J. van der Hoek, M. Sherris Insurance: Mathematics and Economics 28 2001 69–82 73
4. Investment selection
4.1. Expected utility The expected utility-based portfolio theory approach to investment selection is to assume that investors maximize
expected utility subject to constraints. Panjer et al. 1998, Chapter 8 gives more details. It will be assumed that the investor has a single period investment horizon and that an appropriate utility function exists. Assume that there are
N + 1 assets. The return on the ith asset, i = 0, 1, 2, . . . , N, is given by R
i
. The investor with initial wealth W selects a portfolio x
T
= x , x
1
, . . . , x
N
and this selection provides wealth of W
1
= W 1 + R
x
at the end of the period, where R
x
= P
N i=0
x
i
R
i
. The problem is then max
E [U W
[1 + R
x
]], subject to
P
N i=0
x
i
= 1, and possibly other constraints such as non-negativity and perhaps a shortfall constraint. Substituting x
= 1− P
N i=1
x
i
into the objective to be maximized, we obtain max E
U W
1 + R +
N
X
i=1
x
i
R
i
− R .
Differentiating with respect to each x
i
gives the first-order conditions for an optimum as E
R
i
− R ∂
∂W
1
U W
1
= 0 for i = 1–N. We assume that U is strictly increasing ∂∂W
1
U 0 and risk aversion so that U is concave ∂
2
∂W
2 1
U 0.
The first-order conditions are necessary and sufficient for a maximum. We also have that E
[U W [1 + R
x
]] ≤ UE[W [1 + R
x
]], and therefore
U
−1
E [U W
[1 + R
x
]] ≤ E[W [1 + R
x
]], where U
−1
E [U W
[1 + R
x
]] is the certainty equivalent of the random end of period wealth. 4.1.1. Quadratic utility
A common assumption underlying mean-variance portfolio selection models is that the utility function is quadratic. In this case
U W = W − 1
2b W
2
with b 0, W b, ∂
∂W U W = 1 −
1 b
W, and the first-order conditions are
E
R
i
− R
1 −
1 b
W
1 + R
+
N
X
j =1
x
j
R
j
− R
= 0 for i = 1–N,
74 J. van der Hoek, M. Sherris Insurance: Mathematics and Economics 28 2001 69–82
which simplifies to
1 −
1 b
W 1 + R
E [R
i
− R ] −
1 b
W
N
X
j =1
x
j
E [R
j
− R R
i
− R ]
= 0 for i = 1–N. If we let
C =
E [R
1
− R R
1
− R ]
· · · E
[R
N
− R R
1
− R ]
E [R
1
− R R
2
− R ]
· · · E
[R
N
− R R
2
− R ]
.. .
. . .
.. .
E [R
1
− R R
N
− R ]
· · · E[R
N
− R R
N
− R ]
,
E =
ER
1
− R ER
2
− R ..
. ER
1
− R
,
x =
x
1
x
2
.. .
x
N
,
then the solution to the first-order conditions can be written as
x = kC
−1
E ,
where k =
b W
− 1 + R .
Note that the constraint P
N i=0
x
i
= 1 is automatically satisfied. 4.1.2. Gaussian returns and exponential utility
In this case we assume that the returns R , R
1
, . . . , R
N
have a multivariate normal distribution. Therefore, R
x
= P
N i=0
x
i
R
i
has a normal distribution as does W
1
= W 1 + R
x
. Assume the vector of expected returns is µ
T
= µ , µ
1
, µ
2
, . . . , µ
N
with E[R
i
] = µ
i
and the covariance matrix is Σ = σ
ij
with σ
ij
= CovR
i
, R
j
. With x
T
= x , x
1
, . . . , x
N
, where x
i
is the proportion in asset i such that P
N i=1
x
i
= 1, we then have E
[W
1
] = E W
1 +
N
X
i=0
x
i
R
i
= W 1 + µ
T
x, Var[W
1
] = W
2
VarR
x
= W
2 N
X
i=1 N
X
j =1
x
i
σ
ij
x
j
= W
2
x
T
Σ x. We assume that
U W = −exp−bW with b ≥ 0, W 0, so that
∂ ∂W
U W = b exp−bW 0.
J. van der Hoek, M. Sherris Insurance: Mathematics and Economics 28 2001 69–82 75
The optimization problem is max
E [−exp−bW
[1 + R
x
]], subject to
P
N i=0
x
i
= 1. Using the moment generating function of the normal distribution we then have the objective
max[−exp−bW 1 + µ
T
x +
1 2
b
2
W
2
x
T
Σ x ],
which simplifies to max
2 bW
µ
T
x − x
T
Σ x ,
subject to P
N i=0
x
i
= 1. Note that for the multivariate normal distribution assumption
PrW [1 + R
x
] ≤ 0 0, so that unlimited liability is assumed in this case.
In Panjer et al. 1998 the objective for mean-variance optimization is expressed as max{2τ µ
T
x − x
T
Σ x}. The objective for exponential utility with Gaussian returns is therefore the same as for the standard mean-variance
problem with τ = 1bW . Note that as initial wealth W
increases the risk tolerance parameter decreases. This is often considered to be an undesirable feature in determining asset allocations since tolerance to risky investments
decreases with increasing initial wealth. 4.2. Dual theory and distortion functions
For asset allocation purposes, we would normally use H
g
W 1 + R
x
as the certainty equivalent of the random end of period wealth.
In order to be consistent with expected utility and risk aversion, we require H
g
W 1 + R
x
≤ E[W [1 + R
x
]], which holds if g is convex. This contrasts with the insurance case, where the random variable under consideration
is a non-negative loss variable, i.e. a negative change in wealth. For the insurance non-negative loss variable the function g is concave.
Some special cases with convex g are as follows: The PH transform
: gx = x
r
, r ≥ 1.
The dual-power transform :
gx = 1 − [1 − x]
1r
, r ≥ 1.
Under the dual theory the investor is assumed to solve the following problem: max H
g
W 1 + R
x
,
76 J. van der Hoek, M. Sherris Insurance: Mathematics and Economics 28 2001 69–82
subject to the same constraints as for the standard portfolio selection problem. Assuming that H
g
R
x
is defined as in this paper, where we require that R
x
≥ 0, then from the properties of the distortion functions H
g
W 1 + R
x
= W + W
H
g
R
x
. To apply this risk measure to asset allocation we will need to define H
g
R
x
over all possible values of R
x
since asset returns can in general be negative as well as positive. One approach to this problem is to use the extension of
H
g
X to random variables taking both positive and negative values proposed by Wang et al. 1997 based on the
Choquet integral. In this approach, for any random variable X with decumulative distribution S
X
t −∞ t ∞ the Choquet integral is given by
H
g
X = Z
−∞
{g[S
X
t ] − 1} dt +
Z
∞
g [S
X
t ] dt.
5. A new class of risk measure
