Problem Set 1
Topics in Quantitative Finance
Instructions: This problem set is due on 9/5 at 11:59 pm CST and is an individual assignment. All problems must be handwritten. Scan your work and submit a PDF file.
Problem 1 Suppose your production opportunity set in a world with perfect certainty consists of the following investment projects:
| Project | Maximum Investment | IRR |
|---|---|---|
| A | $1,000,000 | 8% |
| B | $1,000,000 | 25% |
| C | $2,000,000 | 4% |
| D | $3,000,000 | 40% |
Consider an investor with utility function U(C_{0}, C_{1}) = C_{0} + 0.9 C_{1}. This investor has $7,000,000 in wealth
- If the investor has no access to capital markets.
- What is the value of her portfolio of projects?
- How much should she consume today and how much next period?
- Suppose now that a new bank comes to town, that allows the investor to borrow or lend at the interest rate of 10% per period.
- What is the value of her portfolio of projects?
- How much should she consume today and how much next period?
- If the bank was forced to close, how much the government would have to pay the investor to leave her indifferent?
Problem 2 Consider an investor with utility U(C_{0}, C_{1}) = \ln(C_{0}) + 2 \ln(C_{1}) for consumption today and next period. The investor has initial wealth W = 1 and can invest K = 1 - C_{0} in a technology that produces f(K) = \sqrt{K} next period.
- Compute the optimal consumption at dates 0 and 1.
Assume now that the investor has access to capital markets and can borrow or lend at r.
- Compute the present value and the net present value of the technology.
- If the investor sells the company at the value computed in b., what is her optimal consumption now?
Problem 3 You are offered the possibility to participate at the following lottery:
| Gain | Probability |
|---|---|
| 2 | 0.5 |
| 0 | 0.5 |
The cost at participating at the lottery is 1 unit of consumption. If you chose not to participate, you keep your unit of consumption.
- Is this a fair gamble?
- Show that the decision not to participate at the gamble implies that your utility function is concave.
Problem 4 Consider gamble A:
| Gain | Probability |
|---|---|
| -2 | 0.09 |
| 4 | 0.30 |
| 10 | 0.40 |
| 16 | 0.21 |
From A, we can construct another gamble B by adding white noise to a number of outcomes. Indeed, we can replace the outcome 4 by the gamble A':
- 3 with probability 1/2
- 5 with probability 1/2
with \operatorname{E}(A') = 4. In the same manner, we can replace outcome 16 with gamble A'':
- 12 with probability 1/3
- 18 with probability 2/3
with \operatorname{E}(A'') = 16.
Show formally why any risk averse individual prefers gamble A to gamble B.
Problem 5 A distant relative in Europe has recently passed away, leaving behind an estimated fortune of $1 million. This has left two grieving but competing close relatives: Peter, who currently has no wealth, and Paul, who has $10,000. With the will missing, they can pursue legal action, but the winner will incur legal costs amounting to 10% of the inheritance. Both Peter and Paul share the same utility function: U(W) = W^{1/4} Peter and Paul both agree that Peter has a 60% chance of winning the million, while Paul has a 40% chance. The judge cannot issue a split decision; the entire amount must go to one of them.
- Should Peter and Paul take their dispute to court, or is there a mutually beneficial agreement they could reach instead?
- Does the conclusion change if the heirs disagree on the probabilities? For instance, what if Peter believes he has an 80% chance of winning, while Paul thinks he has an 80% chance of winning?