Financial economics

1. (20) Automobile dealerships often advertise promotions such as “no payments for 90 days!” or similar deals that sound too good to be true. The goal, of course, is to entice buyers with the ever-tantalizing prospect of getting something now without having to worry about how to pay for it until later. Suppose that you are looking to buy a car priced at $35,000 and are offered a loan with a down payment of $5,000 and an APR of 6% over 60 months.

(a) (5) Calculate the monthly payment for this loan.

(b) (5) Now suppose that the dealer offers you “no money down, no interest, and no payments for 90 days!” and you naively interpret this to imply that the remaining payments are unchanged. In present value terms, what price do you think that you are paying for the car?

(c) (5) Somewhat coming to your senses, you realize that your monthly payment will go up to offset the missing initial payments, but still believe that you will see some savings because of the time value of avoiding interest for three months. What monthly payment do you calculate and what do price do you think you are paying?

(d) (5) Finally, you realize that if the dealer had any intention of offering you a discount, he would have let you know how much you’d be saving, and that “no interest” simply means that any unpaid interest is added to the principal. What is your actual monthly payment?

2. (20) One of the shortcomings of the dividend discount model is that it assumes that the firm can grow at the same rate indefinitely, and projecting a company’s future prospects based only on a brief snapshot of its current status can lead to wildly unrealistic estimates of its growth opportunities. Suppose that shares of Fly By Night Inc. are currently priced at P0 = 100, compared to a book value of B0 = 20 per share, with forecasted earnings of E1 = 8 and a scheduled dividend payment of D1 = 3 per share.

(a) (5) Using the constant growth model, estimate Fly By Night’s growth rate g and market capitalization rate r. Do these numbers seem plausible to you?

(b) (5) Upon closer inspection, you observe that all of Fly By Night’s growth opportunities consist only of a single investment project this year. After this year it cannot repeat this or undertake any other positive NPV project, and must pay out all subsequent earnings to shareholders. Suppose you believe that you are the only one who is aware of this, while all other investors are convinced that the stock will continue growing at the same rate forever. What should the stock be priced at?

(c) (10) You now realize that all investors are well aware of Fly By Night’s limited growth opportunities, and that this is already reflected in its price. What is the correct discount rate?

3. (30) Consider a portfolio choice problem with a risk-free asset with return rf and two risky assets, the first with mean return µ1 = 0.12 and standard deviation σ1 = 0.4 and the second with mean µ2 = 0.08 and standard deviation σ2 = 0.3, with correlation ρ12 = 0. For any stock portfolio let λ denote the proportion invested in stock 1.

(a) (10) Find the weight λ˜ that minimizes portfolio standard deviation σp.

(b) (5) Consider the tangency portfolio and let λ ∗ denote the weight it places on stock 1. Find the condition that defines this value, but do not solve for it, and explain how it would compare to λ˜.

(c) (5) Now consider varying the risk-free rate rf . Again without solving anything, explain how you would expect λ ∗ to vary as rf increases.

(d) (10) Show how the slope of the tangent line changes with rf . Recall a useful theorem that allows you to do this without ever actually solving for λ ∗ .

4. (30) Let Cara be a consumer who she exhibits no time preference δ = 1 and receives an endowment of W in each of two periods. There exists a single risky asset currently priced at P0 = 1 that next period has payoffs of P1 = ( 3 w/ prob. 2 3 0 w/ prob. 1 3 1 Consider Cara’s two-period utility maximization problem. max ξ u(C0) + δE[u(C1)] s.t. C0 = W − ξP0 C1 = W + ξP1

(a) (10) Find the first order condition for Cara’s optimization problem.

(b) (10) Predictably, Cara’s utility function exhibits constant absolute risk aversion u(C) = 1 − e −αC . Show that the number of shares she buys does not depend on her endowment W.

(c) (10) Suppose that Cara has a twin sister, Cora, who faces the same optimization problem but whose utility function instead exhibits constant relative risk aversion u(C) = C 1−ρ 1−ρ . Show that the number of shares she buys is proportional to her endowment W.