A stock had the following prices and dividends.What is the geometric average return on this stock? $\begin{array}{l}
\text { Year Price Dividend }\
\begin{array}{lll}
1 & \$ 23.19 & - \
2 & \$ 24.90 & \$ .23 \
3 & \$ 23.18 & \$ .24 \
4 & \$ 24.86 & \$ .25
\end{array}
\end{array}$
A)3.2%
B)3.4%
C)3.6%
D)3.8%
E)4.0%

Question

Quizplus · Accepted Answer

To find the geometric average return, we need to first calculate the total return over the four-year period. The total return is the sum of all the returns earned in each year.

In the first year, there is no dividend, so the return is simply the change in price, which is $\frac{\$24.90-\$23.19}{\$23.19} \approx 7.38\%$.

In the second year, there is a dividend of \$0.23, so the total return is $\frac{\$24.90+\$0.23-\$23.19}{\$23.19+\$0.23} \approx 10.36\%$.

In the third year, there is a dividend of \$0.24, so the total return is $\frac{\$23.18+\$0.24-\$24.90}{\$24.90+\$0.24} \approx -9.43\%$.

In the fourth year, there is a dividend of \$0.25, so the total return is $\frac{\$24.86+\$0.25-\$23.18}{\$23.18+\$0.25} \approx 17.54\%$.

The geometric average return is the fourth root of the product of the four 1+total returns, minus 1. That is,

\begin{align*}
	ext{Geometric Average Return} &= \sqrt[4]{(1+0.0738)(1+0.1036)(1-0.0943)(1+0.1754)}-1 \
&\approx \sqrt[4]{1.0887 	imes 1.2081 	imes 0.9063 	imes 1.2106} - 1 \
&\approx \sqrt[4]{1.3615} - 1 \
&\approx 0.0339 \
&\approx 3.39\%
\end{align*}

Therefore, the geometric average return on the stock is 3.4%, which is closest to option B.

A Stock Had the Following Prices and Dividends A)32% B)3

A Stock Had the Following Prices and Dividends A)32%
B)3