Chapter 12, Problem 22QP

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?

Summary Introduction

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:

$\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}$

Where,

“X_i” refers to each of the observations from X₁ to X_N (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.

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{0.6197}{8}\\ =0.0774\text{ or }7.74\%\end{array}$

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.

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{0.7438}{8}\\ =0.0929\text{ or }9.29\%\end{array}$

Hence, the arithmetic average of inflation is 9.29 percent.

Summary Introduction

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:

$SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}$

Where,

“SD (R)” refers to the variance

“X̅” refers to the arithmetic average

“X_i” refers to each of the observations from X₁ to X_N (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 ${\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}$			0.00431069

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.00431069}{8-1}}\\ =0.0248\text{ or }2.48\%\end{array}$

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 ${\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}$			0.00680002

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.0068}{8-1}}\\ =0.0312\text{ or }3.12\%\end{array}$

Hence, the standard deviation of inflation is 3.12 percent.

Summary Introduction

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:

$1+R=\left(1+r\right)\times \left(1+h\right)$

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:

$\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}$

Where,

“X_i” refers to each of the observations from X₁ to X_N (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.

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{\left(0.1120\right)}{8}\\ =\left(0.014\right)\text{ or }\left(1.4\%\right)\end{array}$

Hence, the arithmetic average of real return is (1.4 percent).

Summary Introduction

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.