Transcript Nessun titolo diapositiva

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
0, 
 
0, 
 be a function satisfying the following conditions:
Let  : 
• F is strictly increasing
• F is convex
• F(0)=0, F(1)=1 and F(+)= +
(F is a Young function)
Given X  L 
 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:
X
E[F(
)] = 1   , for X  0
H ( X )
H (0) = 0 by convention
with   [0,1)
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:
X  inf a 0 : E 
|X|
a
1
In general, if X is not essentially bounded or F 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 F 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/F1(1)
• H(X) is strictly monotone
• H (X)E[X]/ F1(1)
• H (X+Y) H (X)+H (Y)
• H (aX)=aH (X) for each a 0.
• H is convex
The proofs are straightforward and can be found in Haezendonck et al.
(1982). The simplest example is
H ( X ) =
X
p
(1   )
1
p
if Φ(x) = x p
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.
Haezendonck risk measure/1
For this reason in Bellini and Rosazza Gianin (2008) we considered
  ( X ) = inf x + H  (( X  x) + )
xR
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 infconvolution of risk measures (see Barrieu and El Karoui (2005)). Indeed,
letting
f
XH 
X 
g
X
we get
x if X x a.s.

 otherwise
 ( X ) = ( f  g )(X )
where
( f  g )( X ) = inf  f ( X  Y ) + g (Y )
xL
Haezendonck risk measure/2
Moreover, since (see again Bellini and Rosazza Gianin (2008) for the details)
( f  g )* = f * + g *
and
if E[ ] = 1
0
g * ( ) = 
+  if E[ ]  1
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
E

Xx


Xinf
x


1
x
that coincides with the CVaR in the Rockafellar and Uryasev (2000)
formulation.
Numerical computation of H(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 Hˆ  ,n ( X )
defined as the unique solution of
xi
1 n
F
(
) = 1

ˆ
n i =1
H , n ( X )
We start with a numerical experiment with
1 
Xqx 
1 q
x2
=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 Hˆ  ,n ( X ) by means of fsolve in Matlab environment.
Simulation results
Basic statistics of
ˆ
H
0 ,1000 ( X )
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, HFdFx0
is termed an M-functional (see for example Serfling (2001)). Orlicz premia are Mfunctionals 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 F is strictly increasing , then
ˆ ( X )  H ( X ) for n  
H
 ,n

Orlicz premia as M-functionals
Asymptotic normality
If F is strictly increasing and differentiable, and if
E[F 2 (
X
)]  + for t  U(H  ( X ))
t
then
ˆ ( X )  H ( X ) ) d N (0,  2 )
n (H
 ,n

where
 2 ( H  ( X ), X ) =
X
)]  (1   ) 2 ]
H ( X )
X
X
( E[
F' (
)])2
H ( X )
H ( X )
( H  ( X ))2 [ E[F 2 (
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
IC
x, F, T lim
T

1
F
x 
T
F

0

and if F is differentiable it becomes
x
)  (1   )]
H ( X )
X
x
E[
F' (
)]
H ( X )
H ( X )
H  ( X )[F (
IC( x, X , H  ( X )) =
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 (X)
We now deal with the properties of
ˆ ( X ) = inf ˆ ( X , x)
xR
where
ˆ ((X  x)+ ) + x
ˆ ( X , x) = H

We considered the Young functions
x + x2
ex 1
F1 ( x) = x, F 2 ( x) =
, F 3 ( x) =
2
e 1
and simulated from again from Uniform, Exponential, Pareto and Normal
distributions.
The parameter  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 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.
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