since y −
1 2
γ
+
y
2
0 for all y
1 γ
+
. Finally we obtain ∂
Cx ∂λ
= − ∂π
x ∂λ
0.
Corollary 7. The certainty equivalent decreases as the decision maker’s risk aversion in the domain of either gains or losses increases. Conversely, the risk premium increases
as the decision maker’s risk aversion in the domain of either gains or losses increases.
Proof. The first-order derivatives of the certainty equivalent and risk premium with respect to γ
+
∂ Cx
∂γ
+
= − ∂π
x ∂γ
+
= − 1
2λ U P M
2
x, 0 0, since U PM
2
x, 0 0. The first-order derivatives of the certainty equivalent and risk premium with respect to γ
−
∂ Cx
∂γ
−
= − ∂π
x ∂γ
−
= − 1
2 LP M
2
x, 0 0, since LPM
2
x, 0 0.
6. Impact of the Decision Maker’s Skewness Preferences on Risk Measurement
In this section we discuss briefly the impact of the decision maker’s skewness preferences on risk measurement in our generalised behavioural framework. Recall that according
to mean-partial moments utility 17 the decision maker distinguishes between three sources of risk. Suppose that W = W
+ x and x is a pure risk such that E[x] = 0 and there is no probability distortion. Then the total risk of x, as measured by the risk
premium, is given by π
x = λ
− 1 λ
LPM
1
x, 0 + 1
2 γ
−
LPM
2
x, 0 + γ
+
λ U PM
2
x, 0 .
26 Observe that if γ
−
= γ
+
= γ and λ = 1 as in EUT, then LPM
2
x, 0 and UPM
2
x, 0 have equal weights in the computation of risk which means that the risk is proportional
to variance, in particular, π x =
1 2
γ Var[x]. That is, within EUT variance has a first-
order impact on risk measurement. By contrast, in many of the alternative theories we generally have γ
−
= γ
+
and λ 1 so that LPM
2
x, 0 and UPM
2
x, 0 have different weights in the computation of risk. However, if the probability distribution
of x is symmetric, then LPM
2
x, 0 = U PM
2
x, 0 =
1 2
Var[x] and the risk is again proportional to variance, at least when the value of λ is not markedly different
from 1 π
x = λ
− 1 λ
LPM
1
x, 0 + 1
4 γ
−
+ γ
+
λ Var[x].
If, on the other hand, the probability distribution of x is not symmetric, then LPM
2
x, 0 = UPM
2
x, 0. In particular, if the distribution of x is skewed to the left then LPM
2
x, 0 UPM
2
x, 0, whereas if the distribution of x is skewed to the right then LPM
2
x, 0 UPM
2
x, 0. In this case skewness has a first-order impact on risk measurement when the value of λ is notably different from 1 andor the values
of γ
−
and γ
+
are markedly different from each other. If, for example, γ
−
≫ γ
+
then the main source of risk is the term with LPM
2
x, 0 which reflects the level of
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Table 1 Probability distributions of the three gambles: A, B, and C
Each gamble gives an uncertain outcome which depends on the future state of the world. The probabilities of the states are equal.
State 1
2 3
4 Objective probability
0.25 0.25
0.25 0.25
Payoff gamble A −10
−5 5
10 Payoff gamble B
−12 −3
5 10
Payoff gamble C −9
−6 3
12
negative skewness. By contrast, if γ
−
≪ γ
+
then the main source of risk is the term with UPM
2
x, 0 which reflects the level of positive skewness. In other words, if the decision maker’s degrees of risk aversions below and above the reference point are
substantially different, then, depending on the signs and the values of γ
−
and γ
+
, either the downside or the upside part of variance is a more proper risk measure than
variance. Which part is actually a proper risk measure? The answer depends on the theory being used. Whereas in most models
15
the decision maker is more risk averse in the domain of losses than in the domain of gains and, therefore, the downside part of
variance might be a proper risk measure, in PTCPT the decision maker is risk seeking in the domain of losses and, thus, a proper risk measure might be the upside part of
variance.
Next we provide an example constructed to demonstrate that a decision maker equipped with the generalised behaviour utility may exhibit strong preferences for
skewness. The data for the example is provided in Table 1. In short, we would like to determine which gamble, A, B, or C, is considered to be the least risky. Each gamble gives
an uncertain outcome which depends on the future state of the world. Observe that the probabilities of the states are equal so the presented results of comparison of the
riskiness of the gambles do not depend on the probability distortion where a decision weight distorted probability of a state is computed using the objective probability of
the state only as, for example, in PT. Table 2 presents the descriptive parameters of the distributions of the three gambles. Note that all gambles have zero expected payoff.
Gambles B and C have higher variance than gamble A. Gamble B has greater downside variance than gamble A, but lower upside variance. By contrast, gamble C has lower
downside variance than gamble A, but greater upside variance. Note that the probability distribution of gamble A has zero skewness, whereas the probability distribution of
gamble B is skewed to the left and the probability distribution of gamble C is skewed to the right.
In this example we use the piecewise-quadratic utility function 18 which produces different preferences depending on the set of parameters λ, γ
−
, γ
+
. We compute the certainty equivalents of the gambles for six different decision makers represented by
the distinct shapes of the piecewise-quadratic utility presented in Section 4. We compute the certainty equivalent using equation 21. Observe that the higher the certainty
15
The examples are: mean-semivariance utility of Markowitz 1959, the utility function of Markowitz 1952, the utility function of Fishburn 1977, Disappointment Theory, Regret
Theory, etc.
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Table 2 Descriptive parameters of the distributions of the three gambles presented in Table 1
Observe that for each gamble the expected loss equals the expected gain so that all gambles have zero expected payoffs. Note that gambles B and C have higher variance than gamble A. However, gamble
B has greater downside variance than gamble A, but lower upside variance. By contrast, gamble C has lower downside variance than gamble A, but greater upside variance. Note that the probability
distribution of gamble A has zero skewness, whereas the probability distribution of gamble B is skewed to the left and the probability distribution of gamble C is skewed to the right.
Parameter Gamble A
Gamble B Gamble C
LPM
1
3.75 3.75
3.75 UPM
1
3.75 3.75
3.75 Expected payoff, UPM
1
− LPM
1
LPM
2
31.25 38.25
29.25 UPM
2
31.25 31.25
38.25 Variance, LPM
2
+ UPM
2
62.50 69.50
67.50 Skewness
−0.2718 0.3651
Table 3 The certainty equivalents of three gambles for decision makers with different shapes of the
piecewise-quadratic utility function
For each shape of the utility function the table reports the corresponding values of λ, γ
−
, and γ
+
. Note that for Behavioural I and Behavioural III utilities the values of γ
−
and γ
+
are distinctly different. The highest certainty equivalent for every decision maker is underlined. That is, the underlined certainty
equivalent marks out the gamble which is considered to be the least risky for a distinct decision maker.
Certainty Equivalent Utility
λ γ
−
γ
+
Gamble A Gamble B
Gamble C Quadratic
1 0.04
0.04 −2.8125
−2.9525 −2.8425
Behavioural I 1
0.04 0.00
−2.5000 −2.6400
−2.4600 Behavioural II
2 −0.04
0.04 −1.5625
−1.4225 −1.6725
Behavioural III 2
0.10 0.02
−3.5938 −3.9437
−3.5288 Behavioural IV
2 0.00
0.00 −1.8750
−1.8750 −1.8750
Behavioural V 2
0.04 −0.04
−2.1875 −2.3275
−2.0775
equivalent of a gamble for a particular decision maker, the less risky is the gamble for this decision maker given that the gambles under question have the same expected
payoff. The results of the computations of the certainty equivalents are presented in
Table 3. Obviously, for a decision maker with bilinear Behavioural IV utility the riskiness of the gambles is the same because this decision maker is indifferent to the
second moments of distribution. Observe that gamble A, which has the lowest variance, is the least risky gamble only for a decision maker with Quadratic utility. Even though
gamble C has greater variance than gamble A, gamble C is the least risky gamble if a decision maker is risk averse in the domain of losses and the measure of risk aversion
in the domain of gains either has the opposite sign or is substantially lower than the measure of risk aversion in the domain of losses this is true for the parameters of
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Behavioural I, Behavioural III, and Behavioural V utilities in Table 3. We can say that the decision makers, for whom gamble C is the least risky, exhibit strong preference for
positive skewness. Finally, the decision maker with convex loss function Behavioural II considers gamble B, which has highest downside variance or negative skewness, to be
the least risky.
7. Optimal Capital Allocation in the Generalized Framework
