Question

4. Assume there are three possible future states for the economy (Boom, Stagnant, and Recession) with associated probabilities of 20%, 45%, nd 35%. For each future stae of the economy, a security pays either $40.00 or $20.00 with equal probability (i.e., a 50% chance of either payoff occuring).

a. What is the expected future cash flow for any given future state of the economy?

b. What is the expected future cash flow for the security?

c. Further assuming the future outcomes are one year into the future, what is the current price of the security assuming a 5% annual discount rate?

d. What is the conditional expected cash flow for each future state of nature? (Note: It will be the same for each state.)

e. Assuming the given any future state of nature, the $40.00 payoff is Event 1, and the $20.00payoff is event 2. What is the "conditional mean-zero idiosyncratic component" for the security based on the future payoffs in Event 1 and Event 2? (Note: It will be the same for each state.)

f. What is the CAPM beta for this security?

Answer 1

a. Expected future cash flow for any given future state of the economy = 0.5*40+0.5*20 = $ 30

b. The expected future cash flow for the security = 20%*30+45%*30+35%*30 = $30

c. Current price of the security = 30/1.05 = $28.57

d. Conditional expected cash flow for each future state of nature

Expected cash flow given Boom = 0.5*40+0.5*20 = $ 30

Similarly, for Stagnant and Recession, it is $30

e. Since payoffs are not dependent on the economic state, so conditional mean-zero idiosyncratic component accounts for 100% variation

f. CAPM Beta should be 0, as payoffs are not dependent on the economic state and market return