RSM332 Problem Set 2 Solutions
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 first order condition to figure out the weights of assets A and B in the market portfolio:
E[RA ] − RF
E[RB ] − RF
=
σAM σBM E[RA ] − RF
E[RB ] − RF
⇒
=
2
2 wA σA + (1 − wA )σAB wA σAB + (1 − wA )σB
0.021 − 0.02
0.05 − 0.02
⇒
=
.
wA × 0.004389 + (1 − wA ) × (−0.00099) wA × (−0.00099) + (1 − wA )0.00594
1
This can be solved to obtain wA = 0.2118 and thus wB = 1 − wA = 0.7882. Expected return and standard deviation of the market portfolio are:
E[RM ] = 0.2118 × 0.021 + 0.7882 × 0.05 = 0.04386 = 4.386%
√
σM =
0.21182 × 0.004389 + 0.78822 × 0.00594 − 2 × 0.2118 × 0.7882 × 0.00099
= 0.05964 = 5.964%
(c) Betas of stock A and B can be found from the CAPM equation (or alternatively you can calculate σAM and σBM and use the definition of beta):
0.021 = 0.02 + βA (0.04386 − 0.02)
0.05 = 0.02 + βB (0.04386 − 0.02)
Solving this gives βA = 0.042 and βB = 1.257. The portfolio this investor wants to create has to have a beta of 1.5:
1.5 = wA × 0.042 + (1 − wA ) × 1.257
Solving this gives wA = −0.2 and thus wB = 1.2. So the investor needs to short-sell
$200 of asset A and invest the $1,200 in asset B. The expected return on this portfolio is E[Rp ] = RF + βp (E[RM ] − RF ) = 0.02 + 1.5 × (0.04386 − 0.02) =
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 first order condition to figure out the weights of assets A and B in the market portfolio:
E[RA ] − RF
E[RB ] − RF
=
σAM σBM E[RA ] − RF
E[RB ] − RF
⇒
=
2
2 wA σA + (1 − wA )σAB wA σAB + (1 − wA )σB
0.021 − 0.02
0.05 − 0.02
⇒
=
.
wA × 0.004389 + (1 − wA ) × (−0.00099) wA × (−0.00099) + (1 − wA )0.00594
1
This can be solved to obtain wA = 0.2118 and thus wB = 1 − wA = 0.7882. Expected return and standard deviation of the market portfolio are:
E[RM ] = 0.2118 × 0.021 + 0.7882 × 0.05 = 0.04386 = 4.386%
√
σM =
0.21182 × 0.004389 + 0.78822 × 0.00594 − 2 × 0.2118 × 0.7882 × 0.00099
= 0.05964 = 5.964%
(c) Betas of stock A and B can be found from the CAPM equation (or alternatively you can calculate σAM and σBM and use the definition of beta):
0.021 = 0.02 + βA (0.04386 − 0.02)
0.05 = 0.02 + βB (0.04386 − 0.02)
Solving this gives βA = 0.042 and βB = 1.257. The portfolio this investor wants to create has to have a beta of 1.5:
1.5 = wA × 0.042 + (1 − wA ) × 1.257
Solving this gives wA = −0.2 and thus wB = 1.2. So the investor needs to short-sell
$200 of asset A and invest the $1,200 in asset B. The expected return on this portfolio is E[Rp ] = RF + βp (E[RM ] − RF ) = 0.02 + 1.5 × (0.04386 − 0.02) =