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K. Two Random Variables.
1. Regression (Summary).
2. Covariance ( and )
3. The Correlation Coefficient ( and )
4. Functions of Two Random Variables.
5. Sums of Random Variables,Independence.
In the following problems (i) check for independence, (ii) Compute , and, (iii) Compute and : D&C pg. 221 3, 4, 7,14. In problem 3, find the following: ,
Downing and Clark (formerly pg. 348 now posted at end of 251hwkadd) Old Computational Problem 1: For the sample data below b) Compute and . c) Compute the mean of and .
34 26 9 30 47 10 34 34 45 10 47 32 47 8 45
6 57 89 60 95 42 31 28 90 25 45 23 52 95 48
Text 5.8 (Compute ), 5.9, 5.11, 5.16, 5.17 [5.8, 5.9, 5.10, 5.13] (5.8, 5.10), K1-K4 .
This document includes Exercises 5.8, 5.9, 5.11, 5.16, 5.17 and Problems K1 to K4.
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Exercise 5.8 (5.8 in 8th edition): For the joint probabilities below, compute: a) and ; b) and ; c) and d) and .
Joint Probability / X / Y.20 / -100 / 50
.40 / 50 / 30
.30 / 200 / 20
.10 / 300 / 20
Solution:
We are given and can expand this to the table below.
a)
b) . So .
. So .
c) .
Note that .
e) From the outline. So .
From the outline.
So and .
Exercise 5.9 (5.9 in 8th edition): If we assign the weight to investment X (and .6 to investment Y in a portfolio of two stocks), and , find a) the portfolio expected return and b) the portfolio risk.
Solution: a) If we use the formulas in the outline or the syllabus supplement, b) If we measure risk by the standard deviation, , so . It is really more appropriate to compute the coefficient of variation. or 128.45%.
Exercise 5.11 [5.10 in 9th] (5.10 in 8th edition):In the problem in the text, Dow-Jones Fund has the Weak Economy Fund has and their covariance is Compute the portfolio expected return and risk if 50% is invested in the Dow-Jones Fund (and the rest in the Weak Economy Fund), a) 30% is invested in the Dow-Jones Fund, b) 70% is invested in the Dow-Jones Fund and c) Explain which of these strategies you would recommend.
Solution:From the outline and syllabus supplement, we know that and . If represents the total return of a portfolio and is the return on asset X and is the return on asset Y, is the proportion of our portfolio in asset X and is the proportion of our portfolio in Asset Y, our return on the portfolio is Note that because is a proportion, it must be between 1 and 0. If we let replace and replace , we find that and or .
Originally , and So If we measure risk by the standard deviation,, so .The coefficient of variation is or 14.29%
a) ., so . The coefficient of variation is or 66.91%
b) .
, so or 66.93%
c) The Instructor’s Solutions Manual says “Investing 50% in the Dow Jones index fund will yield the
lowest risk per unit average return ….”. In the 8th edition, where the coefficients of variation were not
reported someone screwed up the numbers for this problem in the Instructor’s Solutions Manual . The
person who wrote the answer book had,and , which gave a much
more interesting problem. The problem here is a no-brainer. The first stock has a higher expected return
and lower variance. We would not take the 30% strategy, because it has lower returns and higher variance
than the others. The question is whether the higher risk with the 70% strategy rather than the 50% strategy
is offset by the higher return.
Exercise 5.16 [5.13a-f in 9th]: We are setting up a portfolio that consists of a corporate bond fund and a common stock fund. The numbers below represent the annual return per thousand dollars of the two funds under various economic conditions and the probability that these conditions will occur.
Probability / State of the Economy / Corporate Bond Fund / Common Stock Fund.10 / Recession / -$30 / -$150
.15 / Stagnation / $50 / -$20
.35 / Slow Growth / $90 / $120
.30 / Moderate Growth / $100 / $160
.10 / High Growth / $110 / $250
Find a) Expected returns for both funds, b) the standard deviation of the return for both funds and c) the covariance between the two funds. d) In which of the two funds would you invest? Explain.
Solution: Let = return of the corporate bond fund and = return of the common stock fund. I will use the same format that I have used for previous problems, though the absence of off-diagonal elements means that a more concise format is possible.
We are given and can expand this to the table below.
a)
b) So and .
So and .
c) The text answer book fails to note that the correlation is , which means that the securities move so similarly, that there may not be much advantage in putting them in the same portfolio.
d) The Instructor’s Solutions Manual says that the ‘Common stock fund gives the investor a higher expected return than corporate bond fund, but also has a standard deviation better than 2.5 times higher than that for corporate bond fund. An investor should carefully weigh the increased risk.’
Exercise 5.17 [5.13g-k in 9th]: Compute the portfolio expected return and portfolio risk for the following percentages invested in the corporate bond fund as in Problem 5.17: a) 30%; b) 50% and c) 70%. d) On the basis of results in a)-c), which portfolio would you recommend?
Solution: Recall that and
We know from a previous problem that, if is our total return and is the fraction of our portfolio in security and or . So we can state the following.
a) If and . The coefficient of variation is or 96.47%
b) If and . The coefficient of variation is or 84.80%.
c) If and . The coefficient of variation is or 72.19%.
d) Here is what the Instructor’s Solutions Manual says:
Based on the results of (a)-(c), you should recommend a portfolio with 70% of
corporate bonds and 30% of common stocks because it has the lowest risk per unit average return.
I say Nonsense! If the lowest risk per unit average return is our goal, clearly we should put all our money in the bond fund. Other than that since higher risk seems to give higher returns, the choice is a personal one based on the age and preferences of the investor.
Note that the 9th edition version is slightly longer and the answers given in the Instructor’s Solutions Manual are:
(a)E(X) = $77(b)E(Y) = $97
(c) = 39.76(d) = 108.95
(e) = 4161
(f) Stock Y gives the investor a higher expected return than stock X, but also has a standard deviation better than 2.5 times higher than that for stock X. An investor should carefully weigh the increased risk.
I have added coefficients of variation to the answers given.
(g)E(R) = $95, = 101.88 and .
(h)E(R) = $91, = 87.79and .
(i)E(R) = $87, = 73.78and .
(j)E(R) = $83, = 59.92and .
(k)E(R) = $79, = 46.35and .
(l) Based on the results of (g)-(k), an investor should recognize that as the expected return increases, so does the portfolio risk.
Problem K1: Let us assume that has a mean of 2 and a standard deviation of 4, and that has a mean of 3 and a standard deviation of 6. Also assume that the correlation between and is .9. If and , find the covariance and correlation between and .
Solution: The problem says , and , so that . The Outline says that and if ,. If we use the first formula from the outline with ,, and , we get . Now we know that and that . So .
Much better, use the second formula from the outline,.
Problem K2: The table below shows average Fahrenheit temperature and yield in lbs./acre for an industrial crop.
F Ya. Find the covariance and correlation between Fahrenheit
70 15temperature and yield.
75 17b. If the conversion formula for Celsius temperature is 79 16 C = 5/9(F-32), find the covariance and correlation between
80 20Celsius temperature and yield.
Solution: a) These are sample data, so we first compute the sample variances and covariance. ,,, and ,,,
, The formula for the sample covariance is so .
b) The formula relating Celsius and Fahrenheit is . The Outline says that if ,and . If we let ,and , use the first formula from the outline with ,, and , we get . Now we know that and that . So .
Much better, use the second formula from the outline,.
Problem K3:a. What is ?
b. What is (i) and (ii) ?
c. So what is ?
d. If represents the per item markup and is fixed cost, and we sell units of an item, the profit is . Assume that we have two goods, good X and Good
Y, and that the quantity of good X sold is and the quantity of good Y sold is . Let be the profits on good X and be the profits on good Y. Assume that , and that data concerning
Good X Good Ysales are as in the table at left. Find the
Markup 5 2variance of profits on each of the two goods.
Fixed Cost 100 50Then find the covariance and correlation between
Mean 90 90profits on the two goods.
Variance 50 50
Solution: a) What is ? The formula far the population variance is , so if we substitute for , we get . If, instead, we want to work with the sample variance , we find that . So we conclude .
b) What is (i) ? The formula we have been using is . If we substitute for , we get . Let ,, and , then )
What is (ii) ? The formula we have been using is . If we substitute for , we get .
c) So what is ? The formula for the population correlation is . If we substitute for , we get . The formula for the sample correlation is . If we substitute for , we get , so if we go back to the previous part of the problem,.
d) If represents the per item markup and is fixed cost, and we sell units of an item, the profit is . Assume that we have two goods, good X and Good Y, and that the quantity of good X sold is and the quantity of good Y sold is . Let be the profits on good X and be the profits on good Y. Assume that , and that data concerning sales is as in the table above. Find the
variance of profits on each of the two goods. Then find the covariance and correlation between the profits on the two goods.
If we look at the table above we find that the profit on X is and the profit on Y is . We know and that . . So . Better, observe that
.
Problem K4: Find for a) and b)
Solution: a)
To summarize (a check),,,, and .
To complete what we have done, write ,, .
So that
b)
To summarize (a check),,,, and .
To complete what we have done, write ,, .
So that In general, joint probability tables with only the diagonals filled produce correlations close to +1 or -1.