Question

You are given the following information on probabilities of events happening, and rates for return for possible projects A and B.

Prob	A	B
0.2	2%	8%
0.2	9%	9%
0.2	10%	10%
0.2	11%	16%
0.2	13%	18%

1. Calculate the expected returns of A (E[rA]) and B (E[rB]) .

2. Calculate the absolute risk levels of A and B.

3. Calculate the relative risk levels of A and B. If you have to invest in one asset which asset would you invest in and why?

4. A portfolio is formed with a w_A invested in project A and w_B invested in project B. Calculate the expected return E[rp] and standard deviation σp for the portfolio when w_A=0.5 and w_B=0.5. The correlation between A and B is 0.5.

5. Assume that there is another possible project C. You decide to create a new portfolio with all three projects A, B, and C. The return for the new portfolio with three assets will be the weighted average of the expected returns of the three assets. A variance of a portfolio with three assets is:

σp2=wA2σA2+wB2σB2+wC2σC2+2wAwBσAσBρA,B+2wAwCσAσCρA,C+2wBwCσBσCρB,C

where w denotes the weight, σ denotes the standard deviation, and ρ denotes the correlation between two denoted assets of the subscript.

The expected return of C is 13% and the variance is 16. The correlation between A and C is 0.7, and the correlation between B and C is -0.1. The correlation between A and B is again 0.5. You allocate 50% of your capital into A, 25% into B, and 25% into C.

For the new portfolio with three assets, calculate (1) the expected return and (2) standard deviation. Then briefly discuss the relationship between the risk and the number of holdings in a portfolio based on your findings from this question and from (d) .

Answer 1

A
			Probability Adjusted Return (return*probability)
Prob	A	B	A	B
0.2	2%	8%	0%	2%
0.2	9%	9%	2%	2%
0.2	10%	10%	2%	2%
0.2	11%	16%	2%	3%
0.2	13%	18%	3%	4%
Expected Return			9%	12%
B.
Absolute risk is the standard deviation of project returns
	A	B
Stdev of projects	4%	4%	Calcualted using excel formula
C.
Coefficient of variation gives the relative risk	Formula	St. Deviation/Expected Return
	A	B
Relative Risk	0.41574	0.329504
D
Expected return with 50%A and 50% B				Expected return*wA+Expected Return*wB
	A	B
Return	9%	12%
Relative weight	50%	50%
Expected Return	10.600%
Expected Standard Deviation	A	B		Variance of portfolio
Variance of Stocks	0.0014	0.001616		σp2=wA2σA2+wB2σB2+2wAwBσAσBρA
Weight square	0.25	0.25
Variance of portfolio	0.001116
St.Dev of portfolio	3.34%