Chapter 8, Problem 20P

a.

Summary Introduction

To determine: The average rate of return for each stock during the given period.

Portfolio:

The portfolio refers to a group of financial assets like bonds, stocks, and equivalents of cash. The portfolio is held by investors and financial users. A portfolio is constructed based on the risk tolerance and the objectives of the company.

The Required Rate of Return:

The required rate of return is the rate, which should be minimum earned on an investment to keep that investment running in the market. When the required return is earned, only then the users and the companies invest in that particular investment.

Explanation of Solution

Calculate the average rate of return for both of the stock.

The formula to calculate the average rate of return is,

$\text{Average}\text{rate}\text{of}\text{return}=\frac{\text{Sum}\text{of}\text{value}\text{of}\text{stocks}}{\text{Number}\text{of}\text{stocks}}$

Calculation of the average rate of return for stock A:

Substitute (18%), 33%, 15%, (0.5%) and 27% for value of stock A and 5 for the number of stocks in the above formula.

$\begin{array}{c}\text{Average}\text{rate}\text{of}\text{return}=\frac{\left[\left(-18\%\right)+33\%+15\%+\left(-0.5\%\right)+27\%\right]}{5}\\ =11.30\%\end{array}$

The average rate of return for stock A is 11.3%.

Calculate the average rate of return for stock B.

Substitute (14.5%), 21.8%, 30.5%, (7.6%) and 26.3% for value of stock B and 5 for the number of stocks in the above formula.

$\begin{array}{c}\text{Average}\text{rate}\text{of}\text{return}=\frac{\left[\left(-14.5\%\right)+21.8\%+30.5\%+\left(-7.6\%\right)+26.3\%\right]}{5}\\ =11.3\%\end{array}$

The average rate of return for stock B is 11.3%.

Conclusion

The average return for the stock A and B both is 11.3%.

b.

Summary Introduction

To determine: The realized rate of return on the portfolio for each year and the average return on portfolio for the given assumptions.

Explanation of Solution

The portfolio for different year has the following value:

Year	Stock A	Stock B	Portfolio
2011	(18%)	(14.5%)	$\frac{\left(-18\%\right)+\left(-14.5\%\right)}{2}=-16.25\%$
2012	33%	21.8%	$\frac{33\%+21.8\%}{2}=27.4\%$
2013	15%	30.5%	$\frac{15\%+30.5\%}{2}=22.75\%$
2014	(0.5%)	(7.6%)	$\frac{\left(-0.5\%\right)+\left(-7.6\%\right)}{2}=-4.05\%$
2015	27%	26.3%	$\frac{27\%+26.3\%}{2}=26.65\%$

Table (1)

So, the realized rate of return on portfolio for each year is (16.25%), 27.4%, 22.75%, (4.05%) and 26.65%.

Calculation of the average rate of return for the portfolio:

The formula to calculate the average rate of return is,

$\text{Average}\text{rate}\text{of}\text{return}=\frac{\text{Sum}\text{of}\text{value}\text{of}\text{stocks}}{\text{Number}\text{of}\text{stocks}}$

Substitute (16.25%), 27.4%, 22.75%, (4.05%) and 26.65% for value of portfolio for different years and 5 for the number of stocks in the above formula.

$\begin{array}{c}\text{Average}\text{rate}\text{of}\text{return}=\frac{\left[\left(-16.25\%\right)+27.4\%+22.75\%+\left(-4.05\%\right)+26.65\%\right]}{5}\\ =11.3\%\end{array}$

The average rate of return for the portfolio is 11.3%.

Conclusion

The realized rate of return on the portfolio for different years is as calculated above and the average rate of return on the portfolio is 11.3%.

c.

Summary Introduction

To determine: The standard deviation of returns for each stock and for the portfolio.

Explanation of Solution

The formula to calculate the standard deviation is,

$\sigma =\frac{\sqrt{{\displaystyle \sum _{\text{i}=1}^{\text{N}}{({\text{r}}_{\text{i}}-\text{r})}^2}}}{\text{N}-1}$ (I)

Where,

• $\stackrel{}{\text{r}}$ is the expected rate of return.
• ${\text{r}}_{\text{i}}$ is the estimated rate of return.
• $\text{N}$ is the number of stocks.
• $\sigma$ is the standard deviation.

Calculation of the standard deviation of stock A:

${\text{r}}_{\text{i}}$	$\left({\text{r}}_{\text{i}}-\text{r}\right)$	${\left({\text{r}}_{\text{i}}-\text{r}\right)}^2$
(18%)	(29.3)	858.49
33%	21.7	470.89
15%	3.7	13.69
(0.5%)	(11.8)	139.24
27%	15.7	246.49
		1,728.8

Table (2)

Where,

${\text{r}}_{\text{i}}$ is the required rate of return.

$\text{r}$ is the average rate of return.

Substitute 1,728.8 for ${\displaystyle \sum _{\text{i}=1}^{\text{N}}{({\text{r}}_{\text{i}}-\text{r})}^2}$ and 4 for $\text{N}-1$ in equation (I).

$\begin{array}{c}\sigma =\sqrt{\frac{1,728.8}{4}}\\ =20.79\end{array}$

The standard deviation of stock A is 20.79%

Calculation of the standard deviation of stock B:

${\text{r}}_{\text{i}}$	$\left({\text{r}}_{\text{i}}-\text{r}\right)$	${\left({\text{r}}_{\text{i}}-\text{r}\right)}^2$
(14.5%)	(25.8)	665.64
21.8%	10.5	110.25
30.5%	19.2	368.64
(7.6%)	(18.9)	357.21
26.3%	15	225
		1,726.74

Where,

${\text{r}}_{\text{i}}$ is the required rate of return.

$\text{r}$ is the average rate of return.

Substitute 1,726.74 for ${\displaystyle \sum _{\text{i}=1}^{\text{N}}{({\text{r}}_{\text{i}}-\text{r})}^2}$ and 4 for $\text{N}-1$ in equation (I).

$\begin{array}{c}\sigma =\sqrt{\frac{1,726.4}{4}}\\ =20.78\end{array}$

The standard deviation of stock B is 20.78%.

Calculation of the standard deviation of the portfolio:

${\text{r}}_{\text{i}}$	$\left({\text{r}}_{\text{i}}-\text{r}\right)$	${\left({\text{r}}_{\text{i}}-\text{r}\right)}^2$
(16.25%)	(27.55)	759.00
27.4%	16.1	259.21
22.75%	11.45	131.10
(4.05%)	(15.35)	235.62
26.65%	15.35	235.62
		1,620.55

Where,

${\text{r}}_{\text{i}}$ is the required rate of return.

$\text{r}$ is the average rate of return.

Substitute 1,620.55 for ${\displaystyle \sum _{\text{i}=1}^{\text{N}}{({\text{r}}_{\text{i}}-\text{r})}^2}$ and 4 for $\text{N}-1$ in equation (I).

$\begin{array}{c}\sigma =\sqrt{\frac{1,620.55}{4}}\\ =20.13\end{array}$

The standard deviation of the portfolio is 20.13%.

Conclusion

Thus, the standard deviation of the stock A, stock B and the portfolio is 20.79%, 20.78% and 20.13% respectively.

d.

Summary Introduction

To determine: The coefficient of variation for each stock and for the portfolio.

The Coefficient of Variation:

The coefficient of variation is a tool to determine the risk. It determines the risk per unit of return. It is used for measurement when the expected returns are same for two data.

Explanation of Solution

Calculated,

For stock A,

Expected return is 11.3% (refer part a).

The standard deviation is 20.79% (refer part c).

For stock B,

Expected return is 11.3% (refer part a).

The standard deviation is 20.78% (refer part c).

For the portfolio,

Expected return is 11.3% (refer part b).

The standard deviation is 20.13% (refer part c).

The formula to calculate the coefficient of variation is,

$\text{Coefficient}\text{of}\text{variation}=\frac{\text{Standard}\text{deviation}\left(\sigma \right)}{\text{Expected}\text{return}\left(\stackrel{\wedge }{\text{r}}\right)}$

Calculate coefficient of variation for stock A.

Substitute 20.79% for standard deviation and 11.3% for expected return on stock.

$\begin{array}{c}\text{Coefficient}\text{of}\text{variation}=\frac{20.79\%}{11.3\%}\\ =1.84\end{array}$

The coefficient of variation for stock A is 1.84.

Calculate coefficient of variation for stock B.

Substitute 20.78% for standard deviation and 11.3% for expected return on stock.

$\begin{array}{c}\text{Coefficient}\text{of}\text{variation}=\frac{20.78\%}{11.3\%}\\ =1.84\end{array}$

The coefficient of variation for stock B is 1.84.

Calculate coefficient of variation for the portfolio.

Substitute 20.13% for standard deviation and 11.3% for expected return on stock.

$\begin{array}{c}\text{Coefficient}\text{of}\text{variation}=\frac{20.13\%}{11.3\%}\\ =1.78\end{array}$

The coefficient of variation for stock A is 1.78.

Conclusion

Thus, the coefficient of variation on stocks A, stock B and for the portfolio is 1.84, 1.84 and 1.78 respectively.

e.

Summary Introduction

To determine: The stock or the portfolio to be chosen and the reason for it.

Answer to Problem 20P

A risk averse investor will choose that investment which has a lower coefficient of variation. The portfolio has the lowest coefficient of variation, so the risk averse investor should choose this.

Explanation of Solution

• A risk averse investor is that investor who does not wants to take risk even if there are higher returns.
• The lower coefficient of variation means that the risk level is lower for that investment.
• The higher coefficient of variation means that the risk level is higher for that investment.

Conclusion

Thus, the risk averse investor will choose the stock of lower coefficient of variation.