a)
To determine: The arithmetic average for large-company stocks and Treasury bills.
Introduction:
Arithmetic average return refers to the returns that an investment earns in an average year over different periods. Variance refers to the average difference of squared deviations of the actual data from the mean or average. Standard deviation refers to the deviation of the observations from the mean.
a)
Answer to Problem 8QP
The arithmetic average of large-company stocks is 3.24%, and the arithmetic average of Treasury bills is 6.55%.
Explanation of Solution
Given information:
Refer Table 10.1 in the chapter. Extract the data for large-company stocks and Treasury bills from 1973 to 1978 as follows:
Year |
Large co. stock return (A) |
T-bill return (B) |
Risk premium (A)-(B) |
1973 | –14.69% | 7.29% | –21.98% |
1974 | –26.47% | 7.99% | –34.46% |
1975 | 37.23% | 5.87% | 31.36% |
1976 | 23.93% | 5.07% | 18.86% |
1977 | –7.16% | 5.45% | –12.61% |
1978 | 6.57% | 7.64% | –1.07% |
Total | 19.41% | 39.31% | –19.90% |
The formula to calculate the arithmetic average return:
Where,
“∑Xi” refers to the total of observations,
“Xi” refers to each of the observations from X1 to XN (as “i” goes from 1 to “N”),
“N” refers to the number of observations.
Compute the arithmetic average for large-company stocks:
Hence, the arithmetic average of large-company stocks is 3.24 %.
Compute the arithmetic average for Treasury bill return:
The total of the observations is 36.24%. There are 6 observations.
Hence, the arithmetic average of Treasury bills is 6.55%.
b)
To determine: The standard deviation of large-company stocks and Treasury bills.
Introduction:
Arithmetic average return refers to the returns that an investment earns in an average year over different periods. Variance refers to the average difference of squared deviations of the actual data from the mean or average. Standard deviation refers to the deviation of the observations from the mean.
b)
Answer to Problem 8QP
The standard deviation of large-company stocks is 24.11%, and the standard deviation of Treasury bills is 1.24%.
Explanation of Solution
Given information:
Refer Table 10.1 in the chapter. The arithmetic average of Treasury bills is 6.55%.
Extract the data for large-company stocks and Treasury bills from 1973 to 1978 as follows:
Year |
Large co. stock return (A) |
T-bill return (B) |
Risk premium (A)-(B) |
1973 | –14.69% | 7.29% | –21.98% |
1974 | –26.47% | 7.99% | –34.46% |
1975 | 37.23% | 5.87% | 31.36% |
1976 | 23.93% | 5.07% | 18.86% |
1977 | –7.16% | 5.45% | –12.61% |
1978 | 6.57% | 7.64% | –1.07% |
Total | 19.41% | 39.31% | –19.90% |
The formula to calculate the standard deviation:
“SD (R)” refers to the variance,
“X̅” refers to the arithmetic average,
“Xi” refers to each of the observations from X1 to XN (as “i” goes from 1 to “N”),
“N” refers to the number of observations.
Compute the squared deviations of large-company stocks:
Large-company stocks | |||
Actual return | Average return (B) | Deviation (A)–(B)=(C) | Squared deviation (C)2 |
(A) | |||
−0.1469 | 0.0324 | −0.1793 | 0.0321485 |
−0.2647 | 0.0324 | −0.2971 | 0.0882684 |
0.3723 | 0.0324 | 0.3399 | 0.115532 |
0.2393 | 0.0324 | 0.2069 | 0.0428076 |
−0.0716 | 0.0324 | −0.104 | 0.010816 |
0.0657 | 0.0324 | 0.0333 | 0.0011089 |
Total of squared deviation | 0.05813 | ||
| |||
Compute the standard deviation:
Hence, the standard deviation of large-company stocks is 24.111%.
Compute the squared deviations of Treasury bill:
Large-company stocks | |||
Actual return | Average return (B) | Deviation (A)–(B)=(C) | Squared deviation (C)2 |
(A) | |||
0.0729 | 0.0655 | 0.0074 | 0.00005 |
0.0799 | 0.0655 | 0.0144 | 0.00020736 |
0.0587 | 0.0655 | −0.0068 | 0.00004624 |
0.0507 | 0.0655 | −0.0148 | 0.00021904 |
0.0545 | 0.0655 | −0.011 | 0.000121 |
0.0764 | 0.0655 | 0.0109 | 0.00011881 |
| 0.000154 |
Compute the standard deviation:
Hence, the standard deviation of Treasury bills is 1.24%.
c)
To determine: The arithmetic average and the standard deviation of observed risk premium.
Introduction:
Arithmetic average return refers to the returns that an investment earns in an average year over different periods. Variance refers to the average difference of squared deviations of the actual data from the mean or average. Standard deviation refers to the deviation of the observations from the mean.
c)
Answer to Problem 8QP
The arithmetic average is −3.32%, and the standard deviation is 24.92%
Explanation of Solution
Given information:
Refer Table 10.1 in the chapter. Extract the data for large-company stocks and Treasury bills from 1973 to 1978 as follows:
Year |
Large co. stock return (A) |
T-bill return (B) |
Risk premium (A)-(B) |
1973 | –14.69% | 7.29% | –21.98% |
1974 | –26.47% | 7.99% | –34.46% |
1975 | 37.23% | 5.87% | 31.36% |
1976 | 23.93% | 5.07% | 18.86% |
1977 | –7.16% | 5.45% | –12.61% |
1978 | 6.57% | 7.64% | –1.07% |
Total | 19.41% | 39.31% | –19.90% |
The formula to calculate the arithmetic average return:
Where,
“∑Xi” refers to the total of observations,
“Xi” refers to each of the observations from X1 to XN (as “i” goes from 1 to “N”),
“N” refers to the number of observations.
The formula to calculate the standard deviation:
“SD (R)” refers to the variance,
“X̅” refers to the arithmetic average,
“Xi” refers to each of the observations from X1 to XN (as “i” goes from 1 to “N”),
“N” refers to the number of observations.
Compute the arithmetic average for risk premium:
Hence, the arithmetic average of risk premium is −3.32%.
Compute the squared deviations of risk premium:
Risk premium | |||
Actual return (A) | Average return (B) | Deviation (A)–(B)=(C) | Squared deviation |
(C)2 | |||
−0.2198 | −0.0332 | −0.1866 | 0.034820 |
−0.3446 | −0.0332 | −0.3114 | 0.096970 |
0.3136 | −0.0332 | 0.3468 | 0.120270 |
0.1886 | −0.0332 | 0.2218 | 0.049195 |
−0.1261 | −0.0332 | −0.0929 | 0.008630 |
−0.0107 | −0.0332 | 0.0225 | 0.000506 |
| 0.062078 |
Compute the standard deviation:
Hence, the standard deviation of risk premium is 24.92%.
d)
To determine: Whether the risk premium can be negative before and after the investment.
Introduction:
Arithmetic average return refers to the returns that an investment earns in an average year over different periods. Variance refers to the average difference of squared deviations of the actual data from the mean or average. Standard deviation refers to the deviation of the observations from the mean.
d)
Explanation of Solution
The risk premium cannot be negative before the investment because the investors require compensation for assuming the risk. They will invest when the stock compensates for the risk. The risk premium can be negative after the investment, if the nominal returns are very low compared to the risk-free returns.
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