Math Meth Oper ResDOI 10.1007/s00186-008-0244-7

ORIGINAL ARTICLE

Estimating allocations for Value-at-Risk portfoliooptimization

Arthur Charpentier · Abder Oulidi

Received: 20 June 2007 / Revised: 15 March 2008© Springer-Verlag 2008

Abstract Value-at-Risk, despite being adopted as the standard risk measure infinance, suffers severe objections from a practical point of view, due to a lack ofconvexity, and since it does not reward diversification (which is an essential featurein portfolio optimization). Furthermore, it is also known as having poor behavior inrisk estimation (which has been justified to impose the use of parametric models,but which induces then model errors). The aim of this paper is to chose in favor oragainst the use of VaR but to add some more information to this discussion, espe-cially from the estimation point of view. Here we propose a simple method not only toestimate the optimal allocation based on a Value-at-Risk minimization constraint, butalso to derive—empirical—confidence intervals based on the fact that the underlyingdistribution is unknown, and can be estimated based on past observations.

Keywords Value-at-Risk · Optimization · Portfolio · Non-parametrics

1 Introduction and motivations

The problem of allocating capital among a set of risky assets can be understood asfinding a portfolio which maximizes return, and minimizes risk. If diversificationeffects were intuited early, Markowitz initiated a formal mathematical model. In thisapproach, return is measured by the expected value of the portfolio return, and risk

A. Charpentier (B)CREM-Université Rennes 1, 7 Place Hoche, 35065 Rennes cedex, Francee-mail: arthur.charpentier@univ-rennes1.fr

A. OulidiUCO/IMA, place André Leroy, 49000 Angers, Francee-mail: abder.oulidi@uco-ima.fr

123

A. Charpentier, A. Oulidi

is quantified by the variance. Thus is the so-called mean-variance risk managementframework.

In Markowitz (1952) was considered the problem of selecting some optimal port-folio with a static perspective: at time 0, money is invested among several risky assets,so that at time T , the return should be high with low risk. The optimal allocation isthe one which minimizes the variance of the return between now and time t , giventhat its expected return exceeds a given return target η. This idea has been extend in amultiperiod model, with multiple possible reallocations (note that it can also be donein a continuous time framwork, see e.g. Merton 1992). Consider for simplicity a twoperiod model. At time 0, money is invested among several risky assets, so that at timeT = 2, the return should be high with low risk. But here, the investor, we can sellsome assets in order to buy others at time t = 1. This yield the notion of optimal strat-egy, which is a set of optimal portfolios, the first one being the one obtained at timet = 0, and the second one being the reallocation at time t = 1, based on additionalinformation observed during period of time [0, 1].

If those models are relevant from a mathematical point of view, it appears quicklythat there are serious drawback from a practical point of view. Optimal allocationis related to the joint distribution of returns, or, at least, the first two moments:the expected returns, and the variance-covariance matrix. But those parameters areunknown, and should be estimated from the past. Based on 4 years of past obser-vations, only a 1,000-sample can be used and therefore, all the parameters can beestimated with a significant confidence interval. And thus, an investor can only con-sider estimated optimal portfolio. In the two period model, assume that at time t = 0,30% of the money should be invested in the first asset, and that, after calculations attime t = 1, it appears that 32% of the money should be invested in the first asset: is this2 point difference significant or is it simply due to estimation issue? How confidenceshould we be in the optimality of those portfolios?

In this paper, we will use the following notations. Consider d risky assets, anddenote by Pt = (Pt,1, . . . , Pt,n) the random vector of prices at time t (the timeinterval is a week). Let Xt = (Xt,1, . . . , Xt,n) denote weekly log-returns, Xt,i =log Xt,i − log Xt−1,i . We will assume in this paper that random vectors Xt ’s areindependent and identically distributed. This assumption is the underlying assumptionin the Black and Scholes framework, but might appears as too strong with regards torecent literature on GARCH processes (see e.g.). Anyway, since the goal here is simplyto allocate capital from time t to t + 1 (static portfolios), we might assume that—as afirst order approximation—the dependence among log-returns is more important thenpossible time dependence. X = (X1, . . . , Xn) will then denote the standard weeklylog return, and we assume that portfolio allocation at time t = n can be performedbased on n past observations, X1, . . . , Xn.

In Sect. 2, we will redefine properly this notion of optimal portfolio, and highlightstatistical issues in the simple mean-variance model in Sect. 3. Section 4 will thenfocus on statistical issues in a more general setting. In Sect. 5, VaR estimation will bebriefly studied, from parametric and semiparametric models (based either on a normalassumption or on extreme value results), to nonparametric (kernel based estimators).And finally, a simulation study will be performed in Sect. 6, followed by a real-dataanalysis in Sect. 7.

123

Estimating allocations for Value-at-Risk portfolio optimization

2 Optimal portfolio

Markowitz developed mean-variance analysis in the context of selecting a portfolioof common stocks, and it has been increasingly applied to asset allocation. But if thismean-variance approach was historically the first idea to formalize optimal allocations,severall extentions in more general settings have been considered. One of the drawbackof this mean-variance approach is the symmetric attitude towards risks, which initiatedreaseach work on downside risk measures.

Definition 1 An optimal allocation among n risky assets (i.e. d random log-returnX = (X1, . . . , Xd)), for a given risk measure R, is a vector of proportions ω∗ =(ω∗

1, . . . , ω∗d) ∈ Ω ⊂ R

n is a solution of the following optimisation program,

ω∗ ∈ argmin

R (ω′X

), ω ∈ Ω

,

under constraints E(ω′X

) ≥ η and ω′1 = 1(1)

Based on the approach initiated by Von Neumann and Morgenstern (in game the-ory), economic theory postulates that individuals make decisions under uncertainty bymaximizing the expected value of an increasing concave utility function of consump-tion (see von Neumann and Morgenstern 1947 or Fishburn 1970). Markowitz assertedthat if the utility function can be approximated closely enough by a second-order Tay-lor expansion over a wide range of returns, then expected utility will be approximatelyequal to a function of expected value (the mean) and variance of returns. This allowsthe investor’s problem to be restated as a mean-variance optimization problem wherethe objective function is a quadratic function of portfolio weights.

The argument in von Neuman and Morgenstein’s theory is that any preferenceordering defined on the set of random risks, satisfying come axioms (namely com-pleteness, transitivity, continuity and independence, see von Neumann and Morgen-stern 1947), can equivalently be represented by some utility function u such thatE(u(X)) ≤ E(u(Y )) holds if and only if X Y . Hence, the formalization of thissecond approach needs first to select a certain utility function u and to formulatethe following optimization problem, ω∗ ∈ argmaxE(u(ω′X)), ω ∈ Ω, where it isusually required that the function u is concave and non-decreasing, thus representingpreferences of a risk-averse decision maker (see e.g. von Neumann and Morgenstern1947 or Fishburn 1970). An alternative formulation can be induced from Kroll et al.(1984): define a risk measure as R(X) = E(u(E(X) − X)) (so that the risk measureis location free) and then ω∗ ∈ argminR(ω′X), ω ∈ Ω.

And finally a third method can be derived from Yaari’s dual approach (see Yaari1987). From early works of de Moivre or Pascal, we know that the price of a gameis the scalar product of the probabilities and the gains, formulated later as the fairprice (from Feller 1968), i.e. the expected value. In the expected value framework,risk measures were defined as EP(u(Y )), which is the scalar product between theprobabilities (P) and utility of gains (u(Y )). Note that this function can be writtenthrough the following integral form

123

A. Charpentier, A. Oulidi

EP(u(Y )) =0∫

−∞P(u(Y ) ≤ y)dy +

+∞∫0

P(u(Y ) > y)dy. (2)

In the dual approach (see Yaari 1987; Quiggin 1993; Wang 1996; among others) isconsidered the scalar product bewteen a distorted version of probabilities (denotedg P) and the gains (Y ),

EgP(Y ) =0∫

−∞g(P(Y ≤ y)) +

+∞∫0

g(P(Y > y))dy, (3)

for some distortion function g : [0, 1] → [0, 1] (increasing with g(0) = 0 andg(1) = 1).

Remark 1 In the case where g is an indicator function, g(x) = 1(x > p), thenEgP(Y ) = V a R(Y, p). Further, if g(x) = g(x) = minx/p, 1, i.e. linear from (0, 0)

to (p, 1), then EgP(Y ) = T V a R(Y, p). Those two measures are defined properly inDefinition 3.

On a formal point of view, Yaari (1987) proposed a more general theoretical frame-work than those distorted measures, also called spectral risk measures in Acerbi andTasche (2002).

3 Estimation issues in the classical mean-variance framework

Several authors have pointed out that the Markowitz optimal portfolio is extremelysensitive to parameter perturbations, and amplifies the estimation errors (see Klein andBawa 1976 or Coles and Loewenstein 1988; among others). Consider d risky assets,with weekly returns X = (X1, . . . , Xd). Denote µ = E(X) and Σ = variance(X).Let ω = (ω1, . . . , ωd) ∈ Ω ⊂ R

d denote the weights in all risky assets.The expected return of the portfolio is E(ω′X) = ω′µ, and the the variance of the

portfolio is var(ω′X) = ω′Σω. The optimization problem is here

ω∗ ∈ argmin

var

(ω′X

), ω ∈ Ω

,

under constraints E(ω′X

) ≥ η and ω′1 = 1(4)

Recall that this mean-variance program can be formulated dually (as the maximizationof the expected value, with a risk constraint), since it is an optimization with aninequality is a quadratically constrained quadratic programming problem.

Remark 2 The set of possible allocation is either the simplex of Rd or the subset of R

d

such that the sum of the component is 1. In this paper, we will assume that Ω = Rd ,

i.e. short sales are allowed. This assumption will have no impact on the optimizationprogram, since we will estimate risk measures on a grid of possible portfolio alloca-tions, and the advantage is that kernel based estimators can be considered since valueswill not be bounded.

123

Estimating allocations for Value-at-Risk portfolio optimization

In practice, µ = [µi ] and Σ = [Σi, j ] as noticed in Coles and Loewenstein (1988),“the parameters governing the central tendency and dispersion of returns are usuallynot known, however, and are often estimated or guessed at using observed returns andother available data. In empirical applications, the estimated parameters are used asif they were the true value”. A natural idea is to use empirical estimated, based onsample X1, . . . , Xn. Hence

µ = [µi ] where µi = 1

n

n∑t=1

Xi,t , (5)

and

Σ = [Σi, j ] where µi = 1

n

n∑t=1

(Xi,t − µi )(X j,t − µ j ), (6)

Recall that µ and Σ have respectively a Gaussian and a Wishart distribution, and thatgiven Σ , those two estimators are independent.

As mentioned in the introduction, there is a difference between theoretical optimalallocation (based on µ and Σ) and the sample based optimal allocation (based on µ

and Σ).

4 Mean-Value-at-Risk optimization

4.1 Value-at-Risk as a risk measure

Current regulations from finance (Basle II) or insurance (Solvency II) business formu-lates risk and capital requirements in terms of quantile based measures (see e.g. Dowdand Blake 2006). The upper quantile of the loss distribution is called Value-at-Risk(VaR) : the 95% VaR is an upper estimate of losses which is exceeded with a 5% prob-ability. For a comprehensive introduction, we refer to Jorion (1997), or Föllmer andSchied (2004) for financial motivations, or for a some economic motivation of VaR,and more generally distorted risks measures. Hence, VaR provides a common riskmeasure that can be used for any type of portfolio (market risk, credit risk, insurancerisk), and a a simply interpretation as possible “lost money”.

Definition 2 For a probability level α ∈ (0, 1), the α-quantile of a random variableY is defined as

Q(Y, α) = infy ∈ R, P(Y ≤ y) > α. (7)

Definition 3 For a probability level α ∈ (0, 1), the Value-at-Risk at level α for log-returns X is defined as

VaR(X, α) = Q(−X, α). (8)

123

A. Charpentier, A. Oulidi

The Tail Value-at-Risk at level α is defined as

TVaR(X, α) = E(−X | − X > VaR(X, α)). (9)

Tail Value-at-Risk (as called in Artzner et al. 1999) is called expected-shortfall inAcerbi and Tasche (2002), conditional tail expectation in Wirch and Hardy (1999) orconditional Value-at-Risk in Rockafellar and Uryasev (2000).

Historically, mostly approaches to calculate VaR relied on linear approximation ofthe portfolio risks, assuming also a joint normal distribution of market parameters (seee.g. Duffie and Pan (1997), or the Cornish Fiher approximation in Sect. 6.1). The ideaof Markowitz is to find the optimal portfolio allocation, defined as a solution of

ω∗ ∈ argmin

R (ω′X

), ω ∈ Ω

,

under constraints E(ω′X

) ≥ η and ω′1 = 1(10)

for some risk measure R (the variance in Markowitz (1952), or the VaR in this paper).Hence, the investor want to find a allocation which minimizes the risk for a givenexpected return, with a budget constraint. Discussions on optimization issues involvingVaR minimization can be found in Litterman (1997) or Kast et al. (1998).As mentioned in the context of coherent risk measures by Artzner et al. (1999), VaRsuffers a major drawback because of nonsubadditivity: the VaR of a sum might exceedthe sum of VaR’s. Therefore, the VaR is neither convex (see e.g. Artzner et al. (1999)or Föllmer and Schied (2004). for a survey on convex risk measures), and thus, severaloptimization theorems and results can be be used. As mentioned in Rockafellar andUryasev (2000), an alternative is to consider a convex risk measure R in (10), suchas the Tail VaR (also called expected shortfall or conditional VaR). This measure isdefined as the conditional expected loss given that it exceeds VaR. For almost nor-mal distributions, those two measures are different, but the optimal allocations almostcoincide. As mentioned in Ingersoll (1987), they even coincide with mean-varianceoptimal allocation for elliptical distributions.For very skewed distributions, it mightnot be the case anymore (see e.g. Gaivoronski and Pflug 2000).

Rockafellar and Uryasev (2000) proposed also to use a stochastic programmingproblem, since the Value-at-Risk, since

VaR(X, α) = argmin

β + 1

αE((X + β)+), β ∈ R

.

But in such a case, Value-at-Risk minimization will involve two minimizationprograms.

4.2 Mean-Value-at-Risk optimality and the safety first principle

The fact that the VaR is not convex (while the variance was) yields a theoretical (andtherefore a practical) issue. In the mean variance problem, the two following programswere equivalent,

123

Estimating allocations for Value-at-Risk portfolio optimization

⎧⎨⎩

ω∗ ∈ argminω′Σω u.c. ω′µ ≥ η and ω′1 = 1 convex

ω∗ ∈ argmaxω′µ u.c. ω′Σω ≤ η′ and ω′1 = 1

But this duality is no longer valid here,

⎧⎨⎩

ω∗ ∈ argminVaR(ω′X, α) u.c. E(ω′X) ≥ η and ω′1 = 1 nonconvex

ω∗ ∈ argmaxE(ω′X) u.c. VaR(ω′X, α) ≤ η′,ω′1 = 1

As pointed out in Cohen and Zinbarg (1967), “we believe strongly that for mostinvestors the proper order of investigation is to consider the quality of a bond orpreferred stock before considering the yield. For most investors, if quality is inadequate,yield is irrelevant”. Therefore, even if most of actual literature focuses on the secondproblem (maximizing the return, see e.g. Basak and Shapiro 2001), we also believethat minimizing risk is the most appropriate way to express investors behavior (see alsoLevy and Sarnat 1972 or Bawa 1978). This approach, also motivated more recently inAlexander and Baptista (2002), will also be considered in this paper.

And so far, in this paper, we have introduced portfolio optimization based onMarkowitz’s approach. But historically, the mean-variance has not the only princi-ple used by investors. As mentioned in Roy (1952),“the optimal bundle of assets(investment) for investors who employ the safety first principle is the portfolio thatminimizes the probability of disaster”. Roy (1952) relates the disaster to the idea ofruin in insurance, which can also be related, with a modern perspective of financialproblems, to Value-at-Risk. In this “safety first principle”, investors are looking forportfolio which minimize disaster probabilities, with some expected return constraint.

Note finally that only a few paper focus explicitly in mean-VaR optimal portfolio,as in Basak and Shapiro (2001). Most of them prefer to focus on TVaR (which presentthe advantage of beeing a convex risk measure), as in Lemus (1999) or Rockafellarand Uryasev (2000).

5 Inference issues

In the mean variance framework, the optimal allocation could be expressed as a func-tion of the expected return, and the variance matrix of the log-returns. A naturalestimation of the optimal allocation is then obtained substituting the average and theempirical variance to the theoretical values.

A practical issue is about estimation of risk measures, and more specifically VaR.All or parts of the distribution have to be estimated. In practice, two techniques areusually considered. Parametric methods estimate quantiles under the assumption thata loss distribution has a particular parametric form. The fist task is then to determinewhat it can be. For instance, in the case of normal distribution, the 95% quantile isVaR(X, 95%) = E(X) + 1.64

√var(X), and can be estimated using an estimation of

the expected value and the variance. Note that the choice of the distribution shouldbe guided by some diagnostics, such as graphical tests (quantile-quantile plot for

123

A. Charpentier, A. Oulidi

instance) and goodness of fit tests. Some additional properties might also be requiredfor modeling issues, such as nice aggregation properties (stable or elliptical distribu-tions), or extreme value distribution (Pareto distribution). Using maximum likelihoodtechniques, least squares, methods of moments, etc, we estimate the parameters ofthe underlying distribution and then plug the parameter estimated into the quantileequation to obtain quantile estimates. Note that in the case of multiple risks (whichis the case in this paper), a multivariate joint distribution is necessary. If the vari-ance of a sum of risks can just be expressed through variances and correlations, thefull joint distribution is necessary when calculating the VaR of a sum. In that case, avery popular idea is find parametric distribution for individual risks, and to model thedependence structure using a parametric copula. Parametric technique are simple touse, and usually do not require not a long computation time. But in order to obtainconsistent estimate, only recent data should be used, and therefore, because of thesmall sample size (e.g. 250 trading days as we shall see on a Monte Carlo study). Butat least, this estimation error can be controlled, under the assumption that the para-metric model is correct : the main issue for risk managers is test whether the modelis relevant. Reliability of VaR estimates is then based on faith practitioners can havein their internal models (see for an interesting discussion on model errors). A bench-mark can be obtained using model-free estimation of VaR : this is precisely the goalof nonparametric techniques, where no (strong) assumption about the distribution ismade. The idea to model distribution is to use histograms, or more more sophisticatedkernel based estimators.

6 Quantile and Value-at-Risk estimation

Recall that the quantile at level

Q(X, α) = infx, P(X ≤ x) > α = F−1X (α) (11)

where F−1X denotes the generalized inverse of the distribution function FX (x) =

P(X ≤ x).

6.1 The Gaussian model: a fully parametric approach

In the Gaussian model, if X ∼ N (µ, σ 2), recall that Q(X, α) = µ + uασ , whereuα = Φ−1(α), Φ standing for the cumulative distribution function of the N (0, 1)

distribution.If X is not Gaussian, it may still be possible to use a Gaussian approximation. If the

variance is finite, (X − E(X))/σ might be closer to the Gaussian distribution, and theconsider the so-called Cornish-Fisher approximation (see Cornish and Fisher 1937 orHill and Davis 1968), i.e. Q(X, α) ≈ E(X) + zα

√X where

zα = Φ−1(α) + ζ1

6[Φ−1(α)2 − 1] (12)

+ ζ2

24[Φ−1(α)3 − 3Φ−1(α)] − ζ 2

1

36[2Φ−1(α)3 − 5Φ−1(α)], (13)

123

Estimating allocations for Value-at-Risk portfolio optimization

where ζ1 and ζ1 are given by

ζ1 = E([X − E]3)

E([X − E]2)3/2 and ζ2 = E([X − E]4)

E([X − E]2)2 − 3. (14)

6.2 Using extreme value results, a semi-parametric approach

Quantile estimation based on extreme value results can be seen as semi parametric.Given a n-sample Y1, . . . , Yn, let Y1:n ≤ Y2:n ≤ · · · ≤ Yn:n denotes the associatedorder statistics. One possible method is to use Pickands-Balkema-de Haan theorem[see Sect.6.4. in Embrechts et al. (1997) or Sect. 4.6. in Beirlant et al. (2006) forinstance]. The idea is to assume that for u large enough, Y − u given Y > u has aGeneralized Pareto distribution with parameters ξ and β, which can be estimatedusing standard maximum likelihood techniques. And therefore, if u = Yn−k:n for klarge enough, and if ξ > 0, denote by βk and ξk the maximum likelihood estimators ofthe Generalized Pareto distribution of sample Yn−k+1:n −Yn−k:n, . . . , Yn:n −Yn−k:n,

Q(Y, α) = Yn−k:n + βk

ξk

((n

k(1 − α)

)−ξk − 1

)(15)

An alternative is to use Hill’s estimator if ξ > 0,

Q(Y, α) = Yn−k:n(n

k(1 − α)

)−ξk, (16)

where ξk = 1k

∑ki=1 log Yn+1−i :n − log Yn−k:n (if ξ > 0).

6.3 Some nonparametric quantile estimates

The most convenient quantile estimator is based on order statistics. For instance, givena sample of size n = 200, the 190th largest value should be a relevant estimator of the95% quantile. More generally, the raw estimator of the αth-Value-at-Risk is then (X) isthe minimal price of a hedging action which added to the risk makes i ≤ αn < i + 1,where Fn is the empirical distribution sample. If αn is not an integer, it could beinteresting to consider a weighted average VaR(Y, α) = λYi :n +(1−λ)Yi+1:n . Or moregenerally, some weighted average of order statistics can be considered,

∑ni=1 λi Yi :n =∫ 1

0 λu F−1n (u)du, where the weight function depends on n and α. This technique has

been developed intensively (see e.g. Harrel and Davis 1982; Padgett 1986 or Park2006).

A completely different approach will be to consider a smoothed version of thecumulative distribution function, and to invert it, Q(Y, α) = F−1

K (α) where FK (y) =1

nh

∑ni=1 K (

y−Yih ).

123

A. Charpentier, A. Oulidi

Remark 3 Note that those two approaches can be formally by presented throughthe following analytical expression,

∫ 10 λu F−1

K (u)du, which is a weighted sum ofsmoothed estimators.

6.4 Transformation based estimates

In order to derive better estimates, Charpentier and Oulidi (2007) pointed out that itcould be more interesting, instead of estimating quantiles of Y , based on a sampleY1, . . . , Yn, to estimate the quantile of g(Y ), based on a sample g(Y1), . . . , g(Yn),where g is a nondecreasing function. Since g(Q(Y, α)) = Q(g(Y ), α), a naturalestimator of the quantile is then Q(Y, α) = g−1 Q(g(Y ), α), where the estimatorQ(g(Y ), α) can be any of the previous estimator.

Gustafsson et al. (2006) suggested to use this technique with g being the cumulativedistribution function of some generalized Champernowne distribution.

6.5 From VaR estimation to allocation estimation

The problem here can be written as follows: a estimated optimal allocation is defined asω∗

n = argminRn(ω), droping the constraints convience and understanding, whereRn(ω) is an estimator of the α-quantile of random variable ω′X , estimated usingsample X1, . . . , Xn. If ω∗ = argminR(ω), if Rn(ω) converges to R(ω) for everyω, it seems reasonable to expect that ω∗

n will converge to ω∗ as n → ∞.Theoretically, convergence—in probability—of the estimator of the risk measure

is too weak to insure the convergence of the optimal allocation, and functional conver-gence is necessary, strengthening the pointwise convergence (see e.g. Theorem 5.7. vander Waart 1998). Since quantile estimators can be obtained as L-statistics (wieghtedsums of order statistics), it can be possible to derive conditions such that functionalconvergence can be satisfied.

7 Application to portfolio optimization: a real-data based study

Consider the dataset of weekly log-returns, obtained from daily closing prices of 4European stock index, from 1991 to 1998: German DAX (Ibis), Switz SMI, FrenchCAC, and British FTSE. The data are sampled in business time, i.e., weekends andholidays are omitted. Table 1 summarizes statistics of the dataset, including the mean-variance optimal allocation, when the target expected weekly return is 0.35% (thistarget will remain unchanged in this study).

The algorithm in order to derive properties of the estimators of allocations is thefollowing. First, we need to generate samples of log-returns,

− semiparametric monte-carlo based estimation: assuming that sample X1,. . . , Xnhas been generated using distribution Lθ , θ can be estimated from those obser-vations, and then generate samples from Lθ : a monte carlo based estimator ofquantiles can be derived, for all possible allocation,

− bootstrap (nonparametric) estimation: since this first approach can induce modelerror if the underlying distribution family (Lθ , θ ∈ Θ) is misspecified,a fully

123

Estimating allocations for Value-at-Risk portfolio optimization

Table 1 Descriptive statistics of Eurostocks dataset

1 2 3 4Asset 1 Asset 2 Asset 3 Asset 4

Average returns −0.0032 −0.0040 −0.0021 −0.0022

Standard deviation 0.024 0.023 0.026 0.019

1.00 0.71 0.75 0.61 Asset 1

Pearson’s correlation 1.00 0.64 0.60 Asset 2

1.00 0.63 Asset 3

1.00 Asset 4

Optimal allocation 0.2277 0.5393 −0.2516 0.4846

Fig. 1 Scatterplot of weekly log-returns

nonparametric approach can also be considered. Estimators of quantiles can bederived using bootstraped samples (Figs. 1 and 2).

In this paper, we focus on the second approach, since we wish to avoid model errors.Given a scenario (a generated sample Xb

1, . . . , Xbn)

– for each possible allocation ω of a finite grid G, determine an estimation ofVaR(Z , α), where Z is the log-return associate to portfolio allocation ω, whichcan be estimated from sample Zb

1 , . . . , Zbn = ω′Xb

1, . . . ,ω′Xbn

– the optimal allocation is obtained at the (empirical) minimum Value-at-Risk forall the points of the allocation of the grid G.

123

A. Charpentier, A. Oulidi

Allo

catio

n (3

rd a

sset

)Allocation (4th asset)

Quantile

Value−at−Risk (75%) on the grid Value−at−Risk (75%) on the grid

Allocation (3rd asset)

Allo

catio

n (4

th a

sset

)−3 −2 −1 0 1 2

−3

−2

−1

01

2

Allo

catio

n (3

rd a

sset

)

Allocation (4th asset)Q

uantile

Value−at−Risk (97.5%) on the grid Value−at−Risk (97,5%) on the grid

Allocation (3rd asset)

Allo

catio

n (4

th a

sset

)

−3 −2 −1 0 1 2

−3

−2

−1

01

2

Fig. 2 Value-at-Risk for all possible allocations on the grid G (surface and level curves), with α = 75%on the left and α = 97.5% on the right

7.1 Optimal allocations as a function of the probability level α

Table 2 and Fig. 3 present the output of 10,000 simulated portfolio returns of the sameperiod of time, with optimal allocation based on a Value-at-Risk criteria for severalvalue for α. Note that the higher the quantile level the more we sell asset 3 to buy asset 4.

7.2 Optimal allocations based on different quantile estimates

In this section, five estimators of quantiles are considered,

1. raw estimator Q(Y, α) = Y[α·n]:n2. mixture estimator Q(Y, α) = ∑n

i=1 λi (α)Yi :n , which is the standard quantileestimate in R (see Hyndman and Fan 1996),

123

Estimating allocations for Value-at-Risk portfolio optimization

Table 2 Mean and standard deviation of estimated optimal allocation, for different quantile levels

Mean variance 75% 77.5% 80% 82.5% 85% 87.5% 90% 92.5% 95% 97.5%

Asset 1 0.2277 0.222(0.253)

0.206(0.244)

0.215(0.259)

0.251(0.275)

0.307(0.276)

0.377(0.241)

0.404(0.243)

0.394(0.224)

0.402(0.214)

0.339(0.268)

Asset 2 0.5393 0.550(0.141)

0.558(0.136)

0.552(0.144)

0.530(0.152)

0.500(0.154)

0.460(0.134)

0.444(0.135)

0.448(0.124)

0.441(0.121)

0.467(0.151)

Asset 3 −0.2516 −0.062(0.161)

−0.083(0.176)

−0.106(0.184)

−0.139(0.187)

−0.163(0.215)

−0.196(0.203)

−0.228(0.163)

−0.253(0.141)

−0.310(0.184)

−0.532(0.219)

Asset 4 0.4846 0.289(0.162)

0.319(0.179)

0.339(0.191)

0.357(0.204)

0.357(0.221)

0.359(0.205)

0.380(0.175)

0.410(0.153)

0.466(0.170)

0.726(0.200)

Fig. 3 Optimal allocations for different probability levels (α = 75, 77.5, 80, . . . , 95, 97.5%), with alloca-tion for the first asset (top left) up to the fourth asset (bottom right)

3. Gaussian estimator Q(Y, α) = Y + zαsd(Y ), where sd stands for the empiricalstandard deviation.

4. Hill’s estimator, with k = [n/5], Q(Y, α) = Yn−k:n( nk (1 − α))−ξk , where ξk =

1k

∑ki=1 log Yn+1−i :n

Yn−k:n (assuming that ξ > 0),

123

A. Charpentier, A. Oulidi

Fig. 4 Optimal allocations for different 95% quantile estimators, with allocation for the first asset (top left)up to the fourth asset (bottom right)

5. kernel based estimator is obtained as a mixture of smoothed quantiles, derived asinverse values of a kernel based estimator of the cumulative distribution function,i.e. Q(Y, α) = ∑n

i=1 λi (α)F−1(i/n).

From Fig. 4 we can observe that the Gaussian approximation is not efficient in thecase the returns are not normally distributed. Nonparametric estimators are interestingsince they are distribution free (and therefore, there should be no model error), andthe uncertainty is rather small, compared with parametric estimators. And from acomputational point of view, the raw estimator performs rather well, compared withmore robust estimators (which are time consuming due to multiple inversion andsmoothing).

8 Conclusion

We have seen in that paper that it was possible to obtain from a finite sample an estima-tion of the optimal mean-VaR portfolio (minimizing Value-at-Risk with a constrainton expected return). An algorithm to derive bounds on the estimation of optimal allo-cations. On the simulated dataset, we have seen that if calculation is much longer than

123

Estimating allocations for Value-at-Risk portfolio optimization

the optimal mean-variance portfolio, confidence bounds are quite similar. An otherpoint is that n = 250 past observed data is not sufficently large to estimate optimalallocation with a small confidence interval.

If estimation was somehow robust and efficient for 4 risky assets and 250 obser-vations, in real life with 100 (or even more) assets, much more past observations arenecessary.

Among possible extensions, we have assumed here that log-returns Xt ’s were inde-pendent. As pointed out in litterature, this assumption is not validated by stylized fact.One idea might be to look forward where the Xi ’s are driven by some stochasticprocess (e.g. ARCH or GARCH process).

References

Acerbi C, Tasche D (2002) On the coherence of expected shortfall. J Bank Financ 26:1487–1503Alexander G, Baptista A (2002) Economic implications of using a mean-VaR model for portfolio selection:

a comparison with mean-variance analysis. J Econ Dyn Control 26:1159–1193Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Finance 9:203–228Basak S, Shapiro A (2001) Value-at-risk management: optimzl policies and asset prices. Rev Financ Stud

14:371–405Bawa VS (1978) Safety-first, stochastic dominance, and optimal portfolio choice. J Financ Quant Anal

13:255–271Beirlant J, Goegebeur Y, Segers J, Teugel J (2006) Statistics of extremes. Wiley/Interscience, New YorkCharpentier A, Oulidi A (2007) Nonparametric quantile estimation. (submitted)Cohen JB, Zinbarg ED (1967) Investment analysis and portfolio management. Homewood, Ill.:

Richard D. Irwin, Inc.,Coles JL, Loewenstein U (1988) Equilibrium pricing and portfolio composition in the presence of uncertain

parameters. J Financ Econ 22:279–303Cornish EA, Fisher RA (1937) Moments and cumulants in the specification of distributions. Rev Int Stat

Inst 5:307–320Dowd K, Blake D (2006) After VaR : the theory, estimation, and insurance applications of quantile-based

risk measures. J Risk Insur 73:193–229Duffie D, Pan J (1997) An overview of value at risk. J Deriv 4:7–49Embrechts P, Kluppelberg C, Mikosh T (1997) Modeling extremal events. Springer, BerlinFeller GW (1968) Generalized asymptotic expansions of Cornish-Fisher type. Ann Math Stat 39:1264–1273Fishburn PC (1970) Utility theory for decision-making. Wiley, New YorkFöllmer H, Schied A (2004) Stochastic finance: an introduction in discrete time. Walter de Gruyter,

New YorkGaivoronski AA, Pflug G (2000) Value-at-Risk in portfolio optimization: properties and computational

approach. Working Paper, 00-2, Norwegian University of Sciences & TechnologyGustafsson J, Hagmann J, Nielsen JP, Scaillet O (2006) Local Transformation Kernel Density Estimation

of Loss. Cahier de Recherche 2006-10, HEC GenveHarrel FE, Davis CE (1982) A new distribution free quantile estimator. Biometrika 69:635–670Hill GW, Davis AW (1968) Generalized asymptotic expansions of Cornish-Fisher type. Ann Math Stat

39:1264–1273Hyndman RJ, Fan Y (1996) Sample quantiles in statistical packages. Am Stat 50:361–365Ingersoll Jonathan E (1987) Theory of financial decision making. Rowman & Littlefield, 496 pp.

ISBN:0847673596–9780847673599Jorion P (1997) Value at risk : the new benchmark for controlling market risk. McGraw-Hill, New YorkKast R, Luciano E, Peccati L (1998) VaR and optimization: 2nd international workshop on preferences and

decisions. Proceedings of the conference, Trento, 1–3 July 1998Klein RW, Bawa VS (1976) The effect of estimation risk on optimal portfolio choice. J Financ Econ

3:215–231Kroll Y, Levy H, Markowitz HM (1984) Mean-variance versus direct utility maximization. J Financ 39:

47–61

123

A. Charpentier, A. Oulidi

Levy H, Sarnat M (1972) Safety first–an expected utility principle. J Financ Quant Anal 7:1829–1834Lemus G (1999) Portfolio optimization with quantile-based risk measures. Ph.D. Thesis. MITLitterman R (1997) Hot spots and edges II. Risk 10:38–42Markowitz HM (1952) Portfolio selection. J Financ 7:77–91Merton RC (1992) Continuous time finance 2nd edn. Blackwell Publishers, OxfordPadgett WJ (1986) A kernel-type estimator of a quantile function from right-censored data. J Am Stat Assoc

81:215–222Park C (2006) Smooth nonparametric estimation of a quantile function under right censoring using beta

kernels. Technical Report (TR 2006-01-CP), Departement of Mathematical Sciences, Clemson Uni-versity

Quiggin J (1993) Generalized expected utility theory: the rank-dependent expected utility model.Kluwer/Nijhoff, Dordrecht/The Hague

Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21–41Roy AD (1952) Safety first and the holding of assets. Econometrica 20:431–449van der Waart AW (1998) Asymptotic statistics. Cambridge University Press, Londonvon Neumann J, Morgenstern O (1947) Theory of games and economic behaviour. Princeton University

Press, PrincetonWang S (1996) Premium calculation by transforming the layer premium density. ASTIN Bull 26:71–92Wirch J, Hardy M (1999) A synthesis of risk measures for capital adequacy. Insur Math Econ 25:337–348Yaari M (1987) A dual theory of choice under risk. Econometrica 55:95–115

123