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
The set up of the problem is the same as that described in Section 2. That is, we consider an investor who wants to allocate the wealth between a risk-free and a risky asset. The
returns on the risky and the risk-free assets over a small time interval t are given by equations 7 and 8 respectively. The investor’s initial wealth is W
I
and the investor’s objective is to maximise the expected utility of his future wealth.
Since the resulting expression for the investor’s expected utility depends on whether the investor buys and holds or sells short the risky asset, we need to consider these two
cases separately. In particular, if the investor uses the amount a ≥ 0 to buy the risky asset, his future wealth is given by equation 9. If the investor sells short the risky asset
such that the proceedings are a ≥ 0, his future wealth is given by
W = ar − x + W
I
1 + r. Note that in both cases the value of a is non-negative. This is necessary to be able to
distinguish between the probability distortion of x − r and r − x. Before attacking the optimal capital allocation problem, we need to choose a suitable
reference point W to which gains and losses are compared. One possible reference
point is the ‘status quo’, that is, the investor’s initial wealth W
I
. Unfortunately, with this choice it is not possible to arrive at a closed-form solution for the optimal capital
allocation problem unless W
I
= 0. However, according to Markowitz 1952, Kahneman and Tversky 1979, Loomes and Sugden 1986, and others, the investor’s initial wealth
does not need to be the reference point. Following Barberis et al. 2001 we assume that the reference point is W
= W
I
1 + r. This is the investor’s initial wealth scaled up by the risk-free rate. This choice of reference point is sensible for several reasons: a
this level of wealth serves as a ‘benchmark’ wealth. The idea here is that the investor is likely to be disappointed if the risky asset provides a return below the risk-free rate of
return; b with this reference point the performance measure does not depend on the investor’s wealth see the subsequent section. That is, the investor’s utility of returns
is independent of wealth; c this choice is also justified if we require that all partial moments should exhibit the homogeneity property in a for explanation, see footnote 8.
For example, if the investor is equipped with the Fisburn’s mean-lower partial moment of order two utility, then the risk of investing the amount a, as measured by downside
deviation, equals a
LP M
2
x − r, 0, which seems reasonable. Having decided on the investor’s reference wealth, we are ready to state and prove the
following theorem.
Theorem 8. If either utility functions U
−
and U
+
are at most quadratic in wealth or the investment risk is small, then the investor’s optimal capital allocation problem has
the following solution: If
E [x − r] − λ − 1LPM
1
x − r, 0 0 27
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and λγ
−
LP M
2
x − r, 0 + γ
+
U P M
2
x − r, 0 0, 28
then the optimal amount invested in the risky asset is given by
a = E [x − r] − λ − 1LPM
1
x − r, 0 λγ
−
LP M
2
x − r, 0 + γ
+
U P M
2
x − r, 0 ,
29 and the investor’s maximum equivalent expected utility from the buy-and-hold BH
strategy is given by E
B H
[U
∗
W ] = 1
2 E[x − r] − λ − 1LPM
1
x − r, 0
2
λγ
−
LP M
2
x − r, 0 + γ
+
U P M
2
x − r, 0 .
30 If
E [r − x] − λ − 1LPM
1
r − x, 0 0 31
and λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0 0, 32
then the optimal amount of the risky asset that should be sold short is given by
a = E [r − x] − λ − 1LPM
1
r − x, 0 λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0 ,
33 and the investor’s maximum equivalent expected utility from the short selling SS
strategy is given by E
S S
[U
∗
W ] = 1
2 E[r − x] − λ − 1LPM
1
r − x, 0
2
λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0 .
34 If all the conditions 27, 28, 31, and 32 are satisfied, then the maximisation
problem has two local maxima. In this case the optimal policy is the one which gives the highest expected utility either buy-and-hold or sell short.
If neither 27 nor 31 is satisfied, then the investor’s optimal policy is to avoid the risky asset and invest all in the risk-free asset.
Proof. If either utility functions U
−
and U
+
are at most quadratic in wealth or the investment risk is small, then, according to Theorem 1, the investor’s preferences can
be represented by the piecewise-quadratic utility 18. First, we consider the buy-and-hold strategy. In this case the investor’s expected utility
is given by E
B H
[U W ] = a E[x − r] − λ − 1LPM
1
x − r, 0 −
1 2
a
2
λγ
−
LP M
2
x − r, 0 + γ
+
U P M
2
x − r, 0 , 35
where the expectation and the partial moments are computed using the cumulative distorted probability distribution function Q
x−r
. Observe that the investor’s expected utility 35 is a quadratic function in a. To guarantee the existence of a local maximum
interior solution, the investor’s expected utility should be concave in a, which means that condition 28 should be satisfied. The first-order condition of optimality of a gives
the solution 29. Since we assume that a ≥ 0, the solution for the optimal a is a valid solution only if condition 27 is satisfied. If not, the optimal solution is a = 0. Finally,
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inserting expression 29 for the optimal value of a into 35, we obtain the solution for the investor’s maximum expected utility 30.
Second, we consider the short selling strategy. In this case the investor’s expected utility is given by
E
S S
[U W ] = a E[r − x] − λ − 1LPM
1
r − x, 0 −
1 2
a
2
λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0 , 36
where the expectation and the partial moments are computed using the cumulative distorted probability distribution function Q
r −x
. Observe again that the investor’s expected utility 36 is a quadratic function in a. To guarantee the existence of a local
maximum, the investor’s expected utility should be concave in a, which means that condition 32 should be satisfied. The first-order condition of optimality of a gives
the solution 33. Since we assume that a ≥ 0, the solution for the optimal a is a valid solution only if condition 31 is satisfied. Otherwise, the solution is a = 0. Finally,
inserting expression 33 for the optimal value of a into 36, we obtain the solution for the investor’s maximum expected utility 34.
Remark 3. Observe that if, for example, condition 27 is satisfied whereas condition 28 is not, then the higher a the higher the investor’s expected utility. In other words,
in this case the investor is willing to borrow an infinite amount at the risk-free interest rate to invest in the risky asset. In this case the optimal capital allocation problem does
not have a solution. For example, conditions 28 and 32 are violated if the investor appreciates uncertainty in losses, γ
−
0, but is neutral to uncertainty in gains, γ
+
= 0. The other example when these conditions are violated occurs when γ
−
= γ
+
= 0, that is, the investor is neutral to both uncertainties. In the latter case the investor’s utility
can be represented by so-called ‘bilinear’ utility. For bilinear utility the solution to the optimal capital allocation problem generally does not exist. This was noted by Sharpe
1998.
Corollary 9. In the EUT framework, the solution for the optimal amount that should be invested in the risky asset or sold short reduces to the Arrow’s solution given by
11.
Proof. This follows from the fact that for a von Neumann-Morgenstern utility function λ
= 1 and γ
−
= γ
+
= γ . Remark 4. Observe from Arrow’s solution 11 that in the EUT framework it is always
optimal to undertake a risky investment when E[x] = r. If E[x] r, it is optimal for the investor to buy some amount of the risky asset, whereas if E[x] r , it is optimal for
the investor to sell short some amount of the risky asset. By contrast, in our generalised framework the investor avoids the risky asset if
E [x − r] − λ − 1LPM
1
x − r, 0 0 and E [r − x] − λ − 1LPM
1
r − x, 0 0. In particular, the investor does not invest in the risky asset if the perceived risk premium
is too low. In other words, the perceived risk premium provided by the risky asset must be rather high to induce the investor to undertake a risky investment. Note that
the investor might avoid the risky asset due to loss aversion andor some particular
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probability distortion. For example, the higher the investor’s loss aversion, the higher the risk premium must be in order to motivate the investor to undertake a risky investment.
It is easy to show that without any probability distortion a loss averse investor does not invest if
1 λ
LPM
1
x, r UPM
1
x, r λLPM
1
x, r . However, some particular distortion of probabilities might cause the investor’s avoidance
of the risky asset even with no loss aversion. In this case we must have E [x − r] 0 and E[r − x] 0.
This may occur with a pessimistic probability distortion. In Appendix B we provide an example which demonstrates how a probability distortion might cause either avoidance
of the risky asset or the existence of two local maxima in the optimal capital allocation problem. This result may help explain why many investors do not invest in equities.
Furthermore, the equity premium puzzle may be explained by either loss aversion or pessimistic probability distortion or a combination of these effects. The explanation lies
in the fact that to induce an investor with either loss aversion or pessimistic beliefs to buy equities, stock returns must be rather high relative to bond returns.
Remark 5. Somewhat surprisingly, an investor with the original PTCPT utility function will always invest in the risky asset. Given the expression for the investor’s value function
in PTCPT 19, it is rather straightforward to arrive at the following solution for the optimal capital allocation problem: If it is optimal for the investor to buy and hold the
risky asset, then the optimal amount invested in the risky asset is given by
a = αU PM
α
x − r, 0 λβLPM
β
x − r, 0
1 β
−α
. 37
If, on the other hand, the short selling strategy is optimal, then the optimal amount that should be sold short is given by
a = αU PM
α
r − x, 0 λβLPM
β
r − x, 0
1 β
−α
. 38
The solution for the optimal capital allocation problem exists under condition that β α
. Observe that both values for a in 37 and 38 are always non-negative which means that there are two local maxima interior solutions in the optimal capital allocation
problem within the PTCPT framework. The investor chooses the strategy that gives the highest utility, but he never avoids the risky investment even when E[x] = r or
E [x] = r.
Next we present a brief comparative static analysis of expressions 29 and 33 for the optimal amount invested in the risky asset.
Corollary 10. The optimal amount that should be invested in the risky asset or sold short decreases as the investor’s risk aversion in the domain of either gains or losses
increases.
Proof. If it is optimal for the investor to buy-and-hold the risky asset, then the first-order derivative of a with respect to γ
+
is ∂
a ∂γ
+
= − E[x − r] − λ − 1LPM
1
x − r, 0 UPM
2
r − x, 0 λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0
2
0.
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If it is optimal for the investor to sell short the risky asset, then the first-order derivative of a with respect to γ
+
is ∂
a ∂γ
+
= − E[r − x] − λ − 1LPM
1
r − x, 0 UPM
2
r − x, 0 λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0
2
0. Similarly, if it is optimal for the investor to buy-and-hold the risky asset, then the first-
order derivative of a with respect to γ
−
is ∂
a ∂γ
−
= − λ
E[x − r] − λ − 1LPM
1
x − r, 0 LPM
2
r − x, 0 λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0
2
0. If it is optimal for the investor to sell short the risky asset, then the first-order derivative
of a with respect to γ
−
is ∂
a ∂γ
−
= − λ
E[r − x] − λ − 1LPM
1
r − x, 0 LPM
2
r − x, 0 λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0
2
0. Next consider the dependence of the optimal amount that should be invested in the
risky asset or sold short on the loss aversion parameter λ. The computation of the first-order derivative of a, given by 29, with respect to λ gives
∂ a
∂λ = −
A × LPM
1
x − r, 0 + γ
−
B × LPM
2
r − x, 0 λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0
2
, where
A = λγ
−
LP M
2
r − x, 0 + γ
+
U P M
2
r − x, 0, B = E[x − r] − λ − 1LPM
1
x − r, 0. Since A 0 due to 28 and B 0 due to 27, the sign of
∂ a
∂λ
is obviously negative if γ
−
≥ 0. However, if γ
−
0, then the sign of
∂ a
∂λ
might be positive. The latter means that when the investor appreciates uncertainty in losses, then an increase in loss
aversion might increase the optimal amount invested in the risky asset. This is obviously counter-intuitive. We believe that the explanation for this paradox lies in the fact that the
piecewise-quadratic utility function 18 is not always increasing for all values of W − W
. This might result in a number of paradoxes similar to those for the standard quadratic utility function. Note that if γ
−
0, then by condition 28 we must have γ
+
0. In this case the piecewise-quadratic utility is increasing only in the range
1 γ
−
W − W
1 γ
+
. For example, if W − W
decreases below
1 γ
−
, then the investor’s utility begins to increase which is not sensible. Generally, when the piecewise-quadratic utility function
is not increasing for all W − W , we can stumble upon some paradoxes. Some of these
paradoxes will be due to the violation of the first-order stochastic dominance principle. For the standard quadratic utility function this type of paradoxes were first noted by
Borch 1969 and Feldstein 1969.
8. Performance Measurement in the Generalized Framework
