Chapter 3.1, Problem 20E

To determine

Identify the mutual fund that performs better.

Answer to Problem 20E

The Mutual Fund T performs better.

Explanation of Solution

Calculation:

The investigator has invested $10,000 at the beginning of 2004 in mutual fund S and $5,000 in Mutual Fund T. The value of each investment at the end of each subsequent year is given.

The geometric mean is used to find the mean growth rate.

The general formula to obtain growth factor for a year is given below:

$\text{Growth factor of year }=\frac{\text{Investment value at the end of year}}{\left(\begin{array}{l}\text{Investment value of the previour year }\left(\text{or}\right)\\ \text{Investment value at the beginning of year}\end{array}\right)}$

Denote the values of growth factors of Mutual Fund S for year 2004 to year 2011 as $x_1,x_2,\mathrm{..},x_8$.

The growth factor of Mutual Fund S for year 2004 is obtained as given below:

$\begin{array}{c}\text{Growth rate of year 2004},x_1=\frac{11,000}{10,000}\\ =1.100\end{array}$

Denote the values of growth factors of Mutual Fund T for year 2004 to year 2011 as $y_1,y_2,\mathrm{..},y_8$.

The growth factor of Mutual Fund T for year 2004 is obtained as given below:

$\begin{array}{c}\text{Growth rate of year 2004,}{y}_1=\frac{5,600}{5,000}\\ =1.120\end{array}$

Similarly, growth factors of Mutual Fund S and Mutual Fund T for the remaining years are obtained in the table given below:

Year	Mutual Fund S		Mutual Fund T
	End of Year Value	Growth Factor	End of Year Value	Growth Factor
2004	$11,000	1.100	$5,600	1.120
2005	$12,000	1.091	$6,300	1.125
2006	$13,000	1.083	$6,900	1.095
2007	$14,000	1.077	$7,600	1.101
2008	$15,000	1.071	$8,500	1.118
2009	$16,000	1.067	$9,200	1.082
2010	$17,000	1.063	$9,900	1.076
2011	$18,000	1.059	$10,600	1.071

The general formula to obtain geometric mean is given below:

${\overline{x}}_g=\sqrt[n]{\left(x_1\right)\left(x_2\right)\mathrm{...}\left(x_n\right)}$

Mutual Fund S:

The mean growth factor of Mutual Fund S over the year 2004 to year 2011 is obtained below:

$\begin{array}{c}{\overline{x}}_g=\sqrt[8]{\left[\left(x_1\right)\left(x_2\right)\mathrm{...}\left(x_8\right)\right]}\\ =\sqrt[8]{\left(1.1\right)\left(1.091\right)\left(1.083\right)\left(1.077\right)\left(1.071\right)\left(1.067\right)\left(1.063\right)\left(1.059\right)}\\ =\sqrt[8]{1.8007}\\ =\sqrt[8]{1.8}\\ =1.07624\end{array}$

Thus, the mean growth factor of Mutual Fund S over the years 2004 to 2011 is 1.07624.

The mean annual return of Mutual Fund S over the years 2004 to 2011 is obtained below:

$\begin{array}{c}\text{Mean annual return}=\left(\text{mean annual growth factor}-1\right)\times 100\%\\ \left(1.07624-1\right)100\%=7.624\%\end{array}$

Thus, the mean annual return of Mutual Fund S over the years 2004 to 2011 is 7.624%.

For the Mutual Fund T:

The mean growth factor of Mutual Fund T over the years 2004 to 2011 is obtained below:

$\begin{array}{c}{\overline{y}}_g=\sqrt[8]{\left[\left(y_1\right)\left(y_2\right)\mathrm{...}\left(y_8\right)\right]}\\ =\sqrt[8]{\left(1.12\right)\left(1.125\right)\left(1.095\right)\left(1.101\right)\left(1.118\right)\left(1.082\right)\left(1.076\right)\left(1.071\right)}\\ =\sqrt[8]{2.1176}\\ =\sqrt[8]{2.12}\\ =1.09848\end{array}$

Thus, the mean growth factor of Mutual Fund T over the years 2004 to 2011 is 1.09848.

The mean annual return of Mutual Fund T over the years 2004 to 2011 is obtained below:

$\begin{array}{c}\text{Mean annual return}=\left(\text{mean annual growth factor}-1\right)\times 100\%\\ \left(1.09848-1\right)100\%=9.848\%\end{array}$

Thus, the mean annual return of Mutual Fund T over the years 2004 to 2011 is 9.848%.

The mean annual return or mean annual growth rate of Mutual Fund T is 9.848% and mean annual return or mean annual growth rate of Mutual Fund S is 7.624%.

Since $9.848\%>7.624\%$, the mean annual return of Mutual Fund T is greater than the mean annual return of Mutual Fund S.

Hence, it can be concluded that Mutual Fund T performs better than Mutual Fund S.