Question

Over the past four years, a stock produced returns of 13 percent, −9 percent, 8 percent, and 14 percent, respectively. Based on these four years, what range of returns would you expect to see 99 percent of the time?

Answer 1

Answer:

The range of returns that would be expected to be seen 99 percent of the time is (-25.5%, 38.5%).

Explanation:

Let the random variable X represent the percentage of returns produced by a stock.

The data for the return produced over the past four year is:

S = {13%, -9%, 8%, 14%}

Compute the average return as follows:

$\text{Average Return}=\bar X$

                         $=\frac{1}{4}\times [0.13-0.09+0.08+0.14]\\\\=0.065$

Compute the standard deviation of returns as follows:

$\text{Standard deviation of Returns}(s)=\sqrt{\frac{1}{n-1}\cdot\ \sum(X-\bar X)^{2}}$

$=\sqrt{\frac{1}{4-1}\cdot\ [(0.13-0.065)^{2}+(-0.09-0.065)^{2}\\+(0.08-0.065)^{2}+(0.14-0.065)^{2}]}\\\\=0.1066$

The 99% probability range is given by:

$\text{99 percent range}=\bar X\pm 3\cdot s$

Compute the range of returns that would be expected to be seen 99 percent of the time as follows:

$\text{99 percent range}=\bar X\pm 3\cdot s$

                          $=0.065\pm 3\times 0.1066\\\\=0.065\pm 0.3198\\\\=(-0.2548,0.3848)\\\\\approx (-0.255,0.385)$

Thus, the range of returns that would be expected to be seen 99 percent of the time is (-25.5%, 38.5%).