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Optimal portfolios with Haezendonck risk measures

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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

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

Orlicz premium principle/1

Let

be a function satisfying the following conditions:

- F is strictly increasing
- F is convex
- F(0)=0, F(1)=1 and F(+)= +
- (F is a Young function)

Given

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:

Orlicz premium principle/2

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

1-a.

Orlicz premium principle/3

- Some properties of H(X) are the following:
- it depends only on the distribution of X
- if X=K than Ha (X)=K/F-1(1-a)
- Ha(X) is strictly monotone
- Ha (X)E[X]/ F-1(1-a)
- Ha (X+Y) Ha (X)+Ha (Y)
- Ha (aX)=aHa (X) for each a 0.
- Ha is convex
- The proofs are straightforward and can be found in Haezendonck et al. (1982). The simplest example is

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.

Haezendonck 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 get

where

Haezendonck 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 F(x)=x, we get

that coincides with the CVaR in the Rockafellar and Uryasev (2000)

formulation.

Numerical computation of Ha(X)

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

- a=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 a=4
- and computed numerically

by means of fsolve in Matlab environment.

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.

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 F is strictly increasing , then

Orlicz premia as M-functionals

Asymptotic normality

If F is strictly increasing and differentiable, and if

then

where

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 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.

Numerical computation of pa(X)

We now deal with the properties of

where

We considered the Young functions

and simulated from again from Uniform, Exponential, Pareto and Normal

distributions.

The parameter a was set to 0.95 in all cases.

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)

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 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.

Comparison of optimal portfolios

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

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

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