Question

2. (2*5) Consider a market with many risky assets and a risk-free security. Asset’s returns are not perfectly correlated. All the CAPM assumptions hold and the market is in equilibrium. The risk-free rate is 5%, the expected return on the market is 15%. Mr. T and Mrs. R are two investors with mean-variance utility functions and different risk-aversion coefficients. They both invest into efficient portfolios composed of the market portfolio and the risk-free security.
   1. Mr. T’s portfolio has an expected return of 11%. What are the weights of his efficient portfolio? What is the beta of his portfolio?
   2. Mrs. R’s portfolio has a beta of 2.0. What are the weights of her efficient portfolio?
   3. (iii)Plot the Security Market Line and place portfolios of Mr. T and Mrs. R on the same graph together with the market portfolio.
   4. (iv)Find the coefficients of risk aversion for Mr. T. and Mrs. R. Are your results in line with part (iiii) Explain.
   5. A particular stock X’s expected return is 25%. What is its beta? If the stock X returns’ standard deviation is 70%, what is the coefficient of correlation between stock X and the market portfolio?

Answer 1

i).

Let Wm be the weight of Market and then the weight of risk free asset will be 1-Wm.

Given that portfolio's expected return is 11%.

So, 11%= Wm*Rm+(1-Wm)*Rf

11%= Wm*15%+(1-Wm)*5%

11%= 10%*Wm+5%

Wm= 60% and 1-Wm= 40%.

Using CAPM, required rate can be calculated as Rf+Beta*(Rm-Rf); where Rf is risk free rate, Rm is market return and Rm-Rf is market risk premium.

So, 11%= 5%+Beta*(15%-5%).

Beta= 0.6

ii).

Given that Beta is 2.

So, Using CAPM,

E(R)= 5%+2*(15%-5%)= 25%.

So, similar to the above method, we have,

25%= Wm*15%+(1-Wm)*5%

Wm= 200% and (1-Wm)= -100%

So, R short sells risk free asset and invest those proceeds in market portfolio.

iii).

SHIL EC" 11 2500 15% 117 5% 2 B

iv).

Coefficient of risk aversion for T can be given as: E(Rm)-E(Rf)/(Wm)*sd(m)^2

= (15%-5%)/(0.6*sd(m)^2)= 0.1/(0.6*s^2)= 0.17/s^2

Coefficient of risk aversion for R can be given as:

E(Rm)-E(Rf)/(Wm)*sd(m)^2

= (15%-5%)/(2*sd(m)^2)= 0.1/(2*s^2)= 0.05/s^2

In the information given in the question, Standard deviation of the market is not given. If given, we can just substitute it at s in the equations (0.17/s^2 and 0.05/s^2).

Comparitively, it is clear that coefficient of risk aversion of T>R. (as 0.17>0.05 and s is same for both).

This makes sense as T is more risk averse than R.

v).

Given that return on Stock X is 25%.

Using CAPM, 25%= 5%+Beta*(15%-5%)

Beta= 2.

Beta can also be given as Correlation*Standard deviation of X*Standard deviation of Market/Variance of the market.

2= Correlation*0.7*s/s^2= Correlation*0.7/s

Correlation= (2*s)/0.7

The information about standard deviation of the market is missing. We can find the correlation by substituting value of Standard deviation of market in s in the equation (2*s)0.7.