Fundamentals of Corporate Finance
Fundamentals of Corporate Finance
11th Edition
ISBN: 9780077861704
Author: Stephen A. Ross Franco Modigliani Professor of Financial Economics Professor, Randolph W Westerfield Robert R. Dockson Deans Chair in Bus. Admin., Bradford D Jordan Professor
Publisher: McGraw-Hill Education
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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?

Expert Solution & Answer
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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:

Total returns(In percentage)}=Dividend income+(The closing priceThe beginning price)The beginning price of the investment

The formula to calculate the arithmetic average return:

Arithmetic average(X¯)=i=1NXiN

Where,

“Xi” refers to each of the observations from X1 to XN (as “i” goes from 1 to “N”)

“N” refers to the number of observations

The formula to calculate the geometric average return:

Geometric average return=[(1+R1)×(1+R2)××(1+RT)]1T1

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.

Total percentage returnfor Year 2}=Dividend income+(The closing priceThe beginning price)The beginning price of the investment=$0.65+($59.32$51.50)$51.50=$8.47$51.50=0.1645 or 16.45%

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.

Total percentage returnfor Year 3}=Dividend income+(The closing priceThe beginning price)The beginning price of the investment=$0.70+($64.13$59.32)$59.32=$5.51$59.32=0.0929 or 9.29%

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.

Total percentage returnfor Year 4}=Dividend income+(The closing priceThe beginning price)The beginning price of the investment=$0.77+((57.86)$64.13)$64.13=($5.51)$64.13=(0.0857) or (8.57)%

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.

Total percentage returnfor Year 5}=Dividend income+(The closing priceThe beginning price)The beginning price of the investment=$0.86+($65.19$57.86)$57.86=$8.19$57.86=0.1415 or 14.15%

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.

Total percentage returnfor Year 6}=Dividend income+(The closing priceThe beginning price)The beginning price of the investment=$0.95+($74.86$65.19)$65.19=$10.62$65.19=0.1629 or 16.29%

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.

Arithmetic average(X¯)=i=1NXiN=(0.1645+0.0929+(0.0857)+0.1415+0.1629)5=0.47615=0.0952 or 9.52%

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

Compute the geometric average return:

Geometric average return=[(1+R1)×(1+R2)××(1+RT)]1T1=[(1+0.1645)×(1+0.0929)×(1+(0.0857))×(1+0.1415)×(1+0.1629)]151=[1.1645×1.0929×0.9143×1.1415×1.1629]151=1.5446151

=1.09091=0.0909 or 9.09%

Hence, the geometric average return is 9.09 percent.

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Chapter 12 Solutions

Fundamentals of Corporate Finance

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