Question

Assume a setting in which the risk-free rate is 4% and the CAPM holds. The market portfolio has a mean return of 16% and a return standard deviation of 30%. The data on two stocks that exist in this market are as follows. Stock X has a mean return of 10% and a standard deviation of 40%. Stock Y has a mean return of 20% and a standard deviation of 50%. The pair of stocks have a return correlation of 0.75. (Note that X and Y are not the only stocks traded in this market). a) What is the maximal attainable Sharpe ratio in this market and how is it achieved? Explain the logic behind your answer. b) Estimate the CAPM betas of the two stocks and comment on their implications for the risk of each stock. c) What is the efficient way to construct a portfolio position that generates a mean return identical to that of stock Y? What is that portfolio's return volatility. Based on your computations, explain why the portfolio dominates stock Y.

Answer 1

Answer. a) By definition, Risk Free Return means a return which does not have any risk or Standard Deviation/Volatility of Return.

In a world where CAPM holds, the Capital Market Line (CML) would hold true. It implies that the market portfolio should have the highest attainable sharpe ratio. So in this case the maximal attainable sharpe ratio would be that of the market portfolio i.e. (16 - 4)/30 = 0.40 [Sharpe Ratio = (Expected Return_i - Risk Free Return)/Std. Devaition]

b) When CAPM holds, the mean expected return of a risky security = Risk Free Return + βi * (Market Return - Risk Free Return)

ERi​ = Rf ​+ βi ​(ERm​−Rf​)

where:

ERi ​= expected return of investment

Rf ​= risk-free rate

βi ​= beta of the investment

(ERm​−Rf​) = market risk premium​

Putting the given value in this formula, the Betas of Stock X & Y would be calculated as follows:

Stock X: 10 = 4 + βi (16 - 4)

Therefore, βi of X = 0.5

Stock Y: 20 = 4 + βi (16 - 4)

Therefore, βi of Y = 1.33

Beta measures the systematic risk of a security. A higher Beta implies a higher systematic risk. Whereas, the average risk of Market i.e. Market Beta is always 1

Therefore, The risk of Stock X is a below avarage or below market risk as its Beta is less than 1. Whereas, the risk of Stock Y is a above avarage or above market risk as its Beta is more than 1

c) As the CAPM holds, the market portfolio will be the most efficient portfolio with the highest sharpe ratio. So any investment in a single stock would be below the efficient frontier and would not give optimum return. This is because the single security will have undiversified risk. So, a better alternative would be to invest in market portfolio and match the return of Security Y by borrowing at risk free return and investing more than 100% in market portfolio. Let x be the weight in market portfolio and y be the weight of risk free borrowing. So to achieve a return of 20% the weights will be calculated as follows: 20 = x(16) + y(4).

As x + y = 1, we can subtitute y = (1 - x), So 20 = x(16) + [(1 - x)(4)]

Solving this we get, x = 1.3333 or 133.33%

So, y = -0.3333 or -33.33%

In our ex. the weight in market portfolio should be 133.33% and we will have to borrow this extra 33.33% at risk free retrun, so the weight in risk free investment will be -33.33%. The retun of this combination would be (16 * 1.3333) + (4 * -0.3333) = 20% which matches the return of Stock Y. Whereas the risk of this combination will be 30% * 1.3333 = 40% (As risk free borrowing has no risk) which is less than the risk of Stock Y. So this combination dominates the investment in Stock Y alone.