The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 494 Wednesday, February 4, 2004 1:09 PM

494 The Mathematics of Financial Modeling and Investment Management

1 = E[( 1 + Rit,)Mt ]

where

U′(Ct + 1 )

Mt + 1 = d-------------------------

U′()Ct

is a random variable known as the stochastic discount factor.

APPLICATION TO THE ASSET ALLOCATION DECISION^17

One of the most direct and widely used applications of MPT is asset

allocation. Because the asset allocation decision is so important, almost

all financial advisors determine an optimal portfolio for their clients—

be they institutional or individual—by performing an asset allocation

analysis using a set of asset classes.^18 They begin by selecting a set of

asset classes (e.g., domestic large cap and small cap stocks, long-term

bonds, international stocks, etc.). To obtain estimates of the returns and

volatilities and correlations they generally start with the historical per-

formance of the indexes representing these asset classes.^19 Exhibit 16.6

shows the major asset classes and an index commonly used to represent

the performance characteristics of that asset class (i.e., mean and stan-

dard deviation of return). These estimates are used as inputs in the

mean-variance optimization which results in an efficient frontier. Then

using some criteria (for instance, using Monte Carlo simulations to

compute the wealth distributions of the candidate portfolios), they pick

an optimal portfolio allocation. Finally, this portfolio is implemented

using either index or actively managed funds.

(^17) This illustration draws from Frank J. Fabozzi, Francis Gupta, and Harry M.
Markowitz, “Applying Mean-Variance,” Chapter 3 in Frank J. Fabozzi and Harry
M. Markowitz (eds.), The Theory and Practice of Investment Management (Hobo-
ken, NJ: John Wiley & Sons, 2002).
(^18) The following two studies conclude that asset allocation is a major determinant of
portfolio performance: Gary L. Brinson, Randolph Hood, and Gilbert Beebower,
“Determinants of Portfolio Performance,” Financial Analysts Journal (July/August
1986), pp. 39–44 and Gary L. Brinson, Randolph Hood, and Gilbert Beebower,
“Determinants of Portfolio Performance II: An Update,” Financial Analysts Journal
(May/June 1991), pp. 40–48.
(^19) Not all institutional asset managers use this method to obtain estimates of expect-
ed returns.