Chapter 3, Problem 3.47E

(a)

To determine

To obtain: The proportion of months with returns greater than 0 and the proportion of months with returns greater than 4%.

Answer to Problem 3.47E

The proportion of months with returns greater than 0 is 0.5832 and proportion of months with returns greater than 4% is 0.2206.

Explanation of Solution

Given info:

The distribution of the 369 monthly returns follows a normal distribution with mean of 0.84% and standard deviation of 4.097%.

Calculation:

For proportion of months with returns greater than 0:

Define the random variable x as percentage monthly returns.

The formula for the standardized score is,

$z=\frac{x-\mu }{\sigma }$

The months with returns greater than 0 is denoted as $x>0$.

Subtract the mean and then divide by the standard deviation to transform the value of x into standard normal z.

$\begin{array}{c}\frac{x-0.84}{4.097}>\frac{0-0.84}{4.097}\\ >-\frac{0.84}{4.097}\\ z>-0.21\end{array}$

Where, the standardized score $z=\frac{x-0.84}{4.097}$

The proportion of months with returns greater than 0, is obtained by finding the area to the right of –0.21 but, the Table A: Standard normal cumulative proportions apply only for cumulative areas from the left.

Use Table A: Standard normal cumulative proportions to find the area to the left of –0.21.

Procedure:

• Locate –0.2 in the left column of the A-2 Table.
• Obtain the value in the corresponding row below 0.01.

That is, $P\left(z<-0.21\right)=0.4168$

The area to the right of –0.21 is,

$\begin{array}{c}P\left(z>-0.21\right)=1-P\left(z<-0.21\right)\\ =1-0.4168\\ =0.5832\end{array}$

Thus, the proportion of months with returns greater than 0 is 58.32%.

For proportion of months with returns greater than 4%:

The months with returns greater than 4% is denoted as $x>4$.

Subtract the mean and then divide by the standard deviation to transform the value of x into standard normal z.

$\begin{array}{c}\frac{x-0.84}{4.097}>\frac{4-0.84}{4.097}\\ >\frac{3.16}{4.097}\\ z>0.77\end{array}$

Where, the standardized score $z=\frac{x-0.84}{4.097}$

The proportion of months with returns greater than 4% is obtained by finding the area to the right of 0.77. But, the Table A: Standard normal cumulative proportions apply only for cumulative areas from the left.

Use Table A: Standard normal cumulative proportions to find the area to the left of 0.77.

Procedure:

• Locate 0.7 in the left column of the A-2 Table.
• Obtain the value in the corresponding row below 0.07.

That is, $P\left(z<0.77\right)=0.7794$

The area to the right of 0.77 is,

$\begin{array}{c}P\left(z>0.77\right)=1-P\left(z<0.77\right)\\ =1-0.7794\\ =0.2206\end{array}$

Thus, the proportion of months with returns greater than 4% is 22.06%.

(b)

To determine

To obtain: The proportion of actual returns greater than 0 and the proportion of actual

returns greater than 4%.

To check: The whether the results suggest that $N\left(0.84,\text{ 4}\text{.097}\right)$ provides a good approximation to the distribution of returns over this period.

Answer to Problem 3.47E

The proportion of actual returns greater than 0 is 0.6264 and the proportion of actual returns greater than 4% is 0.2213.

Yes, results suggest that $N\left(0.84,\text{ 4}\text{.097}\right)$ provides a good approximation to the distribution of returns over this period.

Explanation of Solution

Given info:

The data shows the percentage of returns on common stocks. From the data, the total number of returns is 348, the actual returns greater than 0 is 218 and the actual returns greater than 4% is 77.

Calculation:

For proportion of actual returns greater than 0:

The formula to find the proportion of actual returns greater than 0 is,

$\text{Proportion of actual returns greater than 0 = }\frac{\text{Actual returns greater than 0}}{\text{Total number of returns}}$

Substitute 218 for ‘Actual returns greater than 0’, 348 for ‘Total number of returns’.

$\begin{array}{c}\text{Proportion of actual returns greater than 0 = }\frac{218}{348}\\ =0.6264\end{array}$

Thus, the proportion of actual returns greater than 0 is 0.6264.

For proportion of actual returns greater than 4%:

The formula to find the proportion of actual returns greater than 4% is,

$\text{Proportion of actual returns greater than 4\% = }\frac{\text{Actual returns greater than 4\%}}{\text{Total number of returns}}$

Substitute 77 for ‘Actual returns greater than 4%’, 348 for ‘Total number of returns’.

$\begin{array}{c}\text{Proportion of actual returns greater than 4\% = }\frac{77}{348}\\ =0.2213\end{array}$

Thus, the proportion of actual returns greater than 4% is 0.2213.

Comparison:

The percentage of months with returns greater than 0 is 58.32% and the percentage of months with returns greater than 4% is 22.06%.

Using normal distribution, the percentage of months with returns greater than 0 is 62.64% and the percentage of months with returns greater than 4% is 22.13%. Therefore, the percentages are approximately equal.

Thus, the results suggest that $N\left(0.84,\text{ 4}\text{.097}\right)$ provides a good approximation to the distribution of returns over this period.