The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 478 Wednesday, February 4, 2004 1:09 PM

478 The Mathematics of Financial Modeling and Investment Management

The investor’s problem is again a constrained optimization problem

that can be stated as

min () σ a^2 = min( wR ′ ΣΣΣΣwR)

subject to the constraints

μ a = wR ′ μμμμ+ ( 1 – wR ′ ιιι) Rf

This problem can be solved again with the method of Lagrange multipli-

ers. The Lagrangian is

L = wR ′ ΣΣΣΣwR + d[μ a – wR ′ μμμμ– ( 1 – wR ′ ιιι) Rf]

Equating to zero the derivatives of the Lagrangian with respect to

the weights and to the Lagrange multiplier d, we obtained the solution

of the constrained minimization problem. The solution of this problem

has an interesting feature that leads to the CAPM as we will see in the

next chapter. In fact, developing the lengthy computations, the optimal

portfolio weights can be written as

wR = CΣΣΣΣ–^1 (μμμμ– Rfιιιι)

μ a – Rf

C = --------------------------------------------------------

(μμμμ– Rfιιιι)′ ΣΣΣΣ–^1 (μμμμ– Rfιιιι)

The above formula shows that the weights of the risky assets of any

minimum-variance portfolio are proportional to the same vector. The

proportionality constant is C. Therefore, with a risk-free asset, all mini-

mum variance portfolios are a combination of the risk-free asset and of a

given risky portfolio. This risky portfolio is called the tangency portfolio.

With the exception of the tangency portfolio, the minimum variance

portfolios that are a combination of the tangency portfolio and the risk-

free asset are superior to the portfolio on the Markowitz efficient frontier

that has the same level of risk.

Deriving the Capital Market Line

To derive the Capital Market Line (CML), we begin with the efficient fron-

tier. In the absence of a risk-free asset, Markowitz efficient portfolios can

be constructed as a constrained minimum problem based on expected