Chapter 10, Problem 8QP

Risk Premiums Refer to Table 10.1 in the text and look at the period from 1973 through 1978.

1. a. calculate the arithmetic average returns for large-company stocks and T-bills over this period.
2. b. calculate the standard deviation of the returns for large-company stocks and T-bills over this period.
3. c. calculate the observed risk premium in each year for the large-company stocks versus the T-bills. What was the arithmetic average risk premium over this period? What was the standard deviation of the risk premium over this period?

Summary Introduction

To determine: The average return, variance, and standard deviation

Introduction:

The average return is the average amount of money made by a particular investment. The variance determines the variances among the yearly returns of the stock. The standard deviation is simply the square root of calculated variance that measures the volatility of an investment.

Answer to Problem 8QP

Solutions:

The average return of large company stock is 3.24% and the average return of t-bill is 6.55%.

Explanation of Solution

Given information:

The returns of large company stock are -14.69%, -26.47%, 37.23%, 23.93%, -7.16%, and 6.57. The returns of T-bill return are 7.29%, 7.99%, 5.87%, 5.07%, 5.45%, and 7.64%. The returns of risk premium are -21.98%, -34.46%, 31.36%, 18.86%, -12.61%, and -1.07%.

The formula to calculate the average return:

$\text{Average}\text{return}\left(\text{R}\right)\text{=}\left[\frac{\begin{array}{l}\text{Return}\text{of}\text{year}\text{1+Return}\text{of}\text{year}\text{2+Return of}\\ \text{year}\text{3 +Return}\text{of}\text{year}\text{4+Return}\text{of}\text{year}\text{5}\end{array}}{\text{Total}\text{number}\text{of}\text{returns}\left(\text{T}\right)}\right]$

Compute the average return of large company:

$\begin{array}{c}\text{Average}\text{return}\left(\text{R}\right)\text{for}\text{large}\text{company=}\left[\frac{\begin{array}{l}\text{Return}\text{of}\text{year}\text{1+Return}\text{of}\text{year}\text{2+Return of}\\ \text{year}\text{3 +Return}\text{of}\text{year}\text{4+Return}\text{of}\text{year}\text{5+Return}\text{of}\text{year}6\end{array}}{\text{Total}\text{number}\text{of}\text{returns}\left(\text{T}\right)}\right]\\ =\left[\frac{\left(-14.69-26.47+37.23+23.93-7.16+6.57\right)}{6}\right]\\ =3.24\%\end{array}$ Hence, the average return of large company is 3.24%

Compute the average return of t-bills

$\begin{array}{c}\text{Average}\text{return}\left(\text{R}\right)\text{of}\text{t-bills=}\left[\frac{\begin{array}{l}\text{Return}\text{of}\text{year}\text{1+Return}\text{of}\text{year}\text{2+Return of}\\ \text{year}\text{3 +Return}\text{of}\text{year}\text{4+Return}\text{of}\text{year}\text{5+Return}\text{of}\text{year}6\end{array}}{\text{Total}\text{number}\text{of}\text{returns}\left(\text{T}\right)}\right]\\ =\left[\frac{\left(7.29\%+7.99\%+5.87\%+5.07\%+5.45\%+7.64\%\right)}{6}\right]\\ =6.55\%\end{array}$

Hence, the average return of t-bills is 6.55%.

Summary Introduction

To determine: The variance and standard deviation.

Answer to Problem 8QP

Solutions:

The variance for large company stock is 0.058136 and the variance for t-bills is 0.000153. The standard deviation for large company stock is 24.11% and the standard deviation for t-bills is 1.24%.

Explanation of Solution

Given information:

The returns of large company stock are -14.69%, -26.47%, 37.23%, 23.93%, -7.16%, and 6.57. The returns of T-bill return are 7.29%, 7.99%, 5.87%, 5.07%, 5.45%, and 7.64%. The returns of risk premium are -21.98%, -34.46%, 31.36%, 18.86%, -12.61%, and -1.07%.

The formula to calculate the variance of each stock:

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

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 of large company:

$\begin{array}{c}\text{Variance for large company =}\frac{1}{\text{T}-1}\left[\begin{array}{l}{\left({\text{R}}_1-\overline{\text{R}}\right)}^2+{\left({\text{R}}_2-\overline{\text{R}}\right)}^2+{\left({\text{R}}_3-\overline{\text{R}}\right)}^2\\ +{\left({\text{R}}_4-\overline{\text{R}}\right)}^2+{\left({\text{R}}_5-\overline{\text{R}}\right)}^2+{\left({\text{R}}_6-\overline{\text{R}}\right)}^2\end{array}\right]\\ \text{=}\frac{1}{6-1}\left[\begin{array}{l}{\left(0.1469-0.0324\right)}^2+{\left(0.2647-0.0324\right)}^2+{\left(0.3723-0.0324\right)}^2\\ +{\left(0.2393-0.0324\right)}^2+{\left(-0.0716-0.0324\right)}^2+{\left(0.0657-0.0324\right)}^2\end{array}\right]\\ \text{= 0}\text{.058136}\end{array}$

Hence, the variance of large company is 0.058136.

The formula to calculate the standard deviation:

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

Compute the standard deviation of large company:

$\begin{array}{c}\text{Standard deviation}\left({\text{σ}}_{\text{X}}\right)\text{ =}\sqrt{\text{Variance}\text{of large company}}\\ \text{=}\sqrt{\text{0}\text{.058136}}\\ \text{=0}\text{.2411}\text{or}\text{24}\text{.11\%}\end{array}$

Hence, the standard deviation of large company is 24.11%.

Compute the variance for t-bills:

$\begin{array}{c}\text{Variance of Stock Y=}\frac{1}{\text{T}-1}\left[\begin{array}{l}{\left({\text{R}}_1-\overline{\text{R}}\right)}^2+{\left({\text{R}}_2-\overline{\text{R}}\right)}^2+{\left({\text{R}}_3-\overline{\text{R}}\right)}^2\\ +{\left({\text{R}}_4-\overline{\text{R}}\right)}^2+{\left({\text{R}}_5-\overline{\text{R}}\right)}^2\end{array}\right]\\ \text{=}\frac{1}{6-1}\left[\begin{array}{l}{\left(0.729-0.0655\right)}^2+{\left(0.0799-0.0655\right)}^2+{\left(0.0587-0.0655\right)}^2\\ +{\left(0.0507-0.0655\right)}^2+{\left(0.0545-0.0655\right)}^2+{\left(0.0764-0.0655\right)}^2\end{array}\right]\\ \text{=0}\text{.000153}\end{array}$

Hence, the variance of t-bills is 0.000153.

Compute the standard deviation of t-bills:

$\begin{array}{c}\text{Standard deviation}\left({\text{σ}}_Y\right)\text{ =}\sqrt{\text{Variance}\text{of}\text{t-bills}}\\ \text{=}\sqrt{\text{0}\text{.000153}}\\ \text{=0}\text{.0124 or}\text{1}\text{.24\%}\end{array}$

Hence, the standard deviation of t-bills is 1.24%.

Summary Introduction

To determine: The average return, variance, and standard deviation

Introduction:

Risk premium is an extra or additional return of any risky assets, which is higher than the actual expected rate of return.

Answer to Problem 8QP

Solutions:

The average observed risk premium is -3.32%, variance is 0.062078, and the standard deviation is 24.92%.

Explanation of Solution

Given information:

The returns of large company stock are -14.69%, -26.47%, 37.23%, 23.93%, -7.16%, and 6.57. The returns of T-bill return are 7.29%, 7.99%, 5.87%, 5.07%, 5.45%, and 7.64%. The returns of risk premium are -21.98%, -34.46%, 31.36%, 18.86%, -12.61%, and -1.07%.

Compute the average observed risk premium:

$\begin{array}{c}\text{Average}\text{return}\left(\text{R}\right)\text{for}\text{large}\text{company=}\left[\frac{\begin{array}{l}\text{Return}\text{of}\text{year}\text{1+Return}\text{of}\text{year}\text{2+Return of}\\ \text{year}\text{3 +Return}\text{of}\text{year}\text{4+Return}\text{of}\text{year}\text{5+Return}\text{of}\text{year}6\end{array}}{\text{Total}\text{number}\text{of}\text{returns}\left(\text{T}\right)}\right]\\ =\left[\frac{\left(-21.98-34.46+31.36+18.86-12.61-1.07\right)}{6}\right]\\ =-3.32\%\end{array}$ Hence, the average return of large company is 3.24%

Compute the variance for t-bills:

$\begin{array}{c}\text{Variance of observed risk premium =}\frac{1}{\text{T}-1}\left[\begin{array}{l}{\left({\text{R}}_1-\overline{\text{R}}\right)}^2+{\left({\text{R}}_2-\overline{\text{R}}\right)}^2+{\left({\text{R}}_3-\overline{\text{R}}\right)}^2\\ +{\left({\text{R}}_4-\overline{\text{R}}\right)}^2+{\left({\text{R}}_5-\overline{\text{R}}\right)}^2\end{array}\right]\\ \text{=}\frac{1}{6-1}\left[\begin{array}{l}{\left(0.2198-0.0332\right)}^2+{\left(-0.3446-\left(-0.0332\right)\right)}^2+{\left(0.03136-0.0332\right)}^2\\ +{\left(0.1886-\left(-0.0332\right)\right)}^2+{\left(0.1261-0.0332\right)}^2+{\left(-0.0107-\left(-0.0332\right)\right)}^2\end{array}\right]\\ \text{= 0}\text{.062078}\end{array}$

Hence, the variance of the observed risk-premium is 0.062078.

Compute the standard deviation of observed risk premium:

$\begin{array}{c}\text{Standard deviation}\left({\text{σ}}_Y\right)\text{ =}\sqrt{\text{Variance}\text{of}\text{observed risk premium}}\\ \text{=}\sqrt{\text{0}\text{.06278}}\\ \text{=0}\text{.2492 or}\text{24}\text{.92\%}\end{array}$

Hence, the standard deviation of observed risk premium is 24.92%.