Need help with B & C! Thank You. Suppose that there are many stocks in the secur

Question

Need help with B & C! Thank You.

Suppose that there are many stocks in the security market and that the characteristics of Stocks A and B are given as follows Suppose that it is possible to borrow at the risk-free rate, r_f. What must be the value of the risk-free rate? Explain. Calculate the expected return and standard deviation of an equally weighted portfolio of Stock A, Stock B and the risk-free asset. What is the optimal portfolio? Assume the investors risk aversion coefficient is given by A=4. Provide all the relevant portfolio weights for all three assets. The utility function is given below. U_i = E(r_i) - 0.5Asigma_i^2

Explanation / Answer

Since Stocks A and B are perfectly negatively correlated, a risk-free portfolio can be constructed and its rate of return in equilibrium will be the risk-free rate.

To find the proportions of this portfolio (wA invested in Stock A and wB = 1 - wA in Stock B), set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to:

p = | wAA - wBB|

=> 0 = | 5wA - 10 (1-wA) |

=> wA =0.6667

The expected rate of return on this risk-free portfolio is:

E(R) = 0.6667 x 10% + (0.3333 x 15%) 11.67%.

Thus, to avoid arbitrage, the risk-free rate must also be 11.67%.

Expected return for stock A = 10%

Expected return for stock B = 15%

Standard deviation for stock A = 5%

Standard deviation for stock B = 10%

In the c) part :

We have to calculate Ui :

U1 = E(r1) - 0.5A12

A = 4

U1 = 10% - 0.5*4*5%2 = 0.1 - 0.5*4*0.052 = 0.095

Similarly U2 = E(r2) - 0.5A22

U2 = 15% - 0.5*4*10%2 = 0.15 - 0.5*4*0.12 = 0.13

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