A Robust Portfolio Technique for Mitigating the Fragility of CVaR Minimization

Transcription

1 A Robust Portfolio Technique for Mitigating the Fragility of CVaR Minimization Jun-ya Gotoh Department of Industrial and Systems Engineering Chuo University Kasuga, Bunkyo-ku, Tokyo 2-855, Japan Keita Shinozaki Department of Industrial and Systems Engineering Chuo University Kasuga, Bunkyo-ku, Tokyo 2-855, Japan k Abstract The conditional value-at-risk (CVaR) has gained growing popularity in financial risk management due to the coherence property and tractability in its optimization. However, optimal solutions to the CVaR minimization are highly susceptible to estimation error of the risk measure because the estimate depends on only a small portion of sampled scenarios. In this paper, by employing a robust optimization modeling for minimizing coherent risk measures, we present a simple and practical way for making the solution robust over a certain range of estimation errors. More specifically, we show that a worst-case coherent risk measure minimization leads to a penalized minimization of the empirical risk estimate. With an uncertainty set based on the factor model, which is often employed in practice for estimating the asset return distribution, the minimization becomes a tractable convex optimization problem, in which the penalty term can be represented as a standard deviation of portfolio residuals. Numerical results show that the robust CVaR minimization equipped with a parameter tuning technique achieves dominantly better performance than the existing CVaR minimizations. Keywords: robust portfolio; CVaR (conditional value-at-risk); coherent risk measure; factor model; regularization Introduction In the history of risk measurement for financial risk management, the conditional value-at-risk (CVaR) was originally introduced as an alternative risk measure to the value-at-risk (VaR), which had been used in practice for capturing a risk of large loss with small probability. While VaR has been criticized for its theoretical deficiency, CVaR is known to have nice properties such as the coherence (Artzner et al. 999) and the consistency with the second-order stochastic dominance (Ogryczak and Ruszczýnski 2002). Also, Rockafellar and Uryasev (2000) show that the minimization of CVaR results in a tractable optimization problem. For example, when the loss is defined as the minus return and a finite number of historical observations of returns are used in estimating the objective function, the CVaR minimization can be written as a linear program (LP) and solved efficiently. Due to these preferable properties, it has obtained growing popularity in practice as well, and we can say that CVaR is the most successful coherent risk measure so far. On the other hand, since the estimate of CVaR is computed by using only an upper tail part of the loss distribution, a large number of samples are required for assuring the statistical reliability of the estimate. Especially when CVaR is employed as the objective of a portfolio optimization, a much larger number of samples are required for ensuring the accuracy of the Corresponding author

2 optimal portfolio. For example, Takeda and Kanamori (2009) show the estimation error in the minimal CVaR is inversely proportional to β. This indicates that estimation error in CVaR is much severer than that of the mean loss since β 0.90 is usually applied while the mean loss is equal to CVaR with β = In addition, employing a large number of scenarios makes the resulting LP dense, i.e., to contain a prohibitively large number of nonzero entries. In order to obtain a sparser LP formulation of a large-scale problem, Konno, Waki and Yuuki (2002) propose to introduce a factor-model representation to the CVaR minimization as an analogy to the factor representation of the meanvariance model developed by Perold (984). Although they show through numerical results that the computation time can be significantly decreased by employing the factor representation, the out-of-sample performance is not examined. In practice, however, the number of samples which is available for the estimation is limited, and the estimated CVaR and the resulting optimal portfolio may contain considerable estimation error. To alleviate the estimation error of the CVaR minimization, Gotoh and Takeda (2008, 2009) employ a regularization technique so that the optimal portfolio vector will be less sensitive to the data, as in the mean-variance minimization by DeMiguel et al. (2009), Brodie et al. (2009). For the last decade, robust optimization techniques have been introduced in the field of financial portfolio selection, and shown to achieve a nice performance. See, e.g., Ben-Tal, El- Ghaoui and Nemirovski (2009) for the comprehensive survey of recent development in robust optimization. The robust portfolio takes into account the least favorable scenario or estimate. The set of possible scenarios (or estimates) is called uncertainty set. It should be noted that specifying the shape and size of uncertainty set is critical in the sense that robust formulation with improper selection of the uncertainty set can result in a useless solution. However, little has been examined how to specify the uncertainty sets in practical situation. Among such, Goldfarb and Iyengar (2003) presented a practical robust portfolio selection model, employing the confidence region of the factor model as an uncertainty set. Since their uncertainty set does not capture all the possible cases in the set, it may not be adequate to call it a robust optimization in the sense that a worse case may happen with a very small probability. Nevertheless, the uncertainty modeling based on the factor model is one of the most promising approaches since, in reality, we cannot actually recognize the worst case in a definite manner, but can do only in a probabilistic manner. The purpose of this article is to present a simple and practical approach for providing a CVaR minimizing portfolio with a robust property by employing the ordinary regression of the factor model similarly to Goldfarb and Iyengar (2003). In addition to the difference of the employed risk measures, we examine a more nonparametric way of exploiting the factor model, employing a statistical learning approach for determining the size of the uncertainty set. A robust portfolio model for the CVaR minimization has already been proposed by Zhu and Fukushima (2009). They focus on the uncertainty in the probability distribution used for defining CVaR. Such a modeling is called distributionally robust modeling. It is true that the probability estimation itself is under uncertainty and we cannot know the true one. However, it is not easy to imagine what form of uncertainty set is proper for the probability measure. In this sense, employing the uncertainty of probability distribution may not provide investors with a satisfactory solution. In contrast, we provide a more practical and simple way for making the CVaR minimizing portfolio robust against estimation errors. Another good point on our formulation is that our robust modeling is compatible with the distributionally robust modeling of, e.g., Zhu and Fukushima (2009). Therefore, if a reasonable uncertainty set for the probability distribution is given, then the two uncertainty sets can be taken into account simultaneously. The characteristics of the approach we take can be summarized as follows: Unlike Zhu and Fukushima (2009), we consider the uncertainty associated with the per- 2

3 turbation in observed return vectors in order to mitigate the fragility of the CVaR minimization. In particular, we apply a perturbation set associated with the distribution of the residual vectors arising from the ordinary least square method for multi-factor models. Since it is often assumed that the residual follows a normal distribution, it seems reasonable to employ the ellipsoidal uncertainty in the perturbation. Including CVaR as a special case, our robustifying technique is applicable to any coherent risk measure. In general, the minimization of the worst-case coherent risk measure turns out to be the minimization of the sum of the empirical risk measure and a penalty term. Especially when the perturbation is given by a norm term, the penalty term is represented by the dual norm of portfolio vector. This indicates that the norm-constrained CVaR minimization in Gotoh and Takeda (2009) is interpretable from the worst-case minimization. Another interesting fact is that the worst-case formulation presented in this paper can be combined with the distributionally robust minimization of the coherent risk, e.g., the CVaR minimization in Zhu and Fukushima (2009). Numerical results show that the developed robust CVaR model achieves significantly smaller CVaRs out-of-sample than the existing CVaR minimizations and two benchmark portfolios. Also, the regularization brings an effect of lowering the turnover of the portfolio vectors. The structure of this paper is as follows. The next section will be devoted to describing two existing implementation techniques of the CVaR minimization. Then a robust counterpart of the ordinary CVaR minimization will be provided in Section 3. Section 4 reports some numerical results, showing how the robust CVaR minimization improves the out-of-sample performance of non-robust CVaR minimizations. Finally, some remarks will be provided in the final section. 2 Formulations of CVaR Minimization Let there be n investable assets, and let R := (R,..., R n ) be their random return and π := (π,..., π n ) be a portfolio vector. Each component of π represents the investment ratio into an asset, and e n π = is fulfilled by definition, where e n := (,..., ) IR n. All the conditions on π are denoted by a set Π π IR n : e n π =. A portfolio optimization problem seeks to find a portfolio π Π so that the distribution of portfolio return R π would have a shape which is desirable to the investor. Typical criteria for tailoring the distribution are to minimize the risk and to maximize the mean return. It has been pointed out, however, that estimating the mean return in a satisfactory manner is a hard task. In fact, Jagannathan and Ma (2003) conclude that ignoring the mean return does not lose so much. Therefore, we in this paper limit the discussion to the minimization of a risk as in DeMiguel et al. (2009) where variance minimization is examined. Among many candidates for the risk measure, the conditional value-at-risk (CVaR) associated with the portfolio return is minimized since CVaR is important both in theory and practice but has fragility in estimation and optimization, as pointed out in Introduction. With a nonincreasing function L, let us define a portfolio loss by L(R π). The β-conditional value-at-risk (β-cvar) associated with the loss is defined by ϕ β (π) := min α α + β E[maxL(R π) α, 0] where β [0, ) is a significance level, and E[ ] is the operator of mathematical expectation. In the following part of this paper, a loss of the form R π is only considered. Also, we assume that the probability associated with R is independent of π as in Rockafellar and Uryasev (2002). 3,

4 CVaR can be roughly considered as the conditional expectation of the loss R π which exceeds the 00β-percentile of the loss, i.e., the β-value-at-risk (VaR). Following a custom of statistics, β is usually fixed at a value near.0, e.g., β = In order to minimize ϕ β (π), we have to estimate the risk measure since the true distribution of R is never known in practice. When only a collection of T historical observations R,..., R T of R are available, the empirical distribution is used to estimate the risk measure. More precisely, the empirical β-cvar, ϕ e β, is defined by ϕ e β (π) := min α α + ( β)t max R t π α, 0, () and the minimization is reduced to a linear programming (LP) problem: min α + ( β)t e T y s.t. y t R t π α, t =,..., T y 0, π Π. (2) as long as the set Π is given by a polyhedron. An optimal solution to (2) is an estimate of the portfolio that minimizes ϕ β (π). It is easy to see that ϕ e β (π) depends only on a portion of loss scenarios R [] π,..., R [τ] π with τ = ( β)t, where R [k] π indicates the k-th largest loss scenario and x is the smallest integer that is no less than x. See, e.g., Proposition 8 of Rockafellar and Uryasev (2002) for the details. This indicates that a perturbation in those large loss scenarios can make a big impact on the estimate of CVaR. The things become worse when we compute a portfolio that minimizes the empirical CVaR since the portfolio seeks to fit the perturbed sample. A practical way to alleviate the effect of such a perturbation is to employ a statistical model. Konno, Waki and Yuuki (2002) replace the observed returns R,..., R T in () with values estimated by a regression approach. Let us suppose that the structure of asset returns is described as a linear multi-factor model: K R j = β 0j + β kj F k + E j j =,..., n, (3) k= where F k are K factors used for explaining the asset returns R j, β kj are intercepts (k = 0) and factor loadings (k =,..., K) to be estimated, and E j are the residuals. The parameters β kj, k = 0,..., K, are usually estimated via the ordinary least squares (OLS) method for each asset j. Let F t,k be T observed values of the k-th factor, t =,..., T, and let β k := ( β k,..., β kn ) be the estimated intercept (k = 0) and factor loading (k =,..., K) vectors. Namely, T ( β 0j, β j,..., β K Kj ) arg min (R tj (β 0j + β kj F t,k )) 2, j =,..., n. β 0j,...,β Kj The use of the OLS method implies that any factor F k and residual E j are independent of each other, and satisfies E[E j ] = 0. The vectors R t := β 0 + K β k= k F t,k, t =,..., T, can be considered as T scenarios generated by the linear model (3). Let us define another CVaR by replacing the historical returns R t with the theoretical returns R t, and denote it by ϕ f β. ϕ f β (π) := min α α + ( β)t k= max R t π α, 0. 4

5 This estimate eliminates the residual return from the empirical CVaR, ϕ e β. Konno, Waki and Yuuki (2002) introduce this estimate, motivated by the fact that n j= E jπ j = 0 is expected to hold for a well-diversified portfolio π, and show that this simplification reduces the computation time for solving the LP (2). Let ε,..., ε T denote the realized residuals which are defined by ε t := R t R t = R t β 0 K β k F t,k. k= Then, the factor model-based CVaR can be rewritten by ϕ f β (π) = min α + max (R t ε t ) π α, 0. (4) α ( β)t When Π is represented by a system of linear inequalities or equalities, its minimization can be rewritten as a linear programming problem: min α + ( β)t e T y s.t. y t (R t ε t ) π α, t =,..., T y 0, π Π. (5) Since the variance of R t is larger than that of R t ε t while the mean values meet due to the nature of the OLS method, the factor-based CVaR ϕ f β (π) is expected to be (probabilistically) less than or equal to the empirical CVaR ϕ e β. In particular, if R and E follow normal distributions N(µ, V ) and N(0, Σ), respectively, one then has ϕ F β (π) := min α α + β E[ (R E) π α] + = µ π + C β π (V Σ)π µ π + C β π V π = ϕ β (π) where C β := exp( 2 F (β) 2 ) ( β). Since, for any ϵ > 0, lim 2π T P ϕ f β (π) ϕf β (π) > ϵ = 0 and lim T P ϕ e β (π) ϕ β(π) > ϵ = 0, one has lim T P ϕ f β (π) ϕe β (π)+c β π Σπ > ϵ = 0. In such a sense, the factor representation in Konno, Waki and Yuuki (2002) is expected to underestimate the empirical CVaR ϕ e β. 3 Robust CVaR Minimization Based on Multifactor Model 3. A Worst-case CVaR Recalling that both of the estimated CVaRs, ϕ e β in () and ϕf β in (4), are determined by only a small portion of T samples, R t and R t ε t, respectively, they can be highly affected by a small number of perturbed samples when the number of available samples, T, is small relatively to the number of assets, n. To overcome the fragility in ϕ e β, we below introduce a robust CVaR minimizing portfolio model in a different manner from Zhu and Fukushima (2009). Let us define an uncertainty set U by U := (r,..., r T ) : (r,..., r T ) = (R,..., R T ) (ε,..., ε T ) for some (ε,..., ε T ) Z. with a perturbation set of the form Z := (ε,..., ε T ) : ε t V, t =,..., T, 5

6 where V IR n is a compact set containing the origin in its interior. Among examples of V are () a finite set of possible vectors, denoted by V I := v,..., v k, so that the convex hull of this set contains the origin in its interior, and (2) a box-shaped polytope, denoted by V B := v : v L v v U with v Lj < 0 and v Uj > 0, j =,..., n. In what follows, we especially pay attention to a norm-based perturbation set of the form Z := (ε,..., ε T ) : ε t δ, t =,..., T, (6) where is a norm in IR n and δ is a parameter defining the size of the uncertainty set. Apparently, the corresponding uncertainty set is centered at the observed return vectors and the shape and size are determined by and δ, respectively. For example, if δ = 0, then U shrinks to the observed return vectors (R,..., R T ). Obviously, if ε follows a multivariate distribution, its confidence region may be employed as the perturbation set Z. For example, if ε follows N(0, Σ), the use of a norm of the form ε Σ := ε Σ ε is reasonable since a confidence region of the residual vector forms an ellipsoidal region. With the uncertainty set U, a worst-case CVaR, ϕ w β (π), of π is defined by ϕ w β (π) := max (r,...,r min α + T ) U α = max (ε,...,ε min T ) Z α α + ( β)t ( β)t max r t π α, 0 max (R t ε t ) π α, 0. (7) It is easy to see that ϕ e β (π) ϕw β (π). Combining the observation in the previous section, we can expect that ϕ f β (π) ϕw β (π) holds in a probabilistic sense. In particular, if we employ the norm-based perturbation (6) and set δ = max ε t : t =,..., T, one has ϕ f β (π) ϕw β (π). Also, it is contrastive that the worst-case CVaR (7) assumes the residual vectors, ε t, are variables, whereas the factor model-based CVaR (4) assumes they are fixed values, i.e., ε t = ε t. It should be noted that the worst-case CVaR, ϕ w β (π), is different from the worst-case CVaR proposed by Zhu and Fukushima (2009), where the (discrete distribution version of) worst-case CVaR is defined by max min p P α α + β p t max R t π α, 0 where P is a compact convex set in IR T, representing the uncertainty set of the probability distribution. Our worst-case CVaR takes care of the worst scenario in the return vectors (r,..., r T ) and keeps the empirical probability, p = (/T,..., /T ), intact, whereas that in Zhu and Fukushima (2009) do in the probability distributions (p,..., p T ) and keeps the observed return, (R,..., R T ), intact. Although Zhu and Fukushima (2009) describe how to specify the uncertainty set P of the probability distribution and present some numerical example showing how to apply their risk measure to practical data, it seems more difficult to agree about an uncertainty set of the probability distribution than that of the return vectors. In fact, a more commonly used idea in statistic analysis can be employed in specifying the uncertainty set of the return vector, as will be seen in Subsection 3.3. Despite the different assumptions on the origin of uncertainty, the resulting optimization can also be transformed into a convex optimization problem as the worst-case CVaR of Zhu and Fukushima (2009). 6

7 3.2 Decomposition Property in Worst-case Coherent Risk Measures In order to discuss the tractability of the worst-case CVaR ϕ w β (π) in a general framework, let us first introduce the coherent risk measures (Artzner et al. 999). Let ρ(π) := ρ[ R π] be a coherent risk measure of portfolio π. Namely, ρ[ ] satisfies the following properties for random variables X, Y which are assumed to represent portfolio loss variables: (i) ρ[x ] ρ[y] if X Y; (ii) ρ[ax + b] = aρ[x ] + b for all a 0, b IR; (iii) ρ[x + Y] ρ[x ] + ρ[y]. It is known that any coherent risk measure can be written by a dual representation as follows: ρ[x ] = supep[x ] : p P where Ep[X ] is the mathematical expectation of X under probability distribution p, and P is a compact convex. Indeed, the empirical CVaR is a coherent measure with P = p IR T : e T p =, 0 p e T /(( β)t ). The empirical version of the coherent risk measure, ρ e (π), is then represented by ρ e (π) := max p t R t π : p P for some P. Also, the worst-case coherent risk, ρ w (π), can be similarly defined by ρ w (π) := max (ε max T p t (R t ε t ) π : p P. t ) Z With the generalized perturbation set, the worst-case coherent risk measure can be decomposed into the empirical version plus a penalty term. Theorem The worst-case coherent risk measure with the generalized perturbation can be represented by ρ w (π) = ρ e (π) + max s s π : s V. (8) See the appendix for the proof of the theorem. In the above sense, the worst-case CVaR does not lose the coherent property. Also, the worst-case coherent risk measure ρ w (π) is a convex function since both ρ e (π) and τ(π) := maxs π : s V are convex functions. Since CVaR is a coherent risk measure, the worst-case CVaR, ϕ w β (π), can also be written as the sum of the empirical version and a convex penalty term. Corollary The worst-case CVaR (7) can be represented by ϕ w β (π) = ϕe β (π) + max s s π : s V. In particular, if the norm-based perturbation (6) is employed, the worst-case CVaR can be decomposed into the empirical CVaR and a regularization term: ϕ w β (π) = ϕe β (π) + δ π (9) where π denotes the dual norm of π, i.e., π := maxss π : s. The second statement of Corollary states that when the norm-based perturbation is employed, the worst-case CVaR (7) can be decomposed into the empirical CVaR ϕ e β (π) and a norm of a portfolio. Similarly, the worst-case CVaRs with the finite perturbation set V I and the box perturbation set V B can be shown to be tractable convex functions. 7

8 Consequently, the robust counterpart of the CVaR minimization, which is written by a minmax-min optimization problem: min π Π ϕw β (π) = min max π Π (ε,...,ε min α + max (R t ε t ) π α, 0 T ) Z α ( β)t can be rewritten as a regularized minimization of empirical CVaR. Proposition Let Π be given as a system of linear inequalities and equalities. If the norm-based perturbation set is employed, the min-max-min robust counterpart of the CVaR minimization (2) can be reformulated as a convex optimization problem as follows: min α + ( β)t e T y + δ π s.t. y t R t π α, t =,..., T y 0, π Π. (0) Similarly, if the finite perturbation V I is employed, it results in an LP: min α + ( β)t e T y + ζ s.t. y t R t π α, t =,..., T y 0, π Π ζ v h π, h =,..., k. If the box perturbation V B is employed, it results in an LP: min α + ( β)t e T y + (v U v L ) w + v L π s.t. y t R t π α, t =,..., T y 0, π Π w π, w 0. The first and second statements are proved in a straightforward manner. See the appendix for the proof of the third one. It is noteworthy that the role of the norm-based regularization term in the CVaR minimization can also be justified by the generalization bounds for the ν-support vector machine, which is a collective term for statistical methods including classification and regression methods. As Gotoh and Takeda (2009) show, imposing a norm constraint can improve the performance of the CVaR type minimization. The discussion above explains that the regularized (or, equivalently, norm-constrained) CVaR minimization can be justified also in the context of robust optimization. Although Proposition 2 in Gotoh and Takeda (2009) also shows a constraint-robust formulation can be equivalent to the regularized formulation (0), we restate the statement above in order to clarify the relation between the usual min-max-min type formulation and the regularized formulation. On the other hand, Theorem also indicates that the robustification technique developed in this article can be applied to any coherent risk measure. In this sense, the worst-case CVaR ϕ w β is compatible with the worst-case CVaR proposed in Zhu and Fukushima (2009) since the latter is proved in their paper to be a coherent risk measure. For example, let us consider the box uncertainty set for the underlying probability: P = p IR T : p = e T T + η, for some η s.t. e T η = 0, η L η η U, 8

9 where η L and η U are given lower and upper bounds on probability distribution, respectively. Combining the argument in Zhu and Fukushima (2009) and Proposition with the normuncertainty (6) for the return vector, a worst-case CVaR which cares both the observation uncertainty and the distribution uncertainty can be written by the following convex program: min θ + δ π s.t. α + e T z ( β)t + β (η L ξ + η U ω) θ e T u + ξ + ω = z, ξ 0, ω 0 z t R t π α, z t 0, t =,..., T. Similarly, it is easy to see that the worst-case CVaR minimization (0) can cohabit with the worst-case CVaR minimization of Zhu and Fukushima (2009) with the ellipsoidal uncertainty P = p IR T : p = e T T + Aη 0 for some η s.t. e T Aη = 0, η η, where A IR T T. Needless to say, any combination of the above-mentioned uncertainty sets of perturbation and distribution can results in a convex optimization. Nevertheless, in what follows, we concentrate only on the single application of the norm-based perturbation set in order not to obscure the outline of our argument. 3.3 Formulation with Factor Model-based Uncertainty In order to implement the robust CVaR minimization (0), we have to set up how to determine the norm (or, equivalently, ) and the parameter δ. Goldfarb and Iyengar (2003) show how uncertainty sets of the mean return vector and the covariance matrix are constructed on the basis of the factor model. Although the same line can be traced for the CVaR minimization, we here give a simpler way for constructing the perturbation set Z. Suppose that the OLS method is applied to the estimation of the factor model (3). Then the statistical confidence regions of the residual vectors can be exploited as a perturbation set Z. In fact, the t-tests on the factor loadings of the factor model are often conducted under a normality assumption. Therefore, it is reasonable to employ a norm induced from the covariance matrix of the residual vectors, ε t, of the factor model. Let Σ := ε ε n T E E, E :=.., ε T ε T n and let us consider the norm of the form ε Σ := ε Σ ε for the perturbation set Z. As already mentioned, the use of this norm can be justified if the residual follows an elliptical distribution (not necessarily normal distribution). By Corollary, the worst-case CVaR becomes a convex function of the form ϕ w β (π) = min α α + ( β)t max R t π α, 0 + δ π Σπ. Noting that the term π Σπ indicates the variance of the residual of the portfolio return, reducing the term π Σπ is expected to improve the goodness-of-fit of the factor model. Accordingly, 9

10 it can be regarded that the worst-case CVaR minimization seeks to reduce the empirical CVaR and to improve the predictability simultaneously. The robust counterpart (0) can then be rewritten by a second order cone optimization problem, which can be efficiently solved by an interior point algorithm, of the form: min α + ( β)t e T y + δv s.t. y t R t π α, t =,..., T, y 0, π Π. v z z, z = /T Eπ, as long as Π is represented by a system of linear inequalities and equalities or second order cones. It is noteworthy that this optimization is well-defined even for the case where the number of samples is smaller than the size of the universe, i.e., T < n. Although such a case does not correspond to a norm in the sense that Σ is no longer a positive definite matrix, the quadratic term, π Σπ, certainly defines a regularization term of the portfolio vector. As for the parameter δ, we can apply a quantile of the distribution of the residual, corresponding to a confidence level, e.g., 0.95 and In the numerical experiment in the next section, however, the value of δ is determined by a data-driven approach so as to avoid arbitrariness in assuming a specific distribution or in determining the confidence level. 4 Numerical Experiments In this section, numerical results are presented for comparing the robust formulation () with two existing implementations (2) and (5) of the minimization of CVaR ϕ β (π) subject to π Π = π : e n π =, π 0. We use the 202 monthly return vectors of 85 stocks listed in the Nikkei225 index through February 993 to November The linear models (3) are estimated by the OLS method using the three factors advocated in Fama and French (993). The used factor data are computed along the line of the paper Kubota and Takehara (2007) and are available at the web site of Nikkei Media Marketing, Inc. (200). A rolling horizon scheme is employed for constructing portfolios and evaluating the performances. For the t-th time window of length T = 20, the factor model is estimated by using (R τ, F τ,k ) : τ = t, t +,..., t + 9, and an optimal portfolio, π t, is computed, t =,..., 82(= ). The out-of-sample portfolio returns are accordingly simulated by R t+20π t, t =,..., 82. The out-of-sample performance can be gauged by the following measures: Out-of-sample CVaR ˆϕβ := min α α + 82( β) 82 Out-of-sample turnover of portfolio vectors π := 8 where π t+,j := ( + r t+,j)πt,j n i= ( + r t+,i)πt,i. Out-of-sample mean ˆµ := R 20+tπ t Out-of-sample standard deviation ˆσ := max R 20+tπ t α, 0 8 j= (R 20+tπ t ˆµ)2 n π t+,j πt+,j ()

11 n : size of universe n : size of universe (a) β = 0.90 (b) β = 0.95 Figure : Out-of-sample CVaR Left figure shows the average of the achieved CVaRs with β = 0.90 while right one shows that with β = Each value indicates the average of ten CVaRs, each series of portfolios is constructed using randomly chosen universe. In computing an optimal portfolio π t, the parameter δ > 0 must already be given only on the basis of data available before time t We dynamically tune the value of δ by employing a six-fold cross validation (see, e.g., Hastie, Tibshirani and Friedman 200) in the following manner. For the t-th time window, the data set of size 20 is divided into six subsets of size 20, i.e., (R τ, F τ,k ) : τ = t, t +,..., t + 9, 2 (R τ, F τ,k ) : τ = t + 20,..., t + 39,..., 6 (R τ, F τ,k ) : τ = t + 00,..., t + 9. A portfolios, denoted by π k δ,t, is constructed for some δ by using the union of five subsets out of the six subsets, i.e.,,..., 6 \ k, and the performance of the optimal portfolio is simulated over the remaining subset k which is unused for computing the portfolio. Repeating the construction and simulation for all the six combinations of the subsets, 20 simulated portfolio returns are obtained, i.e., R t π δ,t,..., R t+9π 6 δ,t, and let µ δ,t := τ=0 k= R 20(k )+τ π k δ,t and σ δ,t := τ=0 k= (R 20(k )+τ π k δ,t µ δ,t) 2. The value of δ at time window t is chosen from k : k = 0,, 2, 3, 4 so that the value µ δ,t + C β σ δ,t is the smallest, and the optimal portfolio π t is computed using that δ and all the data,..., 6. This is repeated by sliding the time window. We computed portfolios for various asset universes of size n = 20, 40,..., 60. For each n, ten sets of assets are randomly chosen from the 85 assets contained in the Nikkei225 index. Figure shows the average of the out-of-sample CVaRs, ˆϕ β, with β = 0.90 and We see that the robust model achieves the smallest CVaRs out-of-sample for any size of universe. On the other hand, the minimization of the factor model-based CVaR ϕ f β (π) constantly results in the largest CVaR values. In addition, the difference between ϕ f β (π) and ϕw β (π) becomes larger as the size of universe grows. We guess that this is because minimizing ϕ f β (π) excessively fit the past data, while the other models successfully reduced the out-of-sample CVaR by diversification. We see from Figure 2 that the turnover of the robust CVaR model is smaller than the two CVaR alternatives, while worse than the equally weighted portfolio. The regularization term works for decreasing the turnover as well as improving the out-of-sample CVaR. Figures 3 and 4 show the out-of-sample mean and the standard deviation, respectively, of the out-of-sample returns. Although the mean return of the robust model is smaller than the index and the equally weighted portfolio on average, the robust model achieves higher mean return than the other two alternatives except the case of (β, n) = (0.95, 20). On the other hand, the robust model achieves the smallest standard deviation among all the models. In addition, the value decreases as the

12 n : size of universe n : size of universe (a) β = 0.90 (b) β = 0.95 Figure 2: Out-of-sample Turnover n : size of universe n : size of universe (a) β = 0.90 (b) β = 0.95 Figure 3: Mean of Out-of-sample Returns number of assets decreases. Such a low standard deviation is partly led by the small CVaR values in Figure, and besides, the tuning of δ seems to contribute to a certain degree. Table reports the p-values of the Wilcoxon signed rank test on the out-of-sample performance of the robust model in comparison with the nominal CVaR minimization model. From this table, we see that the robust CVaR model combined with the tuning of δ outperforms the ordinary CVaR models in various aspects. In particular, when the number of assets is large, the difference is significantly supported. 5 Concluding Remarks In this paper, we present a new formulation for the CVaR minimization by introducing a robust optimization technique. When a norm-based perturbation is employed, it results in a convex optimization whose objective function consists of the usual nominal CVaR and a regularization term. The distribution of the residual vectors of the factor model is exploited so as to define the regularization term. More generally, the CVaR can be replaced with any coherent measure of risk. What is interesting here is that the robust modeling employed in the present paper is compatible with the distributionally robust model for coherent risk measure minimization. 2

13 n : size of universe n : size of universe (a) β = 0.90 (b) β = 0.95 Figure 4: Standard Deviation of Out-of-sample Returns Table : p-values of the Wilcoxon signed rank test of the out-of-sample performance of the robust model relative to the nominal model CVaR ˆϕ β Turnover π Mean ˆµ Standard deviation ˆσ β = 0.90 β = 0.95 β = 0.90 β = 0.95 β = 0.90 β = 0.95 β = 0.90 β = n Each value indicates the p-value of the Wilcoxon signed rank test for the out-of-sample performance of the robust model versus that of the nominal model over the ten trials, each using a randomly chosen asset universe of size n. The null hypothesis H 0 : The robust model achieved as good performance as the nominal model; The alternate hypothesis H : The robust model is superior to the nominal model. 3

14 Through some numerical experiments, the robust model constantly achieves smaller CVaRs, turnover, standard deviation and higher mean than the other CVaR minimization models. In particular, the simple factor representation developed in Konno, Waki and Yuuki (2002) results in the worst performance in all aspects among the several models, so that it should not be used. In contrast, employing the regularization term in minimizing CVaR mitigates the estimation error or the effect of outlying data. It should be recalled that the use of the regularization term can also be justified from the viewpoint of the generalization theory for machine learning (Schölkopf et al. 2000, Gotoh and Takeda 2008, 2009). In this paper, we omit the mean maximizing term according to DeMiguel et al. (2009), which focuses on the variance minimization instead of the mean-variance model. It is easy to see that the robust modeling developed in this article can also be applied to the mean maximizing term. The investigation on the effect of the regularization term in the mean-coherent risk minimization remains as a future research. Acknowledgment The research of the first author is partly supported by a MEXT Grant-in-Aid for Young Scientists (B) A Proof of Theorem Proof. The worst-case coherent risk measure ρ w can be rewritten as ρ w (π) = max q ε t R t,q t π + q t ε t π : q P, ε t δ, t =,..., T. Note that for q P, we have max (ε t) Z q t ε t π = so the proof is complete. q t max (ε ε t π = max t) Z ε ε π : ε V q t = max ε ε π : ε V, B Proof of Proposition Proof. When the box perturbation V B is employed, the penalty term is represented by an LP of the maximization form: maxπ s : v L s v U for some π Π. By the duality theorem, it can be replaced with the dual LP of the minimization form: min(v U v L ) w + v L π : w π, w 0, and the formulation of the third statement is obtained. References Artzner P, Delbaen F, Eber JM, Heath D (999) Coherent Measures of Risk. Mathematical Finance 9: Ben-Tal A, ElGhaoui L, Nemirovski A (2009) Robust Optimization. Princeton Univ. Press. 4

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