Question

1. Using the following returns, calculate the arithmetic average returns, the standard deviations, and the geometric average returns for the stock A, B, and C. Assume the returns of stocks are normally distributed. If Mr. Wong owns a stock C, based on the historical return data, there is only a 2.5 percent chance that the stock C will produce a return less than _____ percent in any one year.   If Mr. Wong owns a stock B, which range of returns would Mr. Wong expect to see approximately two-thirds of the time?

Year	A %	B %	C %
1	2.54	4.84	1.5
2	-3.27	-0.97	-1.97
3	-4.54	-2.24	-2.4
4	2.28	-4.58	1.38
5	-1.54	1.76	-2.44
6	4.54	6.84	4.84
7	-2.54	-1.24	-4.44
8	2.35	4.65	1.45
9	2.94	6.24	7.24

Answer 1

Standard deviation

Stock A = square root of [ { (2.54-0.306)^2 + (-3.27-0.306)^2 + (-4.4-0.306)^2 + (2.28-0.306)^2 + (-1.54-0.306)^2 + (4.54-0.306)^2 + (-2.54-0.306)^2 + (2.35-0.306)^2 + (2.94-0.306)^2 }/8 ]

= 3.273

Stock B = square root of [ { (4.84-1.7)^2 + (-0.97-1.7)^2 + (-2.24-1.7)^2 + (-4.58-1.7)^2 + (1.76-1.7)^2 + (6.84-1.7)^2 + (-1.24-1.7)^2 + (4.65-1.7)^2 + (6.24-1.7)^2 }/8 ]

= 4.128

Stock C = square root of [ { (1.5-0.573)^2 + (-1.97-0.573)^2 + (-2.4-0.573)^2 + (1.38-0.573)^2 + (-2.44-0.573)^2 + (4.84-0.573)^2 + (-4.44-0.573)^2 + (1.45-0.573)^2 + (7.24-0.573)^2 }/8 ]

= 3.782

Arithmetic average return

Stock A = (2.54-3.27-4.54+2.28-1.54+4.54-2.54+2.35+2.94)/9

= 0.306

Stock B = (4.84-0.97-2.24-4.58+1.76+6.84-1.24+4.65+6.24)/9

= 1.7

Stock C = (1.5-1.97-2.4+1.38-2.44+4.84-4.44+1.45+7.24)/9

= 0.573

Geometric return

Stock A = (1.0254*0.9673*0.9546*1.0228*0.9846*1.0454*0.9746*1.0235*1.0294)^(1/9) - 1

= 0.26%

Stock B = (1.0484*0.9903*0.9776*0.9542*1.0176*1.0684*0.9876*1.0465*1.0624)^(1/9) - 1

= 1.625%

Stock C = (1.015*0.9803*0.976*1.0138*0.9756*1.0484*0.9556*1.0145*1.0724)^(1/9) - 1

= 0.511%

The confidence interval of 2.5% is = Arithmetic mean - 3*(std deviation)

For Stock C it would be = 0.573 - 3*3.782 = -10.773%

Two third of the time means confidence interval of 68% for Stock B ranges from :

Arithmetic mean - (std deviation) to Arithmetic mean + (std deviation) = 1.7 - 4.128 to 1.7 + 4.128 = -2.428% to 5.828