Efficient and robust portfolio optimization in the

Quantitative Finance,Vol.??,No.?,Month??2009,pp.1–14

Efficient and robust portfolio optimization in the multivariate Generalized Hyperbolic framework 5MARTIN HELLMICH yz and STEFAN KASSBERGER*x

y Frankfurt School of Finance and Management,Sonnemannstraße9–11,60314Frankfurt,Germany

z DekaBank Deutsche Girozentrale,Mainzer Landstraße16,60325Frankfurt am Main,Germany

x Institute of Mathematical Finance,Ulm University,Helmholtzstraße18,89069Ulm,Germany

(Received14October2007;in final form18May2009)

10 1.Introduction

Modern portfolio optimization is a far cry from the

classical mean–variance approach pioneered by

Markowitz(1952).The departure from the time-honored

Markowitz framework has been spurred by two inti-15mately related insights.First,the use of a Gaussian distribution to describe the returns of financial assets will

inevitably lead to what can at best be called a rough

approximation to reality.Second,variance can be an

inadequate risk measure if a more flexible,non-Gaussian 20return distribution is adopted.

It has become a generally accepted fact,supported by

numerous empirical studies,that empirical asset return

distributions are non-normal.In fact,they are almost

always found to exhibit asymmetry)and 25excess kurtosis,which renders the normal(Gaussian) distribution an inadequate model(Prause1999,Raible

2000,Schoutens2003,Cont and Tankov2004).Thus,

realistic modelling calls for alternative probability dis-

tributions.In recent years,several viable alternatives to 30the Gaussian distribution,capable of capturing com-monly observed empirical features,have been proposed

for use in financial modelling.For example,Madan

and Seneta(1990)suggest the Variance Gamma distribu-

tion,Eberlein and Keller(1995)and Bingham and 35Kiesel(2001)advocate the use of the Hyperbolic distri-bution,Barndorff-Nielsen(1997)proposes the Normal

Inverse Gaussian distribution,Eberlein(2001)applies the

Generalized Hyperbolic distribution,and Aas and Hobæk

Haff(2006)find that the Generalized Hyperbolic Skew 40Student’s t distribution matches empirical data very well.

While these studies document the superior capabilities

of the Generalized Hyperbolic class and its subclasses

when it comes to realistically describing univariate

financial data,recent empirical studies conducted in a

45 multivariate setting make a convincing case for the

multivariate Generalized Hyperbolic(mGH)distribution

and its subclasses as a model for multivariate financial

data as well.For instance,McNeil et al.(2005)calibrate

the mGH model and its subclasses to both multivariate

50 stock-and multivariate exchange-rate returns.In a likelihood-ratio test against the general mGH model,

the Gaussian model is always rejected.Aas et al.(2005)

and Kassberger and Kiesel(2006)employ the multivariate

NIG(Normal Inverse Gaussian)distribution successfully

55 for risk management purposes.The latter study demon-strates that the NIG distribution provides a much better

fit to the empirical distribution of hedge fund returns than

the normal distribution.The Gaussian distribution is

found to seriously understate the probability of tail

60 events,while the heavier tails of the mGH class seem

to describe actual tail behavior well.Tail-related risk

measures such as Value at Risk(VaR)and Conditional

Value at Risk(CVaR)are shown to be severely mislead-

ing when calculated on the basis of the Gaussian

65 distribution.This problem is found to carry over into

the portfolio context.

All the aforementioned distributions have two impor-

tant features in common.First,they can all be considered

as marginal distributions of(multivariate)Le vy processes

70 (i.e.processes with independent and stationary,but not necessarily Gaussian increments).Second,they all belong

to the class of(multivariate)Generalized Hyperbolic

distributions.Just as the class of Le vy processes

encompasses(multivariate)Brownian motion as a special

75 case,the class of mGH distributions encompasses the Gaussian distribution as a limiting case.Therefore,the

mGH class offers a natural generalization of the multi-

variate Gaussian class.

The departure from the normal distribution and the

80 adoption of more realistic distributions,however,call

for adequate risk measures and computational tools.

Portfolio optimization using non-Gaussian distributions

*Corresponding author.Email:stefan.kassberger@uni-ulm.de

Quantitative Finance

ISSN1469–7688print/ISSN1469–7696onlineß2009Taylor&Francis

www.informaworld

DOI:10.1080/14697680903280483

should not be performed in a mean–variance framework,

because in the non-Gaussian case it is inappropriate to 85describe the riskiness of a financial asset solely by the variance of its returns(thereby ignoring higher moments).

In recent years,CVaR,also known as Expected Shortfall

or Tail VaR,has been embraced by academics and

practitioners alike as a tractable and theoretically 90well-founded alternative to classical risk measures such as VaR or variance.

In addition to being based on realistic distributional

assumptions and an informative risk measure,an alter-

native portfolio optimization approach should be com-95putationally feasible—even for problems involving a large number of assets—in order to be applicable to real-world

situations.Moreover,it should be amenable to a robust

formulation of the portfolio optimization problem.

Robust formulations are based on the insight that optimal 100portfolios can be remarkably sensitive to only slight variations in the input parameters,which are often

fraught with estimation error.The combined effect of

the uncertainties in the parameters can render the result

of a portfolio optimization procedure highly unreliable. 105To counteract this phenomenon,robust approaches rely on uncertainty sets that contain the‘true’parameters for

a specific confidence level,instead of point estimates

of the parameters,thereby taking parameter uncertainty

into account.For a survey of robust optimization,the 110interested reader is referred to Bertsimas et al.(2008).

A modern portfolio optimization approach is thus

characterized by the following desirable features:allow-

ance for realistic return distributions,use of a realistic risk

measure,computational tractability,and admissibility 115of a tractable robust formulation.We contribut

e to the literature by proposing a portfolio optimization approach

that incorporates all of these features.Our approach is

based on the mGH distribution,relies on CVaR,leads to

a convex optimization problem,and allows for a robust 120formulation that can be solved just as efficiently as the original problem.

The remainder of this paper is structured as follows.

Section2introduces CVaR as an alternative risk measure

and gives an overview of several standard forms of the 125portfolio optimization problem.In section3,the multi-variate Generalized Hyperbolic class of distributions is

introduced.In addition,results relating to the determi-

nation of the CVaR for mGH portfolios are established,

and a decomposition formula for the CVaR of a portfolio 130is presented and proved.These results,

while interesting in their own right for risk management purposes,form

the foundation for an efficient formulation of the port-

folio optimization problem in the mGH framework,

which is the subject of section4.Furthermore,section4 135introduces a robust formulation of the portfolio optimi-zation problem,which relies on Worst Case Conditional

Value at Risk(WCVaR)as a risk measure.It is shown

that the robust portfolio optimization problem can be

solved as efficiently as the original problem.Section5is 140devoted to a numerical study in which the methodologies developed in the paper are applied to empirical data.

Section6sums up the main insights and concludes.2.Risk measures,performance measures,and portfolio

optimization

145 2.1.Coherent measures of risk

Since a non-Gaussian distribution cannot be character-

ized solely in terms of its means and its covariance matrix,

a deviation from the multivariate Gaussian paradigm

of portfolio optimization has to be supported by the

150 adoption of alternative risk measures,such as Value at

Risk or CVaR.In their seminal paper,Artzner et al.

(1999)specify a number of desirable properties a risk

measure should have and introduce the notion of a

coherent risk measure(see also Malevergne and Sornette

155 (2006)).In the following definition,L1and L2can be

interpreted as random losses.A risk measure that

maps a random loss to a real number is said to be coherent

if it satisfies the following axioms.

.(A1)Translation invariance: (Lþl)¼

160 (L)þl,for all random losses L and all l2R.

.(A2)Subadditivity: (L1þL2) (L1)þ (L2),

for all random losses L1,L2.

.(A3)Positive homogeneity: ( L)¼ (L),for

all random losses L and all 40.

165 .(A4)Monotonicity: (L1) (L2),for all

中国股市记忆

random losses L1,L2with L1L2almost

surely.

It is worth noting that subadditivity and positive homo-

geneity imply convexity,whereas the converse generally

170 does not hold.

Value at Risk(VaR)has become an industry standard

for measuring financial risks.VaR has derived much of its

popularity from the fact that it gives a handy and

easy-to-understand representation of potential losses.

175 If X is the random return associated with an asset,then

L¼ÀX is the relative loss,and the VaR at level 2(0,1),

denoted by VaR (L),is defined as VaR (L)X inf{l2R:

P(L4l)1À }¼inf{l2R:F

(l)! }.Hence,VaR (L)

is the smallest relative loss level whose probability of

180 being exceeded is at most1À .For continuous,strictly increasing loss distribution functions(which we will

assume throughout the paper),VaR can be more simply

expressed as the -quantile of the loss distribution

function F L:

VaR ðLÞ¼FÀ1

ð Þ:

185 Of course,VaR can also be defined in terms of absolute losses.However,as we are going to model returns rather

than prices,the above definition is more appropriate for

our purposes.

190 VaR suffers from several shortcomings that become particularly evident when it is to be used as a risk measure

in the portfolio context.As Artzner et al.(1999)point out,

VaR can lack subadditivity when applied to non-elliptical

distributions,which amounts to ignoring the benefits

195 of portfolio diversification.Moreover,VaR is generally

a non-convex function of portfolio weights.As non-

convexity normally leads to multiple local extrema,

2Feature

it renders portfolio optimization a computationally expen-sive problem.

200

Recently,there has been increasing interest in CVaR as a closely related alternative to the VaR approach.CVaR does not suffer from any of the above-mentioned short-comings;in particular,it is a coherent risk measure (,Acerbi and Tasche (2002))and thus has several 205

desirable properties that VaR lacks,such as subadditivity and convexity.The CVaR at level 2(0,1)is defined as the expectation of the relative loss conditional on the relative loss being at least VaR (L ):

CVaR ðL Þ¼4

E ½L j L !VaR ðL Þ :

210

A straightforward consequence of this definition is the relation CVaR (L )!VaR (L ).CVaR is more inf

ormative than VaR,as CVaR (L )takes the loss distribution beyond the point VaR (L )into account and thus also measures the severity of losses that exceed VaR (L ).VaR,215

in contrast,ignores losses beyond VaR (L )and thus discards information implicit in the loss distribution.CVaR is well suited as a risk measure in the context of portfolio optimization,for reasons that will be elaborated on in what follows.

220

2.2.Portfolio optimization using CVaR

Portfolio optimization problems appear in various guises.The following result,which is proved by Krokhmal et al.(2002),establishes a link among three of the most common formulations.Let :X }R be a convex risk 225

measure,and let R :X }R be a concave reward function,

both defined on the convex set X &R d .Let x 2X be a vector of portfolio assume that P

d i ¼1x i ¼1.Then th

e following three optimization problems lead to the same efficient frontiers when varying the parameters 230

, ,and !,respectively:

min

x 2X

ðx ÞÀ R ðx Þ,

subject to !0,

ðP1Þmin x 2X

ðx Þ,

subject to R ðx Þ! ,

ðP2Þmax x 2X

R ðx Þ,

subject to ðx Þ !:

ðP3Þ

In other words,a portfolio that is efficient for one of these three problem formulations will also be efficient for the other two.In our subsequent considerations,we will 235

identify R (x )with the expected portfolio return,which is a linear (and thus concave)function of portfolio weights,and (x )with the portfolio CVaR,which is convex in the portfolio weights.

While the above formulations involving the minimiza-240

tion of a linear functional of risk and reward are very common in the literature,other formulations of the portfolio optimization problem entail the maximization of a reward–risk ratio.For instance,the use of Return-on-Risk-Capital (RORC for short),defined as

245

R (x )/ (x ),is motivated by Fischer and Roehrl (2005).Rachev et al.(2007)provide an overview of various other reward–risk ratios.

3.Beyond Gaussian mean–variance optimization:Using the mGH distribution for portfolio modelling 250

3.1.Modelling multivariate returns with the mGH

distribution As already pointed out,there is compelling empirical evidence that returns or log-returns of financial assets are not Gaussian.As a consequence,a more realistic model 255

is called for.Because of its great generality and relatively high numerical tractability,the mGH distribution is an ideal candidate.

3.1.1.The mGH distribution as a normal mean–variance mixture.A random variable Y 2R þis said to have a 260

Generalized Inverse Gaussian (GIG)distribution with parameters , ,and ,denoted by W $GIG ( , , ),if its density is given by

f GIG ðy ; , , Þ

¼ À ð Þ

=22K ðﬃﬃﬃﬃﬃﬃﬃ p Þy À1exp À y À1

þ y 2 ,

y 40,0,

y 0,

ð1Þ

8<:where,for x 40,K (x )is the modified Bessel function of 265

the third kind with index :

K ðx Þ¼12Z

10y À1exp Àx ðy þy À1Þ

2 d y :

The parameters in (1)are assumed to satisfy 40and

这6张中国名片震撼世界

0if 50; 40and 40if ¼0;and !0and 40if 40.The expected value of Y can be expressed as

E ðY Þ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

= p K þ1ðﬃﬃﬃﬃﬃﬃﬃ p Þ

K ðﬃﬃﬃﬃﬃﬃﬃ p Þ:

ð2Þ270

The class of mGH distributions can now be introduced as the class of normal mean–variance mixtures with a GIG-distributed mixing variable.A random variable X ¼(X 1,...,X d )0is said to follow a d -dimensional mGH 275

distribution with parameters , , , , ,and Æ,denoted by X $GH d ( , , , , ,Æ),if

X ¼d

þW þﬃﬃﬃﬃﬃW p AZ ,where , 2R d are deterministic,Z $N k (0,I k )follows

a k -dimensional normal distribution,W $GIG ( , , )280

is a positive,scalar random variable independent of Z ,A 2R d Âk denotes a d Âk matrix,and Æ¼AA 0.

We find that X |W ¼w $N d ( þw ,w Æ),i.e.that the conditional distribution of X given W is normal,which explains the name normal mean–variance mixture.The 285

mixing variable W can be thought of as a stochastic volatility factor.From the above definition,it follows directly that E (X )¼ þE (W ) and Cov(X )¼E (W )ÆþVar(W ) 0.It is interesting to note that the absence

Feature

of correlation of the components of X implies indepen-290dence if and only if W is almost surely if X is multivariate normal.For ¼0,the class of normal

variance mixture distributions is obtained.These distribu-

tions fall into the class of elliptical distributions,which

will be formally introduced later.

295For non-singularÆ,it can be shown that the following

representation for the density f GH

d of a d-dimensional

GH d( , ,, , ,Æ)distributed random variable holds: f GH

南京 imax

ðy; , ,, , ,ÆÞ

¼cÁ

K Àd=2ð

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð þðyÀ Þ0ÆÀ1ðyÀ ÞÞðþ 0ÆÀ1 Þ

ÞexpððyÀ Þ0ÆÀ1 Þ

()ð

ð þðyÀ Þ0ÆÀ1ðyÀ ÞÞðþ 0ÆÀ1 Þ

Þd=2À

ð3Þ

with c a normalizing constant,

c¼ð ÞÀ =2 ðþ 0ÆÀ1 Þd=2À

ð2pÞd=2jÆj1=2K ð

ﬃﬃﬃﬃﬃﬃﬃ

,ð4Þ

300where jÁj denotes the determinant.Observe that,for every a40,the distributions GH d( , ,, , ,Æ)and

GH d( , /a,a, ,a ,aÆ)coincide,since,for all y2R,

f GH

d ðy; , ,, , ,ÆÞ¼f GH

ðy; , =a,a, ,a ,aÆÞ:

ð5Þ

305This non-uniqueness gives rise to an identifiability problem when trying to calibrate the parameters.

However,this problem can be addressed in several

ways,for example by requiring the determinant ofÆto

assume a pre-specified value,or by fixing the value of 310either or.

3.1.2.Subclasses of the mGH class.The mGH class

of distributions is very general and accommodates

many subclasses that have become popular in financial

modelling.The purpose of this section is to provide a brief 315survey of some of these subclasses.Compare also McNeil et al.(2005),who provide a discussion of the tail behavior

of these classes.

Hyperbolic distributions.For ¼1

2ðdþ1Þ,one arrives

at the d-dimensional Hyperbolic distribution.However, 320the univariate marginals of a d-dimensional Hyperbolic distribution with d!2are not univariate Hyperbolic

distributions.See Eberlein and Keller(1995)for an

三棵树苏童

application of univariate Hyperbolic distributions to

financial modelling.

325Normal Inverse Gaussian(NIG)distributions.For

¼À1

2,one obtains the class of NIG distributions,

which has become widely applied to financial data(see,

<,Aas et al.(2005)and Kassberger and Kiesel(2006)

for recent accounts).The tails of NIG distributions are 330slightly heavier than those of the Hyperbolic class.

Variance Gamma(VG)distributions.For 40and ¼0,

one obtains a limiting case which is known as the

Variance Gamma class.See Madan and Seneta(1990)for an application of univariate VG distributions to

335 equity return modelling.

Skew Student’s t distributions.For ¼À1

and¼0, another limiting case is obtained,which is often called the

Skew Student’s t distribution.The interesting aspect of

this distribution is that,in contrast to the aforementioned

340 ones,it is able to account for heavy-tailedness.See Aas

and Hobæk Haff(2006)for an application in a univariate setting.

Elliptically symmetric mGH(symGH)distributions.For

¼0,one obtains the subclass of elliptically symmetric

345 mGH distributions,which are henceforth called symmet-

ric mGH or symGH distributions.Compared with the general mGH distribution,the density of a symGH distribution simplifies considerably:

f symGH

ðy; , ,, ,ÆÞ

ð ÞÀ =2d=2

ð2pÞjÆj1=2K ð

ﬃﬃﬃﬃﬃﬃﬃ

K Àd=2ð

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð þðyÀ Þ0ÆÀ1ðyÀ ÞÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð þðyÀ Þ0ÆÀ1ðyÀ ÞÞ

Þd=2À

:ð6Þ

350 The symGH class belongs to the class of elliptical distributions.

3.2.Distributional properties,CVaR,and portfolio risk

decomposition of mGH portfolios

In this section,we take advantage of the analytical

355 tractability of the mGH class to state the distribution of a portfolio whose constituents follow an mGH distribution.

This result will be used to derive analytical expressions

for the portfolio’s CVaR and for the risk contribution

of a single asset to overall portfolio risk.

360 3.2.1.Distribution of portfolio returns and CVaR.From

the parametrization of the mGH class,it can easily be inferred that it is closed under linear transformations.激光整形

More precisely,let X$GH d( , ,, , ,Æ)and

Y¼BXþb,where B2R kÂd and b2R k.Then

Y¼BXþb¼B þbþWB þ

ﬃﬃﬃﬃﬃ

BAZ

$GH kð , ,,B þb,B ,BÆB0Þ:

365 Thus,linear transformations of mGH random vari-

ables leave the distribution of the GIG mixing variable unchanged.In particular,it follows that every component X i of X is governed by a univariate GH

370 distribution:X i$GH1( , ,, i, i,Æii).Furthermore,

for x¼(x1,...,x d)02R d,

x0X¼

X d

i¼1

x i X i$GH1ð , ,,x0 ,x0 ,x0ÆxÞ:

If,in addition,the x i are required to sum to hat

10x¼1,where1¼(1,...,1)02R d),and thus can be

375 interpreted as the weights of the individual assets in a

4Feature

portfolio,we can conclude that if the returns of the

constituents of a portfolio follow an mGH distribution,

then the return of the portfolio is univariate GH

distributed.

380Now,we derive the univariate GH density of portfolio returns,which will turn out to be considerably simpler

than its multivariate counterpart.Using(5),we can

represent a univariate GH density of the form f GH

1(y; , ,

, , ,Æ)as f GH

(y; , Æ,/Æ, , /Æ,1).This shows 385that,in the univariate case,without loss of generality, the dispersion parameterÆ2Rþcan be assumed to be1,

and the density thus simplifies(compare(3)and(4))

f GH

ðy; , ,, , Þ

¼ð ÞÀ =2 ðþ 2Þ1=2À ð2pÞ1=2K ð

ﬃﬃﬃﬃﬃﬃﬃ

ÁK À1=2ð

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð þðyÀ Þ2Þðþ 2Þ

ÞexpððyÀ Þ Þð

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð þðyÀ Þ2Þðþ 2Þ

Þ1=2À

Now assume that the returns of d assets X¼(X1,...,X d)0 390are distributed according to GH d( , ,, , ,Æ).Then the return x0X of a portfolio with asset weights x with

10x¼1follows a GH1( , ,,x0 ,x0 ,x0Æx)distribution.

Denoting by f GH

1(y)the density of the portfolio return x0X

evaluated at y,CVaR can be computed as follows: CVaR ðÀx0XÞ¼E½Àx0X jÀx0X!VaR ðÀx0XÞ

¼ÀE½x0X j x0X VaR1À ðx0XÞ

¼À

1À

Z GHÀ1

ð1À Þ

À1

yÁf GH

ðyÞd y:ð7Þ

395The quantile-function GHÀ11ðÁÞof the portfolio return distribution can be calculated using standard numerical

root-finding methods.Having the portfolio distribution

available in closed form is of great advantage,as it allows 400fast,exact,analytical computation of risk figures—or of moments of portfolio returns—without having to fall

back on typically time-consuming Monte Carlo simula-

tions.This key feature makes it possible to set up efficient

portfolio optimization algorithms,as will be demon-405strated later.

3.2.2.Decomposition of portfolio risk.When investigat-

ing the risk profile of a portfolio,not only are its

aggregated risk characteristics of interest,but so too are

the contributions of the individual constituents to its 410overall risk.This issue arises,for instance,when calcu-lating regulatory capital requirements induced by indi-

vidual positions.The CVaR framework provides a very

intuitive decomposition of overall risk into its individual

building blocks.Such a decomposition was presented by 415Panjer(2001)for multivariate normal distributions,and generalized to elliptical distributions by Landsman and

Valdez(2003).Here,we prove a decomposition formula

for the mGH distribution.

Let the portfolio returns be X$GH d( , ,, , ,Æ),

420 and let x2R d denote the asset weights(with10x¼1).

By additivity of conditional expectation,

CVaR ðÀx0XÞ¼E½Àx0X jÀx0X!VaR ðÀx0XÞ

X d

i¼1

E½Àx i X i jÀx0X!VaR ðÀx0XÞ ,

which can be interpreted as follows.The portfolio CVaR

is the sum of the risk contributions of the individual assets

425 in case a shortfall event in case the relative

portfolio loss exceeds VaR (Àx0X).It is important to

note,however,that in general the portfolio CVaR is

different from the sum of the CVaR values for the

individual assets.The following proposition shows how to

430 compute the individual CVaR contribution of a position

in a specific asset.

Proposition3.1:Let X$GH d( , ,, , ,Æ),and let

x2R d with10x¼1.Then,the CVaR contribution of the

position in asset i is

E½Àx i X i jÀx0X!VaR ðÀx0XÞ

¼À

1À

À1

Z GHÀ1

废墟上的鲜花1

ð1À Þ

À1

y1Áf GH

ðy1,y2Þd y2d y1,

435 where GHÀ1

is the quantile function of a GH1( , ,,x0 ,

x0 ,x0Æx)distribution,and f GH

is the density function of a

GH2 , ,,

x i i

ii x i

x j ij

x i

x j ij x0Æx

, x i i

distribution.

440 Proof:See appendix A.œ

4.CVaR-based portfolio optimization in the mGH

framework

4.1.The general case

First,we study a portfolio optimization problem of the

445 class(P2),which is representative of the class of problems

(P1)through(P3).We choose(P2)because of its

similarity to the classical Markowitz problem,which

involves minimizing risk(as measured by portfolio

variance)under a minimum constraint for the expected

450 return.

Consider the portfolio optimization problem(P20):

min

CVaR ðÀx0XÞ,

subject to x2X¼f x2R d

: 0x! ,10x¼1g,

ðP20Þ

where X$GH d( , ,, , ,Æ), ¼( 1,..., d)02R d,and

i¼E(X i)is the expected return of asset i.Hence,the

455 objective is to minimize CVaR under the condition that

the expected portfolio return is at least .

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