Chapter 18, Problem 5PS

Summary Introduction

(a)

To calculate:

The return of arithmetic average on the two stocks named ABC and XYZ.

Introduction:

Arithmetic average return is simple return which is calculated by adding the return for sub periods and then it is divided by total number of periods.

Answer to Problem 5PS

	ABC	XYZ
Arithmetic average return on stock	$10\%$	$10\%$

Explanation of Solution

Given:

The rate of returns of the two stocks:

Year	rABC	Rxyz
1	20%	30%
2	12	12
3	14	18
4	3	0
5	1	-10

The formula for calculating arithmetic average return is:

  $\text{Arithmetic average return}=\frac{\text{Sum of Individual Return}}{\text{Total Number of Returns}}$

The arithmetic return for stock ABC is:

  $\begin{array}{c}\text{Arithmetic average return}=\frac{20\%+12\%+14\%+3\%+1\%}{5}\\ =\frac{50\%}{5}\\ =10\%\end{array}$

The arithmetic return for stock XYZ is:

  $\begin{array}{c}\text{Arithmetic average return}=\frac{30\%+12\%+18\%+0\%-10\%}{5}\\ =\frac{50\%}{5}\\ =10\%\end{array}$

Summary Introduction

(b)

To determine:

The stock which has higer dispersion on the basis of the returns.

Introduction:

In grammatical terms, Dispersion means the size of the variety of value predictable for a particular stock.

Explanation of Solution

Dispersion is analyzing the about the variations of returns from a specific strategy or investment portfolio. It can be said that it is to measure the degree of uncertainty and risk associated with the security.

Both the stock gives equal average rate of return. But, stock XYZ gives zero and negative return at the end of the 4^th and 5^th year which means this stock has higher dispersion as there is a big variation in the returns i.e. negative to positive and high to $30\%$ .

Conclusion

Thus, stock XYZ is the stock with high dispersion as per the returns.

Summary Introduction

(c)

To calculate:

The return of geometric average of both the stocks.

Introduction:

Geometric return is explicitly used to calculate the average rate of return per period on an investment that can be compounded over multiple periods.

Answer to Problem 5PS

	ABC	XYZ
Geometric average return on stock	$9.77\%$	$9.11\%$

From the above analysis, it can be concluded that stock ABC has a higher geometric average return.

Explanation of Solution

Given:

The rate of returns of the two stocks:

Year	rABC	Rxyz
1	20%	30%
2	12	12
3	14	18
4	3	0
5	1	-10

The formula for calculating geometric average return is:

  $\text{Geometric return}={\left[\left(1+r_1\right)\times \left(1+r_2\right)\times \mathrm{...........}\times \left(1+r_n\right)\right]}^{\frac{1}{n}}-1$

The geometric return for stock ABC is:

  $\begin{array}{c}\text{Geometric return}={\left[\left(1+r_1\right)\times \left(1+r_2\right)\times \mathrm{...........}\times \left(1+r_n\right)\right]}^{\frac{1}{n}}-1\\ ={\left[\left(1+20\%\right)\times \left(1+12\%\right)\times \left(1+14\%\right)\left(1+3\%\right)\left(1+1\%\right)\right]}^{\frac{1}{5}}-1\\ ={\left(1.5939\right)}^{\frac{1}{5}}-1\\ =1.0977-1\\ =0.0977\text{ or 9}\text{.77\%}\end{array}$

The geometric return for stock XYZ is:

  $\begin{array}{c}\text{Geometric return}={\left[\left(1+r_1\right)\times \left(1+r_2\right)\times \mathrm{...........}\times \left(1+r_n\right)\right]}^{\frac{1}{n}}-1\\ ={\left[\left(1+30\%\right)\times \left(1+12\%\right)\times \left(1+18\%\right)\left(1+0\%\right)\left(1+\left(-10\%\right)\right)\right]}^{\frac{1}{5}}-1\\ ={\left(1.5462\right)}^{\frac{1}{5}}-1\\ =1.0911-1\\ =0.0911\text{ or 9}\text{.11\%}\end{array}$

Summary Introduction

(d)

To calculate:

The expected return of stock ABC when the returns of this stock is likely to be earned equally in each year.

Introduction:

Expected return is expected value in terms of probability distribution of expected returns which can be providing to investors.

Answer to Problem 5PS

The expected return of stock ABC is $10\%$

Explanation of Solution

Given:

The rate of returns of the two stocks:

Year	rABC	Rxyz
1	20%	30%
2	12	12
3	14	18
4	3	0
5	1	-10

The probability is to be divided equally in each year i.e. $20\%$ in each year.

For computing the expected return, sum of product of probabilities and their respective returns is to be done.

  

Summary Introduction

(e)

To calculate:

The expected return of stock XYZ when the returns of this stock is likely to be earned equally in each year.

Introduction:

Expected return is expected value in terms of probability distribution of expected returns which can be providing to investors.

Answer to Problem 5PS

The expected return of stock XYZ is $10\%$

Explanation of Solution

Given:

The rate of returns of the two stocks:

Year	rABC	Rxyz
1	20%	30%
2	12	12
3	14	18
4	3	0
5	1	-10

The probability is to be divided equally in each year i.e. $20\%$ in each year.

For computing the expected return, sum of product of probabilities and their respective returns is to be done.

  

Summary Introduction

(f)

To determine:

The best average return among arithmetic and geometric for predicting future performance of the two stocks .

Introduction:

Arithmetic average return is simple return which is calculated by adding the return for sub periods and then it is divided by total number of periods.

Geometric return is explicitly used to calculate the average rate of return per period on an investment that can be compounded over multiple periods.

Explanation of Solution

Given:

The rate of returns of the two stocks:

Year	rABC	Rxyz
1	20%	30%
2	12	12
3	14	18
4	3	0
5	1	-10

As on the basis of computation of arithmetic and geometric average return of the stocks, it has been seen that the arithmetic average return of the both the stocks is equal i.e. $10\%$ and this average is also same and equal to the expected return i.e. $10\%$ computed by taking equal probability of earning the returns in the future.

Conclusion

Thus, the arithmetic average return is more appropriate for the prediction of future performance.