The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 507 Wednesday, February 4, 2004 1:09 PM

Portfolio Selection Using Mean-Variance Analysis 507

EXHIBIT 16.17 The Benefits and Costs of Constraining an Efficient Frontier

Maximum Allocation to

EAFE International

Unconstrained Equity = 10.0%

Asset Class A′ B′ A′′ B′′

U.S. bonds 40.4% 15.1% 43.1% 20.1%

U.S. large cap equity 15.8 27.8 26.9 45.1

U.S. small cap equity 16.1 18.6 20.0 24.8

EAFE international equity 27.7 38.5 10.0 10.0

Expected return 9.39% 10.61% 9.20% 10.26%

Standard deviation 9.00% 12.00% 9.00% 12.00%

Cost of constraint — — 19 bps 35 bps

Note: Assumes annual rebalancing.

Source: Exhibit 3.14 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. 57.

worth it for an investor whose optimal appetite for risk is 9%. The

more aggressive portfolio pays more for the constraint (10.61% –

10.26% = 35 basis points).^28

EXTENSIONS OF THE BASIC ASSET ALLOCATION MODEL

In mean-variance analysis, the variance (standard deviation) of returns

is the proxy measure for portfolio risk. As a supplement, the probability

of not achieving a portfolio expected return can be calculated. This type

of analysis, referred to as risk-of-loss analysis, would be useful in deter-

mining the most appropriate mix from the set of optimal portfolio allo-

cations.^29 In the context of setting investment strategy for a pension

fund that has a long-term normal asset allocation policy established, the

(^28) For a discussion on the benefits and costs of constraints, see Francis Gupta and
David Eichhorn, “Mean-Variance Optimization for Practitioners of Asset Alloca-
tion,” in Frank J. Fabozzi (ed.), Handbook of Portfolio Management (New York:
John Wiley & Sons, 1998), pp. 57–74.
(^29) Risk of loss analysis as well as the multiple scenario analysis and short-term/long-
term analysis described next, were developed by Gifford Fong Associates in the early
1980s. See Chapter 4 and Appendix B in H. Gifford Fong and Frank J. Fabozzi,
Fixed Income Portfolio Management (Homewood, IL: Dow-Jones-Irwin, 1985).