Chapter 4.1, Problem 11E

(a)

To determine

Estimate the year of return.

Explanation of Solution

Table-1 shows the price for a 6 year as follows:

Table-1
Year	Price
1	10
2	14
3	15
4	22
5	30
6	25

The year of return for 6 years can be calculated using Table-1:

Year 1 rate of return can be calculated as follows:

$\begin{array}{c}\text{Year 1 rate of return}=\frac{{\text{Price}}_1-\text{Old price}}{\text{Old price}}\\ =\frac{10-12}{12}\\ =\frac{-2}{12}\\ =-0.167\end{array}$

Thus, year 1 rate of return is -0.167.

Year 2 rate of return can be calculated as follows:

$\begin{array}{c}\text{Year 2 rate of return}=\frac{{\text{Price}}_2-{\text{Price}}_1}{{\text{Price}}_1}\\ =\frac{14-10}{10}\\ =\frac{4}{10}\\ =0.4\end{array}$

Thus, year 2 rate of return is 0.4

Year 3 rate of return can be calculated as follows:

$\begin{array}{c}\text{Year 3 rate of return}=\frac{{\text{Price}}_3-{\text{Price}}_2}{{\text{Price}}_2}\\ =\frac{15-14}{14}\\ =\frac{1}{14}\\ =0.071\end{array}$

Thus, year 3 rate of return is 0.071.

Year 4 rate of return can be calculated as follows:

$\begin{array}{c}\text{Year 4 rate of return}=\frac{{\text{Price}}_4-{\text{Price}}_3}{{\text{Price}}_3}\\ =\frac{22-15}{15}\\ =0.467\end{array}$

Thus, year 4 rate of return is 0.467.

Year 5 rate of return can be calculated as follows:

$\begin{array}{c}\text{Year 5 rate of return}=\frac{{\text{Price}}_5-{\text{Price}}_4}{{\text{Price}}_4}\\ =\frac{30-22}{22}\\ =0.364\end{array}$

Thus, year 5 rate of return is 0.364.

Year 6 rate of return can be calculated as follows:

$\begin{array}{c}\text{Year 6 rate of return}=\frac{{\text{Price}}_6-{\text{Price}}_5}{{\text{Price}}_5}\\ =\frac{25-30}{30}\\ =-0.167\end{array}$

Thus, year 6 rate of return is -0.167.

(b)

To determine

Determine the mean and median.

Explanation of Solution

Table–2 shows the investment over a 6-year period as follows:

Table-2
Year	Rate of return
1	-0.167
2	0.4
3	0.071
4	0.467
5	0.364
6	-0.167

The mean can be obtained by summing all observations and dividing the number of observations. The mean can be calculated as follows:

$\begin{array}{c}\text{Mean}=\frac{\sum {\text{x}}_{\text{i}}}{\text{n}}\\ =\frac{-0.167+0.4+0.071+0.467+0.364-0.167}{6}\\ =\frac{0.968}{6}\\ =0.161\end{array}$

Thus, the value of mean is 0.161.

Median:

Table-3 shows the sample in the ascending order as follows:

Table-3
Year	Rate of return
1	-0.167
2	-0.167
3	0.071
4	0.364
5	0.4
6	0.467

The median can be calculated by arranging all the observations in order of ascending or descending and the observation that falls in the middle is the median.

$\begin{array}{c}\text{Median}=\frac{0.071+0.364}{2}\\ =\frac{0.435}{2}\\ =0.2175\end{array}$

Thus, the value of median is 0.218.

(c)

To determine

Determine the geometric mean.

Explanation of Solution

The geometric mean can be calculated as follows:

$\begin{array}{c}{\text{R}}_{\text{g}}=\sqrt[6]{\left(1+{\text{R}}_1\right)\left(1+{\text{R}}_2\right)\left(1+{\text{R}}_3\right)\left(1+{\text{R}}_4\right)\left(1+{\text{R}}_5\right)\left(1+{\text{R}}_6\right)}-1\\ =\sqrt[4]{\left(1-0.167\right)\left(1+0.4\right)\left(1+0.071\right)\left(1+0.467\right)\left(1+0.364\right)\left(1-0.167\right)}-1\\ =1.130-1\\ =0.13\end{array}$

Thus, the value of geometric mean is 0.13.

(d)

To determine

Describe the return over a 4-year period.

Explanation of Solution

From the above calculation, the geometric mean is best over the 6-year period because $25\left(12{\left(1+0.1306\right)}^6\right)$