statistique, théorie et gestion de portefeuille - Docs at ISFA

436 14. Gestion de Portefeuilles multimoments et équilibre de marché
can prove that
so that
∀p > q,
(E [|X| p ]) 1/p ≥ (E [|X| q ]) 1/q , (20)
µ − µ0
(E [|X| p ≤
1/p
])
µ − µ0
(E [|X| q . (21)
1/q
])
On the contrary, when the cumulants are used as risk measures, the generalized Sharpe ratios are not monotonically
decreasing, as exhibited by Procter & Gamble for instance. This can be surprising in view of our
previous remark that the larger is the order of the moments involved in a risk measure, the larger are the fluctuations
it is accounting for. Extrapolating this property to cumulants, it would mean that Procter & Gamble
presents less large risks according to C6 than according to C4, while according to the centered moments, the
reverse evolution is observed.
Thus, the question of the coherence of the cumulants as measures of fluctuations may arise. And if we accept
that such measures are coherent, what are the implications on the preferences of the agents employing such
measures ? To answer this question, it is informative to express the cumulants as a function of the moments.
For instance, let us consider the fourth order cumulant
C4 = µ4 − 3 · µ2 2 , (22)
= µ4 − 3 · C2 2 . (23)
An agent assessing the fluctuations of an asset with respect to C4 presents aversion for the fluctuations
quantified by the fourth central moment µ4 – since C4 increases with µ4 – but is attracted by the fluctuations
measured by the variance - since C4 decreases with µ2. This behavior is not irrational since it remains
globally risk-averse. Indeed, it depicts an agent which tries to avoid the larger risks but is ready to accept
the smallest ones.
This kind of behavior is characteristic of any agent using the cumulants as risk measures. It thus allows us to
understand why Procter & Gamble is more attractive for an agent sentitive to C6 than for an agent sentitive
to C4. From the expression of C6, we remark that the agent sensitive to this cumulant is risk-averse with
respect to the fluctuations mesured by µ6 and µ2 but is risk-seeker with respect to the fluctuations mesured
by µ4 and µ3. Then, is this particular case, the later ones compensate the former ones.
It also allows us to understand from a behavioral stand-point why it is possible to “have your cake and eat
it too” in the sense of (Andersen and Sornette 2001), that is, why, when the cumulants are choosen as risk
measures, it may be possible to increase the expected return of a portfolio while lowering its large risks, or in
other words, why its generalized Sharpe ratio may increase when one consider larger cumulants to measure
its risks. We will discuus this point again in section 9.
4.2 Marginal risk of an asset within a portofolio
Another important question that arises is the contribution of a given asset to the risk of the whole portfolio.
Indeed, it is crucial to know whether the risk is homogeneously shared by all the assets of the portfolio or if
it is only held by a few of them. The quality of the diversification is then at stake. Moreover, this also allows
for the sensitivity analysis of the risk of the portfolio with respect to small changes in its composition 2 ,
which is of practical interest since it can prevent us from recalculating the whole risk of the portfolio after a
small re-adjustment of its composition.
2 see (Gouriéroux et al. 2000, Scaillet 2000) for a sensitivity analysis of the Value-at-Risk and the expected shortfall.
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