Chapter 10, Problem 7QP

Calculating Returns and Variability Using the following returns, calculate the average returns, the variances, and the Standard deviations for X and Y:

Returns

Year	X	Y
1	9%	12%
2	21	27
3	–27	–32
4	15	14
5	23	36

Summary Introduction

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

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.

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 7QP

Solution: The average return for stock X is 8.20% and the average return for stock Y is 11.40%. The variance for X is 0.04172 and the variance for Y is 0.06848. The standard deviation for X is 20.43% and the standard deviation for Y is 26.17%

Explanation of Solution

Given information:

The returns for X are 9%, 21%, -27%, 15%, and 23%. The returns for Y are 12%, 27%, -32%, 14%, and 36%.

The formula to calculate the average return:

$\text{Average return (R)}=\frac{\left(\begin{array}{l}\text{Return of year1+Return of year2}\\ \text{+Return of year3+Return of year4}\\ \text{+Return of year5}\end{array}\right)}{\text{Total number of returns(T)}}$

Compute the average return for X:

$\begin{array}{c}\text{Average return (R)}=\frac{\left(\begin{array}{l}\text{Return of year1+Return of year2}\\ \text{+Return of year3+Return of year4}\\ \text{+Return of year5}\end{array}\right)}{\text{Total number of returns(T)}}\\ =\frac{0.09+0.21-0.27+0.15+0.23}{5}\\ =0.0820\text{ or 8}\text{.20\%}\end{array}$

Hence, the average return of Stock X is 8.20%.

Compute the average return for Y:

$\begin{array}{c}\text{Average return (R)}=\frac{\left(\begin{array}{l}\text{Return of year1+Return of year2}\\ \text{+Return of year3+Return of year4}\\ \text{+Return of year5}\end{array}\right)}{\text{Total number of returns(T)}}\\ =\frac{0.12+0.27-0.32+0.14+0.36}{5}\\ =0.1140\text{ or 11}\text{.40\%}\end{array}$

Hence, the average return of Stock Y is 11.40%.

The formula to calculate the variance of each stock:

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

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 Stock X:

$\begin{array}{c}\text{Variance of stock}=\frac{1}{\text{T}-\text{1}}\left[\begin{array}{l}{\left({\text{R}}_1-\overline{\text{R}}\right)}^2+{\left({\text{R}}_2-\overline{\text{R}}\right)}^2+{\left({\text{R}}_3-\overline{\text{R}}\right)}^2\\ +{\left({\text{R}}_4-\overline{\text{R}}\right)}^2+{\left({\text{R}}_5-\overline{\text{R}}\right)}^2+\end{array}\right]\\ =\frac{1}{5-1}\left[\begin{array}{l}{\left(0.09-0.082\right)}^2+{\left(0.21-0.082\right)}^2+{\left(0.27-0.082\right)}^2\\ ++{\left(0.15-0.082\right)}^2+{\left(0.23-0.082\right)}^2\end{array}\right]\\ =0.04172\end{array}$

Hence, the variance for X is 0.04172.

Compute the variance of Stock Y:

$\begin{array}{c}\text{Variance of stock}=\frac{1}{\text{T}-\text{1}}\left[\begin{array}{l}{\left({\text{R}}_1-\overline{\text{R}}\right)}^2+{\left({\text{R}}_2-\overline{\text{R}}\right)}^2+{\left({\text{R}}_3-\overline{\text{R}}\right)}^2\\ +{\left({\text{R}}_4-\overline{\text{R}}\right)}^2+{\left({\text{R}}_5-\overline{\text{R}}\right)}^2+\end{array}\right]\\ =\frac{1}{5-1}\left[\begin{array}{l}{\left(0.12-0.114\right)}^2+{\left(0.27-0.114\right)}^2+{\left(0.32-0.114\right)}^2\\ ++{\left(0.14-0.114\right)}^2+{\left(0.36-0.114\right)}^2\end{array}\right]\\ =0.06848\end{array}$

Hence, the variance for Y is 0.06848.

The formula to calculate the standard deviation:

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

Compute the standard deviation of X:

$\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance of stock}}\\ =\sqrt{0.04172}\\ =0.2043\text{ or 20}\text{.43\%}\end{array}$

Hence, the standard deviation for X is 20.43%.

Compute the standard deviation of Y:

$\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance of stock}}\\ =\sqrt{0.06848}\\ =0.2617\text{ or 26}\text{.17\%}\end{array}$

Hence, the standard deviation for Y is 26.17%.