Question

21.

Page 345Arithmetic and Geometric Returns. A stock has had the following year-end prices and dividends:

Year	Price	Dividend
1	$58.27	—
2	67.32	$1.10
3	61.46	1.25
4	69.32	1.45
5	75.18	1.60
6	84.32	1.75

What are the arithmetic and geometric returns for the stock?

Answer 1

Arithmetic Mean is the average of the returns and is calculated by adding all the returns and dividing it by the number of periods.

Geometric mean is the average mean which has been compounded over the period. It is calculated as follows:

GM = [ (1 + R2) * (1 + R3) * (1 + R4) * (1 + R5) * (1 + R6) ] ^1/5 -1

Return for a period is both the capital gain and dividends

Year t = ( P_t - P_t-1 + D_t) / P_t-1

Year	Price	Dividend	( Pt - Pt-1 + Dt) / Pt-1
1	58.27
2	67.32	1.1	17.42%
3	61.46	1.25	-6.85%
4	69.32	1.45	15.15%
5	75.18	1.6	10.76%
6	84.32	1.75	14.49%
Sum			50.97%
Average = Sum / 5			10.19%

Arithmetic returns = 10.19%

Geometric returns
GM = [ (1 + R2) * (1 + R3) * (1 + R4) * (1 + R5) * (1 + R6) ] ^1/5 -1

        = [ ( 1+ 0.1742) * (1 + (0.0685)) * ( 1+ 0.1515) * (1+0.1076) * (1+ 0.1449) ]^1/5 - 1

       = (1.5971267 )^1/5 - 1

       = 1.0981657 - 1

       = 0.0981657

      = 9.82%

Geometric returns is 9.82%