51 per ounce and each contract is for 5000 ounces so

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Unformatted text preview: uals: W = (.8889) × Call(S = $17.50, K = $20) W = (.8889)($2.24) W = $1.99 16. To calculate the number of warrants that the company should issue in order to pay off $18 million in six months, we can use the Black-Scholes model to find the price of a single warrant, then divide this amount into the present value of $18 million to find the number of warrants to be issued. So, the value of the liability today is: PV of liability = $18,000,000e–.06(6/12) PV of liability = $17,468,019.60 The company must raise this amount from the warrant issue. The value of the company’s assets will increase by the amount of the warrant issue after the issue, but this increase in value from the warrant issue is exactly offset by the bond issue. Since the cash inflow from the warrants offsets the firm’s debt, the value of the warrants will be exactly the same as if the cash from the warrants were used to immediately pay off the debt. We can use the market value of the company’s assets to find the current stock price, which is: Stock price = $210,000,000 / 3,200,000 Stock price = $65.63 485 CHAPTER 24 B-486 The value of a single warrant (W) equals: W = [# / (# + #W)] × Call(S, K) W = [3,200,000 / (3,200,000 + #W)] × Call($65.63, $75) Since the firm must raise $17,468,019.60 as a result of the warrant issue, we know #W × W must equal $17,468,019.60. Therefore, it can be stated that: $17,468,019.60 = (#W)(W) $17,468,019.60 = (#W)([3,200,000 / (3,200,000 +#W)] × Call($65.63, $75) Using the Black-Scholes formula to value the warrant, which is a call option, we find: d1 = [ln(S/K) + (R + ½σ 2)(t) ] / (σ 2t)1/2 d1 = [ln($65.63 / $75) + {.06 + ½(.502)}(6 / 12) ] / (.502 × 6 / 12)1/2 d1 = –0.1161 d2 = d1 – (σ 2t)1/2 d2 = –0.1161 – (.502 × 6 / 12)1/2 d2 = –0.4696 Next, we need to find N(d1) and N(d2), the area under the normal curve from negative infinity to d 1 and negative infinity to d2, respectively. N(d1) = N(–0.1161) = 0.4538 N(d2) = N(–0.4696) = 0.3193 According to the Black-Scholes formula, the price of a European call option (C) on a non-dividend paying common stock is: C = SN(d1) – Ke–RtN(d2) C = ($65.63)(0.4538) – ($75)e–0.06(6/12)(0.3193) C = $6.54 Using this value in the equation above, we find the number of warrants the company must sell is: $17,468,019.60 = (#W)([3,200,000 / (3,200,000 +#W)] × Call($65.63, $75) $17,468,019.60 = (#W) [3,200,000 / (3,200,000 +#W)] × $6.54 #W = 16,156,877 486 CHAPTER 25 DERIVATIVES AND HEDGING RISK Answers to Concepts Review and Critical Thinking Questions 1. Since the firm is selling futures, it wants to be able to deliver the lumber; therefore, it is a supplier. Since a decline in lumber prices would reduce the income of a lumber supplier, it has hedged its price risk by selling lumber futures. Losses in the spot market due to a fall in lumber prices are offset by gains on the short position in lumber futures. Buying call options gives the firm the right to purchase pork bellies; therefore, it must be a consumer of pork bellies. While a rise in pork belly prices is bad for the consumer, this risk is offset by the gain on the call options; if pork belly prices actually decline, the consumer enjoys lower costs, while the call option expires worthless. Forward contracts are usually designed by the parties involved for their specific needs and are rarely sold in the secondary market, so forwards are somewhat customized financial contracts. All gains and losses on the forward position are settled at the maturity date. Futures contracts are standardized to facilitate liquidity and to allow them to be traded on organized futures exchanges. Gains and losses on futures are marked-to-market daily. Default risk is greatly reduced with futures since the exchange acts as an intermediary between the two parties, guaranteeing performance. Default risk is also reduced because the daily settlement procedure keeps large loss positions from accumulating. You might prefer to use forwards instead of futures if your hedging needs were different from the standard contract size and maturity dates offered by the futures contract. The firm is hurt by declining oil prices, so it should sell oil futures contracts. The firm may not be able to create a perfect hedge because the quantity of oil it needs to hedge doesn’t match the standard contract size on crude oil futures, or perhaps the exact settlement date the company requires isn’t available on these futures. Also, the firm may produce a different grade of crude oil than that specified for delivery in the futures contract. The firm is directly exposed to fluctuations in the price of natural gas since it is a natural gas user. In addition, the firm is indirectly exposed to fluctuations in the price of oil. If oil becomes less expensive relative to natural gas, its competitors will enjoy a cost advantage relative to the firm. Buying the call options is a form of insurance policy for the firm. If cotton prices rise, the firm is protected by the call, while if prices actually decline, they can just allow the call to expir...
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This note was uploaded on 07/10/2010 for the course FIN 6301 taught by Professor Eshmalwi during the Spring '10 term at University of Texas-Tyler.

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