129
described as being “fuzzy” or vague, in terms of both the direction and the size of market moves. Nevertheless, such fuzzy views are the ones needed to structure
portfolios so that the target return, which is assumed to be higher than the risk-free theory to select optimal portfolios that target returns above the risk-free rate by taking
only market risk.
6.2 A Mean VaR portfolio selection multi-objective model with transaction costs
A value-at-risk VaR model measures market risk by determining how much the value of a portfolio could decline over a given period of time with a given probability as a result of
changes in market prices or rates. The two most important components of VaR models are the length of time over which market risk is to be measured and the confidence level at
which market risk is measured. The choice of these components by risk managers greatly affects the nature of the value-at-risk model.
We begin by using the rates of return of the risky securities in the economic have a multivariate normal distribution. In practice, this is a popular assumption when
computing a portfolio’s VaR see Hull and White [74]. In this section, we will study a possibilistic mean VaR multi-objective model with
transaction costs.
6.2.1 Case of Mean downside-risk
In this section we extended Chen et al [31], Inuiguchi and Ramik [75] for n assets. In practice investors are concerned about the risk that their portfolio value falls
below a certain level. That is the reason why different measures of downside-risk are considered in the multi asset allocation problem. Denoted the random variable
i
ν
,
q i
, 1
=
the future portfolio value, i.e., the value of the portfolio by the end of the planning period, then the probability
i i
VaR P
ν
,
q i
, 1
=
130
that
i
ν
the portfolio value falls below the
i
VaR
level, is called the shortfall probability. The conditional mean value of the portfolio given that the portfolio value has fallen
below VaR
i
, called the expected shortfall, is defined as
i i
i
VaR E
ν ν
. Other risk measures used in practice are the mean absolute deviation
{ }
i i
i i
E E
E ν
ν ν
ν −
, and the semi-variance
2 i
i i
i
E E
E
ν ν
ν ν
−
, where we consider only the negative deviations from the mean.
Let
, 1
n j
x
j
=
represents the proportion of the total amount of money devoted to security
j
and
j
M
1
and
j
M
2
represent the minimum and maximum proportion of the total amount of money devoted to security
j
, respectively. For
n j
, 1
=
,
q i
, 1
=
let
ji
r
be a random variable which is the rate of the i return of security j. Then we have
∑
=
=
n j
j ji
i
x r
1
ν
. Assume that an investor wants to allocate hisher wealth among n risky securities. If the
risk profile of the investor is determined in terms of VaR
i
,
q i
, 1
=
, a mean-VaR efficient portfolio will be a solution of the following .
Multi-objective optimization problem
[ ]
, ,
max
1 q
R x
E E
n
ν ν
L
∈
6.1
q i
VaR to
subject
i i
i
, 1
, }
Pr{ =
≤ ≤
β ν
, 6.2
∑
=
=
n j
j
x
1
1
, 6.3
n j
M x
M
j j
j
, 1
,
2 1
= ≤
≤
. 6.4 In this model, the investor is trying to maximize the future value of portfolio,
which requires the probability that the future value of his portfolio falls below VaR
i
not to be greater than
i
β
,
q i
, 1
= .
131
6.2.2. Case of the proportional transaction costs model
The introduction of transaction costs adds considerable complexity to the optimal portfolio selection problem. The problem is simplified if one assumes that the transaction
costs are proportional to the amount of the risky asset traded, and there are no transaction costs on trades in the riskless asset
. Transaction cost is one of the main sources of concern to managers see Arnott and Wagner [1], Zhou and Li [218] are found that ignoring
transaction costs would result in efficient portfolio and some conclusion. Assume the rate of transaction cost of security j
n j
, 1
=
and allocation of i,
q i
, 1
=
assets is
ji
c
, thus the transaction cost of security j and allocation of i assets is
j ji
x c
. The transaction cost of portfolio
,...,
1 n
x x
x =
is
q i
x c
n j
j ji
, 1
,
1
=
∑
=
. Considering the proportional transaction cost and the shortfall probability constraint, we purpose the
following mean VaR portfolio selection model with transaction costs:
⎥ ⎦
⎤ ⎢
⎣ ⎡
− −
∑ ∑
= =
∈ n
j j
jk k
n j
j j
R x
x c
v E
x c
v E
Max
n
1 1
1 1
...., ,
6.5
i i
i
VaR v
to subject
β ≤
} Pr{
,
q i
, 1
=
, 6.6
∑
=
=
n j
j
x
1
1
, 6.7
n j
M x
M
j j
j
, 1
,
2 1
= ≤
≤
. 6.8
6.3 Possibilistic mean Var portfolio selection model.
