16-Port Selection Mean Var Page 492 Wednesday, February 4, 2004 1:09 PM
492 The Mathematics of Financial Modeling and Investment ManagementTheory (EVT) might help to determine the tails of some distributions. In
this way, as we have seen in Chapter 13, it becomes possible to manage
the risk associated with large movements. As observed by Jobst and
Zenios^14 the tails of the return distribution significantly affect portfolio
performance.
A new framework for portfolio selection with arbitrary distribu-
tions was proposed by Malevergne and Sornette.^15 Their framework is
based on transforming arbitrary variables into normal variables. The
distribution of the transformed variables is then determined via the
principle of entropy maximization.^16 They showed that the new trans-
formed variables conserve the structure of correlation of the original
variables as measured by copula functions. In this way they recovered
the multivariate distribution of the original variables.MULTIPERIOD STOCHASTIC OPTIMIZATION
The factor market models explored thus far are static linear regressions
with an underlying dynamic that is either exogenously given or consists
of the assumption of IID returns; these optimization models are myopic
one-period optimization models. From the point of view of investor
behavior, one-period models are based on the assumption that wealth is
consumed at the end of the period.
An investor must solve the problem of optimal portfolio selection.
This means that at every trading moment the investor has to revise the
selected portfolio and to decide what fraction of wealth is consumed
and what fraction is reinvested. Suppose that an investor is character-
ized by a time-separable utility function defined over the consumption
process. A time-separable utility function is such that the total utility is
the sum of utility in different periods, each discounted by an appropri-
ate time-discount factor. It is implicitly assumed that the utility derived
by the consumption of one unit at some future date is less than the util-
ity derived from the same consumption at the present date.
Call Ct consumption at time t. The investor’s consumption of period t
is a fraction of his or her wealth at the beginning of period t. The remaining(^14) Norbert J. Jobst and Stavros A. Zenios, “The Tail That Wags the Dog: Integrating
Credit Risk in Asset Portfolios,” The Journal of Risk Finance (Fall 2001), pp. 31–44.
(^15) Y. Malevergne and D. Sornette, “Higher-Moment Portfolio Theory with Multi-
variate Weibull Distributions,” unpublished paper.
(^16) The Principle of Entropy Maximization chooses the distribution that has the max-
imum entropy among those compatible with a set of constraints. In general, con-
straints are given by the values of empirically determined moments.