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 hb18!!


First of all, we need to calculate the STD of the market portfolio
using the Capital Market Line (CML):
The risk-free rate asset rate is 5% and a STD equal to zero; the
portfolio has an expected return of 25% and a STD equal to 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 that:
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

What we know is:
- the expected return on the market portfolio is 20%,
- Rf is 5%, 
- 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%

Now we can use the above result to find requested beta (for a security that has
a correlation with the market portfolio of 0.5 and a STD of 2%):

Beta_S = [Correlation * STD of Security)] / STDm =
       = (0.5 * 0.02) / 0.03 =
       = 0.3333 or 1/3

According to the CAPM we have that:

E(Rs) = Rf + Beta_S * [E(Rm) - Rf]

where 
expected return on the security
Rf = risk-free rate
Beta_S = beta of the security
E(Rm) = expected return on the market portfolio

For this problem we have:
E(Rs) = unknown
Rf = 0.05
Beta_S = 0.3333 or 1/3
E(Rm) = 0.20

E(Rs) = Rf + Beta_S * [E(Rm) - Rf] =
      = 0.05 + 1/3 * (0.20 - 0.05) =
      = 0.10  or 10%

The expected rate of return of such security is 10% .

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I hope that this helps you. If you need a clarification, feel free to
request for it, I will be glad to give you further assistance on this
question if you find difficult to understand my answer.

Regards,
livioflores-ga