Research in International Business and Finance 23 (2009) 107–116

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Research in International Businessand Finance

journal homepage: www.elsevier.com/locate/r ibaf

Portfolio optimization with CVaR under VG process�

Jinping Yu, Xiaofeng Yang, Shenghong Li ∗

Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China

a r t i c l e i n f o

Article history:Received 11 October 2007Received in revised form 18 July 2008Accepted 29 July 2008Available online 5 August 2008

Keywords:PortfolioCVaRVariance GammaCopulaMonte Carlo

a b s t r a c t

Formal portfolio optimization methodologies describe the dynam-ics of financial instruments price with Gaussian Copula (GC).Without considering the skewness and kurtosis of assets returnrate, optimization with GC underestimate the optimal CVaR of port-folio. In the present paper, we develop the approach for portfoliooptimization by introducing Lévy processes. It focuses on describ-ing the dynamics of assets’ log price with Variance Gamma copula(VGC) rather than GC. A case study for three Indexes of Chinese StockMarket is performed. On application purpose, we calculate the besthedge positions of Shanghai Index (SHI), Shenzhen Index (SZI) andSmall Cap Index (SCI) with the performance function CVaR underVG model. It can be combined with Monte Carlo Simulation andnonlinear programming techniques. This framework is suitable forany investment companies.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Since Markowitz published his seminal work which introduces mean/variance risk managementframework in 1952, there has been lots of theoretical and empirical work on portfolio optimizationwith different utility functions, risk measures and constraints.

Merton (1969, 1971) pioneered in applying continuous-time stochastic models to the study of finan-cial markets (without transaction costs). He showed that the optimal investment policy of a constantrelative risk aversion (CRRA) investor is to keep a constant fraction of total wealth in the risky assetduring the whole investment period. The introduction of proportional transaction costs to Merton’smodel was first accomplished by Magill and Constantinides (1976), Davis and Norman (1990) stud-

� Project (No. Y604137) supported by ZheJiang Natural Science Foundation.∗ Corresponding author.

E-mail addresses: yujinping@yeah.net (J. Yu), gatty6879@163.com (X. Yang), shli@zju.edu.cn (S. Li).

0275-5319/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.ribaf.2008.07.004

108 J. Yu et al. / Research in International Business and Finance 23 (2009) 107–116

ied the problem with transaction costs for an infinite time. A further work carried out by Shreve andSoner (1994) fully characterizes the optimal polices relying on the concepts of viscosity solution toHamilton–Jacobi–Bellman (HJB) equations. Liu and Loewenstein (2002) considered an optimal prob-lem with a stochastic time horizon following Erlang distribution. Some other papers about this are Daiand Yi (2006), Sun et al. (2007).

Measures of risk have a crucial role in optimization under uncertainty, especially in coping with thelosses that might be incurred in finance of the insurance industry. Value at Risk, or VaR for short, is oneof the most popular measures due to its simplicity, which has achieved the high status of being writteninto industry regulations. But this risk measure is not always sub-additive, nor convex. Artzner et al.(1999) proposed the main properties that a risk measures must satisfy, thus establishing the notion ofcoherent risk measure.

Conditional Value-at-Risk, or CVaR for short, is defined as the weighted average of VaR and lossesstrictly exceeding VaR for general distributions, see Rockafellar and Uryasev (2002). The CVaR riskmeasure has been proved to be a coherent risk measure in Pflug (2000); see also Rockafellar andUryasev (2001), Acerbi et al. (2001), Acerbi and Tasche (2002). After that, other classes of measures havebeen proposed, each with distinctive properties: Conditional Drawdown-at-risk (CDaR) in Chekhlov etal. (2000), ES in Acerbi et al. (2001), convex measures in Follmer and Shied (2002), spectral measuresin Acerbi and Simonetti (2002), and deviation measures in Rockafellar et al. (2006).

A simple description of the approach for minimizing CVaR and optimization problems with CVaRconstraints can be found in Chekhlov et al. (2000). Gaivoronski Pflug (2000) have found that in somecases optimization of VaR and CVaR may lead to quite different portfolios. Rockafellar and Uryasev(2000) demonstrated that linear programming techniques can be used for optimization of the Con-ditional Value-at-Risk (CVaR) risk measure. Several case studies showed that risk optimization withthe CVaR performance function and constraints can be done for large portfolios and a large number ofscenarios with relatively small computational resources. A case study on the hedging of a portfolio ofoptions using the CVaR minimization technique is included in Rockafellar and Uryasev (2000). Also,the CVaR minimization approach was applied to credit risk management of a portfolio of bonds, seeAndersson et al. (1999). This paper extends the CVaR minimization approach in Rockafellar and Uryasev(2000) to other classes of problems with CVaR functions. Further moer, CVaR minimization approachwas extended to derivative portfolio hedging, see Alexander et al. (2003), and with transaction cost inAlexander et al. (2006).

In those papers, they focus on describing the dynamics of assets log price with multiple Weinerprocess which is continuous and normal distribution. Unfortunately, as documented in a considerablenumber of papers written by academics and practitioners, both normality and continuity assumptionsare contradicted by the data in several pieces of evidence. As noted by Fama (1965), return distributionsof financial instruments are more leptokurtic than normal distributions and tend to be exhibit “fattails”. This phenomenon becomes particularly clear on high frequency data and be more accentuatedwhen the holding period becomes shorter. In these aspects the VG process, which was first introducedin financial modeling by Madan and Seneta (1990) to cope with shortcomings of Black–Scholes model,is superior to the Weiner process.

By introducing extra parameters, Variance Gamma (VG) process has a number of good mathemat-ical properties and has been proven to explain a number of economic findings. Mathematically, thedistributions have nice properties such as leptokurtic and fat tails. Economically, Madan et al. (1998)shows that their model is able to explain the well documented biases “volatility smile” in equityoptions. Moreover, Cariboni and Schoutens (2004) shows that their VG model for CDOs pricing fits toa variety of single name credit curves.

In the present paper, we drop the limitations of normality and continuity assumptions and completethe foundations for our methodology. We extend the portfolio optimization framework by describingthe dynamics of assets’ log price with VG copula rather than Gaussian copula. We find that CVaRbased on the classical Multiple Normal Distribution underestimate the risk of financial instruments.As a result, We suggest considering skewness and kurtosis into portfolio optimization framework byintroducing Variance Gamma copula to describe instrument dynamics.

The outline of this paper is as follows. The Variance Gamma process is summarized in Section 2 andits properties are presented and discussed in detail. In Section 3, we reformulate the frameworks of

J. Yu et al. / Research in International Business and Finance 23 (2009) 107–116 109

portfolio optimization with VG copula(VGC) rather than Gaussian copula(GC). Empirical findings andthe analysis of difference between VGC and GC models are presented in Section 4. Section 5 concludes.

2. The Variance Gamma process

Madan and Seneta (1990) provide a model (VG model) for log-return distribution that offers physicalinterpretation and incorporates both the long-tailedness and skewness feature in a log-return distri-bution. This section summarized the Variance Gamma process and its properties as well as proposedin Madan et al. (1998).

2.1. Definition and properties of the VG process

Definition 1. The VG process X(t;�,�,�), is defined in term of the Brownian motion with drift b(t;�,�)and the gamma process with unit mean rate �(t;1,�) as

X(t; �, �, �) = b(�(t; 1, �); �, �) (1)

in which b(t;�,�) = �t + �W(t), W(t) is a standard Brownian motion.

The process has three parameters: (1) � is the volatility of the Brownian motion; (2) � is the variancerate of the gamma time change; (3) � is the drift in the Brownian motion. Under this process, onecontrols volatility, kurtosis and skewness by the three parameters.

Proposition 1. The density function for the VG process at time t can be expressed conditional on therealization of the time change g as a normal density function.

P(Xt = x) =∫ ∞

0

gt/�−1 exp(−g/�)�t/�� (t/�)

· 1

�√

2�gexp

(− (x − �g)2

2�2g

)dg (2)

Proposition 2. The characteristic function for the VG process is

E[exp (iuXt)] =[

11 − i��u + �2u2�/2

]t/�

(3)

Proposition 3. Expressions for the first four central moments of the VG process are as follows:

E[Xt] = �t, (4)

E[(Xt − E[Xt])2] = (�2� + �2)t, (5)

E[(Xt − E[Xt])3] = (2�3�2 + 3�2��)t, (6)

E[(Xt − E[Xt])4] = (3�4� + 12�2�2�2 + 6�4�3)t + (3�4 + 6�2�2� + 3�4�2)t2. (7)

2.2. Describing the dynamics of asset’s log price with VG process

In the option pricing framework introduced by Madan et al. (1998), the stock price dynamics satisfiesSt = S0 exp (rt + X(t;�,�,�) + ωt), where ω = (1/�) ln (1 − �� − �2�/2) ensures the Martingale property ofstock price. We drop the martingale property of stock price in the present paper, because of the differentbasic assumptions between option pricing framework and portfolio optimization framework.

The new specification for the statistical stock price dynamics is obtained by replacing the Brownianmotion in the original Black–Scholes model by the VG process. Let the statistical process for the stockprice be given by

St = S0 exp(t + X(t; �, �, �)), (8)

Yt = t + X(t; �, �, �), (9)

where ,�,�,� are parameters, Yt denotes the log-return of asset, X(t;�,�,�) is a VG process defined asSection 2.1.

110 J. Yu et al. / Research in International Business and Finance 23 (2009) 107–116

3. The multiple VG model

There are a great number of research papers on CDOs pricing by introducing different copula ratherthan Gaussian copula, such as double t-copula in Hull and White (2004), multiple NIG copula in McNeilet al. (2005), etc.

Unfortunately, as noticed before, most researches on portfolio optimization focus on multiple nor-mal distributions. In this section, we proposed the multiple VG model to describe the price dynamicsof financial instruments. Under this assumption, we give the algorithm of portfolio optimization withthe performance function CVaR by using VG copula.

3.1. A portfolio molded by the VG copula

Consider a portfolio consisting of n instruments S1,S2,. . .,Sn, we denote the log-return of ith assetover time interval t as Yi

t , which is described by multiple VG copula as follows:

Sit = Si

0 exp(it +n∑

k=1

aikXkt ), (10)

that is

Yit = it +

n∑k=1

aikXkt , (11)

where {Xkt , k = 1,2,. . .,n} are multiple VG factors as follows, which are independent with the given t.

Xkt ∼ VG(t; �k, �k, �k), (12)

In the empirical performance, we assume there are 252 trade days in a year and t = (1/252) denotesone trade day. For convenience, we denote A = [aik] as the coefficient matrix, and Yi

t by Yi for short toexpress the daily log-return for ith instrument, see Eq. (11).

3.2. Framework of portfolio optimization with CVaR under VG process

Let m = (m1,m2,. . .,mn) be positions of a portfolio and Y = (Y1,Y2,. . .,Yn)T are sample instrumentreturns modeled by the multiple VG copula. Portfolio loss f(m,Y) = −mTY is defined as negative return.We consider optimization problem with the performance function CVaR, which is defined as

CVaRˇ = E[f (m, Y)|f (m, Y) ≥ VaRˇ]. (13)

Then the optimization problem becomes

min CVaRˇ(−mT Y)

s.t. E[−mT Y] ≤ −r0 (14)

Yi = i +n∑

k=1

aikXk (15)

Xk ∼ VG(t; �k, �k, �k) (16)

n∑j=1

mj = 1 (17)

mj ≥ 0, j = 1, n (18)

J. Yu et al. / Research in International Business and Finance 23 (2009) 107–116 111

In which, ˇ is the probability level, that means CVaRˇ is the average of the 5% worst losses if ˇ equalto 95%, Y is the instrument vector, m is the weight vector of instruments, r0 is the riskless rate on timeinterval length t, Xk is the kth VG factor, �k, k, �k are the three parameters of VG process.

This framework is similar to Rockafellar and Uryasev (2002) except describing the log-return offinancial instruments by multiple VG copula rather than multiple Gaussian copula.

3.3. Algorithm of portfolio optimization

It is difficult to solve the optimization problem above because of the nonsmooth of CVaR andnon-normal distribution of VG Copula. In the present paper, we solve it by combining Monte CarloSimulation and nonlinear programming techniques proposed in Rockafellar and Uryasev (2002).

• Step 1: Coefficient matrix estimationIn this paper, we estimate the parameters of multiple VG model by Moment Estimation instead

of MLE approach employed in Madan et al. (1998). According to the increment independency ofVG process, explicit expressions for the first four central moments and covariance of the returndistribution over an interval of length t as followes, which can be proved briefly by Proposition 3 andassumptions (10)–(12).

E[Yi] = (i +n∑

k=1

aik�k) · t (19)

Cov(Yi, Y j) =n∑

k=1

aikajk(�2k �k + �2

k ) · t, (20)

E[(Yi − EYi)3] =

n∑k=1

a3ik(2�3

k �2k + 3�k�k�2

k ) · t, (21)

E[(Yi − EYi)4] =

n∑k=1

a4ik[(3�4

k �k + 12�2k �2

k �2k + 6�4

k �3k )t + (3�4

k + 6�2k �2

k �k + 3�4k �2

k )t2]

+∑k /= j

a2ika2

ij(�2k �k + �2

k )(�2j �j + �2

j )t2 (22)

We calculate the coefficient matrix A and variance of independent factor variable X1,X2,. . .,Xn bycovariance matrix of sample log-return data. Denote

V �Cov(Yi, Y j)n×n, A� (aik)n×n, S �

⎛⎝

s1 0. . .

0 sn

⎞⎠

n×n

.

in which, s2k� (�2

k�k + �2

k)t.

We obtain V/t = A(SST)AT by Eq. (20) in Step 2. Without losing generality, we choose A and S be theeigenvalue and eigenvector of positive definite matrix V/t, which can be obtained by MATLAB easily.

• Step 2: Factor parameters estimationWe can calculate the first four moments of sample data and obtain the nonlinear equations as fol-

lows with factors estimated in step 1. Following equations can be solved by Gauss–Newton algorithm

112 J. Yu et al. / Research in International Business and Finance 23 (2009) 107–116

in MATLAB.⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

i +n∑

k=1

aik�k = E[Yi]t

�2k�k + �2

k= s2

k

tn∑

k=1

a3ik(2�3

k �2k + 3�k�k�2

k ) = E[(Yi − EYi)3]

t

n∑k=1

a4ik[(3�4

k �k + 12�2k �2

k �2k + 6�4

k �3k ) · t + (3�4

k + 6�2k �2

k �k + 3�4k �2

k ) · t2] +∑k /= j

a2ika2

ij(�2k �k

+�2k

) · (�2j�j + �2

j) · t2 = E[(Yi − EYi)

4]

• Step 3: Scenario generationWith the parameters estimated in step 1 and step 2, we simulate the multiple VG distribution fac-

tors X1,X2,. . .,Xn by Monte Carlo Method. According to assumptions (10)–(12), sceneries of financialinstruments log return are obtained.

• Step 4: LinearizationIn this paper, we impose the technique introduced by Rockafellar and Uryasev (2002) to solve

the nonlinear optimization problem and calculate the optimal CVaRˇ in Section 3.2. As proposed inRockafellar and Uryasev (2002), CVaRˇ is equal to the minimization of function F(m,˛), that is

CVaRˇ = min˛ ∈ R

F(m, ˛) (23)

in which,

F(m, ˛) = ˛ + 11 − ˇ

∫Y ∈ Rn

(−mT Y − ˛)+

dF(Y) ≈ ˛ + 1(1 − ˇ)J

J∑j=1

(−mT Yj − ˛)+

(24)

Here, we approximate the integral by discrete points Yj, j = 1,. . .,J in the space Rn which we simu-lated in step 3. We can reduced problem (14)–(18) to a linear programming problem using dummyvariables Zj, which can be solved by Lingo 8.0 easily.

MinZ,˛,m

˛ + 1(1 − ˇ) · J

J∑j=1

Zj

s.t. Zj ≥ −mT Yj − ˛ (25)

Zj ≥ 0 (26)

Yi = i +n∑

k=1

aikXk (27)

Xk ∼ VG(t; �k, �k, �k) (28)

E[−mT Y] ≤ −r0 (29)

n∑j=1

mj = 1 (30)

mj ≥ 0, j = 1, n (31)

J. Yu et al. / Research in International Business and Finance 23 (2009) 107–116 113

Table 1Mean, skewness and kurtosis of portfolio day return

Mean Skewness Kurtosis

SCI 0.0012462727 −0.56170625 6.23436164SZI 0.0011808493 −0.69082357 7.45079086SHI 0.0011072027 −0.86036593 7.53126513

Table 2Covariance matrix of portfolio instruments

SCI (×10−4) SZI (×10−4) SHI (×10−4)

SCI 0.69526318 0.42613848 0.26203772SZI 0.42613848 0.51821187 0.24710096SHI 0.26203772 0.24710096 0.42010475

Table 3Estimation of coefficient matrix A

0.53536969 0.47452624 0.69871606−0.81104582 0.05791121 0.58210907

0.235762544 −0.8783343 0.41586643

4. A case study for Chinese market

Let’s take Chinese Stock Market as a case study. To ensure sufficient representative and diversity, wechoose Shang Hai Index (SHI), Shen Zhen Index (SZI) and Small Cap Index (SCI) as portfolio instruments.The data employed was the 448 daily observations of log returns covering the period from June 8th2005 to April 12th 2007.

We begin the analysis by calculating the mean, skewness, kurtosis and covariance of the sampledata Tables 1 and 2. As shown in Table 1, the skewness and kurtosis of each Index are significantlydifferent from which of the normal distribution with skewness 0 and kurtosis 3. In other words, theportfolio optimization framework under multiple normal distributions fits the Chinese Market badly,which encourage us to import the VG copula from CDOs pricing into portfolio optimization.

Tables 3 and 4 are the numerical results of moment estimation corresponding every portfolio instru-ments and factors. With these parameters, J sceneries of financial instruments log return with the VGcopula are obtained, which can be substituted into the linear programming (25)–(31).

We solved the linear optimization problem with constraints by using Lingo 8.0 on Pentium-IV,2.66 GHz. In order to compare the difference between portfolio optimization under two different cop-ulas, we considered two types of random numbers to approximate the integral: the Gaussian copularandom sequence of numbers (same with Rockafellar and Uryasev approach) and the VG copula ran-dom sequence of numbers simulated with parameters estimated in Section 2, both are simulatedby Monte Carlo approach. The calculation results with given ˇ and r0 = 0.0012 are documented inTables 5 and 6.

As shown in Tables 5 and 6, portfolio positions converge quite slowly when the number of samplesbecomes large. However, when the number of sceneries is larger than 10000, the relative differencebetween calculated VaR and CVaR is less than 1%.

Table 4Estimation of parameters i , �k , �k , �k (i,k = 1,2,3)

� �2 �

Y1 0.60137967 X1 0.02820237 0.004109714 0.06352992Y2 0.61847366 X2 0.13655965 0.006193513 0.02225693Y3 0.61087473 X3 −0.5255622 0.028819414 0.00571285

114 J. Yu et al. / Research in International Business and Finance 23 (2009) 107–116

Table 5The portfolio, VaR and CVaR with minimum CVaR approach under multiple Weiner process under the constraint on meanportfolio loss less than −r0 = −0.0012 (ˇ value, number of samples in a simulation run, three positions, calculated VaR, calculatedCVaR, number of iterations of lingo, processor time: pentium IV, 2.66 GHz)

ˇ J SCI SZI SHI VaR CVaR Itr Time (s)

0.9 1000 0.469011 0.374382 0.156608 0.007640 0.010726 238 00.9 3000 0.544769 0.231325 0.223906 0.007557 0.010843 466 00.9 5000 0.490076 0.334604 0.175320 0.007340 0.010668 824 20.9 10000 0.494902 0.325490 0.179608 0.007491 0.010697 1495 60.9 20000 0.513524 0.290325 0.196151 0.007385 0.010467 2471 230.95 1000 0.546276 0.228479 0.225245 0.010014 0.012402 85 00.95 3000 0.497068 0.321401 0.181532 0.009750 0.012346 200 10.95 5000 0.491980 0.331008 0.177012 0.009972 0.012806 421 10.95 10000 0.525813 0.267120 0.207067 0.009788 0.012471 570 30.95 20000 0.505183 0.306076 0.188741 0.009814 0.012675 1296 120.99 1000 0.564956 0.193205 0.241839 0.013569 0.015279 48 00.99 3000 0.466670 0.378803 0.154528 0.013823 0.016030 69 10.99 5000 0.526029 0.266712 0.207259 0.014767 0.016696 212 00.99 10000 0.490695 0.333436 0.175870 0.013937 0.015955 124 10.99 20000 0.535057 0.249664 0.215279 0.014542 0.016761 370 5

Simulations are conducted with Monte Carlo approach.

Table 6The portfolio, VaR and CVaR with minimum CVaR approach under multiple VG process under the constraint on mean portfolioloss less than −r0 = −0.0012 (ˇ value, number of samples in a simulation run, three positions, calculated VaR, calculated CVaR,number of iterations of lingo, processor time: Pentium IV, 2.66 GHz)

ˇ J SCI SZI SHI VaR CVaR Iter Time (s)

0.9 1000 0.439566 0.429983 0.130451 0.006671 0.012811 274 00.9 3000 0.480882 0.351966 0.167153 0.007211 0.013365 376 10.9 5000 0.496845 0.321822 0.181334 0.006755 0.012917 903 20.9 10000 0.535156 0.249477 0.215367 0.006675 0.012496 1462 70.9 20000 0.525834 0.267080 0.207086 0.006653 0.012380 2756 260.95 1000 0.547579 0.226018 0.226403 0.010144 0.015454 175 00.95 3000 0.589087 0.147636 0.263276 0.010786 0.017269 335 00.95 5000 0.587523 0.150591 0.261887 0.010344 0.016561 406 10.95 10000 0.537491 0.245067 0.217442 0.010370 0.016276 902 70.95 20000 0.525602 0.267519 0.206879 0.010435 0.016422 1526 150.99 1000 0.550602 0.220309 0.229089 0.016214 0.026269 210 00.99 3000 0.663662 0.006814 0.329524 0.020944 0.028572 274 10.99 5000 0.667270 0.000000 0.332730 0.020537 0.025898 176 10.99 10000 0.538792 0.242611 0.218597 0.020312 0.026355 435 30.99 20000 0.557440 0.207397 0.235163 0.020239 0.026112 560 7

Simulations are conducted with Monte Carlo approach.

Table 7Compare the portfolio, VaR and CVaR with minimum CVaR approach under both Gaussian copula (G) and VG copula (VG)(J = 20,000, r0 = 0.0012)

ˇ Copula VaR CVaR Dif (%)

0.9G 0.007385 0.010467

0.182765VG 0.006653 0.01238

0.95G 0.009814 0.012675

0.295621VG 0.010435 0.016422

0.99G 0.014542 0.016761

0.557902VG 0.020239 0.026112

J. Yu et al. / Research in International Business and Finance 23 (2009) 107–116 115

Table 8The portfolio, VaR and CVaR with minimum CVaR approach under multiple VG process with one parameter of Shanghai Index(SHI) changes each time in which ˇ = 0.99, J = 20,000

Parameter Value SCI SZI SHI VaR CVaR

– – 0.557440 0.207397 0.235163 0.020239 0.026112Mean 0.002107 0.064017 0.313175 0.622807 0.015910 0.020318Mean 0.000507 0.301500 0.697646 0.000852 0.020420 0.025918Skewness 0.360366 0.549873 0.221686 0.228441 0.019678 0.025240Skewness −1.060366 0.531131 0.257077 0.211792 0.019341 0.024256Kurtosis 27.531265 0.538169 0.243788 0.218043 0.019297 0.024473r0 0.001150 0.111737 0.370120 0.518143 0.017417 0.021311r0 0.001100 0 0.400736 0.599266 0.017068 0.021008

For instance, Table 7 makes a comparison between the results for optimal CVaR with Gaussiancopula (CVaRG) and CVaR with VG copula (CVaRVG) when the number of samples is equal to 20000.We see that optimal CVaRVG is significant larger than CVaRG, see line 7 in Table 5. We consider therelative difference between CVaRG and CVaRVG by Dif, which is defined as

Dif = CVaRVG − CVaRG

CVaRG(32)

Considering the situation ˇ = 0.9, optimal CVaRVG equals 0.01238, which is obviously larger thanGaussian copula approach with CVaRG = 0.010467. As a result, we conclude that CVaR with Gaussiancopula underestimate the risk of portfolio. It becomes even seriously when ˇ converges to 1 as shown inTable 7. Comparing to the previous situation, the relative difference of optimal CVaR between Gaussianand VG copula becomes 29.5621% with ˇ = 0.95 and even larger as 55.7902% with ˇ = 0.99.

Without losing generally, we change the mean, skewness and kurtosis of Shanghai Index (SHI)separately to estimate the sensitivity of the estimation parameters and the stability of the optimizationalgorithm. The optimal portfolio, VaR and CVaR with Minimum CVaR Approach under multiple VGprocess are shown in Table 8. The first line is the result with original parameters in Tables 1 and 2,others are results with changing mean, skewness and kurtosis parameter. We found that optimalportfolio, VaR and CVaR keep stable while changing one parameter of sample’s skewness or kurtosiseach time, see line 4–6 of Table 8. But the optimal portfolio changed significantly if when we increaseof decrease sample’s mean as shown in line 2 and line 3 of Talbe 8. Also, if we minus the meanportfolio constraint r0 from 0.0012 to 0.00115 and 0.0011, position of SHI climbs rapidly, see line 7and line 8.

These phenomenons not only demonstrate that different copula lead to different optimal CVaR,but also remind all researchers and practitioners to consider fat-tailedness and kurtosis in portfoliooptimization framework.

5. Conclusion

This paper considered a new approach for portfolio optimization by describing the dynamics ofassets’ log price with VG copula rather than Gaussian Copula. We obtained the optimal positions offinancial instruments and CVaR by introducing the linearization techniques proposed in Rockafellarand Uryasev (2002). Via a case study of three indexes in Chinese Stock Market, we demonstrated thatVG copula can be efficiently overcome the shortcomings of Gaussian copula which underestimate theCVaR of portfolio. Further more, the optimal algorithm is stable with skewness and kurtosis parameters.As a result, We suggest describing instrument dynamics by VG copula and modeling the portfoliooptimization framework with it.

There is a lot of room for improvement of the suggested VG copula approach, such as change thedynamics of assets’ log price description by different copulas like NIG or other Lévy process, or extendthe framework to contimuous-time stochastic models with CVaRVG constraints.

116 J. Yu et al. / Research in International Business and Finance 23 (2009) 107–116

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