statistique, théorie et gestion de portefeuille - Docs at ISFA

400 13. Gestion de portefeuille sous contraintes de capital économique
3 Value-at-Risk
3.1 Calculation of the VaR
We consider a portfolio made of N assets with all the same exponent c and scale parameters χi, i ∈
{1, 2, · · · , N}. The weight of the i th asset in the portfolio is denoted by wi. By definition, the Valueat-Risk
at the loss probability α, denoted by VaRα, is given , for a continuous distribution of profit and loss,
by
Pr{W (τ) − W (0) < −VaRα} = α, (47)
which can be rewritten as
Pr S < − VaRα
= α. (48)
W (0)
In this expression, we have assumed that all the wealth is invested in risky assets and that the risk-free
interest rate equals zero, but it is easy to reintroduce it, if necessary. It just leads to discount VaRα by the
discount factor 1/(1 + µ0), where µ0 denotes the risk-free interest rate.
Now, using the fact that FS(x) ∼ λ− FZ(x), when x → −∞, and where Z ∼ W(c, ˆχ), we have
1
Pr S < − VaRα
√2
c/2
VaRα
1 − Φ
, (49)
W (0)
W (0) ˆχ
λ−
as VaRα goes to infinity, which allows us to obtain a closed expression for the asymptotic Value-at-Risk
with a loss probability α:
VaRα W (0) ˆχ
21/c
Φ −1
1 − α
2/c , (50)
λ−
ξ(α) 2/c W (0) · ˆχ, (51)
where the function Φ(·) denotes the cumulative Normal distribution function and
ξ(α) ≡ 1
2 Φ−1
1 − α
. (52)
λ−
In the case where a fraction w0 of the total wealth is invested in the risk-free asset with interest rate µ0, the
previous equation simply becomes
VaRα ξ(α) 2/c (1 − w0) · W (0) · ˆχ − w0W (0)µ0. (53)
Due to the convexity of the scale parameter ˆχ, the VaR is itself convex and therefore sub-additive. Thus, for
this set of distributions, the VaR becomes coherent when the considered quantiles are sufficiently small.
The Expected-Shortfall ESα, which gives the average loss beyond the VaR at probability level α, is also
very easily computable:
α
ESα = 1
VaRu du (54)
α 0
= ζ(α)(1 − w0) · W (0) · ˆχ − w0W (0)µ0, (55)
where ζ(α) = 1
α
α 0 ξ(u)2/c du . Thus, the Value-at-Risk, the Expected-Shortfall and in fact any downside
risk measure involving only the far tail of the distribution of returns are entirely controlled by the scale
parameter ˆχ. We see that our set of multivariate modified Weibull distributions enjoy, in the tail, exactly the
same properties as the Gaussian distributions, for which, all the risk measures are controlled by the standard
deviation.
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