Major Points
RISK AND RETURN: PROBLEMS & DETAILED SOLUTIONS
(copyright © 2016 Joseph W. Trefzger)
Work These for Sure
1. After comparing the annual returns earned on the stock market and those earned by common stockholders
of various corporations over a series of prior years, Wall Street analysts have determined that the beta for Baja California Sunblock’s common stock is .875. They expect the holders of “risk-free” short term U.S. government bonds to earn a 3.75% average annual rate of return, and expect the holders of Baja California common stock to earn a 10.5% average annual return, in future years. Based on the security market line (SML) equation, what average annual rate of return would we expect investors to earn on the overall stock market in future years?
Type: Security market line. Here we are working with the security market line equation:
ke = krf + (km – krf) β .
ke is the average annual rate of return that common stockholders in a given company should expect to earn on their equity investment in future years; krf is a “risk-free” annual rate of interest that
is expected, on average, by holders of short-term U.S. government bonds in future years; km is the average annual rate of return expected on the overall stock market in future years; and β (beta) is
a measure of the degree to which the stock in question is expected to react to economic factors that affect the overall stock market, based on the assumption that historical observations offer
a reasonable guide to what will happen in the future. [The beta for the overall stock market is 1.0. Thus a stock whose beta is less than 1 has historically reacted less severely to market-wide changes than have other stocks, on average; a stock with beta greater than 1 has historically reacted more severely to market-wide changes – its annual returns have hit higher highs and lower lows – than have other stocks, on average.] The quantity (km – krf), called the “market risk premium,” represents the extra (premium) annual return that stock market investors expect to receive, on average, above what buyers of short-term government bonds, with their assured payments, expect to receive.
The logic underlying the SML equation is that the holder of common stock in a given company should expect an annual rate of return that consists of:
• the lowest annual rate of return that any investor would accept (a pure measure of the reward for giving up the ability to spend money today), plus something extra based on
• the average risk of investing in the stock market, and
• the specific risk that accompanies an investment in the given company’s common stock.
The intent of questions 1 – 4 is to show how we can use this relationship to solve for any unknown: ke, krf, km, or β. In this first example we are asked to compute km if krf = .0375, ke = .105, and β = .875. We solve as
ke = krf + (km – krf) β
.105 = .0375 + (km – .0375) ( .875
.105 = .0375 + (km ( .875) – (.0375 ( .875)
.0675 = (km ( .875) – .032813
.100313 = km ( .875
.100313 ÷ .875 = km = .114643, or about 11.5%.
[With km = about 11.5%, we have a market risk premium of (km – krf) = (.115 – .0375) = .0775, or 7.75%.] It should make sense that if a stock like Baja, with below-average risk (a beta less than 1), is expected to provide a ke = 10.5% annual rate of return, then the expected annual rate of return on the stock market overall should be something greater than 10.5%, here km = about 11.5%.
2. After analyzing the annual returns earned on the overall stock market and those earned by various companies’ common stockholders over many previous years, professional investment analysts have measured the beta for Campeche Camouflage Corporation’s common stock to be 1.43. They expect the average annual rate of return on the stock market overall to be 8.75% in future years, and expect the average annual return earned by holders of “risk-free” short term U.S. government bonds to be 3.25%. Using the security market line (SML) equation, estimate the expected average annual rate of return we would expect holders of Campeche common stock to earn. What if Campeche moved to a more stable mix of business activities, causing its beta to fall to 1.05?
Type: Security market line. Again we are working with the security market line equation:
ke = krf + (km – krf) β .
But now we are asked to compute ke if krf = .0325, km = .0875, and β = 1.43. We solve as
ke = .0325 + (.0875 – .0325) ( 1.43
ke = .0325 + .055 ( 1.43
ke = .0325 + .07865 = .11115, or 11.115%.
(Note that the market risk premium here is km – krf = .0875 – .0325 = .055, or 5.5%.) It should make sense that a stock whose returns have, historically, been more volatile than other stocks on average (thus a more risky investment, if we can expect that same high volatility to persist in the future) should have an expected annual rate of return (here ke = 11.115%) greater than the annual rate of return expected on the stock market overall (here km = 8.75%). If Campeche’s beta were
to be reduced to 1.05, then the annual return expected by holders of the company’s common stock would be a considerably lower
ke = .0325 + (.0875 – .0325) ( 1.05
ke = .0325 + .055 ( 1.05
ke = .0325 + .07865 = .09025, or 9.025%.
3. After making a historical comparison of annual returns earned by Chiapas Chair Covers, Inc. common stockholders and those earned by stock market investors on average, professional analysts have computed a beta
of 1.12 for Chiapas common shares. If the average annual rate of return for holders of Chiapas common stock is expected to be 9.875% in future years, while the annual rate of return on the stock market overall is expected to average 9.25%, what average annual “risk-free” rate of return would we expect the holders of short term U.S. government bonds to earn in future years, according to the security market line equation?
Type: Security market line. Here we are asked to compute krf if ke = .09875, β = 1.12, and km = .0925.
We solve as
ke = krf + (km – krf) β
.09875 = krf + (.0925 – krf) ( 1.12
.09875 = krf + (.0925 ( 1.12) – (krf ( 1.12)
.09875 = krf + .1036 – (krf ( 1.12)
– .00485 = krf – (krf ( 1.12)
– .00485 = 1 krf – 1.12 krf
– .00485 = – .12 krf
krf = .0404 or 4.04%.
4. Economists expect that, in the future, the average annual “risk-free” interest rate earned by holders of short term U.S. government bonds will be 3.95%, while stock market investors will earn an additional 5.9%, on average, as a “market risk premium.” If observers who base their analysis on the security market line (SML) equation expect the common stockholders of Chihuahua Cheese Company to earn an 8.25% average annual rate of return on their equity investment in future years, what do these observers estimate Chihuahua’s beta to be?
Type: Security market line. Finally, we are asked to compute β if ke = .0825, krf = .0395, and (km – krf) = .059 (we are given the market risk premium, which is the entire term in parentheses, and not simply the km value that represents the average expected annual return on the entire stock market). We solve as
ke = krf + (km – krf) β
.0825 = .0395 + (.059) ( β
.043 = .059 ( β
β = .043 ÷ .059 = .73 .
With (km – krf) = (km – .0395) = .059, it must be that the expected average annual rate of return on the stock market is km = .059 + .0395 = 9.85%. It should make sense that if a stock provides an expected annual rate of return (here 8.25%) less than the average annual return expected on the overall stock market (here 9.85%), it must be a stock of less-than-average risk – a stock whose returns have, historically, been less volatile than other stocks, on average (as evidenced by a β less than 1 for Chihuahua).
5. The average annual “risk-free” interest rate that holders of short term U.S. government “T-Bills” will earn is expected to be 3.45% in the future, while the average annual rate of return on the stock market is expected to be
9%. A historical comparison of past returns earned on some individual companies’ common stocks with those of
the overall stock market show the beta for Coahuila Coatracks to be 1.10, the beta for Colima Coolers to be 1.20,
and the beta for Durango Durables to be 1.30. After analyzing the economy and each of the three companies,
you believe that the average annual returns for the three will actually be: 9.85% for Coahuila, 10.11% for Colima, and 10.25% for Durango. Is each of your estimates consistent with the security market line (SML) equation?
Type: Security market line. Using the historically-computed betas, we would compute the expected annual return on equity for each of the three companies’ common stocks, according to the security market line equation, to be:
ke = krf + (km – krf) β
= .0345 + (.09 – .0345) ( 1.10
= .0345 + .0555 ( 1.10 = .09555 or 9.555% for Coahuila;
= .0345 + (.09 – .0345) ( 1.20
= .0345 + .0555 ( 1.20 = .1011 or 10.11% for Colima; and
= .0345 + (.09 – .0345) ( 1.30
= .0345 + .0555 ( 1.30 = .10665 or 10.665% for Durango.
Only in the case of Colima is your estimate equal to what the security market line indicates. For Coahuila, you feel that the SML (9.555%) underestimates the 9.85% true average annual return
to be expected (maybe you feel that the historical beta measure understates the stock’s future volatility of returns, which could happen if Coahuila’s business has recently changed in ways that would cause the historic pattern of returns to understate the expected future volatility). For Durango, you feel that the SML (10.665%) overestimates the 10.25% true average annual return to be expected (perhaps Durango’s business activity has recently changed in ways that would cause the historic pattern of returns to overstate the expected future volatility).
Work These for Additional Insights
6. An investor wants to create a portfolio consisting of four companies’ common stocks: Guanajuato Garments, Guerrero Gaming, Jalisco Halogen, and Juarez Warehouses. Information regarding expected annual rates of return on equity for these companies, and on the proportion of the portfolio to be represented by each of the four stocks, is shown below.
Company (stock) Proportion of Portfolio Expected Annual Return
Guanajuato Garments 35% 14.5%
Guerrero Gaming 25% 9.5%
Jalisco Halogen ? 10.5%
Juarez Warehouses ? 8.0%
If the investor wants the portfolio’s expected annual rate of return to be 11.25%, what proportions of the portfolio should Jalisco and Juarez be?
Type: Computing expected return for portfolio. If we already know the expected annual rate of return
for each asset in a portfolio, then the expected annual rate of return on the portfolio is simply the weighted average of the individual assets’ expected returns. Here we happen to know the expected return the investor is targeting, and must solve for missing weight figures. We can solve with the equation:
kPortfolio = .1125 = .35 (.145) + .25 (.095) + ProportionJalisco (.105) + ProportionJuarez (.08) .
The solution involves simple algebra, the key to which is remembering that the four stocks’ proportions must sum to 100%. 35% + 25% = 60% of the portfolio is already accounted for through Guanajuato and Guerrero, so Jalisco and Juarez should account, together, for the other 40%. If we think of Jalisco’s proportion as z%, then Juarez’s proportion must be 40% – z, and we can restate the above equation as
.1125 = .35 (.145) + .25 (.095) + z (.105) + (.40 – z) (.08)
and solve as
.1125 = .35 (.145) + .25 (.095) + z (.105) + .40 (.08) – z (.08)
.1125 = .05075 + .02375 + .105 z + .032 – .08 z
.006 = .025 z ( .24 = z.
We see that 24% of the portfolio should consist of Jalisco. Therefore .40 – .24 = .16 or 16% of the portfolio should consist of Juarez. Let’s double-check our figures:
.35 (.145) + .25 (.095) + .24 (.105) + .16 (.08)
= .05075 + .02375 + .0252 + .0128 = .1125 or 11.25%. (
7. Based on measurements taken over several decades, the standard deviation of annual returns on the stock market has been about 20%. The covariance of returns for Michoacan Methane Corporation’s common stock with returns on the stock market overall has been about 3.85%. Compute the beta for Mexico’s common stock.
Type: Computing beta for a stock. Beta is computed as the covariance of the stock’s historical returns with those of the overall stock market, divided by the variance in the market’s historical returns. Recall that the standard deviation is the square root of the variance, such that the variance is the standard deviation squared, so the variance of the market’s returns is .202 = .04. Thus we compute:
β = [pic] = .9625, or about .96.
8. A $12,000,000 investment portfolio contains 5 different stocks: Nuevo Leon Electronics ($3,480,000 worth,
with a beta of 1.75), Oaxaca Discount Stores ($2,760,000 worth, with a beta of .82), Sinaloa Heavy Manufacturing ($2,520,000 worth, with a beta of 1.38), Sonora Food Products ($1,920,000 worth, with a beta of .65), and Tabasco Bank ($1,320,000 worth, with a beta of 1.15). What is the beta for the portfolio? If the annual risk-free rate of return is expected to average 3.80% in future years and the return on the overall stock market is expected to average 9.75% per year, what is the expected annual rate of return on this portfolio?
Type: Computing beta for a portfolio. A portfolio’s beta, which is a measure of the portfolio’s non-diversifiable (or “systematic” or “market”) risk, is simply the weighted average of the betas of
the individual securities in the portfolio. Here we have a $12,000,000 portfolio with the following breakdown:
$3,480,000/$12,000,000 = 29% of the portfolio has a beta of 1.75
$2,760,000/$12,000,000 = 23% of the portfolio has a beta of .82
$2,520,000/$12,000,000 = 21% of the portfolio has a beta of 1.38
$1,920,000/$12,000,000 = 16% of the portfolio has a beta of .65
$1,320,000/$12,000,000 = 11% of the portfolio has a beta of 1.15
$12,000,000 100%
Therefore the portfolio’s beta is
.29 (1.75) + .23 (.82) + .21 (1.38) + .16 (.65) + .11 (1.15)
= .5075 + .1886 = .2898 + .104 + .1265 = 1.2164, or about 1.22.
The portfolio has about 22% more systematic risk than does the stock market overall (with the market beta defined as 1.0). Thus the portfolio is expected to earn higher returns in good years, and lower returns in bad years, than the market overall, but to provide a long-run average annual return of:
KPortfolio = krf + (km – krf) β
= .0380 + (.0975 – .0380) ( 1.22
= .0380 + .0595 ( 1.22
= .0380 + .07259 = .11059, or about 11.06%.
Another way to look at the situation is: the person who holds this portfolio is accepting a greater than average level of systematic (market) risk, and therefore will be disappointed if the portfolio’s rate of return does not average at least 11.06% per year in the long run, which is more than the expected average return on the stock market overall. (Note that here the expected average market return is km = 9.75% per year, for a market risk premium of km – krf = .0975 – .0380 = 5.95%).
9. Based on historical observations over many years, it seems that the annual rate of return earned on Tamaulipas Vitamin Company’s common stock depends on the state of the economy. If economic conditions are excellent (an outcome that tends to occur about 5% of the time), the annual return should be 12%. If the economy is good (which occurs about 20% of the time), the return should be 10%. Average economic performance (which has occurred about 50% of the time in the past) should be accompanied by a return of 8%; a poor economy (20% of the time) should lead to 6% earned; and a terrible economic state (5% historical frequency) should bring about a 4% annual return. Compute the expected (average, or mean) annual rate of return for Tamaulipas’s common stock, along with the standard deviation and coefficient of variation for the returns.
Type: Standard deviation of returns. The expected return is the average annual rate of return that investors would expect to earn over a long-term holding period. It is computed as a weighted average of the possible outcomes:
.05 (.12) + .20 (.10) + .50 (.08) + .20 (.06) + .05 (.04)
= .006 + .02 + .04 + .012 + .002 = .08 or 8.00%.
The standard deviation measures how far the true observations tend to fall from the mean, or expected value:
State of Return minus Squared difference
economy Return expected return Squared Probability x Probability
Excellent .12 .12 - .08 = .04 .0016 .05 .00008
Good .10 .10 – .08 = .02 .0004 .20 .00008
Average .08 .08 – .08 = .00 .0000 .50 .00000
Poor .06 .06 – .08 = – .02 .0004 .20 .00008
Terrible .04 .04 – .08 = – .04 .0016 .05 .00008
Variance σ2 .00032
The standard deviation is the square root of the variance: σ = [pic] = .017889, or about 2%.
Thus while the average expected return is 8% per year, it would not be at all surprising, based on historical observations, if a particular year’s return were up to 2% higher or lower (recall that if data are approximately normally distributed, actual observations should fall within one standard deviation of the mean about two thirds of the time).
The coefficient of variation is the standard deviation divided by the mean or expected return:
CV = .017889 ÷ .08 = .223613.
The standard deviation is used as a measure of investment risk; a low standard deviation indicates that the past returns have not been tremendously erratic, such that the mean is a reasonably good representation of the “typical” year’s return. The coefficient of variation refines the standard deviation to adjust for size; here we see that the standard deviation is only about 22% of the mean, so the standard deviation is fairly low in relative, and not just absolute, terms.
To understand the coefficient of variation’s usefulness, think of a case in which we compute dollar returns. Project A has a standard deviation of $1,000, while Project B’s is $10,000. It might seem that B shows greater variation. However, what if Project A’s mean expected return is only $2,000, whereas Project B’s is $100,000? With A’s coefficient of variation $1,000/$2,000 = .50 and B’s coefficient of variation $10,000/$100,000 = .10, we can see that B has the lower risk despite its higher standard deviation of dollar returns.
10. Based on historical observations over many years, it seems that the annual rate of return earned on Veracruz Holiday Resorts’ common stock depends on the state of the economy. If economic conditions are excellent (an outcome that tends to occur about 5% of the time), the annual return should be 33%. If the economy is good (which occurs about 20% of the time), the return should be 16%. Average economic performance (which has occurred about 50% of the time in the past) should be accompanied by a return of 9%; a poor economy (20% of the time) should lead to -3% earned; and a terrible economic state (5% historical frequency) should bring about a -15% annual return. Compute the expected (average, or mean) annual rate of return for Veracruz’s common stock, along with the standard deviation and coefficient of variation for the returns.
Type: Standard deviation of returns. The expected return is the average annual rate of return that investors would expect to earn over a long-term holding period. It is computed as a weighted average of the possible outcomes:
.05 (.33) + .20 (.16) + .50 (.09) + .20 (-.03) + .05 (-.15)
= .0165 + .032 + .045 – .006 – .0075 = .08 or 8.00%.
The standard deviation measures how far the true observations tend to fall from the mean, or expected value:
State of Return minus Squared difference
economy Return expected return Squared Probability x Probability
Excellent .33 .33 - .08 = .25 .0625 .05 .003125
Good .16 .16 – .08 = .08 .0064 .20 .001280
Average .09 .09 – .08 = .01 .0001 .50 .000050
Poor – .03 – .03 – .08 = – .11 .0121 .20 .002420
Terrible – .15 – .15 – .08 = – .23 .0529 .05 .002645
Variance σ2 .009520
The standard deviation is the square root of the variance: σ = [pic] = .097570, or about 10%.
Thus while the average expected return is 8% per year, it would not be at all surprising, based on historical observations, if a particular year’s return were up to 10% higher or lower (recall that if data are approximately normally distributed, actual observations should fall within one standard deviation of the mean about two thirds of the time). The coefficient of variation is the standard deviation divided by the mean or expected return:
CV = .097570 ÷ .08 = 1.219625.
The standard deviation is used as a measure of investment risk; a high standard deviation indicates that the past returns have been erratic, such that the mean is not a very good representation of the “typical” year’s return. The coefficient of variation refines the standard deviation to adjust for size; here we see that the standard deviation is about 122% of the mean, so the standard deviation is fairly high in relative, and not just absolute, terms. Compare Veracruz’s stock with Sinaloa’s, as shown in question 9 above; each has an 8% expected annual return, but if historical patterns persist Sinaloa’s stock will deliver returns that are always within a few percentage points of 8%, whereas Veracruz’s will be somewhat-to-very far from 8% a high proportion of the time.
The standard deviation and coefficient of variation are used as measures of investment risk, both for individual common stocks and for portfolios. However, there is a problem with judging the risk of a single investment based on the standard deviation of its past returns. Specifically, a given “risky” stock might be well positioned to offset the risk of another “risky” stock if the historical (and expected future) returns earned on the two have experienced their highs and lows at offsetting times. Recall our “stock M and stock W” example from class; also see the next problem.
11. Based on observations over many years, the common stockholders of the companies shown below have earned the following annual returns (and would expect to earn the same pattern of returns in the future):
State of Economy Deluxe Cruise Lines Uppity Custom Clothiers Cheep Mart
Good (18% of the time) .3350 .3407 -.1025
Average (64% of the time) .1184 .1070 .1100
Poor (18% of the time) -.1450 -.1100 .3225
As an analyst at Yucatan Investment Advisors, you have been asked to compute the mean and standard deviation
of each company’s returns to its common equity investors, and to comment on the risk of each stock as a stand-alone investment. You have also been asked to comment on whether any particular pair would work well in a portfolio together, so you want to compute the standard deviations of returns for portfolios consisting of each pair of stocks.
Type: Standard deviation of portfolio returns. First we compute the means and standard deviations of returns for the individual stocks.
For Deluxe: Expected (mean) annual return = .18 (.3350) + .64 (.1184) + .18 (-.1450) = 11.00%.
State of Return minus Squared difference
economy Return expected return Squared Probability x Probability
Good .3350 .3350 – .11 = .2250 .0506 .18 .009113
Average .1184 .1184 – .11 = .0084 .00007 .64 .000045
Poor – .1450 – .1450 – .11 = – .2550 .0650 .18 .011705
Variance σ2 .020863
The standard deviation is the square root of the variance: σ = [pic] = 14.444%.
For Uppity: Expected (mean) annual return = .18 (.3407) + .64 (.1070) + .18 (-.1100) = 11.00%.
State of Return minus Squared difference
economy Return expected return Squared Probability x Probability
Good .3407 .3407 – .11 = .2307 .0532 .18 .009580
Average .1070 .1070 – .11 = – .0030 .000009 .64 .000006
Poor – .1100 -.1100 – .11 = – .2200 .0484 .18 .008712
Variance σ2 .018298
The standard deviation is the square root of the variance: σ = [pic] = 13.527%.
Finally, for Cheep Mart: Expected annual return = .18 (-.1025) + .64 (.1100) + .18 (.3225) = 11.00%.
State of Return minus Squared difference
economy Return expected return Squared Probability x Probability
Good -.1025 -.1025 – .11 = – .2125 .0452 .18 .008128
Average .1100 .1100 – .11 = .0000 .0000 .64 .000000
Poor .3225 .3225 – .11 = .2125 .0452 .18 .008128
Variance σ2 .016256
The standard deviation is the square root of the variance: σ = [pic] = 12.750%.
So each stock, if held in isolation, would appear to be risky, delivering annual returns that could easily differ from its 11% long-run expected average by about 13 to 14% per year (recall that if data are approximately normally distributed, actual observations should fall within one standard deviation of the mean about two thirds of the time). The reason is that, if the stock is held in isolation, we have not diversified away any company-specific, or “unsystematic,” risk (reflecting things like material cost increases, labor problems, or natural disasters that hit individual firms
in different ways and at different times).
But what if we hold a risky stock in a portfolio with other risky stocks; will some of the individual company-specific risks offset each other? It depends on the patterns of returns. Our goal should be to construct a portfolio of stocks with expected returns that are not highly correlated, as measured by the correlation coefficient ρ, computed for hypothetical stocks A and B as:
ρAB = [pic]
(σA and σB are the standard deviations of the historical returns on A and B, and σAB is the covariance of historical returns on A and B). The correlation coefficient can have a value ranging from 1 (perfect positive correlation, with returns identical from year to year) to – 1 (perfect negative correlation, with returns just the opposite from year to year). [Ideally the correlations would
be negative, although in a real-world sense the returns on many common stocks tend to go up or down together, to some extent, with the perceived strength of the economy.] We compute the covariance by 1) computing the difference, or deviation, between the first stock’s possible return under each state of the economy and its expected return, 2) computing the deviation between the second stock’s possible return under each economic state and its expected return, 3) multiplying
to get the product of the two deviations under each state of the economy, and 4) computing the weighted average of these products.
For a portfolio of Deluxe and Uppity:
Economic State Deluxe Deviation Uppity Deviation Product of Deviations
Good (.18) .3350 – .11 = .2250 .3407 – .11 = .2307 .2250 x .2307 = .051908
Average (.64) .1184 – .11 = .0084 .1070 – .11 = .0030 .0084 x .0030 = .000025
Poor (.18) – .1450 – .11 = – .2550 –.1100 – .11 = – .2200 –.2550 x –.2200 = .056100
σDeluxe,Uppity = .18 (.051908) + .64 (.000025) + .18 (.056100) = .019457 .
ρDeluxe,Uppity = [pic] = .995854 .
Deluxe and Uppity would not be expected to form a good combination, because their historical patterns of annual returns are almost 100% positively correlated. Each delivers high returns in
a good economy and low returns in a poor economy, so there is little diversification benefit to be gained from combining the two into a portfolio (they are like stocks M and M’ in our class example). In fact, we can compute the standard deviation for a portfolio of two assets A and B as:
σPortfolio of A and B = [pic] .
In this equation, XA and XB are the proportions that A and B constitute of the portfolio (XA + XB must equal 100%), σA2 and σB2 are the variances of the returns on A and B, and σAB is the covariance of the returns earned on A and B. The standard deviation for a portfolio thus increases as the individual assets’ variances increase, and then increases even more if the covariance of returns is positive – or declines if the covariance of returns is negative. For a portfolio consisting 50% of Deluxe and 50% of Uppity [expected return = .50 (.11) + .50 (.11) = 11%], we compute
σPortfolio of Deluxe and Uppity = [pic]
= [pic] = .139712 or about 14%.
However, a portfolio of Deluxe and Cheep Mart makes more sense:
Economic State Deluxe Deviation Cheep Mart Deviation Product of Deviations
Good (.18) .3350 – .11 = .2250 -.1025 – .11 = – .2125 .2250 x –.2125 = –.047813
Average (.64) .1184 – .11 = .0084 .1100 – .11 = .0000 .0084 x .0000 = .000000
Poor (.18) – .1450 – .11 = – .2550 .3225 – .11 = .2125 –.2550 x .2125 = –.054188
σDeluxe,Cheep Mart = .18 (–.047813) + .64 (.000000) + .18 (–.054188) = –.018360 .
ρDeluxe,Cheep Mart = [pic] = –.996954 .
Deluxe and Cheep Mart form an excellent combination, because their historical patterns of annual returns approach 100% negative correlation. Deluxe delivers high returns in a good economy and low returns in a poor economy, while Cheep Mart delivers low returns in a good economy and high returns in a poor economy, so there is much diversification benefit to be gained from combining the two into a portfolio (they are like stocks M and W in our class example). For a portfolio consisting 50% of Deluxe and 50% of Cheep Mart [expected return = .50 (.11) + .50 (.11) = 11%], we compute
σPortfolio of Deluxe and Cheep Mart = [pic]
= [pic] = .01 or about 1%.
In a similar manner, for a portfolio of Uppity and Cheep Mart:
Economic State Uppity Deviation Cheep Mart Deviation Product of Deviations
Good (.18) .3407 – .11 = .2307 -.1025 – .11 = – .2125 .2307 x –.2125 = –.049024
Average (.64) .1070 – .11 = .0030 .1100 – .11 = .0000 .0030 x .0000 = .000000
Poor (.18) –.1100 – .11 = – .2200 .3225 – .11 = .2125 –.2200 x .2125 = –.046750
σUppity, Cheep Mart = .18 (–.049024) + .64 (.000000) + .18 (–.046750) = –.017239 .
ρUppity,Cheep Mart = [pic] = –.999540 .
Uppity and Cheep Mart form an even better (by a tiny margin) combination than Deluxe and Cheep Mart, because their historical patterns of annual returns are even closer to being 100% negatively correlated. Again we see one company (here, Uppity) delivering high returns in a good economy and low returns in a poor economy, while the other (again, Cheep Mart) delivers low returns in a good economy and high returns in a poor economy, so there is much diversification benefit to be gained from combining the two into a portfolio (again, like stocks M and W in our class example). For a portfolio consisting 50% of Uppity and 50% of Cheep Mart [expected return = .50 (.11) + .50 (.11) = 11%], we compute
σPortfolio of Uppity and Cheep Mart = [pic]
= [pic] = .004416 or about .44%.
In summary, a 50/50 portfolio of Deluxe and Uppity has an expected return of 11% and a standard deviation of 14% (about the same as the standard deviations for the individual stocks’ returns), whereas a 50/50 portfolio of Deluxe and Cheep Mart has the same expected return of 11% but
a standard deviation of only 1% (much less than the average of the standard deviations for the individual stocks’ returns), while a 50/50 portfolio of Uppity and Cheep Mart has that same 11% expected return but a standard deviation of less than one half of 1% (again, much less than the average of the standard deviations for the individual stocks’ returns). Finding the right combination of stocks for a portfolio allows us to achieve the same expected return, but with much more stability in the returns from year to year.
In a real world sense, we would try to eliminate company-specific risks by creating a portfolio not of just two stocks, but of many stocks, making sure the historic (and expected future) return patterns were not highly positively correlated (finding high negative correlations can be difficult – even companies that sell low priced goods tend to do a little better when the economy is strong and to suffer, though not as much as luxury goods sellers, when the economy is weak). Then, with company-specific risks no longer a concern, we could focus on the degree to which our stocks were, on average, subject to the market-wide, or “systematic,” risks of investing in the stock market – as measured by the beta of the portfolio (the weighted average of the individual stocks’ betas).
12. All of your money is currently invested in the common stock of Steady Stores, Inc., which you selected because the chain’s reasonably stable sales and profits have led to fairly stable annual returns for its owners – albeit higher in years when the economy is strong (and shoppers buy a range of Steady Stores merchandise) than when the economy is weak (and shoppers go to Steady Stores primarily to buy “staple” necessities). You know it is unwise to “put all your eggs in one basket,” but have limited your investment holdings to Steady Stores because you wanted to avoid the common stocks of firms that have, historically, delivered erratic returns. One such company is Used Suit Outlets, Ltd., whose sales and profits skyrocket when the economy is weak, but plummet when the economy is strong and few people buy second-hand clothing. Observations over many recent years show the two companies to have delivered the following annual returns to their common stockholders (and the same pattern would be expected in the future):
State of Economy Steady Stores, Inc. Used Suit Outlets, Ltd.
Excellent ( 5% of the time) .19 –.40
Good (20% of the time) .16 –.025
Average (50% of the time) .13 .11
Poor (20% of the time) .10 .35
Terrible ( 5% of the time) .07 .60
A trust officer at Zacatecas Bank suggests that you could actually create more predictable future investment returns for yourself by creating a “portfolio,” selling 15% of your Steady stock and replacing it with Used Suit stock. But based on the stability (always in the 7% to 19% range) of Steady Stores annual returns, and the severe instability (anywhere from +60% to – 40%) of Used Suit returns in the past, you feel it is safer to keep 100% of your money invested in Steady Stores. Are you correct? Hint: compute the mean and standard deviation of each company’s historical returns to its common equity investors, and the mean and standard deviation of returns for the portfolio.
Type: Covariance of investment returns. The primary intent of this problem is to explain what a covariance is. First, note that the mean (expected) annual return for Steady Stores is .05 (.19) + .20 (.16) + .50 (.13) + .20 (.10) + .05 (.07) = 13%. Now think of a “portfolio” consisting only of Steady Stores common stock. The investor says, “I’ll put some of my money in Steady Stores stock, and then put the rest of it into Steady Stores, too.” In such a case, the variance (and, in turn, the standard deviation) of annual returns depends on how Steady Stores return on equity has varied historically, relative to its own expected value, as the economy has experienced its ups and downs:
Deviation of Deviation of
Returns for Steady Returns for Steady Dev. of Steady x
State of (Return minus (Return minus Deviation of Steady x Dev. of Steady
Economy expected return) expected return) Deviation of Steady Prob. x Prob.
Excellent .19 – .13 = .06 .19 – .13 = .06 .06 x .06 = .0036 .05 .00018
Good .16 – .13 = .03 .16 – .13 = .03 .03 x .03 = .0009 .20 .00018
Average .13 – .13 = .00 .13 – .13 = .00 .00 x .00 = .0000 .50 .00000
Poor .10 – .13 = -.03 .10 – .13 = -.03 -.03 x -.03 = .0009 .20 .00018
Terrible .07 – .13 = -.06 .07 – .13 = -.06 -.06 x -.06 = .0036 .05 .00018
Variance σ2 of returns for Steady Stores .00072
Std. Deviation σ (sq. root of variance) .02683
So a portfolio consisting only of Steady Stores common stock (or the stocks of two identical twin companies that have both historically delivered the same annual returns on equity as Steady Stores) would have a 13% expected annual rate of return, with a standard deviation of about 3%. Thus while the returns would be reasonably stable from year to year, at least in relative terms, we would not
be surprised to see that a given year’s return were something like 16% (3% higher than the 13% long-run average) or 10% (3% lower than the 13% long-run average). Recall that if data are normally distributed, then the observed outcome should be within one standard deviation of the mean about 2/3 of the time (the historical return pattern here is not truly normally distributed, but viewing it as normal may not be too bad an approximation).
Two points to note: first, the standard deviation is small, because the observed outcomes have never been all that far from the 13% expected value (at worst 6% above or below). Second, in computing a variance/standard deviation we typically find each outcome’s deviation from the mean and square it. Here, instead, we multiplied “Deviation of Returns for Steady” by “Deviation of Returns for Steady.” Multiplying something by itself is just squaring it, of course. But we did it that way to better show the relationship between a variance and a covariance (which we will see below).
Now note that the mean (expected) annual return for Used Suit Outlets is .05 (-.40) + .20 (-.025) + .50 (.11) + .20 (.35) + .05 (.60) = the same 13% as the expected return for Steady Stores. Think of
a “portfolio” consisting only of Used Suit Outlets common stock. The investor says, “I’ll put some
of my money in Used Suit stock, and the rest in Used Suit stock, too.” In such a case, the variance (and, in turn, standard deviation) of returns depends on how Used Suit’s return on equity has varied historically, relative to its own expected value, as the economy has fluctuated:
Deviation of Deviation of
Returns for Used Returns for Used Dev. of Used x
State of (Return minus (Return minus Deviation of Used x Dev. of Used
Economy expected return) expected return) Deviation of Used Prob. x Prob.
Excellent -.400 – .13 = –.530 -.400 – .13 = –.530 –.53 x –.53 = .280900 .05 .014045
Good -.025 – .13 = –.155 -.025 – .13 = –.155 –.155 x –.155 = .024025 .20 .004805
Average .110 – .13 = –.020 .110 – .13 = –.020 –.020 x –.020 = .000400 .50 .000200
Poor .350 – .13 = .220 .350 – .13 = .220 .220 x .220 = .048400 .20 .009680
Terrible .600 – .13 = .470 .600 – .13 = .470 .470 x .470 = .220900 .05 .011045
Variance σ2 of returns for Used Suit Outlets .039775
Std. Deviation σ (sq. root of variance) .199437
So we can see that a portfolio consisting only of Used Suit Outlets common stock (or the stocks of two identical twin companies that have both historically delivered the same annual returns on equity as Used Suit) would have a 13% expected annual rate of return – just like Steady Stores – but with a much bigger standard deviation of about 20%. (We could have told just by “eyeballing” Used Suit Outlets’ widely-dispersed historical returns that the standard deviation would be pretty big.) Thus returns are not expected to be stable from year to year; we would not be surprised to see a given year’s return somewhere in the neighborhood of 33% (20% higher than the 13% long-run average)
or –7% (20% lower than the 13% mean). Again recall that if data are normally distributed (here the historical return pattern approximates normality even though it is not truly normal), the observed outcome should be within one standard deviation of the mean about 2/3 of the time. And as above, when computing a variance/standard deviation we typically find each outcome’s deviation from the mean and square it, but here we again very methodically multiplied “Deviation of Returns” by itself to better illustrate the relationship between a variance and a covariance (we’re getting there).
Looking at the stable returns your Steady Stores common stock has delivered, you find it hard to believe that a portfolio mixing the predictable Steady Stores with the volatile Used Suits could provide more stable returns than you get from Steady Stores alone. But if you wanted to relate the historical return on equity for Steady Stores to that of Used Suit Outlets, you would compute this relationship – the covariance – as:
Deviation of Deviation of
Returns for Steady* Returns for Used** Dev. of Steady
State of (Return minus (Return minus Deviation of Steady x x Dev. of Used
Economy expected return) expected return) Deviation of Used Prob. x Prob.
Excellent .19 – .13 = .06 -.400 – .13 = –.530 .06 x –.530 = –.03180 .05 –.00159
Good .16 – .13 = .03 -.025 – .13 = –.155 .03 x –.155 = –.00465 .20 –.00093
Average .13 – .13 = .00 .110 – .13 = –.020 .00 x –.020 = .00000 .50 .00000
Poor .10 – .13 = –.03 .350 – .13 = .220 –.03 x .220 = –.00660 .20 –.00132
Terrible .07 – .13 = –.06 .600 – .13 = .470 –.06 x .470 = –.0282 0 .05 –.00141
Covariance σSteady,Used of returns for Steady, Used –.00525
* From first (Steady Stores) grid above
** From second (Used Suit Outlets) grid above
Notice the similarity of the structure for computing a variance and a covariance. (This approach shows why it is sometimes stated that something’s variance is its covariance with itself.) Since
the covariance is negative, we can see that when the economy changes, the yearly returns earned
by Steady Stores and Used Suit Outlets common stockholders change, on average, in opposite directions. When the covariance of returns is negative, so is the more easily interpreted correlation coefficient, computed as
ρSteady,Used = [pic] = –.98114 .
A correlation coefficient’s value can range from 1 (perfect positive correlation, with past returns identical from year to year) to –1 (perfect negative correlation, with returns just the opposite from year to year). If the correlation coefficient for historical returns is negative, then the two stocks should be good candidates for combining into a portfolio; the ups and downs in their future returns can be expected to have an offsetting effect on each other. Here the patterns of past returns on equity for Steady Stores and Used Suit Outlets are almost completely negatively correlated.
So what happens if you follow the banker’s advice, and replace some of your Steady Stores common stock with Used Suit Outlets common stock? Unfortunately, the standard deviation of returns for a portfolio of two assets is not simply the square root of the covariance in the assets’ returns. We compute the standard deviation for a portfolio combining Steady Stores and Used Suit Outlets’ common stocks using the more complicated formula:
σPortfolio of Steady and Used = [pic] ,
with XS and XU representing the proportions of Steady Stores and Used Suit Outlets in the portfolio (XS + XU must equal 100%), σS2 and σU2 representing the variances of the returns on Steady Stores and Used Suit, and σS,U representing the covariance of the returns earned on Steady and Used. The standard deviation for a portfolio thus is determined by the individual assets’ proportions, the individual assets’ variances, and the covariance of the individual assets’ returns (the standard deviation increases if the covariance of returns is positive, declines if that covariance is negative). For a portfolio consisting 85% of Steady Stores and 15% of Used Suit Outlets [expected return = .85 (.13) + .15 (.13) = 13%], we compute
σPortfolio of 85% Steady and 15% Used = [pic]
= [pic] = .008738 or about .9%.
Because of the negative correlation, you can reduce the volatility of your expected investment returns – even though they are not all that volatile to begin with – by combining a small amount of erratic (but in the opposite direction) Used Suits with relatively stable Steady Stores. A portfolio containing only Steady Stores has an expected annual return of 13% but a standard deviation of about 3%. A portfolio consisting 85% of Steady Stores and 15% of Used Suit Outlets has the same 13% expected annual return, but with a standard deviation of less than 1%. If historical patterns persist into the future, then this portfolio can be expected to provide annual returns that average 13% and never stray very far from 13% (hardly ever more than about 15% or less than about 11%).
Does this result mean that your instincts were wrong? Not completely. The benefit works here only if the amount of predictable Steady Stores replaced by erratic Used Suit is fairly small. What would happen, for example, if you replaced 40% of the Steady Stores common stock with Used Suit Outlets common stock (60%/40% proportions)? Then the portfolio’s standard deviation would be a considerably higher
σPortfolio of 60% Steady and 40% Used = [pic]
= [pic] = .064056, or more than 6%.
Here, replacing a small amount of the stable stock with the negatively correlated erratic stock causes more predictability of returns, but replacing a lot of the stable asset with the volatile one leaves you with less predictability (approximately twice the volatility; 6% vs. 3% standard deviation) than if you had simply kept all your eggs in the relatively safe Steady Stores basket.
But what if you were instead starting out with all of your money invested in Caviar Hut, a stock with historical returns much more volatile than Steady Stores: .42, .25, .13, .01, and -.16 as the economy moves from Excellent to Terrible. The expected annual return on equity for Caviar Hut common stockholders would be 13%, but the 11.904% standard deviation (1.417% variance) of returns would be much higher than that for Steady Stores, while Caviar Hut’s covariance with Used Suit Outlets would be -.02350 (you might double-check these figures for practice). Replacing 15% of the Caviar Hut stock with Used Suit Outlets would keep the expected annual return at 13% while reducing the portfolio standard deviation from 11.904% to
σPortfolio of 85% Caviar and 15% Used = [pic]
= [pic] = .071696, or about 7.2%,
a nice reduction. But replacing an even greater portion, for example half, of the volatile Caviar Hut common stock with volatile-but-in-the-opposite-direction Used Suit Outlets stock would reduce the portfolio’s standard deviation even more, to
σPortfolio of 50% Caviar and 50% Used = [pic]
= [pic] = .041668, or about 4.2%.
[Note that a portfolio invested half in Used Suit Outlets and the other half in Used Suit outlets also (or in another company with returns identical to Used) would have a standard deviation of
σPortfolio of 50% Used and the Other 50% Used Also = [pic]
= [pic] = .199437, as found for Used alone.]
In a real world sense, we would try to eliminate company-specific risks by creating a portfolio not
of just two stocks, but of many stocks, making sure that the historical (and expected future) return patterns were not highly positively correlated. [Finding high negative correlations can be difficult. In fact, finding low negative correlations is no picnic; for example, even companies that sell bargain priced goods or services tend to sell somewhat more of them when the economy is strong and other firms are earning high annual returns for their owners, as well.] Then, with company-specific risks no longer a concern, we could focus on the degree to which our stocks were, on average, subject to the market-wide, or “systematic,” risks of investing in the stock market – as measured by the beta of the portfolio (the weighted average of the individual stocks’ betas).
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