Chapter 10, Problem 21QP

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 21QP

The arithmetic average return is 17.68% and the geometric average return is 16.15%.

Explanation of Solution

Given information:

Refer Question and Problems 21 for price and dividend details.

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 the total of observations,

“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.

Formula:

The formula to calculate the geometric average return:

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

Where,

“R” is the annual returns for the investment,

“T” is the years of returns.

Compute the percentage return for year 1:

Note: The opening price of stock of year 1 is the closing price of stock of year 2 that is $107.11 and the opening stock of price is $99.15. The dividend received in the year 2 is $1.60.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 1}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$1.60+\left(\$107.11-\$99.15\right)}{\$99.15}\\ =\frac{\$1.60+\left(\$7.96\right)}{\$99.15}\\ =0.096\text{or}9.6\%\end{array}$

Hence, the percentage return for Year 1 is 9.6%.

Compute the percentage return for year 2:

 Note: The opening price of stock of year 2 is the closing price of stock of year 3 that is $91.65 and the opening stock of price is $107.11. The dividend received in year 3 is $2.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 2}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$2+\left(\$91.65-\$107.11\right)}{\$107.11}\\ =\frac{\$2-\$15.46}{\$107.11}\\ =-0.12567\text{or}-12.567\%\end{array}$

Hence, the percentage return for Year 2 is −12.567%

Compute the percentage return for year 3:

Note: The opening stock of year 4 is the closing price of stock of year 3 that is $127.16 and the opening stock of price is $91.65. The dividend received in year 4 is $2.20.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 3}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$2.20+\left(\$127.16-\$91.65\right)}{\$91.65}\\ =0.4115\text{or}41.15\%\end{array}$

Hence, the percentage return for Year 3 is 41.15%.

Compute the percentage return for Year 4:

Note: The opening stock of year 5 is the closing price of stock of year 4 that is $162.15 and the opening stock of price is $127.16. The dividend received in year 5 is $2.60.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 4}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$2.60+\left(\$162.15-\$127.16\right)}{\$127.16}\\ =\$0.2957\text{or 29}\text{.57\%}\end{array}$

Hence, the percentage return for Year 4 is 29.57%.

Compute the percentage return for Year 5:

Note: The opening stock of year 6 is the closing price of stock of year 5 that is $192.60 and the opening stock of price is $162.15. The dividend received in year 5 is $3.

$\begin{array}{c}\begin{array}{l}\text{Total percentage return}\\ \text{for Year 5}\end{array}\}=\frac{\text{Dividend income}+\left(\text{The closing price}-\text{The beginning price}\right)}{\text{The beginning price of the investment}}\\ =\frac{\$3+\left(\$192.60-\$162.15\right)}{\$162.15}\\ =0.2063\text{or 20}\text{.63\%}\end{array}$

Hence, the percentage return for Year 5 is 20.63%.

Compute the arithmetic average return:

$\begin{array}{c}\text{Arithmetic average}\left(\overline{X}\right)=\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}\\ =\frac{0.096-0.12567+0.4115+0.2957+0.2063}{5}\\ =\frac{0.88383}{5}\\ =0.1768\text{or }17.68\%\end{array}$

Hence, the arithmetic average return is 17.68%.

Compute the geometric average return:

$\begin{array}{c}\text{Geometric average return}={\left[\left(1+R_1\right)\times \left(1+R_2\right)\times \mathrm{...}\times \left(1+R_T\right)\right]}^{\frac{1}{T}}-1\\ ={\left[\begin{array}{l}\left(1+0.096\right)\times \left(1-0.12567\right)\times \left(1+0.4115\right)\times \\ \left(1+0.2957\right)\times \left(1+0.2063\right)\end{array}\right]}^{\frac{1}{5}}-1\\ ={\left[1.096\times 0.87433\times 1.4115\times 1.2957\times 1.2063\right]}^{\frac{1}{5}}-1\\ ={\left(2.1141\right)}^{\frac{1}{5}}-1\end{array}$

$\begin{array}{l}=1.1615-1\\ =0.1615\text{ or 16}\text{.15\%}\end{array}$

Hence, the geometric average return is 16.15%.