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RSM332 Problem Set 2 Solutions

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UNIVERSITY OF TORONTO
Joseph L. Rotman School of Management
RSM332

PROBLEM SET #2

SOLUTIONS

1. (a) Expected returns are:
E[RA ] = 0.3 × 0.07 + 0.4 × 0.06 + 0.3 × (−0.08) = 0.021 = 2.1%,
E[RB ] = 0.3 × 0.14 + 0.4 × (−0.04) + 0.3 × 0.08 = 0.05 = 5%.
Variances are:
2
σA = 0.3 × (0.07)2 + 0.4 × (0.06)2 + 0.3 × (0.08)2 − (0.021)2 = 0.004389,
2
σB = 0.3 × (0.14)2 + 0.4 × (0.04)2 + 0.3 × (0.08)2 − (0.05)2 = 0.00594.

Standard deviations are:
√
0.004389 = 6.625%, σA =
√
0.00594 = 7.707%. σB =
Covariance is: σAB = 0.3 × 0.07 × 0.14 + 0.4 × 0.06 × (−0.04) + 0.3 × (−0.08) × 0.08 − 0.021 × 0.05
= −0.00099.
Correlation is: ρAB =

σAB
−0.00099
=
= −0.19389. σA σB
0.06625 × 0.07707

(b) We can use the following …show more content…

The second one is efficient, so the investor should invest $369.35 in asset A and the remaining $630.65 in asset B. The expected return of this portfolio is
E[Rp ] = 0.36935 × 0.021 + 0.63065 × 0.05 = 3.93%.

For the third investor, we let wTb to be the weight of his portfolio that is invested in
Tb , we have σp = wTb σTb ⇒ wTb =

σp
7
= 1.1735.
=
σTb
5.964

b
Therefore, the third investor should invest wTb × wA = 1.1735 × 0.2118 = 0.24855, or b $248.55 in asset A, wTb × wB = 1.1735 × 0.7882 = 0.92495, or $924.95 in asset B.
In addition, he needs to borrow $173.5 at the risk-free borrowing rate of RF,b . The expected return of the portfolio is

E[Rp ] = (1 − wTb )RF,b + wTb E[RTb ] = (1 − 1.1735) × 0.02 + 1.1735 × 0.04386 = 4.80%.

2. (a) B and D are not minimum variance efficient portfolios. D is not efficient because
A offers same mean for less variance. As long as A and C are not perfectly correlated,
B will also not be minimum-variance efficient. This is because some combination of
A and C will offer the same mean return yet less variance than B. This is pictured below. 3

0.25

Minimum Variance Efficient Portfolio

C

Expected Return

0.2

B

0.15

ρAC = 1
0.1

A

D ρAC = 0.5

0.05
0.1

0.15

0.2
0.25
Standard Deviation of Return

0.3

0.35

(b) False, the higher is a security’s beta (and not its variance), the higher is its expected return. A security’s variance is made up of two components: (i) market

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