Question

Using the following returns, calculate the arithmetic average returns, the variances, and the standard deviations for X and Y.

  

	Returns
Year	X	Y
1	6 %	19 %
2	24	40
3	13	-10
4	-14	-24
5	15	48

   

Calculate the arithmetic average return for X.

   

Calculate the arithmetic average return for Y.

  

Calculate the variance for X.

   

Calculate the variance for Y.

  

Calculate the standard deviation for X.

   

Calculate the standard deviation for Y.

Answer 1

Dear Student,

Here is the solution

1) Concept

> Arithmetic average return is the return on investment calculated by simply adding the returns for all sub-periods and then dividing it by total number of periods.

> Variance of returns In portfolio theory, the variance of return is the measure of risk inherent in investing in a single asset or portfolio.

> Standard deviation is a measure of how much an investment's returns can vary from its average return. It is a measure of volatility and in turn, risk.

2) Formualas

> Arithmetic average return =    $m = \frac{\text{sum of the terms}}{\text{number of terms}}$

> Variance of returns = (Standard deviation)2

> Standard deviation is = $\sigma={\sqrt {\frac {\sum(x_{i}-{\mu})^{2}}{N}}}$

3) Calculations

> Stock X

Year	Return Stock X	(X-Mean)2
1	6	7.84
2	24	231.04
3	13	17.64
4	-14	519.84
5	15	38.44
Total		814.8

- Mean = [6+24+13-14+15] / 5

               = 8.8 %

- Standard Deviation = [814.8/5]1/2

                                          = 12.77 %

- Variance = (12.77)2

                       = 163.07 %2

> Stock Y

Year	Return Stock Y	(Y-Mean)2
1	19	19.36
2	40	645.16
3	-10	605.16
4	-24	1489.96
5	48	1115.56
Total	73

- Mean = 73 / 5

               = 14.60 %

- Standard Deviation = [3875.2/5]1/2

                                          = 27.84 %

- Variance = (27.84)2

                       = 775.04 %2

Hope you understand the solution.