Quantitative Risk Management

Homework #2
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.
(b) Find the value of the hedged unlevered firm.
(c) Find the value of the unhedged firm if it issues an optimally chosen quantity of safe debt.
(d) Find the value of the hedged firm that issues an optimally chosen quantity of safe debt.
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.
(f) If the unhedged firm chooses the face value of risky debt. Find the face value of debt, the yield
on the risky debt and the value of the unhedged firm.
Now suppose that the firm has issued $250 of risky debt and just before the debt matures the firm
has an investment opportunity (gamble) that costs zero, and with 99.5% probability will provide
a loss of $20, and with a 0.5% chance will provide a gain of $1000.
(g) Will the equity holders take this opportunity if their value has a) increased to $500, b) decreased
to $200? You will need to compute the yield on the risky debt assuming answers to a) and b) and
then verify them for this yield. Find the value of the firm with the possibility of the gamble.
(h) Suppose the firm were to hedge. Would equity holders take the gamble and what would be the
value of the firm?
2. Problems 9.18 and 9.19 in the textbook problems posted in Course Documents.
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