Question

1. Find the mean and standard deviation for option 1 and option 2

Option 1:

Year Return

1 15 %

2 -12 %

3 8 %

4 11 %

Option 2:

Prob. Return

0.05 16 %

0.40 8 %

0.25 -6 %

0.30 7 %

Answer 1

Calculation of Mean & Standard Deviation for Option1:

Formula:

Mean = Sum of all the returns / total number of returns

Sum of all the returns = 15% + (-12%) + 8% + 11% = 22%

Total number of returns = 4

therefore, Mean of returns = 22% / 4 = 5.5%

Now, Standard deviation:

Formula: (xi - )^2 / N

where, xi = returns = 15%, -12%, 8%, 11%

= mean of returns = 5.5%

N = number of returns = 4

therefore, standard deviation = [(15% - 5.5%)^2 + (-12% - 5.5%)^2 + (8% - 5.5%)^2 + (11% - 5.5%)^2 ] / 4

= [(9.5)^2 + (-17.5)^2 + (2.5)^2 + (5.5)^2 ] / 4 = 108.25 = 10.40

Hence, the Mean return is 5.5% and standard deviation is 10.40%.

Option2:

Probability	returns	returns with probabilities
0.05	16%	0.8%
0.40	8%	3.2%
0.25	-6%	-1.5%
0.30	7%	2.1%

Total returns = 0.8% + 3.2% + -1.5% + 2.1% = 4.6%

Mean returns = total or sum of returns / number of returns = 4.6% / 4 = 1.15%

Standard deviation of returns:

Formula: (xi - )^2 / N

where, xi = returns = 0.8%, 3.2%, -1.5%, 2.1%

= mean of returns = 1.15%

N = number of returns = 4

therefore, standard deviation = [(0.8% - 1.15.%)^2 + (3.2% - 1.15%)^2 + (-1.5% - 1.15%)^2 + (2.1% - 1.15%)^2 ] / 4

= [(-0.35)^2 + (2.05)^2 + (-0.35)^2 + (0.95)^2 ] / 4 = 1.3375 = 1.157%

Hence, the mean return is 1.15% and standard deviation is 1.157%