16-Port Selection Mean Var Page 472 Wednesday, February 4, 2004 1:09 PM
472 The Mathematics of Financial Modeling and Investment Managementpricing theory such as the capital asset pricing model, which we discuss
in the next chapter, goes on to formalize the relationship that should
exist between asset returns and risk if investors constructed and selected
portfolios according to mean-variance analysis. In contrast to a norma-
tive theory, asset pricing theory is a positive theory—a theory that
hypothesizes how investors behave rather than how investors should
behave. Based on that hypothesized behavior of investors, we derive an
asset pricing model that provides the expected return is derived.
Our objective in this chapter is to explain the principles of mean-vari-
ance analysis and present a formal mathematical treatment for determin-
ing “efficient portfolios.” The extensions of Markowitz’s formulation
includes the case where a risk-free asset is available in the capital mar-
ket. This leads to efficient portfolio’s that dominate efficient portfolios
that can be constructed in a capital market in which there is no risk-free
asset. We then provide an application of how M-V analysis is used in
portfolio selection. While there have been many applications of M-V
analysis in the areas of finance and insurance, we present an application
to the asset allocation problem. This decision involves deciding how to
allocate funds across major asset classes.DIVERSIFICATION AS A CENTRAL THEME IN FINANCE
Conventional wisdom has always dictated “not putting all your eggs in
one basket.” In more technical terms, this old adage is addressing the
benefits of diversification. Markowitz quantified the concept of diversifi-
cation, or “undiversification” through the statistical notion of covari-
ance, or correlation. In essence, the old adage is saying that putting all
your money in investments that may all perform poorly at the same
time—that is, whose returns are highly correlated—is not a very prudent
investment strategy—no matter how small the chance is that any one
single investment will perform poorly. This is because if any one single
investment performs poorly, it is very likely, due to its high correlation
with the other investments, that the other investments are also going to
perform poorly, leading to the poor performance of the portfolio.
The concept of diversification is so intuitive and so strong that it has
been continuously applied to different areas within finance. Indeed, a
vast number of the innovations surrounding finance have either been an
application of the concept of diversification, or the introduction of new
methods of obtaining improved estimates of the variances and covari-
ances, thereby, allowing for a more precise measure of diversification,
and consequently, for a more precise measure of risk.