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

Authors: Le Tang; Aifan Ling
Journal: Mathematical Problems in Engineering
 (2014)

---

## 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.


---

## Body

## 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 Section2 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 ben available risky assets in the market and let their random returns vector be denoted by r=(r1,…,rn)T∈Rn. The expected returns and covariance matrix are denoted, respectively, by μ=E[r]=(μ1,…,μn)T∈Rn and Σ=(σij)n×n, where σii=σi2=var(ri) is variance of asset i(i=1,…,n). The rate of return of risky-free asset is denoted by rf. The portfolio vector is w=(w1,…,wn)T∈Rn with wi the proportion of wealth invested in asset i. The weight of wealth invested in risky-free asset is denoted by wf. 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
(1)Ψ(w,α)=P{f(w,r)≤α}=∫f(w,r)≤α‍dF(r).
Let β∈(0,1); then, with confidence level β, VaRβ can be expressed as
(2)VaRβ(w)=min⁡{α∈R∣Ψ(w,α)≥β}.
Rockafellar and Uryasev [2] defined CVaR as the conditional expectation of loss f(w,r) greater than VaRβ. With the definition of CVaR and the given confidence level β, the mathematic formulation of CVaR can be written as
(3)CVaRβ(w)=E[f(w,r)∣f(w,r)≥VaRβ(w)]=11-β∫f(w,r)≥VaRβ(w)f(w,r)dF(r).
Let
(4)Hβ(w,α)=α+11-βE[(f(w,r)-α)+].
Then CVaRβ(w) can be expressed further as [2]
(5)CVaRβ(w)=min⁡α⁡Hβ(w,α),
where
(6)(a)+={a,a>0;0,a≤0.
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
(7)D={r∣E[r]=μ,Cov(r)=Σ≻0},
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 CVaRβ when the worst-case probability distribution in D occurs. To this end, we define worst-case CVaR as follows.Definition 1.
Letβ∈(0,1), worst-case CVaR (WCCVaR) of portfolio w under uncertainty set D is defined by
(8)WCCVaRβ(w)=sup⁡F∈DCVaRβ(w)=sup⁡F∈Dmin⁡αHβ(w,α).Noticing thatHβ(w,α) is the convex function of α [14], then we have from max-min theorem [15]
(9)WCCVaRβ(w)=sup⁡F∈Dmin⁡αHβ(w,α)=min⁡αsup⁡F∈DHβ(w,α).
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
(10)WCCVaRβ(w)≥CVaRβ(w)≥VaRβ(w).Hence,WCCVaRβ(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
(RP1)max⁡wμTws.t.iiWCCVaRβ(w)≤τ,keTw=1,
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]).
Letξ 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
(11)sup⁡ξ~(μ,σ)E[(ρ-ξ)+]=ρ-μ+σ2+(ρ-μ)22.Lemma 4 (see [14, 16]).
Let for any vectora∈Rn,
(12)S1={aTr∣r∈D},S2={η∣E(η)=aTμ,Var⁡(η)=aTΣa}.
Then S1=S2.Lemma3 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 aTμ and variance aTΣa, 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.
Iff(w,r)=-rTw, then for any r∈D and α∈R, we have
(13)sup⁡r∈DE[(f(w,r)-α)+]=sup⁡r∈DE[(-α-rTw)+]=12(wTΣw+(μTw+α)2-(μTw+α)).Proof.
Leta=w and η=wTr. Then, it follows from Lemma 4 that
(14)sup⁡r∈DE[(f(w,r)-α)+]=sup⁡r∈DE[(-α-aTr)+]=sup⁡wTr∈S1E[(-α-wTr)+]=sup⁡η∈S2E[(-α-η)+]=sup⁡η~(aTμ,aTΣa)E[(-α-η)+]=sup⁡η~(wTμ,wTΣw)E[(-α-η)+]=12(wTΣw+(μTw+α)2-(μTw+α)).
The final equality follows from Lemma 3; this is the desired result.Hence, from Lemma5, WCCVaRβ(w) can be expressed by
(15)WCCVaRβ(w)hhh=min⁡α∈R{α+12(1-β)(wTΣw+(μTw+α)2hhhhhhhhhhhhhhhhhhhhhh-(μTw+α)wTΣw+(μTw+α)2)α+12(1-β)(wTΣw+(μTw+α)2}.
Clearly, the right side of the equation above is convex function in α and we can prove that WCCVaRβ(w) can be attained at
(16)α*=-μTw+2β-12β(1-β)wTΣw;
that is,
(17)WCCVaRβ(w)=-μTw+β1-βwTΣw.
Then, robust mean-WCCVaR portfolio problem (RP1) can be expressed further as
(RP2)max⁡wμTws.t.i-μTw+β1-βwTΣw≤τ,ieTw=1.
In the rest of this paper, we discuss mainly the solution of (RP2). To this end, let a=eTΣ-1e, b=μTΣ-1e, c=μTΣ-1μ, a0=(ac-b2)/a, and β0=(1-β)/β. Then, the following result gives feasible conditions of problem (RP2).Lemma 6.
Ifa0β02<1 and
(18)τ≥max⁡{τ*,0},
then problem (RP2) is feasible, where τ*=1-a0β02/(β0a)-b/a.Proof.
For any givenβ∈(0,1), consider the problem
(19)τ*=min⁡w{-μTw+β1-βwTΣw∣s.t.eTw=1}.
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:
(20)-μ+λ′e+2λ′′Σ-1w^=0,2λ′′v=β1-β,eTw^=1,w^TΣw^=v2,
where λ′,λ′′∈R are the Lagrange multipliers. It is not hard to obtain from the first and third equations of (20)
(21)w^=12λ′′(Σ-1μ-baΣ-1e)+1aΣ-1e.
Substituting w^ in the fourth equation and combining the second equation of (20), then we can get the optimal solution of (19) when a0β02<1(22)w^=β0a(1-a0β02)(Σ-1μ-baΣ-1e)+1aΣ-1e.
The optimal value
(23)τ*=1-a0β02β0a-ba.
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.Lemma6 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.
Ifa0β02<1 and τ>max⁡{τ*,0}, the optimal solution of the problem (RP2) can explicitly be expressed
(24)w*=f(τ)(Σ-1μ-baΣ-1e)+1aΣ-1e,
where
(25)f(τ)=a0bβ02(τ+b/a)+ba0β02(τ+b/a)2-(a0/a)(1-a0β02)a0(1-a0β02).Proof.
Letv=wΣw; the optimization problem (RP2) can be rewritten as
(26)max⁡w,v>0E[rTw]s.t.v-β0μTw≤β0τ,eTw=1,wTΣw=v2.
Let w* be the optimal solution of problem (26). Then from KKT condition, w* satisfies that
(27)(1+β0λ1)μ-λ2e-2λ3Σw*=0,-λ1+2λ3v=0,eTw*=1,(w*)TΣw*=v2,v-β0μTw*-β0τ=0,
where λ1≥0, λ2∈R and λ3∈R are Lagrange multipliers. It follows from the first and third equations in (27) that
(28)w*=(1+β0λ1)2λ3(Σ-1μ-baΣ-1e)+1aΣ-1e.
Substituting w* in the fourth and fifth equation in (27), we have
(29)a0[(1+β0λ1)2λ3]2+1a=v2,a0β0(1+β0λ1)2λ3=v-β0(τ+ba).
Eliminating v from (29), it follows that the quadratic equation with respect to (1+β0λ1)/2λ3 is
(30)a0(1-a0β02)((1+β0λ1)2λ3)2-2a0β02(τ+ba)((1+β0λ1)2λ3)+1a-β02(τ+ba)2=0.
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
(31)(1+β0λ1)2λ3=a0β02(τ+b/a)+a0β02(τ+b/a)2-(a0/a)(1-a0β02)a0(1-a0β02)>0.
Then the optimal solution of problem (RP2) can be obtained by substituting it in (27). This finishes the proof.We need that conditiona0β02<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 a0β02<1, we only require that β satisfies
(32)β>a01+a0.
If β∈(1/2,1), the condition a0β02<1 can be satisfied while
(33)β>max⁡{12,a01+a0}.


## 3. Two-Fund Separation Theorem

In our analysis of this section, we vieww* as a function of input parameter τ; that is, we denote it by w*=w(τ). Let
(34)S(w)={w(τ):max⁡{τ*,0}≤τ<∞}.
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.
Letw(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
(35)w(τ)=θw(τ1)+(1-θ)w(τ2);
that is, the two-fund separation theorem holds.Proof.
Notingf(τ) in Theorem 7, the optimal solution of mean-WCCVaR portfolio problem (RP2) can be written in the simple form
(36)w(τ)=f(τ)wA+(1-f(τ))wσ,
where wσ=(Σ-1e)/a is the portfolio with the minimum variance and wA=(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
(37)θ=f(τ)-f(τ2)f(τ1)-f(τ2);
we have
(38)f(τ)=θf(τ1)+(1-θ)f(τ2).
Thus
(39)w(τ)=f(τ)wA+(1-f(τ))wσ=[θf(τ1)+(1-θ)f(τ2)]wA+[1-[θf(τ1)+(1-θ)f(τ2)]]wσ=θ[f(τ1)wA+(1-f(τ1))wσ]+(1-θ)[f(τ2)wA+(1-f(τ2))wσ]=θw(τ1)+(1-θ)w(τ2).
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
(40)R(τ)=a0β02(τ+b/a)+a0β02(τ+b/a)2-(a0/a)(1-a0β02)(1-a0β02)+ba.
Rearranging this equality, we have the quadratic equation in R(τ) and τ(41)(1-a0β02)2(R(τ)-ba)2-2a0β02(1-a0β02)(R(τ)-ba)(τ+ba)+a0β02(a0β02-1)(τ+ba)2+a0a(1-a0β02)=0.
The determinant of quadratic term coefficient can be expressed as
(42)|(1-a0β02)2-a0β02(1-a0β02)-a0β02(1-a0β02)a0β02(a0β02-1)|=-a0β02(1-a0β02)2<0.
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
(43)R(τ)=1+a0β021-a0β02(a0β02τ+ba),
the intercept at R(τ) axis is
(44)Rb=ba·1+a0β021-a0β02,
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:(45)max⁡{E[μTw]:wTΣw≤τ2,eTw=1}.
It is not hard to compute that the optimal solution of MV model is
(46)wMV*=wMV(τ)=(1-bτ2-1/aa0)Σ-1ea+τ2-1/aa0Σ-1μ.
The corresponding expected value at the optimal solution is
(47)RMV(τ)=μTwMV*=ac-b2aτ2-1/aa0+ba.
In (τ,RMV(τ)) plane, function RMV(τ) plots the efficient frontier of MV:
(48)aτMV2-aa0(RMV(τ)-ba)2=1,
whose slope of asymptotic line is a0. Let
(49)1+a0β021-a0β02a0β02=a0.
Then,
(50)β0=1+4(a0+a0)-12(a0+a0)=:β1.
If β0≥β1, then for the case of τ*<τMV, 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*<τMV; see Figure 2(b).The relationship of efficient frontiers of two modelsβ0≥β1.
(a)
β
0
>
β
1
(b)
β
0
=
β
1The relationship of efficient frontiers of two models in the case ofβ0<β1.
(a)
β
0
<
β
1, τ*>τMV
(b)
β
0
<
β
1, τ*<τMVGenerally speaking, it is more conservative for mean-WCCVaR model than MV model. But the conservative performance can be improved by adjusted the confidence levelβ. 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(RPF)max⁡w,wfE[rTw+wfrf]s.t.-μTw-wfrf+β1-βwTΣw≤τ,eTw+wf=1.
Clearly, (w,wf)=(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(51)β02[a0+(b-arf)2a]<1,
then problem (RPF) has the optimal solution
(52)w*=β0(τ+rf)a0+(b-arf)2/a(1-β0a0+(b-arf)2/a)×(Σ-1μ-rfΣ-1e),
where
(53)wf=1-β0(τ+rf)(b-arf)a0+(b-arf)2/a(1-β0a0+(b-arf)2/a).Proof.
Letv=wΣw; similar to (RP2), optimization problem (RPF) can be rewritten as
(54)max⁡w,v>0E[rTw+rfwf]s.t.v-β0(μTw+rfwf)≤β0τ,eTw+rfwf=1,wTΣw=v2.
Then, from KKT condition, the optimal solution (w*,wf*) must satisfy the first-order condition
(55)(1+β0λf′)μ-λf′′e-2λf′′′Σw*=0,-λf′+2λf′′′v=0,eTw*=1-wf*,(w*)TΣw*=v2,v-β0(μTw*+rfwf*)=β0τ,(1+λf′β0)rf-λf′′=0,
where λf′≥0, λf′′∈R and λf′′′∈R are Lagrangian multipliers. From the first and third equalities, we have
(56)w*=(1+β0λf′)2λf′′′(Σ-1μ-baΣ-1e)+1-wfaΣ-1e.
Substituting w* into the fourth and fifth equalities in (55), then
(57)a0[(1+β0λf′)2λf′′′]2+1-wfa=v2,(58)a0β0(1+β0λf′)2λf′′′+β0(τ+ba(1-wf)+rfwf)=v.
Combining (56), (57), and the third and sixth equations in system of equations (55), we have
(59)(1+β0λf′)2λf′′′=1-wfb-arf.
Eliminating v and 1-wf from (56), (57), and (58), we obtain a quadratic equation in (1+β0λf′)/2λf′′′:
(60)(a0+(b-arf)2a)[1-β02(a0+(b-arf)2a)]×((1+β0λf′)2λf′′′)2-2β02(τ+rf)(a0+(b-arf)2a)((1+β0λf′)2λf′′′)-β02(τ+rf)2=0.
Solving directly this equation, we have that
(61)(1+β0λf′)2λf′′′=β0(τ+rf)a0+(b-arf)2/a(1-β0a0+(b-arf)2/a).
And further, we can obtain wf from (58). Then, substituting (1+β0λf′)/2λf′′′ and wf in (55), we get the results of this theorem.It is not hard to compute the relationship between the expectation returnEf of portfolio with risky-free asset and the parameter τ:
(62)Ef(τ)=β0(c-arf2-2brf)Aτ+[β0(c-arf2-2brf)A+1]rf,
where A=a0+(b-arf)2/a(1-β0a0+(b-arf)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 Table1. For simplicity, we take the risky-free annual interest rate is rf=3%.Table 1
The mean and covariance matrix of returns for the chosen 9 indexes.
II
CI1
PI
UI
CI2
HS
N225
FTSE
DJIA
μ
0.1813
0.1875
0.1924
0.2214
0.1636
0.1018
−0.0096
0.0503
0.0849
Σ
0.2676
0.2785
0.3052
0.2909
0.2416
0.0992
0.0210
0.0330
0.0363
0.2785
0.3109
0.3246
0.2859
0.2533
0.0983
0.0184
0.0330
0.0301
0.3052
0.3246
0.3924
0.3177
0.3063
0.1147
0.0309
0.0314
0.0324
0.2909
0.2859
0.3177
0.3611
0.2287
0.0964
0.0081
0.0315
0.0391
0.2416
0.2533
0.3063
0.2287
0.3115
0.0881
0.0296
0.0370
0.0382
0.0992
0.0983
0.1147
0.0964
0.0881
0.0950
0.0539
0.0332
0.0369
0.0210
0.0184
0.0309
0.0081
0.0296
0.0539
0.0526
0.0248
0.0208
0.0330
0.0330
0.0314
0.0315
0.0370
0.0332
0.0248
0.0262
0.0246
0.0363
0.0301
0.0324
0.0391
0.0382
0.0369
0.0208
0.0246
0.0290The rolling procedure is used to test the proposed model as follows.(i)
We estimate first the parametersμ 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 parametersμ 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 parametersμ 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.
Model
Mean
Std
Max.
Min.
tv
cv
WCCVaR
0.0377
0.0131
0.0707
−0.0002
1.0269
1.26%
VaR
0.0037
0.1169
0.2342
−0.1987
0.8141
11.65%
1
/
N
−0.1238
0.0759
0.0127
−0.2485
0.8369
8.66%
MV
0.0046
0.0986
0.0457
−0.1254
0.8433
8.53%


## 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.



---

*Source: 494575-2014-07-02.xml*