What is the standard deviation of the returns on a stock given the following information? State of Economy Probability of State of Economy Rate of Return if State Occurs Boom .28 .175 Normal .67 .128 Recession .05 .026

3.28%

Explanation:

Calculation for the standard deviation of the returns on a stock

The first step is to find the Expected rate of return using this formula

Expected Return = E[R] = p1*R1 + p2*R2 + p3*R3

Let plug in the formula

Expected Return= 0.28*0.175 + 0.67*0.128 + 0.05*0.026

Expected Return = 0.049 + 0.08576 + 0.0013

Expected Return= 0.13606

Second step is to find the Variance using this formula

Variance = σ2 = p1*(R1-E[R])2 + p2*(R2-E[R])2 + p3*(R3-E[R])2

Let plug in the formula

Variance = σ2 = 0.28*(0.175-0.13606)2 + 0.67*(0.128-0.13606)2 + 0.05*(0.026-0.13606)2

Variance = 0.000424570608 + 0.0000435256119999998 + 0.00060566018

Variance= 0.0010737564

Last step is to find Standard Deviation of the returns on a stock

Note that the Standard Deviation is square-root of variance

Using this formula

Standard Deviation =√Variance

Let plug in the formula

Standard Deviation = σ =√ (0.0010737564)

Standard Deviation= 0.032768222411*100

Standard Deviation= 3.2768222411%

Standard Deviation =3.28% Approximately

Therefore the standard deviation of the returns on a stock will be 3.28%