Chapter 10, Problem 22QP

a)

Summary Introduction

To determine: The Treasury bills average return and the average inflation rate for the period from 1973 to 1980.

Introduction:

The rate at which the inflation increases is the inflation rate. The Fisher effect helps to establish a relationship between the nominal rate of return, inflation, and the real rate of return.

Answer to Problem 22QP

The average return for Treasury bills is 7.75% and the average inflation rate for the period from 1973 to 1980 is 9.30%.

Explanation of Solution

Given information:

The T bill return for the years 1973 is 0.729, 1974 is 0.799, 1975 is 0.0587, 1976 is 0.507, 1977 is 0.0545, 1978 is 0.0764, 1979 is 0.1056, 1980 is 0.1210.

The inflation rates as follows:

For the year 1973 is 0.087, 1974 is 0.1234, 1975 is 0.0694, 1976 is 0.0486, 1977 is 0.0670, 1978 is 0.0902, 1979 is 0.1329, 1980 is 0.1252.

The formula to compute the average return of the Treasury bills:

$\text{Average return of treasury bills}=\frac{\text{Total treasury bill return}}{\text{Total number of years}}$

Compute the average real return of the Treasury bills:

$\begin{array}{c}\text{Average real return of treasury bills}=\frac{\text{Total T}\text{.bill return}}{\text{Total number of years}}\\ =\frac{\left(\begin{array}{l}0.0729+0.0799+0.0587+0.0507\\ +0.545+0.0764+0.1056+0.1210\end{array}\right)}{8}\\ =\frac{0.6197}{8}\\ =0.0775\text{or 7}\text{.75}\end{array}$

Hence, the average return of Treasury bill is7.75%.

Determine the average inflation rate:

The formula to calculate the averageinflation rate of the treasury bills:

$\begin{array}{c}\text{Average inflation rate}=\frac{\text{Total inflation}}{\text{Total number of years}}\\ =\frac{\left(\begin{array}{l}0.087+0.1234+0.0694+0.0486\\ +0.0670+0.0902+0.1329+0.1252\end{array}\right)}{8}\\ =\frac{0.7437}{8}\\ =0.093\text{or}\text{9}\text{.30\%}\end{array}$

Hence, the average inflation rate is 9.30%.

b)

Summary Introduction

To determine: The standard deviation of Treasury bill returns and inflation over this time period.

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 observations from the mean.

Answer to Problem 22QP

The standard deviation of Treasury bill returns is 2.48% and inflation over this time period is 3.12%

Explanation of Solution

Given information:

The T bill return for the years 1973 is 0.729, 1974 is 0.799, 1975 is 0.0587, 1976 is 0.507, 1977 is 0.0545, 1978 is 0.0764, 1979 is 0.1056, 1980 is 0.1210.

The inflation rates as follows:

For the year 1973 is 0.087, 1974 is 0.1234, 1975 is 0.0694, 1976 is 0.0486, 1977 is 0.0670, 1978 is 0.0902, 1979 is 0.1329, 1980 is 0.1252.

The formula to calculate the standard deviation of Treasury bill returns:

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

 “$SD(R)$” refers to the variance,

“$\overline{X}$” refers to the arithmetic average,

“$Xi$ ” 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 sum of squared deviations of T-bill:

T-bill
Year (A)	Actual return (B)	Average return (C)	Deviation (B)–(C)=(D)	Squared deviation(D)2
1	0.0729	0.0775	-0.7021	0.49294441
2	0.0799	0.0775	-0.6951	0.48316401
3	0.0587	0.0775	-0.7163	0.51308569
4	0.0507	0.0775	-0.7243	0.52461049
5	0.0545	0.0775	-0.7205	0.51912025
6	0.0764	0.0775	-0.6986	0.48804196
7	0.1056	0.0775	-0.6694	0.44809636
8	0.121	0.0775	-0.654	0.427716
Total of squared deviation $\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}$				0.004310

Hence, the squared deviation of the T-bill is 0.004310.

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.00431067}{8-1}}\\ =\sqrt{0.000616}\\ =0.0248\text{ or 2}\text{.48}\%\end{array}$

Hence, the standard deviation of the T-bill is 0.0248 or 2.48%.

Compute the standard deviation of inflation:

The formula to calculate the standard deviation of inflation:

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

 “$SD(R)$” refers to the variance,

“$\overline{X}$” refers to the arithmetic average,

“$Xi$ ” 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 sum of squared deviations of inflation:

Stock
Year (A)	Actual return (B)	Average return (C)	Deviation (B)–(C)=(D)	Squared deviation(D)2
1	0.0871	0.093	-0.0059	0.00003481
2	0.1234	0.093	0.0304	0.00092416
3	0.0694	0.093	-0.0236	0.00055696
4	0.0486	0.093	-0.0444	0.00197136
5	0.067	0.093	-0.026	0.000676
6	0.0902	0.093	-0.0028	0.0000078
7	0.1329	0.093	0.0399	0.00159201
8	0.1252	0.093	0.0322	0.00103684
Total of squared deviation ${\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}$				0.006799

Hence, the squared deviation is 0.006799.

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.006799}{8-1}}\\ =\sqrt{0.0009714}\\ =0.031\text{or 3}\text{.11\%}\end{array}$

Hence, the standard deviation is 0.031 or 3.11%.

c)

Summary Introduction

To determine: The real return for every year.

Introduction:

The real rate of return refers to the rate of return on an investment after adjusting the inflation rate.

Answer to Problem 22QP

The average real return for Treasury bill is −1.40%.

Explanation of Solution

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 risk-free rate of return,

“r” is the real rate of return,

“h” is the inflation rate.

The formula to calculate the average real return of the treasury bills:

$\text{Average real return of treasury bills}=\frac{\text{Total real return}}{\text{Total number of years}}$

Compute the real return for each year:

For 1973:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.0729}{1+0.0871}-1\\ =-0.0131\end{array}$

For 1974:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.0799}{1+0.1234}-1\\ =-0.0387\end{array}$

For 1975:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.0587}{1+0.0694}-1\\ =-0.0100\end{array}$

For 1976:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.0507}{1+0.0486}-1\\ =0.00200\end{array}$

For 1977:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.0545}{1+0.0670}-1\\ =0.0117\end{array}$

For 1978:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.0764}{1-0.0902}-1\\ =-0.0127\end{array}$

For 1979:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.1056}{1-0.1329}-1\\ =-0.0241\end{array}$

For 1980:

$\begin{array}{c}1+R=\left(1+r\right)\times \left(1+h\right)\\ r=\frac{1+R}{1+h}-1\\ =\frac{1+0.1210}{1-0.1252}-1\\ =-0.0037\end{array}$

Compute the average real return:

$\begin{array}{c}\text{Average real return}=\frac{\text{Total real return}}{\text{Total number of years}}\\ =\frac{\left(\begin{array}{l}-0.0131-0.0387-0.0100+0.00200+0.0117\\ -0.0127-0.0241-0.0037\end{array}\right)}{8}\\ =\frac{-0.112}{8}\\ =-0.014\text{or }-\text{1}\text{.40\%}\end{array}$

Hence, the average real return is -1.40%.

d)

Summary Introduction

To discuss: The potential risks of Treasury bills.

Explanation of Solution

T-bills are usually risk free items as they are normally short-term. They also held with limited interest rate risk. If the investors are earning positive nominal return, which will decline the inflation risk ultimately.