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
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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
And finally, we calculate the total uncertainty in our single mean value which is given by the equation below. δ ¯ Vi 2 r2 I = 1
5.Comparing two projects, Project B appears riskier because it has a larger standard deviation ($125,000) than Project A, but that does not consider relative risk. Actually, Project A is riskier because it has a larger coefficient of variation than Project B does.
A sample of 54 day-shift workers showed that the mean number of units produced was 345, with a standard deviation of 21. A sample of 60 afternoon-shift workers showed that the mean number of units produced was 351, with a standard deviation of 28 units. At the .05 significance level, is the number of units produced on the afternoon shift larger?
Two sets of data both have a mean of 17. Set A has a standard deviation of 3.5. Set B has a standard deviation of 6.8. Explain specifically what the different standard deviation measurements tell you as a researcher about the two data sets?
b. What is the expected return on a portfolio that is 40% invested in A and 60% invested in B? 12.08%
5. (TCO C) A tool manufacturing company wants to estimate the mean number of bolts produced per hour by a specific machine. A simple random sample of 9 hours of performance by this machine is selected and the number of bolts produced each hour is noted. This leads to the following results.
Standard deviation: sq. root ((1-3.5)^2 •(1⁄6) + (2-3.5)^2•(1⁄6) + (3-3.5)^2•(1⁄6) + (4-3.5)^2•(1⁄6) + (5-3.5)^2•(1⁄6) + (6-3.5)^2•(1⁄6))= sq. root2.916=σ 1.707
2.68 0.75 69.69 4.83 105.26 50.56 94.69 135.65 6.58 1.24 0.71 0.64 4.84 1.30 41.97% 16.13% 8.32% 12.99% 35.22%
Based on the sample of gas prices within the area, the Average cost is 3.32 and the standard deviation is 0.281791 for the sample given.
A manufacturer of chocolate candies uses machines to package candies as they move along a filling line. Although the packages are labeled as 8 ounces, the company wants the packages to contain a mean of 8.17 ounces so that virtually none of the packages contain less than 8 ounces. A sample of 50 packages is selected periodically, and the packaging process is stopped if there is evidence that the mean amount packaged is different from 8.17 ounces. Suppose that in a particular of 50 packages, the mean amount dispensed is 8.159 ounces, with a sample standard deviation of 0.051
= .7794. c) Now we are dealing with variation of the sample mean, so the standard error is 10.75/sqrt(100) = 1.075 Z
You have a $1,000 portfolio which is invested in stocks A and B plus a risk-free asset.
this scenario is 0.90. The analysis with the lowest MAPE will help us determine which