We use the relationship with all rates expressed as decimals that: Real rate = 1 Asset class Nominal Return Inflation Real Rate 6.. On the other hand, the real expected rate of return
Trang 1Solutions to Chapter 10 Introduction to Risk, Return, and the Opportunity Cost of Capital
1 Return =
= = 15 = 15%
Dividend yield = dividend / initial price = 2/40 = 05 = 5%
Capital gains yield = capital gains / initial price
= 4/40 = 10 = 10%
2 Dividend yield = 2/40 = 05 = 5% The dividend yield is unaffected; it is based on the initial price, not the final price
Capital gain = $36 – $40 = $4
Capital gains yield = –4/40 = –.10 = – 10%
3 a Rate of return =
= = 0 Real rate = 1 = 1 = –.0291 = –2.91%
b Rate of return = = 05 = 5%
Real rate = 1 = 1 = 0194 = 1.94%
c Rate of return = = 10 = 10%
Real rate = 1 = 1 = 0680 = 6.80%
Trang 24 Real return = 1
Costaguana: Real return = 1 = 0833 = 8.33%
Canada: Real return = 1 = 1067 = 10.67%
Canada provides the higher real return despite the lower nominal return Notice that the approximation
real rate nominal rate – inflation rate
would incorrectly suggest that the Costaguanan real rate was higher than the Canadian real rate The approximation is valid only for low rates
5 We use the relationship (with all rates expressed as decimals) that:
Real rate = 1
Asset class Nominal Return Inflation Real Rate
6 The nominal interest rate cannot be negative If it were, investors would choose to hold cash (which pays a return of zero) rather than buy a bill providing a negative
return On the other hand, the real expected rate of return is negative if the inflation
rate exceeds the nominal return
7
Quarter
Average price of stocks in market
Index (using DJIA method)
Total market value of stocks
Index (using S&P method)
Trang 33 888.33 98.43 607,760 96.64
8
Quarterly Rates of Return Tanzani
Tanzania Tanzania Cigarette portfolio Quarter Breweries, TOL PackersTea Company Simba Dahaco 1/6 of each stock Sep
2003 -0.0317 0 -0.1667 -0.0423 -0.0143 0.04 -0.0397 Dec
2003 -0.0164 -0.019 0.16 0.0118 0.0145 0.0962 0.025 Mar
Average
rate of
return -0.0605 -0.006 -0.008 -0.0102 0.062 0.0512 -0.0093 Standard
deviation
of return 0.0635 0.0109 0.1635 0.0284 0.1081 0.0405 0.0325
The simple average of the individual stocks’ standard deviation, is 0692 or 6.92% The standard deviation of the equal-weighted portfolio, shown in the table, is 3.25% This is striking evidence of the benefits of diversification
Note: Since the question works with observed data, the sample standard deviations are calculated Thus for each stock the average rate of return is calculated Then, for each quarter, the squared difference between the quarter’s’s return and the average rate of return for all quarters is calculated The squared deviations are summed and divided by 2 (the number of quarterly returns s minus 1) This gives the sample variance The sample standard deviation is the square root of the sample variance 9
TSX T-Bill Long Bond premiumTSX risk Long bond riskpremium
Trang 4b The average TSX risk premium was 15.26 % The average long bond risk premium was 4.4% for these five years These results are largely due to the very good performance of the TSX in 2003 to 2007
c A fast way to calculate standard deviation of a sample of data is using a
spreadsheet, such as Excel In Excel, use the STDEV function Alternatively, the standard deviation can be calculated by hand First, calculate the sample variance, then take the square root The sample variance is the sum of the squared deviations from the mean, divided by the number of observations minus 1 We illustrate with the TSX risk premium:
Variance of TSX risk premium
= [1/(5-1)] × [(23.68 – 15.26)2 + (12.34 – 15.26)2 + (21.74 – 15.26)2
+ (13.0 – 15.26)2 + (5.55 – 15.26)2
= 55.20
Standard deviation of TSX risk premium = = 7.43%
We would expect that the risk premium standard deviation would be higher for the TSX than for the Long Bond portfolio This is what we find: the TSX risk premium has a 7.43% standard deviation and the Long Bond risk
premium has a 5.56% standard deviation There is more variation in the TSX risk premium because there is more variation in the TSX return than for the Long Bond portfolio
10 In 2007, the S&P/TSX was more than four times its 1990 level Therefore a 40-point movement was far less significant in percentage terms than in 1990 We
would expect to see more 40-point days even if market risk as measured by
percentage returns is no higher than in 1990
11 Investors would not have invested in bonds if they had expected to earn negative
average returns Unanticipated events must have led to these results For example,
inflation and nominal interest rates during this period rose to levels not seen for decades These increases, which resulted in large capital losses on long-term bonds, were almost surely unanticipated by investors who bought those bonds in prior
years
The results from this period demonstrate the perils of attempting to measure “normal” maturity (or risk) premiums from historical data While experience over long periods may be a reasonable guide to normal premiums, the realized premium over short periods may contain little information about expectations of future premiums
12 If investors become less willing to bear investment risk, they will require a higher
risk premium for holding risky assets Security prices will fall until the expected rates of return on those securities rise to the now-higher required rates of return.
Trang 513 Based on the historical risk premium of the TSX (7.0 percent), and the current level
of the risk-free rate (about 2.75 percent), one would predict an expected rate of
return of 9.75 percent If the stock has the same systematic risk, it also should
provide this expected return Therefore, the stock price equals the present value of
cash flows for a one-year horizon
50
2
= $47.38
Expected return = 3 122.22 + 5 13.33 + 2 (100)= 23.33%
Variance = 0.3 (122.22 23.33)2 + 5 (13.3323.33)2 + 2 (10023.33)2 = 6025.8 Standard deviation = = 77.63%
15 The bankruptcy lawyer does well when the rest of the economy is floundering, but
does poorly when the rest of the economy is flourishing and the number of
bankruptcies is down Therefore, the Tower of Pita is a good hedge When the
economy does well and the lawyer’s bankruptcy business suffers, the stock return is excellent, thereby stabilizing total income The owner of the gambling casino
probably does well when the economy is flourishing and less well when it is doing poorly For the casino owner, holding Tower of Pita stock will not stabilize total
income as much as it does for the bankruptcy lawyer
Expected return = 3 (28%) + 5 8% + 2 48% = 5.2%
Variance = .3 (28 – 5.2)2 + 5 (8 – 5.2)2 + 2 (48 – 5.2)2 = 700.96
Trang 6Standard deviation = = 26.5%
Portfolio Rate of Return
Normal (8 + 13.33)/2 = 10.665%
Recession (48 –100)/2 = –26.0%
Expected return = 3 47.11% + 5 10.665% + 2 (-26.0%) = 14.27%
Variance = .3 (47.11 – 14.27)2 + 5 (10.665 – 14.27)2 + 2 (-26.0 – 14.27)2 = 654.4
Standard deviation = = 25.6%
Standard deviation is lower than for either firm individually because the variations
in the returns of the two firms serve to offset each other When one firm does
poorly, the other does well, which reduces the risk of the combination of the two
17 a Interest rates tend to fall at the outset of a recession and rise during boom
periods Because bond prices move inversely with interest rates, bonds will provide higher returns during recessions when interest rates fall
b rstock = 2 (5%) + 6 15% + 2 25% = 13%
rbonds = 2 14% + 6 8% + 2 4% = 8.4%
Variance(stocks) = 2 (513)2 + 6 (1513)2 + 2 (25 – 13)2 = 96 Standard deviation = = 9.80%
Variance(bonds) = 2 (148.4)2 + 6 (88.4)2 + 2 (48.4)2 = 10.24 Standard deviation = = 3.20%
c Stocks have higher expected return and higher volatility More risk averse investors will choose bonds, while others will choose stocks
18 a Recession (5% 6) + (14% 4) = 2.6%
Normal (15% 6) + ( 8% 4) = 12.2%
b Expected return = 2 2.6% + 6 12.2% + 2 16.6% = 11.16%
Trang 7Variance = 2 (2.6 – 11.16)2 + 6 (12.2 – 11.16)2 + 2 (16.6 – 11.16)2
= 21.22
Standard deviation = 22.= 4.61%
c The investment opportunities have these characteristics:
Mean Return Standard Deviation
The best choice depends on the degree of your aversion to risk Nevertheless,
we suspect most people would choose the portfolio over stocks since it gives almost the same return with much lower volatility This is the advantage of diversification
d To calculate the correlation coefficient, rearrange the formula for the portfolio standard deviation as we did in Check Point 10.7
Correlation between bond and stock returns
= (σp – xs2 σs2 – xb σb ) / ( 2 xs xb σs σb)
= (.04612 – 62× 0982 – 42 × 0322) / ( 2 × 6 × 4 × 098 × 032) = -.995 The stocks and bonds are almost perfectly negatively correlated
19 If we use historical averages to compute the “normal” risk premium, then our
estimate of “normal” returns and “normal” risk premiums will fall when we include
a year with a negative market return This makes sense if we believe that each additional year of data reveals new information about the “normal” behaviour of the market portfolio We should update our beliefs as additional observations about the market become available
20 Risk reduction is most pronounced when the stock returns vary against each other When one firm does poorly, the other will tend to do well, thereby stabilizing the return of the overall portfolio By contrast stock returns that move together provide
no risk reduction If stock returns are independent, some risk reduction (variability reduction) occurs but it is less than if the stock returns vary against each other
21 a General Steel ought to have more sensitivity to broad market movements
Steel production is more sensitive to changes in the economy than is food consumption
Trang 8b Exotic World Tours Agency sells a luxury good (expensive vacations) while General Cinema sells movies, which are less sensitive to changes in the
economy Exotic World Tours Agency will have greater market risk
22 a Expected return = 5 × (-20%) + 5 × 30% = 5%
Standard deviation = [ 5 × (-20% - 5%)2 + 5 × (30% - 5%)2]1/2 = 25%
The expected rate of return on the stock is 5 percent The standard deviation
is 25 percent
b Because the stock offers a risk premium of zero (its expected return is the same as for Treasury bills), it must have no market risk All the risk must be diversifiable, and therefore of no concern to investors
23 Sassafras is not a risky investment to a diversified investor Its return is better when
the economy enters a recession Therefore, the company risk offsets the risk of the
rest of the portfolio It is a portfolio stabilizer despite the fact that there is a 90
percent chance of loss
(Compare Sassafras to purchasing an insurance policy Most of the time, you will lose money on your insurance policy But the policy will pay off big if you suffer losses elsewhere — for example, if your house burns down For this reason, we
view insurance as a risk-reducing hedge, not as speculation Similarly, Sassafras may be viewed as analogous to an insurance policy on the rest of your portfolio
since it tends to yield higher returns when the rest of the economy is faring poorly.)
In contrast, the Leaning Tower of Pita has returns that are positively correlated with the rest of the economy It does best in a boom and goes out of business in a
recession For this reason, Leaning Tower would be a risky investment to a
diversified investor since it increases exposure to the macroeconomic or market risk
to which the investor is already exposed
24 a Portfolio expected return = 3 × 9% + 7 × 8% = 8.3%
Portfolio standard deviation = [.32 × 22 +.72 × 252 + 2 × 3 × 7 × 2 × 2 × 25]1/2
= 196 = 19.6%
b With correlation of 7, the portfolio standard deviation is
= [.32 × 22 +.72 × 252 + 2 × 3 × 7 × 7 × 2 × 25]1/2
= 221 = 22.1%
c The higher is the correlation between two variables, the less potential for
diversification In (a), with correlation of only 2, the portfolio standard
deviation is less than the standard deviation of return of either of the two
stocks in the portfolio However, with the higher correlation of 7, the stocks’ return move more closely together and forming a portfolio only somewhat
Trang 9reduces total variability
25 a
The following table contains the annual rates of return, the five-year average rate of return and the standard deviation of the rates of return for each index and the portfolio with one-third in each of the indexes:
TSX T-Bill Long Bond Portfolio
b The table summarizes the calculations from (a):
Average return (%) deviation (%)Standard
The average standard deviation of the three securities is 4.23% = (6.95+4.85+0.88)/3, higher than the portfolio standard deviation of 3.40%, showing the benefit of
diversification If there were no benefits from diversification, the portfolio standard deviation would simply be the average of the standard deviations of each of the
securities in the portfolio, weighted by their portfolio weights (here the weights are each 1/3)
26 The correlation coefficients between the 3 quarterly rates of return on Tanzania
Breweries and each of the stocks are as follows:
Tanzania
Breweries TOL Tea Packers Company Simba Dahaco Correlation with TB 1.0000 (0.6009) 0.1694 (0.1927) (0.9678) 0.7990
As expected, the correlation of Tanzania Breweries with itself is 1 The stock offering the best diversification benefit is Simba Its return is most negatively correlated with
Trang 10Tanzania Breweries’ rate of return.
27 Internet:
a From the above tables, the overall risk premium is bigger when using the Treasury
Bill as the risk free security than using Treasury Bonds as the risk free security This makes sense: Treasury Bills are less risky than Treasury Bonds, making the difference
in risk between Treasury Bills and the market index bigger than the difference in risk between Treasury Bonds and the market index
b The risk premium becomes smaller over time
28 Internet:
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3yr risk: 15.17
TD Precious Metals
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3yr risk: 31.66
TD Energy
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