3. Mixture Model of Cash Flows (20 points) You know that you will receive a cash flow C at some point in the future, but you do not know when it will happen. All you know is the following structure: There are two possible realizations, good and bad, denoted by g and b. If the good state occurs, then the cash flow will be distributed as Cg 10 is a standard deviation). ~ N(100, 102) (where If the bad state occurs, then the cash flow will be distributed as Cg ~ N(50, 10) (where again 10 is a standard deviation). The later the cash flow actually realizes, the higher the chance it will be of the bad type. To model this, suppose that if the cash flow is realized at time t, the cash flow C = Cg with probability exp{\t} and C = with probability 1 exp{\t} for some > 0. You may assume that Cg, Cb, and the identity of which cash flow is realized (b or g) are all independent of each other. Recall that for X ~ N(, o) variable: 1 1 E[X] = var(X) = 0 f(x) = = exp 2 2 {{}=("")"} F(x) = $ = (-") Recall that for Y ~ Bern(p): E[Y] = p var(Y) = p(1 - p) Recall that for Z~ Exp(X): E[Z] = 1 var(Z) = 1-2 f(x) = e-Ax F(x)=1-e-A Finally, recall the law of iterated expectations and law of total variance (where here X and Y are just two random variables unrelated to the normal and Bernoulli r.v.'s above): E[X] = E[E[X | Y]] var(X) = var(E[X | Y]) + E[var(X | Y)] (a) (5 points) Suppose you know for sure that the cash flow will occur at time t. What is its expected value? (b) (8 points) Again, suppose you know for sure that the cash flow will occur at time t. What is the variance of the cash flow? (c) (4 points) Sketch your answer to part (b) as a function of time t. That is, put t on the horizontal axis, and the variance on the vertical axis. Make sure to mark the point t = In(2)/ which is the point where exp{\t} = 1 exp{\t} = 0.5. Note that this is the point where et (1-e) is maximized. = (d) (3 points) In at most two sentences, explain the pattern you see in (c).