Question

Consider the following five monthly​ returns:

7​%

−3​%

5​%

11​%

−1​%

a. Calculate the arithmetic average monthly return over this period and express your answer as a percentage per month.

b. Calculate the geometric average monthly return over this period.

c. Calculate the monthly variance over this period.

d. Calculate the monthly standard deviation over this period.

Answer 1

Solution:
a.	Arithmetic Average return	3.80	%
b.	Geometric average return	3.67	%
c.	monthly variance	0.00332
d.	monthly standard deviation	5.76	%
Working Notes:
	Stock returns for the last six years
Year 1	7%   or 0.07          r1
Year 2	-3% or -0.03                r2
Year 3	5% or 0.05                r3
Year 4	11% or 0.11             r4
Year 5	-1% or   -0.01                   r5
a.	Arithmetic Average return
	=(sum of all returns/ No period taken)
	= (r1 + r2 + r3 + r4 + r5 )/5
	= (7% -3% + 5% + 11% - 1% )/5
	= 19%/5
	=3.80%
b.	Geometric average return
	=((1+r1) x ( 1+ r2) x ( 1 +r3) x (1+ r4) x (1+r5) )^(1/5) - 1
	=(( 1+ 0.07) x ( 1 - 0.03) x (1+ 0.05) x (1+ 0.11) x (1 - 0.01) )^(1/5) - 1
	=1.036717899 - 1
	=0.036717899
	=3.67%
Notes:	Both the average returns are relevant in different cases like arithmetic method is used expected return of next period while Geometric is better measure of the change in wealth over the period.
	a	b = a - 3.80%	b^2
Year	Annual total return (%) R	(R-r)	(R-r)^2
1	7	3.20	10.240
2	-3	-6.80	46.240
3	5	1.20	1.440
4	11	7.20	51.840
5	-1	-4.80	23.040
Sum of annual return	19.00	Sum of (R-r)^2	132.80
	Mean (r) = sum of returns / no of returns
	Mean (r) = 19%/5
	Mean (r) =3.80%
	Where n = no of return considered = 5
c.	Variance = Sum of (R-r)^2 /(n-1)
	Variance = 132.80 %^2 /(5-1)
	Variance = 132.80 %^2 /4
	Variance = 33.20 %^2
	OR
	Variance = 0.00332
d.	standard deviation (s.d.) = Square root of (Sum of (R-r)^2 /(n-1))
	standard deviation (s.d.) = Square root of ( 132.80 %^2 /(5-1))
	standard deviation (s.d.) = Square root of ( 132.80 %^2/4)
	standard deviation (s.d.) = Square root of (33.20 %^2)
	standard deviation (s.d.) = (33.20 %^2)^(1/2)
	standard deviation (s.d.) = 5.761944%
	standard deviation (s.d.) = 5.76%
Please feel free to ask if anything about above solution in comment section of the question.