Question

Consider the following three investments.   Listed are the possible returns on each. For simplicity we’ll assume that there are only three possibilities, and that they are equally likely.

Probability	Asset M	Asset I	Asset A
1/3	30%	-18%	39%
1/3	-15%	34%	-24%
1/3	18%	5%	18%

1. What is the expected return on each asset?

2. What is the expected return on a portfolio with 50% of funds in M and 50% in I?

3. What is the expected return on a portfolio with 50% of funds in M and 50% in A?

Based on this probability distribution, the standard deviation of Asset M returns is 19.03%, the standard deviation of asset I returns is 21.28%, and the standard deviation of Asset A returns is 26.19%.   The correlation between M and I returns is -.981 while the correlation between M and A returns is .997.   

4. Consider two portfolios. The first has 50% of funds in Asset M and 50% in Asset I. The second has 50% of funds in Asset M and 50% in Asset A. Which portfolio is less risky, and why? (Answer this question based on principles discussed in class, without referring to actual risk outcomes – these will be addressed in the next part of the question).

5. What, specifically, are the return standard deviations on the two portfolios described in (d)?

Answer 1

Probability	Asset M	Asset I	Asset A
1/3	30%	-18%	39%
1/3	-15%	34%	-24%
1/3	18%	5%	18%

(a) Expected return = Σ(P*Return)

Asset M = (1/3*30)+1/3*(-15)+(1/3*18)

=11%

Asset I = 1/3*(-18)+(1/3*34)+(1/3*5)

= 7%

Asset A = (1/3*39)+1/3*(-24)+(1/3*18)

= 11%

(b) Expected return of the portfolio with 50% of Asset M and 50% in Asset I = 0.5*11+0.5*7

= 9%

(c) Expected return of the portfolio with 50% of Asset M and 50% in Asset A = 0.5*11+0.5*11

= 11%

(d) Portfolio I – 50% in Asset M and 50% in Asset I

Portfolio II – 50% in Asset M and 50% in Asset A

Standard deviation of Portfolio I = {(X2M Sd2M)+ (X2I Sd2I )+(2 XM XI(SdM SdI rMI))}1/2

= {(0.52 *19.032)+(0.52*21.282)+(2*0.5*0.5*19.03*21.28*(-0.981))}1/2

=(0.25*362.14)+ (0.25*452.84)+(202.48*(-0.981))}1/2

=(90.54+113.21-198.63)1/2

= (5.12)1/2

= 2.26%

Standard deviation of Portfolio II = {(X2M Sd2M)+ (X2A Sd2A)+(2 XM XA(SdM SdA rMA))}1/2

= {(0.52 *19.032)+(0.52*26.192)+(2*0.5*0.5*19.03*26.19*0.997)}1/2

=(0.25*362.14)+ (0.25*685.92)+(248.45))}1/2

=(90.54+171.48+248.45)1/2

= (510.47)1/2

= 22.59%

Coefficient of Variation: Sd/Expected return*100

Portfolio I = 2.26/9*100

= 25.11%

Portfolio II = 22.59/11*100

= 205.36%

Since Coefficient of Variation of Portfolio I is less than Portfolio therefore Portfolio I is less risky.

(e) As calculated in the part (d), Standard deviation of the Portfolio I = 2.26%

and Standard deviation of the Portfolio II = 22.59%