Estimation risk for the VaR of portfolios driven by semi-parametric multivariate models Christian Francq

Jean-Michel Zakoïan

CREST and University of Lille, France

CFE 2017, London 17 December 2017

Supported by the ANR via the Project MultiRisk (ANR-16-CE26-0015-02) Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Objectives Setup: A portfolio of assets with time-varying composition, whose vector of individual returns follows a general dynamic model. Aims: Estimate the conditional risk of the portfolio (market risk). Evaluate the accuracy of the estimation (model risk): ⇒ quantify simultaneously the market and estimation risks. Compare univariate and multivariate approaches. Crystallized portfolios; Optimal (conditional) mean-variance portfolios; Minimal VaR porfolios.

Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Risk factors Dynamic model Conditional VaR parameter

Risk factors pt = (p1t , . . . , pmt )0 vector of prices of m assets yt = (y1t , . . . , ymt )0 vector of log-returns, yit = log(pit /pi,t−1 ) Vt value of a portfolio composed of µi,t−1 units of asset i, for i = 1, . . . , m: Vt =

m X

µi,t−1 pit ,

i=1

where the µi,t−1 are measurable functions of the past prices.

Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Risk factors Dynamic model Conditional VaR parameter

Return of the portfolio The return of the portfolio over the period [t − 1, t], assuming Vt−1 6= 0, is m X Vt − 1 = ai,t−1 exp(yit ) − 1 ≈ rt Vt−1 i=1 where rt =

m X i=1

with

ai,t−1 yit = a0t−1 yt ,

µi,t−1 pi,t−1

ai,t−1 = Pm

j=1 µj,t−2 pj,t−1

and at−1 = (a1,t−1 , . . . , am,t−1 )0 ,

Francq, Zakoian

,

i = 1, . . . , m,

yt = (y1t , . . . , ymt )0 .

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Risk factors Dynamic model Conditional VaR parameter

Conditional VaR of the portfolio’s return The conditional VaR of the portfolio’s return rt at risk level α ∈ (0, 1) is defined by h

i

Pt−1 rt < −VaR(α) t−1 (rt ) = α, where Pªt−1 denotes the historical distribution conditional on © pu , u < t . Consequence The evaluation of the portfolio’s conditional VaR requires either a dynamic model for the vector of risk factors yt , or a dynamic univariate model for the portfolio’s return rt .

Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Risk factors Dynamic model Conditional VaR parameter

Dynamic model for the vector of log-returns Let (yt ) be a strictly stationary and non anticipative solution of the multivariate model with conditional mean and GARCH-type errors: yt = mt (θ 0 ) + ²t , ²t = Σt (θ 0 )ηt iid

where ηt ∼ (0, Im ),

θ 0 ∈ Rd and

mt (θ 0 ) = m(yt−1 , yt−2 , . . . , θ 0 ),

Σt (θ 0 ) = Σ(yt−1 , yt−2 . . . . , θ 0 ). Examples of MGARCH

Thus, the portfolio’s return satisfies rt = a0t−1 yt = a0t−1 mt (θ 0 ) + a0t−1 Σt (θ 0 )ηt , and its conditional VaR at level α is (α) 0 0 VaR(α) t−1 (rt ) = −at−1 mt (θ 0 ) + VaRt−1 at−1 Σt (θ 0 )ηt .

¡

Francq, Zakoian

Conditional VaR of a portfolio

¢

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Risk factors Dynamic model Conditional VaR parameter

A simplification for elliptic conditional distributions In the multivariate volatility model yt = mt (θ 0 ) + Σt (θ 0 )ηt ,

(ηt ) iid (0, Im ),

assume that the errors ηt have a spherical distribution: A1:

d

for any non-random vector λ ∈ Rm , λ0 ηt = kλkη 1t ,

where k · k is the euclidean norm on Rm . Remark: This is equivalent to assuming that the conditional distribution of ²t given its past is elliptic. Under A1 we have VaRt(α) (r ) = −a0t−1 mt (θ 0 ) + °a0t−1 Σt (θ 0 )° VaR(α) η , −1 t °

°

¡ ¢

where VaR(α) η is the (marginal) VaR of η 1t . ¡ ¢

Example of spherical distributions Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Risk factors Dynamic model Conditional VaR parameter

Assumption on the conditional variance model

B1:

There exists a continuously differentiable function G : Rd 7→ Rd such that for any θ ∈ Θ, any K > 0, and any sequence (xi )i on Rm , m(x1 , x2 , . . . ; θ) = m(x1 , x2 , . . . ; θ ∗ ),

and

∗

K Σ(x1 , x2 , . . . ; θ) = Σ(x1 , x2 , . . . ; θ ), where θ ∗ = G(θ, K ). Examples of the CCC and DCC-GARCH

Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Risk factors Dynamic model Conditional VaR parameter

VaR parameter for an elliptic conditional distribution At the risk level α ∈ (0, 0.5), the conditional VaR of the portfolio’s return is 0 VaRt(α) (r ) = −a0t−1 mt (θ 0 ) + VaR(α) −1 t t−1 at−1 Σt (θ 0 )ηt

¡

¢

° ° = −a0t−1 mt (θ 0 ) + °a0t−1 Σt (θ 0 )° VaR(α) (η)

= −a0t−1 mt (θ ∗0 ) + ka0t−1 Σt (θ ∗0 )k,

where, under B1, ³ ´ θ ∗0 = G θ 0 , VaR(α) (η) .

The parameter θ ∗0 can be called conditional VaR parameter. Remark: The conditional VaR parameter does not depend on the portfolio composition summarizes the risk at a given level Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Multivariate estimation under ellipticity Relaxing the ellipticity assumption Univariate approaches

1

General framework

2

Estimating the conditional VaR Multivariate estimation under ellipticity Relaxing the ellipticity assumption Univariate approaches

3

Numerical comparison of the different VaR estimators

Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Multivariate estimation under ellipticity Relaxing the ellipticity assumption Univariate approaches

Estimating the conditional VaR parameter Observations: y1 , . . . , yn (+ initial values ey0 , ey−1 , . . . ). b n : estimator of θ 0 . θ e t (θ) = m(yt−1 , . . . , y1 , e m y0 , ey−1 , . . . , θ), e Σt (θ) = Σ(yt−1 , . . . , y1 , e y0 , ey−1 , . . . , θ), for t ≥ 1 and θ ∈ Θ. −1

b n ){y − m b e t (θ bt = Σ b 1t , . . . , η b mt )0 . Residuals: η t e t (θ n )}) = (η

Under the conditional sphericity assumption, an estimator of the conditional VaR at level α is (α)

∗

∗

0 b ) + ka0 Σ e b e t (θ VaR S,t−1 (r ) = −at−1 m n t−1 t (θ n )k,

where

n ¡ ¢o (α) η , b∗ = G θ b n , VaR θ n n

(α) ¡

VaR n

¢ b it |, 1 ≤ i ≤ m, 1 ≤ t ≤ n}. η = ξn,1−2α : (1 − 2α)-quantile of {|η Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Multivariate estimation under ellipticity Relaxing the ellipticity assumption Univariate approaches

Assumptions A2: (yt ) is a strictly stationary and nonanticipative solution. b n → θ 0 , a.s. and the following expansion A3: We have θ n X ¢ p ¡ b n − θ 0 oP=(1) p1 n θ ∆t−1 V (ηt ),

n t=1

where ∆t−1 ∈ F t−1 , V : Rm 7→ RK for some K ≥ 1, EV (ηt ) = 0, var{V (ηt )} = Υ is nonsingular and E∆t = Λ is full row Example of the Gaussian QML rank. A4: The functions θ 7→ m(x1 , x2 , . . . ; θ) and θ 7→ Σ(x1 , x2 , . . . ; θ) are C 1 . A5: |η 1t | has a density f which is continuous and strictly positive in a neighborhood of ξ1−2α (the (1 − 2α)-quantile of |η 1t |). Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Multivariate estimation under ellipticity Relaxing the ellipticity assumption Univariate approaches

Asymptotic distribution Asymptotic normality Under the previous assumptions p

µ

n

bn − θ0 θ ξn,1−2α − ξ1−2α h©

where Ω0 = E vec Σ−t 1 ¡

¶

¢ª0 n

Ã

L

→ N 0, Ξ :=

∂ vec (Σt ) ∂ϑ0

Ã

Ψ Ξ0θξ

Ξθξ ζ1−2α

!!

,

oi

, W α = Cov(V (ηt ), Nt ), γα = var(Nt ), with Nt = j=1 1{|ηjt | 0 and c0 (θ 0 )ηt admits a density fc which is continuous and strictly positive in a neighborhood of x0 = −b(θ 0 ) + ξα (θ 0 ).

Francq, Zakoian

Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators

Multivariate estimation under ellipticity Relaxing the ellipticity assumption Univariate approaches

Asymptotic distribution Estimator of the quantile of a linear combination of η t Under the previous assumptions (but without the sphericity assumption A1), µ ¶ α(1 − α) L 2 0 0 b n{ξn,α (θ n ) − ξα (θ 0 )} → N 0, σ := ω Ψω + 2ω ΛAα + 2 , fc (x0 )

p

where Aα = Cov(V (ηt ), 1{b(θ0 )−c0 (θ0 )ηt