General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators
Estimation risk for the VaR of portfolios driven by semi-parametric multivariate models Christian Francq
Jean-Michel Zakoïan
CREST and University of Lille, France
Troisièmes Journées d’Econométrie de la Finance, JEF’2016 Rabat, November 18-19, 2016
Francq, Zakoian
Conditional VaR of a portfolio
General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators
Objectives
Estimate the conditional risk of a portfolio of assets (market risk) Setup: the portfolio’s composition is time-varying. The vector of individual returns follows a general dynamic model. Aims: Evaluate the accuracy of the estimation: ⇒ 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
Self-financing constraint
At date t, the investor may rebalance his portfolio in such a way that SF:
Pm
i=1 µi,t−1 pit
=
Pm
i=1 µi,t pit .
The value at time t of the portfolio bought at time t − 1 serves to buy the new portfolio at time t.
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 Under SF, 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−1 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 . Remark: The univariate approach seems simpler, but is likely to be inefficient, or even inconsistent when at is time-varying. 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 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
Conditional VaR parameter The conditional VaR is a stochastic process 0 VaRt(α) (r ) = −a0t−1 mt (θ 0 ) + VaR(α) −1 t t−1 at−1 Σt (θ 0 )ηt .
¡
¢
Depends on i) the mean-volatility parameter θ 0 and ii) the law of ηt . Under certain assumptions, VaR(α) t−1 (rt ) can be related to a so-called conditional VaR parameter θ ∗0 : VaRt(α) (r ) = −a0t−1 mt (θ ∗0 ) + ka0t−1 Σt (θ ∗0 )k −1 t
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 ²t = 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
Multi-step estimators
Remark: For multivariate GARCH models, multi-step estimators are often used: Variance targeting estimators (Francq, Horvath, Zakoian, 2013) Equation-by-Equation estimators of the individual volatilities + Estimation of the Cond. correlation (FZ, 2016) The Bahadur expansion in A3 holds for both 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
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
