X Workshop on Quantitative Finance Milano, 29/01/2009 . Optimal portfolios with Haezendonck risk measures. Fabio Bellini Università di Milano – Bicocca Emanuela Rosazza Gianin Università di Milano - Bicocca.
Optimal portfolios with Haezendonck risk measures
Università di Milano – Bicocca
Emanuela Rosazza Gianin
Università di Milano - Bicocca
be a function satisfying the following conditions:
representing a potential loss, the Orlicz premium principle H(X)
has been introduced by Haezendonck et. al (1982) as the unique solution
of the equation:
From a mathematical point of view it is a Luxemburg norm, that is usually defined on the Orlicz space Lf in the following way:
In general, if X isnotessentiallybounded or F doesnotsatisfy a growthcondition,
the Luxemburgnormcannotbedefinedas the solutionofanequation
(seeforexampleRao and Ren (1991)).
From an economical point of view, this definition is a kind of positively
homogeneous version of the certainty equivalent; if F is interpreted as a loss function, it means that the agent is indifferent between X/H(X) and
Ha (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 a=0.
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
Moreover, since (see again Bellini and Rosazza Gianin (2008) for the details)
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 F(x)=x, we get
that coincides with the CVaR in the Rockafellar and Uryasev (2000)
In order to use Haezendonck risk measures in portfolio problems we need to
compute Orlicz premia Ha 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
by means of fsolve in Matlab environment.
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.
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:
If the Young function F is strictly increasing , then
If F is strictly increasing and differentiable, and if
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 F 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 F, a situation that would be
considered extremely undesirable in robust statistics but perhaps not so inappropriate
in the financial cases.
We now deal with the properties of
We considered the Young functions
and simulated from again from Uniform, Exponential, Pareto and Normal
The parameter a was set to 0.95 in all cases.
The computation of the efficient frontiers is quite time-consuming, since there are
several nested numerical steps: the numerical computation of Ha, the minimization
over x in order to compute pa(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.