1. A portfolio that combines the risk-free asset and the market
portfolio has an expected return of 25 percent and a standard
deviation of 4 percent. The risk-free rate is 5 percent, and the
expected return on the market portfolio is 20 percent. Assume the
capital-asset-pricing model holds. What expected rate of return would
a security earn if it had a 0.5 correlation with the market portfolio
and a standard deviation of 2 percent?

Hi again!!

Start calculating the standard deviation of the market portfolio using
the Capital Market Line (CML):
The risk-free rate asset has a return of 5% and a standard deviation
of zero and the portfolio has an expected return of 25% and a standard
deviation of 4%. These two points must lie on the Capital Market Line.
The slope of the Capital Market Line is:

Slope of CML = Increase in Expected Return / Increase in Standard Deviation
             = (0.25? 0.05) / (0.04 - 0)
             = 5

According to the Capital Market Line we have also:
E(ri) = rf + SlopeCML * STDi

where
E(ri) = the expected return on security i
rf = risk-free rate
SlopeCML = slope of the Capital Market Line
STDi = the standard deviation of security i

We know that the expected return on the market portfolio is 20%, the
risk-free rate is 5%, and the slope of the Capital Market Line is 5;
then we can solve for the standard deviation of the market portfolio
(STDm):
E(rm) = rf + SlopeCML * STDm  ==>
==> 0.20 = 0.05 + 5 * STDm  ==>
==> STDm = (0.20 ? 0.05) / 5 = 0.03  or 3%

Using the STDm we can find the beta of a security that has a
correlation with the market portfolio of 0.5 and a standard deviation
of 2%:

Beta of security = [Correlation * STD of Security)] / STDm
		 = (0.5 * 0.02) / 0.03
		 = 1/3

According to the CAPM we have that:

E(r) = rf + Beta_s * [E(rm) - rf]

where 
E(r) = expected return on the security
rf = risk-free rate
Beta_s = beta of the security
E(rm) = expected return on the market portfolio

In this problem we have that:
rf = 0.05
Beta_s = 1/3
E(rm) = 0.20

E(r) = rf + Beta_s * [E(rm) - rf] =
     = 0.05 + 1/3 * (0.20 - 0.05) =
     = 0.10  or 10%


Hope that this helps you.

Regards,
livioflores-ga

Thank you very much.