The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 495 Wednesday, February 4, 2004 1:09 PM

Portfolio Selection Using Mean-Variance Analysis 495

EXHIBIT 16.6 Asset Classes and Commonly Used Indexes

Index Asset Class Inception Date

U.S. 30 day T-bill U.S. Cash 1/26

Lehman Brothers aggregate bond U.S. Bonds 1/76

S&P 500 U.S. Large Cap Equity 1/26

Russell 2000 U.S. Small Cap Equity 1/79

MSCI EAFE Europe/Japan Equity 1/70

MSCI EM Free Emerging Markets Equity 1/88

Source: Exhibit 3.6 in Frank J. Fabozzi, Francis Gupta, and Harry M. Markow-

itz, “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), p. 49.

Once the funds are allocated to portfolio managers who specialize

in the asset class, each portfolio manager selects the specific securities to

be included in the portfolio. The portfolio can be actively managed or

indexed. In fact, M-V analysis can be employed to construct the specific

securities from within an asset class.

The Inputs

There are a number of approaches that can be used to obtain estimates

of the inputs that are used in a mean-variance optimization, and all

approaches have their pros and cons. Since the use of historical returns

is the approach that is most commonly used, it may be useful to present

a discussion on this method.

As explained in Chapters 11 and 12, in the language of economet-

rics the above means that historical returns (i.e., the empirical average

of past returns), are an estimate of the expected values of returns. This

entails a model of returns, in particular a stationary model of returns.

The assumption that returns are independent and identically distributed

(IID) sequences^20 is the simplest model where historical returns are an

estimate of expected returns.

Exhibit 16.7 uses monthly returns over different and varying time peri-

ods to present the annualized historical returns for four market indexes.

One drawback of using the historical performance to obtain esti-

mates is clearly evident from this exhibit. Historical returns are not sta-

ble, the future does not repeat the past. This is one of the reasons

(^20) See Chapter 6 for the definition of an IID sequence.