The rest of the paper is organised as follows. In Section 2 we briefly review the justification of the mean-variance analysis and present the results we want to
generalise. In Section 3 we present assumptions, definitions, and notation that we will use in our generalised framework. In Section 4 we perform the approximation analysis
and generalise mean-variance utility. In Section 5 we generalise the Arrow-Pratt risk premium. In Section 6 we discuss briefly the impact of the decision maker’s skewness
preferences on risk measurement in our generalised framework. In Section 7 we analyse the optimal capital allocation problem. In Section 8 we derive the expression for a
portfolio performance measure. Section 9 concludes the paper.
2. Expected Utility Theory and the Mean-Variance Analysis
In this section we present a brief justification of the mean-variance analysis as well as some of its most important results. Throughout the paper we consider a decision
maker with random wealth W . In this section we assume that the decision maker has a von Neumann-Morgenstern utility function which is defined over wealth as a single
function U W . The decision maker’s objective is to maximise the expected utility of wealth, E[U W ], where E[.] is the expectation operator.
We suppose that the utility function U is increasing in wealth and is a differentiable function. Then, taking a Taylor series expansion around some deterministic W
, the expected utility of the decision maker can be written as
E[U W ] =
∞ n=0
1 n
U
n
W E
W − W
n
. Our intension now is to keep the terms up to the second derivative of U and disregard
6
the terms with higher derivatives of U . This gives us E[U W ] ≈ U W
+ U
1
W E[W − W
] + 1
2 U
2
W E
W − W
2
. Since a utility function is unique up to a positive linear transformation and U
1
0, then a convenient form of equivalent expected utility is
E[ U W ] = E
U W − U W U
1
W = E[W − W
] − 1
2 γ
E W − W
2
, 1
where γ
= − U
2
W U
1
W 2
is the Arrow-Pratt measure of absolute risk aversion. The equivalent expected utility 1 can be interpreted as mean-variance utility since the term E[W − W
2
] is proportional to variance. Observe that we can arrive at the same expression for the expected utility
as 1 if we assume that the utility function of the decision maker is quadratic U W = W − W
− 1
2 γ
W − W
2
. 3
6
This can be justified when at least one of the following conditions is satisfied: the investor’s utility function is quadratic see Tobin 1969; the probability distribution is normal see
Tobin 1969; the probability distribution belongs to the family of ‘compact’ or ‘small risk’ distributions see Samuelson 1970; the aggregate risk is small compared with the wealth
see Tsiang 1972.
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Consider now a risk averse decision maker with deterministic wealth W . According
to Arrow 1971, p. 90 ‘a risk averter is defined as one who, starting from a position of certainty, is unwilling to take a bet which is actuarially fair’. More formally, if x is a
fair gamble such that E[x] = 0, then
E[U W + x] U W
. 4
Beginning from the landmark papers of Arrow and Pratt, it became standard in economics to characterize the risk aversion in terms of either a certainty equivalent,
Cx, or a risk premium, π x, which are defined by the following indifference condition
E[U W + x] = U W
+ Cx = U W + E[x] − πx.
5 This says that the decision maker is indifferent between receiving x and receiving a
non-random amount of Cx = E[x] − πx. Observe that for a fair gamble Cx = −πx. If the risk of a fair gamble is small, then the decision maker’s preferences can
be approximated by quadratic utility. The use of quadratic utility 3 in the indifference condition 5 for W = W
+ x combined with the assumption of risk aversion 4 gives the following expression
7
for the risk premium and certainty equivalent π
x = −Cx = 1
2 γ
Var[x]. 6
This is the famous result of Pratt 1964. Now consider the optimal capital allocation problem of an investor with the quadratic
utility function. The investor wants to allocate his wealth between a risk-free and a risky asset. The return on the risky asset over a small time interval t is
x = μt + σ √
t ε, 7
where μ and σ are, respectively, the mean and standard deviation of the risky asset return per unit of time, and ε is some normalized stochastic variable such that E[ε] =
0 and Var[ε] = 1. The return on the risk-free asset over the same time interval equals
r = ρt, 8
where ρ is the risk-free interest rate per unit of time. We assume that the risky asset can be either bought or sold short without any limitations and the risk-free rate of return is
the same for both borrowing and lending. We further assume that the investor’s initial wealth is W
I
and he invests a in the risky asset and, consequently, W
I
− a in the risk-free asset. Thus, the investor’s wealth after t is
W = ax − r + W
I
1 + r. 9
The investor’s objective is to choose a to maximise the expected utility E[U
∗
W ] = max
a
E[U W ]. 10
Observe that there is some ambiguity in the choice of the level of wealth W around
which we perform the Taylor series expansion. A reasonable choice is W = W
I
1 + r.
7
We also need to disregard the term with C
2
x. This is valid since we assume that the risk is small.
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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
