A Closed-Form Solution for Robust Portfolio Selection with Worst-Case CVaR Risk Measure

Abstract

With the uncertainty probability distribution, we establish the worst-case CVaR (WCCVaR) risk measure and discuss a robust portfolio selection problem with WCCVaR constraint. The explicit solution, instead of numerical solution, is found and two-fund separation is proved. The comparison of efficient frontier with mean-variance model is discussed and finally we give numerical comparison with VaR model and equally weighted strategy. The numerical findings indicate that the proposed WCCVaR model has relatively smaller risk and greater return and relatively higher accumulative wealth than VaR model and equally weighted strategy.

1. Introduction

VaR (value at risk) has been a popular risk measure in finance industry and academic research and is written in New Basel Accord. But two difficulties are faced by user: (1) the explicit expression of VaR is unavailable unless the normal distribution assumption is done and (2) VaR, as a risk measure tool, does not satisfy the coherent axiom [1]. Hence, an approximation of VaR is often considered in practice by either assuming normal distribution or simulation method based historical data. Rockafellar and Uryasev [2] proposed an alternative risk measure, namely, conditional VaR (CVaR), which is coherent and provided a linear programming approximate with historical data. But the assumption that the return of risky asset follows the normal distribution is usually done when one computes CVaR by parameterized approaches. As we know, normal distribution can usually underestimate the loss of the rare event and is not clearly a very good approximation of the return of risky asset. This is still a challenge for computing an explicit expression of CVaR without any special distribution information. The current paper will explore this problem and establish the mean-CVaR portfolio model without probability distribution assumption.

Robust portfolio problems with parameters uncertainty are recently paid close attention to. Goldfarb and Iyengar [3], for instance, considered a class of robust portfolio problem with risk factors in which they solve numerically robust mean-variance portfolio problem, robust downside risk portfolio problem with normal distribution, and robust Sharpe ratio portfolio problem; see also, Costa and Paiva [4], Halldórsson and Tütüncü [5], Tütüncü and Koenig [6], Lu [7], and Ling and Xu [8] for the relative researches. The uncertainty of models above is only from the parameters under the deterministic distribution and cannot capture the uncertainty in distribution. El Ghaoui et al. [9] proposed the worst-case VaR (WCVaR) risk measure and considered a portfolio selection problem with minimization of WCVaR. Zhu and Fukushima [10] proposed the worst-case CVaR risk measure and discussed a robust mean-CVaR portfolio model with uncertainty discrete distribution. Some similar researches can be found in Zhu et al. [11] and Huang et al. [12]. A richer literature can be referred to in Fabozzi et al. [13].

We define the worst-case CVaR with uncertainty distribution including the continuous and discrete distribution and consider a portfolio selection problem with WCCVaR as risk measure. Our results extend that of Zhu and Fukushima [10] for which they considered only the discrete case to the case including the continuous and discrete distribution. Most of methods for robust portfolio problems are that one converts first the problems into convex cone (e.g., linear programming, second-order cone programming, or positive semidefinite programming) and then solves them numerically. Differently from these numerical methods, we consider an analytic solution approach for the proposed robust mean-WCCVaR problem. We discuss two cases of the proposed robust problems with and without risky-free asset and prove two-fund separable theorem. Numerical results and comparisons with VaR and equally weighted strategy for real market data are reported.

The outline of this paper is arranged as follows. We introduce the definition of worst-case CVaR, establish mean-WCCVaR portfolio model, and give the closed-form solution in Section 2 and Section 3 proves the two-fund separation theorem. The extension of the model with risky-free asset is considered in Section 4. Numerical results are reported in Section 5.

2. Mean-WCCVaR Portfolio Model

We consider mainly an investing and holding strategy in this paper for which the investor allocates his (her) assets at time 0 and collects his (her) returns of portfolio at time 1. Generally speaking, two things must be done at time 0: one is that the investor needs to estimate the returns of risky assets at time 1 using the available information at time 0 and another is that the investor must choose an optimal decision to allocate his (her) wealth.

Let there be 𝑛 available risky assets in the market and let their random returns vector be denoted by 𝐫=(𝑟₁,…,𝑟_𝑛)^𝑇∈ℝ^𝑛. The expected returns and covariance matrix are denoted, respectively, by 𝝁=𝔼[𝐫]=(𝜇₁,…,𝜇_𝑛)^𝑇∈ℝ^𝑛 and Σ=(𝜎_𝑖𝑗)_𝑛×𝑛, where 𝜎_𝑖𝑖=𝜎²_𝑖=var(𝑟_𝑖) is variance of asset 𝑖(𝑖=1,…,𝑛). The rate of return of risky-free asset is denoted by 𝑟_𝑓. The portfolio vector is 𝐰=(𝑤₁,…,𝑤_𝑛)^𝑇∈ℝ^𝑛 with 𝑤_𝑖 the proportion of wealth invested in asset 𝑖. The weight of wealth invested in risky-free asset is denoted by 𝑤_𝑓. Let 𝑓(𝐰,𝐫) be the loss for portfolio vector 𝐰 and satisfy 𝔼[|𝑓(𝐰,𝐫)|]<+∞, and let 𝐹(⋅) be the joint cumulative probability distribution of random vector 𝐫. Then the probability that the loss 𝑓(𝐰,𝐫) is not greater than a given constant 𝛼 is Ψ(𝐰,𝛼)=ℙ𝑓(𝐰,𝐫)≤𝛼=_{𝑓(𝐰,𝐫)≤𝛼}𝑑𝐹(𝐫). (1) Let 𝛽∈(0,1); then, with confidence level 𝛽, VaR_𝛽 can be expressed as VaR_𝛽(𝐰)=min𝛼∈ℝ∣Ψ(𝐰,𝛼)≥𝛽. (2) Rockafellar and Uryasev [2] defined CVaR as the conditional expectation of loss 𝑓(𝐰,𝐫) greater than VaR_𝛽. With the definition of CVaR and the given confidence level 𝛽, the mathematic formulation of CVaR can be written as CVaR_𝛽(𝐰)=𝔼𝑓(𝐰,𝐫)∣𝑓(𝐰,𝐫)≥VaR_𝛽(𝐰=1)1−𝛽_{𝑓(𝐰,𝐫)≥VaR𝛽(𝐰)}𝑓(𝐰,𝐫)𝑑𝐹(𝐫). (3) Let 𝐻_𝛽()1𝐰,𝛼=𝛼+𝔼𝑓()1−𝛽𝐰,𝐫−𝛼₊. (4) Then CVaR_𝛽(𝐰) can be expressed further as [2] CVaR_𝛽(𝐰)=min_𝛼𝐻_𝛽(𝐰,𝛼), (5) where (𝑎)₊=𝑎,𝑎>0;0,𝑎≤0. (6) Clearly, it is not possible to get an exact result of CVaR by (4) if we have not any information on the distribution of random vector 𝐫. Some sampling or simulation methods are used to computes the approximation of CVaR in the literature. We explore a closed-form solution in this paper with only partial distribution assumptions for random vector 𝐫, for which we assume that random vector 𝐫 follows a family of distributions 𝒟 defined by 𝒟={𝐫∣𝔼[𝐫]=𝝁,Cov(𝐫)=Σ≻0}, (7) where Σ≻0 means that Σ is a positive definite matrix and 𝒟 is called the uncertainty set of distribution of random vector 𝐫. Clearly, 𝒟 is a distribution family with given mean value 𝝁 and covariance matrix Cov, where 𝝁, Cov are assumed to be known. We then can compute CVaR_𝛽 when the worst-case probability distribution in 𝒟 occurs. To this end, we define worst-case CVaR as follows.

Definition 1. Let 𝛽∈(0,1), worst-case CVaR (WCCVaR) of portfolio 𝐰 under uncertainty set 𝒟 is defined by WCCVaR_𝛽(𝐰)=sup_𝐹∈𝒟CVaR_𝛽(𝐰)=sup_𝐹∈𝒟min_𝛼𝐻_𝛽(𝐰,𝛼). (8)

Noticing that 𝐻_𝛽(𝐰,𝛼) is the convex function of 𝛼 [14], then we have from max-min theorem [15] WCCVaR_𝛽(𝐰)=sup_𝐹∈𝒟min_𝛼𝐻_𝛽(𝐰,𝛼)=min_𝛼sup_𝐹∈𝒟𝐻_𝛽(𝐰,𝛼). (9) The following results are straightforward and a similar proof can be found in [10].

Theorem 2. WCCVaR_𝛽(𝐰) is a coherent risk measure and satisfies that WCCVaR_𝛽(𝐰)≥CVaR_𝛽(𝐰)≥VaR_𝛽(𝐰). (10)

Hence, WCCVaR_𝛽(𝐰) can be used as a risk measure and if the investor measures the risk of portfolio based on WCCVaR, then we can establish the mean-WCCVaR portfolio model by max_𝐰𝝁^𝑇𝐰s.t.WCCVaR_𝛽(𝐰)𝐞≤𝜏,_𝑇𝐰=1, (RP1) where 𝜏>0 is a preset constant. We solve mainly problem (RP1) by exploring an explicit approach. To this end, for convenience, we denote sometime 𝐫∈𝒟 by 𝐫∼(𝝁,Σ). The following result is helpful for our analysis later.

Lemma 3 (see [15]). Let 𝜉 be a random variable with mean value 𝜇 and variance 𝜎. 𝜌 is any real number. Then, for the supper bound of 𝔼[(𝜌−𝜉)₊], we have sup_{𝜉∼(𝜇,𝜎)}𝔼𝜌−𝜉₊=𝜌−𝜇+𝜎²+𝜌−𝜇²2. (11)

Lemma 4 (see [14, 16]). Let for any vector 𝐚∈ℝ^𝑛, 𝑆₁=𝐚^𝑇,𝑆𝐫∣𝐫∈𝒟₂=𝜂∣𝔼𝜂=𝐚^𝑇𝝁,Var𝜂=𝐚^𝑇Σ𝐚. (12) Then 𝑆₁=𝑆₂.

Lemma 3 provides an upper bound of 1-order lower partial moment for one dimensional random variable and Lemma 4 provides a relationship of uncertainty set between one dimensional and several dimensional random variable with given mean value and variance. Lemma 4 indicates also that 𝑆₁ (or 𝑆₂) is in fact a single variable distribution family with given mean 𝐚^𝑇𝝁 and variance 𝐚^𝑇Σ𝐚, where 𝐚, 𝝁, and 𝚺 are known. Further, we can obtain an explicit expression for sup in (4) if the loss 𝑓(𝐰,𝐫) is linear.

Lemma 5. If 𝑓(𝐰,𝐫)=−𝐫^𝑇𝐰, then for any 𝐫∈𝒟 and 𝛼∈ℝ, we have sup_𝐫∈𝒟𝔼𝑓(𝐰,𝐫)−𝛼₊=sup_𝐫∈𝒟𝔼−𝛼−𝐫^𝑇𝐰₊=12𝐰^𝑇Σ𝐰+𝝁^𝑇𝐰+𝛼²−𝝁^𝑇𝐰+𝛼. (13)

Proof. Let 𝐚=𝐰 and 𝜂=𝐰^𝑇𝐫. Then, it follows from Lemma 4 that sup_𝐫∈𝒟𝔼𝑓(𝐰,𝐫)−𝛼₊=sup_𝐫∈𝒟𝔼−𝛼−𝐚^𝑇𝐫₊=sup_{𝐰𝑇𝐫∈𝑆1}𝔼−𝛼−𝐰^𝑇𝐫₊=sup_𝜂∈𝑆2𝔼−𝛼−𝜂₊=sup_{𝐚𝜂∼𝑇𝝁,𝐚𝑇Σ𝐚}𝔼−𝛼−𝜂₊=sup_{𝐰𝜂∼𝑇𝝁,𝐰𝑇Σ𝐰}𝔼−𝛼−𝜂₊=12𝐰^𝑇Σ𝐰+𝝁^𝑇𝐰+𝛼²−𝝁^𝑇𝐰+𝛼. (14) The final equality follows from Lemma 3; this is the desired result.

Hence, from Lemma 5, WCCVaR_𝛽(𝐰) can be expressed by WCCVaR_𝛽(𝐰)=min_𝛼∈ℝ1𝛼+21−𝛽𝐰^𝑇Σ𝐰+𝝁^𝑇𝐰+𝛼²−𝝁^𝑇𝐰+𝛼. (15) Clearly, the right side of the equation above is convex function in 𝛼 and we can prove that WCCVaR_𝛽(𝐰) can be attained at 𝛼^∗=−𝝁^𝑇𝐰+2𝛽−12𝛽1−𝛽𝐰^𝑇Σ𝐰; (16) that is, WCCVaR_𝛽(𝐰)=−𝝁^𝑇𝐰+𝛽1−𝛽𝐰^𝑇Σ𝐰. (17) Then, robust mean-WCCVaR portfolio problem (RP1) can be expressed further as max_𝐰𝝁^𝑇𝐰s.t.−𝝁^𝑇𝐰+𝛽1−𝛽𝐰^𝑇𝐞Σ𝐰≤𝜏,_𝑇𝐰=1. (RP2) In the rest of this paper, we discuss mainly the solution of (RP2). To this end, let 𝑎=𝐞^𝑇Σ⁻¹𝐞, 𝑏=𝝁^𝑇Σ⁻¹𝐞, 𝑐=𝝁^𝑇Σ⁻¹𝝁, 𝑎₀=(𝑎𝑐−𝑏²)/𝑎, and 𝛽₀=(1−𝛽)/𝛽. Then, the following result gives feasible conditions of problem (RP2).

Lemma 6. If 𝑎₀𝛽²₀<1 and 𝜏≥max𝜏^∗,0, (18) then problem (RP2) is feasible, where 𝜏^∗=1−𝑎₀𝛽²₀/(𝛽₀√𝑎)−𝑏/𝑎.

Proof. For any given 𝛽∈(0,1), consider the problem 𝜏^∗=min_𝐰−𝝁^𝑇𝐰+𝛽1−𝛽𝐰^𝑇Σ𝐰∣s.t.𝐞^𝑇𝐰=1. (19) Let 𝑣=√𝐰Σ𝐰 and let 𝑣 be viewed as a new variable. Denote the optimal solution of problem (19) by 𝐰; then 𝐰 satisfies the first order condition: −𝝁+𝜆^′𝐞+2𝜆^′′Σ⁻¹𝐰=0,2𝜆^′′𝑣=𝛽,𝐞1−𝛽_𝑇𝐰𝐰=1,_𝑇Σ𝐰=𝑣², (20) where 𝜆^′,𝜆^′′∈ℝ are the Lagrange multipliers. It is not hard to obtain from the first and third equations of (20) 1𝐰=2𝜆^′′Σ⁻¹𝑏𝝁−𝑎Σ⁻¹𝐞+1𝑎Σ⁻¹𝐞. (21) Substituting 𝐰 in the fourth equation and combining the second equation of (20), then we can get the optimal solution of (19) when 𝑎₀𝛽²₀<1𝛽𝐰=⁰𝑎1−𝑎₀𝛽²₀Σ⁻¹𝑏𝝁−𝑎Σ⁻¹𝐞+1𝑎Σ⁻¹𝐞. (22) The optimal value 𝜏^∗=1−𝑎₀𝛽²₀𝛽₀√𝑎−𝑏𝑎. (23) If 𝜏=𝜏^∗>0, then 𝐰 is unique solution of problem. This means that the feasible condition is 𝜏≥max{𝜏^∗,0}. The proof is finished.

Lemma 6 means that the portfolio problem (RP2) is well defined if the investor chooses an appropriate risk tolerance parameter 𝜏. The following theorem gives the main results of the current paper.

Theorem 7. If 𝑎₀𝛽²₀<1 and 𝜏>max{𝜏^∗,0}, the optimal solution of the problem (RP2) can explicitly be expressed 𝐰^∗=𝑓(𝜏)Σ⁻¹𝑏𝝁−𝑎Σ⁻¹𝐞+1𝑎Σ⁻¹𝐞, (24) where 𝑓(𝜏)=𝑎₀𝑏𝛽²₀(𝜏+𝑏/𝑎)+𝑏𝑎₀𝛽²₀(𝜏+𝑏/𝑎)²−𝑎₀/𝑎1−𝑎₀𝛽²₀𝑎₀1−𝑎₀𝛽²₀. (25)

Proof. Let 𝑣=√𝐰Σ𝐰; the optimization problem (RP2) can be rewritten as max_𝐰,𝑣>0𝔼𝐫^𝑇𝐰s.t.𝑣−𝛽₀𝝁^𝑇𝐰≤𝛽₀𝐞𝜏,_𝑇𝐰𝐰=1,_𝑇Σ𝐰=𝑣². (26) Let 𝐰^∗ be the optimal solution of problem (26). Then from KKT condition, 𝐰^∗ satisfies that 1+𝛽₀𝜆₁𝝁−𝜆₂𝐞−2𝜆₃Σ𝐰^∗=0,−𝜆₁+2𝜆₃𝐞𝑣=0,_𝑇𝐰^∗𝐰=1,_∗𝑇Σ𝐰^∗=𝑣²,𝑣−𝛽₀𝝁^𝑇𝐰^∗−𝛽₀𝜏=0, (27) where 𝜆₁≥0, 𝜆₂∈ℝ and 𝜆₃∈ℝ are Lagrange multipliers. It follows from the first and third equations in (27) that 𝐰^∗=1+𝛽₀𝜆₁2𝜆₃Σ⁻¹𝑏𝝁−𝑎Σ⁻¹𝐞+1𝑎Σ⁻¹𝐞. (28) Substituting 𝐰^∗ in the fourth and fifth equation in (27), we have 𝑎₀(1+𝛽₀𝜆₁)2𝜆₃²+1𝑎=𝑣²,𝑎₀𝛽₀1+𝛽₀𝜆₁2𝜆₃=𝑣−𝛽₀𝑏𝜏+𝑎. (29) Eliminating 𝑣 from (29), it follows that the quadratic equation with respect to (1+𝛽₀𝜆₁)/2𝜆₃ is 𝑎₀1−𝑎₀𝛽²₀(1+𝛽₀𝜆₁)2𝜆₃²−2𝑎₀𝛽²₀𝑏𝜏+𝑎1+𝛽₀𝜆₁2𝜆₃+1𝑎−𝛽²₀𝑏𝜏+𝑎²=0. (30) Notice that 𝑣>0 and 𝜆₁≥0; then from the second equation in (27), 𝜆₃>0, this means that (1+𝛽₀𝜆₁)/2𝜆₃>0. Solving directly the quadratic equation above in (1+𝛽₀𝜆₁)/2𝜆₃, we have that 1+𝛽₀𝜆₁2𝜆₃=𝑎₀𝛽²₀(𝜏+𝑏/𝑎)+𝑎₀𝛽²₀(𝜏+𝑏/𝑎)²−𝑎₀/𝑎1−𝑎₀𝛽²₀𝑎₀1−𝑎₀𝛽²₀>0. (31) Then the optimal solution of problem (RP2) can be obtained by substituting it in (27). This finishes the proof.

We need that condition 𝑎₀𝛽²₀<1 holds in Lemma 6 and Theorem 7. This condition can in fact be easily attained. Notice that 𝛽₀=(1−𝛽)/𝛽. Thus, for any input data 𝑎₀, in order to have 𝑎₀𝛽²₀<1, we only require that 𝛽 satisfies 𝑎𝛽>⁰1+𝑎₀. (32) If 𝛽∈(1/2,1), the condition 𝑎₀𝛽²₀<1 can be satisfied while 1𝛽>max2,𝑎₀1+𝑎₀. (33)

3. Two-Fund Separation Theorem

In our analysis of this section, we view 𝐰^∗ as a function of input parameter 𝜏; that is, we denote it by 𝐰^∗=𝐰(𝜏). Let 𝑆(𝐰)=𝐰(𝜏)∶max𝜏^∗,0≤𝜏<∞. (34) Then, set 𝑆(𝐰) is the solution space of problem (RP2). Now we are interested in the question that whether the solution of problem (RP2) satisfies the two-fund separation theorem or not.

Theorem 8. Let 𝐰(𝜏₁), 𝐰(𝜏₂)(𝜏₁≠𝜏₂) be two solutions of mean-WCCVaR model; that is, 𝐰(𝜏₁),𝐰(𝜏₂)∈𝑆(𝐰). Then for any 𝜏∈[max{𝜏^∗,0},∞) and the corresponding solution 𝐰(𝜏), there exists a real number 𝜃, such that 𝐰(𝜏)=𝜃𝐰𝜏₁+(1−𝜃)𝐰𝜏₂; (35) that is, the two-fund separation theorem holds.

Proof. Noting 𝑓(𝜏) in Theorem 7, the optimal solution of mean-WCCVaR portfolio problem (RP2) can be written in the simple form 𝐰(𝜏)=𝑓(𝜏)𝐰_𝐴+1−𝑓(𝜏)𝐰_𝜎, (36) where 𝐰_𝜎=(Σ⁻¹𝐞)/𝑎 is the portfolio with the minimum variance and 𝐰_𝐴=(Σ⁻¹𝝁)/𝑏. For given 𝜏₁,𝜏₂∈[max{𝜏^∗,0},∞), notice that 𝜏₁≠𝜏₂; it follows that 𝑓(𝜏₁)≠𝑓(𝜏₂). Then for any 𝜏∈[max{𝜏^∗,0},∞), let 𝑓(𝜏)𝜏𝜃=−𝑓₂𝑓𝜏₁−𝑓𝜏₂; (37) we have 𝑓(𝜏)=𝜃𝑓𝜏₁+(1−𝜃)𝑓𝜏₂. (38) Thus 𝐰(𝜏)=𝑓(𝜏)𝐰_𝐴+1−𝑓(𝜏)𝐰_𝜎=𝜃𝑓𝜏₁+(1−𝜃)𝑓𝜏₂𝐰_𝐴+1−𝜃𝑓𝜏₁+(1−𝜃)𝑓𝜏₂𝐰_𝜎=𝜃𝑓𝜏₁𝐰_𝐴+1−𝑓𝜏₁𝐰_𝜎+(1−𝜃)𝑓𝜏₂𝐰_𝐴+1−𝑓𝜏₂𝐰_𝜎=𝜃𝐰𝜏₁+(1−𝜃)𝐰𝜏₂. (39) This gets the desired conclusion.

4. Efficient Frontier

We discuss the efficient frontier of optimal solution of problem (RP2) and analyze the relationship of the efficient frontier between problem (RP2) and mean-variance (MV) model. For any given parameter 𝜏, clearly, the optimal solution 𝐰^∗=𝐰(𝜏) and the expected return of portfolio are the function of 𝜏. Hence, we have 𝑅(𝜏)=𝑎₀𝛽²₀(𝜏+𝑏/𝑎)+𝑎₀𝛽²₀(𝜏+𝑏/𝑎)²−𝑎₀/𝑎1−𝑎₀𝛽²₀1−𝑎₀𝛽²₀+𝑏𝑎. (40) Rearranging this equality, we have the quadratic equation in 𝑅(𝜏) and 𝜏1−𝑎₀𝛽²₀²𝑏𝑅(𝜏)−𝑎²−2𝑎₀𝛽²₀1−𝑎₀𝛽²₀𝑅(𝜏)−𝑏𝑎𝑏𝜏+𝑎+𝑎₀𝛽²₀𝑎₀𝛽²₀𝑏−1𝜏+𝑎²+𝑎₀𝑎1−𝑎₀𝛽²₀=0. (41) The determinant of quadratic term coefficient can be expressed as 1−𝑎₀𝛽²₀²−𝑎₀𝛽²₀1−𝑎₀𝛽²₀−𝑎₀𝛽²₀1−𝑎₀𝛽²₀𝑎₀𝛽²₀𝑎₀𝛽²₀−1=−𝑎₀𝛽²₀1−𝑎₀𝛽²₀²<0. (42) This means from the theory of quadratic curve that the efficient frontier determined by (41) is a branch of the hyperbola and the portfolio at the efficient frontier has the maximum expected return for given 𝜏. The asymptotic line equation of efficient frontier is 𝑅(𝜏)=1+𝑎₀𝛽²₀1−𝑎₀𝛽²₀𝑎₀𝛽²₀𝑏𝜏+𝑎, (43) the intercept at 𝑅(𝜏) axis is 𝑅_𝑏=𝑏𝑎⋅1+𝑎₀𝛽²₀1−𝑎₀𝛽²₀, (44) and the center is (−𝑏/𝑎,𝑏/𝑎). Hence, the location of the hyperbola is determined by sign of 𝑏.

Now, we will discuss a relationship of efficient frontiers between the mean-variance (MV) model and the proposed mean-WCCVaR model. To this end, we compare the mean-WCCVaR model with the following MV model: max𝐸𝝁^𝑇𝐰∶𝐰^𝑇Σ𝐰≤𝜏²,𝐞^𝑇𝐰=1. (45) It is not hard to compute that the optimal solution of MV model is 𝐰^∗_MV=𝐰_MV(𝜏)=1−𝑏𝜏²−1/𝑎𝑎₀Σ⁻¹𝐞𝑎+𝜏²−1/𝑎𝑎₀Σ⁻¹𝝁. (46) The corresponding expected value at the optimal solution is 𝑅_MV(𝜏)=𝝁^𝑇𝐰^∗_MV=𝑎𝑐−𝑏²𝑎𝜏²−1/𝑎𝑎₀+𝑏𝑎. (47) In (𝜏,𝑅_MV(𝜏)) plane, function 𝑅_MV(𝜏) plots the efficient frontier of MV: 𝑎𝜏²_MV−𝑎𝑎₀𝑅_MV𝑏(𝜏)−𝑎²=1, (48) whose slope of asymptotic line is √𝑎₀. Let 1+𝑎₀𝛽²₀1−𝑎₀𝛽²₀𝑎₀𝛽²₀=√𝑎₀. (49) Then, 𝛽₀=1+4𝑎₀+√𝑎₀−12𝑎₀+√𝑎₀=∶𝛽₁. (50) If 𝛽₀≥𝛽₁, then for the case of 𝜏^∗<𝜏_MV, two asymptotic lines are parallel and therefore efficient frontiers are not intersection; see Figure 1. If 𝛽₀<𝛽₁, then the slope of asymptotic line of efficient frontier for mean-WCCVaR model is less than that of MV model and two efficient frontiers can be intersected when 𝜏^∗<𝜏_MV; see Figure 2(b).

(a) 𝛽0>𝛽1

(b) 𝛽0=𝛽1

Figure 1 

The relationship of efficient frontiers of two models 𝛽0≥𝛽1.

(a) 𝛽0<𝛽1, 𝜏∗>𝜏MV

(b) 𝛽0<𝛽1, 𝜏∗<𝜏MV

Figure 2 

The relationship of efficient frontiers of two models in the case of 𝛽0<𝛽1.

Generally speaking, it is more conservative for mean-WCCVaR model than MV model. But the conservative performance can be improved by adjusted the confidence level 𝛽. For example, if 𝛽 is small, such that 𝛽₀>𝛽₁, then mean-WCCVaR has the higher expectation return at the same 𝜏; see Figure 1. This means that ER/𝜏 is also higher; that is, the expectation return of per unit WCCVaR risk (reflected by parameter 𝜏) is higher. The other case is that if we choose 𝛽, such that 𝛽₀<𝛽₁, then mean-WCCVaR model can obtain still the large expected return at the same 𝜏 if 𝜏 is not large; see Figure 2.

5. An Extension with Risky-Free Asset

We consider the portfolio with risky-free asset in this section and therefore the optimization problem can be expressed as max_{𝐰,𝑤𝑓}𝔼𝐫^𝑇𝐰+𝑤_𝑓𝑟_𝑓s.t.−𝝁^𝑇𝐰−𝑤_𝑓𝑟_𝑓+𝛽1−𝛽𝐰^𝑇𝐞Σ𝐰≤𝜏,_𝑇𝐰+𝑤_𝑓=1. (RPF) Clearly, (𝐰,𝑤_𝑓)=(𝟎,1) is a strictly feasible solution of problem (RPF). Hence, for any 𝜏>0, problem (RPF) is always feasible. The following theorem gives the explicit solution of problem (RPF).

Theorem 9. If 𝛽²₀𝑎₀+𝑏−𝑎𝑟_𝑓²𝑎<1, (51) then problem (RPF) has the optimal solution 𝐰^∗=𝛽₀𝜏+𝑟_𝑓𝑎₀+𝑏−𝑎𝑟_𝑓²/𝑎1−𝛽₀𝑎₀+𝑏−𝑎𝑟_𝑓²×Σ/𝑎₋₁𝝁−𝑟_𝑓Σ⁻¹𝐞, (52) where 𝑤_𝑓𝛽=1−⁰𝜏+𝑟_𝑓𝑏−𝑎𝑟_𝑓𝑎₀+𝑏−𝑎𝑟_𝑓²/𝑎1−𝛽₀𝑎₀+𝑏−𝑎𝑟_𝑓²./𝑎 (53)

Proof. Let 𝑣=√𝐰Σ𝐰; similar to (RP2), optimization problem (RPF) can be rewritten as max_𝐰,𝑣>0𝔼𝐫^𝑇𝐰+𝑟_𝑓𝑤_𝑓s.t.𝑣−𝛽₀𝝁^𝑇𝐰+𝑟_𝑓𝑤_𝑓≤𝛽₀𝐞𝜏,_𝑇𝐰+𝑟_𝑓𝑤_𝑓𝐰=1,_𝑇Σ𝐰=𝑣². (54) Then, from KKT condition, the optimal solution (𝐰^∗,𝑤^∗_𝑓) must satisfy the first-order condition 1+𝛽₀𝜆^′_𝑓𝝁−𝜆_𝑓^′′𝐞−2𝜆_𝑓^′′′Σ𝐰^∗=0,−𝜆^′_𝑓+2𝜆_𝑓^′′′𝐞𝑣=0,_𝑇𝐰^∗=1−𝑤^∗_𝑓,𝐰^∗𝑇Σ𝐰^∗=𝑣²,𝑣−𝛽₀𝝁^𝑇𝐰^∗+𝑟_𝑓𝑤^∗_𝑓=𝛽₀𝜏,1+𝜆^′_𝑓𝛽₀𝑟_𝑓−𝜆_𝑓^′′=0, (55) where 𝜆^′_𝑓≥0, 𝜆_𝑓^′′∈ℝ and 𝜆_𝑓^′′′∈ℝ are Lagrangian multipliers. From the first and third equalities, we have 𝐰^∗=1+𝛽₀𝜆^′_𝑓2𝜆_𝑓^′′′Σ⁻¹𝑏𝝁−𝑎Σ⁻¹𝐞+1−𝑤_𝑓𝑎Σ⁻¹𝐞. (56) Substituting 𝐰^∗ into the fourth and fifth equalities in (55), then 𝑎₀(1+𝛽₀𝜆^′_𝑓)2𝜆_𝑓^′′′2+1−𝑤_𝑓𝑎=𝑣²,𝑎₀𝛽₀1+𝛽₀𝜆^′_𝑓2𝜆_𝑓^′′′+𝛽₀𝑏𝜏+𝑎1−𝑤_𝑓+𝑟_𝑓𝑤_𝑓=𝑣. (57)(58) Combining (56), (57), and the third and sixth equations in system of equations (55), we have 1+𝛽₀𝜆^′_𝑓2𝜆_𝑓^′′′=1−𝑤_𝑓𝑏−𝑎𝑟_𝑓. (59) Eliminating 𝑣 and 1−𝑤_𝑓 from (56), (57), and (58), we obtain a quadratic equation in (1+𝛽₀𝜆^′_𝑓)/2𝜆_𝑓^′′′: 𝑎₀+𝑏−𝑎𝑟_𝑓²𝑎1−𝛽²₀𝑎₀+𝑏−𝑎𝑟_𝑓²𝑎×(1+𝛽₀𝜆^′_𝑓)2𝜆_𝑓^′′′2−2𝛽²₀𝜏+𝑟_𝑓𝑎₀+𝑏−𝑎𝑟_𝑓²𝑎1+𝛽₀𝜆^′_𝑓2𝜆_𝑓^′′′−𝛽²₀𝜏+𝑟_𝑓²=0. (60) Solving directly this equation, we have that 1+𝛽₀𝜆^′_𝑓2𝜆_𝑓^′′′=𝛽₀𝜏+𝑟_𝑓𝑎₀+𝑏−𝑎𝑟_𝑓²/𝑎1−𝛽₀𝑎₀+𝑏−𝑎𝑟_𝑓²./𝑎 (61) And further, we can obtain 𝑤_𝑓 from (58). Then, substituting (1+𝛽₀𝜆^′_𝑓)/2𝜆_𝑓^′′′ and 𝑤_𝑓 in (55), we get the results of this theorem.

It is not hard to compute the relationship between the expectation return 𝐸_𝑓 of portfolio with risky-free asset and the parameter 𝜏: 𝐸_𝑓(𝜏)=𝛽₀𝑐−𝑎𝑟²_𝑓−2𝑏𝑟_𝑓𝐴𝜏+𝛽₀𝑐−𝑎𝑟²_𝑓−2𝑏𝑟_𝑓𝐴+1𝑟_𝑓, (62) where 𝐴=𝑎₀+(𝑏−𝑎𝑟_𝑓)²/𝑎(1−𝛽₀𝑎₀+(𝑏−𝑎𝑟_𝑓)²/𝑎) is independent of 𝜏.

6. Numerical Results

We take five domestic risky assets, industrial index (II), commercial index (CI1), properties index (PI), utilities index (UI), and composite index (CI2), and four overseas risky assets, Hengsheng index (HS), Tokyo Nikkei-225 Index (N225), FTSE Index, and Dow-Jones industrial average index (DJIA). The time interval is from January 2, 1995, to December 31, 2012. The returns and covariance matrix of all risky assets can be found in Table 1. For simplicity, we take the risky-free annual interest rate is 𝑟_𝑓=3%.

Table 1 

The mean and covariance matrix of returns for the chosen 9 indexes.

The rolling procedure is used to test the proposed model as follows.(i)We estimate first the parameters 𝝁 and Σ using the 15-year data from January 2, 1995, to December 31, 2009, and test the out-of-sample performance at the whole 2010 year.(ii)And then we further estimate the parameters 𝝁 and Σ using the next 15-year data from January 2, 1996, to December 31, 2010, and test the out-of-sample performance at the whole 2011 year.(iii)We finally estimate the parameters 𝝁 and Σ using the next 15-year data from January 2, 1997, to December 31, 2011, and test the out-of-sample performance at the whole 2012 year.

In our numerical reports, we compare our WCCVaR model with the classical VaR model under normal distribution assumption, equally weighted strategy [17], and MV model. Table 2 gives the results of three models with 𝛽=95% and 𝜏=0.05. The following observations can be found from Table 2.(1)The standard deviation of portfolio obtained by WCCVaR model is clearly less than that of VaR model and equally weighted strategy [17]; moreover, the expected return of portfolio obtained by WCCVaR model is greater than that of VaR model and equally weighted strategy [17]. This means that the proposed WCCVaR model has the better performance ( such as Sharpe ratio).(2)The real wealth of WCCVaR is at least 0.9998 that is very close to the initial wealth one and the coefficient of variation is only 1.26% which is far less than the coefficient of variation of VaR model and equally weighted strategy. The stable performance of WCCVaR is obvious.(3)At the beginning period, within the 300 trade dates, the accumulation wealth of VaR and MV model is better than that of WCCVaR, but, after about 300 trade dates, the accumulation wealth of VaR and MV falls rapidly while WCCVaR still holds the stable wealth. We find that the 300th trade date is about corresponding to the first two months of 2011; the real market at that time is a bear market. This is the main reason that the accumulation wealth of VaR, equally weighted strategy and MV falls rapidly.

Table 2 

The out-of-sample returns statistics of portfolios obtained by WCCVaR, VaR, MV, and equally weighted strategy (1/𝑁), where 1/𝑁 is the equally weighted strategy. Std: standard deviation, tv: terminal value of wealth, and cv: coefficient of variation in this table.

7. Conclusions

We discuss the worst-case CVaR risk measure without the distribution assumption and consider an application in robust portfolio selection problem. The explicit solution is obtained and two-fund separation theorem is proved for the solutions. The theoretical comparison with classical mean-variance model is first discussed by the efficient frontier and the numerical comparison with VaR, MV model and equally weighted strategy using domestic and overseas assets. The numerical results indicate that the proposed WCCVaR has the better expected return and smaller standard deviation than VaR, MV model and equally weighted strategy and therefore can obtain the better performance, such as Sharpe ratio.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundations of China (71371090), Science Foundation of Ministry of Education of China (13YJCZH160), Natural Science Foundation of Jiangxi Province of China (20114BAB211008), and Jiangxi University of Finance and Economics Support Program Funds for Outstanding Youths.