Portfolio optimization with a copulabased extension of conditional valueatrisk
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Abstract
The paper presents a copulabased extension of Conditional ValueatRisk and its application to portfolio optimization. Copulabased conditional valueatrisk (CCVaR) is a scalar risk measure for multivariate risks modeled by multivariate random variables. It is assumed that the univariate risk components are perfect substitutes, i.e., they are expressed in the same units. CCVaR is a quantile risk measure that allows one to emphasize the consequences of more pessimistic scenarios. By changing the level of a quantile, the measure permits to parameterize prudent attitudes toward risk ranging from the extreme risk aversion to the risk neutrality. In terms of definition, CCVaR is slightly different from popular and wellresearched CVaR. Nevertheless, this small difference allows one to efficiently solve CCVaR portfolio optimization problems based on the full information carried by a multivariate random variable by employing column generation algorithm.
Keywords
Multivariate risk measures Quantile risk measures Portfolio optimization Column generation algorithm1 Introduction
In the business environment enterprises are forced to develop and implement enterprisewide integrated risk management systems. Risks have to be limited and managed from an enterprisewide portfolio perspective. The increasing amount of risks in today’s market has increased the demand for risk measurement models and risk management tools. This paper presents an analytic (quantitative) model for the optimization of a portfolio of risks based on prudent and complete stochastic information.
The portfolio optimization problem considered in this paper relates to the original Markowitz (1952) formulation. The original Markowitz portfolio optimization problem is modeled as a meanrisk bicriteria optimization problem where the portfolio mean rate of return is maximized and the risk measured by standard deviation or variance is minimized. Several other risk measures have been later considered thus creating the entire family of meanrisk models (Mitra et al. 2003 and references therein). It is often argued that the variability of the rate of return above the mean should not be penalized, since the investors are concerned with the underperformance rather than the overperformance of a portfolio. This led Markowitz (1959) to propose downside risk measures such as (downside) semivariance to replace variance as the risk measure. Consequently, one observes growing popularity of downside risk models for portfolio selection (Bawa 1978; Fishburn 1977; Sortino and Forsey 1996).
This paper presents copulabased conditional valueatrisk (CCVaR) as a downside risk measure. The measure is intended for multidimensional risk measurement, where the risk is defined as a multivariate random vector whose elements (coordinates) represent risk components. It is assumed that the risk components depend on each other in a stochastic sense and their dependence structure is given by a copula function. It is also assumed that the risk components are perfect substitutes, i.e., they are expressed in the same units (e.g. monetary).
In order to define the measure, the concept of multivariate quantile is introduced. The multivariate quantile is defined as a cone covering the worst (smallest) realizations of a multivariate random variable with a total probability equaling the level of a quantile (throughout the paper it is assumed that larger outcomes are preferred). CCVaR is a scalar risk measure defined as the worst expectation within multivariate quantile of a given level. The measure can be viewed as a prudent variant of multivariate conditional value at risk (MCVaR) introduced by Prékopa (2012), where the conditional expectation of a scalarized random vector is taken over the entire set of multivariate quantiles.
CCVaR allows one to parameterize prudent attitudes toward risk ranging from the extreme risk aversion (worst case) to the risk neutrality (expectation) by changing the level of a quantile. CCVaR is a pessimistic risk measure and it defines almost the same kind of risk as popular and wellresearched conditional valueatrisk (CVaR) (Rockafellar and Uryasev 2000). Specifically, in the univariate case both measures coincide, but in the multivariate setting they differ due to the different definitions of quantiles. CCVaR uses cones as opposed to CVaR which uses halfhiperplanes.
An important advantage of CCVaR is its portfolio optimization model which permits one to efficiently solve real life problems based on the full information carried by a multivariate random variable. For a discrete multivariate random vector, the CCVaR portfolio optimization model is a linear program with an infinite number of constraints. However, the dual formulation of this model can be efficiently solved by column generation algorithm based on the Dantzig and Wolfe (1961) decomposition.
The paper is organized as follows. Section 2 presents the definition of CCVaR. Section 3 describes how the measure relates to Conditional ValueatRisk. Section 4 shows some properties of the measure in terms of coherent risk measures (Artzner et al. 1997, 1999). Section 5 presents the portfolio optimization model of the measure. Section 6 describes the approximate portfolio optimization algorithm. Section 7 presents the results of the computational analysis. Finally, some concluding remarks are given.
2 The definition of copulabased conditional valueatrisk
Definition 1
3 A relation to conditional valueatrisk
Proposition 1
4 The properties of copulabased conditional valueatrisk
In this section we state and prove some properties of CCVaR.
Proposition 2
 (i)\(\hbox {CCVaR}_\beta \) is translationequivariant, i.e.,$$\begin{aligned} \hbox {CCVaR}_\beta (\mathbf {R}+ \mathbf {c}) = \hbox {CCVaR}_\beta (\mathbf {R}) + \mathbf {1}^T \mathbf {c}. \end{aligned}$$
 (ii)\(\hbox {CCVaR}_\beta \) is positively homogenous, i.e.,if \(\lambda > 0\).$$\begin{aligned} \hbox {CCVaR}_\beta (\lambda \mathbf {R}) = \lambda \,\hbox {CCVaR}_\beta (\mathbf {R}), \end{aligned}$$
 (iii)\(\hbox {CCVaR}_\beta \) in general is not monotonic, i.e., ifthen not always$$\begin{aligned} \mathbf {R}_1(\omega ) \ge \mathbf {R}_2(\omega ) \hbox { for all } \omega \in \Omega \end{aligned}$$$$\begin{aligned} \hbox {CCVaR}_\beta (\mathbf {R}_1) \ge \hbox {CCVaR}_\beta (\mathbf {R}_2). \end{aligned}$$
 (iv)\(\hbox {CCVaR}_\beta \) is superadditive in the following sense:$$\begin{aligned} \hbox {CCVaR}_\beta (\mathbf {R}) \ge \sum _{i=1}^n \hbox {CCVaR}_\beta (R_i). \end{aligned}$$
Proof
\(i\)  \(\mathbf {R}_1(\omega _i)\)  \(\mathbf {R}_2(\omega _i)\)  \(\mathbb {P}(\omega _i)\) 

1  \((1,0)^T\)  \((1,0)^T\)  0.1 
2  \((0,1)^T\)  \((0,1)^T\)  0.1 
3  \((2,2)^T\)  \((1,1)^T\)  0.8 
Artzner et al. (1997, 1999) call a risk measure coherent, if it is translationequivariant, positively homogenous, superadditive, and monotonic. One sees that \(\hbox {CCVaR}_\beta \) is not coherent in this sense, since it is not monotonic.
Let us consider the following special case related to portfolio optimization further discussed in Sect. 5. We are interested in selecting the optimal portfolio of risk components \(R_i\) scaled by portfolio weights \(x_i\), i.e., in maximizing \(\hbox {CCVaR}_\beta (\mathbf {x}\circ \mathbf {R})\), where \(\circ \) is the Hadamard product operator. Under the above assumption the following assertion is valid.
Proposition 3
Proof
Hence CCVaR preserves coherency in problems where linear combinations of risk components are considered.
Finally, let us address the issue of accuracy of the measure determined in the computational process for risk components \(R_i\) initially modeled by continuous random variables. We will consider the following approximations of continuous marginal distributions. Let \([a_i, b_i] \supset R_i\), \(i=1,\ldots ,n\) be the closed subsets partitioned with finite sequences \(a_i = t^{(1)}_i < t^{(2)}_i < \cdots < t^{(k)}_i = b_i\) for some \(k \in \mathbb {N}\). We will assume that \(\mathbb {P}[a_i, t^{(j)}_i]= F_{R_i}(t^{(j)}_i)\) for \(1 \le j \le k\). The approximate random variables will be further denoted by \(R^{(k)}\) and random vectors by \(\mathbf {R}^{(k)}\), respectively.
Proposition 4
Proof
5 The portfolio optimization model
6 The approximate portfolio optimization algorithm
Note that the upper bounds in (15) defined by the function (16) ensure that the largest \(\pi _i\)s in (14) representing nonnegative portfolio shares receive the smallest weights given by \(\hbox {CVaR}_\beta (R_i, \mathbf {u})\). For \(\mathbf {u}^\star \) the value of the objective function of the problem (14) approximates the minimum.
The above way of solving the initial pricing subproblem (9) does not guarantee achieving the optimal result of the problem (7). But surely the calculated value is the upper bound of the optimal value of the problem (7), which follows from the fact that we solve its dual formulation (8). Below is the complete approximate algorithm for CCVaR portfolio optimization.

Step 1 Create \(i=1,\ldots ,n\) initial vectors \(\mathbf {u}_i=(1,\ldots ,\beta ,\ldots 1)^T\) with \(\beta \) placed on the \(i\)th position. Compute values of \(\hbox {CVaR}_\beta (R_i, \mathbf {u}_i)\) using Algorithm 1 and set up the restricted master problem (8).

Step 2 Solve the restricted master problem (8) in order to determine the dual prices \(\pi _i\).
 Step 3 Determine the values of the objective function of the problem (14) for several different vectors \(\mathbf {u}\). Retain the smallest value of the objective function along with the corresponding values of \(\hbox {CVaR}_\beta (R_i, \mathbf {u})\) to be further inserted to the restricted master problem (8) as a new column. In order to solve the problem (14), perform the following steps:

Step 4 If the smallest value of the objective function of the problem (14) obtained in Step 3 is negative, insert a new column to the restricted master problem (8) and go to Step 2. Otherwise the optimal value of \(\hbox {CCVaR}_\beta \) is represented by the coefficient \(\pi _{n+1}\), wheras the optimal portfolio weights are represented by the coefficients \(\pi _i\).
7 Computational results
A PC with a 2 GHz Intel Core Duo processor and 2 GB RAM has been used to run an application written in Matlab by using the Global Optimization Toolbox and the IBM ILOG CPLEX optimizer version 12.2. The computations have been conducted for the following marginal distributions: lognormal, Gaussian, and Student’s \(t\) with 4 degrees of freedom. The values of realizations have been limited to the range \([1, 3]\) so as to cover the typical asset returns. The distributions had different expectations and standard deviations.
CCVaR has been calculated for the parameter \(\beta \in \{0.01,\,0.1\}\) and the following copula functions: Clayton, Frank, and Gumbel. The marginal distributions have been approximated by discrete distributions with 100 and 500 realizations. The pricing subproblem has been evaluated 15 times in each iteration of column generation algorithm. In turn CVaR has been calculated for identical betas and 100,000 random variates drawn from the considered discrete multivariate distributions. Both measures have been used to determine optimal portfolios for 10 and 100 assets.
CCVaR optimization results for lognormal marginal distributions
Copula  \(\theta \)  \(\beta \)  \(n\)  \(m\)  CCVaR  Div.  Shares  Time (s)  

#  Max  Min  
Clayton  2  0.1  10  100  0.234  3  0.950  0.018  6.0 
100  100  0.322  2  0.530  0.470  54.3  
100  500  0.251  11  0.099  0.084  267.7  
0.01  10  100  0.160  1  1.000  1.000  1.3  
100  100  0.163  2  0.862  0.138  46.6  
100  500  0.100  4  0.289  0.213  222.5  
10  0.1  10  100  0.218  2  0.661  0.339  1.0  
100  100  0.239  1  1.000  1.000  22.8  
100  500  0.213  1  1.000  1.000  109.2  
0.01  10  100  0.120  1  1.000  1.000  0.7  
100  100  0.160  1  1.000  1.000  17.7  
100  500  0.103  1  1.000  1.000  81.3  
Frank  9  0.1  10  100  0.265  6  0.256  0.088  10.9 
100  100  0.327  8  0.160  0.099  57.5  
100  500  0.322  15  0.074  0.061  292.0  
0.01  10  100  0.195  8  0.207  0.044  8.0  
100  100  0.286  11  0.128  0.073  58.5  
100  500  0.279  16  0.070  0.055  286.4  
Gumbel  5  0.1  10  100  0.215  3  0.705  0.107  5.8 
100  100  0.268  3  0.428  0.280  48.1  
100  500  0.243  7  0.174  0.118  237.9  
0.01  10  100  0.092  3  0.540  0.144  6.5  
100  100  0.175  3  0.597  0.195  37.2  
100  500  0.125  7  0.154  0.135  178.9 
CVaR optimization results for lognormal marginal distributions
Copula  \(\theta \)  \(\beta \)  \(n\)  \(m\)  CVaR  Div.  Time (s)  

Mean  Max  Min  Mean  
Clayton  2  0.1  10  100  0.771  0.773  0.769  10.0  24.6 
100  100  1.320  1.334  1.230  95.0  197.8  
100  500  1.316  1.317  1.314  100.0  182.1  
0.01  10  100  0.559  0.561  0.556  10.0  6.2  
100  100  1.175  1.177  1.172  99.0  78.5  
100  500  1.178  1.181  1.173  100.0  79.8  
10  0.1  10  100  1.014  1.015  1.009  10.0  27.4  
100  100  1.441  1.443  1.440  100.0  202.2  
100  500  1.303  1.304  1.302  100.0  196.9  
0.01  10  100  0.742  0.749  0.738  10.0  6.2  
100  100  1.294  1.299  1.291  100.0  82.4  
100  500  1.168  1.170  1.165  100.0  77.9  
Frank  9  0.1  10  100  0.816  0.818  0.813  100.0  29.8 
100  100  1.236  1.237  1.235  100.0  182.1  
100  500  1.200  1.201  1.199  100.0  196.6  
0.01  10  100  0.620  0.624  0.615  10.0  6.8  
100  100  1.108  1.111  1.105  100.0  74.2  
100  500  1.073  1.074  1.072  100.0  76.6  
Gumbel  5  0.1  10  100  0.863  0.865  0.859  10.0  30.6 
100  100  1.326  1.327  1.325  100.0  204.7  
100  500  1.331  1.333  1.329  100.0  206.9  
0.01  10  100  0.630  0.633  0.624  10.0  6.3  
100  100  1.200  1.203  1.196  100.0  79.1  
100  500  1.174  1.177  1.169  100.0  74.9 
CVaR and CCVaR optimization results for \(n = 5\) and \(m = 10\)
Measure  \(\beta \)  Value  Div. #  Time (s) 

CVaR  0.1  \(0.077\)  5  7.8 
CCVaR  0.235  5  3.2  
CVaR  0.01  \(0.340\)  5  4.1 
CCVaR  \(0.261\)  4  1.9  
CVaR  0.001  \(0.508\)  5  3.8 
CCVaR  \(0.391\)  3  1.4 
Another observation is an excessive diversification of CVaR portfolios, which is not the case for CCVaR. For each multivariate distribution the mean diversification of CVaR portfolios is almost 100 %.
The computation time has also been taken into account as an important performance criterion. The CCVaR optimal portfolios for the largest tested multivariate distributions (\(n=100\) and \(m=500\)) have been determined in 3.5 min on average. The computation time for all the tested CCVaR models never exceeded 12 min.
8 Concluding remarks
In this paper, we presented an analytic model for the optimization of a portfolio of risks based on prudent and complete stochastic information. The model uses a copulabased extension of CVaR. CVaR gained popularity in many practical applications, because it is coherent and, as a downside risk measure, allows one to emphasize the consequences of more pessimistic scenarios. In portfolio selection problems, CVaR leads to linear programming optimization models. In typical real life problems, the high computationally efficient formulations of these models can account only for a small amount of information upon which decisions are made, and consequently, they may be far from being optimal. CCVaR solves this problem as it allows one to efficiently determine the optimal portfolio of risks based on the full information carried by a multivariate random variable.
Notes
Acknowledgments
The research conducted by A. Krzemienowski was supported by the Grant N N111 453440 from the Polish National Science Centre.
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