Chapter 12, Problem 7QP

Summary Introduction

To discuss: The arithmetic average, standard deviation, and variance of Security X and Security Y

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 the observations from the mean.

Answer to Problem 7QP

Security X has an arithmetic average of 8.40 percent, standard deviation of 14.22 percent, and variance of 0.02023. Security Y has an arithmetic average of 11.40 percent, standard deviation of 25.60 percent, and variance of 0.06553.

Explanation of Solution

Given information:

The returns of Security X for Year 1, Year 2, Year 3, Year 4, and Year 5 are 17 percent, 22 percent, 8 percent, (15percent), and 10 percent respectively. The returns of Security Y for Year 1, Year 2, Year 3, Year 4, and Year 5 are 23 percent, 34 percent, 11 percent, (32 percent), and 21 percent respectively.

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

The formula to calculate the variance:

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

Where,

“Var (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

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

Security X:

Compute the arithmetic average return:

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{\left(0.17+0.22+0.08+\left(0.15\right)+0.10\right)}{5}\\ =\frac{0.42}{5}\\ =0.084\end{array}$

Hence, the arithmetic average of Security X is 0.084 or 8.40 percent.

Compute the sum of squared deviations of Security X:

Security X
Year (A)	Actual return (B)	Average return (C)	Deviation (B)–(C)=(D)	Squared deviation (D)2
1	0.17	0.084	0.086	0.007396
2	0.22	0.084	0.136	0.018496
3	0.08	0.084	-0.004	0.000016
4	-0.15	0.084	-0.234	0.054756
5	0.10	0.084	0.016	0.000256
Total of squared deviation ${\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}$				0.08092

Compute the variance:

$\begin{array}{c}\text{Var}(R)={\sigma }^2=\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}\\ =\frac{0.08092}{5-1}\\ =\frac{0.08092}{4}\\ =0.02023\end{array}$

Hence, the variance of Security X is 0.02023.

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.08092}{5-1}}\\ =\sqrt{0.02023}\\ =0.1422\text{ or }14.22\%\end{array}$

Hence, the standard deviation of Security X is 14.22 percent.

Security Y:

Compute the arithmetic average return:

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{\left(0.23+0.34+0.11+\left(0.32\right)+0.21\right)}{5}\\ =\frac{0.57}{5}\\ =0.114\end{array}$

Hence, the arithmetic average of Security Y is 0.114 or 11.40 percent.

Compute the sum of squared deviations of Security X:

Stock Y
Year (A)	Actual return (B)	Average return (C)	Deviation (B)–(C)=(D)	Squared deviation (D)2
1	0.23	0.114	0.116	0.013456
2	0.34	0.114	0.226	0.051076
3	0.11	0.114	-0.004	0.000016
4	-0.32	0.114	-0.434	0.188356
5	0.21	0.114	0.096	0.009216
Total of squared deviation ${\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}$				0.26212

Compute the variance:

$\begin{array}{c}\text{Var}(R)={\sigma }^2=\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}\\ =\frac{0.26212}{5-1}\\ =\frac{0.26212}{4}\\ =0.06553\end{array}$

Hence, the variance of Security Y is 0.06553.

Compute the standard deviation:

$\begin{array}{c}SD(R)=\sigma =\sqrt{\frac{{\displaystyle \sum _{i=1}^N{(X_i-\overline{X})}^2}}{N-1}}\\ =\sqrt{\frac{0.26212}{5-1}}\\ =\sqrt{0.06553}\\ =0.2560\text{ or 25}\text{.60}\%\end{array}$

Hence, the standard deviation of Security Y is 25.60 percent.