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Quantitative Risk Management

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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 · …show more content…

If it did not take the gamble the shareholders value would be zero. So shareholders would take the gamble. If the value of the firm rises to 500, and it does not take the gamble, the value of the shareholders would be 500 − 250 · (1.456) − .4 · (200 − 250 · .456) = 101.6. If they took the gamble, then because they will be able to pay back the debt whether they win or lose the gamble. Therefore, their value with the gamble would be 101.6 - 14.9 = 86.7, so they would not take the gamble. The key intuition is simply that now they get both the gain and the loss from the gamble, while when the initial value was 200, the shareholders only got the gain from the gamble. The value of the firm is now Value of Firm = 1 [.5 · [.995 · 160 + .005 · (1200 − .4 · (900 − 250 · .456))] + .5 · (500 − .4 · (200 − 250 · .456))] 1.05

= 299.629. Notice, that the value of the firm with debt overhang is lower than the same problem without debt overhang in part (e). (h) Suppose the firm were to hedge. Would equity holders take the gamble and what would be the value of the firm? (10 points) Answer: If the firm were to hedge by selling gold in a forward contract, it would get a sure cash flow of $350. Then it would not not take the gamble, because even if it lost, it could still make the debt payment of 250 · (1.05) = 262.5 and taxes of .4(50 − 250 · .05) = 15 (we assume the debt is safe, and therefore has a yield of 0.05, and verify that it can make the payment).

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