108156.pdf

5. Portfolio Management 119

Figure 5.11 Efficient frontier for portfolios with a risk-free security

Since every investor will select a portfolio on the capital market line, everyone
will be holding a portfolio with the same relative proportions of risky securities.
But this means that the portfolio with standard deviationσMand expected
returnμMhas to contain all risky securities with weights equal to their rela-
tive share in the whole market. Because of this property it is called themarket
portfolio. In practice the market portfolio is approximated by a suitable stock
exchange index.
The capital market line joining the risk-free security and the market port-
folio satisfies the equation

μ=rF+

μM−rF

σM

σ. (5.18)

For a portfolio on the capital market line with riskσthe termμMσ−MrFσis called
therisk premium. This additional return above the risk-free level provides
compensation for exposure to risk.

Example 5.13

We shall apply Proposition 5.12 to compute the market portfolio for a toy
market consisting of the three securities in Example 5.10 and a risk-free security
with returnrF=5%.Theweightswin the market portfolio, which belongs to
the efficient frontier, satisfy condition (5.17), which implies that

γw=(m−μu)C−^1.

From the proof of Proposition 5.12 we know thatμ=rFbecause the capital
market line, tangent to the efficient frontier at the point representing the market
portfolio, intersects theμaxis atrF. Substituting the numerical values from
Example 5.10, we find that

γw∼=

[

0 .293 1.341 2. 061

]

.