Joint inference on market and estimation risks in dynamic portfolios

Munich Personal RePEc Archive

Joint inference on market and estimation risks in dynamic portfolios

Francq, Christian and Zakoian, Jean-Michel

November 2015

Online at https://mpra.ub.uni-muenchen.de/68100/

MPRA Paper No. 68100, posted 27 Nov 2015 15:25 UTC

Joint inference on market and estimation risks in dynamic portfolios

Christian Francq^∗and Jean-Michel Zakoïan^†

Abstract

We study the estimation risk induced by univariate and multivariate methods for evaluating the conditional Value-at-Risk (VaR) of a portfolio of assets. The composition of the portfolio can be time-varying and the individual returns are assumed to follow a general multivariate dynamic model. Under sphericity of the innovations distribution, we introduce in the multi- variate framework a concept of VaR parameter, and we establish the asymptotic distribution of its estimator. A multivariate Filtered Historical Simulation method, which does not rely on sphericity, is also studied. We derive asymptotic confidence intervals for the conditional VaR, which allow to quantify simultaneously the market and estimation risks. The particular case of minimal variance and minimal VaR portfolios is considered. Potential usefulness, feasibility and drawbacks of the different approaches are illustrated via Monte-Carlo experiments and an empirical study based on stock returns.

Keywords: Confidence Intervals for VaR, DCC GARCH model, Estimation risk, Filtered Historical Simulation, Optimal Dynamic Portfolio.

1 Introduction

A large strand of the recent literature on quantitative risk management has been concerned with risk aggregation (see for instance Embrechts and Puccetti (2010) and the references therein). For a vector of one-period profit-and-loss random variablesǫ= (ǫ1, . . . , ǫm)^′, risk aggregation concerns the

∗CREST and University Lille 3, BP 60149, 59653 Villeneuve d’Ascq cedex, France. E-Mail: christian.francq@univ- lille3.fr

†Corresponding author: Jean-Michel Zakoïan, University Lille 3 and CREST, 15 boulevard Gabriel Péri, 92245 Malakoff Cedex, France. E-mail: zakoian@ensae.fr, Phone number: 33.1.41.17.77.25. The second author gratefully thanks the IDR ACP "Régulation et risques systémiques" for financial support.

risk implied by an aggregate financial position defined as a real-valued function of ǫ. For instance, under the terms of Basel II, banks often measure the risk of a vector ǫ of financial positions by the Value-at-Risk (VaR) ofa1ǫ1+· · ·+amǫm where theai’s define the composition of a portfolio.

Exact calculation of the risk associated with an aggregate position can represent a difficult task, as it requires knowledge of the joint distribution of the components of ǫ.

It is even more difficult, in a dynamic framework, to evaluate theconditional risk of a portfolio of assets or returns. The current regulatory framework for banking supervision (Basel II and Basel III), allows large international banks to develop internal models for the calculation of risk capital.

The so-called advanced approaches are based on conditional distributions, that is, conditional on the past, rather than marginal ones. In this article, we will focus on the VaR, arguably the most popular risk measure in finance and insurance due to its importance within the Basel II capital adequacy framework.

1.1 Conditional VaR of a dynamic portfolio

Let p_t = (p_1t, . . . , p_mt)^′ denote the vector of prices of m assets at time t. Let ǫ_t = (ǫ_1t, . . . , ǫ_mt)^′ denote the corresponding vector of log-returns, with ǫ_it = log(p_it/p_i,t−1) for i= 1, . . . , m. Let V_t denote the value at timet of a portfolio composed ofµ_i,t−1 units of asseti, fori= 1, . . . , m:

V₀ = Xm

i=1

µ_ip_i0, V_t= Xm

i=1

µ_i,t−1p_it, fort≥1 (1.1) where theµi,t−1 are measurable functions of the prices up to time t−1, and theµi are constants.

The return of the portfolio over the period[t−1, t]is, fort≥1, assuming that V_t−1 6= 0, V_t

V_t−1 −1 = Xm

i=1

a_i,t−1e^ǫit −1≈ Xm

i=1

a_i,t−1ǫ_it+a_0,t−1 where

a_i,t−1= µ_i,t−1p_i,t−1 Pm

j=1µ_j,t−2p_j,t−1, i= 1, . . . , m and a_0,t−1 =−1 + Xm

i=1

a_i,t−1.

We assume that, at date t, the investor may rebalance his portfolio under a "self-financing" con- straint.

SF: The portfolio is rebalanced in such a way that Pm

i=1µi,t−1pit=Pm

i=1µi,tpit.

In other words, the value at time tof the portfolio bought at time t−1equals the value at time t of the portfolio bought at time t. An obvious consequence of the self-financing assumption SF, is

that the change of value of the portfolio between t−1 and tis only due to the change of value of the underlying assets:

Vt−Vt−1= Xm i=1

µi,t−1(pi,t−pi,t−1).

Another consequence is that the weights a_i,t−1 sum up to 1, that is a_0,t−1 = 0. Thus, underSF we have _V^V_t−1^t −1≈ǫ^(P_t ⁾, where

ǫ^(P_t ⁾= Xm

i=1

a_i,t−1ǫ_it=a^′_t

−1ǫ_t, a_i,t−1 = µ_i,t−1p_i,t−1 Pm

j=1µ_j,t−1p_j,t−1, (1.2) for i = 1, . . . , m, and a_t−1 = (a1,t−1, . . . , am,t−1)^′. The conditional VaR of the portfolio’s return process(ǫ^(P_t ⁾) at risk levelα∈(0,1), denoted by VaR^(α)_t−1(ǫ^(P)), is defined by

Pt−1

h

ǫ^(P_t ⁾<−VaR^(α)_t−1(ǫ^(P))i

=α, (1.3)

where P_t−1 denotes the historical distribution conditional on {p_u, u < t}.

1.2 Univariate vs multivariate modeling of the portfolio’s dynamic

In order to estimate the conditional risk of the portfolio’s returnǫ^(P)_t from observationsǫ₁, . . . ,ǫ_n, two strategies can be advocated. A multivariate strategy requires a dynamic model for the vector of risk factors ǫ_t, while a univariate approach will be based on a dynamic model for the portfolio’s return (ǫ^(P_t ⁾). According to Bauwens, Laurent and Rombouts (2006), "it is probably simpler to use the univariate framework if there are many assets, but we conjecture that using a multivariate specification may become a feasible alternative. Whether the univariate "repeated" approach is more adequate than the multivariate one is an open question." These issues were tackled, by means of Monte-carlo experiments and real data analysis, by McAleer and da Veiga (2008), and Santos, Nogales and Ruiz (2013).

In fact, deriving a univariate model for the portfolio’s return may raise several difficulties.

i) Without further constraints on the past-dependent weights ai,t−1, the resulting process (ǫ^(P_t ⁾) might not be stationary (details will be given below). Needless to say that developing statistical inference procedures in this situation can be cumbersome.

ii) By embedding the weights into the stochastic process, the univariate approach does not fa- cilitate portfolio comparison. For instance the determination of an optimal portfolio in the mean-variance sense requires knowledge of the first two conditional moments of the vector process.

iii) More importantly, the univariate approach provides a VaR defined by P_t^∗₋₁h

ǫ^(P_t ⁾<−VaR^(α)_t−1^∗(ǫ^(P))i

=α, (1.4)

where P_t^∗₋₁ denotes the distribution conditional on {ǫ^(Pu ⁾, u < t}, which is different from the VaR defined in (1.3). The latter takes into account the full information brought by the past prices.

We now describe more thoroughly the multivariate approach.

1.3 Multivariate modeling of the risk factors

The multivariate approach is based on a model which is independent of the weight sequence. Con- sider a general multivariate model of the form

ǫ_t=m_t(ϕ₀) +Σ_t(ϑ0)η_t, (1.5) where (η_t) is a sequence of independent and identically distributed (iid)R^m-valued variables with zero mean and identity covariance matrix; them×mnon-singular matrixΣ_t(ϑ₀)and them×1vector m_t(ϕ₀) are specified as functions parameterized by a d-dimensional parameter θ₀ = (ϕ^′₀,ϑ^′₀) ∈ R^d1 ×R^d2 of the past values of ǫ_t:

m_t(ϕ₀) =m(ǫ_t−1,ǫ_t−2, . . . ,ϕ₀), Σ_t(ϑ₀) =Σ(ǫ_t−1,ǫ_t−2, . . . ,ϑ₀). (1.6) For the sake of generality, we do not consider a particular specification of the conditional meanm_t, or the conditional varianceΣ_t.¹

In view of (1.2)-(1.5), the portfolio’s return satisfies ǫ^(P_t ⁾ =a^′

t−1m_t(ϕ₀) +a^′

t−1Σ_t(ϑ₀)η_t, (1.7)

from which it follows that its conditional VaR at level α is given by VaR^(α)_t−1(ǫ^(P)) =−a^′

t−1m_t(ϕ₀) +VaR^(α)_t−1 a^′

t−1Σ_t(ϑ₀)η_t

. (1.8)

The VaR formula can be simplified if we assume that the errors η_t have a spherical distribution, that is,P η_t andη_thave the same distribution for any orthogonal matrix P. This is equivalent to assuming that

1 The most widely used specifications of multivariate GARCH models are discussed in Bauwens, Laurent and Rombouts (2006), Silvennoinen and Teräsvirta (2009), Francq and Zakoïan (2010, Chapter 11), Bauwens, Hafner and Laurent (2012), Tsay (2014, Chapter 7). Model (1.6) also includes multivariate extensions of the double-autoregressive models studied by Ling (2004).

A1: for any non-random vector λ∈R^m,λ^′η_t=^d kλkη1t,

where k · k denotes the euclidian norm onR^m,ηit denotes thei-th component ofη_t, and =^d stands for the equality in distribution. ² Under the sphericity assumption A1we have

VaR^(α)_t−1(ǫ^(P)) =−a^′_t

−1m_t(ϕ₀) +a^′_t

−1Σ_t(ϑ₀)VaR^(α)(η), (1.9) where VaR^(α)(η) is the (marginal) VaR of η_1t.

1.4 Estimation risk

Estimation risk refers to the uncertainty implied by statistical procedures in the implementation of risk measures. Uncertainty affects the estimation of risk measures, as well as the backtesting proce- dures used to assess the validity of risk measures. As far as the VaR of a portfolio is concerned, as defined in (1.9), it is clear that uncertainty results from the estimation of the model parameterθ₀, as well as from the estimation of the VaR ofη_1t. The econometric literature devoted to the estimation risk in dynamic models is scant. Christoffersen and Gonçalves (2005), and Spierdijk (2014) used re- sampling techniques to account for parameter estimation uncertainty in univariate dynamic models.

Escanciano and Olmo (2010, 2011) proposed corrections of the standard backtesting procedures in presence of estimation risk (and also of model risk). Gouriéroux and Zakoïan (2013) showed that estimation induces an asymptotic bias in the coverage probabilities and proposed a corrected VaR.

Francq and Zakoïan (2015a) introduced the notion of risk parameter (to be discussed below) and derived asymptotic confidence intervals for the conditional VaR of univariate returns.

1.5 Aims of the paper

The first aim of this paper is to study the asymptotic properties of different multivariate approaches for estimating the conditional VaR of a portfolio of risk factors (returns). One approach for estimat- ing conditional VaR’s requires sphericity of the innovations distribution. Based on formula (1.9), it consists in estimating parameter θ₀ in the first step, and replacing the VaR of η_t by an empirical quantile of the residuals. An alternative approach, known as the Filtered Historical Simulation (FHS) method in the literature (see Barone-Adesi, Giannopoulos and Vosper (1999), Mancini and Trojani (2011) and the references therein), is assumption-free on the innovations distribution. The second aim is to provide a method for constructing confidence intervals for the conditional VaR of

2Note that the choice of any other norm in this assumption would not be compatible with the assumed unit covariance matrix forη_t. Indeed, underA1we have Var(λ^′η_t) =λ^′λ=kλk²Var(η1t) =kλk².

portfolios, that is, a way to visualize the estimation risk. The third aim is to provide a framework for selecting portfolios, on the basis of their estimated conditional risks. The goal is to estimate the composition, as well as the risk, of dynamic "optimal portfolios" (in the sense of minimal con- ditional variance or minimal conditional VaR). The last aim is to compare, from a practical point of view, the univariate and multivariate approaches. Despite the previously underlined difficulties, the univariate approach is popular among practitioners because of its simplicity, and may provide good results in certain situations.

The rest of this paper is organized as follows. Section 2is devoted to the asymptotic properties of the estimators of the conditional VaR under the sphericity assumption. This assumption is relaxed in Section 3. Comparisons of the different approaches are proposed in Section 4. Proofs and complementary results are collected in the Appendix.

2 Conditional VaR estimation under sphericity

Under the sphericity assumption A1, a natural strategy for estimating the conditional VaR of a portfolio is to estimateθ₀by some consistent estimator bθ_n= (ϕb^′_n,ϑb^′_n)^′ in a first step, to extract the residuals and to estimate VaR^(α)(η) in a second step. For the first step, we will consider a general estimator satisfying some regularity conditions. For the second step, the sphericity assumption will allow us to interpret VaR^(α)(η) as the (1−2α)-quantile ξ_1−2α of the absolute residuals, and to estimate this quantile by an empirical quantile using all components of the first-step residuals.

LetΘ = Θ_ϕ×Θ_ϑ denote the parameter space, and assumeθ₀ ∈Θ. Letbθ_n = (ϕb^′_n,ϑb^′_n)^′ denote an estimator of parameterθ₀, obtained from observations ǫ₁, . . . ,ǫ_n and initial values eǫ₀,eǫ₋₁, . . .. The vector of residuals is defined by ηb_t =Σe⁻¹

t (ϑb_n){ǫ_t−fm_t(ϕb_n)}) = (bη_1t, . . . ,ηb_mt)^′.Let fm_t(ϕ) = m(ǫt−1, . . . ,ǫ₁,eǫ₀,eǫ₋₁, . . . ,ϕ), Σe_t(ϑ) = Σ(ǫt−1, . . . ,ǫ₁,eǫ₀,eǫ₋₁, . . . ,ϑ), for t≥ 1 and (ϕ^′,ϑ^′)^′ ∈ Θ. For α ∈ (0,1), let q_α(S) denote the α-quantile of a set S ⊂ R. In view of (1.9), an estimator based on the spherical assumptionof the conditional VaR at level α is

VaRd^(α)_S,t−1(ǫ^(P)) =−a^′

t−1fm_t(ϕb_n) +ka^′

t−1Σe_t(bϑ_n)kξ_n,1−2α, (2.1) where ξ_n,1−2α = q_1−2α({|bη_it|,1≤i≤m,1≤t≤n}). The latter estimator takes advantage of the fact that the components ofη_tare identically distributed under A1.

2.1 Asymptotic joint distribution of bθ_n and a quantile of absolute returns We start by introducing the assumptions that are employed to establish the asymptotic distribution of(bθ^′_n, ξn,1−2α).

A2: (ǫ_t) is a strictly stationary and nonanticipative³ solution of Model (1.5)-(1.6).

This assumption can be made explicit for particular classes of MGARCH models satisfying Model (1.5)-(1.6). We now assume that the estimator bθ_n admits a Bahadur representation.

A3: We have bθ_n→θ₀, a.s. Moreover, the following expansion holds

√n

bθ_n−θ_{0 oP(1)}

= 1

√n Xn t=1

∆_t−1V(η_t), (2.2)

where V(·) is a measurable function, V :R^m 7→ R^K for some positive integer K, and ∆_t−1 is ad×K matrix, measurable with respect to the sigma-field generated by {η_u, u < t}. The variables ∆_t and V(η_t) belong to L² with EV(η_t) = 0, var{V(η_t)} = Υ is nonsingular andE∆_t=Λ=



 Λ_ϕ Λ_ϑ



 is full row rank.

Assumption A3 holds for a variety of MGARCH models and estimators⁴ (see Appendix A for examples).

A4: For all x∈R^K,y∈R^m,

x^′V(η_t) +y^′ν_α(η_t) = 0, a.s. =⇒ x=0_K, y=0_m, where ν_α(η_t) = (1

{|η1t|<ξ1−2α}−1 + 2α, . . . ,1

{|ηmt|<ξ1−2α}−1 + 2α)^′.

Assumption A4 will be used to ensure the invertibility of the asymptotic covariance matrix of (bθ^′_n, ξn,1−2α). It is, in particular, satisfied if the random vectors η_t and V(η_t) have a positive density over R^m and R^K, respectively. The next assumption imposes smoothness of the functions mand Σwith respect to the parameter.

3In the sense thatǫtis a measurable function of the variablesη_uwithu≤t.

4In the univariate setting, the asymptotic theory of estimation for GARCH parameters has been extensively studied, in particular for the QMLE by Berkes, Horváth and Kokoszka (2003) and for the LAD (Least Absolute Deviation) estimator by Ling (2005). In the multivariate setting, the asymptotic properties of the QMLE or alternative estimators were established, for particular classes, by Comte and Lieberman (2003), Boswijk and van der Weide (2011), Francq and Zakoian (2012), Pedersen and Rahbek (2014), Francq, Horváth and Zakoian (2015) among others.

A5: The functionsϕ7→m(x1, x2, . . .;ϕ)andϑ7→Σ(x1, x2, . . .;ϑ)are continuously differentiable over Θ_ϕ andΘ_ϑ respectively.

The next theorem establishes the asymptotic normality of (bθ^′_n, ξ_n,1−2α). Let Ψ = E(∆_tΥ∆^′_t) =



 Ψ_ϕϕ Ψ_ϕϑ Ψ_ϑϕ Ψ_ϑϑ



 = (Ψ_·ϕ Ψ_·ϑ), Ω = Eh

vec Σ⁻¹

t

′ _∂

∂ϑ^′vec(Σ_t) i ,

W_α = Cov(V(η_t), Nt), γα = var(Nt), with Nt = Pm j=1

1{|ηjt|<ξ1−2α}−1 + 2α , and, denoting by f the density of |η_1t|, Ξ_θξ = ⁻_m¹n

ξ_1−2αΨ_·ϑΩ^′+_f(ξ¹

1−2α)ΛW_αo

, ζ_1−2α =

1 m²

n

ξ²_1−2αΩΨ_ϑϑΩ^′+_f^2ξ_(ξ^1−2α

1−2α)ΩΛ_ϑW_α+_f2(ξ^γ1−2α^α )

o .

Theorem 2.1. Assume that A1-A5 hold. Let α ∈ (0,0.5). Suppose that |η1t| admit a density f which is continuous and strictly positive in a neighborhood of ξ_1−2α. Then

√n



 bθ_n−θ₀ ξ_n,1−2α−ξ_1−2α



 → N^L



0,Ξ:=



 Ψ Ξ_θξ Ξ^′

θξ ζ_1−2α







. (2.3)

Moreover, Ξis nonsingular.

Details on how to estimate the asymptotic covariance matrix Ξcan be found in AppendixC.

2.2 Conditional VaR parameter

The notion of VaR parameter, introduced for univariate GARCH models by Francq and Zakoïan (2015a), allows to summarize the conditional risk, that is the joint effects of the volatility coefficients and the tails of the innovation process, in a single vector of coefficients. Its extension to the multivariate framework requires the following assumption.

A6: There exists a continuously differentiable function G : R^d2+1 7→ R^d2 such that for any ϑ∈Θ_ϑ, any K >0, and any sequence (x_i)_i onR^m

KΣ(x₁,x₂, . . .;ϑ) =Σ(x₁,x₂, . . .;ϑ^∗), where ϑ^∗ =G(ϑ, K).

In other words, a change of the scale in the components of η can be compensated by a change of the variance parameter. This assumption is obviously satisfied for all commonly used parametric forms of Σ_t(ϑ).⁵ Under sphericity and the stability-by scale assumption A6 on the volatility functionΣ_t(·), the conditional VaR can be expressed in function of the expected returns vector and

5For instance, in the case of the BEKK-GARCH(1,1) model (C.1), withϑ= (vec(A)^′,vec(B)^′,vec(C)^′)^′, we find ϑ^∗= (Kvec(A)^′,vec(B)^′, K²vec(C)^′)^′.

a reparameterized volatility matrix. Letα <1/2, so that VaR^(α)(η)>0under A1. It follows from A6 that

VaR^(α)_t−1(ǫ^(P)) =−a^′_t

−1m_t(ϕ₀) +ka^′_t

−1Σ_t(ϑ^∗₀)k (2.4) where

ϑ^∗₀=Gn

ϑ₀,VaR^(α)(η)o

. (2.5)

The new parameter θ^∗₀ = (ϕ^′₀,ϑ^∗₀^′)^′ is referred to as the conditional VaR parameter, for a given risk level. It does not depend on the portfolio composition. An estimator of the conditional VaR parameter can be defined as

θb^∗_n= (ϕb^′_n,ϑb^∗

′

n)^′ where ϑb^∗_n=G

ϑb_n,VaRd^(α)_n (η)

with obvious notations. The asymptotic properties of bθ^∗_n are a direct consequence of Theorem 2.1.

Corollary 2.1 (CAN of the VaR-parameter estimator). Under the assumptions of Theorem 2.1,

√n

bθ^∗_n−θ^∗_{0 L}

→ N

0,Ξ^∗:= ˙GΞG˙^′ where

G˙ =





I_d1 0_d

1×(d+1)

0_(d+1)

×d1

h∂G(ϑ,ξ)

∂(ϑ^′,ξ)

i

(ϑ0,ξ1−2α)



.

The asymptotic distribution of bθ^∗_n provides a quantification of the estimation risk.

2.3 Asymptotic confidence intervals for the VaR’s of portfolios

In view of (2.4), the estimator in (2.1) of the conditional VaR of the portfolio at levelα writes VaRd^(α)_S,t−1(ǫ^(P)) =−a^′_t

−1m_t(ϕb_n) +ka^′_t

−1Σ_t(ϑb^∗_n)k. (2.6) Let Ξb^∗ denote a consistent estimator of Ξ^∗. By the delta method, an approximate (1 −α0)%

confidence interval (CI) forVaR_t(α) has bounds given by VaRd^(α)_S,t−1(ǫ^(P))± 1

√nΦ⁻¹(1−α₀/2)n

δ^′_t−1Ξb^∗δ_t−1o1/2

, (2.7)

where

δ^′_t−1 = a^′

t−1

∂m(˜ ϕb_n)

∂ϕ^′

1 2ka^′

t−1Σe_t(ϑb^∗_n)k(a^′

t−1⊗a^′

t−1)∂vecfH_t(bϑ^∗_n)

∂ϑ^′

! ,

fH_t(·) = Σe_t(·)Σe^′_t(·), and Φ⁻¹(u) denotes the u-quantile of the standard Gaussian distribution, u ∈ (0,1). Drawing such CIs allows to take into account the estimation risk inherent to the

700 720 740 760 780 800

0.200.250.30

Time

True and estimated VaR

Figure 1: True 1%-VaR (full black line) and estimated 95%-confidence interval (dotted blue line) for the 1%-VaR, on a simulation of a fixed portfolio of a bivariate BEKK.

evaluation of the VaR of the portfolio. Note that the levelα₀ of risk estimation is independent from the market risk levelα.

An illustration is displayed in Figure 1, for the simulation of a bivariate BEKK model. The model parameters were estimated on 700 observations. The figure provides the true and estimated conditional 1%-VaRs, fort >700, as well a CIs at 95% for the true conditional VaR, of a portfolio with fixed composition. This graph allows to visualize simultaneously the market risk (through the magnitude of the VaR) and the estimation risk (through the width of the CIs).

2.4 Optimal dynamic portfolios

The portfolio with the smallest variance (the mean-variance efficient portfolio, that we call hereafter Markowitz’s portfolio) is

ǫ^(P_t ^)∗ =ǫ^′_ta^∗_0,t−1, a^∗_0,t−1 = Σ⁻²

t (ϑ₀)e e^′Σ⁻²

t (ϑ₀)e. (2.8)

The theoretical conditional VaR of this portfolio is obtained by computing the opposite of the α-quantile of a^∗_0,t^′₋₁Σ_t(ϑ₀)η₁, which is simply given by

VaR^(α)_t−1 ǫ^(P_t ^)∗

=a^∗_0,t^′ ₋₁Σ_t(ϑ₀)ξ_1−2α = 1 q

e^′Σ⁻²

t (ϑ₀)e

ξ_1−2α (2.9)

under the sphericity assumption. Different alternative types of optimal portfolios have been intro- duced in the finance literature. In particular, several papers developed portfolio selection based on

VaR (see for instance Alexander and Baptista (2002), Campbell, Huisman and Koedijk (2001)). In the following, we derive the optimal dynamic composition of a portfolio that minimizes the VaR at levelα. Such a portfolio can be called optimal-VaR portfolio at levelα.

Under the sphericity assumptionA1, the conditional VaR of the portfolio’s return process(ǫ^(P_t ⁾) at risk levelα is given by (1.9) which, omitting the parameter, writes

VaR^(α)_t−1(ǫ^(P)) =−a^′

t−1m_t+a^′

t−1Σ_tξ_1−2α:=q_t−1(a_t−1),

where a_t−1 satisfies e^′a_t−1 = 1. Let a^∗_α,t−1 := arg min_{a|e′a=1}q_t−1(a), the composition of the optimal-VaR portfolio in the spherical case. Let

∆^(α)_t−1 = e^′Σ⁻²

t m_t2

− e^′Σ⁻²

t e

m^′_tΣ⁻²

t m_t

+ e^′Σ⁻²

t e

ξ₁²_−2α. (2.10) Proposition 2.1. Under the sphericity assumption A1, the optimal-VaR portfolio at time t exists and is unique if and only if∆^(α)_t−1 >0. The optimal composition is given by

a^∗_α,t−1= Σ⁻²

t (mt+λe) e^′Σ⁻²

t (mt+λe) where λ= −e^′Σ⁻²

t m_t+ q

∆^(α)_t−1 e^′Σ⁻²

t e (2.11)

and the optimal VaR is q_t−1(a^∗_α,t−1) =λ.

In the particular case where m_t and e are colinear, that is m_t =m_te where m_t ∈ R, we find that a^∗_α,t−1 reduces to ^{Σ−2t e}

e^′Σ⁻²_t e := a^∗_0,t−1, which is the optimal composition in the mean-variance sense. Note thata^∗_0,t−1 = lim_α→0a^∗_α,t−1. In this case, the optimal-VaR portfolio coincides with the Markowitz portfolio and this portfolio does not depend onα. Interestingly, this property no longer holds when m_t 6= m_te: the optimal portfolio in (2.11) clearly depends on the risk level α. More precisely, the difference between the VaRs of the optimal-VaR and the Markowitz portfolios is

q_t−1(a^∗_0,t−1)−q_t−1(a^∗_t−1) = e^′Σ⁻²

t e

m^′_tΣ⁻²

t m_t

− e^′Σ⁻²

t m_t2

e^′Σ⁻²

t e

τ_t−1 ≥0,

where τ_t−1 = e^′Σ⁻²

t e1/2

ξ_1−2α+ q

∆^(α)_t−1. The nonnegativity of the numerator follows from the Cauchy-Schwarz inequality. This inequality is strict unless if m_t and e are colinear. Notice that the difference between the two VaRs increases with the non colinearity of theses two vectors. On the other hand, whenα tends to 0, the difference vanishes.

3 Conditional VaR estimation without the sphericity assumption

In this section, we develop a method which does not require symmetries of the conditional distri- bution, inherent to the sphericity assumption.

3.1 FHS estimator and asymptotic CIs

To derive asymptotic results, we slightly modify the statistical framework by assuming that the estimator bθ_n is based on past observations ǫ_t−n, . . . ,ǫ_t−1. We will use the FHS approach which relies on

i) interpreting the conditional VaR at timetas theα-quantile of a linear combination (depending ont) of the components of the innovations;

ii) replacing the innovations by the GARCH residuals and computing the empirical α-quantile of the estimated linear combination.

The conditional VaR of the portfolio return is VaR^(α)_t−1(ǫ^(P)) = VaR^(α)_t−1

a^′

t−1m_t(ϕ₀) +a^′

t−1Σ_t(ϑ₀)η_t = −a^′

t−1m_t(ϕ₀) − q_α(t;ϑ₀) where q_α(t;ϑ) denotes the theoreticalα-quantile of c^′_t(ϑ)η₁, with the (considered as) non random vector c^′_t(ϑ) =a′

t−1Σ_t(ϑ).

The conditional VaR at time t can thus be interpreted as the sum of the conditional mean and a quantile of a time-varying linear combination of the components of the iid noise. It can be estimated by

VaRd^(α)_{F HS,t−1}(ǫ^(P)) =−a^′

t−1m_t(ϕb_n)−q_n,α(t;θb_n), where qn,α(t;bθ_n) =qα

{c^′_t(ϑb_n)ηb_s, t−n≤s≤t−1} .

Let c :Θ_ϑ 7→R^m and b:Θ_ϕ 7→ R denote continuously differentiable vector-valued functions.

Letξα(θ)denote the theoreticalα-quantile ofb(ϕ)+c^′(ϑ)η_t(θ), whereη_t(θ) =Σ⁻¹

t (ϑ){ǫ_t−m_t(ϕ)}. Letξ_n,α(θ) =q_α({b(ϕ) +c^′(ϑ)η_t(θ),1≤t≤n}). We need to introduce the following identifiability assumption.

A7: For all x∈R^K, y∈R, x^′V(η_t) +y(1_b(ϕ

0)+c^′(ϑ0)η_t<ξα(θ0)−α) = 0, a.s. =⇒ x= 0_K, y= 0.

LetA_α =Cov(V(η_t),1

{b(ϕ₀)+c^′(ϑ0)η_t<ξα(θ0)}), ω^′ =

c^′(ϑ₀)E(C_t)− ∂b

∂ϕ^′(ϕ₀) d^′_α

(c^′(ϑ₀)⊗I_m)E(Ω^∗_t)− ∂c

∂ϑ^′(ϑ₀)

,

where d_α=E(η_t|b(ϕ₀) +c^′(ϑ0)η_t=ξα(θ0))and

Ω^∗

t =







I_m⊗e^′₁ ... Im⊗e^′_m





(Im⊗Σ⁻¹

t ) ∂

∂ϑ^′ {vec(Σ_t)},

C_t =

I_m⊗vec^′ ∂m_t

∂ϕ^′







I_d1 ⊗Σ⁻¹

t e₁ ... I_d1 ⊗Σ⁻¹

t e_m





.

The following result establishes the asymptotic distribution ofξ_n,α(bθ_n).

Theorem 3.1. Assume thatA2, A3, A7hold. Suppose that the variablec^′(ϑ₀)η_t admits a density fc which is continuous and strictly positive in a neighborhood of x0 =ξα(θ0)−b(ϕ₀). Then

√n{ξ_n,α(θb_n)−ξ_α(θ₀)} → N^L

0, σ² :=ω^′Ψω+ 2ω^′ΛA_α+α(1−α) f_c²(x₀)

. Moreover σ² >0.

This theorem can be used to derive CIs for the conditional VaR at timetof the portfolio return, with b(ϕ) = a′

t−1m_t(ϕ) and c^′(ϑ) = a′

t−1Σ_t(ϑ). A Nadaraya-Watson estimator of d_α is, with standard notation,

bd_α,t= P_t−1

s=t−nηb_sK_h

b(ϕb_n) +c^′(ϑb_n)ηb_s−ξ_n,α(bθ_n) P_t−1

s=t−nK_h

b(ϕb_n) +c^′(ϑb_n)ηb_s−ξ_n,α(bθ_n) .

A consistent estimator σˆ_t²₋₁ of σ² (based on the n observations anterior to time t−1) can be ob- tained by replacing the other theoretical quantities introduced before the theorem by their empirical counterparts, and by using the approach described in AppendixCto compute the derivatives ofΣ_t andm_t for particular models. An approximate (1−α₀)%CI for VaR^(α)_t−1(ǫ^(P)) is thus given by

VaRd^(α)_{F HS,t−1}(ǫ^(P))± 1

√nΦ⁻¹(1−α₀/2)ˆσ_t−1. (3.1) 3.2 Efficiency comparisons in the static case

In this section, we compare the efficiencies of the multivariate and univariate approaches for es- timating the VaR of a simplistic portfolio. We consider a static framework in which, in (1.5), m(·) = 0 and the matrix Σ_t(ϑ₀) is constant and diagonal, Σ_t(ϑ₀) =Σ(ϑ₀) =diag(σ₀₁, . . . , σ_0m), withϑ₀ = (σ²₀₁, . . . , σ²_0m)^′. Moreover, the portfolio satisfies

a_t−1=a= (a₁, . . . , a_m)^′, where a₁, . . . , a_m≥0, and Xm j=1

a_j = 1.

Such a portfolio can be called static and it is obtained by taking in (1.1) the dynamic weights µ_i,t−1 =V_t−1a_i/p_i,t−1.⁶ The return’s portfolio a′ǫ_t is thus iid and its conditional VaR is constant.

Under the sphericity assumptionA1, we have

VaR^(α)_t−1(ǫ^(P)) =ka^′Σ(ϑ₀)kξ_1−2α={ea^′ϑ₀}^1/2ξ_1−2α,

where ea = (a²₁, . . . , a²_m)^′. Let ϑb_n = (σb²_n1, . . . ,bσ_nm² )^′ the Gaussian QMLE of ϑ₀, with bσ_ni² =

1 n

Pn

t=1ǫ²_it.Under the sphericity assumption, the (constant) conditional VaR can be estimated by VaRd^(α)_S,t−1(ǫ^(P)) =ka^′Σ(bϑ_n)kξ_n,1−2α ={ea^′ϑb_n}^1/2ξ_n,1−2α.

On the other hand, the FHS method, without the sphericity assumption, reduces to a univariate method in this setting. Indeed,

c^′_t(ϑb_n)ηb_s=a^′Σ(ϑb_n)Σ⁻¹(ϑb_n)ǫ_s=a^′ǫ_s,

and the estimator VaRd^(α)_{F HS,t−1}(ǫ^(P)) is simply −q_α({a^′ǫ₁, . . . ,a^′ǫ_n}). An alternative uni- variate method exploits the symmetry of the distribution of a^′ǫ_t: let VaRd^(α)_U,t−1(ǫ^(P)) = q_1−2α({|a^′ǫ₁|, . . . ,|a^′ǫ_n|}).

The following result compares the asymptotic distributions of those three estimators of VaR^(α)_t−1(a′ǫ_t), when the distribution ofη_tis Gaussian. Letφdenote the probability density function of the standard normal law.

Corollary 3.1. For the static model ǫ_t = Σ(ϑ₀)η_t, where Σ(ϑ₀) = diag(σ₀₁, . . . , σ_0m) and η_t ❀ N(0,I_m) we have

√n

VaRd^(α)_S,t−1(ǫ^(P))−VaR^(α)_t−1(a^′ǫ_t) L

→ N 0, σ_S²(α,a) ,

√n

VaRd^(α)_{F HS,t−1}(ǫ^(P))−VaR^(α)_t−1(ǫ^(P)) L

→ N 0, σ_{F HS}² (α,a) ,

√n

VaRd^(α)_U,t−1(ǫ^(P))−VaR^(α)_t−1(ǫ^(P)) L

→ N 0, σ_U²(α,a) , where

σ²_S(α,a) = ξ₁²_−2α 2

Pm i=1a⁴_iσ_0i⁴ Pm

i=1a²_iσ_0i² − Pm

i=1a²_iσ²_0i m

+2α(1−2α) 4φ²(ξ_1−2α)

Pm i=1a²_iσ_0i²

m ,

σ²_{F HS}(α,a) = α(1−α) φ²(ξ1−2α)

Xm i=1

a²_iσ_0i², σ_U²(α,a) = 2α(1−2α) 4φ²(ξ1−2α)

Xm i=1

a²_iσ_0i².

Moreover, σ_S²(α,a)< σ²_U(α,a)< σ²_{F HS}(α,a) whenm≥2.

6Symmetrically, it is possible to take fixed units of each asset in the composition of the portfolio. A portfolio will be calledcrystallizedif, for eachi= 1, . . . m, we haveµi,t−1=µifor allt.

Remark 3.1. For the static Gaussian model withm≥2, the multivariate estimator is thus asymp- totically strictly more efficient than the univariate estimator, and the efficiency ratio is given by

σ²_S(α,a) σ_U²(α,a) = 1

m

"

1 + ξ₁²_−2αφ²(ξ_1−2α) α(1−2α)

( 1 m

P_m

i=1a⁴_iσ⁴_0i

1 m

Pm

i=1a²_iσ_0i²2 −1 )#

.

It follows that

1

m ≤ σ_S²(α,a) σ²_U(α,a) ≤ 1

m

1 + (m−1)ξ₁²_−2αφ²(ξ_1−2α) α(1−2α)

. (3.2)

The lower bound is reached when the weight of each assetiin the portfolio is proportional to1/σ_i. The upper bound is reached when the portfolio reduces to one asset (aoi = 0for all except one i).

It is maximized, forα= 0.069..., by 0.652 + ^0.348_m .

Remark 3.2. The computations required to obtain the asymptotic variance σ_S²(α,a) are hardly extendable to the case whereη_tfollows another spherical distribution than the Gaussian. Simulation experiments reported in Appendix E show that for some fat tailed distributions the univariate method may be more accurate than the multivariate method.

Remark 3.3(Estimation effect on the asymptotic accuracies). In the multivariate estimation of the VaR, the estimation ofϑ₀ occurs in two places: in the estimation of{ea^′ϑ₀}^1/2 and in the estimator ξ_n,1−2αof the residuals quantile. To separate the two effects, let us introduce the infeasible estimator of the VaR

VaRg^(α)_t−1(a^′ǫ_t) ={ea^′ϑ₀}^1/2ξ_n,1−2α. The asymptotic variance eσ_S²(α,a) of √

n

VaRg^(α)_t−1(a′ǫ_t)−VaR^(α)_t−1(a′ǫ_t)

is given by

e

σ_S²(α,a) = 1 m

Xm i=1

a²_iσ_0i²

−ξ₁²_−2α

2 +2α(1−2α) 4φ²(ξ_1−2α)

and the ratio of asymptotic efficiency of the univariate estimator with respect to this theoretical estimator is independent of the portfolio,

e σ_S²(α,a) σ_U²(α,a) = 1

m

1−ξ²_1−2αφ²(ξ_1−2α) α(1−2α)

.

Unsurprisingly, this ratio varies in 1/m, the quantile ξ_n,1−2α being based on m times more obser- vations than the univariate estimator of the VaR. The negative second term in the bracket comes from the fact that, in the Gaussian framework, quantiles based on residuals are more accurate than