The Mathematics of Financial Modelingand Investment Management

16-Port Selection Mean Var Page 481 Wednesday, February 4, 2004 1:09 PM

Portfolio Selection Using Mean-Variance Analysis 481

var(R^2

p) = wM var(RM)

In other words, the variance of the portfolio is represented by the

weighted variance of portfolio M. We can solve for the weight of portfo-

lio M by substituting standard deviations for variances. Since the stan-

dard deviation is the square root of the variance, we can write

SD(Rp) = wMSD(RM)

and therefore

SD(Rp )

wM = ---------------------

SD(RM )

If we substitute the above result and rearrange terms we get the CML:

ER( M ) – Rf

ER( p ) = Rf + ------------------------------ SD(Rp )

SD(RM )

What is Portfolio M?

Now we know that portfolio M is pivotal to the CML; we now need to

know what portfolio M is. That is, how does an investor construct port-

folio M? Eugene Fama demonstrated that portfolio M must consist of

all assets available to investors, and each asset must be held in propor-

tion to its market value relative to the total market value of all assets.^9

That is, tangency portfolio M is the “market portfolio.” So, rather than

referring to the market portfolio, we can simply refer to the “market.”

Recall that using Lagrange multipliers we formally demonstrated in

a previous section that in the presence of risk-free lending and borrow-

ing the optimal portfolio held by investors is made up of the risk-free

asset and of one special portfolio called the tangency portfolio. This

important property is called separation. We can now complete the previ-

ous demonstration: if risk-free lending and borrowing is allowed the

market is M-V efficient and each investor holds the risk-free asset plus a

portfolio proportional to the market.

(^9) Eugene F. Fama, “Efficient Capital Markets: A Review of Theory and Empirical
Work,” Journal of Finance (May 1970), pp. 383–417.