Question

A stock will have a loss of 13.6 percent in a recession, a return of 12.3 percent in a normal economy, and a return of 27 percent in a boom. There is 33 percent probability of a recession, 36 percent probability of normal economy, and 31 percent probability of boom. What is the standard deviation of the stock's returns?

Answer 1

State of economy	Probability	Stock Return
Recession	33%	-13.6%
Normal	36%	12.3%
Boom	31%	27%

We have the following data:

Probability: p1 = 33%, p2 = 36%, p3 = 31%

Return: R1 = -13.6%, R2 = 12.3%, R3 = 27%

Expected return of the stock is calculated using the formula:

Expected return = E[R] = p1*R1 + p2*R2 + p3*R3 = 33%*(-13.6%) + 36%*12.3% + 31%*27% = 8.31%

The variance of the returns is calculated using the formula:

Variance = σ2 = p1*(R1 - E[R])2 + p2*(R2 - E[R])2 + p3*(R3 - E[R])2​​​​​​​ = 33%*(-13.6%-8.31%)2 + 36%*(12.3%-8.31%)2+ 31%*(27%-8.31%)2 =  0.0158415873+0.0005731236+0.0108287991 = 0.02724351

Standard deviation is the square root of variance

Standard deviation of stock's return = σ = (0.02724351)1/2 = 16.5056081378421% ~ 16.51% (Rounded to two decimals)

Standard deviation of stock's return = 16.51%

Answer -> 16.51%