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	13%	23%
2	31	44
3	20	-10
4	- 21	- 24
5	22	52

Answer 1

a) Arithmatic Return

Arithmatic Return = (x1 + x2 + x3 + ............ + xn)/n

In this question, for X,

Arithmatic return = (13% + 31% + 20% -21% + 22%)/5 = 13%

In this question, for Y,

Arithmatic return = (23% + 44% - 10% - 24% + 52%)/5 = 15%

b) Variance

Mathematically, variance is calculated using mathematical relation:

So, s² is the variance and s is the standard deviation, implying standard deviation is square root of variance.

X	(X - Mean)^2
13%	= (0.13 - 0.13)^2 = 0
31%	= (0.31 - 0.13)^2 = 0.0324
20%	= (0.2 - 0.13)^2 = 0.0049
-21%	= (-0.21 - 0.13)^2 = 0.1156
22%	= (0.22 - 0.13)^2 = 0.0081

Variance = (0 + 0.0324 + 0.0049 + 0.1156 + 0.0081)/(5 - 1) = 0.1610/4 = 0.04025 (402.5% in % form - - since it is a square of percentage number so multiplied by 100 * 100)

Y	(Y - Mean)^2
23%	= (0.23 - 0.17)^2 = 0.0036
44%	= (0.44 - 0.17)^2 = 0.0729
-10%	= (-0.1 - 0.17)^2 = 0.0729
-24%	= (-0.24 - 0.17)^2 = 0.1681
52%	= (0.52 - 0.17)^2 = 0.1225

Variance = (0.0036 + 0.0729 + 0.0729 + 0.1681 + 0.1225)/(5-1) = 0.11 (1100% in % form - since it is a square of percentage number so multiplied by 100 * 100)

c) Standard deviation

Since it is a ssquare root of variance we just calculated,

For X, standard deviation = sqr root(0.04025) = 20.06%

For Y, standard deviation = sqr root(0.11) = 33.17%