CH-23-25 Options and Corporate Finance: Extensions and Applications – Corporate Finance : Book Guide

This Document Contains Chapters 23 to 25 Chapter 23 options and corporate finance: Extensions and Applications 1. One of the purposes to give executive share options to CEOs (instead of cash) is to tie the performance of the firm’s share price with the compensation of the CEO. In this way, the CEO has an incentive to increase shareholder value. 2. Most businesses have the option to abandon under bad conditions and the option to expand under good conditions. 3. Undoubtedly, entrepreneurs do make projects seem better than they possibly are. However, there are several offsetting factors that mitigate against this activity. First, why would any rational entrepreneur increase investor expectations only for them to fail in the future? Also, funders recognise that this may be a problem and, as a result, they give funding in stages rather than a lump sum. This makes it easier to drop out of a project if it fails to show promise. 4. The binomial model is more appropriate because it allows the analyst to incorporate specific business decisions (such as abandonment or expansion) into the analysis. Black & Scholes does not do this. 5. Real option analysis is a very advanced capital budgeting technique that should be known to Finance specialists. However, many executives have other specialisms such as marketing, science, and management. As a result, the methodology can be difficult to intuitively understand even though it is clearly important. A general rule in any quantitative area is that parsimony is best. The same applies to corporate finance where payback period is still regularly used and real option analysis is not popular in many companies. 6. Virtually all projects have embedded options, which are ignored in NPV calculations and likely leads to undervaluation. 7. As the volatility increases, the value of an option increases. As the volatility of coal and oil increases, the option to burn either increases. However, if the prices of coal and oil are highly correlated, the value of the option would decline. If coal and oil prices both increase at the same time, the option to switch becomes less valuable since the company will likely save less money. 8. The advantage is that the value of the land may increase if you wait. Additionally, if you wait, the best use of the land other than sale may become more valuable. 9. The company has an option to sell the warehouse temporarily, which is an American put. 10. Traditional capital budgeting techniques are useful because they are simple and intuitive. However, their simplicity means that they can sometime miss drivers of value via future events that could arise from taking a project. Real options explicitly consider these and therefore can provide a more accurate reflection of a project’s value. This Document Contains Chapters 23 to 25 11. Insurance is a put option. Consider your homeowner’s insurance. If your house were to burn down, you would receive the value of the policy from your insurer. In essence, you are selling your burned house (“putting”) to the insurance company for the value of the policy (the strike price). 12. In a market with competitors, you must realize that the competitors have real options as well. The decisions made by these competitors may often change the payoffs for your company’s options. For example, the first entrant into a market can often be rewarded with a larger market share because the name can become synonymous with the product (think of Jacuzzi and Kleenex). Thus, the option to become the first entrant can be valuable. However, we must also consider that it may be better to be a later entrant in the market. Either way, we must realize that the competitors’ actions will affect our options as well. 13. a. The inputs to the Black-Scholes model are the current price of the underlying asset (S), the strike price of the option (K), the time to expiration of the option in fractions of a year (t), the variance of the underlying asset, and the continuously-compounded risk-free interest rate (R). Since these options were granted at-the-money, the strike price of each option is equal to the current value of one share, or £50. We can use Black-Scholes to solve for the option price. Doing so, we find: d1 = [ln(S/K) + (R + 2/2)(t) ] / (2t)1/2 d1 = [ln(£50/£50) + (.06 + .552/2)  (4)] / (.55  ) = .7682 d2 = .7682 – (.55  ) = –.3318 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. Doing so: N(d1) = N(0.7682) = 0.7788 N(d2) = N(–0.3318) = 0.3700 Now we can find the value of each option, which will be: C = SN(d1) – Ke-–RtN(d2) C = £50(0.7788) – (£50e–.06(4))(0.3700) C = £24.39 Since the option grant is for 20,000 options, the value of the grant is: Grant value = 20,000(£24.39) Grant value = £487,747.66 b. Because he is risk-neutral, you should recommend the alternative with the highest net present value. Since the expected value of the executive share option package is worth more than £450,000, he would prefer to be compensated with the options rather than with the immediate bonus. c. If he is risk-averse, he may or may not prefer the executive share option package to the immediate bonus. Even though the share option package has a higher net present value, he may not prefer it because it is undiversified. The fact that he cannot sell his options 4 4 prematurely makes it much more risky than the immediate bonus. Therefore, we cannot say which alternative he would prefer. 14. The total compensation package consists of an annual salary in addition to 10,000 at-the- money stock options. First, we will find the present value of the salary payments. Since the payments occur at the end of the year, the payments can be valued as a three-year annuity, which will be: PV(Salary) = €400,000(PVIFA9%,3) PV(Salary) = €1,012,517.87 Next, we can use the Black-Scholes model to determine the value of the executive share options. Doing so, we find: d1 = [ln(S/K) + (R + 2/2)(t) ] / (2t)1/2 d1 = [ln(€40/€40) + (.05 + .682/2)  (3)] / (.68  ) = .7163 d2 = .7163 – (.68  ) = –.4615 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. Doing so: N(d1) = N(0.7163) = 0.7631 N(d2) = N(–0.4615) = 0.3222 Now we can find the value of each option, which will be: C = SN(d1) – Ke-–RtN(d2) C = €40(0.7631) – (€40e–.05(3))(0.3222) C = €19.43 Since the option grant is for 10,000 options, the value of the grant is: Grant value = 10,000(€19.43) Grant value = €194,303.19 The total value of the contract is the sum of the present value of the salary, plus the option value, or: Contract value = €1,012,517.87 + €194,303.19 Contract value = €1,206,821.05 15. Since the contract is to sell up to 5 million litres, it is a call option, so we need to value the contract accordingly. Using the binomial mode, we will find the value of u and d, which are: u = e/ u = e.46/ u = 1.26 3 3 n 12/3 d = 1 / u d = 1 / 1.26 d = 0.79 This implies the percentage increase if gasoline increases will be 26 percent, and the percentage decrease if prices fall will be 21 percent. So, the price in three months with an up or down move will be: PUp = €1.65(1.26) PUp = €2.08 PDown = €1.65(0.79) PDown = €1.31 The option is worthless if the price decreases. If the price increases, the value of the option per gallon is: Value with price increase = €2.08 – €1.85 Value with price increase = €0.23 Next, we need to find the risk neutral probability of a price increase or decrease, which will be: .06 / (12/3) = 0.26(Probability of rise) + –0.21(1 – Probability of rise) Probability of rise = 0.4751 And the probability of a price decrease is: Probability of decrease = 1 – 0.4751 Probability of decrease = 0.5249 The contract will not be exercised if gasoline prices fall, so the value of the contract with a price decrease is zero. So, the value per gallon of the call option contract will be: C = [0.4751(€0.23) + 0.5249(0)] / [1 + 0.06/(12 / 3)] C = €0.106 This means the value of the entire contract is: Value of contract = €0.106(5,000,000) Value of contract = €530,516.17 16. When solving a question dealing with real options, begin by identifying the option-like features of the situation. First, since the company will exercise its option to build if the value of an office building rises, the right to build the office building is similar to a call option. Second, an office building would be worth £10 million today. This amount can be viewed as the current price of the underlying asset (S). Third, it will cost £10.5 million to construct such an office building. This amount can be viewed as the strike price of a call option (K), since it is the amount that the firm must pay in order to ‘exercise’ its right to erect an office building. Finally, since the firm’s right to build on the land lasts only 1 year, the time to expiration (t) of the real option is one year. We can use the two-state model to value the option to build on the land. First, we need to find the return of the land if the value rises or falls. The return will be: RRise = (£12,500,000 – 10,000,000) / £10,000,000 RRise = .25 or 25% RFall = (£8,000,000 – 10,000,000) / £10,000,000 RFall = –.20 or –20% Now we can find the risk-neutral probability of a rise in the value of the building as: Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) 0.025 = (ProbabilityRise)(0.25) + (1 – ProbabilityRise)(–.2) ProbabilityRise = 0.5 So, a probability of a fall is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 – 0.5 Value of an Office Building (in millions) Webber's Real Call Option with a Strike of £10.5 (in millions) Today 1 Year Today 1 Year 12.5 2 = max(0, 12.5-10.5) 10 ? 8 0 = max(0, 8-10.5) ProbabilityFall = 0.5 Using these risk-neutral probabilities, we can determine the expected payoff of the real option at expiration. Expected payoff at expiration = (.5)(£2,000,000) + (.50)(£0) Expected payoff at expiration = £1,000,000 Since this payoff will occur 1 year from now, it must be discounted at the risk-free rate in order to find its present value, which is: PV = (£1,000,000 / 1.025) PV = £975,609.8 Therefore, the right to build an office building over the next year is worth £975,609.8 today. Since the offer to purchase the land is less than the value of the real option to build, the company should not accept the offer. 17. When solving a question dealing with real options, begin by identifying the option-like features of the situation. First, the company can choose to negotiate and experience an uncertain price in three months or they can purchase the planes today with certainty for £9 million. In each case, the planes will be bought and so this is not a classical real options problem. Here, you are comparing the certain price of £9 million today with the expected present value of the plan cost. The expected cost of each plane in 3 months is 0.8(£2.7 million) + 0.2(3.6 million) = £2.88 million Given that the cost of a plane from purchasing now is £3 million per plane and £2.88 million per plane in 3 months (the PV of £2.88 million is even less), then the company should negotiate. 18. When solving a question dealing with real options, begin by identifying the option-like features of the situation. First, since the company will only choose to drill and excavate if the price of oil rises, the right to drill on the land can be viewed as a call option. Second, since the land contains 125,000 barrels of oil and the current price of oil is $55 per barrel, the current price of the underlying asset (S) to be used in the Black-Scholes model is: “Share” price = 125,000($55) “Share” price = $6,875,000 Third, since the company will not drill unless the price of oil in one year will compensate its excavation costs, these costs can be viewed as the real option’s strike price (K). Finally, since the winner of the auction has the right to drill for oil in one year, the real option can be viewed as having a time to expiration (t) of one year. Using the Black-Scholes model to determine the value of the option, we find: d1 = [ln(S/K) + (R + 2/2)(t) ] / (2t)1/2 d1 = [ln($6,875,000/$10,000,000) + (.065 + .502/2)  (1)] / (.50  ) = –.3694 d2 = –.3694 – (.50  ) = –.8694 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. Doing so: N(d1) = N(–0.3694) = 0.3559 N(d2) = N(–0.8694) = 0.1923 Now we can find the value of call option, which will be: C = SN(d1) – Ke-–RtN(d2) C = $6,875,000(0.3559) – ($10,000,000e–.065(1))(0.1923) C = $644,800.53 This is the maximum bid the company should be willing to make at auction. 19. When solving a question dealing with real options, begin by identifying the option-like features of the situation. First, since Sardano and Sons will only choose to manufacture the steel rods if the price of steel falls, the lease, which gives the firm the ability to manufacture steel, can be viewed as a put option. Second, since the firm will receive a fixed amount of money if it chooses to manufacture the rods: Amount received = 4,800 steel rods(£360 – 120) Amount received = £1,152,000 The amount received can be viewed as the put option’s strike price (K). Third, since the project requires Sardano and Sons to purchase 400 tons of steel and the current price of steel is £3,600 per ton, the current price of the underlying asset (S) to be used in the Black-Scholes formula is: “Share” price = 400 tons(£3,600 per ton) “Share” price = £1,440,000 Finally, since Sardano and Sons must decide whether to purchase the steel or not in six months, the firm’s real option to manufacture steel rods can be viewed as having a time to expiration (t) of six months. In order to calculate the value of this real put option, we can use the Black-Scholes model to determine the value of an otherwise identical call option then infer the value of the put using put-call parity. Using the Black-Scholes model to determine the value of the option, we find: 1 1 d1 = [ln(S/K) + (R + 2/2)(t) ] / (2t)1/2 d1 = [ln(£1,440,000/£1,152,000) + (.045 + .452/2)  (6/12)] / (.45  ) = .9311 d2 = .9311 – (.45  ) = .6129 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. Doing so: N(d1) = N(0.9311) = 0.8241 N(d2) = N(0.6129) = 0.7300 Now we can find the value of call option, which will be: C = SN(d1) – Ke-–RtN(d2) C = £1,440,000(0.8241) – (£1,152,000e–.045(6/12))(0.7300) C = £364,419.87 Now we can use put-call parity to find the price of the put option, which is: C = P + S – Ke–Rt £364,419.87 = P + £1,440,000 – £1,152,000e–.045(6/12) P = £50,789.29 This is the most the company should be willing to pay for the lease. 20. In one year, the company will abandon the technology if the demand is low since the value of abandonment is higher than the value of continuing operations. Since the company is selling the technology in this case, the option is a put option. The value of the put option in one year if demand is low will be: Value of put with low demand = €7,000,000 – €6,000,000 Value of put with low demand = €1,000,000 Of course, if demand is high, the company will not sell the technology, so the put will expire worthless. We can value the put with the binomial model. In one year, the percentage gain on the project if the demand is high will be: Percentage increase with high demand = (€10,000,000 – €9,100,000) / €9,100,000 Percentage increase with high demand = .0989 or 9.89% And the percentage decrease in the value of the technology with low demand is: Percentage decrease with low demand = (€6,000,000 – €9,100,000) / €9,100,000 Percentage decrease with low demand = –.3407 or –34.07% Now we can find the risk-neutral probability of a rise in the value of the technology as: Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall) 6 / 12 6 / 12 0.06 = (ProbabilityRise)(0.0989) + (1 – ProbabilityRise)(–.3407) ProbabilityRise = 0.9115 So, a probability of a fall is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 – 0.9115 ProbabilityFall = 0.0885 Using these risk-neutral probabilities, we can determine the expected payoff of the real option at expiration. With high demand, the option is worthless since the technology will not be sold, and the value of the technology with low demand is the €1 million we calculated previously. So, the value of the option to abandon is: Value of option to abandon = [(.9115)(0) + (.0885)(€1,000,000)] / (1 + .06) Value of option to abandon = €83,490.57 21. Using the binomial model, we will find the value of u and d, which are: u = e/ u = e.35/ u = 1.1063 d = 1 / u d = 1 / 1.1063 d = 0.9039 This implies the percentage increase if the share price increases will be 10.6 percent, and the percentage decrease if the share price falls will be 9.96 percent. The monthly interest rate is 0.0029. Next, we need to find the risk neutral probability of a price increase or decrease, which will be: 0.0029 = 0.1063(Probability of rise) - 0.0996(1 – Probability of rise) Probability of rise = 0.4949 And the probability of a price decrease is: Probability of decrease = 1 – 0.4949 Probability of decrease = 0.5051 The following figure shows the share price and put price for each possible move over the next two months: n 12 Share price (D) €16.16 Put price €0 Share price (B) €14.60 Put price €0 Share price (A) €13.20 Share price (E) €13.2 Put price €.80 Put price €.80 Share price (C) €11.93 Put price €2.07 Share price (F) €10.78 Put price €3.22 The share price at node (A) is the current share price. The share price at node (B) is from an up move, which means: Share price (B) = €13.20(1.1063) Share price (B) = €14.60 And the share price at node (D) is two up moves, or: Share price (D) = €13.20(1.1063)(1.1063) Share price (D) = €16.16 The share price at node (C) is from a down move, or: Share price (C) = €13.20(0.9039) Share price (C) = €11.93 And the share price at node (F) is two down moves, or: Share price (F) = €13.20(0.9039)(0.9039) Share price (F) = €10.78 Finally, the share price at node (E) is from an up move followed by a down move, or a down move followed by an up move. Since the binomial tree recombines, both calculations yield the same result, which is: Share price (E) = €13.20(1.1063)(0.9039) Share price (E) = €13.20 Now we can value the put option at the expiration nodes, namely (D), (E), and (F). The value of the put option at these nodes is the maximum of the strike price minus the stock price, or zero. So: Put value (D) = Max(€14 – €16.16, £0) Put value (D) = £0 Put value (E) = Max(€14 – €13.20, £0) Put value (E) = €.80 Put value (F) = Max(€14 – €10.78, £0) Put value (F) = €3.22 The value of the put at node (B) is the present value of the expected value. We find the expected value by using the value of the put at nodes (D) and (E) since those are the only two possible share prices after node (B). So, the value of the put at node (B) is: Put value (B) = [.4949(€0) + .5051(€.80)] / 1.0029 Put value (B) = €0.4029 Similarly, the value of the put at node (C) is the present value of the expected value of the put at nodes (E) and (F) since those are the only two possible share prices after node (C). So, the value of the put at node (C) is: Put value (C) = [.4949(€.8) + .5051(€3.2151)] / 1.0029 Put value (C) = €2.014 Notice, however, that the put option is an American option. Because it is an American option, it can be exercised any time prior to expiration. If the share price falls next month, the value of the put option if exercised is: Value if exercised = €14– €11.93 Value if exercised = €2.069 This is greater than the present value of waiting one month, so the option will be exercised early in one month if the share price falls. This is the value of the put option at node (C). Using this put value, we can now find the value of the put today, which is: Put value (A) = [.4949(€0) + .5051 (€2.069)] / 1.0029 Put value (A) = €1.041 22. Using the binomial model, we will find the value of u and d, which are: u = e/ u = e.25/ u = 1.19 n 2 d = 1 / u d = 1 / 1.19 d = 0.84 This implies the percentage increase if the building value increases will be 19 percent, and the percentage decrease if the building value falls will be 16 percent. The six month interest rate is: Six month interest rate = (1.0816)1/2 - 1 Six month interest rate = 0.04 Next, we need to find the risk neutral probability of a price increase or decrease, which will be: 0.04 = 0.19(Probability of rise) + –0.16(1 – Probability of rise) Probability of rise = 0.5685 And the probability of a price decrease is: Probability of decrease = 1 – 0.5685 Probability of decrease = 0.4315 The following figure shows the building price and call price for each possible move over the each of the six month steps: (values) Value (D) £63,488,674 Call price £16,488,674 Value pre-payment £53,701,406 Value post-payment (B) £53,201,406 Call price £9,012,812 Value (E) £44,581,017 Call price £0 Value (A) £45,000,000 Call price £4,926,460 Value (F) £44,403,318 Call price £0 Value pre-payment £37,708,510 Value post-payment (C) £37,208,510 Call price £0 Value (G) £31,179,499 Call price £0 First, we need to find the building value at every step along the binomial tree. The building value at node (A) is the current building value. The building value at node (B) is from an up move, which means: Building value (B) = £45,000,000(1.1934) Building value (B) = £53,701,406 At node (B), the accrued rent payment will be made, so the value of the building after the payment will be reduced by the amount of the payment, which means the building value at node (B) is: Building value (B) after payment = £53,701,406 – £500,000 Building value (B) after payment = £53,201,406 To find the building value at node (D), we multiply the after-payment building value at node (B) by the up move, or: Building value (D) = £53,201,406(1.1934) Building value (D) = £63,488,674 To find the building value at node (E), we multiply the after-payment building value at node (B) by the down move, or: Building value (E) = £53,201,406(0.8380) Building value (E) = £44,581,017 The building value at node (C) is from a down move, which means the building value will be: Building value (C) = £45,000,000(0.8380) Building value (C) = £37,708,510 At node (C), the accrued rent payment will be made, so the value of the building after the payment will be reduced by the amount of the payment, which means the building value at node (C) is: Building value (C) after payment = £37,708,510 – £500,000 Building value (C) after payment = £37,208,510 To find the building value at node (F), we multiply the after-payment building value at node (C) by the down move, or: Building value (F) = £37,208,510(1.1934) Building value (F) = £44,403,318 Finally, the building value at node (G) is from a down move from node (C), so the building value is: Building value (G) = £37,208,510(0.8380) Building value (G) = £31,179,499 Note that because of the accrued rent payment in six months, the binomial tree does not recombine during the next step. This occurs whenever a fixed payment is made during a binomial tree. For example, when using a binomial tree for an executive share option, a fixed dividend payment will mean that the tree does not recombine. With the expiration values, we can value the call option at the expiration nodes, namely (D), (E), (F), and (G). The value of the call option at these nodes is the maximum of the building value minus the strike price, or zero. We do not need to account for the value of the building after the accrued rent payments in this case since if the option is exercised, you will receive the rent payment. So: Call value (D) = Max(£63,488,674 – 47,000,000, £0) Call value (D) = £16,488,674 Call value (E) = Max(£44,581,017 – 47,000,000, £0) Call value (E) =£0 Call value (F) = Max(£44,403,318– 47,000,000, £0) Call value (F) = £0 Call value (G) = Max(£31,179,499 – 47,000,000, ¥0) Call value (G) = £0 The value of the call at node (B) is the present value of the expected value. We find the expected value by using the value of the call at nodes (D) and (E) since those are the only two possible building values after node (B). So, the value of the call at node (B) is: Call value (B) = [.5685(£16,488,674) + .4315(£0)] / 1.04 Call value (B) = £9,012,812 Note that you would not want to exercise the option early at node (B). The value of the option at node (B) is exercised if the value of the building including the accrued rent payment minus the strike price, or: Option value at node (B) if exercised = £53,201,406 – £45,000,000 Option value at node (B) if exercised = £8,201,406 Since this is less than the value of the option if it left “alive”, the option will not be exercised. With a call option, unless a large cash payment (dividend) is made, it is generally not valuable to exercise the call option early. The reason is that the potential gain is unlimited. In contrast, the potential gain on a put option is limited by the strike price, so it may be valuable to exercise an American put option early if it is deep in the money. We can value the call at node (C), which will be the present value of the expected value of the call at nodes (F) and (G) since those are the only two possible building values after node (C). Since neither node has a value greater than zero, obviously the value of the option at node (C) will also be zero. Now we need to find the value of the option today, which is: Call value (A) = [.5685(£9,012,812) + .4315(£0)] / 1.04 Call value (A) = £4,926,459 23. This question is for advanced students and it is expected that they will have read the paper and understood the main insights. Effectively, the viability of oil exploration projects is largely determined by the investment environment, exchange rate and oil price. Therefore, modelling their effect on the value of an investment within a real options framework makes sense. 24. Downsizing is a variant of the abandonment option. To value this, one should use a similar methodology to quantitative questions in this chapter. 25. This question allows students to develop their own real options example. It is for advanced students only. 26. Research & Development is like an abandonment and expansion option. If the value of the product or service under R&D is found to have value, it will be developed further. However, if it doesn’t have enough value, it will be abandoned. Further research can then be carried out on a successful project and this may lead to even more value in the future. Chapter 23 Case Study Exotic Cuisine Employee Share Options 1. We can use the Black-Scholes equation to value the employee share options. We need to use the risk-free rate that is the same as the maturity as the options. So, assuming expiration in three years, the value of the share options per share of equity is: d1 = [ln(£24.38/£50) + (.038 + .602/2)  3] / (.60  ) = –.0618 d2 = –.0618 – (.60  ) = –1.1011 N(d1) = .4753 N(d2) = .1354 Putting these values into the Black-Scholes model, we find the option value is: C = £24.38(.4753) – (£50e–.038(3))(.1354) = £5.55 Assuming expiration in ten years, the value of the share options per share of equity is: d1 = [ln(£24.38/£50) + (.044 + .602/2)  10] / (.60  ) = .8020 d2 = .8020 – (.60  ) = –1.0953 N(d1) = .7887 3 3 10 10 N(d2) = .1367 Putting these values into the Black-Scholes model, we find the option value is: C = £24.38(.7887) – (£50e–.044(10))(.1367) = £14.83 2. Whether you should exercise the options in three years depends on several factors. A primary factor is how long you plan to stay with the company. If you are planning to leave next week, you should exercise the options. A second factor is how the option exercise will affect your taxes. 3. The fact that the employee share options are not traded decreases the value of the options. A basic way to understand this is to realize that an option always has value since, ignoring the premium, it can never lose money. The right to sell an option also has to have value. If the right to sell is removed, it decreases the price of the option. 4. The rationale for employee stock options is to reduce agency costs by better aligning employee and shareholder interests. Vesting requires employees to work at a company for a specified time, which means the employee actions are actually part of the company performance. Vesting is also a “golden handcuff”. The employee is less likely to leave the company if in-the-money employee share options will vest soon. 5. The evaluation of the argument for or against repricing is open-ended. There are valid reasons on both sides of the discussion. Repricing increases the value of the employee share option. Consider an extreme: A company announces the employee share options will be worth a minimum of £10 at expiration. Since all values less than £10 are no longer possible, the value of the option increases. Employee share options increase in value if the share price increases; however, the share price can increase because of a general market increase. Consider a company of average risk in a bull market that has a large return for several years. The company’s equity should closely mirror the market return, even though most of the share price increase is due to the general market increase. Similarly, if the market falls, the company’s equity will likely fall as well, even if the company is doing well. A better method of valuing employee share options might be to reward employees for company performance in excess of the market performance, adjusted for the company’s level of risk. Chapter 24 Warrants and convertibles 1. Warrants are often called ‘equity kickers’ because they are issued together with privately placed bonds. 2. When a warrant is exercised, the number of shares increases. A call option is a contract between investors and does not affect the number of shares of the firm. 3. The dilution factor is the proportionate increase in the number of outstanding shares after the warrants have been exercised. This will cause the share price to fall. Clearly, the higher the dilution factor, the more the share price will fall. 4. If bonds are priced appropriately, there will be no advantage to either issuer or investor. Clearly the option has value to the holder and they would be willing to pay more for the convertible to have that option. 5. There are three potential reasons: 1) To match cash flows, that is, they issue securities whose cash flows match those of the firm. 2) To bypass assessing the risk of the company (risk synergy). For example, the risk of company start-ups is hard to evaluate. 3) To reduce agency costs associated with raising money by providing a package that reduces bondholder- stockholder conflicts. The trade-off theory would argue that convertible bonds reduce the tax burden of firms and therefore would increase the value of the firm. Clearly, the market would also incorporate the probability of conversion in the valuation and this would reduce the effective tax shield. With respect to the pecking order theory, companies would choose straight debt, then convertible debt, then equity. This is because convertibles are a hybrid of debt and equity. Finally, the market timing theory would argue that firms will issue convertible debt when it is being traded at a premium in the markets. 6. Young, small, high-growth firms cannot usually issue debt on reasonable terms due to high financial distress costs. However, the owners may be unwilling to issue equity if current share prices are too low. Convertible bonds can be viewed as a compromise situation, hence the name, backdoor equity. 7. Theoretically conversion should be forced as soon as the conversion value reaches the call price because other conversion policies will reduce shareholder value. If conversion is forced when conversion values are above the call price, bondholders will be allowed to exchange less valuable bonds for more valuable equity. In the opposite situation, shareholders are giving bondholders the excess value. 8. a. If the share price is less than the exercise price of the warrant at expiration, the warrant is worthless. Prior to expiration, however, the warrant will have value as long as there is some probability that the share price will rise above the exercise price in the time remaining until expiration. Therefore, if the share price is below the exercise price of the warrant, the lower bound on the price of the warrant is zero. b. If the share price is above the exercise price of the warrant, the warrant must be worth at least the difference between these two prices. If warrants were selling for less than the difference between the current share price and the exercise price, an investor could earn an arbitrage profit (i.e. an immediate cash inflow) by purchasing warrants, exercising them immediately, and selling the shares. c. If the warrant is selling for more than the shares, it would be cheaper to purchase the shares than to purchase the warrant, which gives its owner the right to buy the shares. Therefore, an upper bound on the price of any warrant is the firm’s current share price. 9. An increase in share price volatility increases the bond price. If the share price becomes more volatile, the conversion option on the equity becomes more valuable. 10. The two components of the value of a convertible bond are the straight bond value and the option value. An increase in interest rates decreases the straight value component of the convertible bond. Conversely, an increase in interest rates increases the value of the equity call option. Generally, the effect on the straight bond value will be much greater, so we would expect the bond value to fall, although not as much as the decrease in a comparable straight bond. With a putable bond, the interest rate will decrease the value of the put option because the present value of the exercise price will be lower. Together, an increase in interest rates would cause the putable bond value to fall. 11. When warrants are exercised, the number of shares outstanding increases. This results in the value of the firm being spread out over a larger number of shares, often leading to a decrease in value of each individual share. The decrease in the per-share price of a company’s stock due to a greater number of shares outstanding is known as dilution. 12. In an efficient capital market the difference between the market value of a convertible bond and the value of straight bond is the fair price investors pay for the call option that the convertible or the warrant provides. 13. It is not a bad decision. By having equity linked components to the issue, investors in the convertible bond can capture recovery opportunities from a potential increase in the value of the company. 14. Because the holder of the convertible has the option to wait and perhaps do better than what is implied by current share prices 15. No, the market price of the warrant will not equal zero. Since there is a chance that the market price of the equity will rise above the €23 per share exercise price before expiration, the warrant still has some value. Its market price will be greater than zero. As a practical matter, warrants that are far out-of-the-money may sell at 0, due to transaction costs. 16. The Risk Synergy explanation would appeal to a risk averse investor who wishes to hedge against future uncertainties in a risky company. 17. The conversion price is the par value divided by the conversion ratio. Assuming a €1,000 par value, Conversion price = €1,000 / 16.4 Conversion price = €60.98 18. The conversion ratio is the par value divided by the conversion price. Conversion ratio = Skr10,000 / 365 Conversion ratio = 17.40 19. The conversion value is the number of shares that the bond can be converted to times the share price. So, the conversion value for this bond is: Conversion value = (100) £9.20 Conversion value = £920 20. First, we need to find the conversion price, which is the par value divided by the conversion ratio, or: Conversion price = RMB100,000 / 420 Conversion price = RMB238.10 The conversion premium is the necessary increase in share price to make the bond convertible. So, the conversion premium is: Conversion premium = (RMB238.10- 124) /RMB124 Conversion premium = 0.9202 or 92.02% 21. a. The conversion ratio is defined as the number of shares that will be issued upon conversion. Since each bond is convertible into 2,425 shares of Hannon’s equity, the conversion ratio of the convertible bonds is 2,425. b. The conversion price is defined as the face amount of a convertible bond that the holder must surrender in order to receive a single share. Since the conversion ratio indicates that each bond is convertible into 2,425 shares, the conversion price is: Conversion price = £100,000 / 2,425 Conversion price = £41.24 c. The conversion premium is defined as the percentage difference between the conversion price of the convertible bonds and the current share price. So, the conversion premium is: Conversion premium = (£41.24 – 31.25) /£31.25 Conversion premium = 0.3196 or 31.96% d. The conversion value is defined as the amount that each convertible bond would be worth if it were immediately converted into equity. So, the conversion value is: Conversion value =£31.25(2,425) Conversion value = £75,781 e. If the share price increases by £2, the new conversion value will be: Conversion value =£33.25(2,425) Conversion value = £80,631 22. The total exercise price of each warrant is shares each warrant can purchase times the exercise price, which in this case will be: Exercise price = 3(SKr32) Exercise price = SKr96 Since the shares are selling at SKr39, the value of three shares is: Value of shares = 3(SKr39) Value of shares = SKr117 Therefore, the warrant effectively gives its owner the right to buy SKr117 worth of shares for SKr96. It follows that the minimum value of the warrant is the difference between these numbers, or: Minimum warrant value = SKr117 – SKr 96 Minimum warrant value = SKr21 If the warrant were selling for less than SKr21, an investor could earn an arbitrage profit by purchasing the warrant, exercising it immediately, and selling the shares. Here, the warrant holder pays less than SKr21 while receiving the SKr21 difference between the price of three shares and the exercise price. 23. Since a convertible bond gives its holder the right to a fixed payment plus the right to convert, it must be worth at least as much as its straight value. Therefore, if the market value of a convertible bond is less than its straight value, there is an opportunity to make an arbitrage profit by purchasing the bond and holding it until expiration. In Scenario A, the market value of the convertible bond is €1,000. Since this amount is greater than the convertible’s straight value (€900), Scenario A is feasible. In Scenario B, the market value of the convertible bond is €900. Since this amount is less than the convertible’s straight value (€950), Scenario B is not feasible. 24. a. Using the conversion price, we can determine the conversion ratio, which is: Conversion ratio = £1,000 / £20 Conversion ratio = 50 So, each bond can be exchanged for 50 shares. This means the conversion price of the bond is: Conversion value = 50(£18) Conversion value = £900 Therefore, the minimum price the bond should sell for is £900. Since the bond price is higher than this price, the bond is selling at the straight value, plus a premium for the conversion feature. b. A convertible bond gives its owner the right to convert his bond into a fixed number of shares. The market price of a convertible bond includes a premium over the value of immediate conversion that accounts for the possibility of increases in the share price of the firm before the maturity of the bond. If the share price rises, a convertible bondholder will convert and receive valuable shares of equity. If the share price decreases, the convertible bondholder holds the bond and retains his right to a fixed interest and principal payments. 25. You can convert or tender the bond (i.e., surrender the bond in exchange for the call price). If you convert, you get shares worth: 500 × 22 = 11,000 If you tender, you get 105,000 (105 percent of par). It’s a no-brainer: convert. 26. a. Since the share price is currently below the exercise price of the warrant, the lower bound on the price of the warrant is zero. If there is only a small probability that the firm’s share price will rise above the exercise price of the warrant, the warrant has little value. An upper bound on the price of the warrant is £33, the current share price. One would never pay more than £33 to receive the right to purchase a share of the company’s equity if the firm’s shares were only worth £33. b. If the equity is trading for £39 per share, the lower bound on the price of the warrant is £4, the difference between the current share price and the warrant’s exercise price. If warrants were selling for less than this amount, an investor could earn an arbitrage profit by purchasing warrants, exercising them immediately, and selling the shares. As always, the upper bound on the price of a warrant is the current share price. In this case, one would never pay more than £39 for the right to buy a single share of equity when he could purchase a share outright for £39. 27. a. The minimum convertible bond value is the greater of the conversion price or the straight bond price. To find the conversion price of the bond, we need to determine the conversion ratio, which is: Conversion ratio = €1,000 / €125 Conversion ratio = 8 So, each bond can be exchanged for 8 shares. This means the conversion price of the bond is: Conversion price = 8(€32) Conversion price = €256 And the straight bond value is: P = €70({1 – [1/(1 + .12)]30 } / .12) + €1,000[1 / (1 + .12)30] P = €597.24 So, the minimum price of the bond is €597.24 b. If the share price were growing by 15 percent per year forever, each share would be worth approximately €32(1.15)t after t years. Since each bond is convertible into 8 shares, the conversion value of the bond equals (€32)(8)(1.15)t after t years. In order to calculate the number of years that it will take for the conversion value to equal €1,100, set up the following equation: (€32)(8)(1.15)t = €1,100 t = 10.43 years 28. a. The percentage of the company shares currently owned by the CEO is: Percentage of shares = 500,000 / 4,000,000 Percentage of shares = .1250 or 12.50% b. The conversion price indicates that for every £20 of face value of convertible bonds outstanding, the company will be obligated to issue a new share upon conversion. So, the new number of shares the company must issue will be: New shares issued = £20,000,000 / £20 New shares issued = 1,000,000 So, the new number of shares outstanding will be: New total shares = 4,000,000 + 1,000,000 New total shares = 5,000,000 After the conversion, the percentage of company shares owned by the CEO will be: New percentage of shares = 500,000 / 5,000,000 New percentage of shares = .10 or 10% 29. a. Before the warrant was issued, the firm’s assets were worth: Value of assets = 5 oz of platinum(€1,000 per oz) Value of assets = €5,000 So, the price per share is: Price per share = €5,000 / 3 Price per share = €1,666.67 b. When the warrant was issued, the firm received €1,000, increasing the total value of the firm’s assets to €6,000 (= €5,000 + €1,000). If the three shares were the only outstanding claims on the firm’s assets, each share would be worth €2,000 (= €6,000 / 3 shares). However, since the warrant gives warrant holder a claim on the firm’s assets worth €1,000, the value of the firm’s assets available to shareholders is only €5,000 (= €6,000 – €1,000). Since there are three shares outstanding, the value per share remains at €1,666.67 (= €5,000 / 3 shares) after the warrant issue. Note that the firm uses warrant price of €1,000 to purchase one more ounce of platinum. c. If the price of platinum is €1,100 per ounce, the total value of the firm’s assets is €6,600 (= 6 oz of platinum × €1,100 per oz). If the warrant is not exercised, the value of the firm’s assets would remain at €6,600 and there would be three shares of equity outstanding. If the warrant is exercised, the firm would receive the warrant’s €2,100 strike price and issue one share of equity. The total value of the firm’s assets would increase to €8,700 (= €6,600 + €2,100). Since there would now be 4 shares outstanding and no warrants, the price per share would be €2,175.00 (= €8,700 / 4 shares). Since the €2,175 value of the share that the warrant holder will receive is greater than the €2,100 exercise price of the warrant, investors will expect the warrant to be exercised. The firm’s share price will reflect this information and will be priced at €2,175 per share on the warrant’s expiration date. 30. The value of the company’s assets is the combined value of the equity and the warrants. So, the value of the company’s assets before the warrants are exercised is: Company value = 10,000,000(£17) + 1,000,000(£3) Company value = £173,000,000 When the warrants are exercised, the value of the company will increase by the number of warrants times the exercise price, or: Value increase = 1,000,000(£15) Value increase = £15,000,000 So, the new value of the company is: New company value = £173,000,000 + 15,000,000 New company value = £188,000,000 This means the new share price is: New share price = £188,000,000 / 11,000,000 New share price = £17.09 Note that since the warrants were exercised when the price per warrant (£3) was above the exercise value of each warrant (£2 = £17 – 15), the stockholders gain and the warrant holders lose. 31. The straight bond value today is: Straight bond value = £68(PVIFA10%,25) + £1,000/1.1025 Straight bond value = £709.53 And the conversion value of the bond today is: Conversion value = £44.75(£1,000/£150) Conversion value = £298.33 We expect the bond to be called when the conversion value increases to £1,300, so we need to find the number of periods it will take for the current conversion value to reach the expected value at which the bond will be converted. Doing so, we find: £298.33(1.12)t = £1,300 t = 12.99 years. The bond will be called in 12.99 years. The bond value is the present value of the expected cash flows. The cash flows will be the annual coupon payments plus the conversion price. The present value of these cash flows is: Bond value = £68(PVIFA10%,12.99) + £1,300/1.1012.99 = £859.80 32. Step 1. Value the call option component of the warrant. Calculate d1 and d2. Next, we need to find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. 2 2 1 2 2 1 ln ( /2) €22 ln (.07 .04/2)1 .04 €20 .9266 .7266 S d R t t E d d t        =   + +         =   + +     = = − = N(d1) = N(0.9266) = 0.8229 N(d2) = N(0.7266) = 0.7662 According to the Black-Scholes formula, the price of a European call option (C) on a non- dividend paying equity is: C = SN(d1) – Ke–RtN(d2) C = (€22)(0.8229) – (20)e–0.07(1) (0.7662) C = €3.82 Step 2. The Value of the Superior Clamps AB warrant is thus: 33. To calculate the number of warrants that the company should issue in order to pay off $10 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 $10 million to find the number of warrants to be issued. So, the value of the liability today is: PV of liability = $10,000,000e–.06(6/12) PV of liability = $9,704,455.34 The company must raise this amount from the warrant issue. The value of 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 share price, which is: Share price = $160,000,000 / 1,500,000 Share price = $106.67 The value of a single warrant (W) equals: W = [# / (# + #W)] × Call(S, K) W = [1,500,000 / (1,500,000 + #W)] × Call($106.67, $95) Since the firm must raise $9,704,455 as a result of the warrant issue, we know #W × W must equal $9,704,455. Therefore, it can be stated that: 1 1 €3.82 # 500,000 1 1 # 4,000,000 0.8333(€3.82) €3.39 w w w = c =  +   +          = = $9,704,455 = (#W)(W) $9,704,455 = (#W)([1,500,000 / (1,500,000 +#W)] × Call($106.67, $95) Using the Black-Scholes formula to value the warrant, which is a call option, we find: d1 = [ln(S/K) + (R + ½2)(t) ] / (2t)1/2 d1 = [ln($106.67 / $95) + {.06 + ½(.652)}(6 / 12) ] / (.652 × 6 / 12)1/2 d1 = 0.5471 d2 = d1 – (2t)1/2 d2 = 0.5471 – (.652 × 6 / 12)1/2 d2 = 0.0875 Next, we need to find N(d1) and N(d2), the area under the normal curve from negative infinity to d1 and negative infinity to d2, respectively. N(d1) = N(0.5471) = 0.7078 N(d2) = N(0.0875) = 0.5349 According to the Black-Scholes formula, the price of a European call option (C) on a non- dividend paying equity is: C = SN(d1) – Ke–RtN(d2) C = ($106.67)(0.7078) – ($95)e–0.06(6/12)(0.5349) C = $26.19 Using this value in the equation above, we find the number of warrants the company must sell is: $9,704,455 = (#W)([1,500,000 / (1,500,000 +#W)] × Call($106.67, $95) $9,704,455 = (#W) [1,500,000 / (1,500,000 +#W)] × $26.19 #W = 492,006 Chapter 24 Case Study S&S Air’s Convertible Bond 1. Kartner is suggesting a conversion price of €25 because it means the share price will have to increase before the bondholders can benefit from the conversion. Even though the company is not publicly traded, the conversion price is important. First, the company may go public in the future. The case does not discuss whether the company has plans to go public, and if so, how soon it might go public. If the company does go public, the bondholders will have an active market for the equity if they convert. Second, even if the company does not go public, the bondholders could potentially have an equity interest in the company. This equity interest can be sold to the original owners, or someone else. The potential problem with private equity is that the market is not as liquid as the market for a public company. This illiquidity lowers the value of the stock. We can use the PE ratio to estimate the current share price. Doing so, we get: P/E = Price/EPS 12.50 = Price/€1.60 Price = €20.00 2. The floor value is the maximum of the conversion value and the intrinsic value. The conversion value of the bond is given as €800. The intrinsic value of the bond is: Intrinsic value = €60(PVIFA7%,20) + €1000(PVIF7%,20) Intrinsic value = €894.06 So, the floor value of the bond is €894.06. This means that if the company offered bonds with the same coupon rate and no conversion feature, they would be able to sell them for €894.06. 3. The conversion ratio the bonds is: Conversion ratio = €800/€25 = 32.00 So, each bond can be converted to 32 shares of equity. 4. The conversion premium is the increase in share price necessary to make the conversion option possible. Since the equity is currently selling for €20, and the conversion price is €25, the conversion premium of the bond is: Conversion premium = (€25 – 20) / €20 Conversion premium = 0.25 or 25% 5. The option value of a convertible bond is defined as the difference between the market value of the bond and the maximum of its straight value and conversion value. Since the bond is sold at par value, the option value is: Option value = Market value – Max[Straight value, Conversion value] Option value = €1,000 – Max[€893.22, €800 ] Option value = €1,000 – 893.22 Option value = €106.78 6. Hans’s argument is wrong because it ignores the fact that if the company does well, bondholders will be allowed to participate in the company’s success. If the share price rises to €40, bondholders are effectively allowed to purchase stock at the conversion price of €25. 7. Stephan’s argument is incorrect because the company is issuing debt with a lower coupon rate than they would have been able to otherwise. If the company does poorly, it will receive the benefit of a lower coupon rate. 8. Reconciling the two arguments requires that we remember our central goal: to increase the wealth of the existing shareholders. Thus, with 20–20 hindsight, we see that issuing convertible bonds will turn out to be worse than issuing straight bonds and better than issuing ordinary equity if the company prospers. The reason is that the prosperity has to be shared with bondholders after they convert. In contrast, if a company does poorly, issuing convertible bonds will turn out to be better than issuing straight bonds and worse than issuing ordinary equity. The reason is that the firm will have benefited from the lower coupon payments on the convertible bonds. Both of the arguments have a grain of truth; we just need to combine them. Ultimately, which option is better for the company will only be known in the future and will depend on the performance of the company. The table below illustrates this point. If the company does poorly If the company prospers Low share price and no conversion High share price and conversion Convertible bonds issued instead of straight bonds Cheap financing because coupon rate is lower (good outcome). Expensive financing because bonds are converted, which dilutes existing equity (bad outcome). Convertible bonds issued instead of common stock Expensive financing because firm could have issued ordinary shares at high prices (bad outcome). Cheap financing because firm issues equity at high prices when bonds are converted (good outcome). 9. The call provision allows the company to redeem the bonds at the company’s discretion. If the company’s equity appears to be poised to rise, the company can call the outstanding bonds. It could be possible that the bondholders would benefit from converting the bonds at that point, but it would eliminate the potential future gains to the bondholders. Chapter 25 Financial Risk Management with Derivatives 1. When a firm hedges with derivatives it is reducing the risk of its future cash flows. When it speculates with derivatives, it is taking a naked bet on how an asset’s value will change in the future. The fact that a risk management division makes a profit from derivatives is not necessarily bad. It could mean that its hedging activity has saved the firm’s valuable cash. However, a firm should not aim to make a profit solely with derivatives. 2. A forward contract is an agreement to buy or sell an asset for a specific value at some point in the future. An example of a forward contract is an agreement to buy new champagne at the end of the wine harvest for a specific price. This normally takes place each year between vintners and wine growers. 3. 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. 4. Hedging activity should take place when a firm expects a commodity or asset value to move against them in the future. Examples are exporters who think that the home currency will appreciate in the future, importers who think that their home currency will depreciate in the future and commodity producers who think that the commodity price will fall in the future. In each case, the fall (increase) in cash flows, resulting in the underlying asset value change, is offset by the increase (fall) in value resulting in the forward contract. 5. An interest rate futures contract pays out the difference between the actual future interest rate and the rate specified in the contract. Thus, if a bank is wishing to hedge against increases in interest rate (because it wishes to limit the interest on liabilities it needs to pay), it would buy one interest rate futures contract. Thus, if interest rates did increase, the extra interest payments that the bank makes will be offset by the cash flow generated by the futures contract. 6. Duration is the effective maturity of a bond or any asset with a stream of cash flows. To immunise your portfolio against interest rate movements, you would match the duration of your assets and liabilities. 7. A swap contract is an agreement between parties to exchange assets over several time intervals in the future. The swap contract is usually an exchange of cash flows, but not necessarily so. Since a forward contract is also an agreement between parties to exchange assets in the future, but at a single point in time, a swap can be viewed as a series of forward contracts with different settlement dates. The firm participating in the swap agreement is exposed to the default risk of the dealer, in that the dealer may not make the cash flow payments called for in the contract. The dealer faces the same risk from the contracting party, but can more easily hedge its default risk by entering into an offsetting swap agreement with another party. 8. By far the most important concern of risk managers is foreign exchange and interest rate risk. However, there are country and regional differences. For example, Latin American firms are also concerned with commodity risk. 9. Since the firm is buying futures, it wants to be able to purchase the lumber; therefore, it is a consumer. Since an increase in lumber prices would decrease the income of a lumber consumer, it has hedged its price risk by buying lumber futures. Losses in the spot market due to growth in lumber prices are offset by gains on the short position in lumber futures. 10. Writing call options gives the holder of the option the right to buy cocoa; therefore, the writer must be a supplier of cocoa. While a rise in cocoa prices is good for the producer, this risk is offset by the loss on the call options; if cocoa prices actually decline, the producer loses on the income but is insured by the call premium it has received. 11. The firm is hurt by declining energy prices, so it should sell energy futures contracts. The firm will not be able to create a perfect hedge because there are no wind energy derivative contracts. This means that they would have to hedge general energy contracts such as oil or gas. 12. 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. 13. 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 expire worthless. However, options hedges are costly because of the initial premium that must be paid. The futures contract can be entered into at no initial cost, with the disadvantage that the firm is locking in one price for cotton; it can’t profit from cotton price declines. 14. The put option on the bond gives the owner the right to sell the bond at the option’s strike price. If bond prices decline, the owner of the put option profits. However, since bond prices and interest rates move in opposite directions, if the put owner profits from a decline in bond prices, he would also profit from a rise in interest rates. Hence, a call option on interest rates is conceptually the same thing as a put option on bond prices. 15. The company would like to lock in the current low rates, or at least be protected from a rise in rates, allowing for the possibility of benefit if rates actually fall. The former hedge could be implemented by selling bond futures; the latter could be implemented by buying put options on bond prices or buying call options on interest rates. 16. The firm will borrow at a fixed rate of interest, receive fixed rate payments from the dealer as part of the swap agreement, and make floating rate payments back to the dealer; the net position of the firm is that it has effectively borrowed at floating rates. A swap may be preferable if there are no stock exchange derivatives available that exactly match the risk the firm wishes to hedge. 17. Transaction exposure is the short-term exposure due to uncertain prices in the near future. Economic exposure is the long-term exposure due to changes in overall economic conditions. There are a variety of instruments available to hedge transaction exposure, but very few long-term hedging instruments exist. It is much more difficult to hedge against economic exposure, since fundamental changes in the business generally must be made to offset long-run changes in the economic environment. 18. The risk is that the euro will strengthen relative to sterling, since the fixed sterling payments in the future will be worth fewer euros. Since this implies a decline in the €/£ exchange rate, the firm should sell sterling futures. The way the interest rate is quoted will affect the calculation of which currency is strengthening. 19. a) The number of futures contracts required is €150 million / €100,000 = 1,500 contracts b) If the firm is receiving euros, then you will want to short a euro futures contract with a futures price of £0.85/€. In September the spot rate is £0.95/€, so the firm will sell the euros at this exchange rate and go long in the December futures with the price of £0.92/€ to close out the contract. The total amount received by the firm is: Total amount received = £0.95(€150 million) + 1500(£0.85 - £.92) = £132,000,000 c) The effective exchange rate is 132/150 = £0.88/€ 20. Syco must have felt that the combination of fixed plus swap would result in an overall better rate. In other words, the variable rate available via a swap may have been more attractive than the rate available from issuing a floating-rate bond. 21. a) The spread differential is (9.85 – 6.35) – (1.5 – 0.5) = 2.5 percent. b) There are many combinations of swaps that could be made and the 2.5 percent can be shared in any combination between the two parties. The following is one outcome: Larss borrows at 6.35 percent from the bank and lends to Sousa at 8.35 percent Sousa borrows at LIBOR plus 1.5 percent and lends to Larss at 1 percent Larss gains (8.35 – 6.35) + (.5 – 1) = 1.5 percent Sousa gains (1 – 1.5) + (9.85 – 8.35) = 1 percent c) A currency swap is an exchange of receivables in one currency for receivables in another currency. Companies may wish to enter into this type of transaction if they wish to have access to one currency with explicitly borrowing or raising new funds. 22. Suntharee will be hurt if the baht loses value relative to sterling over the next eight months. Depreciation in the baht relative to sterling results in a decrease in the baht/£ exchange rate. Since Suntharee is hurt by a decrease in the exchange rate, she should take on a short position in baht per sterling futures contracts to hedge her risk. 23. The initial price is $1.1615 per pound and each contract is for 40,000 lbs, so the initial contract value is: Initial contract value = ($1.1615 per pound)(40,000 lbs per contract) = $46,460 And the final contract value is: Final contract value = ($1.20 per pound)(40,000 lbs per contract) = $48,000 You will have a profit on this futures position of: Profit on futures contract = $48,000 – $46,460 = $1,540. 24. The price quote is $1,648.6 per ounce and each contract is for 100 ounces, so the initial contract value is: Initial contract value = ($1,648.6 per oz.)(100 oz. per contract) = $164,860 At a final price of $1,500 per ounce, the value of the position is: Final contract value = ($1,500 per oz.)(100 oz. per contract) = $150,000 Since this is a short position, there is a net profit of: $164,860 – $150,000 = $14,860 per contract Since you sold five contracts, the net profit is: Net loss = 5($14,860) = $74,300 At a final price of $1,300 per ounce, the value of the position is: Final contract value = ($1,300 per oz.)(100 oz. per contract) = $130,000 Since this is a short position, there is a net gain of $164,860 – $130,000 = $34,860 Since you sold five contracts, the net gain is: Net gain = 5($34,860) = $174,300 With a short position, you make a profit when the price falls, and incur a loss when the price rises. 25. The call options give the manager the right to purchase oil futures contracts at a futures price of £120 per barrel. The manager will exercise the option if the price rises above £120. Selling put options obligates the manager to buy oil futures contracts at a futures price of £120 per barrel. The put holder will exercise the option if the price falls below £120. The payoffs per barrel are: Oil futures price: £115 £120 £125 £130 £135 Value of call option position: 0 0 5 10 15 Value of put option position: –5 0 0 0 0 Total value: –£5 £0 £5 £10 £15 The payoff profile is identical to that of a long position in a forward contract with a £120 strike price. 26. When you purchase the contracts, the initial value is: Initial value = 10(100)(€1,000) Initial value = €1,000,000 At the end of the first day, the value of your account is: Day 1 account value = 10(100)( €1,003) Day 1 account value = €1,003,000 So, your cash flow is: Day 1 cash flow = €1,003,000 – €1,000,000 Day 1 cash flow = €3,000 The day 2 account value is: Day 2 account value = 10(100)( €1,009) Day 2 account value = €1,009,000 So, your cash flow is: Day 2 cash flow = €1,009,000 – €1,003,000 Day 2 cash flow = €6,000 The day 3 account value is: Day 3 account value = 10(100)( €1,012) Day 3 account value = €1,012,000 So, your cash flow is: Day 3 cash flow = €1,012,000 – €1,009,000 Day 3 cash flow = €3,000 The day 4 account value is: Day 4 account value = 10(100)( €1,004) Day 4 account value = €1,004,000 So, your cash flow is: Day 4 cash flow = €1,004,000 – €1,012,000 Day 4 cash flow = -€8,000 You total profit for the transaction is: Profit = €1,004,000 – €1,000,000 Profit = €4,000 27. When you purchase the contracts, your cash outflow is: Cash outflow = 25(42,000)(€1.52) Cash outflow = €1,596,000 At the end of the first day, the value of your account is: Day 1 account value = 25(42,000)(€1.46) Day 1 account value = €1,533,000 Remember, on a short position you gain when the price declines, and lose when the price increase. So, your cash flow is: Day 1 cash flow = €1,596,000 – €1,533,000 Day 1 cash flow = €63,000 The day 2 account value is: Day 2 account value = 25(42,000)(€1.55) Day 2 account value = €1,627,500 So, your cash flow is: Day 2 cash flow = €1,533,000 – €1,627,500 Day 2 cash flow = –€94,500 The day 3 account value is: Day 3 account value = 25(42,000)(€1.59) Day 3 account value = €1,669,500 So, your cash flow is: Day 3 cash flow = €1,627,500 – €1,669,500 Day 3 cash flow = –€42,000 The day 4 account value is: Day 4 account value = 25(42,000)(€1.62) Day 4 account value = €1,701,000 So, your cash flow is: Day 4 cash flow = €1,669,500 – €1,701,000 Day 4 cash flow = –€31,500 You total profit for the transaction is: Profit = €1,596,000 – €1,701,000 Profit = –€105,000 28. The duration of a bond is the average time to payment of the bond’s cash flows, weighted by the ratio of the present value of each payment to the price of the bond. Since the bond is selling at par, the market interest rate must equal 9 percent, the annual coupon rate on the bond. The price of a bond selling at par is equal to its face value. Therefore, the price of this bond is €1,000. The relative value of each payment is the present value of the payment divided by the price of the bond. The contribution of each payment to the duration of the bond is the relative value of the payment multiplied by the amount of time (in years) until the payment occurs. So, the duration of the bond is: Year PV of payment Payment weight 1 €82.5688 0.0826 2 75.7512 0.1515 3 69.4965 0.2085 4 772.1835 3.0887 Price of bond €1,000 Duration = 3.5313 29. The duration of a portfolio of assets or liabilities is the weighted average of the duration of the portfolio’s individual items, weighted by their relative market values. a. The total market value of assets in billions is: Market value of assets = £28 + £580 + £390 + £84 + £315 Market value of assets = £1,397 So, the market value weight of each asset is: Government deposits = £28 / £1,397 = 0.020 Trade receivables = £580 / £1,397 = 0.415 Short-term loans = £390 / £1,397 = 0.279 Long-term loans = £84 / £1,397 = 0.060 Mortgages = £315 / £1,397 = 0.225 Since the duration of a group of assets is the weighted average of the durations of each individual asset in the group, the duration of assets is: Duration of assets = 0.020(0) + 0.415(1.20) + 0.279(2.65) + 0.060(7.25) + 0.225(16.25) Duration of assets = 5.33 years b. The total market value of liabilities in billions is: Market value of liabilities = £520 + £340 + £260 Market value of liabilities = £1,120 Note that equity is not included in this calculation since it is not a liability. So, the market value weight of each liability is: Chequing and savings deposits = £520 / £1,120 = 0.464 Certificates of deposit = £340 / £1,120 = 0.304 Long-term financing = £260 / £1,120 = 0.232 Since the duration of a group of liabilities is the weighted average of the durations of each individual liability in the group, the duration of liabilities is: Duration of liabilities = 0.464(0) + 0.304(2.60) + 0.232(17.8) Duration of liabilities = 4.92 years c. Since the duration of assets does not equal the duration of its liabilities, the bank is not immune from interest rate risk. 30. a. You’re concerned about a rise in cocoa prices, so you would buy July contracts. Since each contract is for 10 tonnes, the number of contracts you would need to buy is: Number of contracts to buy = 75,000/10 = 7,500 By doing so, you’re effectively locking in the settle price in July, 2012 of £1,468 per tonne of cocoa, or: Total price for 75,000 tonnes = 7,500(£1,468)(10) = £110,100,000. b. If the price of cocoa at expiration is £1,500 per tonne, the value of your futures position is: Value of future position = (£1,500 per tonne)(10 tonnes per contract)(7,500 contracts) = £112,500,000. Ignoring any transaction costs, your gain on the futures position will be: Gain = £112,500,000 – £110,100,000 = £2,400,000 While the price of the cocoa your firm needs has become £2,400,000 more expensive since April, your gain from the futures position has netted out this higher cost. 31. a. XYZ has a comparative advantage relative to ABC in borrowing at fixed interest rates, while ABC has a comparative advantage relative to XYZ in borrowing at floating interest rates. Since the spread between ABC and XYZ’s fixed rate costs is only 1%, while their differential is 2% in floating rate markets, there is an opportunity for a 3% total gain by entering into a fixed for floating rate swap agreement. b. If the swap dealer must capture 2% of the available gain, there is 1% left for ABC and XYZ. Any division of that gain is feasible; in an actual swap deal, the divisions would probably be negotiated by the dealer. One possible combination is ½% for ABC and ½% for XYZ: ABC Dealer EURIBOR +1% 10.5% XYZ EURIBOR +2.5% 10.0% Debt Market EURIB OR+1% Debt Market 10% 32. The duration of a liability is the average time to payment of the cash flows required to retire the liability, weighted by the ratio of the present value of each payment to the present value of all payments related to the liability. In order to determine the duration of a liability, first calculate the present value of all the payments required to retire it. Since the cost is €30,000 at the beginning of each year for four years, we can find the present value of each payment using the PV equation: PV = FV / (1 + R)t So, the PV each year of university is: Year 1 PV = €30,000 / (1.10)3 = €22,539.44 Year 2 PV = €30,000 / (1.10)4 = €20,490.40 Year 3 PV = €30,000 / (1.10)5 = €18,627.64 Year 4 PV = €30,000 / (1.10)6 = €16,934.22 So, the total PV of the university cost is: PV of university = €22,539.44 + €20,490.40 + €18,627.64 + €16,934.22 PV of college = €78,591.71 Now, we can set up the following table to calculate the liability’s duration. The relative value of each payment is the present value of the payment divided by the present value of the entire liability. The contribution of each payment to the duration of the entire liability is the relative value of the payment multiplied by the amount of time (in years) until the payment occurs. Year PV of payment Relative value Payment weight 3 €22,539.44 0.286792 0.860375 4 €20,490.4 0.26072 1.042879 5 €18,627.64 0.237018 1.185089 6 €16,934.22 0.215471 1.292825 PV of university €78,591.71 Duration = 4.381168 years 33. The duration of a bond is the average time to payment of the bond’s cash flows, weighted by the ratio of the present value of each payment to the price of the bond. We need to find the present value of the bond’s payments at the market rate. The relative value of each payment is the present value of the payment divided by the price of the bond. The contribution of each payment to the duration of the bond is the relative value of the payment multiplied by the amount of time (in years) until the payment occurs. Since this bond has semi-annual coupons, the years will include half-years. So, the duration of the bond is: Period / Year Net Cash Flow Present Value of NCF Duration 0.5 €40 €38.6695 0.0189 1 €40 €37.3832 0.0366 1.5 €40 €36.1397 0.0531 2 €1040 €908.3763 1.7801 The duration of the bond is 1.889 years. 34. Let R equal the interest rate change between the initiation of the contract and the delivery of the asset. Cash flows from Strategy 1: Today 1 Year Purchase silver –S0 0 Borrow +S0 –S0(1 + R) Total cash flow 0 –S0(1 + R) Cash flows from Strategy 2: Today 1 Year Purchase silver 0 –F Total cash flow 0 –F Notice that each strategy results in the ownership of silver in one year for no cash outflow today. Since the payoffs from both the strategies are identical, the two strategies must cost the same in order to preclude arbitrage. The forward price (F) of a contract on an asset with no carrying costs or convenience value equals the current spot price of the asset (S0) multiplied by 1 plus the appropriate interest rate change between the initiation of the contract and the delivery date of the asset. 35. a. The forward price of an asset with no carrying costs or convenience value is: Forward price = S0(1 + R)T Since you will receive the bond’s face value of £100,000 in 11 years and the 11 year spot interest rate is currently 9 percent, the current price of the bond is: Current bond price = £100,000 / (1.09)11 Current bond price = £38,753 Since the forward contract defers delivery of the bond for one year, the appropriate interest rate to use in the forward pricing equation is the one-year spot interest rate of 5 percent: Forward price = £38,753(1.05) Forward price = £40,691 b. If both the 1-year and 11-year spot interest rates unexpectedly shift downward by 2 percent, the appropriate interest rates to use when pricing the bond is 7 percent, and the appropriate interest rate to use in the forward pricing equation is 3 percent. Given these changes, the new price of the bond will be: New bond price = £100,000 / (1.07)11 New bond price = £47,509 And the new forward price of the contract is: Forward price = £47,509(1.03) Forward price = £48,935 36. a. The forward price of an asset with no carrying costs or convenience value is: Forward price = S0(1 + R)T Since you will receive the bond’s face value of £100,000 in 18 months, we can find the price of the bond today, which will be: Current bond price = £100,000 / (1.0879)3/2 Current bond price = £88,129 Since the forward contract defers delivery of the bond for six months, the appropriate interest rate to use in the forward pricing equation is the six month EAR, so the forward price will be: Forward price = £88,129(1.0742)1/2 Forward price = £91,340 b. It is important to remember that 100 basis points equals 1 percent and one basis point equals 0.01%. Therefore, if all rates increase by 30 basis points, each rate increases by 0.003. So, the new price of the bond today will be: New bond price = £100,000 / (1 + .0879 + .003)3/2 New bond price = £87,765 Since the forward contract defers delivery of the bond for six months, the appropriate interest rate to use in the forward pricing equation is the six month EAR, increased by the interest rate change. So, the new forward price will be: Forward price = £87,765(1 + .0742 + .003)1/2 Forward price = £91,090 37. The financial engineer can replicate the payoffs of owning a put option by selling a forward contract and buying a call. For example, suppose the forward contract has a settle price of £50 and the exercise price of the call is also £50. The payoffs below show that the position is the same as owning a put with an exercise price of £50: Price of coal: £40 £45 £50 £55 £60 Value of call option position: 0 0 0 5 10 Value of forward position: 10 5 0 –5 –10 Total value: £10 £5 £0 £0 £0 Value of put position: £10 £5 £0 £0 £0 The payoffs for the combined position are exactly the same as those of owning a put. This means that, in general, the relationship between puts, calls, and forwards must be such that the cost of the two strategies will be the same, or an arbitrage opportunity exists. In general, given any two of the instruments, the third can be synthesized. Chapter 25 Case Study MCAFEE Mortgages Ltd 1. Finn’s mortgage payments form a 25-year annuity with monthly payments, discounted at the long-term interest rate of 8 percent. We can solve for the payment amount so that the present value of the annuity equals £500,000, the amount of principal that he plans to borrow. The monthly mortgage payment will be: £500,000 = C(PVIFA8%/12,300) C = £3,859.08 2. The most significant risk that she faces is interest rate risk. If the current market rate of interest rises between today and the date the mortgage is sold, the fair value of the mortgage will decrease, and Ian will only be willing to purchase the mortgage for a price less than £500,000. If this is the case, she will not be able to loan Finn the full £500,000 promised. 3. Treasury bond prices have an inverse relationship with interest rates. As interest rates rise, Treasury bonds become less valuable; as interest rates fall, Treasury bonds become more valuable. Since Jennifer will be hurt when interest rates rise, she is also hurt when Treasury bonds decrease in value. In order to protect herself from decreases in the price of Treasury bonds, she should take a short position in Treasury bond futures to hedge this interest rate risk. Since three- month Treasury bond futures contracts are available and each contract is for £100,000 of Treasury bonds, she would take a short position in five 3-month Treasury bond futures contracts in order to hedge her £500,000 exposure to changes in the market interest rate over the next three months 4. a. If the market interest rate is 9 percent on the date that Jennifer meets with Ian, the fair value of the mortgage is the present value of an annuity that makes monthly payments of £3,859.08 for 25 years, discounted at 9 percent, or: Mortgage value = £3,859.08(PVIFA9%/12,300) Mortgage value = £459,854.36 b. An increase in the interest rate will cause the value of the T-bond futures contracts to decrease. The long position will lose and the short position will gain. Since Jennifer is short in the futures, the futures gain will offset the loss in value of the mortgage. 5. a. If the market interest rate is 7 percent on the date that Jennifer meets with Ian, the fair value of the mortgage is the present value of an annuity that makes monthly payments of £3,859.08 for 25 years, discounted at 7 percent, or: Mortgage value = £3,859.08(PVIFA7%/12,300) Mortgage value = £546,009.43 b. An decrease in the interest rate will cause the value of the Treasury bond futures contracts to increase. The long position will gain and the short position will lose. Since Jennifer is short in the futures, the futures lose will be offset by the gain in value of the mortgage. 6. The biggest risk is that the hedge is not a perfect hedge. If interest rates change, the fact that Treasury bond interest is semiannual, while the mortgage payments are monthly, may affect the relative value of the two. Additionally, while a change in one of the interest rates will likely coincide with a change in the other interest rate, the change does not have to be the same. For example, the Treasury rate could increase 20 basis points, and the mortgage rates could increase by 40 basis points. The fact that this is not a perfect hedge simply means that the gain/loss from the futures contracts may not exactly offset the loss/gain in the mortgage. We would expect, especially given the short-term nature of the hedge, that the loss in one instrument would be similar to the gain in the other instrument. Solution Manual for Corporate Finance David Hillier, Stephen Ross, Randolph Westerfield, Jeffrey Jaffe, Bradford Jordan 9780077139148