Chapter 7, Problem 2PS

Standard deviation of returns The following table shows the nominal returns on the U.S. stocks and the rate of inflation.

1. a. What was the standard deviation of the nominal market returns?
2. b. Calculate the arithmatic average real return.

Summary Introduction

To compute: The standard deviation of nominal market return.

Explanation of Solution

The formula to calculate average nominal return is as follows:

$\text{Average nominal return }=\frac{\text{Total amount of nominal return}}{\text{Total number of years}}$

The computation of average nominal return is as follows:

$\begin{array}{c}\text{Average nominal return}=\frac{\left(.172+.010+.161+.331+.127\right)}{5}\\ =.1602,\text{or 16}\text{.02\%}\text{.}\end{array}$

Hence, the average nominal return is 16.02%.

The formula to calculate variance is as follows:

$\text{Variance}=\frac{\left[\begin{array}{l}{\left({\text{Nominal return}}_1-\text{ Average norminal return}\right)}^2+\\ {\left({\text{Nominal return}}_2-\text{ Average norminal return}\right)}^2+\\ {\left({\text{Nominal return}}_3-\text{ Average norminal return}\right)}^2+\\ {\left({\text{Nominal return}}_4-\text{ Average norminal return}\right)}^2+\\ {\left({\text{Nominal return}}_5-\text{ Average norminal return}\right)}^2\end{array}\right]}{\text{Total number of years}}$

The calculation of variance is as follows:

$\begin{array}{c}\text{Variance }=\frac{\left[\begin{array}{l}{\left(.172–.1602\right)}^2+{\left(.010–.1602\right)}^2+{\left(.161–.1602\right)}^2+(.331–\\ .1602{)}^2+{\left(.127–.1602\right)}^2\end{array}\right]}{5}\\ =.010595\end{array}$

Hence, the variance is .010595

The formula to calculate standard deviation is as follows:

$\text{Standard deviation}=\sqrt{\text{Variance}}$

The Computation of standard deviation is as follows:

$\begin{array}{c}\text{Standard deviation}=\sqrt{.010595}\\ =.1029\text{or }10.29\%\end{array}$

Hence, the standard deviation of nominal market return is 10.29%%.

Summary Introduction

To compute: The arithmetic average real return.

Explanation of Solution

The formula to calculate average real return is as follows:

$\text{Average real return}=\frac{\begin{array}{c}\left[\left(\frac{{\text{Norminal return}}_1}{{\text{1+Inflation}}_1}\right)-1\right]+\left[\left(\frac{{\text{Norminal return}}_2}{{\text{1+Inflation}}_2}\right)-1\right]+\\ \left[\left(\frac{{\text{Norminal return}}_3}{{\text{1+Inflation}}_3}\right)-1\right]+\left[\left(\frac{{\text{Norminal return}}_4}{{\text{1+Inflation}}_4}\right)-1\right]+\\ \left[\left(\frac{{\text{Norminal return}}_5}{{\text{1+Inflation}}_5}\right)-1\right]\end{array}}{\text{Total number of years}}$

The computation of average real return is as follows:

$\begin{array}{c}\text{Average real return}=\frac{\left\{\begin{array}{l}\left[\left(\frac{1.172}{1.015}\right)–1\right]+\left[\left(\frac{1.010}{1.030}\right)–1\right]+\\ \left[\left(\frac{1.161 }{1.017}\right) –1\right]+\left[\left(\frac{1.331}{1.015}\right)–1\right]+\left[\left(\frac{1.127}{1.008}\right)–1\right]\end{array}\right\}}{5}\\ =.1412,\text{or 14}\text{.12\%}\text{.}\end{array}$

Hence, the average real return is .1412, or 14.12%.