The common stock of The DownTowne should return 23 percent in a boom, 16 percent in a normal economy, and lose 32 percent in a recession. The probabilities of a boom, normal economy, and recession are 5 percent, 90 percent, and 5 percent, respectively. What is the variance of the returns on this stock?

The variance of the returns on this stock is 0.011345

Explanation:

The returns on a stock is dependent on the market conditions. Variance of returns on a stock refers to the variability in returns of the stock around the mean expected return.

Given data from the question;

Probability of Boom = 5% = 0.05

Return in Boom = 23% = 0.23

Probability of Normal = 90% = 0.90

Return in Normal = 16% = 0.16

Probability of recession = 5% = 0.05

Return in recession = 32% = 0.32

To determine the variance of the returns, first solve for expected return;

Expected return = (Prob. of boom * return in boom) + (Prob. of normal * return in normal) + (Prob. of recession * return in recession)

= (0.05 * 0.23) + (0.90 * 0.16) + [0.05 * (-0.32) ]

= 0.1395

Variance of return = 0.05 (0.23-.1395) 2 + 0.90 (0.16-0.1395) 2 + 0.05 (-0.32 - 0.1395) 2

= 0.011345