Review of Markowitz’s portfolio selection theory

Harry Markowitz (1952, 1959) developed the first and most famous portfolio selection
model which showed how to build a portfolio of assets with the optimal risk and
return characteristics.
Markowitz’s model starts with a collection of assets for which it is assumed one
knows the expected returns and risks as well as all the pair-wise correlation of the
returns. Here risk is defined as the standard deviation of return.
It is a fairly strong assumption to assume that these statistics are known. The
model further assumes that the asset returns are modeled as a standard multivariate
normal distribution, so, in particular, each asset’s return is a standard normal
distribution.
Thus the assets are completely described by their expected return and their pairwise
covariances of returns
E[ri ] and
Covariance(ri , rj )óE[rirj ]ñE[ri ]E[rj ]
respectively, where ri is the random variable of return for the ith asset. Under these
assumptions Markowitz shows for a target expected return how to calculate the exact
proportion to hold of each asset so as to minimize risk, or equivalently, how to
minimize the standard deviation of return. Figure 10.2 depicts the theoretical
workings of the Markowitz model. Two different portfolios of assets held by two
different institutions have different risk and return characteristics.
While one may slightly relax the assumptions in Markowitz’s theory, the assumptions
are still fairly strong. Moreover, the results are sensitive to the inputs; two
users of the theory who disagree on the expected returns and covariance of returns
may calculate widely different portfolios. In addition, the definition of risk as the
standard deviation of returns is only reasonable when returns are a multi-normal
distribution. Standard deviation is a very poor measure of risk. So far there is no
consensus on the right probability distribution when returns are not a multi-normal
distribution.
Nevertheless, Markowitz’s theory survives because it was the first portfolio theory
to quantify risk and return. Moreover, it showed that mathematical modeling could
vastly improve portfolio theory techniques. Other portfolio selection models are
described in Elton and Gruber (1991). 294