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 = –.0476 = –4.76%
b Rate of return = = 05 = 5%
Real rate = − 1 = − 1 = 0
c Rate of return = = 10 = 10%
Real rate = − 1 = − 1 = 0476 = 4.76%
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
Trang 3Week
Average
price of
stocks in
market
% change
in average stock price
Equal Weighted Index
% change
in Equal Weighted Indexa
Total Market value of stocks
% change
in total market value
Value Weighted Indexb
% change in Value Weighted Index
2 85.00 -1.05% 98.95 -1.05% 552,950 -3.23% 96.77 -3.23%
3 84.80 -0.24% 98.72 -0.24% 540,300 -2.29% 94.56 -2.29%
6 86.00 -2.93% 100.12 -2.93% 546,550 -1.00% 95.65 -1.00%
8 83.60 -1.88% 97.32 -1.88% 548,750 -0.93% 96.04 -0.93%
9 78.60 -5.98% 91.50 -5.98% 511,050 -6.87% 89.44 -6.87%
Notes:
a The value of the portfolio for the equal weighted index is the simple average of the prices of the stocks in the index For any week, the value of the index is the current simple average price of the stocks in the index divided by the average stock price for the first week of the index, multiplied by an arbitrary starting value for the index Here we start the index at 100 A second way to calculate the new value of the index is to multiply the previous value by 1 plus the percentage change in average stock price for that week You can see this by noting that the weekly percentage change in the simple average stock price is identical to the percentage change in the equal weighted index
NOTE: the values above have been calculated with Excel and more decimal places were used than are shown If you use the rounded values in the table, your answers will differ slightly than the ones above
b The market value index tracks the value of a portfolio that holds shares in each firm in proportion to the number of outstanding shares The composition of the value-weighted portfolio is identical to that of the entire market Therefore, the new value of the index equals the previous index value increased (or decreased)
by the percentage change in the total value of stocks in the market The market value index can also be calculated as the current market value of the stocks in the index divided by the market value of the stocks in the index for the first week of the index, multiplied by an arbitrary starting value for the index Again
we start the index at 100 Note that the weekly percentage change in the total market value of the stocks is identical to the percentage change in the value weighted index
NOTE: The values above have been calculated with Excel and more decimal places were used than are shown If you use the rounded values in the table, your answers will differ slightly than the ones above
Trang 4Weekly Rates of Return on Stocks
1
Average rate
Standard
deviation of
The average of the individual stocks’ standard deviation, is 0818 or 8.18% The
standard deviation of the equal-weighted portfolio, shown in the table is 3.81%
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 week, the squared difference between the week’s return
and the average rate of return for all weeks is calculated The squared deviations
are summed and dividend by 8 (the number of weeks minus 1) This gives the
sample variance The sample standard deviation is the square root of the sample
variance
9 a
Long bond risk premium
Trang 5b The average TSX risk premium was 0.63 % The average long bond risk premium was 5.4% for these five years These results are largely due to the very poor performance of the TSX in 2001 and 2002 No investor expected
to lose 12% each year on their stock portfolio!
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)] × [(1.78 - 63)2 + (-16.71 - 63)2 + (-14.99 - 63)2
+ (23.79 - 63)2 + (9.25 - 63)2
= 289.21
Standard deviation of TSX risk premium = = 17.01%
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 14.2% standard deviation and the Long Bond risk premium has a 3.33% standard deviation There is a lot more variation in the TSX risk premium because there is a lot more variation in the TSX
return than for the Long Bond portfolio
10 In early 2000, the Dow was more than three 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 613 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
P0 = 09751
50
2+
= $47.38
Expected return = 3 × 122.22 + 5 × 13.33 + 2 × (−100)= 23.33%
Variance = 3 × (122.22 − 23.33)2 + 5 × (13.33−23.33)2 + 2 × (−100−23.33)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
Recession = 48%
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 7Standard deviation = = 26.5%
Portfolio Rate of Return Boom (−28 + 122.22)/2 = 47.11%
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 × (−5−13)2 + 6 × (15−13)2 + 2 × (25 – 13)2 = 96 Standard deviation = = 9.80%
Variance(bonds) = 2 × (14−8.4)2 + 6 × (8−8.4)2 + 2 × (4−8.4)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%
Boom (25% × 6) + ( 4% × 4) = 16.6%
b Expected return = 2 × 2.6% + 6 × 12.2% + 2 × 16.6% = 11.16%
Trang 8Variance = 2 × (2.6 – 11.16)2 + 6 × (12.2 – 11.16)2 + 2 × (16.6 – 11.16)2
= 21.22
Standard deviation = 21.22= 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 9.6
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 9b Club Med sells a luxury good (expensive vacations) while General Cinema sells movies, which are less sensitive to changes in the economy Club Med 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
Trang 10somewhat reduces total variability
25 a and b
Average return (%)
Standard deviation (%)
The average standard deviation of the three securities is 7.23% = (16.8 +3.51+1.39)/3, higher than the portfolio standard deviation of 6.01%, 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)
TSX T-Bill Long Bond Portfolio
26 The correlation coefficients between the 9 weekly rates of return on Tonsil and each of the stocks are as follows:
Tonsil Prochnik Krosno Exbud Kable
Correlation with
As expected, the correlation of Tonsil with itself is 1 The stock offering the best
diversification benefit is Exbud Its return is negatively correlated with Tonsil's rate of