# Quantitative Risk Management

Topics: Finance, Money, Forward contract Pages: 3 (1136 words) Published: May 2, 2013
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....