Portfolio Optimization: Parametric Approach1
Consider a portfolio composed of three stocks. Let the random variable si be the
annual return on stock i, with expected value μi, variance i2 , and standard
deviation i .
The means are: μ1 = 0.14, μ2 = 0.11, μ3 = 0.10.
The variances are: 1 = 0.20, 2 = 0.08, 3 = 0.18.
2 2 2
The correlation between si and sj, for any pair of stocks i and j, is ρij.
The correlations are: ρ12. = 0.8, ρ13. = 0.7, and ρ23. = 0.9.
The covariance between si and sj, for any pair of stocks i and j, is σiσjρij.
(a) Determine the minimum-variance portfolio that attains an expected
annual return of at least 0.12, with no shorting of stocks allowed.
(b) Construct a trade-off curve between the chosen portfolio's expected return
and risk (standard deviation), with risk on the horizontal axis. In finance
terminology this trade-off curve is called the efficient frontier.
(c) Determine the minimum-variance portfolio that attains an expected
annual return of at least 0.12, with shorting of stocks allowed.
Based on 7-49 (p. 393) and 9-24 (p. 471) in Practical Management Science (2nd ed., Winston and
Albright, 2001 Duxbury Press).