Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection

Transcription

1 Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection Li CHEN Simai HE Shuzhong ZHANG July 7, 200 Abstract In this paper we develop tight bounds on the expected values of several risk measures that are of interest to us. This work is motivated by the robust optimization models arising from portfolio selection problems. Indeed the whole paper is centered around robust portfolio models and solutions. The basic setting is to find a portfolio which maximizes respectively minimizes the expected utility respectively disutility values, in the midst of infinitely many possible ambiguous distributions of the investment returns fitting the given mean and variance estimations. First, we show that the single-stage portfolio selection problem within this framework, whenever the disutility function is in the form of Lower Partial Moments LPM, or Conditional Value at Risk CVaR, or Value-at-Risk VaR, can be solved analytically. The results lead to the solutions for single-stage robust portfolio selection models. Furthermore, the results also lead to a multi-stage Adjustable Robust Optimization ARO solution when the disutility function is the second order LPM. Exploring beyond the confines of convex optimization, we also consider the so-called S-shaped value function, which plays a key role in the prospect theory of Kahneman and Tversky. The non-robust version of the problem is shown to be NP-hard in general. However, we present an efficient procedure for solving the robust counterpart of the same portfolio selection problem. In this particular case, the consideration of the robustness actually helps to reduce the computational complexity. Finally, we consider the situation whereby we have some additional information about the chance that a quadratic function of the random distribution reaches a certain threshold. That information helps to further reduce the ambiguity in the robust model. We show that the robust optimization problem in that case can be solved by means of Semidefinite Programming SDP, if no more than two additional chance inequalities are to be incorporated. Keywords: Portfolio selection, robust optimization, S-shaped function, Chebyshev inequality. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. lchen@se.cuhk.edu.hk Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong. simaihe@cityu.edu.hk Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. zhang@se.cuhk.edu.hk. Research supported by Hong Kong RGC Grant CUHK48406.

2 Introduction The recent financial turmoil certainly has reminded us the role played by risk management. Many investors have learned in a hard way that the stability of the investment return does matter, even at the expenses of some occasional loss of the performance. In Operations Research, the relevant keyword in this context is robust optimization, which has been rapidly developed since the pioneering work of Ben-Tal and Nemirovski [5]. On the downside, a possible price one has to pay for the nice robust solution is to overcome a computational hurdle: the so-called robust counterparts are typically infinite-dimensional and, unless proper care is chosen in the choice of uncertainty set, the problem may be intractable. Therefore, careful modeling plays a decisive role in robust optimization. This paper aims to present a few more successful cases in this difficult terrain. In particular, we are concerned with robust portfolio selection. Recall that the usual portfolio selection is to maximize respectively minimize the expected utility value respectively disutility value, subject to some physical constraints, under all prospects of the investment. A most famous disutility function is arguably the convex quadratic function, because the minimization of its expected value leads to the mean-variance paradigm of Markowitz [33] and Roy [48]. Since the variance does not differentiate the gain from the loss, Markowitz [34] later proposed to use the semivariance instead. To better suit different risk profiles of the investors, Bawa [2] and Fishburn [5] introduced a class of downside risk measure known as the lower partial moment LPM: LPM m r = E[r X + m ], where X is the asset return, and r is the return on a benchmark index such as the risk-free rate of return, and m is a parameter, which can take any non-negative value to model the risk attitude of an investor. Specifically, if m = 0, then LPM 0 is nothing but the probability of the asset return falling below the benchmark index; if m =, then LPM is the expected shortfall of the investment, falling below the benchmark index; if m = 2, then LPM 2 is an analog of the semi-variance, where, however, the deviation is in reference to the benchmark return instead of the mean. Nevertheless, not everyone in the field is convinced that the scheme of maximizing minimizing the expected value of a concave convex utility disutility function is all that matters to investors. Kahneman and Tversky [26], for instance, developed an alternative in 979, known as the prospect theory, in which they hypothesized that usually an investor would have a reference point in mind. Judged by the reference point, a loss of a given magnitude matters more than a gain of the same magnitude. Furthermore, the fact that the perception of change in wealth decreases with its distance from the reference point termed diminishing sensitivity can be modeled by a value function that is concave for gains and convex for losses. Such function is also called the S-shaped value function. Tversky and Kahneman [53] established that if the so-called preference homogeneity holds, then the value function of the prospect theory has the following power form: ux = { x α, x 0, λ x β, x 0, with the loss aversion implying that 0 < α β < and λ. While lower partial moments or 2

3 expected S-shaped utility are used to measure the expected loss or utility for some given distribution, Rockafellar and Uryasev [46] proposed to evaluate the risk of an investment by computing the expected loss in the worst q% of the distribution, known as the conditional value-at-risk CVaR. Speaking of different disutility functions and their expected value under various assumptions on the underlying distribution, there is an interesting recent finding. Our study [0] showed that if the return of the investment follows the so-called Q-radial distribution, then LPM m r will lead to the same efficient portfolio as the standard mean-variance model, and vice versa. The assumptions on the distribution, however, are arguably always subjective. Estimation on the moments of the assets returns using the historical data, on the other hand, may be considered more objective measurements. For this reason, it is natural to rely on the knowledge of the moments estimations, rather than on the assumption of the entire distribution. Estimating probability bounds using the information of a few estimated moments is a common practice; in the univariate case, e.g. the famous probability inequalities such as the Chebyshev inequality using the first and second moments and the Markov inequality using the first moment are exactly of this type. For the multivariate case, Zuluaga and Pena [56] derived a special class of generalized Tchebycheff inequalities by transforming them within the framework of conic programming. Beyond the probability bounds, the moment bounds of piece-wise linear utility functions are also very useful, due to the applications in finance and supply chain management. For the simplest two-piece linear utility function fx = max{0, X z}, Jensen s inequality can be used to derive an upper bound based on the first and second moments. Scarf [49] used the bound in a min-max newsvendor model wherein X denotes the random demand for a product and z denotes the order quantity. Likewise, Lo [30] used the bound on a call option price where X denotes the stock price and z denotes the strike price. Natarajan and Zhou [38] obtained the explicit and tight upper bound for three-piece linear utility function using the mean and variance information. All these bounds are based on a univariate random variable. Popescu [44] proved that the problem of evaluating the worst case expected utility by optimizing over an n-variate distribution can be in fact reduced to optimization over a univariate distribution with some appropriate mean and variance. Based on this observation, Natarajan, Sim and Uichanco [37] showed that the upper bound for a class of piecewise linear concave utility functions can be computed by solving a single compact second order cone program. They also derived the closed form expression for the worst-case CVaR and its generalizations. Historically, the approach of using the moment cone, via the conic duality theory, to obtain various bounds by Semidefinite Programs SDP, was due to Nesterov [39], and was formalized by Bertsimas and Popescu [7], and Ben-Tal and Nemirovski [6]. In some cases, such bounds are even explicitly given. For the latter cases, we mention Popescu [42] for a closed-form bound for LPM 0 r when up to the third order moments are known, and He, Zhang and Zhang [22] for a closed-form and tight upper bound for LPM 0 r when the first, second and fourth moments are known. Since the moment information are sometimes estimated based on the limited historical data, Delage and Ye [] demonstrated that, disregarding the uncertainty in these estimates can lead to taking poor decisions; furthermore, they proposed to use sample data to help derive confidence regions for the mean and covariance matrix to deal with the ambiguous distributions. In order to reduce the conservativeness of the robust model, Ben-Tal, Bertsimas and Brown [3] proposed 3

4 a softened framework for robust optimization that relaxes the standard notion of robustness by allowing the decision-maker to vary the protection level in a smooth way across the uncertainty set. Portfolio selection is the problem of allocating capital across a set of assets in a way that maximizes some measure of performance for a given probability distribution. If the underlying distribution is actually ambiguous then it is natural to consider a robust portfolio selection model. In particular, the ambiguous distribution set may be described by the knowledge of its support set and/or its first few moments. There are a number of recent papers along this line, including Natarajan, Pachamanova, and Sim [36], and Delage and Ye []. On a different front, we remark that robust optimization in multiple stage setting is in general difficult to model. The so-called adjustable robust optimization ARO is one such attempt; see [4, 20]. However, how to incorporate the recourse actions when the time progresses and new information arrives is a hard problem in general. As we mentioned before, this paper is about robust portfolio selection. In particular, the certainty regarding the ambiguous distribution is its mean and covariance matrix, and possibly some additional knowledge concerning the probability of some projections of the distribution. Throughout the paper, we study various robust portfolio selection models that involve risk measures such as the lower partial moments LPM m with m = 0,, 2, the conditional value-at-risk CVaR, the Value-at-Risk, nondecreasing convex-concave S-shaped value function, and concave utility function with convex, concave or concave-convex derivative. Moreover, we study single-stage decision models as well as multi-stage adjusted dynamic robust decision models. Specifically, in Subsection 2. we presented a set of tight bounds for the expected value of the lower partial moments and the CVaR, under the condition that the mean and the covariance of the distribution are given. In the rest of Section 2, the results are used to develop various robust portfolio selection models. In Section 3, we give an adjustable robust optimization ARO solution for the multi-stage portfolio selection problem when the disutility function is the second order lower partial moment. In Section 4, we proceed to consider portfolio selection using the so-called S-shaped value function, which plays an important role in the prospect theory developed by Kahneman and Tversky [26]. Our findings are as follows. First, finding a portfolio that maximizes a given power S-shaped function, when the entire distribution is known for example to be normally distributed, is in general NP-hard. Surprisingly, we show that finding a robust portfolio in this context can be reduced to searching along a single parameter; hence it can be done efficiently. Moreover, we generalize the result to more general value functions, such as monotone convex functions and concave functions with convex, concave or concave-convex derivative. In Section 5, we move on to consider the robust optimization models whereby some additional information regarding the distribution is known. For instance, in addition to the mean and the covariance matrix, we also know that the probability for a quadratic function of the random vector above a certain threshold value is bounded by a known constant. Such robust models can also be solved using SDP. This has some immediate implications. For instance, this implies that if we have some probability estimation of a random variable, then this information can be used to estimate the probability of a correlated random variable. In Section 6, we show numerically the relevance of the results that we have developed in the paper. 4

5 2 Robust Portfolio Selection Based on the Lower Partial Moments We shall start our discussion by considering tight bounds on the probabilities and higher order lower partial moments LPM, using the information about the mean and the covariance of the underlying distribution. Such bounds will naturally lead to robust portfolio optimization models as we shall see later. Let us introduce some notations first. For any given positive semidefinite matrix Γ S n ++ and vector µ R n, we denote D = {π E π [ξ] = µ, Cov π [ξ] = Γ 0 } to stand for the set of probability distributions with mean µ and covariance Γ. And we also denote X µ, Γ to represent the fact that the random vector X belongs to the set whose elements have mean µ and covariance matrix Γ. Since it is a one-to-one correspondence between random variable and probability distribution, we will use the notations π D and X µ, Γ interchangeably in our discussion. 2. The Univariate Cases The case for univariate random variable is a classical one. We shall, however, start with this simple case for completeness. Observe that LPM 0 r measures the probability that a random return falls below the target r. The well-known Chebyshev-Cantelli inequality [9] presents its tight moment upper bound exactly. Lemma 2.. It holds that sup X µ, σ 2 Prob {X r} = sup LPM 0 r = X µ, σ 2 {, +r µ 2 /σ 2 if r < µ,, if r µ. LPM r is the expected shortfall of X below the benchmark r. Its upper bound can be derived by Jensen s inequality and has been widely applied: Scarf [49] first introduced this bound to simplify a min-max newsvendor model; Lo [30] used the result to bound European options. Natarajan and Zhou [38] generalized this result to the expected value of any three-piece linear function. Lemma 2.2. It holds that sup X µ, σ 2 E[r X + ] = r µ + σ 2 + r µ 2. 2 As we will see later, the above measurement of risk is highly related to Conditional Valueat-Risk CVaR. Given the mean and variance, the worst-case CVaR can also be found explicitly; see [38]. LPM 2 r is an analog of the semi-variance, where the reference is to the benchmark return r instead of to the mean. A tight bound on LPM 2 r can be established by using the Jensen inequality. 5

6 Proposition 2.3. It holds that sup X µ, σ 2 E[r X 2 +] = [r µ + ] 2 + σ 2. Proof. For any X µ, σ 2, by Jensen s inequality we have E[r X 2 +] = E[r X 2 ] E[r X 2 ] E[r X 2 ] [Er X] 2 = σ 2 + r µ 2 [r µ ] 2 = σ 2 + [r µ + ] 2. To show the tightness of the bound, consider a sequence of distributions { µ + σ X n = n, with probability n n, µ n σ, with probability n. It is easy to check that X n µ, σ 2, and E[r X n 2 +] σ 2 + [r µ + ] 2, as n. This indicates that the upper bound is indeed tight. Remark that sup X µ, σ 2 E[r X m + ] = + for any m > 2. This is evident by observing E[r X n m + ] where the sequence X n is taken as the one in the proof of Proposition 2.3. Therefore, we shall only be concerned with the lower partial moments problem LPM 0, LPM and LPM 2 in this paper. 2.2 Portfolio Selection with LPM as Risk Measure In this section, we will discuss several robust downside risk models. Let us consider the lower partial moments LPM m r, m = 0,, 2, Conditional Value-at-Risk, and Value-at-Risk as possible risk measures. If the parameters are ambiguous, then the tight upper bounds achieved in last section can be used as the worst-case objective. In fact, for up to two moments, all the models have explicit optimal solutions. We assume that the first two moments are known, and the entire distribution is otherwise completely free. The portfolio models are as follows. Denote the n financial assets as S j, j =,..., n. We assume that the total invested wealth is, i.e., x T e =, where x = x,, x n is the invested wealth on each asset. We denote by ξ j the random return rate of the jth financial asset, j =,..., n. The benchmark return rate is denoted as r > 0. Here the risk measures are the lower partial moments LPM m r. The portfolio selection models P m are formulated as P m min x E[r x T ξ m + ], s.t. x T e =, 6

7 where e stands for the vector of all-ones with an appropriate dimension, m = 0,, or 2 represents the degree of the investor s risk aversion. Let us make no assumption on the distribution ξ. Instead, we assume that we have some knowledge regarding the statistical properties of ξ. In this particular case, we assume that we know the first two moments of ξ. In any case, for a given portfolio x, our risk measure is H m x; ξ = E[r x T ξ m + ], the corresponding robust portfolio selection model may be written as RP m = min x sup ξ µ,γ E[r ξ T x m + ] s.t. x T e = where 0 m 2. The bounds developed in Section 2. become attractive, since these are tight bounds and they can be used to get an explicit expression for the objective function in RP m. Of course, the bounds in Section 2. are good only for univariate distributions, while the ambiguous distribution set D in RP m involves multi-dimensional distributions. For any given ξ, we have all the information about the moments for the distribution ξ T x, once we know the moments of ξ. The opposite is in general not true. Fortunately, if we speak only about the first two moments, then there is actually no loss of any information. To be precise, let us consider two sets A := {a T ξ E[ξ] = µ, Covξ = Γ }, B := {η E[η] = a T µ, Varη = a T Γa }. Evidently, A B. The following lemma asserts that the opposite relationship also holds. Lemma 2.4. For any a 0 R n, it holds that A = B. Proof. We shall need only to show B A. For any fixed a 0, let us take an arbitrary η B. We then construct ξ = C 2 β a T Γa + η at µγa a T + µ, Γa where C := a T ΓaΓ Γaa T Γ, and β N0, I n is independent of η. Now C 0 and a T Ca = 0, and so a T C 2 = 0,, 0. Hence a T ξ = η, E[ξ] = µ, Covξ = Γ. In other words, η = a T ξ A. Therefore A = B. Remark that the same equivalence result was established in Popescu [44] by a different method. In light of the equivalence between the two sets, we have sup ξ µ, Γ E[r ξ T x m + ] = sup ζ x T µ, x T Γx E[r ζ m + ]. Hence, the univariate moments bounds in Section 2. can be applied directly. This makes it possible to derive explicit solutions, which we shall present in the theorem below, albeit for the cases m = 0,, 2 only. Let us first denote c 0 := e T Γ e, c := e T Γ µ, c 2 := µ T Γ µ, b 0 := c 0, b c 0 c 2 c 2 := c, b c 0 c 2 c 2 2 := c 2. c 0 c 2 c 2 7

8 Theorem 2.5. Suppose Γ 0. Consider the optimization problem vrp m := min x sup ζ x T µ, x T Γx E[r ζ m + ] s.t. x T e =. For the cases m = 0,, 2 we have the following explicit solutions: a. If b rb 0, then vrp 0 = x RP 0 = + µ re T Γ µ re, Γ µ re e T Γ µ r e T Γ e. Else if b < rb 0, then vrp 0 = + b 0. b. vrp = b 0r b + b 0 r b 2 + b 0 b 2 b 2 b 0 +, 2 b 0 + x RP = Γ µ Γ b 0 b b0 b +r+ b 0 r b 2 +b 0 b 2 b 2 b 0+ e b b 2 b 0 b 0 +. c. Proof. Let us denote vrp 2 = [b 0r b + ] 2 b 0 b 0 + x RP 2 = Γ µ Γ e +, c 0 b 0 b b b 2 b0 r b + b 0 b b b0. f 0 s, t := inf X s,t 2 E[r X0 +] = f s, t := inf X s,t 2 E[r X +] = r s + + r s 2 /t 2 ; t 2 + r s 2 ; 2 f 2 s, t := inf X s,t 2 E[r X2 +] = [r s + ] 2 + t 2. Now our optimization problem can be expressed by function f m, i.e., vrp m = min x {f m x T µ, x T Γx x T e = } = min s R min x {f ms, x T Γx x T e =, x T µ = s} 2. 8

9 For any given s and m, the optimal solution x s of the inner optimization problem in 2. is a mean-variance efficient solution: x s = arg min x {f m s, x T Γx x T e =, x T µ = s} = arg min{x T Γx x T e =, x T µ = s} x = Γ µ Γ b 0 b s e. b b 2 Note that the second equality comes from the increasing property of f m s, t in t for all m = 0,, 2. Since x s T Γx s = b 0 s 2 2b s + b 2, we have vrp m = min s R f m s, b 0 s 2 2b s + b 2. Finally, by solving the above problem we are led to the solutions: s RP 0 = arg min s R f 0 s, b 0 s 2 2b s + b 2 r s = arg max 2 { s r b 0 s 2 2b s+b 2 b2 b r = b b 0 r if b rb 0 + if b < rb 0 s RP = arg min s R f s, b 0 s 2 2b s + b 2 = arg min s R r s+ b 0 s 2 2b s+b 2 +r s 2 2 = b 0 b +r+ b 0 r b 2 +b 0 b 2 b 2 b 0+ b 0 b 0 +. s RP 2 = arg min s R f 2 s, b 0 s 2 2b s + b 2 = arg min s R [r s + ] 2 + b 0 s 2 2b s + b 2 = b 0r b + b 0 b b b0. Remark 2.6. All the above portfolios are mean-variance efficient, with different locations on the same mean-variance efficient frontier. In particular, s RP 0 s RP s RP 2, which means that for a fixed r, the higher the order of lower partial moments, the more conservative the portfolio. Remark 2.7. All the explicit results heavily depend on the single constraint x T e =, as in the classical mean-variance model. Our approach can be adapted to solve more generally constrained portfolio problems. However, the solutions may no longer be explicit. Remark 2.8. In case the mean and the covariance matrix are also uncertain, if we assume that they belong to their respective uncertainty sets S µ and S Γ, then we can recast the problem as a semidefinite program. In particular, for m = or 2, by noticing f m s, t is decreasing in s and increasing in t, we have min x X = min x X f m = min x, y, s, t max f m µ S µ, Γ S Γ min x T µ, µ S µ x T µ, x T Γx max Γ S Γ x T Γx {y y f m s, t, s min µ S µ x T µ, t 2 max Γ S Γ x T Γx, x X, t 0}. 9

10 It is straightforward to recast the first constraint into several linear or second-order cone constraints by adding one or two auxiliary decision variables. The only question left is if the next two constraints are LMI representable, which depends on the particulars of S µ and S Γ. As a matter of fact, the minimum mean and maximum variance in these two constraints have been well studied by many researchers, e.g. Lobo et al. [3], Nestrov and Nemirivski [40], El Ghaoui and Lebret [2], Goldfarb and Iyengar [6]. In each of the above cases, the constraints in question can be converted into a second-order cone constraint see Section 3. of [6] for a summary. If the uncertain mean and covariance matrix are coupled and belong to a confidence region S µ,γ = {µ, Γ µ µ 0 T Γ 0 µ µ 0 γ, Γ + µ µ 0 µ µ 0 T γ 2 Γ 0 } see Delage and Ye [], the robust problem min x X is equivalent to the following dual problem min min Q, q, s, t s + t x X max E[r x T ξ m + ] ξ µ,γ S µ,γ s.t. s [r x T ξ m + ] ξ T Qξ ξ T q, ξ R n t γ 2 Γ 0 + µ 0 µ T 0 Q + µ T 0 q + γ Γ /2 0 q + 2Qµ 0, Q 0. See Lemma of []. By the S-lemma, it is easy to verify that when m = the first constraint of the dual problem can be cast by LMI s, while for m = 0 or 2, this is not the case. Therefore, only for m =, the robust counterpart can be formulated as an SDP. Our model in Theorem 2.5b is similar to Delage and Ye []; we assumed the parameters to be S = R n, γ = 0 and γ 2 =. Because of this, rather than a numerical scheme Lemma of [], we provide an explicit optimal solution for vrp. 2.3 Conditional Value-at-Risk as the Risk Measure Conditional Value-at-Risk CVaR, defined as the mean of the tail distribution exceeding VaR, has attracted much attention in recent years. As a measure of risk, CVaR exhibits far better computational properties than VaR. With the help of the expression of LPM r, for linear loss functions, our robust version CVaR problem can be solved explicitly. As is common in the CVaR analysis, let fx, ξ denote the loss function associated with decision vector x K R n and random vector ξ R n. We assume that the cumulative probability distribution function for ξ is π. We also assume E[ fx, ξ ] < + for each x K. Given a decision x K, the probability of fx, ξ not exceeding a threshold α is given by Ψx, α dπξ. fx, ξ α 0

11 For a given confidence level β usually greater than 0.9 and a fixed x K, the value-at-risk is defined as VaR β x min{α R Ψx, α β}. The corresponding conditional value-at-risk, denoted by CVaR β x, is defined as the expected value of loss that exceeds VaR β x; that is, CVaR β x fx, ξdπξ. β fx,ξ VaR β x Rockafellar and Uryasev [46, 47] demonstrate that the calculation of CVaR can be done by minimizing the following auxiliary function with respect to the variable α R: F β x, α α + [fx, ξ α] + dπξ, 2.2 β ξ R n and subsequently CVaR β x = min α R F βx, α. Recall that in Section 2.2, we assumed the distribution π is uncertain: π D = {π E π [ξ] = µ, Cov π ξ = Γ 0}. For fixed x K, the robust optimization counterpart of the portfolio selection problems with respect to the ambiguity set D using CVaR or VaR as the risk measure, are formulated by and RCVaR β x sup CVaR β x, 2.3 π D RVaR β x sup VaR β x. 2.4 π D Also with the simple constraint x T e =, we find that the robust portfolio problem with CVaR or VaR as risk measure could be solved explicitly. Theorem 2.9. Suppose the loss function is fx, ξ = x T ξ and the random vector ξ has mean µ and covariance matrix Γ 0. Let β 0.5, ]. Consider vrc β = min x sup π D CVaR β x s.t. x T e =, and vrv β = min x sup π D VaR β x s.t. x T e =.

12 Then the solution of worst-case CVaR is: vrc β =, b0 b 2 b 2 βb0 β b 0 b b0, β if β b 0, β if β b 0 <, and when β β b 0, with optimal solution x RC β = Γ µ Γ e b 0 b b b 2 b0 b 2 b 2 + b βb0 b 0 β b0. The solution of worst-case VaR is: RV β =, b0 b 2 b 2 b0 4β β b 0 b 0 b b0, b 0 if 4β β + b 0, b if 0 4β β + b 0. When b 0 4β β + b 0, the optimal solution is x RV β = Γ µ Γ e b 0 b b b 2 b0 b 2 b 2 + b b0 b 0 4β β b b0 0. The proof of the theorem can be found in Appendix A. Remark 2.0. It is interesting to note that when β b b 0 < and 0 4β β + b 0 then the expected downfall measured by CVaR and VaR are unbounded. In all other cases, the optimal portfolios are mean-variance efficient. Remark 2.. In Natarajan, Sim and Uichanco [37], a closed form for the worst-case Conditional Value-at-Risk and its coherent generalization are derived. In Zhu and Fukushima [55], the worstcase CVaR under mixture distribution uncertainty, box uncertainty and ellipsoidal uncertainty is minimized. In El Ghaoui, Oks and Oustry [3], the robust optimization model using worst-case VaR model is investigated. All those results have been applied to robust portfolio optimization and the corresponding problems have been transformed as conic programs see Problem 2.8 in [37], Section 3.2 in [55], and Theorem in [3]. β 3 Multi-Stage Portfolio Selection with Robust LPM 2 It is natural to extend the analysis to the multiple-stage setting. In the spirit of adjustable robust optimization ARO of Ben-Tal et al. [4], we present in this section a tractable ARO model using the LPM 2 r risk measure. 2

13 Let us consider an K-stage portfolio selection model, and let us denote y k to be the k-th stage recourse portfolio decision, k =,..., K. The objective of the investor is to minimize the terminal LPM 2 risk, i.e., E[r y T K ξ K 2 +]. Let ξ k be the rate of return vector for the k-th stage. In this section, we denote the rate of return of an asset to be the ratio of its final value to its initial value. This convention helps simplify the formulas. With regard to the random vectors ξ k, we only know their first and second moment estimation: µ k, Γ k. As ξ k unfolds, we will need to make the k +- th stage recourse decision y k+, before the exact status of ξ k+ is revealed, where k =, 2,..., K. When we select y, we do not know the actual rate of return ξ, and when we select the portfolio y k k 2, ξ,..., ξ k are known but ξ k is unknown, and so y k is an adjustable variable depending on the uncertain data ξ,, ξ k. Our model is thus an adjustable robust optimization model. For general problems, the ARO formulations often lead to intractable optimization problems; see [4]. Below we shall elaborate on the following particular ARO formulation of the multi-stage robust portfolio selection model. Mathematically, the problem is formulated as: V K ω K = min yk sup ξk µ K, Γ K E[r yk T ξ K 2 +] s.t. yk T e = ω K ; V K ω K 2 = min yk sup ξk µ K, Γ K E[V K yk T ξ K ] s.t. yk T e = ω K 2. V = min y sup ξ µ, Γ E[V 2 y Tξ ] s.t. y T e =. Here we denote V k+ ω k as the optimal objective value at stage k +, given the wealth ω k = y T k ξ k as the result of investment at stage k, k =,..., K. We shall present the explicit solutions and values for each stage in the theorem below. Theorem 3.. Suppose that the first and second moments of ξ k are µ k, Γ k, k =,..., K, and furthermore, for k =,..., K, denote c k 0 := et Γ k e, ck := et Γ k µ k, c k 2 := µt k Γ k µ k, b k 0 := c k 0 q k = c k 0 ck 2 ck b k+ b k+ 0 b k , b k := c k, r k = bk+ c k 0 ck 2 ck 0 r k+ b k+ where we assume r K = r, q K =. For k =,..., K, it holds that 2, b k 2 := c k 2 c k 0 ck 2 ck 2,, Γk = Γ k + Γ k+µ T k µ k, c k+ 0 qk 2 V k yk T ξ k = q K q k 2 [r k yk T ξ k 2 ] + + qk q k 2 yt k ξ k 2 ; c k 0 yk = Γ k µ k Γ k e b k 0 b k b k 0 r k b k yt k ξ k + b k b k b k 0 bk bk yt k ξ k b k

14 Proof. Recall that and so sup E[r η 2 +] = r u σ 2, r R η u,σ min x sup ξ µ,γ E[r ξ T x 2 +] s.t. x T e = ω = min x r x T µ x T Γx s.t. x T e = ω = min z r ω z T µ ω 2 z T Γz s.t. z T e = = ω 2 min z r/ω z T µ z T Γz s.t. z T e =. The above problem can be explicitly solved, with the optimal value and solution being v = [b 0r b ω + ] 2 b 0 b 0 + x = Γ µ Γ e + ω2 c 0, b 0 b b b 2 b0 r b ω + b 0 b b ω b 0, where we use the same notations as in Theorem 2.5; in particular, c 0 := e T Γ e, c := e T Γ µ, c 2 := µ T Γ µ, b 0 := c 0, b c 0 c 2 c 2 := c, b c 0 c 2 c 2 2 := c 2. c 0 c 2 c 2 With these relations, we can prove the theorem inductively in a reversed fashion. For the final stage, V K yk T ξ K = min yk sup ξk µ K, Γ K E[r yk T ξ K 2 +] s.t. yk T e = yt K ξ K = min yk [r yk T µ K 2 +] + yk T Γ Ky K s.t. yk T e = yt K ξ K = [bk 0 r K b K yt K ξ K + ] 2 b K 0 bk yt K ξ K 2 c K 0 = qk 2 [r K yk T ξ K + ] 2 + yt K ξ K 2 and y K = Γ K µ K Γ K e b K 0 b K b K b K 2 bk 0 r K b K yt K ξ K + b K 0 bk 0 + c K 0 + bk yt K ξ K b K 0. Assume the formula for the optimal value and solution hold true for the k + th stage, then for the kth stage, we have 4

15 V k yk T ξ k = min yk sup ξk µ k, Γ k E[V k+ yk Tξ k] s.t. yk Te = yt k ξ k { } = min yk sup ξk µ k, Γ k E q K q k 2 [r k yk Tξ k + ] 2 + q K q k+ 2 yt k ξ k 2 s.t. yk Te = yt k ξ k = min yk q K q k 2 {[r k yk Tµ k 2 +] + yk T Γ k y k } s.t. yk Te = yt k { ξ k } = q K q k 2 [b k 0 r k b k yt k ξ k + ] 2 b k 0 bk yt k ξ k 2 = q K q k q k 2 [r k yk T ξ k + ] 2 + q K q k 2 yt k ξ k 2, c k 0 c k 0 c k+ 0 and y k = Γ k µ k Γ k e b k 0 b k b k b k 2 b k 0 r k b k yt k ξ k + b k 0 bk bk yt k ξ k b k 0. The theorem is thus proven by induction. It is crucial that the robust form of the LPM 2 r holds the similar structure as its non-robust one. Therefore it is not clear how to extend the result to other risk measures. In case the objective is simply the variance, Li and Ng [29] derived an interesting analytical solution to the multi-stage portfolio selection problem. Related to our discussions in this section, Hernández-Hernández and Schied [23] proposed a stochastic control approach to the dynamic maximization of robust utility functionals and obtained an explicit PDE characterization of the optimal strategy in an incomplete diffusion market model where the robust utility functional is defined in terms of a logarithmic utility function and a rather general dynamically consistent penalty function γ. They characterized the value function and the optimal investment strategy via the solution of a quasi-linear Hamilton-Jacobi-Bellman PDE. Iyengar [25], and Nilim and El Ghaoui [4] studied robust dynamic programming algorithms for finite-state and finite-action Markovian decision processes with uncertain transition probabilities. They proved that both finite and infinite planning horizon problems can be extended to natural robust counterpart when the measures have a certain rectangularity property. 4 Robust Portfolio Selection using the S-Shaped Value Function As an alternative to the expected utility maximization, Kahneman and Tversky [26], 979 developed the so-called prospect theory to count for psychological factors in economical decision making. This work eventually won Nobel prize in economics in One aspect of the prospect theory is to promote a value function that is S-shaped, in place of an overall concave utility function. The rationale behind such consideration is that, typically a decision maker has a reference point 5

16 in mind, and he/she would exhibit diminishing sensitivity in view of the gain and loss from the reference point. In other words, the value function is concave in the domain of the gains, and is convex in the domain of the losses. Moreover, the loss is more acutely felt than the gain near the reference point. Mathematically, an S-shaped value function is given as: { x α, x > 0 vx = λ x β, x 0, with 0 < α β < and λ. Under the new tenet, an investor is interested in solving the following optimization model: max E[vx T ξ r] s.t. x X, where ξ is the return of the assets, r is the reference point, and x is the portfolio to be selected, and X a convex set represents the constraint that the investor would like to impose on the portfolio. It is perhaps not surprising that portfolio selection based on the S-shaped value function is difficult in general. This fact is formalized in the next theorem. Theorem 4.. It is NP-hard to solve max E[vξ T x] s.t. e T x =, x i, i =,..., n, where vx = { x α, x > 0 λ x β, x 0, and ξ Nµ, Γ. Proof. Let µ = 0 and σx = x T Γx. Then, E[vξ T x] = 2πσx 0 t α e t 2 2σx 2 dt λ 2πσx 0 t β e t 2 2σx 2 dt = σx α 2 E η N0, η α σx β λ 2 E η N0, η β = σx α c σx β c 2, where c := 2 E η N0, η α > 0 and c 2 := λ 2 E η N0, η β > 0. Since c t α c 2 t β = t α αc βc 2 t β α > 0 when t is small note β > α, the quantity E[vξ T x] is monotonically increasing in σx. Therefore, max E[vξ T x] s.t. e T x =, x i, i =,..., n 6

17 is equivalent to MC max x T Γx s.t. e T x =, x i, i =,..., n, if Γ is properly scaled to be sufficiently small. Consider now the problem of finding the max-cut for a weighted graph of n = 2m + nodes, with m nodes on one side and m + nodes on the other side of the cut. Let Γ 0 be the Laplacian matrix of the graph. The objective function of MC is convex, and so its optimal solution is attained at a vertex. Hence, the m, m + max-cut problem can be cast as MC. Since the max-cut problem is NP-hard, it follows that the portfolio selection problem based on the S-shaped value function is NP-hard in general. It is quite unexpected, however, that the robust counter-part of the optimization model turns out to be easy. The robust optimization model in question is: max inf ξ µ,γ E[vx T ξ r] s.t. x X. To see this we need a few intermediate steps. Let us first introduce a function v R µ, σ := inf E[vη]. η µ, σ 2 Lemma 4.2. For any fixed µ, the function v R µ, σ is monotonically non-increasing in σ > 0. Proof. Consider 0 < σ < σ. For any ϵ > 0 we have a distribution η, satisfying E[ η] = µ, Var η = σ 2, and E[v η] v R µ, σ + ϵ. For any fixed n > 0, let us consider a new distribution as follows { η, with probability n2, η n = n 2 n σ 2 σ 2, with probability. n 2 Clearly, as n we have This shows that v R µ, σ v R µ, σ. E[η n ] = µ n2 n σ 2 σ 2 µ, n 2 n 2 Varη n = σ 2 µ 2 + σ 2 /n 2 σ 2, E[vη n ] = n2 E[v η] + v n σ 2 σ 2 E[v η]. n 2 n 2 Next, we shall exactly compute the value v R µ, σ, and its associated optimal solution. Theorem 4.3. It holds that v R µ, σ = s2 s 2 + µ + σ/sα λ s 2 + σs µβ 4.5 and an optimal solution is given by { µ σs, with probability η = s 2 + ; µ + σ/s, with probability s2 s 2 +, 7

18 where s is a root for the following function gx := 2 ασ/x + µ α + λ2 βσx µ β ασx µσ/x + µ α 4.6 λβσ/x + µσx µ β, where x max{0, µ/σ}, µ + µ 2 + σ 2 σ ]. We shall delegate the proof of Theorem 4.3 to Appendix B. The existence of the root can be verified by checking the boundary values of g, which are lim x max{0, µ/σ} gx = and g > 0. Due to the continuity of function g, one may search the root by e.g. the µ+ µ 2 +σ 2 σ golden-section type line-search method see Chapter 8 of [32]. By Lemma 2.4, the robust portfolio selection problem can be reduced to a single-parameter searching problem along the mean-variance efficient frontier: where r is the reference return level and max x X min ξ µ,γ E[vx T ξ r] = max x X v R x T µ r, x T Γx = max t R max x X v R t r, x T Γx s.t. x T µ = t = max t R v R t r, σ t, [σ t] 2 = min x T µ=t, x X x T Γx. 4.7 The last step in the derivation is due to the monotonicity as established in Lemma 4.2. Equivalently, we may pose 4.7 as σ t = Γ /2 x. min x T µ=t, x X If X is convex, for any t, t 2 R and λ [0, ], we have σ λt + λt 2 λσ t + λσ t 2, because of the triangle inequality Γ 2 λ x t + λ x t 2 λ Γ 2 x t + λ Γ 2 x t 2. Therefore, σ t is in general an increasing and convex function if X is convex. On the premise that σ t is easy to compute, the robust portfolio selection model based on the S-shaped value may be considered easy to solve, since it reduces to searching a single parameter value. For all practical purposes, one can always sample the value of v R t r, σ t for various t s, and then select the highest. Theoretically however, the search can be made efficient, in terms of the polynomial-time computational complexity, if v R t r, σ t is unimodal or quasi-concave. Proposition 4.4. a. If σ t + r t [σ t + r] 0 t, then v R t, σ t + r is quasi-concave in t. 8

19 b. Suppose that x 0, r 0 is the optimal solution of min x T µ=t, x X x T Γx, where x, t is the joint decision vector in the above model. Suppose that r r 0 and that the feasible region X is an affine subspace does not contain the origin. Then we have σ t + r t[σ t + r] for all t. Concerning the condition required by Proposition 4.4, the following proposition gives one such example. In fact, we conjecture that v R t, σ t + r is always quasi-concave as long as X is a convex set. Proposition 4.4 may possibly need be further strengthened in that case. We shall point out here that the S-shaped value function is a two-piece power function, hence outside the realm of polynomial functions. This property is interesting. As a matter of fact, if the feasible set of the dual problem is semi-algebraic, i.e. a set given by polynomial inequalities, then Popescu [43] already showed that computing optimal moment bounds boils down to SDP. Popescu [43] also pointed out that if the objective function is piecewise polynomial, then the same SDP formulation is possible. In fact, almost all the known moment bounds focus on piecewise polynomials; see [7, 8, 36, 38, 42, 43]. Other from these moment bounds on polynomials, Popescu [44] also studied moment bounds for general non-polynomial utility function with the given mean and variance information, namely, max x X min ξ µ,γ E[uxT ξ] which is what we study in this section. Popescu proved that if the inner optimization problem is quasi-concave and satisfies a certain monotonicity condition, then the robust portfolio selection problem can be reduced to a parametric quadratic program. In fact, Popescu [44] elaborated on several specific utility classes in Propositions 5, 7, 8 and 0, where min ξ µ,γ E[ux T ξ] indeed satisfies the quasi-concavity. However, the S-shaped function that we investigate here is not included in the classes specified in [44]. Instead of assuming the inner optimization problem to be quasi-concave, we prove the quasi-concavity for our S-shaped value function, and propose a single-parameter searching algorithm to solve its robust portfolio selection problem. Similar to the algorithm of Popescu in [44], our algorithm also requires to solve a parametric quadratic program our parameter is the expected return of assets rather than a general weight parameter. It is worth mentioning that our method, as well as Popescu s [44] algorithms, require that the utility function must have at most two tangent points with any quadratic function from below. Furthermore, Popescu [44] provided necessary and sufficient conditions for this two-point tangency property to hold, and a necessary condition for the one-point tangency property to hold see Lemma and Proposition 6 of [44]. In the context of Popescu s characterizations, it is also possible to get easy-computable bounds even if the objective function is more general than a two-piece power function. It only requires that the objective function has at most two tangent points with the quadratic function. Its proof is almost identical to that of Theorem 4.3. As for the tangent points 9

20 requirement, one may regard Lemma and Proposition 6 of [44] as necessary and/or sufficient conditions. Popescu provided several concrete examples for the conditions to hold; however, they are all monotone convex/concave functions with convex, concave or concave-convex derivative. Note that for any fixed µ, g R µ, σ is monotonically non-increasing in σ > 0, the proof being exactly the same as for Lemma 4.2. With this understanding, we solve Problem GP as follows:. Get σ t by solving [σ t] 2 = min x T µ=t,x X x T Γx. 2. Solve max t R g R t, σ t. However, the quasi-concavity of g R t, σ t is not known in general, although we believe it is true. 5 Robust Portfolio Selection with Chance Information In this paper we are concerned with robust optimization under distributional uncertainties. So far, the informational structure has been the knowledge of the mean and the covariance of the underlying distribution. However, it is in general always possible to obtain more information regarding the distribution, e.g. we may estimate the chance of a projection of the random vector above a certain threshold via some statistical methods. It turns out that in some cases, this additional information can result in a sharpened robust optimization formulation, which can be solved by SDP. We shall present three such cases in this section to showcase the potential of the technique. 5. Additional Chance Constraints Let us consider robust portfolio selection as in Subsection 2.2, with u being a piecewise linear utility function whose expected value is to be maximized. Now, the additional information with regard to ξ is that Prob {ξ T A ξ + a T ξ > r} β is known to hold, where A is a certain n-dimensional symmetric matrix, and a is an n-dimensional vector. Together with the knowledge about the first two moments of ξ, the robust portfolio selection model becomes Assumption 5.. max x X min ξ E[ux T ξ] s.t. ξ µ, Γ and Prob {ξ T A ξ + a T ξ > r} β. a. There exists ξ R n such that ξ T A ξ + a T ξ < r. b. Γ 0 and β 0,. 20

21 Following the moments cone approach developed by Popescu [42, 43, 44], we obtain various bounds by SDP via the conic duality theory. From Assumption 5.b, the strong duality holds. So, the above problem can be recast as max x X max Z,z0,z,z 3 z 0 + z Tµ + Z Γ + µµt βz 3 s.t. z 0 + z Tξ + ξt Zξ z 3 {ξ T A ξ+a T ξ>r} + ux T ξ, ξ R n z 3 0. Suppose that uy := min{c + by, 0} with b, c R. The above problem can be further written as max x X max Z,z0,z,z 3 z 0 + z Tµ + Z Γ + µµt βz 3 s.t. z 0 + z Tξ + ξt Zξ z 3 + c + bx T ξ, ξ R n z 0 + z Tξ + ξt Zξ z 3, ξ R n z 0 + z Tξ + ξt Zξ c + bx T ξ, ξ T A ξ + a T ξ r z 0 + z Tξ + ξt Zξ 0, ξ T A ξ + a T ξ r z 3 0. Using the S-lemma under Assumption 5.a see [6], the constraints can be written as Linear Matrix Inequalities LMI as shown below: max x X max Z,z0,z,z 3,s,t z 0 + z Tµ + Z Γ + µµt βz 3 s.t. z 0 z 3 c z T bxt /2 z bx/2 Z 0 z 0 z 3 z T/2 z /2 Z 0 z 0 c + tr z T bxt ta T /2 z bx ta/2 Z ta z 0 + sr z T sat /2 z sa/2 Z sa 0 s, t, z Therefore, the robust portfolio selection problem can be solved by SDP using e.g. SeDuMi of Sturm [5], or via CVX of Grant, Boyd, and Ye [7, 8] which has a friendly interface. It is possible to extend the method to include one more chance information in the robust optimization formulation, based on the extended S-lemma of Sturm and Zhang [52]. The problem in question is: max x X min ξ µ,γ E[ux T ξ] s.t. Prob {ξ T A ξ + a T ξ > r } β, Prob {d T ξ > r 2 } β 2. To apply the results in [52], we need the following technical assumptions: Assumption 5.2. a. A 0 and there exists ξ R n such that ξ T A ξ + a T ξ < r and d T ξ < r2. 2

22 b. Γ 0 and β, β 2 0,. Similar as before, its dual formulation is max x X max Z,z0,z,z 3,z 4 z 0 + z Tµ + Z Γ + µµt β z 3 β 2 z 4 s.t. z 0 + z Tξ + ξt Zξ z 3 {ξ T A ξ+a T ξ>r } + z 4 {d T ξ>r2 } + uxt ξ, ξ R n which can be further written as z 3, z 4 0, max x X max Z,z0,z,z 3,z 4 z 0 + z Tµ + Z Γ + µµt β z 3 β 2 z 4 s.t. z 0 + z Tξ + ξt Zξ z 3 + z 4 + c + bx T ξ, ξ R n z 0 + z Tξ + ξt Zξ z 3 + c + bx T ξ, d T ξ r 2 z 0 + z Tξ + ξt Zξ z 3 + z 4, ξ R n z 0 + z Tξ + ξt Zξ z 3, d T ξ r 2 z 0 + z Tξ + ξt Zξ c + bx T ξ + z 4, ξ T A ξ + a T ξ r z 0 + z Tξ + ξt Zξ c + bx T ξ, ξ T A ξ + a T ξ r, d T ξ r 2 z 0 + z Tξ + ξt Zξ z 4, ξ T A ξ + a T ξ r z 0 + z Tξ + ξt Zξ 0, ξ T A ξ + a T ξ r, d T ξ r 2 z 3, z 4 0. By Corollary 7 in [52], under the Assumption 5.2a we have the following equivalent conic formu- 22

23 lation: max z 0 + z Tµ + Z Γ + µµt β z 3 β 2 z 4 z 0 z 3 z 4 c z T s.t. bxt /2 0, z bx/2 Z z 0 z 3 c + t r 2 z T bxt t d T /2 0, z bx t d/2 Z z 0 z 3 z 4 z T/2 0, z /2 Z z 0 z 3 + t 2 r 2 z T t 2d T /2 0, z t 2 d/2 Z z 0 z 4 c + t 3 r z T t 3a T bx T /2 0, z t 3 a bx/2 Z t 3 A z z 0 c + t 4 r + 2s r T bxt t 4 a T r 2 v T s d T 0, z bx t 4 a 2 + r 2 v s d Z t 4 A vd T dv T z 0 z 4 + t 5 r z T t 5a T /2 0, z t 5 a/2 Z t 5 A z z 0 + t 6 r + 2s 2 r T t 6a T r 2 w T s 2 d T 0, z t 6 a 2 + r 2 w s 2 d Z t 6 A wd T dw T s r 2 + r 2 0 = v a a 2R T y, s 2 w = r 2 + r 2 0 a a 2R T y, y 2 SOC2 + ranka, z 3, z 4, t,, t 6 0, where the decision variables are: x, Z, z 0, z, z 3, z 4, v, w, y, y 2, s, s 2, t,, t 6, R is a ranka n matrix satisfying A = R T R, and SOC denotes the second-order cone with { } SOCm := x = x,, x m T R m x m x x2 m. This model fits perfectly within the domain of SeDuMi [5] of Sturm. y 2, 5.2 Extending Lo s Option Bound An immediate application of the results in Section 5. is to extend the option bound due to Lo [30], which yields a closed-form upper bound on the expected payoff of a European call option when the first two moments i.e. the mean and variance of the underlying asset price at maturity is known. In practice, it is possible to get more information about the distribution of the underlying asset. For instance, one may estimate the statistics of a correlated asset, in the hope that this information will help to sharpen the bound as given in Lo [30]. In this subsection, we consider such bound when a probability bound of another correlated asset is available. Specifically, we consider two different 23

24 assets, s and s 2, whose mean vector µ R 2 and covariance matrix Γ a 2 by 2 positive semidefinite matrix are given. Suppose that we have an estimation on the probability of stock s 2 above some reference point r. The problem is to get the tightest possible expected value of a European call option on stock s. Mathematically, the problem under consideration is: max s,s 2 µ,γ E[s k + ] s.t. Prob {s 2 > r} β. To avoid trivial cases, using the Chebyshev-Cantelli bound as shown in Proposition 2., we shall consider the parameters to satisfy: [ ] Γ 0, 22 r µ 2 2 +Γ 22, and r µ 2, β [ ] r µ 2 2 r µ 2 2 +Γ 22,, and r µ 2. Similar as before, the dual is given by min Z,z0,z,z 3 z 0 + z Tµ + Z Γ + µµt + βz 3 T s.t. z 0 + z T s s s + Z s 2 s 2 s 2 z 3 0, which, by the S-lemma, can be cast as an SDP: + z 3 {s2 >r} s k +, s, s 2 R 2 min Z,z0,z,z 3,w,t z 0 + z Tµ + Z Γ + µµt + βz 3 z 0 + z 3 + k 0.5 z T, 0 s.t. 0.5 z, 0 T 0, Z z 0 + z z T 0, 0.5 z Z z 0 + k wr 0.5 z T +, w 0.5 z +, w T 0, Z z 0 tr 0.5 z T + 0, t 0.5 z + 0, t T 0, Z z 3, w, t 0. The above bound is computable using an SDP solver, and is tighter than the corresponding bound of Lo which uses only the first two moments of the underlying asset s. The numerical performance of the method will be presented at Section Extending Chebyshev s Probability Bound The same approach can be used to improve the bound on probability if some other probability bound of a related random variable is known. A straightforward application is to strengthen the 24

25 Chebyshev type inequality, using the first two moments. In particular, let us use the same setting as in Subsection 5.2, and find the tightest upper and lower bounds of the probability Prob {s k} with the information that s, s 2 µ, Γ, and that Prob s 2 > r β, where β [ 0, ] Γ 22 r µ 2 2 +Γ 22, and r µ 2, [ ] r µ 2 2 r µ 2 2 +Γ 22,, and r µ 2. Consider the upper bounding problem first, which is now cast as with its dual problem: max s,s 2 µ,γ Prob {s k} s.t. Prob {s 2 > r} β, min Z,z0,z,z 3 z 0 + z Tµ + Z Γ + µµt + βz 3 T s.t. z 0 + z T s s s + Z s 2 s 2 s 2 z 3 0. Writing out the first constraint explicitly, we have min Z,z0,z,z 3 z 0 + z Tµ + Z Γ + µµt + βz 3 T s.t. z 0 + z T s s s + Z s 2 s 2 s 2 z 0 + z T z 0 + z T z 0 + z T z 3 0. s s 2 s s 2 s s s s 2 s s 2 s s 2 T Z T Z T Z Using the S-lemma and the extended S-lemma [52], we have + z 3 {s2 >r} {s k}, s, s 2 R 2 s s 2 s s 2 s s 2 + z 3, s k, + z 3 0, s, s 2 R 2,, s k, s 2 r, 0, s 2 r, min Z,z0,z,z 3,w,t,p,q z 0 + z Tµ + Z Γ + µµt + βz 3 s.t. 0, 0.5 z Z z 0 + z 3 + wk 0.5 z T w, z w, 0 T Z z 0 + z z T 0, z 0 + p k q r z T + p, q 2 0, z + p, q T 2 Z z 0 tr 0.5 z T + 0, t 0.5 z + 0, t T 0, Z z 3, w, t, p, q 0. 25