Question

Suppose that the percentage annual return you obtain when you invest a dollar in gold or the stock market is dependent on the general state of the national economy as indicated below. For example, the probability that the economy will be in "boom" state is 0.15. In this case, if you invest in the stock market your return is assumed to be 25%; on the other hand if you invest in gold when the economy is in a "boom" state your return will be minus 30%. Likewise for the other possible states of the economy. Note that the sum of the probabilities has to be 1--and is.

State of Economy                Probability        MarketReturn     GoldReturn

Boom                                     0.15                 25%                      (-30%)

Moderate Growth                  0.35                  20%                        (-9%)

Weak Growth                         0.25                    5%                       35%

No Growth                             0.25                   (-14%)                    50%

calculate the variance for Gold & Stock Market returns?

Answer 1

Expected value of Gold Return = E(G) = 0.15 * (-30) + 0.35 * (-9) + 0.25 * 35 + 0.25 * 50

= 13.6%

E(G²) = 0.15 * (-30)² + 0.35 * (-9)² + 0.25 * 35² + 0.25 * 50²  = 1094.6

Variance for Gold returns = E(G²) - [E(G)]² = 1094.6 - 13.6² = 909.64

Expected value of Stock Market Return = E(S) = 0.15 * 25 + 0.35 * 20 + 0.25 * 5 + 0.25 * (-14)

= 8.5%

E(S²) = 0.15 * 25² + 0.35 * 20² + 0.25 * 5² + 0.25 * (-14)²  = 289

Variance for Stock Market returns = E(S²) - [E(S)]² = 289 - 8.5² = 216.75