Chapter 10, Problem 22QP

Calculating Returns Refer to Table 10.1 in the text and look at the period from 1973 through 1980.

1. a. calculate the average return for Treasury bills and the average annual inflation rate (consumer price index) for this period.
2. b. calculate the standard deviation of Treasury bill returns and inflation over this period.
3. c. calculate the real return for each year. What is the average real return for Treasury bills?
4. d. Many people consider Treasury bills to be risk-free. What do these calculations tell you about the potential risks of Treasury bills?

Summary Introduction

To determine: The returns

Introduction:

The term return refers to a profit or gain made on an investment that is usually expressed in terms of percentage or dollars. The percentage total return shows the overall performance and efficiency of the amount invested.

Answer to Problem 22QP

Solution: The average return for t-bills over the period of time is 7.75%.

Explanation of Solution

Given information:

The returns of t-bills are 0.0729, 0.0799, 0.0587, 0.0507, 0.0545, 0.764, 0.1056, and 0.1210. The inflations are 0.0871, 0.1234, 0.0694, 0.0486, 0.0670, 0.0902, 0.1329, and 0.1252.

The formula to calculate the real return by using fisher equation:

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

Compute the real return of 1973:

$\begin{array}{c}\left(\text{1+R}\right)\text{=}\left(\text{1+r}\right)\left(\text{1+h}\right)\\ 1+0.0729=\left(\text{1+r}\right)\left(\text{1}+0.0871\right)\\ \left(\text{1+r}\right)=\frac{1.0729}{1.0871}-1\\ =0.0131\end{array}$

Hence, the real return for the year 1973 is 0.0131.

Compute the real return of 1974:

$\begin{array}{c}\left(\text{1+R}\right)\text{=}\left(\text{1+r}\right)\left(\text{1+h}\right)\\ 1+0.0799=\left(\text{1+r}\right)\left(\text{1}+1.1234\right)\\ \left(\text{1+r}\right)=\frac{1.0799}{2.1234}-1\\ \text{r}=-0.0387\end{array}$

Hence, the real return for the year 1974 is -0.0387.

Compute the real return of 1975:

$\begin{array}{c}\left(\text{1+R}\right)\text{=}\left(\text{1+r}\right)\left(\text{1+h}\right)\\ 1+0.0587=\left(\text{1+r}\right)\left(\text{1}+0.0694\right)\\ \left(\text{1+r}\right)=\frac{1.0587}{1.0694}-1\\ \text{r}=-0.0100\end{array}$

Hence, the real return for the year 1975 is -0.0100.

Compute the real return of 1976:

$\begin{array}{c}\left(\text{1+R}\right)\text{=}\left(\text{1+r}\right)\left(\text{1+h}\right)\\ 1+0.0507=\left(\text{1+r}\right)\left(\text{1+0}\text{.0694}\right)\\ \left(\text{1+r}\right)=\frac{1.0507}{1.0694}-1\\ \text{r}=0.0020\end{array}$

Hence, the real return for the year 1976 is 0.0020.

Compute the real return of 1977:

$\begin{array}{c}\left(\text{1+R}\right)\text{=}\left(\text{1+r}\right)\left(\text{1+h}\right)\\ 1+0.0545=\left(\text{1+r}\right)\left(1+0.0694\right)\\ \left(\text{1+r}\right)=\frac{1.0545}{1.0694}-1\\ \text{r}=-0.0117\end{array}$

Hence, the real return for the year 1977 is -0.0117.

Compute the real return for the year 1978:

$\begin{array}{c}\left(\text{1+R}\right)\text{=}\left(\text{1+r}\right)\left(\text{1+h}\right)\\ 1+0.0764=\left(\text{1+r}\right)\left(1+0.0902\right)\\ \left(\text{1+r}\right)=\frac{1.0764}{1.0902}-1\\ \text{r}=-0.0127\end{array}$

Hence, the real return for the year 1978 is -0.0127.

Compute the real return for the year 1979:

$\begin{array}{c}\left(\text{1+R}\right)\text{=}\left(\text{1+r}\right)\left(\text{1+h}\right)\\ 1+0.1056=\left(\text{1+r}\right)\left(1+0.1329\right)\\ \left(\text{1+r}\right)=\frac{1.1056}{1.1329}-1\\ \text{r}=-0.0241\end{array}$

Hence, the real return for the year 1979 is 0.0241.

Compute the real return for the year 1980:

$\begin{array}{c}\left(\text{1+R}\right)\text{=}\left(\text{1+r}\right)\left(\text{1+h}\right)\\ 1+0.1210=\left(\text{1+r}\right)\left(1+0.1252\right)\\ \left(\text{1+r}\right)=\frac{1.1210}{1.1252}-1\\ \text{r}=-0.0037\end{array}$

Hence, the real return for the year 1980 is -0.0037.

Summary Introduction

To determine: The returns

Introduction:

The term return refers to a profit or gain made on an investment that is usually expressed in terms of percentage or dollars. The percentage total return shows the overall performance and efficiency of the amount invested.

Answer to Problem 22QP

The variance for t-bills is 0.000616 and the standard deviation is 2.48%. The

variance for inflation is 0.000971 and the standard deviation is 3.12%.

Explanation of Solution

The formula to calculate the arithmetic average return:

$\text{Average}\text{return}\left(\text{R}\right)\text{=}\left[\frac{{\text{R}}_1+{\text{R}}_2+{\text{R}}_3+{\text{R}}_4+{\text{R}}_5}{\text{T}}\right]$

Where,

R refers to average return,

${\text{R}}_1$ refers to year 1 return,

${\text{R}}_2$ refers to year 2 return,

${\text{R}}_3$ refers to year 3 return,

${\text{R}}_4$ refers to year 4 return,

${\text{R}}_5$ refers to year 5 return,

T refers to number of returns

Compute the arithmetic average return:

$\begin{array}{c}\text{Arithmetic average}\text{return}\left(\text{R}\right)\text{=}\left[\frac{{\text{R}}_1+{\text{R}}_2+{\text{R}}_3+{\text{R}}_4+{\text{R}}_5+{\text{R}}_6+{\text{R}}_7+{\text{R}}_8}{\text{T}}\right]\\ =\frac{\left(-0.013\right)-0.0387-0.0100+0.0020-0.0117-0.0127-0.0241-0.0037}{8}\\ =\frac{0.6197}{8}\\ =0.0775\text{or}7.75\%\end{array}$ Hence, the arithmetic average return is 7.75%.

$\begin{array}{c}\text{Average inflation}\text{=}\left[\frac{{\text{h}}_1+{\text{h}}_2+{\text{h}}_3+{\text{h}}_4+{\text{h}}_5+{\text{h}}_6+{\text{h}}_7+{\text{h}}_8}{\text{T}}\right]\\ =\left[\frac{\begin{array}{l}0.871+0.1234+0.0694+0.0486+\\ 0.0670+0.902+0.1329+0.1252\end{array}}{8}\right]\\ =\frac{0.7438}{8}\\ =0.0930\text{or}9.30\%\end{array}$

The formula to calculate the variance for t-bills:

$\text{Variance of Stock=}\frac{1}{\text{T}-1}\left[\begin{array}{l}{\left({\text{R}}_1-\overline{\text{R}}\right)}^2+{\left({\text{R}}_2-\overline{\text{R}}\right)}^2+{\left({\text{R}}_3-\overline{\text{R}}\right)}^2\\ +{\left({\text{R}}_4-\overline{\text{R}}\right)}^2+{\left({\text{R}}_5-\overline{\text{R}}\right)}^2\end{array}\right]$

Where,

“T” refers to the total number of returns,

“R” refers to the return of each year,

“ $\overline{\text{R}}$” refers to the average return.

Compute the variance for t-bills:

$\begin{array}{c}\text{Variance for t-billss =}\frac{1}{\text{T}-1}\left[\begin{array}{l}{\left({\text{R}}_1-\overline{\text{R}}\right)}^2+{\left({\text{R}}_2-\overline{\text{R}}\right)}^2+{\left({\text{R}}_3-\overline{\text{R}}\right)}^2\\ +{\left({\text{R}}_4-\overline{\text{R}}\right)}^2+{\left({\text{R}}_5-\overline{\text{R}}\right)}^2+{\left({\text{R}}_6-\overline{\text{R}}\right)}^2\\ +{\left({\text{R}}_7-\overline{\text{R}}\right)}^2+{\left({\text{R}}_8-\overline{\text{R}}\right)}^2\end{array}\right]\\ \text{=}\frac{1}{8-1}\left[\begin{array}{l}{\left(0.0729-0.0775\right)}^2+{\left(0.0799-0.0775\right)}^2+{\left(0.0587-0.0775\right)}^2\\ +{\left(0.0507-0.0775\right)}^2+{\left(0.0545-0.0775\right)}^2+{\left(0.0764-0.0775\right)}^2\\ +{\left(0.1056-0.0775\right)}^2+{\left(0.1210-0.0775\right)}^2\end{array}\right]\\ \text{= 0}\text{.000616}\end{array}$

Hence, the variance  for t-bills is 0.000616.

The formula to calculate the standard deviation:

$\text{Standard deviation =}\sqrt{\text{Variance}\text{of}\text{stock}}$

Compute the standard deviation for t-bills:

$\begin{array}{c}\text{Standard deviation}\left({\text{σ}}_{\text{X}}\right)\text{ =}\sqrt{\text{Variance}\text{of t-bills}}\\ \text{=}\sqrt{\text{0}\text{.000616}}\\ \text{=0}\text{.248}\text{or}\text{2}\text{.48\%}\end{array}$

Hence, the standard deviation for t-bills is 2.48%.

Compute the variance for inflation:

$\begin{array}{c}\text{Variance for inflation =}\frac{1}{\text{T}-1}\left[\begin{array}{l}{\left({\text{h}}_1-\overline{\text{h}}\right)}^2+{\left({\text{h}}_2-\overline{\text{h}}\right)}^2+{\left({\text{h}}_3-\overline{\text{h}}\right)}^2\\ +{\left({\text{h}}_4-\overline{\text{h}}\right)}^2+{\left({\text{h}}_5-\overline{\text{h}}\right)}^2+{\left({\text{h}}_5-\overline{\text{h}}\right)}^2\backslash \\ +{\left({\text{h}}_5-\overline{\text{h}}\right)}^2+{\left({\text{h}}_5-\overline{\text{h}}\right)}^2\end{array}\right]\\ \text{=}\frac{1}{6-1}\left[\begin{array}{l}{\left(0.0871-0.0930\right)}^2+{\left(0.1234-0.0930\right)}^2+{\left(0.0694-0.0930\right)}^2\\ +{\left(0.0486-0.0930\right)}^2+{\left(0.0670-0.0930\right)}^2+{\left(0.0902-0.0930\right)}^2\\ +{\left(0.1329-0.0930\right)}^2+{\left(0.1252-0.0930\right)}^2\end{array}\right]\\ \text{= 0}\text{.000971}\end{array}$

Hence, the variance for inflation over the period is 0.000971.

Compute the standard deviation for inflation:

$\begin{array}{c}\text{Standard deviation}\left(\text{σ}\right)\text{ =}\sqrt{\text{Variance}\text{of}\text{inflation}}\\ \text{=}\sqrt{\text{0}\text{.000971}}\\ \text{=0}\text{.0312 or}\text{3}\text{.12\%}\end{array}$

Hence, the standard deviation for inflation over the period is 3.12%.

Summary Introduction

To determine: The returns

Introduction:

The term return refers to a profit or gain made on an investment that is usually expressed in terms of percentage or dollars. The percentage total return shows the overall performance and efficiency of the amount invested.

Answer to Problem 22QP

The average observed risk premium is -1.40%.

Explanation of Solution

Compute the average observed risk premium:

$\begin{array}{c}\text{Average observed real return}\text{=}\left[\frac{{\text{R}}_1+{\text{R}}_2+{\text{R}}_3+{\text{R}}_4+{\text{R}}_5+{\text{R}}_6+{\text{R}}_7+{\text{R}}_8}{\text{T}}\right]\\ =\left[\frac{\begin{array}{l}-0.0131-0.0387-0.0100+0.0020\\ -0.0117-0.0127-0.0241-0.0037\end{array}}{8}\right]\\ =\frac{-0.1122}{8}\\ =-0.0140\text{or}-1.40\%\end{array}$

Summary Introduction

To determine: The returns

Introduction:

The term return refers to a profit or gain made on an investment that is usually expressed in terms of percentage or dollars. The percentage total return shows the overall performance and efficiency of the amount invested.

Answer to Problem 22QP

Treasury bills have no risk.

Explanation of Solution

Since, there is only an enormous small possibility of the government defaulting, treasury bills have no risk or modest default risk. The calculation shows that there is a risk of inflation. The actual purchasing power of the investment has declined over time even though the return is positive.