Question

The rate of return on the common stock of Lancaster Woolens is expected to be 18 percent in a boom economy, 8 percent in a normal economy, and only 2 percent in a recessionary economy. The probabilities of these economic states are 12 percent for a boom and 10 percent for a recession. What is the variance of the returns on this common stock?

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 = (12% * 18%) + (78% * 8%) + (10% * 2%)

Mean = (0.12 * 18%) + (0.78 * 8%) + (0.10 * 2%)

Mean = 2.16 + 6.24 + 0.2

Mean = 0.086 or 8.6%

Steps for calculating variance are:

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

Boom : 18 - 8.6 = 9.4

Normal growth: 8 - 8.6 = -0.6

Recession 2 - 8.6 = -6.6

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

Boom: (9.4)² = 88.36

Normal growth: (-0.6)² = 0.36

Recession : (-6.6)² = 43.56

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

Boom: 88.36 * 12% = 10.6032

Normal growth: 0.36 * 78% = 0.2808

Recession: 43.56 * 10% = 4.356

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

Variance = 10.6032 + 0.2808 + 4.356 = 15.24