Question

Based on the following information, what is the standard deviation of returns?

State of Economy	Probability of State of Economy	Rate of Return if State Occurs
Recession	.23	−	.091
Normal	.46		.106
Boom	.31		.216

Answer 1

First we will calculate the mean return as per below:

The formula for mean is:

Mean = p1 * r1 + p2 * r2 + p3 * r3

where, p1,p2 and p3 are the probabilities and r1,r2 and r3 are the returns.

Putting the given values of the probability in the above formula, we get,

Mean = (0.23 * -.091%) + (0.46 * .106%) + (0.31 * .216%)

Mean = -.02093 + 0.04876 + 0.06696

Mean = 0.09479

Now, steps for calculating standard deviation are:

First we will calculate the deviation of returns from the mean return as per below:

Recession : -0.091 - 0.09479 = 0.18579

Normal : 0.106 - 0.09479 = 0.01121

Boom 0.216 - 0.09479 = 0.12121

In the next step, we will square the deviations computed above, as per below:

Recession: (0.18579)² = 0.0345179241

Normal: (0.01121)² = 0.0001256641

Boom : (0.12121)² = 0.0146918641

In the next step, we will multiply the squared deviations computed above with their probabilities as per below:

Recession: 0.0345179241 * 0.23 = 0.007939122543

Normal:  0.0001256641 * 0.46 = 0.000057805486

Boom: 0.0146918641 * 0.31 = 0.004554477871

In the next step we will add up the values calculated above to find the variance, as per below:

Variance = 0.007939122543 + 0.000057805486 + 0.004554477871 = 0.01255141

In the final step, we will square root the variance calculated above to find the standard deviation:

Standard deviation = (0.01255141)^1/2 = 0.11203306