statistique, théorie et gestion de portefeuille - Docs at ISFA

invested in the risky fund depends on the risk aversion of each agents, which may vary from an agent to
another one.
The composition of the market portfolio for such a heterogenous market is derived in appendix C. We find
that the market portfolio Π is nothing but the weighted sum of the mean-ρα(n) optimal portfolio Πn:
Π =
439
N
γnΠn, (28)
where γn is the fraction of the total wealth invested in the fund Πn by the n th agent.
n=1
Appendix D demonstrates that, for every asset i and for any mean-ρα(n) efficient portfolio Πn, for all n, the
following equation holds
µ(i) − µ0 = β i n · (µΠn − µ0) . (29)
Multiplying these equations by γn/β i n, we get
γn
βi n
for all n, and summing over the different agents, we obtain
γn
· (µ(i) − µ0) =
so that
with
n
β i n
· (µ(i) − µ0) = γn · (µΠn − µ0), (30)
n
γn · µΠn
− µ0, (31)
µ(i) − µ0 = β i · (µΠ − µ0), (32)
β i =
n
γn
βi n
−1
. (33)
This allows us to conclude that, even in a heterogeneous market, the expected excess return of each individual
stock is directly proportionnal to the expected excess return of the market portfolio, showing that
the homogeneity of the market is not a key property necessary for observing a linear relationship between
individual excess asset returns and the market excess return.
6 Estimation of the joint probability distribution of returns of several assets
A priori, one of the main practical advantage of (Markovitz 1959)’s method and its generalization presented
above is that one does not need the multivariate probability distribution function of the assets returns, as the
analysis solely relies on the coherent measures ρ(X) defined in section 2, such as the centered moments
or the cumulants of all orders that can in principle be estimated empirically. Unfortunately, this apparent
advantage maybe an illusion. Indeed, as underlined by (Stuart and Ord 1994) for instance, the error of
the empirically estimated moment of order n is proportional to the moment of order 2n, so that the error
becomes quickly of the same order as the estimated moment itself. Thus, above n = 6 (or may be n = 8) it is
not reasonable to estimate the moments and/or cumulants directly. Thus, the knowledge of the multivariate
distribution of assets returns remains necessary. In addition, there is a current of thoughts that provides
evidence that marginal distributions of returns may be regularly varying with index µ in the range 3-4
(Lux 1996, Pagan 1996, Gopikrishnan et al. 1998), suggesting the non-existence of asymptotically defined
moments and cumulants of order equal to or larger than µ.
15