Concept explainers
Calculating Returns [LO2, 3] Refer to Table 12.1 in the text and look at the period from 1973 through 1980:
a. Calculate the average return for Treasury bills and the average annual inflation rate (consumer price index) for this period.
b. Calculate the standard deviation of Treasury bill returns and inflation over this period.
c. Calculate the real return for each year. What is the average real return for Treasury bills?
d. Many people consider Treasury bills risk-free. What do these calculations tell you about the potential risks of Treasury bills?
a)
To determine: The arithmetic average for Treasury bills and consumer price index (Inflation).
Introduction:
Arithmetic average return refers to the returns that an investment earns in an average year over different periods. Standard deviation refers to the deviation of the observations from the mean. Real return refers to the return after adjusting the inflation rate.
Answer to Problem 22QP
The arithmetic average of Treasury bills is 7.74 percent, and the arithmetic average of inflation rate is 9.29 percent.
Explanation of Solution
Given information:
Refer to Table 12.1 in the chapter. Extract the data for Treasury bills and consumer price index from 1973 to 1980 as follows:
Year | Treasury Bill Return | Consumer price index (Inflation) |
1973 | 0.0729 | 0.0871 |
1974 | 0.0799 | 0.1234 |
1975 | 0.0587 | 0.0694 |
1976 | 0.0507 | 0.0486 |
1977 | 0.0545 | 0.0670 |
1978 | 0.0764 | 0.0902 |
1979 | 0.1056 | 0.1329 |
1980 | 0.1210 | 0.1252 |
Total | 0.6197 | 0.7438 |
The formula to calculate the arithmetic average return:
Where,
“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 Treasury bill return:
The total of observations is 0.6197. There are 8 observations.
Hence, the arithmetic average of Treasury bills is 7.74 percent.
Compute the arithmetic average for inflation rate:
The total of observations is 0.7438. There are 8 observations.
Hence, the arithmetic average of inflation is 9.29 percent.
b)
To determine: The standard deviation of Treasury bills and consumer price index (Inflation).
Answer to Problem 22QP
The standard deviation of Treasury bills is 2.48 percent, and the standard deviation of consumer price index (Inflation) is 3.12 percent.
Explanation of Solution
Given information:
Refer to Table 12.1 in the chapter. Extract the data for Treasury bills and consumer price index from 1973 to 1980 as follows:
Year | Treasury Bill Return | Consumer price index (Inflation) |
1973 | 0.0729 | 0.0871 |
1974 | 0.0799 | 0.1234 |
1975 | 0.0587 | 0.0694 |
1976 | 0.0507 | 0.0486 |
1977 | 0.0545 | 0.0670 |
1978 | 0.0764 | 0.0902 |
1979 | 0.1056 | 0.1329 |
1980 | 0.1210 | 0.1252 |
Total | 0.6197 | 0.7438 |
The formula to calculate the standard deviation:
Where,
“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 Treasury bill:
Treasury bills | |||
Actual return (A) | Average return (B) | Deviation (A)–(B)=(C) | Squared deviation (C)2 |
0.0729 | 0.0774 | -0.0045 | 2.025E-05 |
0.0799 | 0.0774 | 0.0025 | 6.25E-06 |
0.0587 | 0.0774 | -0.0187 | 0.00034969 |
0.0507 | 0.0774 | -0.0267 | 0.00071289 |
0.0545 | 0.0774 | -0.0229 | 0.00052441 |
0.0764 | 0.0774 | -0.001 | 0.000001 |
0.1056 | 0.0774 | 0.0282 | 0.00079524 |
0.1210 | 0.0774 | 0.0436 | 0.00190096 |
Total of squared deviation
| 0.00431069 |
Compute the standard deviation:
Hence, the standard deviation of Treasury bills is 2.48 percent.
Compute the squared deviations of inflation:
Consumer price index (Inflation) | |||
Actual return (A) | Average return (B) | Deviation (A)–(B)=(C) | Squared deviation (C)2 |
0.0871 | 0.0929 | -0.0058 | 0.00003364 |
0.1234 | 0.0929 | 0.0305 | 0.00093025 |
0.0694 | 0.0929 | -0.0235 | 0.00055225 |
0.0486 | 0.0929 | -0.0443 | 0.00196249 |
0.0670 | 0.0929 | -0.0259 | 0.00067081 |
0.0902 | 0.0929 | -0.0027 | 7.29E-06 |
0.1329 | 0.0929 | 0.04 | 0.0016 |
0.1252 | 0.0929 | 0.0323 | 0.00104329 |
Total of squared deviation | 0.00680002 |
Compute the standard deviation:
Hence, the standard deviation of inflation is 3.12 percent.
c)
To determine: The real return for each year and the average real return.
Answer to Problem 22QP
The real return is as follows:
Year (A) | Treasury Bill Return (B) | Inflation (C) | Real return [1+(B)/1+(C)]-1 |
1973 | 0.0729 | 0.0871 | -0.0131 |
1974 | 0.0799 | 0.1234 | -0.0387 |
1975 | 0.0587 | 0.0694 | -0.0100 |
1976 | 0.0507 | 0.0486 | 0.0020 |
1977 | 0.0545 | 0.0670 | -0.0117 |
1978 | 0.0764 | 0.0902 | -0.0127 |
1979 | 0.1056 | 0.1329 | -0.0241 |
1980 | 0.1210 | 0.1252 | -0.0037 |
Total | -0.1120 |
The average real return is (1.4 percent).
Explanation of Solution
Given information:
Refer to Table 12.1 in the chapter. Extract the data for Treasury bills and consumer price index from 1973 to 1980 as follows:
Year | Treasury Bill Return | Consumer price index (Inflation) |
1973 | 0.0729 | 0.0871 |
1974 | 0.0799 | 0.1234 |
1975 | 0.0587 | 0.0694 |
1976 | 0.0507 | 0.0486 |
1977 | 0.0545 | 0.0670 |
1978 | 0.0764 | 0.0902 |
1979 | 0.1056 | 0.1329 |
1980 | 0.1210 | 0.1252 |
Total | 0.6197 | 0.7438 |
The formula to calculate the real rate using Fisher’s relationship:
Where,
“R” is the nominal rate of return
“r” is the real rate of return
“h” is the inflation rate
The formula to calculate the arithmetic average return:
Where,
“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:
The total of observations is (0.1120). There are 8 observations.
Hence, the arithmetic average of real return is (1.4 percent).
d)
To discuss: The risks of Treasury bills
Explanation of Solution
The investors believe that the Treasury bills are risk-free because there is zero default risk on these instruments. Moreover, the bills do not have higher interest rate risk because they maturity period is short. From the above calculations, it is clear that the Treasury bills face inflation risk. If the inflation rises, it will decrease the real return from the Treasury bill.
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