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Operations Research Transactions

Vol.17 No.1

Linear conic optimization models for robust credit risk optimization∗ ZHANG Hongjie1

BAI Yanqin1,†

FANG Chunliang1

Abstract In this paper we deal with the credit risk optimization problem. We present a model based on the worst-case Conditional Value-at-Risk (CVaR) risk measure and the uncertainty for the credit risk loss distribution. Under the box uncertainty, we reformulate the model into a linear optimization problem. Furthermore, under the ellipsoidal uncertainty, we reformulate the model into a seconde-order cone optimization problem. Finally, we show a numerical example to demonstrate the effective of our models. Keywords credit risk optimization, worst-case CVaR, linear optimization, secondorder cone optimization Chinese Library Classification O221.2 2010 Mathematics Subject Classification 90C30

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0

Introduction

In asset allocation, the risk management is a core activity conducted by banks, insurance and investment companies, or any financial institutions that evaluates risks. The credit risk is the risk of a trading partner not fulfilling their obligations in full on the due date. Losses ÂvFϵ2012c3 28F * Supported by the Foundation of National Natural Science Foundation of China (No. 11071158) and Key Disciplines of Shanghai Municipality (No. S30104). 1. Department of Mathematics, Shanghai University, Shanghai 200444, China; þ°ŒÆêÆX§þ ° 200444 † Corresponding author ÏÕŠö, Email: [email protected]

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87

sometimes can result from counterparty default, or from a decline in market value caused by the credit quality migration of an issuer or counterparty. We consider the one-period credit risk optimization problems: a portfolio of N instruments and J scenarios. Let x ∈ Rn denote the amount of portfolio weights expressed as multiples of current holdings, b ∈ Rn be the future values of each instrument with no credit risk, and y ∈ Rn be the future (scenario-dependent) values with credit risk, which is random vector. Then the credit risk loss under the scenario j is then defined as L(x, yj ) =

N X (bi − yji )xi .

(0.1)

i=1

Denote ψ to the set of available portfolio allocation vectors. The initial optimal credit risk target is to choose the x ∈ ψ to maximize the profit on the investment while minimizing the risk losses. Generally, the set ψ depends on how to define and to measure the risk of the loss. Due to using different risk measure methods, several different risk optimization models have been considered in the literature[1−3] . In the traditional mean-variance framework of Markowitz’s approach[4] , the risk measure is quantified by the variance of the losses. But for credit portfolios, there is a large chance of positive returns and a very small probability of large investment losses. The distribution of losses is asymmetric and highly skewed, not satisfying the normal distribution assumptions of the mean-variance framework. Various extensions and improvements of this approach have been proposed since the pioneering work of Markowitz. During the middle of the 1990s, the Value-at-Risk[3] , a new measure of risk, has been developed in financial risk management. The Value-at-Risk can be given as the minimal α with which the probability of the risk loss L(x, y) not exceeding α under confident level β. The Value-at-Risk (VaR) can be written as VaRβ (x) = inf{α ∈ R : Ψ(x, α) > β}

(0.2)

where for fixed x, Ψ(x, α) = Prob{L(x, y) 6 α} is the cumulative distribution function for the loss associated with x. Moreover, β ∈ (0, 1]. The VaR risk measurement has been widely accepted and various estimation techniques have been proposed. However, the VaR has serious limitations as a risk measure for optimization. First of all, the VaR is generally non-convex function of the portfolio weights, and may exist many local extreme for discrete distributions. Moreover, the VaR is not sub-additive. Consequently it is not a coherent risk measure in the sense of Artzner[5] , which may discourage diversification. As an alternative risk measure, the conditional Value-at-Risk(CVaR) is also known as expected shortfall, mean shortfall, mean excess loss, or tail VaR-overcomes various limitations of the VaR. It can be interpreted as the mean of the tail distribution exceeding the VaR.  1 CVaRβ (x) = E Prob{L(x, y) > VaRβ (x)} (0.3) 1−β CVaR has better properties than VaR. It is a coherent risk measure proved by Pflug[6] . Then we need not solve the corresponding VaR before minimizing CVaR showed by Rock-

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afellar and Uryasev[7] . Also, Rockafellar and Uryasev[3,8] transformed the CVaR to convex programs, and sometimes simpler linear programs. Andersson[9] has developed CVaR to solve the credit risk optimization. Besides, Saunders[10] used factor models applying to this problem. Furthermore, in the classical models of credit risk measure, some little perturbation of the inputting data always trigger large differences of the optimal solution. In other words, these classical models require very accurate historical data of marketing. But it is too hard to do sufficient and meaningful statistics to obtain exact historical data. So the robustness of different problems must be considered. So far, some researchers, like Costa and Paiva[11] , Goldfarb and Iyengar[12] , have studied the robust portfolio problems, especially in the mean-variance framework. Some types of uncertainties, such as the box uncertainty and the ellipsoidal uncertainty are introduced for the mean and covariance, which often had some historical data from the market. And the models can be transformed into semidefinite programming or second-order cone programming. Goldfarb and Iyengar[12] also considered the robust VaR portfolio selection problem. The worst-case VaR are used to solve some robust portfolio by El Ghaoui[1] , where incomplete information of the distribution is known. Zhu and Fukushima[13] introduced a worst-case CVaR(WCVaR) method, and transformed the portfolio problem as convex programming problem. In this paper we use the worst-case CVaR (WCVaR) as credit risk measure to optimize the credit risk model. The WCVaR is also a coherent risk measure and could solve the distribution of heavy-tailed risk losses. Besides we reformulated the model as linear programming problem and second-order cone programming problem under the box uncertainty and the ellipsoidal uncertainty, respectively, which can be solved efficiently by the interior-point method. In the aspect of credit risk loss distribution evaluations, we are interested in the losses experienced in the event of counterparty default in the course of a day, a year or other standardized period. Several approaches are available for estimating credit risk distribution. Probably, one of the most influential contribution in this field has been J. P. Morgan Inc’s Credit metrics methodology[14] . The paper is organized as follows. In Section 1 we briefly recall the concepts of the CVaR and the worst-case CVaR. In section 2, we present general form of the robust credit risk optimization model. In section 3, we respectively reformulate the model as a linear programming problem under the box uncertainty and a second-order cone programming problem under the ellipsoidal uncertainty. Finally, we show a numerical example using credit metrics methodology to illustrate our model in Section 4.

1

Preliminary of the worst-case CVaR

In this section, we recall the basic meaning of CVaR and the worst-case CVaR introduced by Rocakafellar[8] and Zhu[13] . We show the form under continuous distribution and

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Linear conic optimization models for robust credit risk optimization

the discrete distribution is similar. Let L(x, y) denote the credit risk losses where x is the decision vector and y is the random vector. Assume that y belongs to a continuous distribution and denote its density function as p(.). The conditional Value-at-Risk, denoted by CVaRβ (x), is defined as the expected value of losses that exceeds V aRβ (x) as follows, Z 1 CVaRβ (x) = L(x, y)p(y)dy. (1.1) 1 − β L(x,y)>V aRβ (x) Denote

1 Fβ (x, α) = α + 1−β

Z

[L(x, y) − α]+ p(y)dy,

(1.2)

where [t]+ = max{t, 0}. For x and p(y), the corresponding CVaR can be defined as[3] CVaRβ (x) = min Fβ (x, α). α∈R

(1.3)

Assume that the density function is only known to belong to a certain set p, i.e., p(.) ∈ p and for fixed x ∈ X, the worst-case CVaR can be defined as WCVaRβ (x) = sup CVaRβ (x),

(1.4)

p(.)∈p

and equivalently, WCVaRβ (x) = sup min Fβ (x, α). p(.)∈p α∈R

Zhu[13] have pointed out that if p is a convex compact set, then WCVaRβ (x) = min max Fβ (x, α). α∈R p(.)∈p

2

(1.5)

Reformulation of the WCVaR model for credit risk optimization

At present we consider the one-period credit risk optimization problems: a portfolio of N instruments and J scenarios. Let x ∈ Rn denote the amount of portfolio weights expressed as multiples of current holdings, b ∈ Rn be future values of each instrument with no credit risk, and y ∈ Rn be the future (scenario-dependent) values with credit risk, which is random vector. Then the credit risk loss under scenario j is then defined as L(x, yj ) =

N X (bi − yji )xi . i=1

And let the expected value of negative L(x, y) denote the profit function P (x, y), i.e. P (x, y) = E[−L(x, y)].

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For the credit risk optimization, the target of the bank is to choose a portfolio, which minimize the credit risk, meanwhile can get more return from obligators. First of all, we consider conditional value-at-risk (CVaR) as the risk measure, and we exactly know the distribution of random vector y. The optimal portfolio x∗ , which minimize the CVaR subject to the profite more than Q1 is the optimal solution of the following program: min CVaR(x)β s.t. P (x, y) > Q1 eT x = 1 x > 0.

(2.1)

The CVaR model (2.1) is assumed on that the distribution is known exactly, however, in practice there exists the situation where only partial information of the distribution is known. The optiaml solution of (2.1) is related to the market parameter and distribution. Moreover sometimes the optimal solution is very sensitive to the market parameters. Since the parameters are often carried out from noisy historic data or belong to some distribution, the model should solve the problem for a wide range of problem data and should be scale invariant, that is, insensitive to any constraint or variable scaling that might be used. So in these cases using CVaR directly is inappropriate, and optimization problems should be solved assuming the worst case behavior of these robustness. Then to overcome the sensitivity from the parameters, we minimize the worst case CVaR with the profit constraints as min s.t.

WCVaR(x)β min P (x, y) > Q1

p(.)∈p

(2.2)

eT x = 1 x > 0.

Since the problem (2.2) consists of minimizing the WCVaR and meanwhile, in the constraints there is a minimization operation, it is not easy to solve this programming problem. Next we present that, under box uncertainty and ellipsoidal uncertainty, this problem can be reformulated as a linear programming problem and a second-order cone programming problem respectively.

3

Credit risk optimization with robustness under discrete distribution

In this section, we describe the formulation under discrete distribution, and the continuous distribution is similar. Now assume the random vector y has a discrete distribution, S P and is given by{y[1] , y[2] , · · · , y[s] }. The probability of {y[i] } is πi , where πi = 1, πi > i=1

0, i = 1, 2, · · · , S, and π = (π1 , π2 , · · · , πs )T . The WCVaR under the discrete distribution turns out to be WCVaRβ (x) = min max Gβ (x, α, π) (3.1) α∈R π∈pπ

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Linear conic optimization models for robust credit risk optimization

Denote Gβ (x, α, π) = α +

S 1 X [L(x, y) − α]+ , 1−β

91

(3.2)

k=1

We formulate the WCVaR credit risk optimization problem under discrete distribution as follows: min max Gβ (x, α, π) x,α

π∈Pπ

s.t.

min P (x, y) > Q1

π∈Pπ T

(3.3)

e x=1 x > 0. Particularly, we are interested in the box uncertainty and the ellipsoidal uncertainty among various uncertainty structures. First of all, these two types of uncertain sets are the simplest and can be reformulated to tractable optimization models. Then we can use familiar methods to solve them. Furthermore, these two uncertainty structures are used most and very typical in robust optimization[3,15−16] .

3.1

Box uncertainty

In this subsection, we consider the WCVaR credit risk programming problem with the box uncertainty and we reformulate it as a linear programming problem. Let π belong to a box, i.e., 0 T π ∈ PB (3.4) π = {π : π = π + η, e η = 0, η1 6 η 6 η2 } where π 0 is a known discrete distribution, e denotes the vectors of ones, η1 and η2 are constant vectors and the condition eT η = 0 ensures that π is a probability distribution. First of all, we simplify the objective function of (3.2). From (1.5), we have WCVaRβ (x) = min max Gβ (x, α, π), α∈R π∈pπ

(3.5)

we set the auxiliary variable, zj > L(x, yj ) − α, zj > 0, j = 1, · · · , S, consequently, we get max Gβ (x, α, π) = max {α +

π∈pB π

π∈pB π

1 1 γ ∗ (z) π T z} = α + (π 0 )T z + , 1−β 1−β 1−β

(3.6)

where γ ∗ (z) is the optimal solution of the following linear programming problem max η

s.t.

z T η, eT η = 0, η1 6 η 6 η2 .

(3.7)

The dual problem of (3.7) is as follows: min

u,ξ,ω

s.t.

η2T ξ − η1T ω eu + ξ − ω = z ξ > 0, ω > 0.

(3.8)

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Now we consider the following minimization problem on (x, α, z, u, ξ, ω) : min

x,α,z,u,ξ,ω

s.t.

α+

1 0 T 1−β (π ) z

+

1 T 1−β (η2 ξ

− η1T ω)

zj > L(x, yj ) − α, zj > 0 eu + ξ − ω = z x > 0, ξ > 0, ω > 0, zj > 0.

(3.9)

Theorem 3.1 If (x∗ , α∗ , u∗ , z ∗ , ξ ∗ , ω ∗ ) solves (3.9), then (x∗ , α∗ , z ∗ ) solves (3.5) when ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ B pπ = pB π ; conversely, if (x , α , z ) solves (3.5) when pπ = pπ , then (x , α , u , z , ξ , ω ) ∗ ∗ ∗ ∗ solves (3.9), and (u , ξ , ω ) is the optimal solution to (3.8) when z = z . Proof First of all, if (x∗ , α∗ , u∗ , z ∗ , ξ ∗ , ω ∗ ) solves (3.9), then (u∗ , ξ ∗ , ω ∗ ) is feasible to (3.8) when z = z ∗ , by the duality theorem of the linear programming, we have γ ∗ (z ∗ ) 6 η2T ξ ∗ − η1T ω ∗ .

(3.10)

Thus max {α∗ +

π∈pB π

1 1 γ ∗ (z ∗ ) π T z ∗ } = α∗ + (π 0 )T z ∗ + 1−β 1−β 1−β 6 α∗ +

1 η T ξ ∗ − η1T ω ∗ (π 0 )T z ∗ + 2 1−β 1−β

(3.11)

implies that (x∗ , α∗ , z ∗ ) is feasible to (3.5) when pπ = pB π. Then we assume (x∗ , α∗ , z ∗ ) is not an optimal solution to (3.5) when pπ = pB π , but ∗ ∗ ∗ ∗ b∗ ∗ there is an optimal solution (b x ,α b , zb ) of (3.5). Now let (b u ,ξ ,ω b ) be the optimal solution to (3.8) when z = zb∗ . By the strong duality theorem of the linear programming, we get 1 γ ∗ (η2T ξb∗ − η1T ω b∗) (π 0 )T zb∗ + 1−β 1−β ∗ ∗ γ (b z ) 1 (π 0 )T zb∗ + =α b∗ + 1−β 1−β 1 = max {b α∗ + π T zb∗ } B 1−β π∈pπ α b∗ +

(3.12)

which implies (b x∗ , α b∗ , zb∗ , u b∗ , ξb∗ , ω b ∗ ) is feasible to (3.9). This contradicts the assumption that (x∗ , α∗ , u∗ , z ∗ , ξ ∗ , ω ∗ ) solves (3.9). ∗ ∗ ∗ Thus we have that (x∗ , α∗ , z ∗ ) solves (3.5) when pπ = pB π . In contrast, let (x , α , z ) ∗ ∗ ∗ ∗ solve (3.5) with pπ = pB π , and let (u , ξ , ω ) be the optimal solution to (3.8) when z = z . ∗ ∗ ∗ ∗ ∗ ∗ Then (x , α , u , z , ξ , ω ) must solve (3.9). Actually, if it is not the case, then there appears b∗ , zb∗ , u b∗ , ξb∗ , ω b ∗ ) of (3.9). From the proof above, (b u∗ , ξb∗ , ω b ∗ ) is an an optimal solution (b x∗ , α ∗ ∗ ∗ optimal solution of (3.8), which contradicts the assumption that (x , α , z ) solves (3.5). Now we complete the proof. Next consider the constraint function of (3.3), that is, min P (x, y) > Q1 .

π∈pB π

(3.13)

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Equivalently, max E[L(x, y)] 6 Q1 .

(3.14)

max E[L(x, y)] = L(x, y)π 0 + max L(x, y)η.

(3.15)

π∈pB π

From (3.4) we have

η

π∈pB π

The second term in (3.15) is a maximizing problem on the right side, i.e. max η

s.t.

L(x, y)η eT η = 0 η1 6 η 6 η2 .

(3.16)

The dual problem of (3.16) is given by min

δ,τ,ν

s.t.

η1T ν − η2T τ eδ + τ − ν = L(x, y) τ > o, ν > 0.

(3.17)

By the strong duality theory of linear programming, there are zero dual gaps between primal problem (3.16) and dual problem (3.17) under the condition of the existence of interior points in the feasible set. So at last the constraint functions are worked out to be the following system of inequalities L(x, y)π 0 + η1T ν − η2T τ 6 Q1 eδ + τ − ν = L(x, y) τ > o, ν > 0.

(3.18)

Combine (3.9) and (3.18), and the credit risk WCVaR model under the box uncertainty can be reformulated as follows: min

x,α,u,ξ,ω

s.t.

α+

1 0 T 1−β (π ) z

+

1 T 1−β (η2 ξ

− η1T ω)

zj > L(x, yj ) − α eu + ξ + ω = z eδ + τ − ν = L(x, y) L(x, y)π 0 + η1 ν − η2 τ 6 Q1 eT x = 1 x > 0, α > 0, ξ > 0, ω > 0, zj > 0, τ > o, ν > 0.

(3.19)

Since L(x, y) is linear function, (3.19) turns out to be a linear programming problem. Moreover, if L(x, y) is a convex function, then it can be a convex programming problem. In particular, if η1 = η2 = 0 (3.19) reduces to the former CVaR problem.

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Ellipsoidal uncertainty Now assume that π belongs to an ellipsoid as

0 T 0 π ∈ PE (3.20) π = {π : π = π + Ah, e Ah = 0, π + Ah > 0, khk 6 1} √ where khk = hT h, π 0 is a nominal discrete distribution, which is the center of the ellipsoid, A ∈ RS∗S is the scaling matrix of the ellipsoid. The condition eT Ah = 0, π 0 +Ah > 0, ensures that π is a probability distribution. Similarily with the box uncertainty, the objective function of (3.3) under the ellipsoidal uncertainty distribution can be rewritten as

max Gβ (x, α, π) = max {α +

π∈pE π

π∈pE π

1 1 γ ∗ (z) π T z} = α + (π 0 )T z + . 1−β 1−β 1−β

(3.21)

We also introduce the auxiliary variable, zj > L(x, yj ) − α, zj > 0, j − 1, · · · , S, where γ (z) is the optimal solution of the following convex programming problem ∗

max h

s.t.

zAh eT Ah = 0, π 0 + Ah > 0 khk 6 1.

(3.22)

The dual problem of (3.22) is a second-order cone programming problem min

u,ξ,ω,λ

s.t.

λ + (π 0 )T ω eAT u − ξ − AT ω = AT z kξk 6 λ, ω > 0.

(3.23)

Now we consider the following minimization problem on (x, α, ξ, λ, ω) : min

x,α,u,ξ,ω,λ

s.t.

α+

1 0 T 1−β (π ) (z

+ ω) +

1 1−β λ

L(x, y) − α 6 zj eAT u − ξ − AT ω = AT z x > 0, α > 0, kξk 6 λ, ω > 0, zj > 0.

(3.24)

The proof of equivalence between (3.21) and (3.24) is in the same way as Theorem 1. Next consider the constraint function: min P (x, y) > Q1 .

(3.25)

max L(x, y) 6 Q1 .

(3.26)

max E[L(x, y)] = L(x, y)π 0 + max L(x, y)Ah.

(3.27)

π

Equivalently, π∈pE π

From (3.20) we have π∈pE π

h

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The second term in (3.27) is a maximizing problem, i.e. max h

s.t.

L(x, y)Ah eT Ah = 0. π 0 + Ah > 0 khk 6 1.

(3.28)

The dual problem of (3.28) is given by min

δ,τ,ν,κ

δ + (π 0 )T τ

s.t. −ν − AT τ + AT eκ = AT L, τ > o, kνk 6 δ.

(3.29)

By using the strong duality theory of linear programming, there are zero dual gaps between primal problem (3.28) and dual problem (3.29) under the condition of the existence of interior points in the feasible set. So at last the constraint functions are worked out to be L(x, y)π 0 + δ + (π 0 )T τ 6 Q1 −ν − AT τ + AT eκ = AT L τ > o, kνk 6 δ.

(3.30)

Combining (3.24) and (3.30), the credit risk WCVaR model under the ellipsoidal uncertainty can be reformulated as follows: min

x,α,u,ξ,ω,λ

s.t.

α+

1 0 T 1−β (π ) (z

+ ω) +

1 1−β λ

L(x, y) − α 6 zj eAT u − ξ − AT ω = AT z L(x, y)π 0 + δ + (π 0 )T τ 6 Q1 −ν − AT τ + AT eκ = AT L x > 0, α > 0, kξk 6 λ, ω > 0, zj > 0, τ > o, kνk 6 δ.

(3.31)

Since L(x, y) is a linear function, (3.31) turns out to be a secnod-order programming prbolem. Furthermore, if L(x, y) is a convex function, then it can be a convex program. In particular, if A = 0 then (3.31) reduces to the original CVaR problem. By means of the uncertainty distribution, not only are we capable to maintain the advantage of CVaR to the credit risk optimization, but also we can reduce the sensitive to the perturbations in the parameters and make the solutions of the optimization more reasonable. And so far we can use the interior-point method to solve the conic programming. In practice, we can use several softwares such as SeDuMi, CVX to implement the modle conveniently.

4

Analysis of a numerical example

We choose the example based on a portfolio consisting of 20 obligors of varying credit quality, which values one billion. Details of this portfolio are given as follows.

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Table 1 Details of the portfolio Name 1 2 3 4 5 6 7 8 9 10

Rate AAA AA AA A A A A A BBB BBB

Interest 4.10 5.00 4.70 5.30 5.90 3.70 5.30 4.30 5.00 5.00

Year 3 5 4 5 4 5 5 3 5 5

Name 11 12 13 14 15 16 17 18 19 20

Rate BBB BBB BB BB B B B B B CCC

Interest 5.50 4.90 7.00 6.00 8.00 6.02 6.02 7.00 8.00 15.00

Year 5 4 5 3 4 5 5 3 4 4

We consider the modeling time period is one year, and by using technology of creditmetrics we can get one year credit loss distribution. In the example, we set minimum expected return rate is 0.052, and give each portfolio weight one upper bound 0.2. Numerical experiments for the original and the robust credit risk optimization are performed via the linear programming model under box uncertainty described above. The first row of Table 2 uses the original CVaR as the risk measure, and on contrary the second row uses the robust WCVaR. The optimization toolbox in MATLAB is employed in our numerical example. Table 2 shows that the various CVaR and the WCVaR computed by setting different values of the confident level. Table 2 The results of the numerical example β CVaR WCVaR

0.950 0.0117 0.0126

0.990 0.0246 0.0258

0.999 0.0701 0.0812

From Table 2, we obtain that for fixed expected return rate the larger confident level β we set, the larger CVaR and WCVaR we get. Also, it is clearly that WCVaR is large than CVaR for the same confident level. This illustrates that we are able to avoid unnecessary credit risk by using WCVaR as the risk measure rather than CVaR.

References [1] Ghaoui L, Oustry F. Worst-case Value-at-Risk and robust portfolio optimization: A conic programming approach [J]. Operations Research, 2003, 51: 543-556. [2] Markowitz H M. Portfolio selection [J]. Journal of Finance, 1952, 7: 77-91. [3] Rockafellar R T, Uryasev S. Optimization of conditional Value-at-Risk [J]. Journal of Risk, 2000, 2: 21-41. [4] Lobo M, Fazel M, Boyd S. Portfolio optimization with linear and fixed transaction costs [J]. Annals of Operations Research, 2007, 152(1): 376-394.

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[5] Artzner P, Delbaen F, Eber J M, et al. Coherence measuresof risk [J]. Mathematical Finance, 1999, 9: 203-228. [6] Lobo M S, Vandenberghe L, Boyd S. Applications of second-order cone programming [J]. Linear Algebra and its Applications, 1998, 284: 193-228. [7] Pflug G. Some remarks on the Value-at-Risk and conditional Value-at-Risk [M]// Proba-bilistic Constrained Optimization: Methodology and Applications, Dordrecht: Ed. S. Uryasev, Kluwer Academic Publishers, 2002. [8] Rockafellar R T, Uryasev S. Conditional Value-at-Risk for general loss distributions [J]. Journal of Banking and Finance, 2002, 26: 1443-1471. [9] Andersson F, Mausser H. Credit risk optimization with Conditional Value-at-Risk criterion [J]. Math.Program, 2001, 89: 273-291. [10] Saunders D, Xiouros C. Credit risk optimization using factor models [J]. Ann Oper Res, 2007, 152: 49-77. [11] Costa O, Paiva A. Robust porfolio selection using linear-matrix inequality [J]. Journal of Ecnomic Dynamics Control, 2002, 26: 889-909. [12] Goldfarb D, Iyengar G. Robust portfolio selection problems [J]. Mathematics of Operations Research, 2003, 28: 1-38. [13] Zhu S S, Fukushima M. Worst-case conditional Value-at-Risk with application to robust portfolio management [J]. Operations Research, 2009, 57: 1155-1168. [14] RiskMetrics Technical Document [R]. New York: J P Morgan Inc, 1996. [15] Black F, Litterman R. Global portfolio optimization [J]. Financial Analysts Journal, 1992, 48: 28-43. [16] Ben-Tal A, Nemirovski A. Robust optimization methodology and applications [J]. Mathematical Programming (Ser. B), 2002, 92: 453-480.