Chapter 3.1, Problem 20E

Suppose that at the beginning of Year 1 you invested $10,000 in the Stivers mutual fund and $5000 in the Trippi mutual fund. The value of each investment at the end of each subsequent year is provided in the table below. Which mutual fund performed better?

To determine

Identify the mutual fund that performed better.

Answer to Problem 20E

The Trippi mutual fund performed better than Stivers.

Explanation of Solution

Calculation:

In the Stivers mutual fund, $10,000 is invested at the beginning of Year 1 and in Trippi mutual fund, $5,000 is invested at the beginning of Year 1. The given table represents the value of each investment at the end of each subsequent year.

The geometric mean is often used for finding the mean growth rate.

The mean growth factor over these eight periods is:

${\overline{x}}_g=\sqrt[8]{\left(x_1\right)\left(x_2\right)\left(x_3\right)\left(x_4\right)\left(x_5\right)\left(x_6\right)\left(x_7\right)\left(x_8\right)}$

Where, $x_i$’s are the growth rate for the i^th year.

The growing factor of each year can be obtained by dividing the value of end of year by the previous year value.

The growth factor of Stivers for the year 1 can be obtained as follows:

$\begin{array}{c}\text{Growth rate of }{x}_1=\frac{11,000}{10,000}\\ =1.100\end{array}$

The growth factor of Trippi for the year 1 can be obtained as follows:

$\begin{array}{c}\text{Growth rate of }{x}_1=\frac{5,600}{5,000}\\ =1.12\end{array}$

Similarly the growth actor for remaining years can be obtained as shown in the table given below:

Year	Stivers		Trippi
	End of Year Value	Growth Factor	End of Year Value	Growth Factor
1	$11,000	1.100	$5,600	1.120
2	$12,000	1.091	$6,300	1.125
3	$13,000	1.083	$6,900	1.095
4	$14,000	1.077	$7,600	1.101
5	$15,000	1.071	$8,500	1.118
6	$16,000	1.067	$9,200	1.082
7	$17,000	1.063	$9,900	1.076
8	$18,000	1.059	$10,600	1.071

For the Stivers mutual fund:

The balance at the end of the year 8 is the initial investment times the product of all 8 growth factors.

$18,000=10,000\left[\left(x_1\right)\left(x_2\right)\mathrm{...}\left(x_8\right)\right]$

That is,

$\begin{array}{c}\left[\left(x_1\right)\left(x_2\right)\mathrm{...}\left(x_8\right)\right]=\frac{18,000}{10,000}\\ =1.8\end{array}$

Substitute $\left[\left(x_1\right)\left(x_2\right)\mathrm{...}\left(x_8\right)\right]=1.8$ in the formula of geometric mean.

$\begin{array}{c}{\overline{x}}_g=\sqrt[8]{1.8}\\ =1.07624\end{array}$

Thus, the mean annual return for the Stivers mutual fund is $\left(1.07624-1\right)100\%=\underset{\_}{7.624\%}$

For the Trippi mutual fund:

The balance at the end of the year 8 is the initial investment times the product of all 8 growth factors.

$10,600=5,000\left[\left(x_1\right)\left(x_2\right)\mathrm{...}\left(x_8\right)\right]$

That is,

$\begin{array}{c}\left[\left(x_1\right)\left(x_2\right)\mathrm{...}\left(x_8\right)\right]=\frac{10,600}{5,000}\\ =2.12\end{array}$

Substitute $\left[\left(x_1\right)\left(x_2\right)\mathrm{...}\left(x_8\right)\right]=2.12$ in the formula of geometric mean.

$\begin{array}{c}{\overline{x}}_g=\sqrt[8]{2.12}\\ =1.09848\end{array}$

Thus, the mean annual return for the Trippi mutual fund is $\left(1.09848-1\right)100\%=\underset{\_}{9.848\%}$

The annual return earned by the Trippi mutual fund is 9.8% and by the Stivers mutual fund is 7.6%. Hence, the annual return earned by the Tripp mutual fund is more than that of Stivers mutual fund.

Thus, it can be concluded that the Trippi mutual fund performed better than Stivers.