13-Fat Tails-Scaling-Stabl Page 392 Wednesday, February 4, 2004 1:00 PM
392 The Mathematics of Financial Modeling and Investment Managementmal to a multivariate t CrVaR drops from 102.9 basis points to 96.6 basis
points but ES grows from 240.3 basis points to 256.2 basis points. This
happens because the t-distribution is more fat-tailed than the normal dis-
tribution. As a consequence, VaR underestimates the risk of large losses.
Though there are still questions as to whether asset prices have a
finite variance, there is little doubt that financial time series are not
Gaussian. Large events happen at a rate incompatible with Gaussian
behavior. This problem must be addressed from the point of view of
both risk management and financial optimization.
Many issues regarding risk management have been discussed in the
literature. A number of key issues are summarized by Mulvey who
points out the need to correctly address problems stemming from conta-
gion phenomena and from the possibility of joint actions such as those
occurring in market crashes.^36 A better understanding of the dynamics
of these events could lead to effective measures to protect market partic-
ipants from unnecessary risk.SUMMARY
■ Fat-tailed laws have been found in many economic variables
■ Fully approximating a finite economic system with fat-tailed laws
depends on an accurate statistical analysis of the phenomena, but also
on a number of the theoretical implications of subexponentiality and
scaling.
■ Modeling financial variables with stable laws implies the assumption of
infinite variance, which seems to contradict empirical observations.
■ Scaling laws might still be an appropriate modeling paradigm given the
complex interaction of distributional shape and correlations in price
processes.
■ Scaling laws might help in understanding not only the sheer size of eco-
nomic fluctuations but also the complexity of economic cycles.(^36) John M. Mulvey, “Risk Management Systems for Long-term Investors: Address-
ing/Managing Extreme Events,” Working Paper, May 2001, Operations Research
and Financial Engineering Department, Bendheim Center for Finance, Princeton
University.