In this paper we consider the problem of optimal portfolio selection within the class of translation-invariant and positive-homogeneous (TIPH) risk measures, popular in actuarial and financial contexts. Important members of this class are the value-at-risk (VaR) and the Tail Condition Expectation (TCE) (also known as Tail VaR (TVaR) or Expected Shortfall), which are suggested in BASEL II and SOLVENCY II. Notice that the variance premium risk measure (expected quadratic utility) associated with the classical mean variance (MV) model (see Markowitz, 1952, Boyle et al., 1998, Section 6; Steinbach, 2001, McNeil et al., 2005, Section 6.1.5) is not a member of the TIPH-class.
For risk returns following the multivariate normal model, optimal portfolio management with the TIPH-class leads to the optimization of a combination of linear and square root of quadratic functionals of portfolio weights. This phenomenon is also preserved for the more general multivariate elliptical class of risk distributions. This class is attractive for modeling heavy-tailed asset or loss returns (see Owen and Rabinovitch, 1983 and Bingham and Kiesel, 2002). Landsman and Makov (2011), who investigated the TIPH-class, provided a simple and feasible condition for the existence of the solution as well as an analytical solution to the optimization problem when none of the returns are riskless. In this paper we provide an analytical closed form expression for the optimal portfolio when one of the returns is riskless.
Let the vector of random variables View the MathML sourceXT=(X1,X2,…,Xn) of asset returns be multivariate elliptically distributed, written as View the MathML sourceX∽En(μ,Σ,g), namely, the density function of vector View the MathML sourceX can be expressed
equation(1.1)
View the MathML sourcefX(x)=cn|Σ|gn[12(x−μ)TΣ−1(x−μ)],
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for some column-vector View the MathML sourceμ,n×n-positive-definite matrix View the MathML sourceΣ=‖σij‖i,j=1n, and for function gn(t)gn(t), called the density generator (see details in Fang et al., 1990).
A risk measure, which may be denoted by ρρ, is defined to be a mapping from the space of risks (random variables) XX to the real line RR. In effect, we have ρ:X∋X→ρ(X)∈Rρ:X∋X→ρ(X)∈R.
The TIPH-class of risk measures is defined as the class of risk measures that satisfies the following two conditions:
1.
Translation invariance: for any constant αα, one has that
ρ(X+α)=ρ(X)+α.ρ(X+α)=ρ(X)+α.
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2.
Positive homogeneity: for any positive constant γ>0γ>0, one has that
ρ(γX)=γρ(X).ρ(γX)=γρ(X).
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This class includes, in addition to the VaR and TCE (TVaR) risk measures, the distorted function-based risk measure Wang (1995), Wang (1996), Dhaene et al. (2006) and the Tail Standard Deviation (TSD) risk measure Furman and Landsman (2006).
Let View the MathML sourceR=xTX be the portfolio return, where View the MathML sourcexT=(x1,…,xn) is the vector of real numbers. Let the risk measure ρρ, a member of the TIPH-class, be related to losses L=−RL=−R and, therefore, is to be minimized. Then we can write
equation(1.2)
View the MathML sourceρ(−R)=−μTx+ρ(xTX−μTxvar(xTX)var(xTX))=−μTx+ρ(Z)xTΣx,
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where View the MathML sourceZ=xT(X−μ)/xTΣx. The property of an elliptical family that
View the MathML sourceAX+b∼Em(Aμ+b,AΣAT,gm)
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for a m×nm×n matrix AA of rank m≤nm≤n, implies that ZZ is distributed as E1(0,1,g1)E1(0,1,g1), and so the distribution of ZZ, a standard univariate elliptical random variable, does not depend on vectors View the MathML sourceμ,x, or matrix ΣΣ, and consequently, nor does ρ(Z)ρ(Z). In a special case, when View the MathML sourceX is nn-dimensional normal distribution, View the MathML sourceX∼Nn(μ,Σ),Z has the standard normal distribution, N(0,1)N(0,1).
For a value-at-risk as a TIPH risk measure, View the MathML sourceρ(⋅)=V aRq(⋅), (1.2) holds with
equation(1.3)
View the MathML sourceρ(Z)=Zq=V aRq(Z)=FZ−1(q).
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If the mean vector exists, it coincides with vector View the MathML sourceμ, and then the tail condition expectation TCEq(X)TCEq(X) exists and is a TIPH risk measure. For details see Landsman and Makov (2011), where ρ(Z)ρ(Z) is also discussed for tail standard deviation risk.
The problem of the minimization of risk measure from the TIPH class is equivalent to that of the problem of the minimization of the functional
equation(1.4)
View the MathML sourceρ(−R)=−μTx+λxTΣx,λ>0,
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where View the MathML sourceμ=EX and View the MathML sourceΣ=cov(X). This is a combination of the linear functional and square root of a quadratic functional with a balance parameter λ>0λ>0, subject to the linear constraint
equation(1.5)
View the MathML source1Tx=1,
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where View the MathML source1 is the vector-column of nn ones. However, if one of the asset returns is riskless, ΣΣ is singular and the considered optimization problem requires a new methodology which is the central topic of this paper.
Section 2 is devoted to the solution of the optimization problem when the model contains the riskless component. In Section 3 we illustrate the results using the data of 10 stocks from the NASDAQ Computer Index and the return of the riskless component.
