Chapter 3, Problem 3.55RE

(a)

To determine

To Explain: the range do the middle 95% of all yearly returns lie.

Explanation of Solution

Given:

  $\begin{array}{l}\mu =12\\ \sigma =16.5\end{array}$

Calculation:

Knowing that 95% of the data lie within two standard deviations of the mean

  $\begin{array}{c}\mu -2\sigma =12-2\times 16.5\\ =-21\\ \mu +2\sigma =12+2\times 16.5\\ =45\end{array}$

Therefore the middle 95% lie between -21% and 45%

Graph:

(b)

To determine

To Calculate: the proportion of years is the market down.

Answer to Problem 3.55RE

23.27%

Explanation of Solution

Given:

  $\begin{array}{c}x=0\\ \text{μ=12\%}\\ \text{σ=}\text{16}\text{.5\%}\end{array}$

Formula used:

  $z=\frac{x-\mu }{\sigma }$

Calculation:

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ z=\frac{0-12}{16.5}\\ z=-0.73\end{array}$

Explanation:

Raw containing the correct integer 10th of z score (-0.7) and the column containing the corresponding 100th of z score (0.03)

0.2327=23.27%

Therefore about 23.27% has an index of less than 0

(c)

To determine

To Calculate: the proportion of years does the index gain between 15% and 25%.

Answer to Problem 3.55RE

21.38%

Explanation of Solution

Given:

  $\begin{array}{c}\text{μ=12\%}\\ \text{σ=}\text{16}\text{.5\%}\end{array}$

Formula used:

  $z=\frac{x-\mu }{\sigma }$

Calculation:

Probability x that the index is between 15% and 25%

For $x=25\%$

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ z=\frac{25\%-12}{16.5}\\ z=0.79\end{array}$

For $x=15\%$

  $\begin{array}{c}z=\frac{x-\mu }{\sigma }\\ z=\frac{15\%-12}{16.5}\\ z=0.18\end{array}$

  $\begin{array}{c}z=0.79\Rightarrow 0.7852\\ =78.52\%\end{array}$

  $\begin{array}{c}z=0.18\Rightarrow 0.5714\\ =57.14\%\end{array}$

The difference between 78.52% and 57.14 is

78.52%-57.14%=21.38%

Therefore about 21.38% has an index between 15% and 25%