Question 1 [24 points]
Consider an economy with three dates (t=0,1,2) and two safe bonds. The payoffs and prices of the bonds are given as follows
Payoff Price
t=1 t=2 t=0
Bond A: 10 110 114
Bond B: 20 120 p
Suppose p=133.
(a) Is there an arbitrage? If so find an arbitrage portfolio. [6p]
(b) Design a portfolio with payoff (1, 0). What is the price of this portfolio? [8p]
(c) Suppose a start-up company wants to go public. The firm has costs of $100,000 at t=1 and $100,000 at t=2. It has sales of $120,000 at t=1 and $140,000 at date 2. The firm wants to issue 1,000 IPO shares. (A share is endowed with a cash flow right of 0.1% of the total profits ofthe firm.) Should the underwriter optimally suggest an IPO price of $30 per share? [4p]
(d) Find the full set {p} such that there is no arbitrage in the above economy. [6p]
An investment company offers put options on a stock with exercise date t=1.
(a) Draw the payoff (at t=1) of a portfolio that consists of the following two put options:
Buy a put with exercise price E=$14
Sell a put with exercise price E=$23
as a function of the underlying stock price at t=1. [3p] Suppose the (t=0) prices of these puts are as follows:
Price
E=$14 $3.00
E=$23 $2.00
(b) Is there an arbitrage? If so, find a profitable trading strategy. [4p] The investment company adds one more option to the above portfolio.
(c) Draw the payoff (at t=1) of a portfolio that consists of the following three put options:
Buy a put with exercise price E=$8
Buy a put with exercise price E=$14
Sell a put with exercise price E=$23
as a function of the underlying stock price at t=1. [3p]
Question 3 [12 points]
Consider an economy with two states which occur with equal probability at t=1. Suppose the CAPM holds. The risk free rate is 1%. The market portfolio has an expected return of 12% and generates the state dependent payoff (100, 200) at t=1. There are two assets. Asset A
generates the state dependent payoff (10, 40) at t=1. Asset B generates the state dependent payoff (30, 20) at t=1.
(a) Determine the beta of each of the assets? [4p]
(b) Suppose an investor has mean variance (CAPM-) preference and can either choose asset A or asset B for free. Which asset should he choose? [6p]
(c) What is the maximum price the investor is willing to pay for the chosen asset? [2p]
Suppose a firm has a profit (EBIT) of $200,000 at t=1. All agents are risk neutral and the interest rate is 25%.
(a) What is the value of the firm with 100% equity? [1p]
(b) What is the value of the firm if it has debt with face value of $100,000 outstanding? [1p]
Now suppose the firm has to pay corporate taxes. The tax rate is 20%.
(c) Determine the value of the firm in (a) and (b) when there is taxation. [4p]
(d) The firm has debt with face value of $100,000 outstanding. Suppose the government increases the tax rate from 20% to 30%. what iS the value of the firm,S equity before and after the tax reform? What is the percent change of the stock price with a 10% increase in tax rate? [8p]
Now suppose the firm can go bankrupt and the probability of bankruptcy is given by
prob= (D/200,000)2
where D is the face value of debt. The cost of bankruptcy is C=$20,000 and the tax rate is 20%.
(e) What is the optimal amount of debt? What is the value of the firm? [6p]
(f) A firm with debt might face financial distress. What are the costs associated with bankruptcy? What types of firms tend to have higher expected cost of bankruptcy? [4p]