162 Optimizing Optimization
may or may not be reasonable (see end notes). Early assumptions were that
distributions were bell shaped, i.e., followed a standard normal distribution.
Common sense says otherwise. Since one cannot lose an infinite amount of
money, even if you are a hedge fund, the distribution must be truncated on the
downside and it must be positively skewed in the long run. Only to the extent
that the estimate of the joint distribution of returns of a portfolio is reason-
able should one put any faith in the veracity of an efficient frontier. We have
expended considerable effort to obtain reasonable estimates of the joint distri-
butions of portfolios, which leads us to the following conclusions:
● Optimal solutions to a mathematical problem are often on the boundary of possible
solutions. This causes problems in applying mathematics to real-world situation as the
mathematical model used to describe the situation is often only an approximation. So
the mathematical solutions to the model will often be extreme and since the model is
only approximate the solution to the model may be far from optimal. Think of the
case in which the returns of the possible investments are thought to be known with
certainty. The optimal solution might be a portfolio of 100% in alternative invest-
ments like oil futures. People who behave as if they know or their model knows with
certainty what is going to happen are unknowingly taking a dangerous amount of risk.
● Even when it is only assumed that the joint probability of returns is known, limiting
solutions to some efficient frontier will give extreme portfolios that may be far from
optimal if the probability model does not fit reality. For example, many probability
models have thin tails that may lead to underestimating probabilities of large losses.
This may, for example, lead to portfolios with too much weight in equities.
● To the extent we can accurately describe the joint distribution of returns, we should get
reasonably reliable estimates of efficient portfolios. If that statement is true, the pioneer-
ing work of the innovators discussed elsewhere should prove beneficial. However, if the
input to the optimizer is seriously flawed so will be the output (GIGO).
7.2 Part 1: The Forsey–Sortino Optimizer
This is a model we built after Brian Rom terminated his relationship with
the Pension Research Institute. It has never been marketed because we have
no interest in becoming a software provider. Neither do we want to keep our
research efforts a secret. Therefore, we will provide an executable version and
the source code for the Forsey – Sortino Optimizer on the Elsevier web site
(http://booksite.elsevier.com/Sortino/ - Password: SAT7TQ46SH27). Be advised,
we have no intention of supporting the software in any way. Our intention is to
provide a starting point from which other researchers around the world can
make improvements and in that way make a contribution to the state of the art.
7.2.1 Basic assumptions
We begin with the assumption that the user wants to maximize the geometric
average rate of return in a multiperiod framework. Therefore, the three-parameter
lognormal distribution suggested by Atchison and Brown should provide a better