The paper discuss the sensitivity to the presence of outliers of the portfolio optimization procedure based on the expected shortfall as a measure of risk. A robust approach based on the forward search is then suggested which seems to give quite good results

Grossi L.

Laurini F.

Scandolo G.

GROSSI, Luigi

SCANDOLO, Giacomo

Catalogo dei prodotti della ricerca

OPTIMAL PORTFOLIO ALLOCATION WITH CVAR: A ROBUSTAPPROACHLuigi Grossi1, Fabrizio Laurini2 and Giacomo Scandolo11 Dipartimento di Scienze EconomicheUniversita` di Verona(e-mail: luigi.grossi@univr.it, giacomo.scandolo@univr.it)2 Dipartimento di Economia Universita` di Parma(e-mail: fabrizio.laurini@unipr.it)ABSTRACT. The paper discuss the sensitivity to the presence of outliers of the portfolio optimizationprocedure based on the expected shortfall as a measure of risk. A robust approach based on the forwardsearch is then suggested which seems to give quite good results.1 INTRODUCTIONThe main objective of portfolio selection is the construction of portfolios that maximize ex-pected returns at a certain level of risk. In the classical Markowitz mean-variance efficientfrontier problem, estimates of the expected returns and the covariance matrix of the assetsare used to calculate the optimal allocation weights (Markowitz, 1959). The drawbacks ofthe classical mean-variance approach have been widely discussed (Michaud, 1989). It is wellknown that asset returns are not normal and, therefore, the mean and the variance alone donot fully describe the characteristics of the joint asset distribution. As a consequence, espe-cially in cases of strong nonnormality, the classical mean-variance approach will not be asatisfactory portfolio allocation model. Indeed, it is sometimes considered useless because itcan lead to financially irrelevant optimal portfolios (Alexander and Baptista, 2002). Amongthe reasons of this drawback a relevant role is played by the influence of extreme returns(Huang et al., 2008). A sensitivity analysis of the mean-variance model, together with a ro-bust alternative, has been carried out by Grossi and Laurini (2011).Another criticism which is commonly made to the Markowitz is the use of the histori-cal standard deviation as a measure of risk. Several alternative measures of risk have beenproposed (e.g., value of risk and expected shortfall, also called conditional value at risk) andoptimizers based on these measures have been implemented (Rockafeller and Uryasev, 2002).Some of these measures are more appropriate than classical volatility (the return standard de-viation) for long-tailed return models and are more realistic than the classical normal model.In addition, the computation of optimal portfolios based on these measures does not evenrequire the return covariance matrix.In this paper we discuss the problem of statistical robustness of optimization methodsbased on Spectral Risk Measures and show that the latter are not robust, meaning that a fewextreme assets prices or returns can lead to “sub-optimal” portfolios. We then introduce arobust estimator based on the forward search (Atkinson et al., 2004) of input parameters inthe maximization procedure and show that it is far more stable than the classical versionbased on maximum likelihood estimator (MLE).2 PORTFOLIO SELECTION WITH DIFFERENT MEASURES OF RISKTo introduce the general problem, let us suppose to have N risky assets, whose observedprices for T periods are pit , t = 1, . . . ,T , i = 1, . . . ,N and let x = (x1, . . . ,xN)′ be the vectorof portfolio weights. The assets returns are given by a matrix Y = (y1, . . . ,yN), where yi =(yi1, . . . ,yit , . . . ,yiT )′ and yit = ln(pit/pit−1)≈ (pit/pit−1)−1 with expected returns given bya N× 1 vector µ and N×N expected covariance matrix Σ. The expected return and varianceof the portfolio can be written as µp = x′µ and σ2p = x′Σx, respectively.For a given level of risk tolerance γ, the classical mean-variance optimization problemcan be formulated asminx(x′Σx− γx′µ), (1)subject to the constraints x ≥ 0 (meaning that all the weights are strictly non negative) andx′ιN = 1, where ιN is a N×1 vector of ones. The constraint of no short-selling (x≥ 0) is veryfrequently imposed as many funds and institutional investors are not allowed to sell stocksshort. We will make use of the no short-selling constraint throughout the paper. When thisconstraint is removed, it is easily proved that, using the Lagrange method, for any γ ≥ 0, themaximization problem has an analytical solution. If we gradually increase γ from zero andfor each instance solve the optimization problem, we end up calculating each portfolio alongthe efficient frontier. Loosely, the efficient frontier is the line connecting the upper boundaryof the set of feasible portfolios that have the maximum return for a given level of risk.In the present paper, we replace, as risk indicator, volatility with the Expected Shortfall(or Conditional VaR, CVaR) of the portfolio. Remind that the Expected Shortfall of V oforder α ∈ (0,1) isESα(V ) =1α∫ α0VaRu(V )du, (2)where VaRu(V ) = − inf{v : P(V ≤ v) ≥ u} is the Value-at-Risk of V of order u. Uryasev etal. (2000) proved a very useful result for the computation of ES:• the functionGα(z,V ) = z+1αE[(−z−V)+] (3)is jointly convex in (z,V ) (z ∈ R and V is in a space of random variables) for any α• In particular(z,x) 7→ z+1αE[(−z− x′Y )+]is jointly convex in (z,x)• the expected shortfall is computed throughESα(V ) = minz∈YGα(z,V )and VaRα(V ) = z∗ is the solution of the minimization problem.The problem 1 becomesminx ESα(x′Y )− γx′µ = minx,z{z+α−1E[(−z− x′Y )+]− γx′Y}∑Ni=1 xi = 1xi ≥ 0, ∀i(4)In the numerical optimization program, the trick of introducing dummy variables is used.Note that we have built the estimated frontier, as the problem (4) is not analytically solvablein general, so that no true frontier can be exactly computed (of course a true frontier exists,in principle).3 SENSITIVITY ANALYSIS OF PORTFOLIO OPTIMIZATIONIn this section, it would be useful to analyze how the CVaR is affected by the presence ofoutliers. It would be a preliminary step to the application of the spectral risk measures inasset allocation problems. At this step of the analysis the non-robust version of the CVARis used to study the influence of units on portfolio weights. Also efficient frontiers shouldbe analyzed on non-contaminated and contaminated data selecting some values of the riskaversion parameter.A deeply studied problem of portfolio allocation (Broadie, 1993) is given by Y Gaussianwith off-diagonal elements ρi j = 0.3 and components given byµY = (0.006,0.01,0.014,0.018,0.022)′ (5)σY = (0.085,0.08,0.095,0.09,0.1)′ (6)We will consider such parameters.In Figure 1 the “true” efficient frontier (bold line) obtained through a Gaussian optimiza-tion is compared with efficient frontiers (thinner lines) estimated on data simulated from aGaussian distribution with mean vector µY in (5) and covariance matrix obtained from thevariance vector σY in (6) and constant correlation ρi j = 0.3. As it can be seen the range of theestimated frontiers in both axes are approximately as large as the domains of the true frontier.Figure 2 has been drawn similarly to Figure 1, but the estimated frontiers has been ob-tained from contaminated data. Contamination of simulated data has been carried out intro-ducing outliers at random positions according to the following scheme. Let U be a discreterandom set of indices belonging to {1, . . . ,T}which gives the positions of the p outliers and Ra multivariate Gaussian distribution with vector of means equal to zero and covariance matrixθ× Ip. The parameter θ gives the magnitude of the contamination and Ip is a p sized identitymatrix. Finally the contaminated data-set is ˜Yt = Yt , for t /∈U and ˜Yt = Rt fot t ∈U , whereYt ∼ N(µ,Σ), and the vector µ and matrix Σ, reported above, are taken from Broadie (1993).Notice that the scale of the axes in the two Figures are the same. In the case of contaminateddata the estimated frontiers are more scattered around the true frontier and the domain ofestimated frontiers has been inflated by the presence of outliers in the data.This simulation experiment proves that a robust estimation procedure of optimal portfolioweights is needed.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.010.020.030.040.050.060.07Efficient FrontiersPortFolio Mean; µpPortfolio CVaRFigure 1. True efficient frontier for the covariance matrix of Broadie (1993) (bold line) comparedwith efficient frontiers estimated on simulated data (thin lines). Data are simulated from a Gaussiandistribution with Broadie’s mean vector and covariance matrix.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.010.020.030.040.050.060.07Efficient FrontiersPortFolio Mean; µpPortfolio CVaRFigure 2. True efficient frontier for the covariance matrix of Broadie (1993) (bold line) comparedwith efficient frontiers estimated on simulated contaminated data (thin lines). Data are simulated from aGaussian distribution with Broadie’s mean vector and covariance matrix and contaminated with additiveoutliers at random positions.4 ROBUST PORTFOLIO OPTIMIZATIONThe last step is to compute forward-search robust weights for each observation. The portfoliooptimization procedure will then be applied to a transformed weighted matrix of returns.Finally, the robust efficient frontier will be compared with the non-robust frontier.Our target is to compute weights wt ∈ [0,1], for each observation in the multiple timeseries yt =(y1t , . . . ,yNt)′, t = 1, . . . ,T , with the forward search method (Atkinson et al., 2004).A similar procedure has been applied in a previous paper by Grossi and Laurini (2011) to geta robust version of the covariance matrix in the classical Markowitz problem. The weightswill then be used to obtain a weighted version of the matrix Y of returns such that the mostoutlying observations get small weight. For multivariate data, standard methods for outlierdetection are based on the squared Mahalanobis distance for the t-th observation: d2t = (yt −µˆ)′ ˆΣ−1(yt − µˆ), where both mean-vector µ and covariance matrix Σ are estimated. One of themain pitfalls of the classical Mahalanobis distance as an outlier detection tool, is the bias onthe estimation of µ and Σ caused by the presence of multiple outliers. This “masking effect”of multiple outliers, is overcome by the forward search (see, Atkinson et al., 2004). The goalof the forward search is the detection of units which are different from the main bulk of theobservations, called Clean Data Set (CDS) and to assess the effect of these units on inferencesmade about the correct model.Given the best subset S(m)∗ of size m ≥ m0 detected at step m, we can calculate a set of Tsquared Mahalanobis distances, defined asd2t(m) = (yt − µˆ∗m)′( ˆΣ∗m)−1(yt − µˆ∗m), t = 1, . . . ,T, (7)where µˆ∗m and ˆΣ∗m are the mean and covariance matrix estimated on the m-sized subset. Thedistance introduced in equation (7) is the forward search version of the Mahalanobis distance.In the second step of the forward search, we increase the size of the initial CDS selectingobservations with small value of (7) and so are unlikely to be outliers. Thus, with the forwardsearch algorithm the data are ordered according to their degree of closeness to the CDS, withobservations furthest from it joining the CDS in the last steps of the procedure. When µ andΣ are estimated by MLE on the whole sample, the classical Mahalanobis distances follow ascaled beta distribution. But in equation (7) the Mahalanobis distances are estimated from asubset of m observations which do not include the observation being tested. In such a case,the reference null distribution would be (see, Riani et al., 2009):d2t(m) ∼ [T/(T − 1)][N(m− 1)/(m−N)]FN,T−N , (8)where N is the number of columns of Y .For the computation of the weights we compare the trajectories of d∗2(t) during the forwardsearch with confidence bands from the F distribution. Formally, for the t-th unit at step m,we define the squared Euclidean distance as: pi(t)m = 0 if d2t(m) ∈ [0,Fδ], pi(t)m = (d2t(m)−Fδ)2if d2t(m) > Fδ, where d2t(m) has been defined in (7) and Fδ is the δ percentile of the FN,T−Ndistribution.Then, for the t-th observation, we havepit =∑Tm=m0 pi(t)mT −m0 + 1. (9)The final outlyingness index piit for each return yit is then obtained as follows:piit = pitθit∑Ni=1 θit(10)where θit = |yit |/MAD(yi) and MAD(yi) is the median absolute deviation from the me-dian of yi.Finally, the main goal of under-weighting the most extreme observations is obtainedthrough the computation of a weight, in the interval [0,1], with the following mapping of(10): wit = exp(−piit). The weights are computed for each observation at the end of the for-ward search.The next step of the procedure is based on building a weighted matrix of returns Y ⋆, withgeneric element y⋆it = yitw1/2it . The weighted matrix Y ⋆ will be finally used as input matrix inthe estimation procedure of optimal portfolio weights.The efficient frontiers robustly estimated on contaminated data are very similar to thoseobtained in Figure 1.5 FINAL REMARKSIn this paper a new robust method for estimating optimal portfolio allocation has been intro-duced based on the forward search, The results are quite promising. One open issue is the ref-erence distribution for determining the threshold used to get the weight for each observation.Moreover a suitable metric to measure the distance between frontiers must be introduced.Grossi and Laurini (2011) suggested a Root Mean Error measuring the distance betweenfrontiers considering different values of the tolerance parameter, but separately computed forthe return and the volatility of portfolios.REFERENCESALEXANDER G.J., BAPTISTA A.M. (2002), Economic implication of using a mean-VaR model forportfolio selection: a comparison with mean-variance analysis. Journal of Economic Dynamicsand Control 26, 1159–1193.ATKINSON A. C., RIANI M., CERIOLI A. (2004), Exploring Multivariate Data with the ForwardSearch. Springer–Verlag, New York.BROADIE M. (1993), Computing efficient frontiers using estimated parameters, Annals of OperationsResearch, 45, pp. 21–58.GROSSI L., LAURINI F. (2011), Robust estimation of efficient mean-variance frontiers, Advances inData Analysis and Classification, 5(1), pp. 3–22.HUANG, D., ZHU, S.S., FABOZZI, F.J., FUKUSHIMA, M. (2008), Portfolio selection with uncertainexit time: A robust CVaR approach. Journal of Economic Dynamics and Control, 32, pp. 594–623.MARKOWITZ, H. (1959), Portfolio Selection: Efficient Diversification of Investments. Wiley, New York.MICHAUD, R. (1989), The Markowitz oprimization enigma: is optimized optimal? Financial AnalystisJournal, 45, pp. 31–42.Riani, M., Atkinson, A. C. and Cerioli, A. (2009), Finding an unknown number of multivariate outliers.Journal of the Royal Statistical Society, Series B, 71(2), pp. 447–466.ROCKAFELLAR R. T., URYASEV, S. (2000), Optimisation of Conditional Value-at-Risk, Journal ofRisk, 2(3), pp. 21–41.

OPTIMAL PORTFOLIO ALLOCATION WITH CVAR: A ROBUSTAPPROACH

https://iris.univr.it/bitstream/11562/390278/1/grossi_etal.pdf

OPTIMAL PORTFOLIO ALLOCATION WITH CVAR: A ROBUSTAPPROACH

Abstract

Similar works

Full text

Available Versions

Catalogo dei prodotti della ricerca