Portfolio Optimization

64 Sharpe Ratios and Implied Risk Free Returns Chap. 4

We now use this expression for µ

m

in the expression for the implied

risk free return in part (a). This gives

r

m

= α

0

−

α

1

β

0

µ

m

− α

0

= α

0

−

α

1

β

0

(R

2

− β

2

)

1/2

α

1

β

1/2

0

= α

0

− [β

0

(R

2

− β

2

)]

1/2

,

and this completes the proof of the theorem. 

In Theorem 4.1(a) recall from (2 .1 2), (2 .1 3) and (2.17) that α

1

> 0

and β

0

> 0. It then follows that r

m

< α

0

and that r

m

tends to α

0

as

µ

m

tends to infinity. See Figure 4.1.

We illustrate Theorem 4 .1 in t he following example.

Example 4.1

For the given data, answer the following questions.

(a) Find the risk free return r

m

, implied by a market expected return

of µ

m

= 1.4. Also find the