Chapter 8, Problem 1OR

Summary Introduction

To discuss:

Annual average return and standard deviation.

Introduction:

Return: In financial context, return is seen as percentage that represents the profit in an investment.

Explanation of Solution

The annual average return for each asset over the period can be calculated using the following formula

$\text{Annual average returns}\left(\overline{\text{r}}\right)={\displaystyle \sum _{\text{i=1}}^{\text{n}}\frac{\left(\text{ Returns}\left({\text{R}}_{\text{i}}\right)\right)}{\text{n}}}$ (1)

Using equation (1) the annual average return of Miller’s Fund (MF) is calculated as follows:

$\begin{array}{c}\text{Annual average returns}\left(\overline{{\text{r}}_{\text{MF}}}\right)\text{ =}\frac{76+16.6-34.9+39.6}{4}\\ =\frac{97.3}{4}\\ =24.325\%\end{array}$

The annual average return of Miller’s Fund (MF) is 24.325%..

Using equation (1) the annual average return of S&P is calculated as follows:

$\begin{array}{c}\text{Expected returns}\left(\overline{{\text{r}}_{\text{S\&P}}}\right)\text{ =}\frac{26.5+15.1+2.1+16}{4}\\ =\frac{59.7}{4}\\ =14.925\%\end{array}$

The annual average return of S&P is 14.925%.

By the annual average returns, Miller’s Fund performed better than the S&P over the given period of time.

If money investment of $1,000 is made in Miller’s Fund in 2009, the money reaped at the end of 2012 would be $1,243.25 $\left(\begin{array}{c}\text{Money at the end of 2012 =Investment }\times \left(1+\text{Return}\right)\\ \text{=1,000}\times \left(1+0.24325\right)\\ =1,000\times 1.24325\\ \text{=1,243}\text{.25 }\end{array}\right)$

If money investment of $1,000 is made in S&P in 2009, the money reaped at the end of 2012 would be $1,149.25 $\left(\begin{array}{c}\text{Money at the end of 2012 =Investment }\times \left(1+\text{Return}\right)\\ \text{=1,000}\times \left(1+0.14925\right)\\ =1,000\times 1.14925\\ \text{=1,149}\text{.25 }\end{array}\right)$

The standard deviation of Miller’s Fund can be calculated as follows using excel functions as in table1.

Table 1

The standard deviation of Miller’s Fund is calculated as follows:

$\begin{array}{c}{\text{σ}}_{\text{MF}}\text{=}\sqrt{{\displaystyle \sum _{\text{i}\text{=}\text{1}}^{\text{n}}\frac{{\left({\text{r}}_{\text{i}}\text{-E(r)}\right)}^{\text{2}}}{\text{(n-1)}}}}\\ =\sqrt{\frac{0.6471}{4-1}}\\ =\sqrt{\frac{0.6471}{3}}\\ =\sqrt{0.2157}\\ =0.464435\end{array}$

The standard deviation of Miller’s Fund is 46.44%.

The standard deviation of S&P can be calculated as follows using excel functions as in table 2.

Table 2

The standard deviation of S&P is calculated as follows:

$\begin{array}{c}{\text{σ}}_{\text{S\&P}}\text{=}\sqrt{{\displaystyle \sum _{\text{i}\text{=}\text{1}}^{\text{n}}\frac{{\left({\text{r}}_{\text{i}}\text{-E(r)}\right)}^{\text{2}}}{\text{(n-1)}}}}\\ =\sqrt{\frac{0.039615}{4-1}}\\ =\sqrt{\frac{0.039615}{3}}\\ =\sqrt{0.013205}\\ =0.114913\end{array}$

The standard deviation of S&P is 11.5%.

By the value of standard deviation, Millers Fund is more volatile than S&P.