# CVaR Robust Mean-CVaR Portfolio Optimization

Authors: Maziar Salahi; Farshid Mehrdoust; Farzaneh Piri
Journal: ISRN Applied Mathematics
 (2013)

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## Abstract

One of the most important problems faced by every investor is asset allocation. An investor during making investment decisions has to search for equilibrium between risk and returns. Risk and return are uncertain parameters in the suggested portfolio optimization models and should be estimated to solve the problem. However, the estimation might lead to large error in the final decision. One of the widely used and effective approaches for optimization with data uncertainty is robust optimization. In this paper, we present a new robust portfolio optimization technique for mean-CVaR portfolio selection problem under the estimation risk in mean return. We additionally use CVaR as risk measure, to measure the estimation risk in mean return. To solve the model efficiently, we use the smoothing technique of Alexander et al. (2006). We compare the performance of the CVaR robust mean-CVaR model with robust mean-CVaR models using interval and ellipsoidal uncertainty sets. It is observed that the CVaR robust mean-CVaR portfolios are more diversified. Moreover, we study the impact of the value of confidence level on the conservatism level of a portfolio and also on the value of the maximum expected return of the portfolio.


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## Body

## 1. Introduction

Portfolio optimization is one of the best known approaches in financial portfolio selection. The earliest technique to solve the portfolio selection problem is developed by Harry Markowitz in the 1952. In his so-called mean-variance (MV) portfolio optimization model, the portfolio return is measured by the expected return of the portfolio, and the associated risk is measured by the variance of portfolio returns [1].Variance as the risk measure has its weaknesses. Controlling the variance does not only lead to low deviation from the expected return on the downside, but also on the upside [2]. Hence, alternative risk measures have been suggested to replace the variance such as Value at Risk (VaR) that manage and control risk in terms of percentiles of loss distribution. Instead of regarding both upside and downside of the expected return, VaR considers only the downside of the expected return as risk and represents the predicted maximum loss with a specified confidence level (e.g., 95%) over a certain period of time (e.g., one day) [3–5].V
a
R is a popular risk measure. However, VaR may have drawbacks and undesirable properties that limit its use [6–8], such as lack of subadditivity; that is, VaR of two different investment portfolios may be greater than the sum of the individual VaRs. Also, VaR is nonconvex and nonsmooth and has multiple local minimum, while we seek the global minimum [4, 8, 9]. So alternative risk measures were introduced such as Conditional Value at Risk (CVaR)—the conditional expected value of loss, under the condition that it exceeds the value at risk [3]. VaR implies that “what is the maximum loss that we realize?” but CVaR asks: “How do we expect to incur losses when situation is undesirable?”. Numerical experiments show that minimum CVaR often leads to optimal solutions close to the minimum VaR, because VaR never exceeds CVaR [5]. CVaR has better properties than VaR. It is a convex optimization problem, and thus it is easy to optimize [4]. It is demonstrated that linear programming techniques can be used for optimization of CVaR risk measure [5, 9].Risk and return are uncertain parameters in portfolio optimization models, and estimating them might lead to large error in the final decision. To deal with such situation, one of the widely used and effective approaches is robust optimization technique. In this paper, we have applied this technique to give the robust counterpart of the mean-CVaR portfolio selection problem under the estimation risk in mean return. Moreover, we have used CVaR as the risk measure to measure the estimation risk in mean return. The rest of the paper is organized as follows. In Section 2, we state the mean-CVaR portfolio selection problem. Then because of the inevitable estimation error of the mean return of the assets, we present robust optimization by CVaR in Section 3. To solve the model efficiently, we use the smoothing technique of Alexander et al. [10]. Finally, in Section 4, we compare the performance of the CVaR robust mean-CVaR model with robust mean-CVaR models using interval and ellipsoidal uncertainty sets on an example. We have observed that the CVaR robust mean-CVaR portfolios are more diversified and they are sensitive to initial data used to generate each set of μ samples. Moreover, we demonstrate that the value of confidence level affects the conservatism level, diversification, and also the value of the maximum expected return of the resulting portfolios.


## 2. Mean-Conditional Value at Risk

Consider assetsS1,…,Sn, n≥2, with random returns. Suppose μi denotes the expected return of asset Si, and also consider xi as the proportion of holding in the ith asset. We can represent the expected return of the resulting portfolio x as follows:
(1)E[x]=μ1x1+⋯+μnxn=μTx.Also, we will assume that the set of feasible portfolios is a nonempty polyhedral set and represent that asΩ={x∣Ax=b,Cx≥d} where A is a m×n matrix, b is an m-dimensional vector, C is a p×n matrix, and d is a p-dimensional vector [4]. In particular, one of the constraints in the set Ω is ∑i=1nxi=1.Letf(x,y) denote the loss function when we choose the portfolio x from a set of feasible portfolios, and y is the realization of the random events (the vector of the asset returns of n assets). We consider the portfolio return loss, f(x,y), the negative of the portfolio return that is a convex (linear) function of the portfolio variables x:
(2)f(x,y)=-yTx=-[y1x1+⋯+ynxn].We assume that the random vectory has a probability density function denoted by p(y). For a fixed decision vector x, the cumulative distribution function of the loss associated with that vector is computed as follows:
(3)ψ(x,γ)=∫f(x,y)≤γp(y)dy.Then, for a given confidence levelα, the α-VaR associated with portfolio x is represented as
(4)VaRα(x)=min{γ∈ℝ∣ψ(x,γ)≥α}.Also, we define theα-CVaR associated with portfolio x as
(5)CVaRα(x)=1(1-α)∫f(x,y)≥VaRα(x)f(x,y)p(y)dy.Theorem 1.
We always haveCVaRα(x)≥VaRα(x), that means CVaR of a portfolio is always at least as big as its VaR. Consequently, portfolios with small CVaR also have small VaR. However, in general minimizing CVaR and VaR are not equivalent.Proof.
See [4].Since the definition ofCVaR implies the VaR function clearly, it is difficult to work with and optimize this function. Instead, the following simpler auxiliary function is considered [5]:
(6)Fα(x,γ)=γ+1(1-α)∫f(x,y)≥γ(f(x,y)-γ)p(y)dy,
and/or
(7)Fα(x,γ)=γ+1(1-α)∫(f(x,y)-γ)+p(y)dy,
where a+=max{a,0}. This function, considered as a function of γ, has the following important properties that makes it useful for the computation of VaR and CVaR [4]:(1)
Fα is a convex function of γ.(2)
VaRα is a minimizer over γ of Fα.(3)
The minimum value overγ of the function Fα is CVaRα.As a consequence of the listed properties, we immediately deduce that, in order to minimizeCVaRα(x) over x, we need to minimize the function Fα(x,γ) with respect to x and γ simultaneously
(8)minxCVaRα(x)=minx,γFα(x,γ).Consequently, we can optimizeCVaR directly, without needing to compute VaR first. Since we assumed that the loss function f(x,y) is the convex (linear) function of the portfolio variables x, Fα(x,γ) is also a convex (linear) function of x. In this case, provided the feasible portfolio set Ω is also convex, the optimization problems in (8) are convex optimization problems that can be solved using well-known optimization techniques for such problems.Instead of using the density functionp(y) of the random events in formulation (7) that it is often impossible or undesirable to compute it, we can use a number of scenarios in the names of yi for i=1,…,T. In this case, we consider the following approximation to the function Fα(x,γ)(9)F¯α(x,γ)=γ+1(1-α)T∑i=1T(f(x,yi)-γ)+.Now, in the problemminxCVaRα(x), we replace Fα(x,γ) with F¯α(x,γ)(10)minx,γγ+1(1-α)T∑i=1T(f(x,yi)-γ)+.To solve this optimization problem, we introduce artificial variableszi to replace (f(x,yi)-γ)+. To do so, we add the constraints zi≥0 and zi≥f(x,yi)-γ to the problem [5]:
(11)minx,z,γγ+1(1-α)T∑i=1Tzis.t.zi≥0,i=1,…,Tzi≥f(x,yi)-γ,i=1,…,Tx∈Ω.It should be noted that risk managers often try to optimize risk measure while expected return is more than a threshold value. In this case, we can represent mean-CVaR model as follows:
(12)minx,z,γγ+1(1-α)T∑i=1Tzis.t.μTx≥R,zi≥0,i=1,…,Tzi≥f(x,yi)-γ,i=1,…,Tx∈Ω,
or
(13)minx,z,γ-μTx+λ(γ+1(1-α)T∑i=1Tzi)s.t.zi≥0,i=1,…,Tzi≥f(x,yi)-γ,i=1,…,Tx∈Ω,
where the first constraint of problem (12) indicates that the expected return is no less than the target value R and λ≥0 used in problem (13) is risk aversion parameter that adapts the balance between expected return and CVaRα(x). It is important to note that there is an equivalence between R and λ so that the problems (12) and (13) generate the same efficient frontiers. Since f(x,y) is linear in x, all the expressions zi≥f(x,yi)-γ represent linear constraints, and therefore the problem is a linear programming problem that can be efficiently solved using the simplex or interior point methods.


## 3.CVaR Robust Mean-CVaR Model

One of the uncertain parameters for mean-CVaR model is μ, and using estimations for this parameter leads to an estimation risk in portfolio selection. In particular, small differences in the estimations of μ can create large changes in the composition of an optimal portfolio. One way to reduce the sensitivity of mean-CVaR model to the parameter estimations is using robust optimization to determine the optimal portfolio under the worst case scenario in the uncertainty set of the expected return. To this end, we represented robust mean-CVaR models with interval and ellipsoidal uncertainty sets in the previous studies that have been demonstrated in formulations (14) and (15), respectively,
(14)minx,z,γγ+1(1-α)T∑i=1Tzis.t.(μL)Tx≥R,zi≥0,i=1,…,Tzi≥f(x,yi)-γ,i=1,…,Tx∈Ω,(15)minx,z,γγ+1(1-α)T∑i=1Tzis.t.∥MTx∥≤μTx-R,zi≥0,i=1,…,Tzi≥f(x,yi)-γ,i=1,…,Tx∈Ω,
where μL is a given vector and M is a n-dimensional matrix.Now, we presentCVaR robust mean-CVaR portfolio optimization problem that estimation risk in mean return is measured by CVaR. The CVaR robust mean-CVaR model specifies that an optimal portfolio based on the tail of the mean loss distribution and the adjustment of the confidence level with regard to the preference of the investor corresponds to the adjustment of the conservative level, considering the uncertainty of the mean return [11].In this model,CVaR is used to measure the risk of the portfolio return as before. In addition, when using the mean-CVaR model, we consider the uncertainty of the expected return that can be considered as estimation risk and use CVaR to measure estimation risk. CVaR with this perspective is denoted as CVaRμ (We use CVaRy to denote the CVaR risk measure discussed in Section 2 in order to differentiate it from CVaRμ and also we use Fαy(x,γ) to denote its associated Fα(x,γ)). Thus, considering the problem (13), a CVaR robust mean-CVaR portfolio will be determined as the solution of the following optimization problem:
(16)minx,z,γCVaRβμ(-μTx)+λ(γ+1(1-α)T∑i=1Tzi)s.t.zi≥0,i=1,…,Tzi≥f(x,yi)-γ,i=1,…,Tx∈Ω.For a portfolio ofn assets, we assume μ∈ℝn is the random vector of the expected returns of the assets with a probability density function p(μ). To determine the mean loss of the portfolio, we define mean loss function, f(x,μ), as follows [11]:
(17)f(x,μ)=-μTx=-[μ1x1+⋯+μnxn].So, for confidence levelβ, CVaRβμ(-μTx) can be defined as follows:
(18)CVaRβμ(-μTx)=minη(η+11-βE([f(x,μ)-η]+)).According to the definition ofCVaRβμ and CVaR robust mean-CVaR model, we find that CVaRβμ will increase as the value of β increases. This corresponds to taking more pessimism on the estimation risk in μ in the model and to optimizing the portfolio under worse cases of the mean loss. Thus, the resulting CVaR robust portfolio is more conservative. Conversely, conservatism of the portfolio is reduced as the value of β decreases [11]. In Section 4, we will illustrate the impact of the value of β on the conservatism level of a portfolio and also on the value of the maximum expected return of the portfolio.As before, we can consider an auxiliary function to simplify the computations:(19)Fβμ(x,η)=η+1(1-β)∫μ∈ℝn(f(x,μ)-η)+p(μ)dμ,
and use the following approximation to the function Fβμ(x,η):
(20)F¯βμ(x,η)=η+1(1-β)m∑i=1m(-μiTx-η)+,
where μ1,…,μm are a collection of m independent samples for μ based on its density function p(μ). We can show that [5]
(21)minx∈ΩCVaRβμ(-μTx)=minx∈Ω,ηF¯βμ(x,η).So, with introducing artificial variablesvi to replace (f(x,μi)-η)+ in F¯βμ(x,η) and adding the constraints vi≥0 and vi≥f(x,μi)-η to the problem to do so, the CVaR robust mean-CVaR portfolio optimization problem becomes
(22)minx,z,v,γ,ηη+1(1-β)m∑i=1mvi+λ(γ+1(1-α)T∑i=1Tzi)s.t.zi≥0,i=1,…,Tzi≥f(x,yi)-γ,i=1,…,Tvi≥0,i=1,…,mvi≥f(x,μi)-η,i=1,…,mx∈Ω.This problem hasO(m+n+T) variables and O(m+n+T) constraints that m is the number of μ-samples, n is the number of assets, and T is the number of y-scenarios. When the number of y-scenarios and μ-samples increase, the approximations are getting closer to the exact values. But the computational cost significantly increases and thus makes the method inefficient.Instead of this method, we can more efficiently determine theCVaR robust mean-CVaR portfolios using the smoothing method suggested by Alexander et al. [10]. Alexander presented the following function to approximate F¯βμ(x,η):
(23)F^βμ(x,η)=η+1(1-β)m∑i=1mρε(-μiTx-η),
where ρε(a) is defined as follows:
(24)ρε(a)={aa≥ε,a24ε+12a+14ε-ε≤a≤ε,0o.w.For a given resolution parameterε>0,ρε(a) is continuous differentiable and approximates the piecewise linear function max(a,0) [10]. We can also use this function to approximate F-αy(x,γ) as follows:
(25)F^αy(x,γ)=γ+1(1-α)T∑i=1Tρε(f(x,yi)-γ).Using smoothing method, theCVaR robust mean-CVaR model is as follows:
(26)minx,γ,ηη+1(1-β)m∑i=1mρε(-μiTx-η)+λ(γ+1(1-α)T∑i=1Tρε(-yiTx-γ))s.t.x∈Ω.This formulation hasO(n) variables and O(n) constraints. Thus, the number of variables and constraints does not change as the size of μ-samples (m) and y-scenarios (T) increases. The efficiency of the smoothing approach is shown in Section 4.


## 4. Numerical Results

In this section, first we will compare the performance of theCVaR robust mean-CVaR model with robust mean-CVaR models using interval and ellipsoidal uncertainty sets by actual data. Then, we will compare time required to compute the CVaR robust portfolios using problems (22) and (26). The dataset used here contains available returns for eight assets that expected return [12] and covariance matrix of the return of assets  which have been given in Tables 1 and 2. In addition, the computations are based on 10,000 μ-samples generated from the Monte Carlo resampling (RS) method introduced in [13] and 96y-scenarios obtained via computer simulation. It should be further noted that the computation is performed in MATLAB version 7.12, and run on a Core i5 CPU 2.40 GHz Laptop with 4 GB of RAM. Problems are solved using CVX [14] and function “fmincon” in Optimization Toolbox of MATLAB.Table 1
Mean return vectorμ.
0.01×
S1
S2
S3
S4
S5
S6
S7
S8
1.0160
0.4746
0.4756
0.4734
0.4742
−0.0500
−0.1120
0.0360Table 2
Covariance matrixQ.
0.01×
S1
S2
S3
S4
S5
S6
S7
S8
S1
0.0980
S2
0.0659
0.1549
S3
0.0714
0.0911
0.2738
S4
0.0105
0.0058
−0.0062
0.0097
S5
0.0058
0.0379
−0.0116
0.0082
0.0461
S6
−0.0236
−0.0260
0.0083
−0.0215
−0.0315
0.2691
S7
−0.0164
0.0079
0.0059
−0.0003
0.0076
−0.0080
0.0925
S8
0.0004
−0.0248
0.0077
−0.0026
−0.0304
0.0159
−0.0095
0.0245


### 4.1. Sensitivity to Initial Data

To show the sensitivity of theCVaR robust portfolios to initial data, we repeat RS sampling technique 100 times. Each of Figures 1, 2 and 3 display 100 CVaR robust actual frontiers (actual frontiers are obtained by applying the true parameters on the portfolio weights derived from their estimated values [15]) for β=99%,90%,75%, respectively.Figure 1
100CVaR robust mean-CVaR actual frontiers (β=99%).Figure 2
100CVaR robust mean-CVaR actual frontiers (β=90%).Figure 3
100CVaR robust mean-CVaR actual frontiers (β=75%).As can be seen from figures, theCVaR robust mean-CVaR actual frontiers change with initial data used to generate μ samples. Also, this changes increase as the confidence level β decreases. Thus, we can regard β as an estimation risk aversion parameter. With these qualities, an investor who is more averse to estimation risk will choose a larger β. On the other hand, an investor who is more tolerant to estimation risk may choose a smaller β.


### 4.2. Portfolio Diversification

As we know, diversification decreases risk [4]. Portfolio diversification indicates distributing investment among assets in the portfolio. We illustrate in the following that compared with the robust mean-CVaR portfolios with interval and ellipsoidal uncertainty sets, the CVaR robust mean-CVaR portfolios are more diversified. In addition, the diversification of the CVaR robust mean-CVaR portfolios decreases as the confidence level β decreases.To do so, we compute theCVaR robust mean-CVaR portfolios (for β=99%,90%,75%) and robust mean-CVaR portfolios with interval and ellipsoidal uncertainty sets for the 8-asset example. The composition graphs of the resulting optimal portfolios are presented in Figures 4, 5, 6, 7, and 8. Considering these figures, when the expected return value increases from left to right, the allocated assets in the portfolios with minimum expected return are replaced by a composition of other assets, gently. Observing the right-most end of each graph, we can conclude the composition of the assets of the portfolio achieved from CVaR robust mean-CVaR model with β=99% is more diversified than that achieved from other models.Figure 4
Composition of robust mean-CVaR portfolio weights with interval uncertainty set.Figure 5
Compositions of robust mean-CVaR portfolio weights with ellipsoidal uncertainty set.Figure 6
Compositions ofCVaR robust mean-CVaR portfolio weights (β=99%).Figure 7
Compositions ofCVaR robust mean-CVaR portfolio weights (β=90%).Figure 8
Compositions ofCVaR robust mean-CVaR portfolio weights (β=75%).In Figures9, 10, and 11, the CVaR robust mean-CVaR actual frontiers for different values of β are compared with robust mean-CVaR actual frontiers with interval and ellipsoidal uncertainty sets and mean-CVaR true efficient frontier. Since portfolios on the robust mean-CVaR actual frontiers with interval and ellipsoidal uncertainty sets are less diversified, they should accept more risk for a given level of expected return and also achieve a lower maximum expected return. Consequently, in the figures, their actual frontiers are more right and lower than the other frontiers and this is one of the disadvantages of the low diversification in the portfolio. Seeing these frontiers, we also deduce that the maximum expected return and the associated return risk increase as the confidence level β decreases. But in this case, the variations on the compositions of the resulting maximum-return portfolios might be large, and so the exact solution will not  be always achieved. Instead, the maximum expected return of the portfolio is low for β=99%, and the variations will be low. So, the probability of having poor performance of the portfolio will be reduced when there is a big estimation risk of μ. Thus, resulting robust portfolios will be too conservative. Consequently, an investor who is more risk averse to estimation risk selects a larger β and obtains a more diversified portfolio. This justifies that it is reasonable to regard β as a estimation risk aversion parameter.Figure 9
Robust mean-CVaR with interval and ellipsoidal uncertainty sets and CVaR robust (β=99%) actual frontiers.Figure 10
Robust mean-CVaR with interval and ellipsoidal uncertainty sets and CVaR robust (β=90%) actual frontiers.Figure 11
Robust mean-CVaR with interval and ellipsoidal uncertainty sets and CVaR robust (β=75%) actual frontiers.


### 4.3. Comparison of the Efficiency of Two Approaches for ComputingCVaR Robust Portfolios

In Section3, we introduced two formulations (22) and (26) to compute CVaR robust portfolios. Now, we show that computing CVaR robust portfolios via the smoothing approach (problem (26)) is more efficient. To do so, we compare the time required to solve problems (22) and (26) with different number of assets and different number of μ-samples and y-scenarios. The results have been given in Table 3.Table 3
Time required to compute maximum-return (λ=0) portfolios for LP and smoothing approach (β=99%, ε=0.005).
Scenarios (T)
Samples (m)
LP (problem (22))
Smoothing (problem (26))
8 assets
50 assets
148 assets
8 assets
50 assets
148 assets
500
5,000
3.01
5.7
23.05
1.13
3.08
15.67
1,000
10,000
4.85
9.95
37.2
1.28
4.67
26.1
3,000
25,000
13.86
45.32
185.49
1.69
12.22
79.31As we see, the time required to computeCVaR robust portfolios via two approaches differs significantly when the sample size and the number of assets increase. For example, the time required to solve the CVaR robust mean-CVaR problem by two approaches for a problem with 8 assets and 5000 samples and 500 scenarios differs slightly. But, when the number of assets is more than 50, the number of scenarios is more than 500, and the sample size is more than 5000, differences become significant. A problem with 148 assets, 3000 y-scenarios, and 25000 μ-samples is solved in less than 80 seconds using smoothing technique, while by (22) it took over 185 seconds. These comparisons show that when the number of scenarios and samples becomes larger, the smoothing approach is more computationally efficient to determine CVaR robust portfolios than other approach.


## 4.1. Sensitivity to Initial Data

To show the sensitivity of theCVaR robust portfolios to initial data, we repeat RS sampling technique 100 times. Each of Figures 1, 2 and 3 display 100 CVaR robust actual frontiers (actual frontiers are obtained by applying the true parameters on the portfolio weights derived from their estimated values [15]) for β=99%,90%,75%, respectively.Figure 1
100CVaR robust mean-CVaR actual frontiers (β=99%).Figure 2
100CVaR robust mean-CVaR actual frontiers (β=90%).Figure 3
100CVaR robust mean-CVaR actual frontiers (β=75%).As can be seen from figures, theCVaR robust mean-CVaR actual frontiers change with initial data used to generate μ samples. Also, this changes increase as the confidence level β decreases. Thus, we can regard β as an estimation risk aversion parameter. With these qualities, an investor who is more averse to estimation risk will choose a larger β. On the other hand, an investor who is more tolerant to estimation risk may choose a smaller β.


## 4.2. Portfolio Diversification

As we know, diversification decreases risk [4]. Portfolio diversification indicates distributing investment among assets in the portfolio. We illustrate in the following that compared with the robust mean-CVaR portfolios with interval and ellipsoidal uncertainty sets, the CVaR robust mean-CVaR portfolios are more diversified. In addition, the diversification of the CVaR robust mean-CVaR portfolios decreases as the confidence level β decreases.To do so, we compute theCVaR robust mean-CVaR portfolios (for β=99%,90%,75%) and robust mean-CVaR portfolios with interval and ellipsoidal uncertainty sets for the 8-asset example. The composition graphs of the resulting optimal portfolios are presented in Figures 4, 5, 6, 7, and 8. Considering these figures, when the expected return value increases from left to right, the allocated assets in the portfolios with minimum expected return are replaced by a composition of other assets, gently. Observing the right-most end of each graph, we can conclude the composition of the assets of the portfolio achieved from CVaR robust mean-CVaR model with β=99% is more diversified than that achieved from other models.Figure 4
Composition of robust mean-CVaR portfolio weights with interval uncertainty set.Figure 5
Compositions of robust mean-CVaR portfolio weights with ellipsoidal uncertainty set.Figure 6
Compositions ofCVaR robust mean-CVaR portfolio weights (β=99%).Figure 7
Compositions ofCVaR robust mean-CVaR portfolio weights (β=90%).Figure 8
Compositions ofCVaR robust mean-CVaR portfolio weights (β=75%).In Figures9, 10, and 11, the CVaR robust mean-CVaR actual frontiers for different values of β are compared with robust mean-CVaR actual frontiers with interval and ellipsoidal uncertainty sets and mean-CVaR true efficient frontier. Since portfolios on the robust mean-CVaR actual frontiers with interval and ellipsoidal uncertainty sets are less diversified, they should accept more risk for a given level of expected return and also achieve a lower maximum expected return. Consequently, in the figures, their actual frontiers are more right and lower than the other frontiers and this is one of the disadvantages of the low diversification in the portfolio. Seeing these frontiers, we also deduce that the maximum expected return and the associated return risk increase as the confidence level β decreases. But in this case, the variations on the compositions of the resulting maximum-return portfolios might be large, and so the exact solution will not  be always achieved. Instead, the maximum expected return of the portfolio is low for β=99%, and the variations will be low. So, the probability of having poor performance of the portfolio will be reduced when there is a big estimation risk of μ. Thus, resulting robust portfolios will be too conservative. Consequently, an investor who is more risk averse to estimation risk selects a larger β and obtains a more diversified portfolio. This justifies that it is reasonable to regard β as a estimation risk aversion parameter.Figure 9
Robust mean-CVaR with interval and ellipsoidal uncertainty sets and CVaR robust (β=99%) actual frontiers.Figure 10
Robust mean-CVaR with interval and ellipsoidal uncertainty sets and CVaR robust (β=90%) actual frontiers.Figure 11
Robust mean-CVaR with interval and ellipsoidal uncertainty sets and CVaR robust (β=75%) actual frontiers.


## 4.3. Comparison of the Efficiency of Two Approaches for ComputingCVaR Robust Portfolios

In Section3, we introduced two formulations (22) and (26) to compute CVaR robust portfolios. Now, we show that computing CVaR robust portfolios via the smoothing approach (problem (26)) is more efficient. To do so, we compare the time required to solve problems (22) and (26) with different number of assets and different number of μ-samples and y-scenarios. The results have been given in Table 3.Table 3
Time required to compute maximum-return (λ=0) portfolios for LP and smoothing approach (β=99%, ε=0.005).
Scenarios (T)
Samples (m)
LP (problem (22))
Smoothing (problem (26))
8 assets
50 assets
148 assets
8 assets
50 assets
148 assets
500
5,000
3.01
5.7
23.05
1.13
3.08
15.67
1,000
10,000
4.85
9.95
37.2
1.28
4.67
26.1
3,000
25,000
13.86
45.32
185.49
1.69
12.22
79.31As we see, the time required to computeCVaR robust portfolios via two approaches differs significantly when the sample size and the number of assets increase. For example, the time required to solve the CVaR robust mean-CVaR problem by two approaches for a problem with 8 assets and 5000 samples and 500 scenarios differs slightly. But, when the number of assets is more than 50, the number of scenarios is more than 500, and the sample size is more than 5000, differences become significant. A problem with 148 assets, 3000 y-scenarios, and 25000 μ-samples is solved in less than 80 seconds using smoothing technique, while by (22) it took over 185 seconds. These comparisons show that when the number of scenarios and samples becomes larger, the smoothing approach is more computationally efficient to determine CVaR robust portfolios than other approach.


## 5. Conclusions

Since risk and return are uncertain parameters in the portfolio optimization, thus their estimation might lead to large error in the final decision. One of the widely used and effective approaches in optimization to deal with data uncertainty is robust optimization. In this paper, we have presented the robust counterpart of mean-CVaR portfolio selection problem under the estimation risk in mean return. We have additionally used CVaR as risk measure to measure the estimation risk in mean return. To solve the model efficiently, the smoothing technique of Alexander et al. [10] is utilized. The performance of the CVaR robust mean-CVaR model with robust mean-CVaR models for both interval and ellipsoidal uncertainty sets is compared. Our experiments have verified that the CVaR robust mean-CVaR portfolios are more diversified. Applying the model for large and real data sets can be considered for future research.



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