Olin School of Business, Washington University

FI 8360

Special Topics in Corporate Finance

Assignment 1

Due Monday, February 14

Consider an economy where there is a risk-free asset that returns 7% per year. There is a risky asset with current value 2. Over one year, only two things can happen: the economy booms and the risky asset goes up in value to 5, or the economy goes into recession and the risky asset goes down in value to 1. Each state is equally likely – the probability of boom or bust is ½. Here’s a picture:

1. There is a derivative that pays off 6 if the economy booms and 0 if the economy goes into recession.
2. Create the hedge portfolio that mimics the derivative, and interpret the portfolio holdings. Do you borrow or lend to create this derivative?
3. What is the value of the derivative?
4. If the beta of the risky asset is 1.0, what is the beta of the derivative?

1. There is a derivative that pays off 4 if the economy booms and 2 if the economy goes bust.
2. Create the hedge portfolio that mimics the derivative, and interpret the portfolio holdings. Do you borrow or lend to create this derivative?
3. What is the value of the derivative?
4. If the beta of the risky asset is 1.0, what is the beta of the derivative?
5. The expected payoff on this derivative is exactly the same as the expected payoff on derivative 1 (.5*6 +.5*0 = 3 = .5*4 + .5*2). Why do the derivatives have different values?

1. There is a derivative that pays off 0 if the economy booms and 6 if the economy goes into recession.
2. Create the hedge portfolio that mimics the derivative, and interpret the portfolio holdings. Do you borrow or lend to create this derivative?
3. What is the value of the derivative?
4. If the beta of the risky asset is 1.0, what is the beta of the derivative?
5. The expected payoff on this derivative is exactly the same as the expected payoff on derivatives 1 and 2 (.5*6 +.5*0 =.5*4 + .5*2 = .5*0 + .5*6 = 3). Why do the derivatives have different values?

1. For this economy, calculate the “risk-neutral probabilities” q and (1 – q).
2. What are q and (1 – q) in this economy?
3. Compare p (the ‘subjective’ probability of the boom state) to q (the ‘risk-neutral’, probability of the boom state), and similarly compare (1 – p) with (1 – q) (the ‘subjective’ and ‘risk-neutral’ probabilities of the recession state, respectively). The morphing from {p, 1 – p} to {q, 1-q} is called ‘change of measure’. What happens in the ‘change of measure’ here? Can you provide an intuitive explanation for this?
4. For the derivative in Problem #1 above

1)Calculate its expected return using the subjective probabilities {p, 1 – p}.

2)Calculate its expected return using the risk-neutral probabilities {q, 1-q}.

1. For the derivative in Problem #2 above

1)Calculate its expected return using the subjective probabilities {p, 1 – p}.

2)Calculate its expected return using the risk-neutral probabilities {q, 1-q}.

1. For the derivative in Problem #3 above

1)Calculate its expected return using the subjective probabilities {p, 1 – p}.

2)Calculate its expected return using the risk-neutral probabilities {q, 1-q}.

1. What is the common result in parts c through e?

1. Consider the following economy pictured below:

Derivatives A and B look like the ‘pink’ and the ‘blue’ derivatives from class (respectively). But if you work out their prices, you’ll find that the price of Derivative A is $0.47, while the price of Derivative B is $0.44. In other words, it appears that a payoff in the ‘up state’ is more valuable than a payoff in the ‘down state’, which contradicts what we learned in class. Can you explain this seeming conundrum?

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