statistique, théorie et gestion de portefeuille - Docs at ISFA

386 12. Gestion des risques grands et extrêmes
returns on the index returns, which shows that there is significantly less
linear correlation between P 1 and the index (correlation coefficient of 0.52
for both the equally weighted and the minimum variance P 1 ) compared
with P 2 and the index (correlation coefficient of 0.73 for the equally weighted
P 2 and of 0.70 for the minimum variance P 2 ). Theoretically, it is possible
to construct two random variables with small correlation coefficient
and large λ and vice versa. Recall that the correlation coefficient and the
tail-dependence coefficient are two opposite end-members of dependence
measures. The correlation coefficient quantifies the dependence between
relatively small moves while the tail-dependence coefficient measures the
dependence during extreme events. The finding that P 1 comes with both
the smallest correlation and the smallest tail-dependence coefficients suggests
that they are not independent properties of assets. This intuition is
in fact explained and encompassed by the factor model since the larger
β is, the larger the correlation coefficient and the larger the tail dependence.
Diversifying away extreme shocks may provide a useful diversification
tool for less extreme dependences, thus improving the potential
usefulness of a strategy of portfolio management based on tail dependence
proposed here.
As a final remark, the almost identical values of the coefficients of tail
dependence for negative and positive tails show that assets that are the
most likely to suffer from the large losses of the market factor are also
those that are the most likely to take advantage of its large gains. This
has the following consequence: minimising the large concomitant losses
between the stocks and the market means renouncing the potential
concomitant large gains. This point is well exemplified by our two portfolios
(see figure 3): P 2 obviously underwent severe negative co-movements
but it also enjoyed large gains with the large positive movements
of the index. In contrast, P 1 is almost completely decoupled from the
large negative movements of the market but is also insensitive to the large
positive movements of the index. Thus, a good dynamic strategy seems
to be: invest in P 1 during bearish or trend-less market phases and prefer
P 2 in a bullish market. ■
Yannick Malevergne is a PhD student at the University of Nice-Sophia
Antipolis and at the ISFA Actuarial School – University of Lyon. Didier
Bingham N, C Goldie and J Teugel, 1987
Regular variation
Cambridge University Press, Cambridge
Boyer B, M Gibson and M Lauretan, 1997
Pitfalls in tests for changes in correlations
International Finance Discussion Paper 597, Board of the Governors of the Federal
Reserve System
Coles S, J Heffernan and J Tawn, 1999
Dependence measures for extreme value analysis
Extremes 2, pages 339–365
Embrechts P, A McNeil and D Straumann, 1999
Correlation: pitfalls and alternatives
Risk May, pages 69–71
Embrechts P, A McNeil and D Straumann, 2002
Correlation and dependence in risk management: properties and pitfalls
In Risk Management: Value at Risk and Beyond, edited by M Dempster, pages
176–223, Cambridge University Press, Cambridge
Hull J and A White, 1987
The option pricing on assets with stochastic volatilities
Journal of Finance 42, pages 281–300
Hult H and F Lindskog, 2002
Multivariate extremes, aggregation and dependence in elliptical distributions
Forthcoming in Advances in Applied Probability 34(3)
Ledford A and J Tawn, 1998
Concomitant tail behaviour for extremes
Advances in Applied Probability 30, pages 197–215
REFERENCES
Appendix: empirical estimation of the
coefficient of tail dependence
We show how to estimate the coefficient of tail dependence
between an asset X and the market factor Y related by the relation
(6) where ε is an idiosyncratic noise uncorrelated with X.
Given a sample of N realisations {X 1 , X 2 , ... , X N } and
{Y 1 , Y 2 , ... , Y N } of X and Y, we first estimate the coefficient β using
the ordinary least square estimator. Let β ^ denote its estimate. Then,
using Hill’s estimator, we obtain the tail index ν^ of the factor Y:
⎡ k 1
ˆ = ⎢ ∑ logY −logY
⎣⎢
k j=
1
νk j, N k, N
where Y 1, N ≥ Y 2, N ≥ ... ≥ Y N, N are the order statistics of the N realisations
of Y. The constant l is non-parametrically estimated with
the formula:
FXu X
l = lim ≃
u→1
−1
F u Y
for k = o(N), which means that k must remain very small with
respect to N but large enough to ensure an accurate determination
of l. Figure 2 presents l ^ as a function of k/N.
Finally, using equation (7), the estimated λ ^ is:
ˆ
max , ˆ
+ 1
λ =
νˆ
{ 1
βˆ
} l
Sornette is a CNRS research director at the University of Nice-Sophia
Antipolis and professor of geophysics at the University of California at
Los Angeles. They acknowledge helpful discussions with Jean-Paul
Laurent. This work was partially supported by the James S McDonnell
Foundation twenty-first century scientist award/studying complex
system. e-mail: Yannick.Malevergne@unice.fr and sornette@unice.fr
Malevergne Y and D Sornette, 2002a
Investigating extreme dependences: concepts and tools
Working paper, available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id =
303465
Malevergne Y and D Sornette, 2002b
Tail dependence of factor models
Working paper, available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id =
301266
Mandelbrot B, 1997
Fractals and scaling in finance: discontinuity, concentration
Springer-Verlag, New York
Nelsen R, 1998
An introduction to copulas
Lectures Notes in Statistics 139, Springer Verlag, New York
Ross S, 1976
The arbitrage theory of capital asset pricing
Journal of Economic Theory 17, pages 254–286
Rubinstein M, 1973
The fundamental theorem of parameter-preference security valuation
Journal of Financial and Quantitative Analysis 8, pages 61–69
Sharpe W, 1964
Capital asset pricing: a theory of market equilibrium under conditions of risk
Journal of Finance 19, pages 425–442
Taylor S, 1994
Modeling stochastic volatility
Mathematical Finance 4, pages 183–204
−1
Y
( )
( )
kN ,
kN ,
⎤
⎥
⎦⎥
WWW.RISK.NET ● NOVEMBER 2002 RISK 133