Investment models based on clustered scenario trees

European Journal of Operational Research xxx (2013) xxx–xxx

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

Investment models based on clustered scenario trees q Man Hong Wong Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong

a r t i c l e

i n f o

Article history: Received 11 October 2011 Accepted 27 November 2012 Available online xxxx Keywords: Conic programming Stochastic programming Interior point methods Robust optimization Scenario tree Portfolio selection

a b s t r a c t Stochastic programming is widely applied in financial decision problems. In particular, when we need to carry out the actual calculations for portfolio selection problems, we have to assign a value for each expected return and the associated conditional probability in advance. These estimated random parameters often rely on a scenario tree representing the distribution of the underlying asset returns. One of the drawbacks is that the estimated parameters may be deviated from the actual ones. Therefore, robustness is considered so as to cope with the issue of parameter inaccuracy. In view of this, we propose a clustered scenario-tree approach, which accommodates the parameter inaccuracy problem in the context of a scenario tree.  2012 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Our motivation In recent years, stochastic programming has gained an increasing popularity in the framework of decision making under uncertainty. Typically, we have to approximate the uncertainties by a limited number of discrete outcomes which are often generated by sampling from the true or assumed probability distribution. The discretization is usually called a scenario tree. For a better tree to represent the data, a lot of research has been done to evaluate the scenario generating methods (see Boender, 1997; Mulvey et al., 1999; Kouwenbery, 2001; Hoyland and Wallace, 2001; Kaut and Wallace, 2003; Tietje, 2005; Latorre et al., 2007; Gabriel et al., 2009; Liesio and Salo, 2012). Meanwhile, some other people (Hiller and Eckstein, 1993; Zenios, 1999; Hu, 2003; Geyer et al., 2010) are dedicated to find out a better tree representation. Focusing on this issue, we would like to propose a tree that is different from the traditional ones. ‘‘Traditional’’ trees are regarded as those representations of a discrete distribution with a limited number of outcomes. Pictorially, there are single dots at the end of the tree. What is new in our tree is that we replace each single dot by an ellipsoid with a uniform density. In other words, each scenario is now represented by infinitely many ‘‘dots’’ within the ellipsoid and the tree becomes q An earlier working paper, titled Conditional Value-at-Risk under ellipsoidal uncertainties, has appeared in the Proceedings of the Computational Finance 2006. It contains the results mainly in Section 4 in this paper. Please refer to Wong (2008). E-mail address: [email protected]

a representation of data somehow between a discrete and continuous distribution. Owing to its clustering nature, we call it a clustered tree. The advantage is that it could reduce the chance of error from the over-estimation or under-estimation of parameters from the tree since we are now considering a range of data instead of a point. Shen and Zhang (2008) introduced the robustness consideration (see Ben-Tal and Nemirovski, 1998; Lutgens and Sturm, 2002; Goldfarb and Iyengar, 2003; Gulpmar and Rustem, 2007; Gregory et al., 2011; Huang et al., 2010; Zymler et al., 2011) into the scenario tree. However, their ‘‘robust tree’’ is a worst-case analysis and we wonder if it can be less conservative. Therefore, in our clustered tree approach, we consider the average case by assuming the uniform ellipsoidal distribution under each ‘‘scenario’’, which, to be more precise according to its nature, should now be called a cluster (see Fig. 1 ). The use of cluster tree can be justified with a simple but realistic concern regarding the traditional scenario tree: Suppose the economy will either boom or bust in the next period. Of course, we cannot tell the exact price of stocks in the future until they are realized. However, we usually have a range estimations based on experience or statistics. For instance, we are more confident and comfortable to say that ‘‘a stock will go 10% to 20% up or down’’, rather than saying that ‘‘a stock will go 15% up or down’’. (The width of the range is another issue.) Given this common circumstance, cluster tree can answer the need in modeling issues. In this paper, portfolio selection through stochastic programming will be our framework. Three financial models will be introduced for the applications of our clustered tree, namely, the chance constrained model, the downside risk model and the conditional value-at-risk model.

0377-2217/$ - see front matter  2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.11.051

Please cite this article in press as: Wong, M.H. Investment models based on clustered scenario trees. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.11.051

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M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

1.3. Paper structure

Fig. 1. Returns happen equally likely in each scenario in this clustered tree.

Here outlines the remaining plot of this paper. In Section 2, the probability expression under our clustered tree will be introduced. The chance constrained programming will then be discussed and we will show it can be transformed to a second-order cone programming (SOCP) in a single scenario case. Under an assumption, we have a save approximation by considering a SOCP in multiple scenario case. Therefore, it can be solved efficiently with the use of SeDuMi or cvx. In Section 3, we will derive the downside risk measure and formulate the model. The formulation will be similar in both single and multiple scenario cases. We will also calculate the derivatives that are necessary for the Interior Point Algorithm. In Section 4, the conditional value-at-risk measure will be derived based on Section 3 in a parallel manner, since conditional value-atrisk measure possesses a component that is close to the downside risk. We will also apply the Interior Point Algorithm to solve the problem. In Section 5, we try to compare the models by some the numerical results. Finally, Section 6 contains the conclusion. 2. The probability model

1.2. Literature review 2.1. Derive the chance constraint Consideration of risk is a major element in financial decision making. Risk generally conflicts with our financial goal, namely, maximizing our return. It can be considered in a variety of ways. Chance constrained programming was originally proposed by Charnes, Cooper and Symonds (see Charnes et al., 1958; Charnes and Cooper, 1959). This has been developed and applied by many others (e.g. Kataoka, 1963; Van De Panne and Popp, 1963; Joshi, 1995; Liu and Iwamura, 1997; Dupacova, 2002; Topaloglou et al., 2008; Castro, 2009; Shapiro, 2011). Nasland and Whinston (1962) gives an introduction to the subject. In the chance constrained model, decision makers (investors) are required to select a prescribed level of risk, b. Risk is in the form of a probability expression. The investor has to state how confident he or she wants the portfolio to exceed the target return. In the downside risk model, the investor’s focus is on the circumstances that the target return cannot be achieved and he or she wants to minimize the loss under these circumstances. This model sounds more appealing to the safety-first investors. Ever since Markowitz (1950, 1952) published his portfolio theory, the construction of mean–variance portfolio has been a rather popular subject both in academic research and practices in the real world. The quantitative analysis between expected return and risk has had a profound impact on the everyday investment activities of portfolio managers. However, there has also been criticism regarding the use of variance for its indifferent attitude toward the upside and the downside potential. The notion of downside risk was then raised (see Grootveld and Hallerbach, 1999; Roy, 1952). Among various measures for the downside risk, the lower partial moment (see Harlow and Rao, 1989) of order one will be of our choice of the risk measure. Because of Rockafellar and Uryasev (2000, 2002), our last model, the conditional value-at-risk (CVaR) model, is an ad hoc issue nowadays. Risk management plays an important role in assets allocation. It entails the exercise of control over some statistical characteristics of the uncertain portfolio return. The aim is to avoid selecting portfolios that may be more susceptible to severe losses. Including a prescribed level of confidence, our model considers the minimization of the CVaR, which we sometimes call the expected shortfall or tailed value-at-risk (VaR). Although the popularity of VaR is comparable to that of the CVaR for the financial decision makers, the latter possesses nicer mathematical properties that must have gained much more appreciation from academic researchers.

We are now going to examine the probability of attaining the target return based on our clustered tree. It will be used in the later sections for the chance-constrained optimization. We first introduce the notations. To demonstrate the idea of our cluster tree, we would assume there is only a single cluster in the next period. Suppose we invest in n stocks. Let.  / 2 Rn be my portfolio,  r 2 Rnþ be the rate of return of the n stocks in the next period. (0 6 r < 1 means a loss while r > 1 a profit.)  R 2 R+ be my target return. For the no short selling case, our target return can be attained only when it is within the possible range of our portfolio return, i.e.

min r i 6 R 6 maxr i

16i6n

16i6n

Within the single cluster, we suppose the return vector lie uniformly inside an ellipsoid E with a positive centre c = (c1, . . . , cn)T, i.e.

r 2 E :¼ fr 2 Rn jðr  cÞT Q ðr  cÞ 6 q2 g where q > 0 and Q is a diagonal matrix with diagonal entries a12 for i some ai > 0 for i = 1, . . . , n. So the cluster can be represented as an ellipsoid. But for the better properties of sphere regarding its symmetry, we want to transform the ellipsoid into a sphere BðcÞ  Rn instead with the centre c = (c1, . . . , cn)T and radius q. This can be achieved without loss of generality by a linear transformation: 1

r ¼ Q 2 ð~r  cÞ þ c where ~r is the vector after transformation. So, we have

~r 2 f~r 2 Rn jk~r  ck 6 qg For there is no confusion, we will keep using the notation r instead of ~r from now onwards. Let f(/, r) be our utility function. We can define a probability function of attaining a certain utility value a by

Wð/; aÞ ¼

Z

pðrÞdr

ð1Þ

f ð/;rÞPa 1 where pðrÞ ¼ VðBðcÞÞ is the constant density function for r being uniformly distributed in BðcÞ, and VðBðcÞÞ is the volume of the sphere.

Please cite this article in press as: Wong, M.H. Investment models based on clustered scenario trees. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.11.051

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M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

In our settings, we choose f(/, r) = /Tr and a = R. This can be interpreted as our desire of achieving the target return R from our portfolio. We also write W(/, a) as Pr(/Tr P R). To further investigate the probability, some mathematical facts are needed to be recalled. Lemma 1. Denote VðBðcÞÞ as the volume of BðcÞ with radius q. Then

VðBðcÞÞ ¼ Xn1

Z

q

ðq2  t 2 Þ

n1 2

Prð/T r P RÞ ¼

volume of the portion lying on /T r P R volume of the whole ball T

cR Using the signed distance function qð/Þ :¼ /k/k and by Lemma 1, we can write out the probability’s expression explicitly as follows.

Definition 2.

8 0; > <

T

Xn1 > Xn qn

Prð/ r P RÞ ¼

dt

:

q

where Xn1 ¼ C Proof. Let r ¼ VðBðcÞÞ ¼

Z

where Xn ¼ C

Þ  r1 , where r0 2 Rn1. r0

Z

dr c 1 þq

Z

c 1 q

¼

Z

c2 þ

c2 

Z

c 1 þq

qð/Þ

2

n1 2

ðq  t Þ

dt; for  q 6 qð/Þ 6 q; for qð/Þ > q;

1;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi q2 ðr1 c1 Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi    q2 ðr1 c1 Þ



c 1 q

Z

cn1 þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 q2 ðr1 c1 Þ ðrn1 cn1 Þ

dr n drn1    dr1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2

cn1 

q2 ðr1 c1 Þ ðrn1 cn1 Þ

!

Z fr 0 2Rn1 :kr 0 k2 ¼q2 ðr1 c1 Þ2 g

p2

ðn2þ1Þ

2.2. Single cluster model

BðcÞ

¼

2

n

n1 2 n1þ1 2

p

ð 

for qð/Þ < q;

Rq

dr n drn1    dr 2 dr 1

Since the variance depends only on the norm of the portfolio (see Section B), the mean–variance model may seem not reasonable. Instead, we consider another objective: Pr(/Tr P R), the probability that we have discussed. For we are going to maximize Pr(/Tr P R), or

F n ðqð/ÞÞ :¼

Z

q

ðq2  t 2 Þ

n1 2

dt

ð2Þ

qð/Þ

So, the integration inside the bracket represents a disc in Rn. But it is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi also a volume in Rn1 with radius q2  ðr1  c1 Þ2 . By (Appendix A.1) we have:

VðBðcÞÞ ¼

Z

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 Xn1 q2  ðr1  c1 Þ2 dr 1

c 1 þq c 1 q

¼ Xn1

Z

q

ðq2  t 2 Þ

n1 2

dt

q

The last step is due to the change of variable t = r1  c1 h Corollary 1. Let q > d > 0. If a ball with radius q in Rn is cut by a hyperplane which has an Euclidean distance of d from the centre, then the volume of the cap (smaller half) not containing the centre is given by

VðcapÞ ¼ Xn1

Z

q

ðq2  t 2 Þ

n1 2

dt

qd

or equivalent, the volume of the other half is given by

Xn1

Z

qþd

ðq2  t 2 Þ

n1 2

dt

q

The two choices in the corollary depend on the orientation of the hyperplane and the sphere. Therefore, to make things clear, a signed distance is necessary. Definition 1. Given vectors / and b 2 R, the signed distance between the centre c of the ball and the hyperplane /Tx = b is given by

/T c  b qð/Þ :¼ k/k Note that q(/) > 0 ) /Tc > b ) c lies on the half-plane /Tx > b) more than half of the ball lies on /Tx > b. The interpretation for q(/) < 0 is similar. Let us go back to the probability Pr(/Tr P R). Geometrically, we want to investigate whether the half-plane /Tr P R covers the ball and to what extent this happens. We can define the probability in this way:

we can just maximize q(/) as, being fortunate enough, we find that Fn(q(/)) is an increasing function in q(/). With a budget constraint and no short selling constraint, our model becomes

ðPÞ max qð/Þ :¼

/T c  R k/k

s:t: /T e ¼1 / P0 It is a single-ratio fractional programming problem. Using the Dinkelbach’s Algorithm, it can be solve through a sequence of concave maximization problems, namely

ðPqj Þ max cT /R  qj k/k s:t: /T e ¼1 / P0 Here qj is the iterative parameter in Dinkelbach’s Algorithm. The optimal solution q⁄ corresponds to that in (P) (see Appendix C). As a matter of fact, it is rare to use the probability as the objective in optimization problems to the modelling aspect. More often, we treat it as a constraint, leading to the following discussion of what is so-called chance-constrained programming. Considering the chance constraints allows investors to evaluate his/her objective in terms of their desire level of probability. If bC is his/her desired preset confidence level, that means a constraint will have a probability satisfaction of bC. In general, the choice of bC is a trade-off between risk and return, and it could be subjective. Like the typical stochastic investment models, we use the expected return Eð/T rÞ as our objective function and form the following model.

ðCCPÞ min  Eð/T rÞ s:t: Prð/T r P RÞ P bC /T e ¼ 1 /P0 Normally speaking, an investor should expect at least more than a half of chance that his/her portfolio return can achieve the target. Hence it would be reasonable for us to further assume bC 2 (0.5, 1). (In fact bC are often set much higher than 0.5 like 0.95 or 0.99.) This would be a very crucial assumption for the

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M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

development of our model, though it looks trivial. The reason is revealed right now. In the probability constraint, for F(q(/)) is increasing in q(/), the inverse of F must exist uniquely for bC 2 (0, 1). On the other hand,  n  Xn q bC bC > 0.5 impliesqn;bC :¼ F 1 must be negative (i.e. n Xn1 n1 Xn1 R q 2 2 2 qn;bC P 0). Otherwise, Xn qn qn;b ðq  t Þ dt 6 0:5, contradicting C

bC > 0.5. Therefore, we have

Xn1 Xn qn bC F n ðqð/ÞÞ P bC ) F n ðqð/ÞÞ P n Xn q Xn1 ) qð/Þ 6

F 1 n





m X



k¼1

Z ðkÞ m X xk Xn1 q  ðkÞ 2 2 n1 2 () q t dt P bC n ðkÞ Þ ðkÞ X ð q q ð/Þ n k¼1 T ðkÞ

r R where qðkÞ ð/Þ :¼ / k/k . This constraint is non-convex and would be difficult to solve in general. However, if we impose the assumption that all q(k)’s are equal, we can still consider the corresponding second-order cone programming as in the single cluster case,

ðSOCPmÞ min 

Xn qn b ¼ qn;bC Xn1



xk Pr /T rðkÞ P R P bC

xk /T cðkÞ

k¼1 T

s:t: 0 T

/T c  R P qn;bC ) k/k  T  / cR ) qn;b / 2 SOCðn þ 1Þ

m X

/ e¼1 1

/ cðkÞ R

@

A 2 SOCðn þ 1Þ 8k ¼ 1; . . . ; m

ðkÞ

qn;b

C

/ /P0

C

Hence, our chance-constrained programming problem is also a second order cone programming problem:

ðSOCP1Þ min  /T c s:t: /T e ¼ 1 ! T / cR qn;b

/

2 SOCðn þ 1Þ where qn;b :¼ F 1 n



Xn qn b Xn1



ðkÞ

where qn;b :¼ F 1 n

(k)

where c is the centre, Q is a diagonal matrix with the (i, i)-entry (k) 1 is some positive parameter. Also, let R 2 for i 2 {1, . . . , n} and q ðaðkÞ Þ be the target return such that ðkÞ

min ri 6 R 6

1 A 2 SOCðn þ 1Þ 8k ¼ 1; . . . ; m

Z m X xk Xn1 ðkÞ Þn

ðq

qðkÞ



2

qðkÞ  t2

n1 2

qðkÞ ð/Þ

dt P Xn bC

Proof. /T cðkÞ R

!

ðkÞ

2 SOCðn þ 1Þ 8k ¼ 1; . . . ; m C /  n  T ðkÞ ðkÞ X n q bC ) qðkÞ ð/Þ ¼ / ck/kR P qn;bC ¼ F 1 8k ¼ 1; . . . ; m n Xn1  ðkÞ  Xn qn b   ) F n q ð/Þ P Xn1 8k ¼ 1; . . . ; m *F is increasing in qðkÞ ð/Þ m X   qn b ) xk F n qðkÞ ð/Þ P XXnn1 qn;b



k¼1

3. The downside risk model

 12   ~r ðkÞ  cðkÞ þ cðkÞ r ðkÞ ¼ Q ðkÞ The m scenarios are then of the form

r ðkÞ 2 frðkÞ 2 Rn jkr ðkÞ  cðkÞ k 6 qg Let x for i = 1, . . . , m be the probability that cluster k will occur (i.e. Pm k k k¼1 xk ¼ 1). We have assumed that xk is independent of r . A natural extension from the single cluster model should be as follows. m X min  xk /T cðkÞ

s:t:

@

ðkÞ qn;b C

ðkÞ

max r i

16i6n 16k6m

There are now m ellipsoids. Like before, we transform each of them into spheres without changing the centres by

ðCCPmÞ

/T cðkÞ R

k¼1

(k)

16i6n 16k6m

0

/ )

 T    2 r ðkÞ 2 E ðkÞ :¼ frðkÞ j r ðkÞ  cðkÞ Q ðkÞ r ðkÞ  cðkÞ 6 qðkÞ g  Rnþ ;

for all k. This problem is not equivalent to

Lemma 2. If all q(k)’s are equal to q, The intersection of the secondorder cone in (SOCPm) implies the probability constraint in (CCPm), i.e.

The solution can computed with SeDuMi or cvx.

We now extend the use of notations to the multiple clusters case. Suppose we invest in n stocks and there will be m possible scenarios in the next period. For each cluster k 2 {1, . . . , m}, let r(k) be the return such that



Xn qn b Xn1

the (CCPm), but it is tighter. Let us put it as a lemma.

/P0

2.3. Multi-clusters model



k¼1 m X





xk Pr /T rðkÞ P R P bC

k¼1

/T e ¼ 1 /P0 in which the probability constraint

3.1. Derive the downside risk measure While the discussion of chance constraint has a long history, downside risk measure has been noted for over a half century. From this perspective, risk is measured in terms of the undesired rate of return which is below the preset target rate of return. The general way to express the downside risk may be concluded by the lower partial moment (LPM), the form of which is given as follows.

LPMh ðR; xÞ :¼ E½R  xhþ :¼

Z

RPx x2S

ðR  xÞh dFðxÞ

where R is the preset target return, x is a random variable (future return) in a space S defined in the measure F() (see Harlow and Rao, 1989). The degree, h, represents the investor’s utility in terms of risk aversion. The investor is considered as a risk seeker when h < 1, while the investor is averse to risk when h > 1. When h = 1, the investor is risk neutral. As shown by Bawa and Lindenberg (1977), LPMh(R, x) is a convex function of x, which is an appealing mathematical properties for optimization problems. In the following discussion, we will choose h to be 1, since this involves a simpler calculation to develop the model that is of our main concern.

Please cite this article in press as: Wong, M.H. Investment models based on clustered scenario trees. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.11.051

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M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

Using the same notations c, r, q and Q, we now do the transformation, 1

1

~r ¼ Q 2 ðr  cÞ to shift the centre to the origin and shear the ellipsoid into a sphere B ¼ f~r j~r T ~r 6 q2 g (since ðr  cÞT Q ðr  cÞ ¼ ~rT ~r 6 q2 ). In this section, our interest is to calculate the downside risk Er ½R  /T rþ , where Er ðÞ is the expectation of r uniformly over E. ~T ~ c / Consider the case when Rk/k > q: ~

Er ½R  /T rþ h  1 i ¼ E~r R  /T Q 2~r þ c

þ

~ T ~r 6 R  / ~ T ~c /

 1

¼

~r T ~r 6 q ~T ~r 6 R~/ ~c

Xn qn

¼

 ~ re1 d~r ~ T ~c  k/k R/

R

det Q 2

~ T ~c minðR/ ~ ;qÞ k/k



q

Xn qn

 1

Xn1 det Q 2 Xn qn



pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin1 ~ re1 Xn1 ~ T ~c  k/k R/ q2  re1 2 d re1

~ ~c /   n1 R minðRk/k ~ ;qÞ ~ T ~c  k/k ~ re1 q2  re1 2 2 d re1 R/ q T

~ ~c /  R minðRk/k ~ ;qÞ T

¼ Dn1

q

~ re1 ~ T ~c  k/k R/



q2  re1 2

n1 2

d re1 ;

where

Dn1 :¼

 1 Xn1 det Q 2

Xn qn

n1 2

p

¼ n1

C

 1   C nþ2 det Q 2 2 nþ1pffiffiffiffi ¼ n

C

2

2

 þ1

n

C 2þ1

 1 det Q 2



pn2



qn

~ ~  / g ð/Þ ~ ¼ ~cg 1 ð/Þ ~ 2 k/k

ð5Þ

~ ~ @F SD ð/Þ ~  /i g ð/Þ ~ ¼  cei g 1 ð/Þ ~ 2 k/k @f / i

Let me remark that (5) requires a few lines of algebra to work it out. For the second derivative,

!2 0 !2 1n1 2 ~i ~ ~ T ~cÞ/ ~ T ~c @F 2SD ð/Þ 1 ðR  / R  / 2 @q  A ei þ c ¼ ~ ~2 ~ ~ 2 k/k @/ k/k k/k i ! ~2 / 1 i ~ þ  g 2 ð/Þ ~ ~ 3 k/k k/k ! ! ~i ~j ~ ~ T ~cÞ/ ~ T ~cÞ/ @F 2SD ð/Þ 1 ðR  / ðR  / cej þ cei þ ¼ ~ ~ 2 ~ 2 k/k k/k k/k @f /i @ f /j n1    ~ T 2 2 ~/ ~ / / ~c i j ~ þ k/k  q2  Rk/k ~ ~ 3 g 2 ð/Þ

3.3. Single cluster model and numerical algorithm Like the chance constraint model (CCP), we consider a portfolio selection model, but now with the downside risk measure. Suppose we have an obligation of meeting a portfolio return R. With the budget constraint and no short selling constraint, we consider:

s:t: /T e ¼ 1

pq

~ Te R/ c ~ k/k

/T c P R /P0

6 q. Replacing

Er ½R  /T rþ 8 T T R R~/~ ~c R R~/~ ~c 2 > n1 n1 < D ðR  / ~ T ~cÞ qk/k ~ qk/k ðq  t 2 Þ 2 dt  Dn1 k/k tðq2  t2 Þ 2 dt; n1 ¼ > : 0; 8 ~ T ~c R/ ~  k/kg ~ ~ ~ T ~cÞg ð/Þ < ðR  / > q; 1 2 ð/Þ; ~ k/k ¼ T~ ~ c R / : 0; 6 q: ~ k/k

~ T ~c R/ ~ k/k

> q;

~ T ~c R/ ~ k/k

6 q:

ð3Þ

ð4Þ

or equivalently

where

~ ¼ g 2 ð/Þ

~ / ~  k/kg ~ 0 ð/Þ ~ g ð/Þ 2 ~ 2 k/k

min Er ½R  /T rþ

Obviously, the downside risk will be zero if the dummy variable re1 by t, we have

~ ¼ g 1 ð/Þ

~ ¼ ~cg ð/Þ ~ þ ðR  / ~ T ~cÞg 0 ð/Þ ~  rF SD ð/Þ 1 1

and for i – j,

k/k

 1

¼



R

2

~ Then Er ½R  /T rþ :¼ Dn1 F SD ð/Þ).

or componentwisely

~ T ~r  /T c ~ ¼ Q 12 / ¼ E~r ½R  / where / h iþ 1 ~ T ~c ~ T ~r  / where ~c ¼ Q 2 c ¼ E~r R  / þ     R ~ T ~c det Q 12  1 d~r ~ T ~r  / ¼ B T f/~ T ~r6R/~ T ~cg R  / VolðBÞ  1   det Q 2 R ~ T ~r d~r where VolðBÞ ¼ Xn qn ~ T ~c  / R/ ¼ Xn qn ~rT ~r 6 q det Q 2

ther mentioning from now on. For the implementation of the interior point method, we are going to calculate its first and sec~ :¼ ðR  / ~ T ~cÞg ð/Þ ~  k/kg ~ ~ ond derivatives. Let F SD ð/Þ ð/Þ (i.e.

Z

~ T ~c R/ ~ k/k

n1 2

ðq 2  t 2 Þ

dt;

~ :¼ ðR  / ~ T ~cÞg ð/Þ ~  k/kg ~ ~ min F SD ð/Þ 1 2 ð/Þ

q

Z

~ T ~e ¼ 1 s:t: / ~ T ~c R/ ~ k/k

tðq2  t2 Þ

n1 2

dt

~ T ~c P R /

q

~ P0 / 3.2. Calculate the first and second derivative of the downside risk According to (4), the downside risk of interest is when ~ T ~c R/ ~ k/k

> q. Hence, this will be the implied assumption without fur-

1

where ~e ¼ Q 2 e We will solve the optimization problem by an interior point ~ nþ1 ; s :¼ algorithm. First of all, we add the slack variables / ðs1 ; . . . ; sn ÞT to the inequalities to become equalities:

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M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

Er ½R  /T rþ m X

¼ xk ErðkÞ R  /T rðkÞ þ

~ T ~c  R  / ~ nþ1 ¼ 0 / ~ /þs¼0

k¼1

and formulate the Langrangian function:

~ nþ1 Þ  zT x ~ þ y ð/ ~ T ~e  1Þ þ y ð/ ~ T ~c  R  / Lðx; y; zÞ ¼ F SD ð/Þ 1 2

ð6Þ

x¼@

1

~ / ~ nþ1 /

y1

A 2 Rnþ1 ; y ¼

!

!

z0 z¼ znþ1

z1



T





T

þ

k¼1

¼

2 R2 ;

m   X ðkÞ ~ ðkÞ Dn1 xk F SDj / k¼1

y2

    T      ~ ðkÞ ~cðkÞ g / ~ ðkÞ kg / ~ ðkÞ :¼ R  / ~ ðkÞ  k/ ~ ðkÞ , and where F SDj / 1k 2k

and 00



xk E~rðkÞ R  /~ ðkÞ ~rðkÞ  /~ ðkÞ ~cðkÞ ;

 12  1 ~ ðkÞ ¼ Q ðkÞ 2 / where ~c ¼ Q ðkÞ c; /

where

0

¼

m X

ðkÞ

Dn1 ; g 1k and g2k are defined similarly as the single-cluster ones,   P ~ ðkÞ as an except now they depend on q(k). We treat m k¼1 xk F SDj /   P ðkÞ ~ ðkÞ . implicit function of / and define F MD ð/Þ :¼ m k¼1 xk Dn1 F SDj /

11

BB CC BB . CC B B .. C C BB CC ¼ B @ A C 2 Rnþ1 C B B zn C A @

The derivatives calculation can be referred to Appendix D. Indeed, the model with multiple clusters and that with a single one are very much alike:

znþ1

min

The first order KKT conditions are

F MD ð/Þ

s:t:

~ þ y ~e þ y ~c  z0 ¼ 0 r/~ F SD ð/Þ 1 2

/T

ðr/~ nþ1 Lðx; y; xÞ ¼Þ  y2  znþ1 ¼ 0

m X

/T e ¼ 1 !

xk cðkÞ

PR

k¼1

~T~

/ e1¼0 T~ ~ ~ nþ1 ¼ 0 / cR/

/P0 We can use the same algorithm as for the single cluster.

xþs¼0 zi si ¼ 0;

i ¼ 1; . . . ; n þ 1

4. The conditional Value-at-Risk model

ðz; sÞ P 0 4.1. Derive the conditional Value-at-Risk Writing the no-shorting selling constraints (with the slack vari~ nþ1 ) as a vector function h(x) = x, the Jacobian of it able / will be Jh(x) = I(n+1)(n+1). Hence, the modified Newton step used for interior-point algorithm for the equality conditions becomes

0

r2x Lðx; y; zÞ

J h ðxÞT

ry ðrx Lðx; y; zÞÞ

B B rx ðry Lðx; y; zÞÞ r2y Lðx; y; zÞ B B J h ðxÞ 0ðnþ1Þ2 @ 0ðnþ1Þðnþ1Þ 0ðnþ1Þ2 1 0 rx Lðx;y;zÞ C B B ry Lðx; y; zÞ C C B ¼B C B ðhðxÞ þ sÞ C A @

1 0 Dx 1 C CB B Dy C 02ðnþ1Þ C C CB CB C C Iðnþ1Þðnþ1Þ AB D z @ A Z Ds

0ðnþ1Þðnþ1Þ

02ðnþ1Þ 0ðnþ1Þðnþ1Þ S

Wð/; rÞ ¼

where

B S¼@

s1

1 ..

0

C B A; Z ¼ @

. snþ1

z1

1 ..

C sT z ; A; l ¼ nþ1

. znþ1

chosen, ry ðrx Lðx;y;zÞÞ¼  2  ~ ðrx ðry Lðx;y;zÞÞÞT ; r2x Lðx;y;zÞ¼ r/~ F SD ð/Þ 0n1 , and 0n1 0 022 ð0; 1Þ

is

arbitrarily

Z

pðrÞdy

f ð/;rÞ6a

SZe þ rle 0

Having derived the downside risk measure, we proceed to a more popular issue: the conditional Value-at-Risk, or CVaR. We can use the same technique to handle the CVaR as for the downside risk. For details of the derivation of optimizing the CVaR, refer to Rockafellar and Uryasev (2000, 2002). We will modify their model to suit our purpose. Let f(/, r) = /Tr be our loss function. Without loss of generality, we assume Q to be the identity matrix so that r lies on a sphere with a density function pðrÞ ¼ Xn1qn (see (Appendix A.1)). The probability of f(/, r) not exceeding a threshold a is then given by



r2

 1n1 c ¼ 0 1

r

2 y Lðx;y;zÞ¼

3.4. Multi-clusters model Using again the notations r(k),c(k),q(k),Q(k) and xk for k = 1, . . . , m, we can define and calculate the downside risk measure in multiple clusters:

As a function of a for fixed /, W is the cumulative distribution function for the loss associated with /. It completely determines the behavior of this random variable and is fundamental in defining VaR and CVaR. With a specified probability level b 2 (0, 1), the bVaR and b-CVaR values for the loss random variable associated with / are given by:

VaRb ð/Þ :¼ ab ð/Þ :¼ minfa 2 RjWð/; aÞ P bg CVaRb ð/Þ :¼ cb ð/Þ :¼

1 1b

Z

f ð/; rÞpðrÞdr

ð7Þ ð8Þ

f ð/;rÞPab ð/Þ

In the first formula ab(/) comes out as the left endpoint of the nonempty interval consisting of the values a such that W(/, r) = b, since W(/, a) is continuous and nondecreasing with respect to a. In the second formula, the probability that f(/, r) P ab(/) is therefore equal to 1  b. Hence cb(/) comes out as the conditional expectation of the loss associated with / relative to that loss being ab(/) or greater.

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M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

Our objective is to minimize the b-CVaR, but solving this directly seems to be difficult owing to the nature of its definition in terms of the b-VaR value (ab(/)) and the often poor mathematical properties of that value. Fortunately, we can handle a far simpler expression introduced by Rockafellar and Uryasev:

F b ð/; aÞ :¼ a þ

1 1b

Z

/2X

r2B

min F b ð/; aÞ

ð9Þ

ð/;aÞ2XR

Since Fb(/, a) is convex with respect to (/, a), assuming X to be a convex set, the joint minimization is a convex programming problem. This is a high motivation leading us to explore this function in our settings. We now go into the details of the integration in Fb. As p(r) is also a constant, we omit this term for the time being. T c Consider the case when aþ/ < q: k/k

R

½f ð/; rÞ  aþ dr ¼ r2E

¼

R

R

r/ g 1 ¼ @q  2

0 !2 1n1 2 @g 1 1 @ 2 a þ /T c A ¼ q  k/k k/k @a

0 !2 1n1 2 @g 2 a þ /T c @ 2 a þ /T c A ¼ q  k/k @a k/k2 The above derivatives will be zero otherwise. So, the first and second derivatives are

r/ F b ¼

@2 Fb @/i @ a

½/T r  aþ dr r2E

ðr  cÞ

¼

¼ T

ð/ ~r  /T c  aÞd~r

~r T ~r 6 q

/T c R ak/k

q

n1 2

ðk/kt  /T c  aÞðq2  t2 Þ

¼ ð/T c þ aÞ

/T c R ak/k

q

ðq2  t 2 Þ

n1 2

dt

dt  k/k

/T c R ak/k

q

n1 2

tðq2  t2 Þ

dt

¼ ð/T c þ aÞg 1 ð/; aÞ  k/kg 2 ð/; aÞ @2 F b @/i @/j

where

g 1 ð/; aÞ ¼

g 2 ð/; aÞ ¼

Z

a/T c k/k

n1 2

ðq2  t2 Þ

dt

q

Z

a/T c k/k

tðq2  t2 Þ

dt

8 T > < a  Ab;n ðð/ c þ aÞg 1 ð/; aÞ þ k/kg 2 ð/; aÞÞ;

aþ/T c k/k T

aþ/ c k/k

< q; P q: ð10Þ

The derivatives of g1 and g2 are similar to the previous section’s, except that one more variable, a, is taken into consideration. For aþ/T c k/k

< q,

T

aþ/ c k/k

aþ/T c k/k

aþ/T c k/k

8 1 < Ab;n @g ; @/

aþ/T c

: 0;

aþ/T c

i

k/k

< q; P q:

< q; P q:

< q;

P q: k/k 8   2 n1  2  > T > < Ab;n q2  aþ/T c ci  ðaþ/ 2cÞ/i ; k/k

k/k

aþ/T c k/k

k/k

> > : 0;

aþ/T c

8 "  T 2 n1  2  > T Ab;n > cÞ/i c > q2  aþ/ ci  ðaþ/ > 2 k/k k/k > k/k > <  i T ¼  ðaþ/ cÞ/j /i /j > þ k/k > 2 g2 ; > cj  k/k2 > > > : 0;

k/k

< q; P q:

aþ/T c k/k T

aþ/ c k/k

aþ/T c k/k

aþ/T c k/k

< q; P q:

< q; P q;

4.2. Single cluster model and numerical algorithm

q

> : a;

k/k

8i – j n1 2

pðrÞ After writing the constant term as Ab;n :¼ 1b ¼ ð1bÞ1Xn qn , the function now becomes

F b ð/; aÞ ¼

: 0;

aþ/T c

8 "  T 2 n1 2 2  > T Ab;n > cÞ/i aþ/ c 2 > q  ci  ðaþ/ > 2 k/k k/k > k/k > @2Fb <   i ¼ /2 > @/2i >  1  k/ki 2 g 2 ; > > > > : 0;

/T ~r 6 /T c  a

¼

¼

ð/ r  aÞdr ðr  cÞ 6 q

/T r 6 a

R

  8 / < Ab;n cg 1 þ k/k g2 ;

8 @F b < 1  Ab;n g 1 ; ¼ @ a : 1;

T

T

!2 1n1 ! 2 c / A  þ ða þ /T cÞ k/k k/k k/k3

a þ /T c

0 !2 1n1 ! 2 a  /T c @ 2 a þ /T c A c / þ ða þ /T cÞ r/ g 2 ¼ q   k/k k/k k/k k/k3

½f ð/; rÞ  aþ pðrÞdr

A crucial feature of Fb is its joint convexity with respect to / and a. A very significant result in their paper is that minimizing CVaRb(/) over the set of possible /’s, say X, is equivalent to minimizing Fb(/,a) over the set (/ ,a) 2 X  R, i.e.

min CVaRb ð/Þ ¼

0

As usual, we would consider the constraint set which includes no shorting selling, meeting (at least) the target return R, and the budget constraint. By (9) and (10), the model of minimizing the b-CVaR becomes

min F b ð/; aÞ s:t: /T e ¼ 1 /T c P R /P0 We will use the interior point algorithm again to solve the problem. As before, we add the slack variables /n+1, s :¼ (s1, . . . , sn)T to the inequalities and obtain:

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M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

/T c  R  /nþ1 ¼ 0

4.3. Multi-clusters model

 / þ s ¼ 0: The Langrangian function is given as T

T

Lðx; a; y; zÞ ¼ F b ð/; aÞ þ y1 ð/ e  1Þ þ y2 ð/ c  R  /nþ1 Þ  zT x

     0  / y1 z where x ¼ 2 Rnþ1 ; y ¼ 2 R2 , and z ¼ ¼ /nþ1 y2 znþ1 00 11 z1 .. C C BB @ B . A C 2 Rnþ1 . The first order KKT conditions are @ A zn znþ1

r/ F b ð/; aÞ þ y1 e þ y2 c  z ¼ 0 ðra F b ð/; aÞ ¼Þ1  Ab;n g 1 ð/; aÞ ¼ 0 ðr/nþ1 Lðx; a; y; xÞ ¼Þ  y2  znþ1 ¼ 0

Parallel to previous sections, we extend this CVaR portfolio selection model for m clusters. With the previous settings, we define

8   ðkÞ ðkÞ  T ðkÞ > a  Ab;n / c þ a g 1 ð/; aÞ > > > <  ðkÞ F b ð/; aÞ :¼ þk/kg 2ðkÞ ð/; aÞ ; > > > > : a; ðkÞ Ab;n ;

xþs¼0 zi si ¼ 0;

i ¼ 1; . . . ; n þ 1

J h ðxÞT

ra ðrx LÞ ry ðrx LÞ 2

ra L

ry ðra LÞ r

021 0ðnþ1Þ1

0ðnþ1Þðnþ1Þ 0ðnþ1Þ1 1 0 rx L B r L C a C B C B C ¼B r L  y C B C B @ ðhðxÞ þ sÞ A

0ðnþ1Þðnþ1Þ

02ðnþ1Þ

0ðnþ1Þ2 0ðnþ1Þðnþ1Þ 0ðnþ1Þ2

10

Dx

S

Z

Ds

where



B S¼B @

1

s1 ..

0

C B C; Z ¼ B A @

.

1

z1 ..

C C; A

.

snþ1 



l¼

sT z ; nþ1

xk F bðkÞ m X

R

xk

r2E ðkÞ

½f ð/; rÞ  aþ pðrÞdr

m X

ðkÞ xk Ab;n

R

r2E ðkÞ

½/T r  aþ dr

k¼1

¼aþ

m X

xk AðkÞ b;n

R a/k/kT cðkÞ  q

 n1 k/kt  /T cðkÞ  a ðq2  t2 Þ 2 dt

k¼1

¼a

h

m X



ðkÞ ðkÞ T ðkÞ xk AðkÞ þ a g 1 ð/; aÞ þ k/kg 2 ð/; aÞ b;n / c

i

k¼1

with the derivatives

  m X @F MC / ðkÞ ðkÞ ðkÞ g2 ¼  xk Ab;n cg 1 þ k/k @/ k¼1 m X @F MC ðkÞ ¼1 xk AðkÞ b;n g 1 @a k¼1

SZe þ rle

0

P q;

m X

¼aþ

1

CB C Da C 02ðnþ1Þ C CB C CB B C 02ðnþ1Þ CB Dy C C CB C Iðnþ1Þðnþ1Þ A@ Dz A

02ðnþ1Þ

2 yL

k/k

k¼1

Writing the no-shorting selling constraints (with the slack variable /n+1) as a vector function h(x) = x, the Jacobian of it will be Jh(x) = I(n+1)(n+1). Hence, the modified Newton step used for interior-point algorithm for the equality conditions becomes

B B rx ðra LÞ B B B rx ðry LÞ B B @ J h ðxÞ

aþ/ c

< q;

ðkÞ g2

1 ¼ a þ 1b

ðz; sÞ P 0

r2x L

T

k¼1

/T c  R  /nþ1 ¼ 0

0

k/k

where and are the same as Ab,n,g1 and g2 respectively except that c and q are replaced by c(k) and q(k) respectively. Then T c our objective function for aþ/ < q is derived as follows. k/k

F MC :¼

/T e  1 ¼ 0

ðkÞ g1

aþ/T c

znþ1

@2 F b

@2 F

 r2a Lðx; a; y; zÞ ¼ @ a2b ,  ry(raL(x,a,y,z)) = 012 = ra(ryL(x,a,y,z))T,

ry ðrx Lðx; a; y; zÞÞ ¼



r2x Lðx; a; y; zÞ ¼





1n1

c

0

1

r2/ F SD ð/Þ 0n1 0n1

and r2y Lðx; a; y; zÞ ¼ 022

@ F MC

20 !2 1n1 2 m X 1 a þ /T cðkÞ A ðkÞ 6@ 2 ¼ xk Ab;n 4 q  k/k k¼1 k/k

1

B @/1 @ a C B . C B . C T . C ra ðrx Lðx; a; y; zÞÞ ¼ B B @2 F C ¼ rx ðra Lðx; a; y; zÞÞ ; B b C @ @/n @ a A 0



2

@/2i

r 2 ð0; 1Þ is arbitrarily chosen, 0

ðkÞ m X @ 2 F MC ðkÞ @g ¼  xk Ab;n 1 @/i @ a @/i k¼1 0 ! !2 1n1 2 m 1 X a þ /T cðkÞ A ða þ /T cÞ/i ðkÞ @ 2 ðkÞ ¼ xk Ab;n q  ci  k/k k¼1 k/k k/k2

0

¼ ðrx ðry Lðx; a; y; zÞÞÞT ; ! ;

ðkÞ ci



 

 !2

a þ /T cðkÞ /i

 1

k/k2

3

!

/2i k/k2

ðkÞ g2 5

20 !2 1n1 2 m T ðkÞ @ 2 F MC 1 X a þ / c ðkÞ 6@ 2 A ¼ xk Ab;n 4 q  @/i @/j k/k k¼1 k/k



ðkÞ ci



ða þ /T cðkÞ Þ/i k/k2

! ðkÞ cj



ða þ /T cðkÞ Þ/j k/k2

! þ

/i /j

ðkÞ g 2 2

k/k

8i – j: We can see that our function is a linear combination of  objective  ðkÞ convex functions F b , which is still convex. Hence we can apply the same techniques as before to solve the model:

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# ;

9

M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

min

F MC ð/Þ

s:t: T

/

m X

xk c

Table 1 Numerical results for the probability model.

/T e ¼ 1 ! ðkÞ

PR

bC

qbC

Portfolio

E½/T r

0.75

0.1389

1.1850

0.80

0.1703

0.85

0.2045

0.90

0.2434

0.95

0.2917

0.6002 0.0000 0.3998 0.6241 0.0000 0.3759 0.6510 0.0000 0.3490 0.6833 0.0000 0.3167 0.7264 0.0000 0.2736

k¼1

/P0 In particular, the numerical procedure will be only differed by the derivatives such that those of Fb is now replaced by those of FMC, and the explicit forms are already shown above. As a reminder, it is worth mentioning that the corresponding b-VaR value (the optimal value of a) comes out as a by-product of the optimization of b-CVaR, but this optimal value a is not equivalent to the result of minimizing ab(/) directly over the same constraint set. However, since b-CVaR(/) P b-VaR(/), solutions to our problem should also be good from the perspective of minimizing ab(/). We will go through the numerical results in the next section.

1.1844

1.1837

1.1829

1.1818

5. Numerical results Table 2 Numerical results for the downside risk model.

5.1. Data set Having derived the three models, we are going to investigate their numerical results. Suppose there are three clustered-scenarios in the next stage. For simplicity, we will assume they are all spheres instead of ellipsoids. Clusters 1, 2 and 3 represent different states of economy. The following table shows the probability of the occurrence of each state, and the expected rates of return for the three stocks in each of the clusters. Clusters (k)

1

2

3

Probability (xk)

1 4

1 2

1 4

Rate of return for A Rate of return for B Rate of return for C

1.6 0.7 0.6

1.3 1.0 1.5

0.5 1.3 1.2

We further set a radius of uncertainties 0.4 for each cluster. In other words, we have.    

c(1) = (1.6, 0.7, 0.6)T; c(2) = (1.3, 1.0, 1.5)T; c(3) = (0.5, 1.3, 1.2)T; q(k) = q = 0.4 " k = 1, 2, 3.

We will use this set of data for the multiple clusters consideration in each of three models.

R

Portfolio

E½/T r

Downside risk

1.00

0.3724 0.4525 0.1751 0.3802 0.4191 0.2007 0.3930 0.3749 0.2320 0.4436 0.1946 0.3619 0.0000 0.0000 1.0000 ⁄⁄⁄

1.1002

0.001967397

1.1067

0.003291075

1.1152

0.005048055

1.1500

0.007361295

1.2000

0.01416250

—

—

1.05

1.10

1.15

1.20

1.25

with infeasible starting points. Given the data set, the only sensitivity measure is the target rate of return R. Table 2 shows our results by varying R. In the table, ‘‘’’ means there is no feasible portfolio. We can see that the downside risk increases with the target rate of return, which meets our intuition. On the other hand, the portfolio return is rather stable and insensitive to R. This may be explained by looking at the expected return of each stock:

Expected rate of the stocks ¼

3 X k¼1

0

1:1750

1

C xðkÞ cðkÞ ¼ B @ 1:0000 A 1:2000

5.2. The probability model In the (SOCPm), a critical part is to determine qn;bC , which depends on the threshold level bC. (Now n = 3 is fixed.) We have used SeDuMi to solve the model. We obtained the following portfolios and expected return with different bC’s (shown in Table 1). In the table, qbC is computed using numerical methods for the equation:

Xn1 Xn qn

Z

q

ðq2  t 2 Þ qb

n1 2

dt ¼ bC

So, we may expect a combination of these stocks (without short selling) would have a return rate ranging from 1 to 1.2. Also by looking at the stock C’s return rate, we can see that the target of 1.2 can only be obtained by investing all your capital into it. This is a very risky move that can be revealed from the big leap of the downside risk from R = 1.15 to R = 1.20 (about 190% riskier!). And we cannot request more, resulting in the infeasibility beyond 1.20 (see the last row).

C

We can see that the model is reasonable in the sense that higher confidence level results in lower expected payoff. 5.3. The downside risk model For the downside risk model, we will use the primal–dual interior-point methods. In particular, we will adapt the algorithm

5.4. The CVaR model Carrying similar mathematical properties with the downside risk model, this model is also implemented by the same primal– dual interior-point algorithm. There is also a threshold parameter b representing the confidence level. Hence the numerical results will be shown in Table 3 according to different values of b.

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M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

For the variance,

Table 3 Numerical results for the CVaR model. 0.85

b Portfolio

E½/T r CVaR VaR

0.90

0.95

0.99

0.3951 0.3686 0.2362 1.1164

0.3762 0.4355 0.1883 1.1035

0.3659 0.4543 0.1798 1.100

0.3572 0.4142 0.2286 1.1082

0.9745 1.1059

0.9289 1.0271

0.8721 0.9384

0.8053 0.8302

Agreeing with the model, the result shows that R is almost an independent factor unless it reaches its the ‘‘boundary’’ value 1.2. Also, the CVaR as well as the associated VaR is decreasing as we increases our confidence level b, which is in line with our common sense that risk is lower when you need more confidence (b). 6. Conclusions After investigating into the numerical results, we may conclude that the validity of the probability model remains a question, while the downside risk and CVaR models are reasonably practical in the sense that the numerical results meet our intuition. However, there is a hinderance for our clustered to develop further to a two-stage tree. The main challenge lies on the recourse problem in the stochastic programming, in which we will suffer from an infinite number of constraints. Still, the projection of second stage’s uncertainties may be reflected by the radius of each cluster. The idea of the cluster tree can be applied to any distribution. An interesting case would be considering the radial distribution in the clusters, since Gaussian distribution also falls into this class. Unfortunately, no neater and simpler form is achieved, which implies we will rely mainly on numerical methods. Appendix A. Calculus for volume of sphere Any n-dimensional ball with radius q is given by

p2n qn 

Vol: of Ball in Rn ¼ n

ðAppendixA:1Þ

C 2þ1

where C() is the Gamma function and q is the radius. n 2 For simplicity, let Xn ¼ C pnþ1 , i.e. Vol. of Ball in Rn = Xnqn ð2 Þ Appendix B. Mean and variance of /Tr in a single-cluster tree Lemma 3. Given the portfolio / and return r 2 BðcÞ, the mean and variance of /Tr are given by

Eð/T rÞ

Varð/T rÞ ¼ Kk/k2 ; Rq

q

t 2 ðq2  t 2 Þ

n1 2

Eð/T rÞ ¼ Xn1qn ¼ Xn1qn ¼ Xn1qn ¼ Xn1qn ¼ XXnn1 qn

BðcÞ

R R R

ð/T r  /T cÞ2 dr

BðOÞ

ð/T rÞ2 dr

ðk/kr1 Þ2 dr Rq 2 2 n1 r ðq  r 21 Þ 2 dr1 q 1

BðOÞ

Xn1 k/k Xn qn

2



¼ Kk/k2 Appendix C. Fractional programming and Dinkelbach’s algorithm Though not of our main concern, we introduce the fractional programming here as we will mention a single-ratio function as one of our objectives in the probability model in the next chapter. Typically, fractional program (Dinkelbach, 1967; Schaible, 1976a; Schaible, 1976b; Barros et al., 1998) is of the form:

ðFPÞ

maxfQ ðxÞ :¼ NðxÞ=DðxÞjx 2 Sg

where S # Rn is nonempty, compact, N; D : S ! R are continuous and D is positive. The problem can be transformed into

maxfNðxÞ  qDðxÞjx 2 Sg

ðAPðqÞÞ

where q 2 R is a parameter. One can see x⁄ is optimal for (FP) if and only if it is optimal for (AP(q⁄)) where q⁄ is the only zero of GðqÞ ¼ maxfNðxÞ  qDðxÞjx 2 Sg; q 2 R. We have q⁄ = Q(x⁄). G(q) is continuous, convex and strictly decreasing on R, G(q) > 0 for q < q⁄ and G(q) < 0 for q > q⁄. If (FP) is a concave-convex fractional program, then (AP(q)) is a convex program for q P 0. Dinkelbach’s Algorithm is a series of steps to solve the (AP(qk)) for subsequent qk. Here outlines the numerical procedure: Step 0 Set q1 = 0 and go to Step 1 with k = 2 or 1Þ Let x1 2 S and q2 ¼ Nðx , go to Step 1 with k = 2 Dðx1 Þ Step 1 By means of any method of concave programming solve the problem:

Fðqk Þ ¼ maxfNðxÞ  qk DðxÞjx 2 Sg and denote any solution point by xk. Step 2 Let d > 0 be any predetermined accuracy. If F(qk) < d, stop and optimal q⁄ = qk and xk is the optimal solution. Otherwise, go to Step 3. kÞ Step 3 Evaluate qkþ1 ¼ Nðx and go to Step 1, replacing qk by qk+1. Dðx Þ k

¼

/T r dr



m X

~ ðkÞ / ~ ðkÞ xk dd/ r/~ ðkÞ F SDj /



k¼1

ð/T r þ /T cÞ dr

BðOÞ

/T r dr þ /T c

BðOÞ

k/kr 1 dr þ /T c

q

¼

R

BðcÞ

r/ F MD ð/Þ

BðOÞ

Rq

¼ Xn1qn

R

dt is a constant.

Proof. Let O be the origin (all zero vector). By definition,

R

¼ Xn1qn

R

Appendix D. First and second derivatives of the downside risk (multiple clusters)

¼ /T c;

where K ¼ XXnn1 qn

Varð/T rÞ ¼ Xn1qn

¼

m X



12



12      T    ~ ðkÞ þ R  / ~ ðkÞ ~ ðkÞ ~cðkÞ g 0 / ~cðkÞ g 1k / 1k

xk Q ðkÞ



~ ðkÞ r/~ ðkÞ F SDj /



k¼1

n1 2

k/kr 1 ðq2  r21 Þ

¼

xk Q ðkÞ

k¼1

dr 1 þ /T c

  ~ ðkÞ ~ ðkÞ  k//~ ðkÞ k g 2k /

¼ /T c The forth line’s integration a rotation on the  is achieved by applying  / U2 . . . Un where U2    Un are sphere, namely, G ¼ k/k / orthogonal to k/k . The last line’s integration is zero since the integrand is an odd function.

m X

¼

m X



xk Q ðkÞ

  ~ ðkÞ ~ ðkÞ kg 0 /  k/ 2k

12      ~ ðkÞ  /~ ðkÞ ~ ðkÞ ~cðkÞ g 1k / ~ ðkÞ k g 2k / k/

k¼1

Or componentwise,

Please cite this article in press as: Wong, M.H. Investment models based on clustered scenario trees. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.11.051

M.H. Wong / European Journal of Operational Research xxx (2013) xxx–xxx

!

m     ~ ðkÞ @F MD ð/Þ X ðkÞ ðkÞ ~ ðkÞ  /i g / ~ ðkÞ : ¼ xk DðkÞ ~ci g 1k / 2k n1 ai ~ ðkÞ

/ @/i k¼1

Then the second derivative will be @F 2MD ð/Þ

¼

@/2i

m X

x

ðkÞ k Dn1

 2 ðkÞ ai

k¼1

¼

~ ðkÞ Þ @F 2SD ð/



"

~ ðkÞ @ / i

!

2

  ðkÞ 2  ~ ~ ðkÞ ÞT ~cðkÞ / Rð/ i ~ciðkÞ þ 2 ~ ðkÞ k / k k¼1 3 !  ðkÞ 2  2 !n1 2   ~ ~ ðkÞ ÞT ~cðkÞ / Rð/ 2 1 ðkÞ 5 i ~ q  þ g / 3  k/~ ðkÞ k k/~ ðkÞ k 2k k/~ ðkÞ k

m X



ðkÞ xk DðkÞ n1 ai

2

1 k/~ ðkÞ k

and for i – j, @F 2MD ð/Þ @/i /j

¼

m X k¼1

¼

m X

ðkÞ ðkÞ ðkÞ xk Dn1 ai aj





~ ðkÞ Þ @F 2SD ð/



~ ðkÞ @ / ~ ðkÞ @/ i j



ðkÞ ðkÞ ðkÞ 1 ~cðkÞ þ xk Dn1 ai aj k/~ ðkÞ k i k¼1





~ ðkÞ ~ ðkÞ ÞT ~cðkÞ / Rð/ i 2 ðkÞ ~ k/ k



  ðkÞ   2 !n1  2 ~ ðkÞ ÞT ~cðkÞ / ~ ~ ðkÞ ÞT ~cðkÞ Rð/ Rð/ j 2 ~cjðkÞ þ q  2 ðkÞ ~ ðkÞ ~ / k k k/ k   ~ ðkÞ / ~ ðkÞ / ~ ðkÞ þ i~ ðkÞ j 3 g 2k / k/ k

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Please cite this article in press as: Wong, M.H. Investment models based on clustered scenario trees. European Journal of Operational Research (2013), http://dx.doi.org/10.1016/j.ejor.2012.11.051