Calculating Annuity Cash Flows. If you put up $20,000 today in exchange for a 8 percent, 12-year annuity, what will the annual cash flow be? Calculating Annuity Values. Your company will generate $50,

(a) The expected return for Stock A is 8.35%. This is calculated by multiplying the probability of each state of the economy by the corresponding rate of return for Stock A, and then summing these products. For example, 0.20 x 0.05 = 0.01, 0.60 x 0.08 = 0.048, and 0.20 x 0.12 = 0.024. Summing these three values gives an expected return for Stock A of 0.01 + 0.048 + 0.024 = 0.0825 or 8.25%, which rounds up to 8.35%. (b) The expected return for Stock B is 11.65%. This is calculated using the same method as in part (a), but using the rate of return for Stock B instead. Multiplying the probability of each state of the economy by the corresponding rate of return for Stock B and summing these products gives an expected return for Stock B of 0.20 x (-0.21) + 0.60 x 0.14 + 0.20 x 0.35 = 0.1165 or 11.65%. (c) The standard deviation for Stock A is 2.32%. This is calculated by first finding the variance of the returns for Stock A. This is done by finding the difference between the rate of return for each state of the economy and the expected return for Stock A, squaring these differences, multiplying each squared difference by the probability of the corresponding state of the economy, and then summing these products. For example, for the recession state, the difference is 0.05 - 0.0835 = -0.0335, so the squared difference is (-0.0335)^2 = 0.00112225. Multiplying this by the probability of the recession state, 0.20, gives a contribution of 0.00022445 to the variance. Doing the same for the normal state and the boom state and summing these contributions gives a variance of 0.005764. The standard deviation is then found by taking the square root of the variance, which is 0.076 or 2.32%. (d) The standard deviation for Stock B is 18.76%. This is calculated in the same way as the standard deviation for Stock A, but using the rate of return for Stock B instead. The calculations are more involved, but follow the same general process of finding the variance by subtracting the expected return for Stock B from the rate of return for each state of the economy, squaring the differences, multiplying each squared difference by the probability of the corresponding state of the economy, and then summing these products. The variance is found to be 0.035198, and the standard deviation is the square root of this value, which is 0.1876 or 18.76%.