Question

There are three assets, A, B and C, where A is the market portfolio and C is the risk-free asset. The return on the market has a mean of 12% and a standard deviation of 20%. The risk-free asset yields a return of 4%. Asset B is a risky asset whose return has a standard deviation of 40% and a market beta of 1. Assume that the CAPM holds.

1. Compute the expected return of asset B and its covariances with asset A (the market portfolio) and asset C (the risk-free asset), respectively.
2. Consider a portfolio of the two risky assets, A and B, with weight w in asset A (the market portfolio) and 1 w in asset B. Compute the expected return and return standard deviation of the portfolio with w being 0, 1/2, and 1, respectively, and enter them into the following table:

Portfolio weight w	0	1/2	1
Expected return
Standard deviation

3. Can you rank the three portfolios in the question above? Explain.
4. Consider a portfolio with equal weights in asset B and C (the risk- free asset). Denote this portfolio as asset D. Compute the return standard deviation and expected return of asset D.
5. Consider a portfolio of asset A (the market portfolio) and C. Find the portfolio weight such that its return standard deviation is the same as that of asset D in Question (d). What is the expected return of this portfolio?
6. What can you say about the mean-variance efficiency of asset A, B and C (i.e., are they efficient portfolios)? Explain.
7. Construct an efficient portfolio from the assets A, B and C with an expected return of 10%.

Answer 1

From CAPM,

Expected Return on Asset B = risk free rate + Beta of Asset B* (market return - risk free rate)

= 4%+1* (12%-4%) = 12%

Beta of Asset B = Covariance of returns of B with returns of A(market) / variance of market returns

=> 1 = Covariance of returns of B with returns of A(market) / 0.2^2

=> Covariance of returns of B with returns of A = 0.04

Asset B's covariance with A = 0.04

Asset B's covariance with C = 0 (as risk free asset has 0 standard deviation and zero covariance with other risky assets)

The return of a portfolio is the weighted return of the two stocks

The standard deviation of a portfolio is given by

п п ор СУ W; * W; *о; *ој * Рај i=1 j=1

Where Wi is the weight of the security i,

$1596291545864_blob.png$ is the standard deviation of returns of security i.

and is the correlation coefficient beltween returns of security i and security j

Pij

Applying these formulas, the Expected Return and Standard deviation of the portfolio is given in the table below

Portfolio weight w	0	0.5	1
Expected Return	12%	12%	12%
Standard deviation	0.4	0.264575131	0.2

Clearly , portfolio with w =1 is the best and that with w=0 is the worst as all the three have same expected returns but the standard deviation of the portfolio with all amount invested in Asset A (market portfolio) is the least and that with all amount invested in Asset B is the highest

Expected Return of portfolio D = 0.5* 4%+ 0.5*12% = 8%

Standard deviation of portfolio D = 0.5*standard deviation of B = 0.5*40% = 20%

As the portfolio D has the standard deviation of 20% (same as Asset A) , the portfolio weights will be 1 in Asset A and 0 in Asset C

Expected Return of the portfolio of A and C = 1*12% +0*4% = 12%