A portfolio that combines the risk-free asset and the market portfolio has an ex

Question

A portfolio that combines the risk-free asset and the market portfolio has an expected return of 7.2 percent and a standard deviation of 10.2 percent. The risk-free rate is 4.2 percent, and the expected return on the market portfolio is 12.2 percent. Assume the capital asset pricing model holds.

What expected rate of return would a security earn if it had a .47 correlation with the market portfolio and a standard deviation of 55.2 percent? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

A portfolio that combines the risk-free asset and the market portfolio has an expected return of 7.2 percent and a standard deviation of 10.2 percent. The risk-free rate is 4.2 percent, and the expected return on the market portfolio is 12.2 percent. Assume the capital asset pricing model holds.

Explanation / Answer

In the given scenario, we will have to use Capital Market Line (CML) to calculate standard deviation of market portfolio:

Formula of CML:

E(Ri) = Rf + SlopeCML x STDi

In which,

E(Ri) = Expected return on security ‘I’

Rf = Risk-free rate of return

SlopeCML = Slope of Capital Market Line

STDi = Standard deviation of security ‘I’

In the given scenario, we know thar Rf is 4.2% and standard deviation for a risk free asset is always zero. Portfolio is having an expected return of 7.2% with standard deviation of 10.2%. The above two points will lie on CML.

Slope of CML = Increase in Expected Return / Increase in Standard Deviation

             = (0.072 - 0.042) / (0.102 - 0)

             = 0.294117647

As per Capital Market Line,

Given in illustration that expected return of market portfolio is 12.2%, the

Rf is 4.2%, and the slope of CML is 0.294117647, standard deviation can be calculated.

E(Rm) = Rf + SlopeCML x STDM

0.122 = 0.042 + 0.294117647 x STDM

STDM = (0.122 - 0.042) / 0.294117647 = 0.272 or 27.2%

Now, beta of security can be calculated using the above calculated standard deviation of market portfolio. Correlation with the market portfolio is 0.47 and standard deviation is 55.2%.

Beta of security = [Correlation x Standard deviation of Security)] / STDM

                               = (0.47 * 0.552) / 0.272

                               = 0.95382353

As per CAPM,

E(R) = Rf + BetaS x [E(Rm) - Rf]

where

E(R) = Expected return on security

Rf = risk-free rate of return

BetaS = Beta of security

E(Rm) = Expected return of market portfolio

Here, we have:

Rf = 0.042

BetaS = 0.95382353

E(Rm) = 0.122

E(R) = Rf + BetaS x [E(Rm) - Rf] =

     = 0.042 + 0.95382353 * (0.122 - 0.042)

     = 0.1183 or 11.83%

Hence, expected rate of return for security of 0.47 correlation with market portfolio and standard deviation of 55.2% is 11.83% .

Thus, answer is 11.83%

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