Introduction Duality Applications Conclusions and Extensions Aggregating Risk Capital, with an Application to Operational Risk 2007 version Paul Embrechts 1 Giovanni Puccetti 2 1 Department of Mathematics ETH Zurich, CH-8092 Zurich, Switzerland 2 Department of Mathematics for Decisions, University of Firenze, 50134 Firenze, Italy available at www.dmd.unifi.it/puccetti P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze Bounding Risk Measures
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Introduction Duality Applications Conclusions and Extensions
Aggregating Risk Capital, with an Application toOperational Risk
2007 version
Paul Embrechts1 Giovanni Puccetti2
1Department of Mathematics ETH Zurich, CH-8092 Zurich, Switzerland
2Department of Mathematics for Decisions, University of Firenze, 50134 Firenze, Italy
available atwww.dmd.unifi.it/puccetti
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
The present talk is mainly based on the following papers:
JMVA Embrechts, P. and G. Puccetti (2006). Bounds for functions ofmultivariate risks.J. Mult. Analysis, 97(2), 526–547.
F&S Embrechts, P. and G. Puccetti (2006b). Bounds for functionsofdependent risks.Finance Stoch. 10(3), 341–352.
GRIR Embrechts, P. and G. Puccetti (2006c). Aggregating risk capital,with an application to operational risk.Geneva Risk. Insur.Rev., 31(2), 71–90.
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
The problem at hand
The problem at hand
On some probability space (Ω,A, P), consider a random vector
X := (X1, . . . ,Xn)
of n one-period financial losses or insurance claims,
and fix its marginal dfsF1, . . . ,Fn.
Thejoint df of the random vectorXis not completely determined by theFi’s.
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
The problem at hand
There are infinitely many distributions for the vectorX which areconsistent with the initial choice of the marginals.
Figure:Two bivariate dfs havingN(0, 1)-marginals and the same correlation
Given an aggregating functionψ : Rn → R, which is the df giving theworst-possible Value-at-Risk (VaR) for ψ(X) ?
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
Value-at-Risk
The Value-at-Risk at probability levelα for ψ(X) is the maximumaggregate loss which can occur with probabilityα, α ∈ [0, 1].
If G (the df ofψ(X)) is strictly increasing,VaRα(ψ(X)) is the unique thresholdt at whichF(t) = α, i.e. F−1(α).
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
Value-at-Risk
Searching for the worst-possible VaR means looking for
mψ(s) := inf P[ψ(X) < s] : Xi v Fi, i = 1, . . . , n, s ∈ R.
Indeed, according to the definition of VaR, we have
VaRα(ψ(X)) ≤ m−1ψ (α), α ∈ [0, 1].
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
Duality theorem
mψ(s) is a linear problem over a convex feasible space of measures.Therefore, it admits adual representation.
Main Duality Theorem (Ruschendorf (1982))
mψ(s) = inf P[ψ(X) < s] : Xi v Fi, i = 1, . . . , n
= 1− inf
n∑
i=1
∫
fidFi : fi ∈ L1(Fi), i ∈ N s.t.
n∑
i=1
fi(xi) ≥ 1[s,+∞)(ψ(x)) for all x ∈ Rn
.
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
Duality theorem
Some remarks on the dual problem
mψ(s), as well as its dual counterpart, is very difficult to solve.Solutions are known only in few cases.
• Whenn = 2; see Ruschendorf (1982).
• Whenn > 2, the only explicit solution we know is givenin Ruschendorf (1982) for the sum of risks uniformly distributedon the unit interval
Even if we do not solve the dual problem,dual admissible functions provide bounds on the solutions which
are conservative from a risk management viewpoint
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
Standard/Dual bounds
We callstandard bounds those bounds obtained by choosingpiecewise-constant dual choices.
• Standard bounds are those typically obtained from elementaryprobability; see: Denuit, Genest, and Marceau (1999). Theyaresharp whenn = 2.
We calldual bounds those bounds obtained by choosingpiecewise-linear dual choices.
• Dual bounds arebetter than standard bounds whenn > 2 but areavailable only forψ = +.
• Extensions are stated for portfolios of vectors; see [JMVA]
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
• Large differences between the two VaR-aggregation methods.
• Starting from a maximum in the independence set-up,∆
becomes smaller as the strength of dependence increases.
• Under the comonotonic assumption,∆ = 0 due to VaR additivity.
• Coming soon: a more sophisticatedsoft model
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
Conclusions
Conclusions
• The worst-possible VaR for a non-decreasing function ofdependent risks can be calculated when the portfolio istwo-dimensional.
• When dealing with more than two risks, the problem gets muchmore complicated and we provide adual bound which we proveto be better than the standard one generally used in the literature.
• OpRisk VaR-aggregation leads to problems anddiversification effects have to be handled with care.
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
Final Conclusions and possible extensions
Extensions (more research is needed!)
• Exact VaR bounds whenn > 2, calculation of bounds for otherportfolio functionsψ
• Basel II has some issues to solve (2008+)
• Problems of scaling when fixing marginal dfs(Market+ Credit+ Op Risk).
• For a textbook treatment, see
Visit the book zone:www.ma.hw.ac.uk/˜mcneil/book/index.html
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
Introduction Duality Applications Conclusions and Extensions
Final Conclusions and possible extensions
Acknowledgements
The second author would like to thank
RiskLab, ETH Zurich
for financial support.
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze
Bounding Risk Measures
References
For Further Reading I
Denuit, M., C. Genest, andE. Marceau (1999). Stochastic bounds on sums of dependent risks.Insurance Math. Econom. 25(1), 85–104.
Dutta, K. and J. Perry (2006). A tale of tails: an empirical analysis of loss distribution modelsfor estimating operational risk capitalWorking Papers, Federal Reserve Bank of Boston
Embrechts, P., A. Hoing, and A. Juri (2003). Using copulae to bound the Value-at-Risk forfunctions of dependent risks.Finance Stoch. 7(2), 145–167.
Embrechts, P., A. Hoing, and G. Puccetti (2005). Worst VaR scenariosInsurance Math.Econom., 37(1), 115–134.
Moscadelli, M. (2004). The modelling of operational risk: experience with the analysis of thedata collected by the Basel Committee. Preprint, Banca d’Italia.
Ruschendorf, L. (1982). Random variables with maximum sums. Adv. in Appl. Probab. 14(3),623–632.
Williamson, R. C. and T. Downs (1990). Probabilistic arithmetic. I. Numerical methods for
calculating convolutions and dependency bounds.Internat. J. Approx. Reason. 4(2), 89–158.
P. Embrechts and G. Puccetti ETHZ Zurich, DMD Firenze