This Document Contains Chapters 21 to 22 Chapter 21 leasing and off-balance sheet financing 1. There are two main types of lease financing: operating leases and capital or financial leases. The main characteristics of an operating lease are 1) Operating leases are usually not fully amortized. 2) Operating leases usually require the lessor to maintain and insure the leased assets. 3) Operating leases tend to have a cancellation option. The main characteristics of a financial lease are 1) Financial leases do not provide for maintenance or service by the lessor. 2) Financial leases are fully amortized. 3) The lessee usually has a right to renew the lease on expiration. 4) Generally, financial leases cannot be cancelled. In other words, the lessee must make all payments or face the risk of bankruptcy. There is another form of leasing called a sale and leaseback. In a sale and leaseback two things happen. One, the lessee receives cash from the sale of the asset and two, the lessee makes periodic lease payments, thereby retaining use of the asset. Finally, a firm can take out a leveraged lease. In a leveraged lease, the lessee uses the assets and makes periodic lease payments. The lessor purchases the assets, delivers them to the lessee, and collects the lease payments. The lenders supply the remaining financing and receive interest payments from the lessor. 2. The term off-balance sheet funding does not adequately describe leasing activity because most leases are shown in a company’s financial statements. Many firms use leasing as a form of financing and prior to 2005, accounting standards in most European countries did not require leasing activity to be reported in a firm’s balance sheet (or statement of financial position). Nowadays, financial leases will be reported on the balance sheet and operating leases will be reported in a firm’s income statement. 3. If a firm was unable to attract financing to purchase an asset and buying was not an option, you should not include the buy decision in your leasing analysis. 4. In a world with taxes, a firm should discount leasing cash flows at the after-tax risk free rate. 5. WACC is not appropriate in a lease-versus-buy decision because the cash flows in a lease are more like debt service cash flows which are of considerably less risk than a firm’s WACC. 6. As long as (1) both parties are subject to the same interest and tax rates and (2) transaction costs are ignored, there can be no leasing deal that benefits both parties. An example is given in Section 21.6. 7. Firms undertake leasing for a number of reasons. These are 1) tax differences may result in a profitable transaction for two parties, 2) the lease contract may reduce certain types of uncertainty, 3) transaction costs may be higher when buying an asset than leasing it. There are also a number of bad reasons for leasing an asset. For more information, see Section 21.7 in the text. 8. The main reason for differences across firms in leasing activity is the availability of capital for purchasing new assets. Small firms have less access to capital and so leasing may prove attractive. This Document Contains Chapters 21 to 22 9. The accounting treatment of operating and finance leases is very different. In an operating lease, lease payments are treated as expenses and appear in a firm’s income statement. In contrast, with a finance lease, the leased asset appears on the balance sheet and is depreciated in the same way as other assets. The value to be recorded in the balance sheet must be the fair or realizable value of the asset or, if lower, the present value of the lease payments. This means that, for all intents and purposes, assets that are funded by a finance lease are regarded in the exact same way as normal assets in a company without the need to undertake a substantial capital expenditure to purchase the asset. 10. To follow 11. Two reasons why a firm may choose to undertake a sale and leaseback is that it has assets but little cash and can raise funding by selling but still retain use of the asset. Another reason is that the firm may follow Islamic principles and thus does not want to raise cash through debt financing. A sale and leaseback transaction provides similar cash flow streams without the perception that interest is being paid. A sale and leaseback is a useful tool for financially distressed firms because it can realise cash during a much needed period. 12. We will calculate cash flows from the depreciation tax shield first. The depreciation tax shield is: Year Starting Value Depreciation Cumulative Depreciation Residual Value Tax Shield 20% 23% 1 £22,000,000 £4,400,000 £4,400,000 £17,600,000 £1,012,000 2 £17,600,000 £3,520,000 £7,920,000 £14,080,000 £809,600 3 £14,080,000 £2,816,000 £10,736,000 £11,264,000 £647,680 4 £11,264,000 £11,264,000 £22,000,000 £0 £2,590,720 The after-tax cost of the lease payments will be: After-tax lease payment = (£6,000,000)(1 – .23) = £4,620,000 So, the total cash flows from leasing are: Year After Tax Lease Cash Flows Depreciation Tax Shield OCF 1 £4,620,000 £1,012,000 £5,632,000 2 £4,620,000 £809,600 £5,429,600 3 £4,620,000 £647,680 £5,267,680 4 £4,620,000 £2,590,720 £7,210,720 The after-tax cost of debt is: After-tax debt cost = .08(1 – .23) = .0616 Using all of this information, we can calculate the NAL as: Year OCF Investment Net Cash Flow PV 0 £22,000,000 £22,000,000 £22,000,000 1 -£5,632,000 -£5,632,000 -£5,305,200 2 -£5,429,600 -£5,429,600 -£4,817,769 3 -£5,267,680 -£5,267,680 -£4,402,878 4 -£7,210,720 -£7,210,720 -£5,677,210 NAL = £1,796,942 The NAL is positive so you should lease. 13. If we assume the lessor has the same cost of debt and the same tax rate, the NAL to the lessor is the negative of our company’s NAL, so: NAL = – £1,790,942 14. To find the maximum lease payment that would satisfy both the lessor and the lessee, we need to find the payment that makes the NAL equal to zero. Using solver, we find: Lease Payment = £6,675,952.65 To see this, note the following analysis. The after-tax cost of the lease payments will be: After-tax lease payment = (£6,675,952)(1 – .23) = £5,140,484 Year After-Tax Lease Cash Flows Depreciation Tax Shield OCF 1 £5,140,484 £1,012,000 £6,152,484 2 £5,140,484 £809,600 £5,950,084 3 £5,140,484 £647,680 £5,788,164 4 £5,140,484 £2,590,720 £7,731,204 Year OCF Investment Net Cash Flow PV 0 £22,000,000 £22,000,000 £22,000,000 1 -£6,152,484 -£6,152,484 -£5,795,482 2 -£5,950,084 -£5,950,084 -£5,279,603 3 -£5,788,164 -£5,788,164 -£4,837,913 4 -£7,731,204 -£7,731,204 -£6,087,002 The sum of the present values is £0. 15. If the tax rate is zero, there is no depreciation tax shield foregone. Also, the after-tax lease payment is the same as the pre-tax payment, and the after-tax cost of debt is the same as the pre-tax cost. So: Cost of debt = .08 Annual cost of leasing = leasing payment = £6,000,000 The NAL to leasing with these assumptions is: Year OCF Investment Net Cash Flow PV 0 £22,000,000 £22,000,000 £22,000,000 1 -£6,000,000 -£6,000,000 -£5,555,556 2 -£6,000,000 -£6,000,000 -£5,144,033 3 -£6,000,000 -£6,000,000 -£4,762,993 4 -£6,000,000 -£6,000,000 -£4,410,179 NAL = £2,127,239 16. We already calculated the breakeven lease payment for the lessor in Problem 14. Since the assumption about the lessor concerning the tax rate have not changed. So, the lessor breaks even with a payment of £6,675,952.65. For the lessee, we need to calculate the breakeven lease payment which results in a zero NAL. Using the assumptions in Problem 15, we find: NAL = 0 = £22,000,000 – PMT(PVIFA8%,4) PMT = £2,094,237.60 So, the range of lease payments that would satisfy both the lessee and the lessor are: Total payment range = £2,094,237 to £2,127,239. 17. We will calculate cash flows from the depreciation tax shield first. The depreciation tax shield is: Year Starting Value Depreciation Cumulative Depreciation Residual Value Tax Shield 20% 28% 1 NKr3,500,000 NKr700,000 NKr700,000 NKr2,800,000 NKr196,000 2 2,800,000 560,000 1,260,000 2,240,000 156,800 3 2,240,000 448,000 1,708,000 1,792,000 125,440 4 1,792,000 358,400 2,066,400 1,433,600 100,352 5 1,433,600 286,720 2,353,120 1,146,880 80,282 After-tax lease payment = (NKr942,000)(1 – .28) = NKr678,240 So, the total cash flows from leasing are: Year After-Tax Lease Cash Flows Depreciation Tax Shield OCF 1 NKr678,240 NKr196,000 NKr874,240 2 678,240 156,800 835,040 3 678,240 125,440 803,680 4 678,240 100,352 778,592 5 678,240 80,282 758,522 The after-tax cost of debt is: After-tax debt cost = .09(1 – .28) = .0648 Using all of this information, we can calculate the NAL as: Year OCF Investment Net Cash Flow PV 0 NKr3,500,000 NKr3,500,000 NKr3,500,000 1 -874,240 -874,240 -821,037 2 -835,040 -835,040 -736,497 3 -803,680 -803,680 -665,701 4 -778,592 -778,592 -605,672 5 -758,522 -80,282 -838,803 -612,802 NAL = NKr58,291 The NAL is positive so you should lease. 18. The pre-tax cost savings are not relevant to the lease versus buy decision, since the firm will definitely use the equipment and realize the savings regardless of the financing choice made. The depreciation tax shield is: Year Starting Value Depreciation Cumulative Depreciation Residual Value Tax Shield 20% 28% 1 £6,000,000 £1,200,000 £1,200,000 £4,800,000 £336,000 2 £4,800,000 £960,000 £2,160,000 £3,840,000 £268,800 3 £3,840,000 £768,000 £2,928,000 £3,072,000 £215,040 4 £3,072,000 £614,400 £3,542,400 £2,457,600 £172,032 5 £2,457,600 £2,457,600 £6,000,000 £0 £688,128 And the after-tax lease payment is: After-tax lease payment = £1,400,000(1 – .28) = £1,008,000 The after-tax cost of debt is: After-tax debt cost = .09(1 – .28) = .0648 or 6.48% With these cash flows, the NAL is: Year After-Tax Lease Cash Flows Depreciation Tax Shield OCF 0 £1,008,000 £1,008,000 1 £1,008,000 £336,000 £1,344,000 2 £1,008,000 £268,800 £1,276,800 3 £1,008,000 £215,040 £1,223,040 4 £1,008,000 £172,032 £1,180,032 5 £688,128 £688,128 Year OCF Investment Net Cash Flow PV 0 -£1,008,000 £6,000,000 £4,992,000 £4,992,000 1 -£1,344,000 -£1,344,000 -£1,262,209 2 -£1,276,800 -£1,276,800 -£1,126,125 3 -£1,223,040 -£1,223,040 -£1,013,063 4 -£1,180,032 -£1,180,032 -£917,955 5 -£688,128 -£688,128 -£502,723 NAL = £169,924 The equipment should be leased. To find the maximum payment, we find where the NAL is equal to zero, and solve for the payment. We do this using solver. The maximum pre-tax lease payment is £1,453,306. A break down of the analysis using this lease payment is given below: Year After-Tax Lease Cash Flows Depreciation Tax Shield OCF 0 £1,046,380 £1,046,380 1 £1,046,380 £336,000 £1,382,380 2 £1,046,380 £268,800 £1,315,180 3 £1,046,380 £215,040 £1,261,420 4 £1,046,380 £172,032 £1,218,412 5 £688,128 £688,128 Year OCF Investment Net Cash Flow PV 0 -£1,046,380 £6,000,000 £4,953,620 £4,953,620 1 -£1,382,380 -£1,382,380 -£1,298,254 2 -£1,315,180 -£1,315,180 -£1,159,977 3 -£1,261,420 -£1,261,420 -£1,044,854 4 -£1,218,412 -£1,218,412 -£947,812 5 -£688,128 -£688,128 -£502,723 The sum of present values is equal to zero. 19. The after-tax residual value of the asset is an opportunity cost to the leasing decision, occurring at the end of the project life (year 5). Also, the residual value is not really a debt-like cash flow, since there is uncertainty associated with it at year 0. Nevertheless, although a higher discount rate may be appropriate, we’ll use the after-tax cost of debt to discount the residual value as is common in practice. Solving for NAL equal to zero, the maximum lease payment is £1,370,800. This is shown below using solver. The depreciation schedule is: Year Starting Value Depreciation Cumulative Depreciation Residual Value Tax Shield 20% 28% 1 £6,000,000 £1,200,000 £1,200,000 £4,800,000 £336,000 2 £4,800,000 £960,000 £2,160,000 £3,840,000 £268,800 3 £3,840,000 £768,000 £2,928,000 £3,072,000 £215,040 4 £3,072,000 £614,400 £3,542,400 £2,457,600 £172,032 5 £2,457,600 £1,957,600 £5,500,000 £500,000 £548,128 The operating cash flow is: Year After-Tax Lease Cash Flows Depreciation Tax Shield OCF 0 £986,976 £986,976 1 £986,976 £336,000 £1,322,976 2 £986,976 £268,800 £1,255,776 3 £986,976 £215,040 £1,202,016 4 £986,976 £172,032 £1,159,008 5 £548,128 £548,128 The cash flow analysis is: Year OCF Investment Net Cash Flow PV 0 -£986,976 £6,000,000 £5,013,024 £5,013,024 1 -£1,322,976 -£1,322,976 -£1,242,464 2 -£1,255,776 -£1,255,776 -£1,107,583 3 -£1,202,016 -£1,202,016 -£995,649 4 -£1,159,008 -£1,159,008 -£901,601 5 -£548,128 -£500,000 -£1,048,128 -£765,727 The sum of the present values is zero. 20. The security deposit is a cash outflow at the beginning of the lease and a cash inflow at the end of the lease when it is returned. The NAL, with these assumptions, is: Year OCF Investment Security Deposit Net Cash Flow PV 0 -£1,008,000 £6,000,000 -£200,000 £4,792,000 £4,792,000 1 -£1,344,000 -£1,344,000 -£1,262,209 2 -£1,276,800 -£1,276,800 -£1,126,125 3 -£1,223,040 -£1,223,040 -£1,013,063 4 -£1,180,032 -£1,180,032 -£917,955 5 -£688,128 £200,000 -£488,128 -£356,610 NAL = £116,037 With the security deposit, the firm should still lease the equipment rather than buy it, because the NAL is greater than zero. 21. The lessee is paying taxes, so will forego the depreciation tax shield if it leases the equipment. The depreciation tax shield for the lessee is: Depreciation tax shield = (€1,500,000 / 6)(.25) Depreciation tax shield = €62,500 The after-tax cost of debt for the lessee is: After-tax debt cost = .08(1 – .25) = .0600 Using all of this information, we can calculate the maximum pre-tax lease payment for the lessee as: NAL = 0 = €1,500,000 – PMT(1 – .25)(PVIFA6.00%,6) + €62,500(PVIFA6.00%,6) PMT = €323,391.92 For the lessor, the depreciation tax shield is: Depreciation tax shield = (€1,500,000 / 6)(.40) Depreciation tax shield = €100,000 The after-tax cost of debt for the lessor is: After-tax debt cost = .08(1 – .40) = .0480 Using all of this information, we can calculate the minimum pre-tax lease payment for the lessor as: NAL = 0 = €1,500,000 – PMT(1 – .40)(PVIFA4.80%,6) + €100,000(PVIFA4.80%,6) PMT = €322,731.18 22. a. Since both companies have the same tax rate, there is only one lease payment that will result in a zero NAL for each company. We will calculate cash flows from the depreciation tax shield first. The depreciation tax shield is: Depreciation tax shield = (£360,000/3)(.34) = £40,800 The after-tax cost of debt is: After-tax debt cost = .10(1 – .34) = .0660 Using all of this information, we can calculate the lease payment as: NAL = 0 = £360,000 – PMT(1 – .34)(PVIFA6.60%,3) + £40,800(PVIFA6.60%,3) PMT = £144,510.96 b. To generalize the result from part a: Let T1 denote the lessor’s tax rate. Let T2 denote the lessee’s tax rate. Let P denote the purchase price of the asset. Let D equal the annual depreciation expense. Let N denote the length of the lease in years. Let R equal the pre-tax cost of debt. The value to the lessor is: ValueLessor = = + − − + − + N 1 2 1 1 [1 R(1 T )] L(1 T ) D(T ) P t t And the value to the lessee is: ValueLessee = = + − − + − N 1 2 2 2 [1 R(1 T )] L(1 T ) D(T ) P t t Since all the values in both equations above are the same except T1 and T2 , we can see that the values of the lease to its two parties will be opposite in sign only if T1 = T2. c. Since the lessor’s tax bracket is unchanged, the zero NAL lease payment is the same as we found in part a. The lessee will not realize the depreciation tax shield, and the aftertax cost of debt will be the same as the pretax cost of debt. So, the lessee’s maximum lease payment will be: NAL = 0 = –£360,000 + PMT(PVIFA10%,3) PMT = £144,761.33 Both parties have positive NAL for lease payments between £144,510.96 and £144,761.33. 23. The decision to buy or lease is made by looking at the incremental cash flows. The loan offered by the bank merely helps you to establish the appropriate discount rate. Since the deal they are offering is the same as the market-wide rate, you can ignore the offer and simply use 9 percent as the pre-tax discount rate. In any capital budgeting project, you do not consider the financing which was to be applied to a specific project. The only exception would be if a specific and special financing deal were tied to a specific project (like a lower-than-market interest rate loan if you buy a particular car). a. The incremental cash flows from leasing the machine are the lease payments, the tax savings on the lease, the lost depreciation tax shield, and the saved purchase price of the machine. The lease payments are due at the beginning of each year, so the incremental cash flows are: Depreciation tax shield: Year Starting Value Depreciation Cumulative Depreciation Residual Value Tax Shield 20% 28% 1 £4,200,000 £840,000 £840,000 £3,360,000 £235,200 2 £3,360,000 £672,000 £1,512,000 £2,688,000 £188,160 3 £2,688,000 £537,600 £2,049,600 £2,150,400 £150,528 4 £2,150,400 £2,150,400 £4,200,000 £0 £602,112 Year Lease Payments Investment Tax savings on lease Lost Depreciation Tax Shield Net Cash Flow 0 -£1,200,000 £4,200,000 £336,000 £3,336,000 1 -£1,200,000 £336,000 -£235,200 -£1,099,200 2 -£1,200,000 £336,000 -£188,160 -£1,052,160 3 -£1,200,000 £336,000 -£150,528 -£1,014,528 4 -£602,112 -£602,112 The after-tax discount rate is: After-tax discount rate = .09(1 – .28) After-tax discount rate = .0648 or 6.48% So, the NAL of leasing is: Year Net Cash Flow PV 0 £3,336,000 £3,336,000 1 -£1,099,200 -£1,032,307 2 -£1,052,160 -£927,995 3 -£1,014,528 -£840,349 4 -£602,112 -£468,387 NAL = £66,962 Since the NAL is positive, the company should lease the equipment. b. The company is indifferent at the lease payment which makes the NAL of the lease equal to zero. The NAL equation of the lease is found using solver: Year Lease Payments Investment Tax savings on lease Lost Depreciation Tax Shield Net Cash Flow 0 -£1,225,484 £4,200,000 £343,135 £3,317,652 1 -£1,225,484 £343,135 -£235,200 -£1,117,548 2 -£1,225,484 £343,135 -£188,160 -£1,070,508 3 -£1,225,484 £343,135 -£150,528 -£1,032,876 4 -£602,112 -£602,112 Year Net Cash Flow PV 0 £3,317,652 £3,317,652 1 -£1,117,548 -£1,049,538 2 -£1,070,508 -£944,178 3 -£1,032,876 -£855,548 4 -£602,112 -£468,387 The sum of the present values is zero when the lease payment is £1,225,484 24. With a four-year loan, the annual loan payment will be £22,000,000 = PMT(PVIFA8%,4) PMT = £6,642,257.70 The after-tax loan payment is found by: After-tax payment = Pre-tax payment – Interest tax shield So, we need to find the interest tax shield. To find this, we need a loan amortization table since the interest payment each year is the beginning balance times the loan interest rate of 8 percent. The interest tax shield is the interest payment times the tax rate. The amortization table for this loan is: Year Beginning balance Total payment Interest payment Principal payment Ending balance 1 £22,000,000 £6,642,258 £1,760,000 £4,882,258 £17,117,742 2 £17,117,742 £6,642,258 £1,369,419 £5,272,838 £11,844,904 3 £11,844,904 £6,642,258 £947,592 £5,694,665 £6,150,239 4 £6,150,239 £6,642,258 £492,019 £6,150,239 £0 So, the total cash flows each year are: Year After-tax loan Payment Operating Cash Flow Total Cash Flow PV 1 £6,237,457.70 £5,632,000 -£605,457.70 -£570,326 2 £6,327,291.24 £5,429,600 -£897,691.24 -£796,536 3 £6,424,311.46 £5,267,680 -£1,156,631.46 -£966,746 4 £6,529,093.31 £7,210,720 £681,626.69 £536,665 So, the NPV with the loan payments is: NPV = £-1,796,942 The NPV is the same because the present value of the after-tax loan payments, discounted at the after-tax cost of capital (which is the after-tax cost of debt) equals £22,000,000. 25. a. The decision to buy or lease is made by looking at the incremental cash flows, so we need to find the cash flows for each alternative. The cash flows if the company leases are: Cash flows from leasing: After-tax cost savings = £6,000(1 – .28) After-tax cost savings = £4,320 The tax benefit of the lease is the lease payment times the tax rate, so the tax benefit of the lease is: Lease tax benefit = £2,100(.28) Lease tax benefit = £588 We need to remember the lease payments are due at the beginning of the year. So, if the company leases, the cash flows each year will be: Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 After-tax savings £4,320 £4,320 £4,320 £4,320 £4,320 Lease payment -£2,100 -£2,100 -£2,100 -£2,100 -£2,100 Tax benefit £588 £588 £588 £588 £588 Net cash flows -£1,512 £2,808 £2,808 £2,808 £2,808 £4,320 The amount the company borrows and the repayment schedule are irrelevant since the company maintains a target debt-equity ratio. So, the cash flows from buying the machine will be: Cash flows from purchasing: After-tax cost savings = £9,000(1 – .28) After-tax cost savings = £6,480 And the deprecation tax shield will be: Year Starting Value Depreciation Cumulative Depreciation Residual Value Tax Shield 20% 28% 1 £15,000 £3,000 £3,000 £12,000 £840 2 £12,000 £2,400 £5,400 £9,600 £672 3 £9,600 £1,920 £7,320 £7,680 £538 4 £7,680 £1,536 £8,856 £6,144 £430 5 £6,144 £6,144 £15,000 £0 £1,720 Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 After-tax savings £6,480 £6,480 £6,480 £6,480 £6,480 Purchase -£15,000 Dep. tax shield £840 £672 £538 £430 £1,720 Net cash flows -£15,000 £7,320 £7,152 £7,018 £6,910 £8,200 Now we can calculate the incremental cash flows from leasing versus buying by subtracting the net cash flows buying from the net cash flows from leasing. The incremental cash flows from leasing are: 0 1 2 3 4 5 Lease Cash Flows -£1,512 £2,808 £2,808 £2,808 £2,808 £4,320 Purchase Cash Flows -£15,000 £7,320 £7,152 £7,018 £6,910 £8,200 –Incremental cash flow -£13,488 £4,512 £4,344 £4,210 £4,102 £3,880 The after-tax discount rate is: After-tax discount rate = .10(1 – .28) After-tax discount rate = .072 or 7.20% So, the NPV of buying is: Purchase - lease -£13,488 £4,512 £4,344 £4,210 £4,102 £3,880 PV -£13,488 £4,209 £3,780 £3,417 £3,106 £2,741 NPV = £3,765 Since the NPV is positive, the company should buy, not lease the equipment. b. As long as the company maintains its target debt-equity ratio, the answer does not depend upon the form of financing used for the direct purchase. A financial lease will displace debt regardless of the form of financing. c. The amount of displaced debt is the PV of the incremental cash flows from year one through five. NPV = £17,253 26. This question is for advanced readers and links in to the literature presented in the references of this chapter. Students will be expected to provide a brief literature review and to present a balanced argument. 27. This question is for advanced readers and links in to the literature presented in the references of this chapter. Students should be expected to provide an explanation of Moral Hazard and present a brief literature review and to present a balanced argument. 28. This question is for advanced readers and links in to the literature presented in the references of this chapter. Students should be expected to discuss Gavazza (2010) in detail. 29. This question is for advanced readers and links in to the literature presented in the references of this chapter. Students should be expected to discuss Beatty et al (2010) in detail and present a balanced critique. Chapter 21 Case Study The Decision to Lease or Buy at Warf Computers 1. The decision to buy or lease is made by looking at the incremental cash flows. The incremental cash flows from leasing the machine are the security deposit, the lease payments, the tax savings on the lease, the lost depreciation tax shield, the saved purchase price of the machine, and the lost salvage value. The depreciation schedule of the equipment is: 1 2 3 4 Initial Cost £5,000,000 £4,000,000 £3,200,000 £2,560,000 Depreciation £1,000,000 £800,000 £640,000 £1,960,000 Balance £4,000,000 £3,200,000 £2,560,000 £600,000 Tax £280,000 £224,000 £179,200 £548,800 This is an opportunity cost to Warf Computers since if the company leases the equipment it will not receive the depreciation tax shield. The incremental cash flows are: Year 0 Year 1 Year 2 Year 3 Year 4 Saved purchase £5,000,000 - £600,000* Lost dep. tax shield -£280,000 -£224,000 -£179,200 -£548,800 Security deposit -£300,000 £300,000 Lease payment -£1,300,000 -£1,300,000 -£1,300,000 -£1,300,000 Tax on lease payment £364,000 £364,000 £364,000 £364,000 Cash flow from leasing £3,764,000 -£1,216,000 -£1,160,000 -£1,115,200 -£848,800 * Lost salvage value The after-tax cost of debt is: After-tax cost of debt = .11(1 – .28) After-tax cost of debt = .0792 or 792% And the NAL of the lease is: 0 1 2 3 4 Cash flow from leasing £3,764,000 -£1,216,000 -£1,160,000 -£1,115,200 -£848,800 PV(Cash Flow) £3,764,000 -£1,126,761 -£995,988 -£887,252 -£625,745 NAL £128,254 The company should lease the equipment. 2. The new depreciation schedule is: 1 2 Initial Cost £5,000,000 £4,000,000 Depreciation £1,000,000 £2,000,000 Balance £4,000,000 £2,000,000 Tax £280,000 £560,000 So, the NAL of the lease under the new terms would be: Year 0 Year 1 Year 2 Saved purchase £5,000,000 Lost Salvage Value -£2,000,000 Lost dep. tax shield -£280,000 -£560,000 Lease payment -£2,300,000 -£2,300,000 Tax on lease payment £644,000 £644,000 Cash flow from leasing £3,344,000 -£1,936,000 -£2,560,000 So, the NAL of the lease under these terms is: 0 1 2 Lease payment £3,344,000 -£1,936,000 -£2,560,000 PV(Cash Flow) £3,344,000 -£1,793,921 -£2,198,043 NAL -£647,964 3. a. The inclusion of a right to purchase the equipment will have no effect on the value of the lease. If the company does not purchase the equipment, it can go on the market and purchase identical equipment at the same price. b. The right to purchase the equipment at a fixed price will have increase the value of the lease. If the company can purchase the equipment at the end of the lease at below market value, it will save money, or at a minimum, can purchase the equipment at the fixed price and resell it in the open market. This is a real option, therefore has value to the lessee. It is a call option on the equipment. As such, it must have a value until it expires or is exercised. It is also important to note that this would likely make the lease contract a capitalized lease. c. The right to purchase the equipment at a bargain price is also a real option for the lessee, and will increase the value of the lease. It is a call option, and therefore will have value until it expires or is exercised. This contract condition will definitely ensure the lease is classified as a capitalized lease. 4. The cancellation option is also a real option. The cancellation option is a put option on the equipment. It will increase the value of the lease since the lessee will only exercise the option when it is to the lessee’s advantage. Chapter 22 Options and Corporate Finance 1. There are many ways to explain how an option works. When a construction company builds houses, they routinely take in deposits for houses long before a brick has even been laid. If, when the house is built, the prospective buyer wishes to go ahead with the purchase, he will pay the agreed price. However, it is possible that the houses do not turn out as planned or as the prospective buyer wished. He can opt not to go ahead with the purchase and lose his deposit. A call option works in a similar way. You pay a premium today to buy an asset in the future for a set price. If the price you agreed is not attractive (i.e. too low), you will choose not to buy and the premium is a sunk cost. If the price you agreed is attractive (i.e. higher than the agreed price), you will buy the asset. 2. A call option gives the owner the right but not the obligation to buy an asset at a specific price during a set time period or on a specific date. Companies would purchase call options if they felt that an asset they wished to purchase in the future was going to grow in value. 3. Buying a put is not the same as writing a call. The maximum loss that can be made with a long call position is the call premium. On the other hand, the maximum loss on a short call position is theoretically infinite. 4. Corporations normally write puts as part of a larger option portfolio hedging strategy. It would be very rare for a firm to enter into a naked short put position. 5. Given that the payoff to a call option is max (0, Share price – strike price), it would be expected that the settlement price (or premium) would rise as the underlying share price gets higher or the strike price gets lower. The converse logic would hold for put options. There are other factors that determine the value of an option and so we may see such a pattern. For example, two options with significantly different terms to maturity way exhibit such a pattern. In the Air France-KLM case, the pattern is evident for put options but not for call options. This does appear anomalous. However, notice that there are no bid quotes for call options and so it may be a result of lack of demand on the buy side. A puzzle! 6. One example is called a straddle, where the company holds one long call and one long put with the same exercise price and term to maturity. In this situation, the company pays two premiums but benefits if the price of the commodity increases by a large amount or falls by a large amount. A company would hold this position if it felt that there was going to be significant volatility and a very high probability of price changes in the future but unsure of the direction of the price change. 7. The factors that determine option values are given in Table 22.2, which is reprinted below: Increase in Call Option* Put Option* Value of underlying asset (share price) + – Exercise price – + Share price volatility + + Interest rate + – Time to expiration + + In addition to the preceding, we have presented the following four relationships for American calls: 1. The call price can never be greater than the share price (upper bound). 2. The call price can never be less than either zero or the difference between the share price and the exercise price (lower bound). 3. The call is worth zero if the underlying equity is worth zero. 4. When the share price is much greater than the exercise price, the call price tends toward the difference between the share price and the present value of the exercise price. 8. You cannot use discounted cash flows to value options because it is impossible to calculate the appropriate discount rate. 9. The Greeks are exceptionally important because the valuation and characteristics of options change over time. It is necessary to quantify how these changes affect the company’s risk. 10. A firm’s equity can be viewed as a call option on a firm because the payoffs to equity holders are the same as the payoff to call option holders. To illustrate this consider a firm that must pay €1,000 in interest every year. If the firm’s cash flows are less than €1,000 all the money must go to the bondholders. If it is more than €1,000, the equity holders receive S - €1,000. Thus, the payoff is the same. 11. Mergers for financial risk reasons are wrong because the benefits of the reduction in risk go to the bondholders, not the equity holders. Most mergers do not occur for this reason and so there is no reason why one should not buy shares in a newly merged bank. For a full discussion of this topic, please see Section 22.11. 12. The factors that determine option values are given in Table 22.2, which is reprinted below: Increase in Call Option* Put Option* Value of underlying asset (share price) + – Exercise price – + Share price volatility + + Interest rate + – Time to expiration + + In addition to the preceding, we have presented the following four relationships for American calls: 1. The call price can never be greater than the share price (upper bound). 2. The call price can never be less than either zero or the difference between the share price and the exercise price (lower bound). 3. The call is worth zero if the underlying equity is worth zero. 4. When the share price is much greater than the exercise price, the call price tends toward the difference between the share price and the present value of the exercise price. The pattern of put and call prices make sense because the call option is out of the money and the put option is in the money. 13. a. The value of the call is the share price minus the present value of the exercise price, so: C0 = £0.97 – [£.97/1.021] = £0.02 The intrinsic value is the amount by which the share price exceeds the exercise price of the call, so the intrinsic value is £0. b. The value of the call is the share price minus the present value of the exercise price, so: C0 = £.97 – [£.7/1.021] = £.284 The intrinsic value is the amount by which the share price exceeds the exercise price of the call, so the intrinsic value is £.27. c. The value of the put option is £0 since there is no possibility that the put will finish in the money. The intrinsic value is also £0. 14. a. The call option strike prices range from £6.40 to £16.00. Those that are below £11.025 are in the money and those that are above £11.025 are out of the money. The intrinsic value is £11.025 minus the exercise price or zero, whichever is higher. For example, the intrinsic value of the £10 call option is £11.025 - £10 = £1.025. b. The put option strike prices range from £6.40 to £16.00. Those that are above £11.025 are in the money and those that are below £11.025 are out of the money. The intrinsic value is the exercise price minus £11.025 or zero, whichever is higher. For example, the intrinsic value of the £10 put option is £0. c. There is more activity in the call options with lower strike price and put options with higher strike prices. This would indicate that the expectation was of a decrease in the price of Xstrata. 15. a. Each contract is for 100 shares, so the total cost is: Cost = 10(100 shares/contract)(€.08) Cost = €80 b. If the share price at expiration is €1.40, the payoff is: Payoff = 10(100)(€1.70 – €1.50) Payoff = €200 If the share price at expiration is €1.35, the payoff is zero. c. Remembering that each contract is for 100 shares, the cost is: Cost = 10(100)(€.05) Cost = €50 The maximum gain on the put option would occur if the share price goes to €0. We also need to subtract the initial cost, so: Maximum gain = 10(100)(€1.20) – €50 Maximum gain = €1,150 If the share price at expiration is €1.14, the position will have a profit of: Profit = 10(100)(€1.20 – €1.14) – €50 Profit = €10 d. At a share price of €1.14 the put is in the money. As the writer, you will make: Net loss = €50 – 10(100)(€1.20 – €1.14) Net loss = –€10 At a share price of €1.32 the put is out of the money, so the writer will make the initial cost: Net gain = €50 At the breakeven, you would recover the initial cost of €50, so: €50 = 10(100)(€1.20 – ST) ST = €1.15 For terminal share prices above €1.15, the writer of the put option makes a net profit (ignoring transaction costs and the effects of the time value of money). 16. a. The value of the call is the share price minus the present value of the exercise price, so: C0 = R1.74 – R 1.75/1.04 C0 = R0.0573 b. Using the equation presented in the text to prevent arbitrage, we find the value of the call as follows: Delta = (R.1 – R0)/R.5 = .2 C0 = R1.74 x .2 – (R.3/1.04) C0 = R.0595 17. a. The value of the call is the share price minus the present value of the exercise price, so: C0 = €1.68 – €1.50/1.07 C0 = €0.278 b. Using the equation presented in the text to prevent arbitrage, we find the value of the call is: Delta = €.06/€.29 = 0.207 C0 = €1.68(0.207) – (€.2768) /1.07 C0 = €.2587 18. Using put-call parity and solving for the put price, we get: £4.29 + £.25 = £4.40e–(.026)(3/12) + C C = €.169 19. Using put-call parity and solving for the stock price we get: S + €.28 = €1e–(.026)(3/12) + €.11 S = €.824 20. Using put-call parity, we can solve for the risk-free rate as follows: €17.03 + €2.03 = €17e–R(4/12) + €2.07 €16.99 = €17–R(4/12) .9994 = e–R(4/12) ln(.9994) = ln(e–R(4/12)) -.000589 = –R(4/12) Rf = .177% 21. Using the Black-Scholes option pricing model, with a ‘share’ price is £6,700,000 and an exercise price is £7,000,000, the price you should receive is: d1 = .0310 d2 = –.1690 N(d1) = .5124 N(d2) = .4329 Putting these values into the Black-Scholes model, we find the call price is: C = £6,700,000(.5124) – (£7,000,000e–.03(1))(.4329) = £492,117 22. Using the call price we found in the previous problem and put-call parity, you would need to pay: Put = £7,000,000e–.03(1) + 492,117 – 6,700,000 = £585,236 You would have to pay £585,236 in order to guarantee the right to sell the land for £7,000,000. 23. Using the Black-Scholes option pricing model to find the price of the call option, we find: d1 = 2.3655 d2 = 2.1905 N(d1) = .9910 N(d2) = .9858 Putting these values into the Black-Scholes model, we find the call price is: C = €.49 24. When the standard deviation of an asset is infinitely large, the value of a call option is simply the value of the underlying asset. 25. We can use the Black-Scholes model to value the equity of a firm. Using the asset value of £20 billion as the share price, and the face value of debt of £12 billion as the exercise price, the value of the firm’s equity is: d1 = 1.5421 d2 = 1.0278 N(d1) = .9385 N(d2) = .8480 Putting these values into the Black-Scholes model, we find the equity value is: Equity = £10.01 billion The value of the debt is the firm value minus the value of the equity, so: D = £20 billion – £10.01 billion = £9.99 billion 26. a. We can use the Black-Scholes model to value the equity of a firm. Using the asset value of £20.7 billion as the share price, and the face value of debt of £12 billion as the exercise price, the value of the firm if it accepts project A is: d1 = 1.1802 d2 = -.0496 N(d1) = .8810 N(d2) = .4802 Putting these values into the Black-Scholes model, we find the equity value is: EA = £13.28 billion The value of the debt is the firm value minus the value of the equity, so: DA = £20.7 billion – £13.28 billion = £7.42 billion And the value of the firm if it accepts Project B is: d1 = 1.9050 d2 = 1.4801 N(d1) = .9716 N(d2) = .9306 Putting these values into the Black-Scholes model, we find the equity value is: EB = £10.99 billion The value of the debt is the firm value minus the value of the equity, so: DB = £21.2 billion – £10.99 billion= £10.21 billion b. Although the NPV of project B is higher, the equity value with project A is higher. While NPV represents the increase in the value of the assets of the firm, in this case, the increase in the value of the firm’s assets resulting from project B is mostly allocated to the debt holders, resulting in a smaller increase in the value of the equity. Shareholders would, therefore, prefer project A even though it has a lower NPV. c. Yes. If the same group of investors have equal stakes in the firm as bondholders and share- holders, then total firm value matters and project B should be chosen, since it increases the value of the firm to £21.2 billion instead of £20.7 billion. d. Shareholders may have an incentive to take on riskier, less profitable projects if the firm is leveraged; the higher the firm’s debt load, all else the same, the greater is this incentive. 27. a. With a standard deviation of 30 per cent, Shire plc equity value is now: d1 = 1.3205 d2 = .6497 N(d1) = .9067 N(d2) = .7421 Putting these values into the Black-Scholes model, we find the equity value is: E = £10.47 billion The value of the debt is the firm value minus the value of the equity, so: D = £20 billion – £10.47 billion = £9.53 billion c. The change in the value of the firm’s equity is: Equity value change = £10.47 billion – £10.01 billion= £.46 billion or £460 million The change in the value of the firm’s debt is: Debt = £9.53 billion – £9.99 billion = -£.46 billion d. As the standard deviation of the assets grows, the value of the equity grows as well. The shareholders will gain exactly the amount the bondholders lose. 28. a. Using Black-Scholes model to value the equity, we get: d1 = [ln(£22,000,000/£30,000,000) + (.06 + .392/2) 10] / (.39 10 ) = .8517 d2 = .8517 – (.39 10 ) = –.3816 N(d1) = .8028 N(d2) = .3514 Putting these values into Black-Scholes: E = £22,000,000(.8028) – (£30,000,000e–.06(10))(.3514) = £11,876,514.44 b. The value of the debt is the firm value minus the value of the equity, so: D = £22,000,000 – £11,876,514.44 = £10,123,485.56 c. Using the equation for the PV of a continuously compounded lump sum, we get: £10,123,485.56 = £30,000,000e–R(10) .33745 = e–R10 RD = –(1/10)ln(.33745) = 10.86% d. Using Black-Scholes model to value the equity, we get: d1 = [ln(£22,750,000/£30,000,000) + (.06 + .392/2) 10] / (.39 10 ) = .8788 d2 = .8788 – (.39 10 ) = –.3544 N(d1) = .8103 N(d2) = .3615 Putting these values into Black-Scholes: E = £22,750,000(.8103) – (£30,000,000e–.06(10))(.3615) = £12,481,437 e. The value of the debt is the firm value minus the value of the equity, so: D = £22,750,000 – £12,481.437 = £10,268,563 Using the equation for the PV of a continuously compounded lump sum, we get: £10,268,563= £30,000,000e–R(10) .34229 = e–R10 RD = –(1/10)ln(.34229) = 10.72% When the firm accepts the new project, part of the NPV accrues to bondholders. This increases the present value of the bond, thus reducing the return on the bond. Additionally, the new project makes the firm safer in the sense that it increases the value of assets, thus increasing the probability the call will end in-the-money and the bondholders will receive their payment. 29. a. In order to solve a problem using the two-state option model, we first need to draw a share price tree containing both the current share price and the share’s possible values at the time of the option’s expiration. Next, we can draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible share price movements. Share Price Call option price with a strike of £110 Today 1 year Today 1 year £125 £15 =Max(0, £125 – £110) £100 ? £80 £0 =Max(0, £80 – £110) The share price today is £100. It will either increase to £125 or decrease to £80 in one year. If the share price rises to £125, the call will be exercised for £110 and a payoff of £15 will be received at expiration. If the share price falls to £80, the option will not be exercised, and the payoff at expiration will be zero. If the share price rises, its return over the period is 25 percent [= (£125/£100) – 1]. If the share price falls, its return over the period is –20 percent [= (£80/£100) –1]. We can use the following expression to determine the risk-neutral probability of a rise in the price of the share: Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall) .025 = (ProbabilityRise)(0.25) + (1 – ProbabilityRise)(–0.20) ProbabilityRise = .50 or 50% This means the risk neutral probability of a share price decrease is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 – .50 ProbabilityFall = .50 or 50% Using these risk-neutral probabilities, we can now determine the expected payoff of the call option at expiration. The expected payoff at expiration is: Expected payoff at expiration = (.50)(£15) + (.50)(£0) Expected payoff at expiration = £7.50 Since this payoff occurs 1 year from now, we must discount it back to the value today. Since we are using risk-neutral probabilities, we can use the risk-free rate, so: PV(Expected payoff at expiration) = £7.50 / 1.025 PV(Expected payoff at expiration) = £7.32 Therefore, given the information about the share price movements over the next year, a European call option with a strike price of £110 and one year until expiration is worth £7.32 today. b. Yes, there is a way to create a synthetic call option with identical payoffs to the call option described above. In order to do this, we will need to buy shares of equity and borrow at the risk-free rate. The number of shares to buy is based on the delta of the option, where delta is defined as: Delta = (Swing of option) / (Swing of equity) Since the call option will be worth £15 if the share price rises and £0 if it falls, the delta of the option is £15 (= 15 – 0). Since the share price will either be £125 or £80 at the time of the option’s expiration, the swing of the equity is £45 (= £125 – £80). With this information, the delta of the option is: Delta = £15 / £45 Delta = 1/3 or .3333 Therefore, the first step in creating a synthetic call option is to buy 1/3 of a share of the equity. Since the equity is currently trading at £100 per share, this will cost £33.33 [= (1/3)(£100)]. In order to determine the amount that we should borrow, compare the payoff of the actual call option to the payoff of delta shares at expiration. Call Option If the share price rises to £125: Payoff = £15 If the share price falls to £80: Payoff = £0 Delta Shares If the share price rises to £125: Payoff = (1/3)(£125) = £41.66 If the share price falls to £80: Payoff = (1/3)(£80) = £26.66 The payoff of his synthetic call position should be identical to the payoff of an actual call option. However, owning 1/3 of a share leaves us exactly £26.66 above the payoff at expiration, regardless of whether the share price rises or falls. In order to reduce the payoff at expiration by £26.66, we should borrow the present value of £26.66 now. In one year, the obligation to pay £26.66 will reduce the payoffs so that they exactly match those of an actual call option. So, purchase 1/3 of a share of equity and borrow £26.02 (= £26.66 / 1.025) in order to create a synthetic call option with a strike price of £110 and 1 year until expiration. c. Since the cost of the equity purchase is £33.33 to purchase 1/3 of a share and £26.02 is borrowed, the total cost of the synthetic call option is: Cost of synthetic option = £33.33 – £26.02 Cost of synthetic option = £7.32 This is exactly the same price as an actual call option. Since an actual call option and a synthetic call option provide identical payoff structures, we should not expect to pay more for one than for the other. 30. a. The company would be interested in purchasing a call option on the price of gold with a strike price of $875 per ounce and 3 months until expiration. This option will compensate the company for any increases in the price of gold above the strike price and places a cap on the amount the firm must pay for gold at $875 per ounce. b. In order to solve a problem using the two-state option model, first draw a price tree containing both the current price of the underlying asset and the underlying asset’s possible values at the time of the option’s expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible gold price movements. Price of gold Call option price with a strike of $875 Today 3 months Today 3 months $900 $25 =Max(0, $900 – $875) $850 ? $825 $0 =Max(0, $825 – $ 875) The price of gold is $850 per ounce today. If the price rises to $900, the company will exercise its call option for $875 and receive a payoff of $25 at expiration. If the price of gold falls to $825, the company will not exercise its call option, and the firm will receive no payoff at expiration. If the price of gold rises, its return over the period is 5.88 percent [= ($900 / $850) – 1]. If the price of gold falls, its return over the period is –2.94 percent [= ($825 / $850) –1]. Use the following expression to determine the risk-neutral probability of a rise in the price of gold: Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) The risk-free rate over the next three months must be used in the order to match the timing of the expected price change. Since the risk-free rate per annum is 16.99 percent, the risk- free rate over the next three months is 4 percent [= (1.1699)1/4 – 1], so: .04 = (ProbabilityRise)(.0588) + (1 – ProbabilityRise)(–.0294) ProbabilityRise = .7868 or 78.68% And the risk-neutral probability of a price decline is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 –.7868 ProbabilityFall = .2132 or 21.32% Using these risk-neutral probabilities, we can determine the expected payoff to of the call option at expiration, which will be. Expected payoff at expiration = (.7868)($25) + (.2132)($0) Expected payoff at expiration = $19.67 Since this payoff occurs 3 months from now, it must be discounted at the risk-free rate in order to find its present value. Doing so, we find: PV(Expected payoff at expiration) = [$19.67 / (1.1699)1/4 ] PV(Expected payoff at expiration) = $18.91 Therefore, given the information about gold’s price movements over the next three months, a European call option with a strike price of $875 and three months until expiration is worth $18.91 today. c. Yes, there is a way to create a synthetic call option with identical payoffs to the call option described above. In order to do this, the company will need to buy gold and borrow at the risk-free rate. The amount of gold to buy is based on the delta of the option, where delta is defined as: Delta = (Swing of option) / (Swing of price of gold) Since the call option will be worth $25 if the price of gold rises and $0 if it falls, the swing of the call option is $25 (= $25 – 0). Since the price of gold will either be $900 or $825 at the time of the option’s expiration, the swing of the price of gold is $75 (= $900 – $825). Given this information the delta of the call option is: Delta = (Swing of option) / (Swing of price of gold) Delta = ($25 / $75) Delta = 1/3 or .3333 Therefore, the first step in creating a synthetic call option is to buy 1/3 of an ounce of gold. Since gold currently sells for $850 per ounce, the company will pay $283.33 (= 1/3 × $850) to purchase 1/3 of an ounce of gold. In order to determine the amount that should be borrowed, compare the payoff of the actual call option to the payoff of delta shares at expiration: Call Option If the price of gold rises to $900: Payoff = $25 If the price of gold falls to $825: Payoff = $0 Delta Shares If the price of gold rises to $900: Payoff = (1/3)($900) = $300 If the price of gold falls to $825: Payoff = (1/3)($825) = $275 The payoff of this synthetic call position should be identical to the payoff of an actual call option. However, buying 1/3 of a share leaves us exactly $275 above the payoff at expiration, whether the price of gold rises or falls. In order to decrease the company’s payoff at expiration by $275, it should borrow the present value of $275 now. In three months, the company must pay $275, which will decrease its payoffs so that they exactly match those of an actual call option. So, the amount to borrow today is: Amount to borrow today = $275 / 1.16991/4 Amount to borrow today = $264.42 d. Since the company pays $283.33 in order to purchase gold and borrows $264.42, the total cost of the synthetic call option is $18.91 (= $283.33 – $264.42). This is exactly the same price for an actual call option. Since an actual call option and a synthetic call option provide identical payoff structures, the company should not expect to pay more for one than for the other. 31. To construct the collar, the investor must purchase the equity, sell a call option with a high strike price, and buy a put option with a low strike price. So, to find the cost of the collar, we need to find the price of the call option and the price of the put option. We can use Black- Scholes to find the price of the call option, which will be: Price of call option with SwFr120 strike price: d1 = [ln(80/120) + (.10 + .502/2) (6/12)] / (.50 (6 /12) ) = –.8286 d2 = –.8286 – (.50 6 /12 ) = –1.1822 N(d1) = .2037 N(d2) = .1186 Putting these values into the Black-Scholes model, we find the call price is: C = 80(.2037) – (120e–.10(6/12))(.1186) = SwFr2.76 Now we can use Black-Scholes and put-call parity to find the price of the put option with a strike price of SwFr50. Doing so, we find: Price of put option with SwFr50 strike price: d1 = [ln(80/50) + (.10 + .502/2) (6/12)] / (.50 (6 /12) ) = 1.6476 d2 = 1.6476 – (.50 6 /12 ) = 1.2940 N(d1) = .9503 N(d2) = .9022 Putting these values into the Black-Scholes model, we find the put price is: C = 80(.9503) – (50e–.10(6/12))(.9022) = SwFr33.11 Rearranging the put-call parity equation, we get: P = C – S + Xe–Rt P = SwFr33.11 – 80 + 50e–.10(6/12) P = SwFr0.67 So, the investor will buy the equity, sell the call option, and buy the put option, so the total cost is: Total cost of collar = SwFr80 – SwFr 2.76 + SwFr .67 Total cost of collar = SwFr77.91 32. a. Using the equation for the PV of a continuously compounded lump sum, we get: PV = £30,000 e–.05(2) = £27,145.12 b. Using Black-Scholes model to value the equity, we get: d1 = [ln(£13,000/£30,000) + (.05 + .602/2) 2] / (.60 2 ) = –.4434 d2 = –.4434 – (.60 2 ) = –1.2919 N(d1) = .3287 N(d2) = .0982 Putting these values into Black-Scholes: E = £13,000(.3287) – (£30,000e–.05(2))(.0982) = £1,608.19 And using put-call parity, the price of the put option is: Put = £30,000e–.05(2) + £1,608.19 – £13,000 = £15,753.31 c. The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so: Value of risky bond = £27,145.12 – £15,753.31 = £11,391.81 Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get: £11,391.81 = £30,000e–R(2) .37973 = e–R2 RD = –(1/2)ln(.37973) = .4842 or 48.42% d. The value of the debt with five years to maturity at the risk-free rate is: PV = £30,000 e–.05(5) = £23,364.02 Using Black-Scholes model to value the equity, we get: d1 = [ln(£13,000/£30,000) + (.05 + .602/2) 5] / (.60 5 ) = .2339 d2 = .2339 – (.60 5 ) = –1.1078 N(d1) = .5925 N(d2) = .1340 Putting these values into Black-Scholes: E = £13,000(.5925) – (£30,000e–.05(5))(.1340) = £4,571.62 And using put-call parity, the price of the put option is: Put = £30,000e–.05(5) + £4,571.62 – £13,000 = £14,935.64 The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so: Value of risky bond = £23,364.02 – £14,935.64 = £8,428.38 Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get: Return on debt: £8,428.38 = £30,000e–R(5) .28095 = e–R5 RD = –(1/5)ln(.28095) = 25.39% The value of the debt declines because of the time value of money, i.e., it will be longer until shareholders receive their payment. However, the required return on the debt declines. Under the current situation, it is not likely the company will have the assets to pay off bondholders. Under the new plan where the company operates for five more years, the probability of increasing the value of assets to meet or exceed the face value of debt is higher than if the company only operates for two more years. 33. a. Using the equation for the PV of a continuously compounded lump sum, we get: PV = £60,000 e–.06(5) = £44,449.09 b. Using Black-Scholes model to value the equity, we get: d1 = [ln(£57,000/£60,000) + (.06 + .502/2) 5] / (.50 5 ) = .7815 d2 = .7815 – (.50 5 ) = –.3366 N(d1) = .7827 N(d2) = .3682 Putting these values into Black-Scholes: E = £57,000(.7827) – (£60,000e–.06(5))(.3682) = £28,248.84 And using put-call parity, the price of the put option is: Put = £60,000e–.06(5) + £28,248.84 – £57,000 = £15,697.93 c. The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so: Value of risky bond = £44,449.09 – £15,697.93 = £28,751.16 Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get: Return on debt: £28,751.16 = £60,000e–R(5) .47919 = e–R(5) RD = –(1/5)ln(.47919) = 14.71% d. Using the equation for the PV of a continuously compounded lump sum, we get: PV = £60,000 e–.06(5) = £44,449.09 Using Black-Scholes model to value the equity, we get: d1 = [ln(£57,000/£60,000) + (.06 + .602/2) 5] / (.60 5 ) = .8562 d2 = .8562 – (.60 5 ) = –.4854 N(d1) = .8041 N(d2) = .3137 Putting these values into Black-Scholes: E = £57,000(.8041) – (£60,000e–.06(5))(.3137) = £31,888.34 And using put-call parity, the price of the put option is: Put = £60,000e–.06(5) + £31,888.34 – £57,000 = £19,337.44 The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so: Value of risky bond = £44,449.09 – £19,337.44 = £25,111.66 Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get: Return on debt: £25,111.66 = £60,000e–R(5) .41853 = e–R(5) RD = –(1/5)ln(.41853) = 17.42% The value of the debt declines. Since the standard deviation of the company’s assets increases, the value of the put option on the face value of the bond increases, which decreases the bond’s current value. e. From c and d, bondholders lose: £25,111.66 – £28,751.16 = –£3,639.51 From c and d, stockholders gain: £31,888.34 – £28,248.84 = £3,639.51 This is an agency problem for bondholders. Management, acting to increase shareholder wealth in this manner, will reduce bondholder wealth by the exact amount by which shareholder wealth is increased. 34. a. Since the equityholders of a firm financed partially with debt can be thought of as holding a call option on the assets of the firm with a strike price equal to the debt’s face value and a time to expiration equal to the debt’s time to maturity, the value of the company’s equity equals a call option with a strike price of £380 million and 1 year until expiration. In order to value this option using the two-state option model, first draw a tree containing both the current value of the firm and the firm’s possible values at the time of the option’s expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible changes in the firm’s value. The value of the company today is £400 million. It will either increase to £500 million or decrease to £320 million in one year as a result of its new project. If the firm’s value increases to £500 million, the equityholders will exercise their call option, and they will receive a payoff of £120 million at expiration. However, if the firm’s value decreases to £320 million, the equityholders will not exercise their call option, and they will receive no payoff at expiration. Value of company (in millions) Equityholders’ call option price with a strike of £380 (in millions) Today 1 year Today 1 year £500 £120 =Max(0, £500 – £380) £400 ? £320 £0 =Max(0, £320 – £380) If the project is successful and the company’s value rises, the percentage increase in value over the period is 25 percent [= (£500 / £400) – 1]. If the project is unsuccessful and the company’s value falls, the percentage decrease in value over the period is –20 percent [= (£320 / £400) –1]. We can determine the risk-neutral probability of an increase in the value of the company as: Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall) 0.07 = (ProbabilityRise)(.25) + (1 – ProbabilityRise)(–.20) ProbabilityRise = .60 or 60% And the risk-neutral probability of a decline in the company value is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 –.60 ProbabilityFall = .40 or 40% Using these risk-neutral probabilities, we can determine the expected payoff to the equityholders’ call option at expiration, which will be: Expected payoff at expiration = (.60)(£120,000,000) + (.40)(£0) Expected payoff at expiration = £72,000,000 Since this payoff occurs 1 year from now, we must discount it at the risk-free rate in order to find its present value. So: PV(Expected payoff at expiration) = (£72,000,000 / 1.07) PV(Expected payoff at expiration) = £67,289,720 Therefore, the current value of the company’s equity is £67,289,720. The current value of the company is equal to the value of its equity plus the value of its debt. In order to find the value of company’s debt, subtract the value of the company’s equity from the total value of the company: VL = Debt + Equity £400,000,000 = Debt + £67,289,720 Debt = £332,710,280 b. To find the price per share, we can divide the total value of the equity by the number of shares outstanding. So, the price per share is: Price per share = Total equity value / Shares outstanding Price per share = £67,289,720 / 500,000 Price per share = £134.58 c. The market value of the firm’s debt is £332,710,280. The present value of the same face amount of riskless debt is £355,140,187 (= £380,000,000 / 1.07). The firm’s debt is worth less than the present value of riskless debt since there is a risk that it will not be repaid in full. In other words, the market value of the debt takes into account the risk of default. The value of riskless debt is £355,140,187. Since there is a chance that the company might not repay its debtholders in full, the debt is worth less than £355,140,187. d. The value of Strudler today is £400 million. It will either increase to £800 million or decrease to £200 million in one year as a result of the new project. If the firm’s value increases to £800 million, the equityholders will exercise their call option, and they will receive a payoff of £420 million at expiration. However, if the firm’s value decreases to £200 million, the equityholders will not exercise their call option, and they will receive no payoff at expiration. Value of company (in millions) Equityholders’ call option price with a strike of £380 (in millions) Today 1 year Today 1 year £800 £420 =Max(0, £800 – £380) £400 ? £200 £0 =Max(0, £200 – £380) If the project is successful and the company’s value rises, the increase in the value of the company over the period is 100 percent [= (£800 / £400) – 1]. If the project is unsuccessful and the company’s value falls, decrease in the value of the company over the period is –50 percent [= (£200 / £400) –1]. We can use the following expression to determine the risk- neutral probability of an increase in the value of the company: Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) 0.07 = (ProbabilityRise)(1) + (1 – ProbabilityRise)(–.50) ProbabilityRise = .38 or 38 percent So the risk-neutral probability of a decrease in the company value is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 – .38 ProbabilityFall = .62 or 62% Using these risk-neutral probabilities, we can determine the expected payoff to the equityholders’ call option at expiration, which is: Expected payoff at expiration = (.38)(£420,000,000) + (.62)(£0) Expected payoff at expiration = £159,600,000 Since this payoff occurs 1 year from now, we must discount it at the risk-free rate in order to find its present value. So: PV(Expected payoff at expiration) = (£159,600,000 / 1.07) PV(Expected payoff at expiration) = £149,158,879 Therefore, the current value of the firm’s equity is £149,158,879. The current value of the company is equal to the value of its equity plus the value of its debt. In order to find the value of the company’s debt, we can subtract the value of the company’s equity from the total value of the company, which yields: VL = Debt + Equity £400,000,000 = Debt + £149,158,879 Debt = £250,841,121 The riskier project increases the value of the company’s equity and decreases the value of the company’s debt. If the company takes on the riskier project, the company is less likely to be able to pay off its bondholders. Since the risk of default increases if the new project is undertaken, the value of the company’s debt decreases. Bondholders would prefer the company to undertake the more conservative project. 35. a. Going back to the chapter on dividends, the price of the equity will decline by the amount of the dividend (less any tax effects). Therefore, we would expect the share price to drop when a dividend is paid, reducing the upside potential of the call by the amount of the dividend. The price of a call option will decrease when the dividend yield increases. b. Using the Black-Scholes model with dividends, we get: d1 = [ln(2.26/2) + (.05 – .02 + .502/2) .5] / (.50 .5 ) = .5649 d2 = .5649 – (.50 .5 ) = .0036 N(d1) = .7139 N(d2) = .5837 C = 2.266e–(.02)(.5)(.7139) – (2e–.05(.5))(.5837) = DKr0.46 36. a. Going back to the chapter on dividends, the price of the equity will decline by the amount of the dividend (less any tax effects). Therefore, we would expect the share price to drop when a dividend is paid. The price of put option will increase when the dividend yield increases. b. Using put-call parity to find the price of the put option, we get: 2.26e–.02(.5) + P = 2e–.05(.5) + .46 P = DKr0.17 37. N(d1) is the probability that “z” is less than or equal to N(d1), so 1 – N(d1) is the probability that “z” is greater than N(d1). Because of the symmetry of the normal distribution, this is the same thing as the probability that “z” is less than N(–d1). So: N(d1) – 1 = –N(–d1) 38. From put-call parity: P = E × e-Rt + C – S Substituting the Black-Scholes call option formula for C and using the result in the previous question produces the put option formula: P = E × e-Rt + C – S P = E × e-Rt + S ×N(d1) – E × e-Rt ×N(d2) – S P = S ×(N(d1) – 1) + E × e-Rt ×(1 – N(d2)) P = E × e-Rt ×N(–d2) – S × N(–d1) 39. Based on Black-Scholes, the call option is worth £50! The reason is that present value of the exercise price is zero, so the second term disappears. Also, d1 is infinite, so N(d1) is equal to one. The problem is that the call option is European with an infinite expiration, so why would you pay anything for it since you can never exercise it? The paradox can be resolved by examining the share price. Remember that the call option formula only applies to a non- dividend paying equity. If the equity will never pay a dividend, it (and a call option to buy it at any price) must be worthless. 40. The delta of the call option is N(d1) and the delta of the put option is N(d1) – 1. Since you are selling a put option, the delta of the portfolio is N(d1) – [N(d1) – 1]. This leaves the overall delta of your position as 1. This position will change one for one in value with the underlying asset. This position replicates the “action” on the underlying asset. Chapter 22 Case Study Clissold Industries Options 1. Since the Black-Scholes model uses the standard deviation of the underlying asset, and there is only one underlying asset no matter how many strike prices are available, we would only expect to see one implied standard deviation. 2. To find the implied volatility for an option, you can set up a spreadsheet to calculate the option price. The Solver function in Excel will allow you to input the desired price and will solve for the desired unknown variable. We did this (the spreadsheet is available), and the implied standard deviation for each of the options is: Strike Price Option Price Implied Standard Deviation £30 £23.00 86.19% 40 16.05 78.59% 50 9.75 66.83% 55 7.95 67.40% 3. There are two possible explanations. The first is model misspecification. Although the Black- Scholes option pricing model is widely acclaimed, it is possible that the model is incorrect. One potential variable that is incorrectly specified in practice is the assumption of constant volatility. In fact, the volatility of the underlying stock is itself volatile, and will increase or decrease over time. The Black-Scholes model may also ignore important variables. For example, Fisher Black describes trades he, Myron Scholes, Robert Merton, and others made when the model was first developed (Black, Fisher, 1989, “How we can up with the option pricing formula,” The Journal of Portfolio Management, Winter, 4-8.) As in any potential arbitrage opportunity, they purchased underpriced assets, in this warrants on National General stock. Unfortunately, soon after they took this position, American Financial announced a tender offer for National General, which sharply reduced the value of the warrants. The market had already priced the potential tender offer in the warrant price, while this variable was not accounted for in the Black-Scholes model. A second possible explanation is liquidity. At- or near-the-money options tend to be more liquid than deep in-the-money or deep out-of-the-money options. Since options that are not near-the- money are less liquid, the price should carry a liquidity premium. 4. The VIX is a benchmark for stock market volatility. The VFTSE is based on option prices, which reflect investors' consensus view of future expected stock market volatility. During periods of market turmoil, both option prices and the VFTSE tend to rise. When the market is calmer, investor fear, option prices, and the VFTSE decline. 5. The implied volatility of the VFTSE option represent the markets views on how variable option volatility will be in the future. Solution Manual for Corporate Finance David Hillier, Stephen Ross, Randolph Westerfield, Jeffrey Jaffe, Bradford Jordan 9780077139148
Download
Close
Close
