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

Summary and outlineSummaryHaezendonck risk measures have been introduced by Haezendonck et al. (1982) andfurtherly 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 : 0, 0, be a function satisfying the following conditions:

• is strictly increasing• is convex• (0)=0, (1)=1 and is a Young function)

X L 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:

)1,0[with

conventionby 0)0(

0for ,1)])(

([

H

XXH

XE

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:

X inf a 0 : E |X|a 1

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-.

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 • His convex

The proofs are straightforward and can be found in Haezendonck et al. (1982). The simplest example is p

p

p xΦ(x)X

XH

if

)1(

)( 1

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.

aezendonck risk measure/1

For this reason in Bellini and Rosazza Gianin (2008) we considered ))((inf)(

xXHxX

Rx

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

fX H X

gX x if X x a.s.

otherwise

we get where ))(()( XgfX )()(inf))(( YgYXfXgfLx

aezendonck risk measure/2

Moreover, since (see again Bellini and Rosazza Gianin (2008) for the details)

***)( gfgf

and

1][ if

1][ if 0)(*

E

Eg

we see that inf convolution with g reduces the generalized scenarios to normalizedprobabilities, thus achieving cash invariance.

The most important example is the CVaR: if (x)=x, we get

X infx

x EX x 1

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

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

)(ˆ, XH n

1))(ˆ

(1

,1 XH

x

n n

in

i

1X qx 1 qx2

=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 =4and computed numerically )(ˆ

, XH n by means of fsolve in Matlab environment.

Simulation results

Basic statistics of )(ˆ1000,0 XH

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

x,HFdFx 0

is termed an M-functional (see for example Serfling (2001)). Orlicz premia are M-functionals with

x, t xt 1

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

nfor )()(ˆ, XHXH n

Orlicz premia as M-functionals

Asymptotic normality

If is strictly increasing and differentiable, and if

then

where

))(for )]([ 2 XU(H tt

XE

),0() )()(ˆ( 2, NXHXHn dn

2

222

2

)]))(

(')(

[(

])1()])(

([[))((

)),((

XHX

XHX

E

XHX

EXH

XXH

Influence functionIt is also possible to compute the influence function of H(F) in anexplicit form. The well known definition for a distribution invariant functional T is

ICx,F,T lim 0

T1 F x TF

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 inappropriatein the financial cases.

)])(

(')(

[

)]1())(

()[(

))(,,(

XHx

XHX

E

XHx

XH

XHXxIC

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.

),(ˆinf)(ˆ xXXRx

where

xxXHxX ))((ˆ),(ˆ

We considered the Young functions

1

1)(,

2)( ,)( 3

2

21

e

ex

xxxxx

x

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 areseveral nested numerical steps: the numerical computation of H, the minimizationover 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.

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