Chapter 12, Problem 16QP

Arithmetic and Geometric Returns [LO1] A stock has had the following year-end prices and dividends:

Year	Price	Dividend
1	$51.50	-
2	59.32	$.65
3	64.13	.70
4	57.86	.77
5	65.19	.86
6	74.86	.95

What are the arithmetic and geometric returns for the stock?

Summary Introduction

To determine: The arithmetic and geometric average return.

Introduction:

Total return refers to the total income from an investment. The total income includes the periodic incomes and the increase or decrease in the value of an asset.

Arithmetic average return refers to the returns that an investment earns in an average year over different periods.

Geometric average return refers to the return after compounding the average returns for multiple years.

Answer to Problem 16QP

The arithmetic average return is 9.52 percent. The geometric average return is 9.09 percent.

Explanation of Solution

Given information:

The price of the stock at the end of first year is $51.50. The price of the stock at the end of the second year is $59.32, and the dividend paid is $0.65. The price of the stock at the end of the third year is $64.13, and the dividend paid is $0.70. The price of the stock at the end of the fourth year is $57.86, and the dividend paid is $0.77.

The price of the stock at the end of the fifth year is $65.19, and the dividend paid is $0.86. The price of the stock at the end of the sixth year is $74.86, and the dividend paid is $0.95.

The formula to calculate the total percentage returns:

$\begin{array}{l}\text{Total returns}\\ \text{(In percentage)}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}$

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 geometric average return:

$\text{Geometric average return}={\left[\left(1+R_1\right)\times \left(1+R_2\right)\times \mathrm{\dots }\times \left(1+R_T\right)\right]}^{\frac{1}{T}}-1$

Where,

“R” is the annual returns for the investment

“T” is the years of returns

Compute the percentage return for the second year:

The closing price of the stock is $59.32, and the beginning price is $51.50. The dividend received in the second year is $0.65.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 2}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$0.65+\left(\$59.32-\$51.50\right)}{\$51.50}\\ =\frac{\$8.47}{\$51.50}\\ =0.1645\text{ or }16.45\%\end{array}$

Hence, the percentage return for Year 2 is 16.45 percent.

Compute the percentage return for the third year:

The closing price of the stock is $64.13, and the beginning price is $59.32. The dividend received in the second year is $0.70.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 3}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$0.70+\left(\$64.13-\$59.32\right)}{\$59.32}\\ =\frac{\$5.51}{\$59.32}\\ =0.0929\text{ or }9.29\%\end{array}$

Hence, the percentage return for Year 3 is 9.29 percent.

Compute the percentage return for the fourth year:

The closing price of the stock is $57.86, and the beginning price is $64.13. The dividend received in the second year is $0.77.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 4}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$0.77+\left(\left(57.86\right)-\$64.13\right)}{\$64.13}\\ =\frac{\left(\$5.51\right)}{\$64.13}\\ =\left(0.0857\right)\text{ or }\left(8.57\right)\%\end{array}$

Hence, the percentage return for Year 4 is (8.57 percent).

Compute the percentage return for the fifth year:

The closing price of the stock is $65.19, and the beginning price is $57.86. The dividend received in the second year is $0.86.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 5}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$0.86+\left(\$65.19-\$57.86\right)}{\$57.86}\\ =\frac{\$8.19}{\$57.86}\\ =0.1415\text{ or }14.15\%\end{array}$

Hence, the percentage return for Year 5 is 14.15 percent.

Compute the percentage return for the sixth year:

The closing price of the stock is $74.86, and the beginning price is $65.19. The dividend received in the second year is $0.95.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 6}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$0.95+\left(\$74.86-\$65.19\right)}{\$65.19}\\ =\frac{\$10.62}{\$65.19}\\ =0.1629\text{ or }16.29\%\end{array}$

Hence, the percentage return for Year 6 is 16.29 percent.

Compute the arithmetic average return:

The return for the second year is 16.45 percent, the return for the third year is 9.29 percent, the return for the fourth year is (8.57 percent), the return for the fifth year is 14.15 percent, and the return for the sixth year is 16.29 percent.

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{\left(0.1645+0.0929+\left(0.0857\right)+0.1415+0.1629\right)}{5}\\ =\frac{0.4761}{5}\\ =0.0952\text{ or }9.52\%\end{array}$

Hence, the arithmetic average return is 0.0952 or 9.52 percent.

Compute the geometric average return:

$\begin{array}{c}\text{Geometric average return}={\left[\left(1+R_1\right)\times \left(1+R_2\right)\times \mathrm{\dots }\times \left(1+R_T\right)\right]}^{\frac{1}{T}}-1\\ ={\left[\begin{array}{l}\left(1+0.1645\right)\times \left(1+0.0929\right)\times \left(1+\left(0.0857\right)\right)\times \\ \left(1+0.1415\right)\times \left(1+0.1629\right)\end{array}\right]}^{\frac{1}{5}}-1\\ ={\left[1.1645\times 1.0929\times 0.9143\times 1.1415\times 1.1629\right]}^{\frac{1}{5}}-1\\ ={1.5446}^{\frac{1}{5}}-1\end{array}$

$\begin{array}{l}=1.0909-1\\ =0.0909\text{ or 9}\text{.09\%}\end{array}$

Hence, the geometric average return is 9.09 percent.