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Quantitative Risk Management

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Quantitative Risk Management
Homework 2 Solution, Fin 500Q, Quantitative Risk Management 1. Assume gold price risk is diversifiable, and the riskless rate is 5%. A firm produces a unit of gold a year from today. Assume all interest is compounded annually and is tax deductible. The price of gold is either $500 or $200, each with probability 0.5. Suppose the firm pays taxes at a rate of 40% for all its cash flow in excess of $300. The value of the firm is the expected discounted value of its cash flow less the expected discounted value of bankruptcy costs and taxes that it pays. The firm can hedge by buying/selling forward contracts on gold. Start by assuming that bankruptcy costs are zero. (a) Find the value of the unhedged unlevered firm. (10 points) Answer: 1 · [350 − 0.5 · 0.4 · (500 − 300)] = 295.238. Value of firm = 1.05 (b) Find the value of the hedged unlevered firm. (10 points) Answer: 1 Value of firm = · [350 − 0.4 · (350 − 300)] = 314.286. 1.05 (c) Find the value of the unhedged firm if it issues an optimally chosen quantity of safe debt. (10 points) Answer: Maximum riskless debt that the firm can issue is 200/(1.05) = 190.476. Value of firm = [ ] 1 200 · 350 − 0.4 · 0.5 · (500 − 300 − · 0.05) = 297.052. 1.05 1.05

(d) Find the value of the hedged firm that issues an optimally chosen quantity of safe debt. (10 points) Answer: Maximum debt that the firm can issue is such that its cash flow can cover debt repayments (principal plus interest) and taxes. Safe debt has a yield of 0.05. Therefore, total payments must satisfy 1.05F + 0.4 · (350 − 300 − F · 0.05) ≤ 350 implying that the highest permissible F = 320.388. Therefore, Value of firm = 1 · [350 − .4 · 50 + .4 · 0.05 · 320.388] = 320.388 1.05

In the remaining parts assume that bankruptcy costs are $20 per unit of gold. (e) If the firm issues $250 of risky debt, find the yield on the risky debt and the value of the unhedged firm. (10 points) Answer: Yield on risky debt satisfies: 250 · (1.05) = 0.5 · 180 + 0.5 · 250 · (1 + x). Therefore, x = 0.38.
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