Bruno's stock should return 14 percent in a boom, 11 percent in a normal economy, and 4 percent in a recession. The probabilities of a boom, normal economy, and recession are 8 percent, 90 percent, and 2 percent, respectively. What is the variance of the returns on this stock?

a. 0.011387

b. 0.000169

c. 0.001506

d. 0.001538

e. 0.011561

option (b) variance of the returns is 0.000169

Explanation:

Given data

boom return = 14% = 0.14

normal economy = 11% = 0.11

recession = 4% = 0.04

boom probability = 8% = 0.08

normal economy probability = 90 % = 0.90

recession probability = 2 % = 0.02

to find out

the variance of the returns on this stock

solution

we know variance of the returns is sum of standard deviation

expected return = boom return * boom probability

expected return boom = 0.14 * 0.08 = 0.0112

expected return = economy * economy probability

expected return economy = 0.11 * 0.9 = 0.099

expected return = recession * recession probability

expected return recession = 0.04 * 0.02 = 0.0008

total expected return = 0.0112 + 0.099 + 0.0008 = 0.111

for boom

standard deviation = probability * (return - 0.111) ²

standard deviation = 0.08 * (0.14 - 0.111) ² = 0.00006728 ... 1

for normal economy

standard deviation = probability * (economy - 0.111) ²

standard deviation = 0.90 * (0.11 - 0.111) ² = 0.00000090 ... 2

for recession

standard deviation = probability * (recession - 0.111) ²

standard deviation = 0.02 * (0.04 - 0.111) ² = 0.00010082 ... 3

variance of the returns is sum of standard deviation

variance of the returns = 0.00006728 + 0.00000090 + 0.00010082

variance of the returns is 0.000169