α and β.
It is worth noting that for a decision maker with mean-variance utility there is only one source of risk, namely, the variance which measures the total uncertainty
or dispersion. In contrast, a decision maker with mean-partial moments utility generally distinguishes between three sources of risk: the lower partial moment of
order one which is related to the expected loss, the lower partial moment of order two which is related to the uncertainty in losses, and the upper partial moment
of order two which is related to the uncertainty in gains. Note also that in mean- partial moments utility the measure of risk aversion in the domain of losses is scaled
up by the measure of loss aversion. This suggests that a decision maker with loss aversion puts more weight on the uncertainty in losses than on the uncertainty in
gains.
Observe that the piecewise-quadratic utility function 18 is very flexible with regard to the possibility of modelling different preferences of a decision maker. In this function
λ controls the loss aversion, γ
+
controls the concavityconvexity of utility for gains, whereas γ
−
controls the concavityconvexity of utility for losses. Some possible shapes of this utility function are presented in Figure 1.
In particular, Figure 1 presents the following six distinct shapes of the piecewise- quadratic utility function:
Quadratic: The shape of this utility function is given by λ = 1 and γ
−
= γ
+
= γ 0. This utility corresponds to quadratic utility in the EUT framework.
Behavioural I: The shape of this utility function is given by λ = 1, γ
−
0, and γ
+
= 0. This utility corresponds to mean-semivariance utility of Markowitz and the utility function of Fishburn 1977 where one uses the lower partial moment
of the second order. The decision maker equipped with this utility exhibits no loss aversion, risk neutrality in the domain of gains, and risk aversion in the domain
of losses.
Behavioural II: The shape of this utility function is given by λ 1, γ
−
0, and γ
+
0. This corresponds largely to the utility function in PTCPT. The decision maker equipped with this utility exhibits loss aversion, risk aversion in the domain of
gains, and risk seeking in the domain of losses.
Behavioural III: The shape of this utility function is given by λ 1, γ
−
0, and γ
+
0. The decision maker equipped with this utility exhibits loss aversion and risk aversions in the domains of losses and gains. This shape may represent a
utility function in DT.
Behavioural IV: The shape of this utility function is given by λ 1, γ
−
= γ
+
= 0. The decision maker equipped with this utility exhibits loss aversion, but risk
neutrality in the domains of losses and gains. This is a so-called ‘bilinear’ utility function.
Behavioural V: The shape of this utility function is given by λ ≥ 1, γ
−
0, and γ
+
0. The decision maker equipped with this utility may exhibit loss aversion, risk aversion in the domain of losses, and risk seeking in the domain of gains.
This is the utility function of Markowitz 1952. This shape may also represent a utility function in DT if elation is strong.
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Behavioural I
Behavioural II
Behavioural IV Behavioural V
Behavioural III b
c i
t a
r d
a u
Q a
d c
f e
Fig. 1. Alternative shapes of the piecewise-quadratic utility function given by equation 18
The decision maker’s wealth versus the utility scores.The intersection of the dotted lines shows the location of the reference point. Quadratic utility corresponds to quadratic utility in the EUT framework.
Behavioural I utility corresponds to mean-semivariance utility of Markowitz. Behavioural II utility corresponds largely to the utility function in PTCPT. Behavioural III utility may represent a utility
function in DT. Behavioural IV utility is a so-called ‘bilinear’ utility function. Finally, Behavioural V utility is the utility function of Markowitz 1952. The latter shape may also represent a utility function
in DT if elation is strong.
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5. Generalisation of the Arrow-Pratt Risk Premium
