Question

Consider the following realized annual returns:

Year	Stock A	Index
2000	23.6%	48.3%
2001	24.7%	28.7%
2002	30.5%	86.9%
2003	9.0%	23.1%
2004	-2.0%	0.2%
2005	-17.3%	-3.2%
2006	-24.3%	-27.0%
2007	32.2%	27.9%
2008	4.4%	-5.1%
2009	7.4%	-11.3%

a. Calculate the average of annual returns of the index.

b. Compute the standard deviation of annual returns of the index.

c. Compute the lower bound of the 95% confidence interval for annual returns of the index. Use the exact value from Excel, not an approximation.

d. Compute the geometric mean of annual returns of the index.

a. The average of annual returns of the index is %. (round to two decimals)

b. The standard deviation of annual returns of the index is %. (round to two decimals)

c. The lower bound of the 95% confidence interval of annual returns of the index is %. (round to one decimal)

d. The geometric mean of annual returns of the index is %. (round to two decimals)

Answer 1

a) Average of annual returns of the index = Sum of all returns / No of years=(48.3+28.7+89.9+23.1+0.2-3.2-27-27.9-5.1-11.3)/10= 168.50/10 = 16.85%.

b) Standard Deviation formula is as follows:

where,

	=	population standard deviation
	=	the size of the population
	=	each value from the population
	=	the population mean

Now let us tabulate some of the values for the formula

Index	Index Mean	Index – Mean	Square of (Index- mean)
48.30%	16.85%	0.3145	989.1025
28.70%	16.85%	0.1185	140.4225
86.90%	16.85%	0.7005	4907.0025
23.10%	16.85%	0.0625	39.0625
0.20%	16.85%	-0.1665	277.2225
-3.20%	16.85%	-0.2005	402.0025
-27.00%	16.85%	-0.4385	1922.8225
27.90%	16.85%	0.1105	122.1025
-5.10%	16.85%	-0.2195	481.8025
-11.30%	16.85%	-0.2815	792.4225

Mean of square of (Index-mean) = (989.1025+140.4225+4907.0025+39.0625+277.2225+402.0025+1922.8225+122.1025+481.8025+792.4225)/10 =1007.3965

Standard deviation = Square root of the mean of square of (Index-mean) = 31.73951 %. Rounding to two decimals standard deviation is  31.74%

Standard Error = Standard deviation / square root of number of observations = 31.74/ square root of 10 = 10.037

c) Lower bound at 95% confidence level = Historical Average Return - (2* Standard Error) = 16.85 - (2*10.037) = -3.224%

Lower bound at 95% confidence level rounded to one decimal = -3.2%

d) Geometric mean = Multiply all the annual returns and take the 10th root as there are annual returns for 10 years

=(48.30*28.7*86.9*23.1*0.2*(-3.2)*(-27)*27.9*(-5.1)*(-11.3)) ^(1/10)

=12.26946%

= 12.27% (rounded to two decimals)