1. Optimal portfolios with Haezendonck risk measures Fabio Bellini Università di Milano – Bicocca Emanuela Rosazza Gianin Università di Milano - Bicocca X Workshop on Quantitative Finance Milano, 29/01/2009

2. Summary and outline Summary Haezendonck risk measures have been introduced by Haezendonck et al. (1982) and furtherly studied in Goovaerts et al. (2004) and Bellini and Rosazza Gianin (2008). They are a class of coherent risk measures based on Orlicz premia that generalizes Conditional Value at Risk. In this paper we investigate the problem of the numerical computation of these measures, pointing out a connection with the theory of M-functionals, and the optimal portfolios that they generate, in comparison with mean/variance or mean/CVaR criteria. Outline: - Definition and properties of Haezendonck risk measures - Numerical computation of Orlicz premia and Haezendonck risk measures - Orlicz premia as M-functionals - A portfolio example - Directions for further research

3. Orlicz premium principle/1 Letbe a function satisfying the following conditions:  is strictly increasing  is convex  (0)=0,  (1)=1 and   is a Young function) representing a potential loss, the Orlicz premium principle H  (X)Given has been introduced by Haezendonck et. al (1982) as the unique solution of the equation:

4. Orlicz premium principle/2 From a mathematical point of view it is a Luxemburg norm, that is usually defined on the Orlicz space L  in the following way: In general, if X is not essentially bounded or  does not satisfy a growth condition, the Luxemburg norm cannot be defined as the solution of an equation (see for example Rao and Ren (1991)). From an economical point of view, this definition is a kind of positively homogeneous version of the certainty equivalent; if  is interpreted as a loss function, it means that the agent is indifferent between X/H(X) and 1- .

5. Orlicz premium principle/3 Some properties of H  (X) are the following: it depends only on the distribution of X if X=K than H  (X)=K/    H  (X) is strictly monotone H  (X)  E[X]/    H  (X+Y)    X)+H  (Y) H  (aX)=aH  (X) for each a  H  is convex The proofs are straightforward and can be found in Haezendonck et al. (1982). The simplest example is H  (X) is not cash additive, that is it does not satisfy the translation equivariance axiom. REMARK: from Maccheroni et al. (2008) it is cash-subadditive in the sense of Ravanelli and El Karoui (2008) if and only if f  =0.

6.  aezendonck risk measure/1 For this reason in Bellini and Rosazza Gianin (2008) we considered and proved that it is a coherent risk measure, that we call Haezendonck risk measure. A similar construction was proposed in Goovaerts et al. (2004) and in Ben-Tal (2007) but subadditivity was proved only under additional assumptions. From a mathematical point of view, it can be seen as an example of inf- convolution of risk measures (see Barrieu and El Karoui (2005)). Indeed, letting we getwhere

7.  aezendonck risk measure/2 Moreover, since (see again Bellini and Rosazza Gianin (2008) for the details) and we see that inf convolution with g reduces the generalized scenarios to normalized probabilities, thus achieving cash invariance. The most important example is the CVaR: if  (x)=x, we get that coincides with the CVaR in the Rockafellar and Uryasev (2000) formulation.

8. Numerical computation of    X) In order to use Haezendonck risk measures in portfolio problems we need to compute Orlicz premia H  that in general don’t have an analytic expression. For this reason we investigate the properties of the natural estimator defined as the unique solution of We start with a numerical experiment with  =0 and q=0.5 that admits an analytical expression for H(X) that will be used for comparison. We simulated n=1000 values from three positive random variables: Uniform (0,1) Exponential with =1 Pareto with  =4 and computed numerically by means of fsolve in Matlab environment.

9. Simulation results Basic statistics of We have unbiasedness in all cases; normality is not rejected at 1% level in the Uniform and Exponential case while it is rejected at 5% level by a JB test in the Pareto case.

10. Orlicz premia as M-functionals If F is a distribution function, a functional H(F) of the form is termed an M-functional (see for example Serfling (2001)). Orlicz premia are M- functionals with hence the results about the asymptotic theory of M-functionals apply. We have the following: Asymptotic consistency If the Young function  is strictly increasing, then

11. Orlicz premia as M-functionals Asymptotic normality If  is strictly increasing and differentiable, and if then where

12. Influence function It is also possible to compute the influence function of H(F) in an explicit form. The well known definition for a distribution invariant functional T is and if  is differentiable it becomes in accordance with the previous expression of the asymptotic variance. The asymptotic behaviour of IC(x,F,H) is the same of  a situation that would be considered extremely undesirable in robust statistics but perhaps not so inappropriate in the financial cases.

13. Numerical computation of   (X) We now deal with the properties of and simulated from again from Uniform, Exponential, Pareto and Normal distributions. The parameter  was set to 0.95 in all cases. where We considered the Young functions

17. A portfolio example 5 main stocks in the SPMIB Index: Unicredito, Eni, Intesa, Generali, Enel 1000 daily logreturns from 14/11/2003 to 18/10/2007 (approx. 4 years)

18. Comparison of efficient frontiers The computation of the efficient frontiers is quite time-consuming, since there are several nested numerical steps: the numerical computation of H , the minimization over x in order to compute   (X), and the minimization over the portfolio weights in order to determine the optimal portfolio. The last two steps can actually be performed in a single one as in the CVaR case.

19. Comparison of optimal portfolios

20. Directions for further research Extension of the Haezendonck risk measure to Orlicz spaces More explicit dual representations Kusuoka representation of the Haezendonck risk measure Asymptotic results and influence function of the Haezendonck risk measure Comparison results

21. References Barrieu, P., El Karoui, N. (2005) “Inf-convolution of risk measures and Optimal Risk Transfer” Finance and Stochastics 9 pp. 269 – 298 Bellini, F., Rosazza-Gianin, E. (2008) “On Haezendonck risk measures”, Journal of Banking and Finance, vol.32, Issue 6, June 2008, pp. 986 – 994 Bellini, F., Rosazza-Gianin, E. (2008) “Optimal portfolios with Haezendonck risk measures”, Statistics and Decisions, vol. 26 Issue 2 (2008) pp. 89 – 108 Ben-Tal, A., Teboulle, M. (2007) “An old-new concept of convex risk measures: the optimized certainty equivalent” Mathematical Finance, 17 pp. 449 – 476 Cerreia-Vioglio, S., Maccheroni, F.,Marinacci, M., Montrucchio, L. (2009) “Risk measures: rationality and diversification”, presented in this conference

22. References El Karoui, N., Ravanelli, C. (2008) “Cash sub-additive risk measures under interest rate risk ambiguity” forthcoming in Mathematical Finance Goovaerts, M.J., Kaas, R., Dhaene, J., Tang, Q. (2003): "A unified approach to generate risk measures", ASTIN Bulletin 33/2, 173-191 Goovaerts, M.J., Kaas, R., Dhaene, J., Tang, Q. (2004): "Some new classes of consistent risk measures", Insurance: Mathematics and Economics 34/3, 505-516 Haezendonck, J., Goovaerts, M. (1982): "A new premium calculation principle based on Orlicz norms", Insurance: Mathematics and Economics 1, 41-53