First we will calculate the mean return as per below:
The formula for mean is:
Mean = p1 * r1 + p2 * r2 + p3 * r3 * p4 * r4
where, p1,p2 , p3 and p4 are the probabilities and r1,r2,r3 and r4 are the returns.
Putting the given values of the probability in the above formula, we get,
Mean = (0.2 * 25%) + (0.2 * 15%) + (0.3 * 10%) + (0.3* -5%)
Mean = 5 + 3 + 3 - 1.5
Mean = 9.5%
2. Steps for calculating standard deviation are:
First we will calculate the deviation of returns from the mean return as per below:
Boom : 25 - 9.5 = 15.5
Good: 15 - 9.5 = 5.5
Level 10 - 9.5 = 0.5
Slump -5 -9.5 = -14.5
In the next step, we will square the deviations computed above, as per below:
Boom: (15.5)2 = 240.25
Good: (5.5)2 = 30.25
Level : (0.5)2 = 0.25
Slump: (-14.5)2 = 210.25
In the next step, we will multiply the squared deviations computed above with their probabilities as per below:
Boom: 240.25 * 0.2 = 48.05
Good: 30.25 * 0.2 = 6.05
Level: 0.25 * 0.3 = 0.075
Slump: 210.25 * 0.3 = 63.075
In the next step we will add up the values calculated above to find the variance, as per below:
Variance = 48.05 + 6.05 + 0.075 + 63.075 = 117.25
In the final step, we will square root the variance calculated above to find the standard deviation:
Standard deviation = (117.25)1/2 = 10.82820
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