Question

Consider the following realized annual returns:

Year End	Market Realized Return	Stock B Realized Return
2000	21.2%	88.3%
2001	30.3%	56.4%
2002	22.3%	114.6%
2003	25.3%	68.4%
2004	-11.0%	-62.8%

1. 2005	-11.3%	52.7%
   2006	-20.8%	-22.0%
   2007	33.1%	6.9%
   2008	13.0%	9.2%
   2009	7.3%	-0.9%

2. Consider the following realized annual returns:
   
   
   
   
   
   
   
   
   

3. Q1 : Suppose that you want to use the 10 year historical average return on the Market to forecast the expected future return on the Market. Calculate the 95% confidence interval for your estimate of the expect return. Q2 : Using the data provided in the table, calculate the average annual return, the variance of the annual returns, and the standard deviation of the average returns for Stock B from 2000 to 2009.

Answer 1

Q - 1
Please see the table below:

Year End	Market Realized Return	Square of deviation from mean
	Rm	(Rm - Mean)2
2000	21.20%	0.01052676
2001	30.30%	0.03748096
2002	22.30%	0.01290496
2003	25.30%	0.02062096
2004	-11.00%	0.04813636
2005	-11.30%	0.04946176
2006	-20.80%	0.10074276
2007	33.10%	0.04910656
2008	13.00%	0.00042436
2009	7.30%	0.00132496
Total	109.40%	0.3307304
Mean = Total / 10	= 109.4 / 10 = 10.94%	0.033073
Standard deviation	= 0.0330731/2 =	18.19%

Number of observations, n = 10

Mean, µ = 10.94%; Standard deviation =  σ = 18.19%

For 95% confidence interval, the z* value is 1.96

Hence our interval will be:

Lower limit = µ - z* x σ / n^1/2 = 10.94% - 1.96 x 18.19% / 10^1/2 = -0.33%

Upper limit = µ + z* x σ / n^1/2 = 10.94% + 1.96 x 18.19% / 10^1/2 = 22.21%

Hence, the desired interval is (-0.33%, 22.21%)

Q -2

Please see the table below:

Year End	Stock B Realized Return	Square of deviation from mean
	RB	(RB - Mean)^2
2000	88.30%	0.32741284
2001	56.40%	0.06411024
2002	114.60%	0.69755904
2003	68.40%	0.13927824
2004	-62.80%	0.88134544
2005	52.70%	0.04674244
2006	-22.00%	0.28174864
2007	6.90%	0.05846724
2008	9.20%	0.04787344
2009	-0.90%	0.10227204
Total	310.80%	2.6468096
Mean	= 310.08% / 10 = 31.08%	=2.6468096 / 10 = 0.264681
Standard deviation	= 0.2646811/2 =	51.45%

The average annual return = 31.08% ,

the variance of the annual returns = Mean of square of deviation from mean = 0.264681,

and the standard deviation of the average returns for Stock B = Variance^1/2 = 0.5145 = 51.45%