Question

Probability distribution for the one year holding-period return of two stocks. Assume that risk-free rate is 1% and the correlation between A and B is 0.8.

State of Economy	Probability	Return Stock A	Return Stock B
Recession	40%	-9%	-5%
Normal	35%	11%	20%
Boom	25%	22%	15%

    

U = E(r) – 0.5As²  

1. Find the expected returns and standard deviation for each stock.
2. Assume 14% in Stock A and 86% in Stock B is the optimal weights for a portfolio P based on these two risky stocks, what is the return and risk for portfolio P?
3. Assume that the client with risk aversion of 6 wants to allocate his money on risk-free asset and on the risky portfolio P. How should the client allocate the money between the risky portfolio P and the risk-free rate?
4. What is the expected rate of return and risk on a complete portfolio C that includes risk-free and a portfolio P?

Answer 1

Step 1: calculate Expected return and standard deviation of both stock as ,

E(R) =

State	P	RA	P*RA	(RA-mean)^2	P*(RA-mean)^2	RB	P*RB	(RB-mean)^2	P*(RB-mean)^2
Rec	0.40	-9%	-0.04	2.176%	0.870%	-5%	-2.00%	1.891%	0.756%
Nor	0.35	11%	0.04	0.276%	0.096%	20%	7.00%	1.266%	0.443%
Boom	0.25	22%	0.06	2.641%	0.660%	15%	3.75%	0.391%	0.098%
		Mean(E( R ))	5.750%	sum	1.627%	Mean(E( R ))	8.750%	sum	1.297%
				Standard Deviation	12.755%			Standard Deviation	11.388%

Step 2: expected return and standard deviation of portfolio P, as

E(R_P) =

E(R_P) = 0.14*5.75%+0.86*8.75% = = 8.330%

= 11.273%

Step 3: Find expected return at A= 6 , using utility function

,

the expected return , at point 'A' makes this investor indifferent between the risky portfolio and the risk-free asset

E(r) = U +0.5AS² = 1%+ 0.5*6* (11.273%)^2= 4.813%

Step 4: Calculation of weight in risk free asset ( W)

4.813% = W*1%+ (1-W)*8.330%

=> W = 3.517%/3.813% = 0.479

Risk free weight = 0.479

Weight in portfolio P = 1-0.479 = 0.521

Step 5: return of Portfolio C= 0.479*1%+ 0.521*8.33% =4.813%

Risk (standard deviation) = 0.521*11.273% = 5.873%