robust portfolio optimization with copulas

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Accepted Manuscript

Continuous Optimization

Robust Portfolio Optimization with Copulas

Iakovos Kakouris, Berc Rustem

PII: S0377-2217(13)01006-0

DOI: http://dx.doi.org/10.1016/j.ejor.2013.12.022

Reference: EOR 12048

To appear in: European Journal of Operational Research

Received Date: 6 August 2012Accepted Date: 14 December 2013

Please cite this article as: Kakouris, I., Rustem, B., Robust Portfolio Optimization with Copulas,European Journal

of Operational Research(2013), doi: http://dx.doi.org/10.1016/j.ejor.2013.12.022

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Robust Portfolio Optimization with Copulas

Iakovos Kakourisa, Berc Rustemb

a Corresponding Author, Department of Computing, Imperial College London,

180 Queens Gate, London, SW7 2AZ, UK, tel: +44 (0) 20 7594 8341, fax: +44 (0) 20 7581 8024,

[email protected]

b Department of Computing, Imperial College London, 180 Queens Gate,

London, SW7 2AZ, UK, tel: +44 (0) 20 7594 8345, fax: +44 (0) 20 7581 8024, [email protected]

Abstract

Conditional Value at Risk (CVaR) is widely used in portfolio optimization

as a measure of risk. CVaR is clearly dependent on the underlying probability

distribution of the portfolio. We show how copulas can be introduced to any

problem that involves distributions and how they can provide solutions for the

modeling of the portfolio. We use this to provide the copula formulation of the

CVaR of a portfolio. Given the critical dependence of CVaR on the underlying

distribution, we use a robust framework to extend our approach to Worst Case

CVaR (WCVaR). WCVaR is achieved through the use of rival copulas. These rival

copulas have the advantage of exploiting a variety of dependence structures,symmetric and not. We compare our model against two other models, Gaussian

CVaR and Worst Case Markowitz. Our empirical analysis shows that WCVaR

can asses the risk more adequately than the two competitive models during

periods of crisis.

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1 Introduction

In this paper we look into the problem of portfolio optimization where the assets

of the portfolio are described by random variables. In this situation the selection of

the optimal portfolio depends on the underlying assumptions on the behaviour of

the assets and the choice on the measure of risk. Usually the objective is to find the

optimal risk-return trade-off.

One of the pioneers in portfolio optimization was Markowitz (1952) who pro-

posed the mean-variance framework for risk return analysis. Although the most

common measure for the estimation of the return of the portfolio remains the ex-

pected return many other ways of calculating the risk have been developed. A

widely used measure of risk is Value at Risk (VaR). VaR is the measure of risk that

is recommended as a standard by the Basel Committee. However, VaR has been

criticized in recent years mainly for two reasons. Firstly, VaR does not satisfy sub-

additivity and hence it is not a coherent measure of risk in the way that is defined

by Artzner et al. (1998). Also, it is not a convex measure of risk and thus it may

have many local extrema which cause technical issues when optimizing a portfolio.

Secondly, it gives a percentile of loss distribution that does not provide an adequate

picture of the possible losses in the tail of the distribution. Szego (2005) uses this

argument to state that VaR does not measure risk. Then he suggests alternative

measures of risk with one of them being Conditional Value at Risk (CVaR).

CVaR is the expectation of the distribution above VaR. Thus, the value of CVaR is

affected by the fatness of the tail of the distribution. Hence, CVaR provides a better

description of the loss on the tail of the distribution. Rockafellar and Uryasev (2000,

2002) proposed a minimization formulation that usually results in a convex or linear

problem. These are desirable aspects of CVaR and have paved the way of its use in

risk management and portfolio optimization. A literature review on CVaR can be

found in Zhu and Fukushima (2009) and the references therein.

Following the formulation of Rockafellar and Uryasev (2000, 2002) in order to

calculate CVaR one has to make some assumptions on the underlying distribution

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of the assets. This can also be in the form of an uncertainty domain like a hy-

percube or an ellipsoidal set in which all feasible uncertainty values lie (Zhu and

Fukushima, 2009). An alternative is assuming some multivariate distribution (Zhu

and Fukushima, 2009). In this paper we focus on the selection of multivariate distri-

butions.

Gaussian distribution is the most commonly used multivariate case. It is easy to

calibrate and also there are very efficient algorithms to simulate Gaussian data. This

also applies to some extent to the elliptical family of distributions with student-

distribution being widely used in Credit Risk (Chan and Kroese, 2010; Clemente and

Romano, 2004; Romano). One disadvantage of using the Gaussian distribution is its

symmetry. This implies that the probability of losses is the same as the probability of

gains. Studies suggest that at least in the context of financial markets, assets exhibit

stronger comovements during a crisis as opposed to prosperity (Ang and Chen,

2002; Hu, 2002, 2006). The second disadvantage is that it uses linear correlation as a

measure of dependence. As the name suggests, linear correlation is characterized by

linear dependencies. Since the observation of asymmetric comovements mentioned

above suggests non-linear dependencies, linear correlation may not be an adequate

measure of dependence (Artzner et al., 1998; Szego, 2005).

One way of addressing the limitations of the symmetry underlying elliptical

distributions is to consider mixture distributions. A linear combination of a set of

distributions is used to fit the given sample by optimizing the combination weights.

Hasselblad (1966, 1969) was one of the first who looked into mixture distributions

and how their parameters can be estimated. Zhu and Fukushima (2009) avoid the

assumptions needed on the set of distributions and their parameters and also avoid

the estimation of the weights. Subsets of historical returns are used to represent data

arising from different distributions and a worst-case scenario approach is applied to

avoid the calibration of the weights. Hu (2002, 2006) and Smillie (2007) use mixture

copulas to fit their data samples for the bivariate case. The work of Hu (2002, 2006)

and Zhu and Fukushima (2009) motivates us to introduce copulas within a worst

case robust scenario framework.

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Copulas are multivariate distribution functions whose one-dimensional margins

are uniformly distributed on the closed interval [0,1] (Cherubini et al., 2004; Nelsen,

2006; Sklar, 1959). The uniform margins can be replaced by univariate cumula-

tive distributions of random variables (Cherubini et al., 2004; Nelsen, 2006; Sklar,

1959). Hence, copulas consider the dependency between the marginal distributions

of the random variables instead of focusing directly on the dependency between the

random variables themselves. This makes them more flexible than standard distrib-

utions because it is possible to separate the selection of the multivariate dependency

from the selection of the univariate distribution. As an extension to that, the calibra-

tion of the multivariate distribution can be separated into two steps (Cherubini et al.,

2004). Also, the fact that copulas describe the dependency between the marginal

distributions which are monotonic makes them invariant under monotonic trans-

formations (Cherubini et al., 2004; Nelsen, 2006). Copulas are associated with many

measures of dependence that measure the monotonic dependencies between two

random variables. Furthermore, like copulas themselves these monotonic measures

are invariant under monotonic transformations (Embrechts et al., 2001; Schweizer

and Wolff, 1981; Wolff, 1980).

In this paper, we mainlyfocus on Archimedian copulas. This is a familyof copulas

that exhibits some interesting characteristics that can be utilized in our distribution

modelling as discussed in the next section.

The paper is structured as follows. In Section 2 we introduce copulas and the

associated measures of dependence together with some theoretical background. In

Section 3 we derive CVaR for copulas. We extend CVaR to WCVaR through the use

of mixture copulas. We conclude the section by stating the generalized optimization

problem for WCVaR. In Section 4 we construct a model based on the theory of the

previous sections. Then, we provide two numerical examples where we asses the

performance of our model. Finally, we close with conclusions.

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2 Copulas

Copulas arise from the theory of probabilistic metric functions and were first intro-

duced by Sklar in 1959 (Sklar, 1959). Copulas are multivariate distribution functions

whose one-dimensional margins are uniformly distributed on the closed interval

I [0, 1]. A more rigorous definition for copulas is given below.

Definition 2.1. An n dimensionalcopula(n-copula) is a function C fromIn to I with the

following properties

1. C(u1,.., ui,..., un) = 0 if any ui =0 for i = 1, 2,..., n (we also describe a function with

this property asgrounded)

2. C (1,..., 1, um< 1,..., 1)= umfor all um Iwhere m= 1, 2,..., n

3. C(u) 0 u In (we also describe a function with this property asn-increasing)

We continue with the definition of distribution functions and joint multivariate

distribution functions. This is used when we discuss the relation between distribu-

tion functions and copulas.

Definition 2.2. A distribution function is a function F from R to I with the following

properties:

1. F is nondecreasing

2. F()= 0 and F()= 1

Definition 2.3. A joint multivariate distribution function is a function from Rn to Iwith

the following properties

1. F is n-increasing

2. F is grounded

3. F()= 1

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4. F(,..., , xm, ,..., )= Fm(xm)

where m= 1, 2,..., n.

We can use copulas to replace probability distributions in all of their applications

thanks to Sklars theorem. Sklars theorem is probably the most important result

that links copulas to probability distributions. This, together with the corollary

that follows provide the relation between n-Copulas and multivariate distributions.

Sklar introduced his theorem in 1959 (Sklar, 1959), where the proof for its bivariate

case can also be found. The multivariate case is discussed by Schweizer and Sklar

(1983) (proofs of the theorem and corollary can be found therein).

Theorem 2.4(Sklars Theorem). Let F be an n-dimensional distribution function with

margins F1,..., Fn. Then there exists an n-copula C such that, for all xRn

,

F(x1,..., xn)= C(F1(x1),..., Fn(xn)). (1)

Furthermore, if F1,..., Fnare continuous, then C is unique; otherwise C is unique on RanF1

... RanFn(RanRange).

Corollary 2.5.Let F be an n-dimensional distribution function with margins F1,..., Fn, and

let C be an n-copula. Then, for anyu In,

C(u1,..., un)= F(F11 (u1),..., F

1n (un)) (2)

where F11

,..., F1n are the quasi-inverses of the marginals.

The marginsF1,..., Fnand the multivariate distribution function F are as defined

by Definitions 2.2 and 2.3. The marginsui can be replaced by Fi(xi), because they

both belong to the domainI and are uniformly distributed, i.e. let u U(0, 1), then

P(F(x)

u)=

P(x

F

1

(u))=

F(F

1

(u))=

u).Using Theorem 2.4 and Corollary 2.5 we can also derive the relation between

the probability density functions and the copulas. In the following definition f is

the multivariate probability density function of the probability distribution F and

f1,...,fnare the univariate probability density functions of the margins F1,..., Fn.

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Definition 2.6. Thecopulas densityof a n-copula C is the function c : In [0, )such

that

c(u1,..., un)nC(u1..., un)

u1...un=

f(x1,..., xn)ni=1 fi(xi)

(3)

Equations (1)-(3) imply that copulas decompose the multivariate probability dis-

tribution from its margins. The margins F1,..., Fncan be any distribution of our choice

while the copula simply describes the monotonic relation between the margins. This

is one of the biggest advantages of copulas because they divide the problem of find-

ing the correct distribution into two parts; first is finding the distribution of the

margins and second the dependency between them. This is much easier than find-

ing directly the multivariate dependency between the random variables. Hence,

the calibration of the copulas become an easier task. The calibration methods for

copulas can be found in Cherubini et al. (2004). Also, an introduction to copulas can

be found in Nelsen (2006) and Schweizer and Sklar (1983) discuss the relationship

of copulas to probabilistic metric spaces and the underlying theory.

2.1 Special cases of copulas and related measures of dependence

We introduce the copulas to be used in our examples. Together with the copulas we

consider their associated measures of dependence. The focus is on a special family

of copulas calledArchimedian. We also consider the Gaussian copula which is the

copula version of the multivariate normal distribution.

Archimedian copulas were firstly introduced by Ling (1965). They belong to the

family of probabilistic metric spaces that have some of the properties of Archimedes

triangle function and hence the name (Schweizer and Sklar, 1983). This is a family

of copulas that arises differently from the rest. Instead of using Theorem 2.4 we

construct them using directly a function, known as a generator, which enables us

to write the expression for the copula in a closed form.

Definition 2.7. Given a function : I [0, )such that(1) = 0and (0) = and

having inverse 1 completely monotone, an n-place Archimedian copula is a function

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C: In Isuch that

C(u)= 1((u1) + ... + (un)).

Extended literature regarding the Archimedian copulas can be found in Cheru-

bini et al. (2004); Nelsen (2006); Schweizer and Sklar (1983). In our analysis we

focus on three Archimedian copulas,Clayton, GumbelandFrank. Our motivation for

using these particular copulas stems from Hu (2002, 2006). Hu (2002, 2006) focuses

on the calibration of bivariate mixture copula (see also Hasselblad (1966, 1969) for

mixture distributions). There are two reasons why these particular copulas have

been chosen. Each copula better describes a different type of dependency. Clayton

and Gumbel are non symmetric copulas that describe more adequately negative

and positive dependencies, i.e. stronger dependence below and above the 50 th per-

centile respectively. The Frank copula is symmetric but it has different properties

to the Gaussian copula. Hence, by using them in a mixture structure we cover a

large spectrum of possible dependencies. Also, these three copulas are very easy to

calibrate.

The definitions of the three Archimedian copulas are the following:

Definition 2.8. Given a generator of the form (u) = u 1 with (0, )then, the

Claytonn-copula is given by

CCl(u)= max[(u1 + ... + u

n n + 1)

1/, 0].

Definition 2.9. Given a generator of the form(u) = ( ln(u)) with (1, )then, the

Gumbeln-copula is given by

CGu(u)= exp

[( ln u1) + ... + ( ln un)

]1/.

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Definition 2.10. Given a generator of the form(u)= ln

exp(u)1

exp()1

with (0, )then,

theFrankn-copula is given by

CFr(u)= 1

ln

1 +

(eu1 1) ... (eun 1)

(e 1)n1

.

For the calibration of the free parameters,, of the Archimedian copulas we will

use Kendals. Kendals is a bivariate measure of dependence and is defined by

the following equation

(X1, X2)= 4

F(x1, x2)dF(x1, x2) 1

=4

10

10

C(u1, u2)dC(u1, u2) 1.

(4)

As we can see from equation (4), Kendalsmeasures the dependency between the

cumulative distributions of random variableX1andX2and does not depend on the

random variables themselves. Thus, is a measure of monotonic dependence and

is invariant under monotonic transformations. This makes it a more robust measure

of dependence when compared to linear correlation. For comparison purposes, we

also define linear correlation as

(X1, X2)= 1

(X1) (X2)

[F(x1, x2) F1(x1)F2(x2)]dx1dx2

= 1(X1) (X2)

10

10

[C(u1.u2) uv]dF11 (u1)dF12 (u2),

(5)

where(Xi) denotes the variance of the random variableXi. It can be seen that in

the copula version of, the dependency on the random variablesX1andX2remains

in form of the volatility (Xi). An extensive literature on monotonic measures of

dependence can be found in Nelsen (2006); Schweizer and Wolff (1981); Smillie

(2007); Wolff(1980) and the references therein.

Let us denote the free parameter, , of each of the Archimedian copulas byCl

forCCl(Definition 2.8),GuforCGu(Definition 2.9) andFrforCFr(Definition 2.10).

For these three cases we have closed form relations with Kendalls equation (4)

(Cherubini et al., 2004). For CClwe have that

= 1 1Cl , (6)

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for CGu

= GuGu + 2

, (7)

and for CFrwe have

= 1 +4[D1(Fr)]

Fr, (8)

where

Dk()= k

k

0

x

exp(x) 1dx for k= 1, 2.

As we can see from Definitions 2.8-2.10 there is only one free parameter to

calibrate regardless of the dimensions of the copula. On the other hand we have

a for each pair of random variables. Our solution to this problem is to calculate

the for all the pairs and select the largest. To do that we calculate(Xi, Xj) for

i,j = 1, 2,..., nand i j. A higher positive implies stronger negative dependence

in the case of Clayton and stronger positive dependence in the case of Gumbel etc.

Hence, by selecting the highest positive we deliberately choose the most extreme

behaviour for our copulas. This seems to be consistent with the use of copulas within

a worst case robust framework. This also allows an easy calibration.

Other ways used to estimate are mentioned in the literature by Genest and

Rivest (1993), and Smillie (2007). The former employ a more sophisticated approach

where they use a semi-parametric methodology for the decomposition of. The

latter follows a similar approach to ours where instead of taking the maximum ,

takes the median from the estimated s. In both cases the focus is to provide the

best possible fit for the data. Hence, a more optimisticestimation will be provided

affecting the robustness of the model. We believe that our method is more suitable

for the worst case framework that we seek.

Finally, we give the definition of the Gaussian copula (Cherubini et al., 2004;

Embrechts et al., 2001; Smillie, 2007).

Definition 2.11. Given a n-place standard multivariate normal distribution function n

parameterized by a dispersion matrix P [1, 1]nn, theGaussian copulais the function

CGa: In Isuch that

CGa(u)= (11 (u1),...,

1n (un)). (9)

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For CGato be called a Gaussian copula all the margins {i}ni=1

have to be normally

distributed but they can have different mean and variance.

3 Worst Case Conditional Value at Risk

Having introduced the theorems that enable us to associate copulas with distribu-

tions we will derive the copula formulation of the Worst Case Conditional Value at Risk

(WCVaR). Zhu and Fukushima (2009) have derived the WCVaR for distributions.

We follow a similar approach to derive WCVaR for copulas. At every step involv-

ing the use of distributions we present the equivalent copula formulation. For the

derivation of the copula formulation we use equations (1)-(3).

In order to define the WCVaR we first have to define Value at Risk(VaR) and

Conditional Value at Risk(CVaR). We first discuss VaR.

Proposition 3.1.Let w W Rm be a decision vector, u In a random vector, g(w, u) the

cost function andF(x)=(F1(x1),..., Fn(xn))a set of marginal distributions where u =F(x).

Also, let us assume that u follows a continuous distribution with copula density function

c(.). ThenVaRfor a confidence levelis defined as

VaR(w) min{ R: C(u| g(w, u) ) }. (10)

Proof. Given a decisionw Wand a random vectorx Rn which follows a contin-

uous distribution with density function f(.), the probability ofg(w, x) not exceeding

a thresholdis represented by

(w, )

g(w,x)

f(x)dx

=

g(w,x)

c(F(x))ni=1

fi(xi)dx

=

g(w,u)

c(u)du

= C(u| g(w, u) ),

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where fi(xi) = Fi(xi)

xiis the univariate probability distribution of the individual el-

ements of the random vector x (see Definition 2.6). g(w, u) = g(w, F1(u)) where

F1(u) =(F11

(u1),..., F1n (un)) maps the domain of the cost function from Rn to In, as

implied by the transformationui =Fi(xi). For the derivation of the copula version of

(w, ) we use equation (3). Having defined (w, ), we consider the VaR. Given a

fixedw Wand a confidence level, VaRis defined as

VaR(w) min{ R: (w, ) }

=min{ R: C(u| g(w, u) ) }.

We continue with the definition of the CVaR with respect to VaR.

Proposition 3.2. Givenw,u,F(x)and g(w, u)as in Proposition 3.1 we define CVaRfora confidence levelas

CVaR(w) 1

1

g(w,u)VaR(w)

g(w, u)c(u)du. (11)

Proof. We start from the equation of CVaR that arises from the probability density

function f(.) and we derive the copula form.

CVaR(w)= 11

g(w,x)VaR(w)

g(w, x)f(x)dx

= 1

1

g(w,x)VaR(w)

g(w, x)c(F(x))n

i=1

fi(xi)dx

= 1

1

g(w,u)VaR(w)

g(w, u)c(u)du.

(12)

Following Rockafellar and Uryasev (2000) we formulate equation (12) as the

following minimization problem

G(w, )= + 1

1

xRn

[g(w, x) ]+f(x)dx

= + 1

1

uIn

[ g(w, u) ]+c(u)du.(13)

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Hence, we have

CVaR(x)= minR

G(w, ). (14)

By solving the minimization problem in equation (14), we directly obtain both the

values of CVaR and VaR. From Proposition 3.1 we have that the value of VaR is the

value of.In order for the above definitions to be computed, exact knowledge of the distri-

bution f(x) or copula densityc(u) and the marginsF(x) is needed. As the aim in this

paper is to represent distributions with copulas, we shall omit using f(x) and use

c(u) instead. The equivalence of the two has been discussed in Section 2. Knowledge

of the copula C(u) and its margins {ui =Fi(xi)}ni=1

implies knowledge of f(x) andc(u).

A copula representation of the distribution ofx cannot be expected to be exact. Thus,

we assume that our copula representation belongs to a set of copulas c(.) C

. Thenotion of robustness with respect to Cinvolves the worst performing copula (or

copulas, since the worst case may not be unique), i.e. the copula for which we obtain

the greatest CVaR. Hence, we define WCVaR.

Definition 3.3. TheWCVaRfor fixedw Wwith respect to Cis defined as

WCVaR(w) supc(.)C

CVaR(w). (15)

It is known that CVaR is a coherent measure of risk (Artzner et al., 1998; Szego,2005; Zhu and Fukushima, 2009). For a measure of riskmapping a random vector

Xto be coherent it has to satisfy the following properties:

(i) Subadditivitty: for all random vectorsXandY,(X+ Y) (X) + (Y);

(ii) Positive homogeneity : for positive constant, (X)= (X);

(iii) Monotonicity: ifX Yfor each outcome, then(X) (Y);

(iv) Translation invariance: for constantm, (X+ m)= (X) + m.

Zhu and Fukushima (2009) prove that WCVaR preserves coherence. They also give

the following lemma from Fan (1953) which allows us to formulate the problem into

a tractable one.

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Lemma 3.4. Suppose that Wand X are nonempty convex sets in Rn and Rm, respectively,

and the function z(w, x)is convex inwfor anyx, and concave inxfor anyw. Then we have

minwW

maxxX

z(w, x)= maxxX

minwW

z(w, x) (16)

We also use Lemma 3.4 to extend the proof from Zhu and Fukushima (2009) to

copulas and eventually formulate our problem as aminmaxproblem.

3.1 Mixture Copula

In this example the distribution of the vector of returns x is described by a mixture

copula

C(F(x))= TC, (17)

where ={: eT= 1, 0, R l} andC= (C1(F(x)),..., Cl(F(x))) is the vector

with copulas and F(x) = (F1(x),..., Fn(x)) is the vector of the cumulative univariate

distributions. We can apply equation (3) to equation (17) to obtain the density of the

mixture copula. Then, we can use this density in Definitions 2.8-2.11 to obtain

G(w, ,)= + 1

1

uIn

[ g(w, u) ]+l

i=1

ici(u)du =l

i=1

iGi

(w, ), (18)

where

Gi(w, )= + 1

1

uIn

[ g(w, u) ]+ci(u)du for i= 1, 2,..., l. (19)

The optimization problem that we need to solve is stated by the following theorem

and corollary from Zhu and Fukushima (2009):

Theorem 3.5. For eachw, WCVaR(w)with respect toC

is given by

WCVaR(w)= minR

max

G(w, , ), (20)

where ={: eT= 1, 0, Rl}.

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Corollary 3.6. Minimizing WCVaR(w)over Wcan be achieved by the following mini-

mization

minwW

WCVaR(w)= minwW

minR

max

G(w, , ). (21)

More specifically, if(w, , )attains the right hand side minimum, then w attains the

left hand side minimum.Zhuand Fukushima (2009) provide the proof for the case of mixture distributions.

The theorems in Section 2, together with Proposition 3.1 and Proposition 3.2, show

that Theorem 3.5 and Corollary 3.6 can be applied to copulas. For the sake of

completeness we give the motivation behind the proof and we continue with the

formulation of the optimization problem.

In order to optimize the portfolio we need to solve

minwW

WCVaR(w) minwW

max

minR

G(w, , ) (22)

Since the mixture copula (17) is linear in , Zhu and Fukushima (2009) use Lemma

3.4 to show that (22) can be written as

minwW

minR

max

G(w, , ). (23)

An epigraph formulation can be used to reduce problem (23) into a minimization

problem as follows

min(w,,)WRR

:l

i=1

iGi

(w, ) , (24)

andmust satisfy

Gi(w, ) , for i= 1, 2,..., l. (25)

Problem (24) can thus be reduced to

min(w,,)WRR

: Gi(w, ) , i= 1, 2,..., l

. (26)

A straightforward approach for evaluating problem (26) is by Monte Carlo sim-ulation. Rockafellar and Uryasev (2000) give an approximation ofG(w, ), where

Monte Carlo simulation can be used. They write G(w, ) as

G(w, )= + 1

S(1 )

Sk=1

[ g(w, u[k]) ]+, (27)

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where u[k] is the kth sample vector (again here we give the copula version where

u[k] =F(x[k])). Thus, using equation (27) we can express problem (26) for evaluation

using Monte Carlo simulations

min(w,,)WRR

: +

1

Si(1 )

Si

k=1

[ g(w, ui[k]) ]+ , i= 1, 2,..., l.

, (28)

whereui[k]is thekth sample arising from copula Ci of the mixture copula (equation

(17)).Si is the size of the sample that arises from Ci.

Following Zhu and Fukushima (2009) we write the minimization problem as

min (29)

s.t. w W, v Rm, R, R (30)

+ 1

Si(1 )(1i)Tvi , i= 1,..., l (31)

vik g(w, ui[k]) , k= 1,..., S

i, i= 1,..., l (32)

vik0, k= 1,..., Si, i= 1,..., l, (33)

wherev = (v1; ...; vl) Rm withm =l

i=1Si and1i =(1; ...; 1) RS

i.

4 Portfolio management under a worst case copula

scenario

In this section we demonstrate how the theory in Section 3 can be used for the

optimization of a portfolio of financial assets. Financial assets can be described by

distributions and their risk can be measured using CVaR.

We consider a portfolio ofnfinancial assetsA1,...,An. We assume that the returns

of the assets are log-normally distributed and they are consistent with the Black and

Scholes (1973) representation, given by

xi =dAi(t)

Ai(t) =idt + idBi(t), (34)

wherei andi is the mean and the standard deviation of the random variable xi

anddBi(t) denotes a Wiener process. We have the return vectorx =(x1,..., xn) Rn

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and u = (u1,..., un) = (1(x1),..., n(xn)). We also define the decision vector w =

(w1,..., wn) Rn, which denotes the amount of investment in each financial asset in

the portfolio. We also define the loss function

g(w, u)= wT1(u), (35)

where1(u) = (11 (u1),..., 1n (un)). Hence, the loss function is the negative of the

portfolio returnwT1(u).

We have selected above the univariate distribution that describes the asset re-

turns. We now consider the selection of the copula that describes the dependency

between these returns. We first solve a simple optimization problem using the

Gaussian copula. Consider the problem

minwW

CVaR(w), (36)

where Wdefines the domain ofwas described by its constrains andu CGa (see

equation (12)). This is equivalent to equations (29)-(33) when l = 1.

The advantage of problem (36) is that it employs the Gaussian copula. The latter

is the most commonly used copula in practice for characterizing multivariate de-

pendencies. It is also easy to use. Furthermore, the Gaussian copula is a desirable

reference point for assessing portfolio performance. There are, however drawbacks

to the Gaussian copula. The first is its symmetry. Studies show that assets have

stronger negative comovements than positive (Ang and Chen, 2002; Hu, 2002, 2006).

A second disadvantage is the linear correlation that can only capture linear depen-

dencies between assets, which may not be realistic (Artzner et al., 1998; Szego, 2005).

Both of these disadvantages may lead to bad performance in the presence of market

shocks.

We aim to compensate for some of these disadvantages by using a mixture copula.

In problem (37), the mixture set Ccontains the copulas from Section 2.1. The aim is

to cover all types of dependencies and thereby use a robust measure of dependence

(Kendalls(4)). This robustness is further augmented by the worst-case approach.

Thus, the second problem we solve is

minwW

WCVaR(w), (37)

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where Wis defined as above andu cand c Cis a set of copulas (see equation

(15))

For further comparison we introduce a third and final model. The third model is

an extension of Markowitz (1952). Markowitz (1952) model is given by

minR,wW{: wT

w }, (38)

where , the covariance matrix, is symmetric positive semidefinite. Problem (38)

constitutes a quadratic convex problem. The extended model takes into consid-

eration the uncertainty that governs . The solution is provided within a worst

case framework where is unknown, but is assumed to belong in a set S+, which

comprises symmetric positive semidefinite matrices (Kontoghiorghes et al., 2002;

Rustem et al., 2000). Worst Case Markowitz (WCM) has the form

minR,wW

supS+

{: wTw }. (39)

Given a discrete set S+problem (39) can be reformulated into a convex optimization

problem

minR,wW

{: wTiw , i S+}. (40)

WCM provides us with nice comparison model since it is an extended, robust (sim-

ilarly to (37)) version of the well-known classic Markowitz model, and uses to

describe the dependence structure, in a similar fashion to problem (36).

Problems (36) and (37) are solved using equations (29)-(33) and WCM using

(40). Also, for all three problems we assumeWto be convex and, without loss of

generality (and for simplicity), we define it by the following constraints:

eTw= 1, (41)

where e is the vector with all-unity elements. Furthermore, to assure portfolio

diversification, the additional constrains

w w w (42)

can be imposed, for example in the case of a portfolio comprising only long positions.

At this instance we may require w, w [0.1, 1]n where w, w are lower and upper

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bounds respectively. This way we prevent the portfolio from having a single asset

position. Of course, such constraints are not compulsory and vary according to the

investors preference.

Finally, since we optimize an asset portfolio we are interested in its performance.

Hence, it is often desirable to impose an additional performance restriction in terms

of the minimum expected returnr

E(wTF1(u)) r. (43)

4.1 Numerical examples

We use the following seven indices: Nikkei225, FTSE100, Nasdaq, DAX30, Sensex,

Bovespa, Gold index. These represent six different stock exchange markets from

different parts of the world and one commodity index. The markets corresponding

to the indices are Japan, UK, USA, Germany, India and Brazil. These markets, with

the inclusion of the commodity, are intended to lead to a diversified portfolio.

The data used covers the period November 1998 - July 2011, during which we

look at the daily returns. This time line includes the dot-com bubble, South American

crisis and Asian crisis. These three events took place between 1998 and 2002, and

they had a large negative impact on the world markets. The data also include the

2008 Global Recession crisis. Both periods of crises can be observed in Figure 1.

In both numerical examples the data between 1998 and 2003 (1200 time steps) is

used for the calibration of the three problems, (36),(37) and (40). This way we include

data from a period of crisis. We expect to show that the risk and the dependencies

between assets during a crisis period can be assessed more efficiently with the use of

WCVaR as opposed to using only Gaussian copula CVaR. The Worst Case Portfolio

(WCP), given by (37) should perform more robustly than the Gaussian Portfolio

(GP), given by (36). Overall, and particularly during a period of crisis, we expect

that the WCP will perform better in the presence of downside shocks. Also, it will be

interesting to see how Worst Case Markowitz Portfolio (WCMP), given by (40), fares

against the other two portfolios, since it combines some of their individual trades.

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Figure 1. Seven indices from 1998 to 2011. All the indices are normalized to 1 on

July 2003

Jan2000 Jan2002 Jan2004 Jan2006 Jan2008 Jan20100

1

2

3

4

5

6

NIKKEI225

FTSE100

NASDAQ

DAX30

SENSEX

BOVESPA

GOLD

Thus, the critical test is the performance of all three portfolios during the 2008 crisis.

4.1.1 Static portfolio

In our first example, we consider a static portfolio in which the weights of the

portfolio are calculated only once. A daily rebalancing is performed using the same

weights throughout the entire lifespan of the portfolio. For the computation of the

weights we calibrate our copulas using the period between 1998 to 2003 and then

solve problems (36) and (37). To create the setS+for WCM we calculate six . In more

detail, 1is calculated from the period between 1998 to 2003.2is the diagonal of1,

with the off-diagonal elements being all equal to zero to assume independence. 3

is calculated through the relation=sin(0.5) (Smillie, 2007) (which holds under

normality assumption but in our case is different to 1). 4 is calculated from a

smaller sample concentrated around the peak of 2000 crisis for more extreme effects.

Finally, 5 and6 are calculated from different sample periods prior to 2003. The

means, variances, and the covariance matrix (1) of the seven assets as estimated

from the period between 1998 to 2003 are given in Table 1. The former two are

used for the univariate distributions of the assets as defined by equation (34). The

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univariate distributions together with the four copulas of Section 2 are used to run

Monte Carlo simulations in order to provide the inputs ui[k]

needed to solve equations

(29)-(33). Problem (40) does not require any simulations because it is not a stochastic

problem.

Table 1. The mean and variance of the seven assets daily returns between November

1998 and June 2003

(103) Mean Variance Covariance

Nikkey225 -0.41 0.21

FTSE100 -0.32 0.12 0.04

Nasdaq -0.22 0.45 0.04 0.12

DAX30 -0.38 0.23 0.04 0.19 0.24

Sensex 0.18 0.31 0.05 0.03 0.02 0.03

Bovespa 0.36 1.14 0.02 0.09 0.27 0.14 0.02

Gold 0.14 0.06 0.01 -0.02 -0.03 -0.03 -0.00 -0.02

Nikkey225 FTSE100 Nasdaq DAX30 Sensex Bovespa

Simulating data from Gaussian copula is straightforward with the use of the

Cholesky decomposition of the correlation matrix. Simulating data from other cop-

ulas can be a difficult task, in general, which makes them less attractive. The

simulation of data from the three Archimedian copulas uses the algorithms found in

Melchiori (2006). Melchiori (2006) provides a summary of the results from Devroye

(1986); Marshall and Olkin (1988); Nolan (2012).

To solve problems (36),(37) and (40), we also have to define the constraints W.

We use as a starting point equations (41)-(43). We specifyw 0. This implies that

we only allow long positions in the portfolio. The upper bound w in equation (42)

is implied by the budget constraint (41).

The optimization problems are solved using the Yalmip Matlab package, together

with the CPLEX solver. The PC used for the implementation of the numerical

examples is an Intel Core 2 Duo, 2.8GHz with 4 GB memory. For each of the four

copulas, we run 10000 simulations for each of the seven assets. Simulating and

solving all three problems takes less than a minute.

All three problems are solved for r = 0 to 0.00025, where r is the required

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Figure 2. Static portfolios: Out of Sample portfolio performance forr = 0 (see

equation (43))

Jan2004 Jan2005 Jan2006 Jan2007 Jan2008 Jan2009 Jan2010 Jan2011

1

1.5

2

2.5

3

3.5

Normal distribution portfolio

Worst case portfolio

Worst case Markowitz portfolio

Jan2004 Jan2005 Jan2006 Jan2007 Jan2008 Jan2009 Jan2010 Jan20110.3

00.30.6

WCP GP, level difference

minimum daily return (see (43)). The results are presented in Figure 2, Table 2 and

Table 3. Table 2 illustrates the performance of the three portfolios (GP, WCP and

WCMP) both for the In Sample(November 1998 - June 2003) and Out of Sample

(June 2003 - July 2011). In Sample performance shows that the lower boundr

is always satisfied at least by one of the three portfolios in the form of AR. Also,the overall (in and out of sample) performance of the WCP portfolio always has

higher volatility (Vol) and CVaR. This is to be expected since the WCP has to satisfy

more constraints in the optimization problem, i.e., the same constraints that exist for

the Gaussian copula in the GP have to be satisfied for all four copulas used in the

WCP. Hence, the CVaR obtained for the WCP is the CVaR of the worst case copula,

which is the equivalent the requirement imposed by inequality (25). These copulas

also have fat tails, and hence higher Vol. On the other hand WCMP, as a variance

minimization problem, has the smallest Vol and CVaR out of the three, at least for

r = 0 to 0.00015. This advantage stops when higher returns are required.

We expect WCP and WCMP to perform better than the GP, at least under the worst

case scenarios. This should apply throughout the Out of Sample testing period.

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Table 2. Comparison of the performance of Gaussian optimal, Worst Case optimal

and Worst Case Markowitz optimal portfolios

GP (I) In Sample Out of Sample

r WCP (II) AR1 Vol2 CVaR0.95 AR1 Vol2 CVaR0.95 MD13 MD24 TR5

(103) WCMP (III) (103) (103) (103) (103) (103) (103) (%) (%) (%)

0.00I 0.00 6.66 14.4 0.56 9.00 22.1 -7.44 -31.8 198II 0.13 9.29 19.2 0.70 12.2 29.1 -7.23 -28.9 277

III 0.01 6.50 14.2 0.55 8.89 21.9 -7.67 -35.9 191

0.10

I 0.10 7.05 15.1 0.66 9.84 23.9 -7.11 -31.4 263

II 0.13 9.29 19.2 0.70 12.2 29.1 -7.23 -28.9 277

III 0.10 6.79 14.7 0.64 9.46 23.1 -7.14 -35.9 254

0.15

I 0.15 7.56 16.2 0.71 10.4 25.3 -7.02 -31.6 300

II 0.15 9.04 18.7 0.71 12.0 28.8 -7.07 -29.6 286

III 0.15 7.18 15.6 0.70 10.0 24.2 -6.95 -35.1 294

0.20

I 0.20 7.95 16.9 0.73 10.2 24.9 -6.59 -39.7 319

II 0.20 8.80 18.1 0.72 11.1 26.8 -6.89 -35.4 300

III 0.20 7.99 17.2 0.73 10.2 24.9 -6.52 -40.6 322

0.25

I 0.25 11.3 24.0 0.74 11.8 28.5 -7.68 -49.8 313

II 0.25 11.8 24.2 0.72 12.0 28.4 -8.07 -44.7 296

III 0.25 11.8 25.9 0.75 12.4 30.0 -7.48 -52.3 314

1. AR : Average daily return over the period

2. Vol : The volatility defined by the standard deviation

3. MD1 : Maximum draw-down, The worst return between two consecutive days

4. MD2 : Maximum draw-down, The worst return between two days within a periodof maximum 6 months

5. TR : The total return from the beginning to the end of the period

Hence, we focus on the Out of Sample period. Table 3 shows that the performance

of WCP up to January 2008 (the beginning of the crisis) was similar or worse than

GP with WCMP consistently having the highest. The better performance of WCMP

may suggest that assuming some uncertainty in the estimation of the covariance

or correlation matrix may improve your judgment of risk, at least during periods

of prosperity. Whiler increases, the difference between the returns of the WCP

and the other two portfolios increases. This can also be verified in Figure 2 for the

case ofr = 0. This is at most evident at the bottom Figure 2 where the difference

between the notional value of the WCP and GP is displayed. The difference up to

January 2008 is very close to zero, after which it starts to increase with the WCP

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outperforming the GP.

Table 3. Average daily return of Gaussian optimal, Worst Case optimal and Worst

Case Markowitz optimal portfolios up to January 2008

r

(103)

0.00 0.10 0.15 0.20 0.25

GP 0.71 0.82 0.88 1.02 1.20

WCP 0.73 0.73 0.76 0.90 1.03

WCM 0.76 0.87 0.92 1.05 1.28

This behaviour changes from 2008 onwards. In Table 2 we see that the perfor-

mance of GP with respect to Average Return (AR) and Total Return (TR) is worse

or similar to WCP for allrtested. The reason is the robustness of the performance

of the WCP during the 2008 crisis. It is surprising that the WCMP does not exhibit

similar robustness. On the contrary its behaviour is similar to the GP. Assessing therisk between the assets through the use of the covariance matrix, similarly to the

correlation of problem (36), does not carry any information about the characteristics

of the tail of the distribution of each asset. Such a model proves inadequate to in

emulating risk during periods of extreme movements. This is evident from the Max-

imum Drawdown (MD2); this measures the greatest loss within a 6 month period.

For the given dataset this occurs at the time of the crisis. We can see that the WCP

MD2 is better for allr. Another observation from Table 2 is that the higher the re-quirement for r is, the more similar the performance of the three portfolios becomes

with respect to AR and TR. This is the result of our demand for higher returns, which

forces all three portfolios to select high return assets. Thus, the number of assets

that can be selected becomes smaller and hence the portfolios become similar. This

becomes more apparent in the case ofr =0.00025, in which case in addition to the

AR and TR, Vol and CVaR are much closer.

With the above in mind, we can conclude that an optimistic portfolio, like theGP, is more suitable during times of prosperity. Also, the addition of uncertainty

with respect to the values of the covariance matrix, like the WCMP, may improve

our assessment of risk during prosperity even further. In contrast, a more robust

portfolio like WCP proves to be beneficial during a crisis period.

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4.1.2 Dynamic portfolio

We consider the case when the optimal weights of the assets are recomputed. In

this more dynamic portfolio, the weights of the assets are recalibrated on a monthly

basis. At every step, we extend the in sample calibration window by a month, then

we solve problems (36) and (37) to obtain the weights. For problem (40), to obtainthe weights, we use the set S+, as described in the previous section, to which we add

7. 7 is evaluated from the expanding sample. This way we make sure that new

information are taken into account. Then, for all three portfolios, we keep the same

weights for the rest of that month. During the month, daily rebalancing is performed

using the constant weights. We do not adopt a moving window of calibration since

we do not want to lose the information from the old crises, in our case the crises

between 1998 and 2003.The expanding calibration window causes the estimated values ofi, used in

equation (34), to change every month. Thus, we have to make sure that the constraint

(43) remains within the feasible set ofw. To do so we replace constraint (43) with

the dynamic constraint

E(wTF1(u)) max[wEqT, 0], (44)

where =(1,...,n) is the vector of the asset return means as calculated using thecalibration period andwEq = (1/n,..., 1/n) i.e. all the weights are equal. The rest of

the constraints remain as in Section 4.1.1.

For comparison purposes, we also include a simple portfolio not based on opti-

mization. The Equally weighted portfolio (EWP) has equal positions in all assets

i.e. we always usewEqas the weights of the portfolio.

The out of sample performance of all four portfolios (GP, WCP, WCMP and

EWP) is shown in Figure 3 and Table 4. The observations are very similar to thoseof Section 4.1.1. The CVaR and Vol of the WCP are higher than the GP but the MD2,

AR and TR of the WCP is significantly higher. WCMP exhibits similar behaviour to

the GP. Also, in the case of the EWP, the Vol and the CVaR lie between those of the

other two portfolios. At the same time its performance is worse with respect to the

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AR, TR and MD2, i.e., the EWP sustains the largest losses during the 2008 Global

Recession crisis.

Figure 3. Dynamic portfolio performance with = max[0, wEqTx] (see equation

(44))


1.5

2

2.5

3

3.5

Normal distribution Portfolio

Worst case portfolio

Worst Case Markowitz

Equal weights portfolio


00.30.6

WCP GP , level difference

5 Summary and Conclusions

In this paper we demonstrated one way of using copulas in a portfolio optimization

framework where the worst-case copula is considered. In particular, we focus on the

derivation of CVaR and WCVaR for copulas. In the case of WCVaR we show how a

mixture copula can be used in order to obtain a convex optimization problem.

By introducing copulas in the CVaR framework we allow more flexibility in the

selection of the distribution. The most commonly used distribution for modelling

multivariate dependencies is the Gaussian copula. This is due to the simplicity of

its construction and the availability of efficient methods of simulation. Its disadvan-

tages are its symmetry and its ability to only describe linear dependencies via the

use of linear correlation in its structure. However symmetric behaviour and linear

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Table 4. Comparison of theperformance of Gaussian optimal, theWorst Case optimal

and Equally Weighted portfolios

Out of Sample

AR1 Vol2 CVaR0.95 MD13 MD24 TR5

(103) (103) (103) (%) (%) (%)

GP 0.53 8.76 21.5 -7.49 -33.0 174WCP 0.71 12.3 29.2 -7.20 -28.8 274

WCMP 0.51 8.77 21.7 -7.60 -36.2 168

EWP 0.45 9.73 23.9 -6.52 -45.4 130

1. AR : Average daily return over the period

2. Vol : The volatility defined by the standard deviation

3. MD1 : Maximum draw-down, The worst return between two consecutive days

4. MD2 : Maximum draw-down, The worst return between two days within aperiod of maximum 6 months

5. TR : The total return from the beginning to the end of the period

dependencies among assets are unrealistic (Ang and Chen, 2002; Artzner et al., 1998;

Hu, 2002, 2006). We discuss alternative distribution functions in the form of copulas

that can exhibit asymmetric behaviour and utilize monotonic measures of depen-

dence in their formulation. These are the three Archimedian copulas in Section 2.1

that are also easy to simulate from using the algorithms given by Melchiori (2006).

The advantage of using non symmetric distribution functions was demonstrated

in the numerical examples of Section 4.1. In Section 4.1 we provide a comparison

between three competitive models, WCVaR, Gaussian copula CVaR and WCM. We

show that for both staticanddynamic strategies, forlowminimum expected portfolio

return, the WCP outperforms both the GP and the WCMP in every statistic except

Vol and CVaR. In particular during the 2008 crisis the WCP performed more robustly

than the other two portfolios, and that was true even for a high minimum expected

return requirement. This shows that the assumption of symmetry and the use of the

correlation or covariance matrix as a measure of dependence provide insufficient

information for assessing the risk during periods of crisis.

We compare the performance of dynamic portfolios with that of the EWP. The

EWP performance shows that following a naive approach in which risk is neglected

is not necessarily the correct way forward. Although the EWP performed relatively

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adequately soon after 2002, it suffered the biggest loss during the 2008 crisis. As a

result, the EWP became the worst performing portfolio among all portfolios in the

numerical examples.

It seems reasonable to conclude that when optimizing a portfolio, the associated

risk needs to be taken into account. All possible dependencies have to be considered

in order to obtain robust results. One way of achieving this is through copulas and

mixture copulas, that allow dependency systems with higher flexibility in their de-

scription than a single distribution. This flexibility allows non symmetric behaviour

and fat tails in the design of the system.

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