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Companion to Corporate Finance by Ivo Welch

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Companion to Corporate Finance by Ivo Welch Companion to Corporate Finance by Ivo Welch Warning Please be aware that the chapters in the companion are not yet formatted cor- rectly. They are however topically correct (unless otherwise marked, e.g., as in the chapter on investment banking, where some tables are still outdated). After the main book has been distributed, I will work on the chapters here. Part I Individual Chapter Appendices Chapter 3: Projects with unequal lives. Proofs. Chapter 11: Sample event study (congressional elec- tions in 2006). Chapter 5: Forward interest rates. Shorting. Dura- tion. Continuous Compounding. Quotes and Chapter 12: Decision trees for real options. STRIPS. Chapter 13: Coca-Cola financials in 2001. Chapter 8: Efficient Frontier. Variance Formula. With Chapter 16: CAPM, WACC, and NPV fit. shorted positions and risk-free asset. Chapter 17: Discount factor on tax obligations and Chapter 9: Beta implications for risk and conditional tax shelters. returns. CAPM in Certainty Equivalence form. Chapter 20: “In a pinch” regressions to assess fixed- CAPM Theory. CAPM Alternatives (APT and cost and variable-cost components of financial Fama-French models) statement variables. 1 3 Stock and Bond Valuation: Annuities and Perpetuities App.3.A Different Lives and Rental Equivalents You have already briefly met the concept of an equivalent annual cost in Question ?? on Page ??. This concept becomes more useful if you know how to work with annuities. Comparing Annual Payments to Multiyear Contracts Let’s work out a first example. Assume that the prevailing interest rate is 10% per annum. Would you rather sign a lease contract that obliges you to pay $1,000, $650, and $650 in consecutive years, or would you rather pay rent of $780 every year? The present value of the lease payments is PV $650 $650 $1, 000 $2, 128.10 + + 2 1.1 1.1 ≈ The proposed alternative rent would be $780 $780 $780 $2, 133.72 + + 2 1.1 1.1 ≈ The 3-year lease is cheaper for you—of course, assuming that you really want to use the building for 3 years. If you really needed the building for only 1 year, then a 1-year rental contract could be much better. Can you work out at what annual rent you would be indifferent between leasing Annual Cost and renting? This is called the equivalent annual cost (EAC). Easy: eac eac eac $2, 128.10 eac $777.95 + + 2 = 1.1 1.1 ) ≈ This tells you that you are indifferent between the ($1, 000, $650, $650) 3-year lease and an annual payment of $777.95, first payment due immediately. Another version of this calculation has you pay the rent at the end of the year. In this case, eac eac eac $2, 128.10 eac $855.74 (3.1) + 2 + 3 = 1.1 1.1 1.1 ) ≈ You would therefore also be indifferent between the 3-year lease, and paying $855.74 for 3 years with rent payments occurring at year-end, not at year-start. Of course, you could have simply multiplied $777.95 by 1.1 to arrive at $855.74. 3 4 Stock and Bond Valuation: Annuities and Perpetuities IMPORTANT To work out the equivalent annual cost of a contract, use a two-step procedure: 1. Determine the present value of the cost of the contract. 2. Use an annuity calculation to translate this cost into regular and equal flows. An equivalent way to Now stare at Formula 3.1. The left-hand side is an annuity with an unknown solve this payment per year “eac,” an interest rate of 10%, and three years duration: eac 1 3 $2, 128.10 1 – = $2, 128.10 eac $855.74 10% · 1.1 ) ≈ 2.48685 ≈ What if payments If you prefer the version where the first payment occurs immediately, simply discount start now? this by 10%: $855.74 $777.95 1.10 ≈ eacbeginning payments immediately = eacdiscount the beginning 1 + r payments next year Don’t get too worked up over this specific example. For three years, you don’t need to use the annuity formula if you prefer working with the long Formula 3.1 instead. However, if you have many payments, the annuity formula quickly becomes more convenient. A more involved For practice, let us work another lease example. A 5-year lease requires a one-time example. upfront payment of $1,600, followed by four payments of $500. The prevailing interest rate is 10%. What is the equivalent annual cost of this lease? First, work out the present value of the lease payments. This is 2 3 4 PV = $1, 600 + $500/1.1 + $500/1.1 + $500/1.1 + $500/1.1 $3, 184.93 ≈ Now you must solve 2 3 4 eac + eac/1.1 + eac/1.1 + eac/1.1 + eac/1.1 = $3, 184.93 which is eac (1 + 0.9091 + 0.8264 + 0.7513 + 0.6830) $3, 184.93 · ≈ eac $3, 184.93/4.1699 $763.80 ) ≈ ≈ Put differently, you would be indifferent between this 5-year lease and payment of $763.80 per month, first payment immediately. Using the annuity formula with five years duration and an interest rate of 10%, you get eac 1 5 $3, 184.93 1 – = $3, 184.93 eac $840.17 10% · 1.1 ) ≈ 3.7908 ≈ with the first payment at the end of the year. App.3.A. Different Lives and Rental Equivalents 5 Ready to move on to a real-world example? My car lease quoted $1,500 due at A real-world example signing, followed by $500 per month for 35 months. What would be the EAC for this contract, assuming the prevailing interest rate was 0.5% per month? The present value cost of this contract was $500 1 35 $1, 500 + 1 – $1, 500 + $16, 018 = $17, 518 0.005 · 1.005 ≈ The equivalent annual cost—i.e., what a rental without an upfront payment would have been—is therefore eac 1 36 $17, 518 1 – $17, 518 eac = $532.93 (3.2) 0.005 · 1.005 ≈ ) 32.8710 ≈ payable only at the end of each month. Comparing Different Multiyear Contracts Let’s now compare two multiyear leases, instead of a multiyear lease versus an annual For EAC, are all years rent. For example, compare the 3-year lease from the previous section to the 5-year the same to you? lease. First, note that before you even ask this question, you should consider your use of the building. If you need it for only 3 years, you should obviously choose the 3-year lease. If you need it for exactly 5 years, you would have to figure out how much it would cost you to obtain leases for 2 more years if you went with the 3-year lease. However, we shall make our lives simple. The particular question that we are interested in assumes that you do not care about whether you sign a 3-year or a 5-year lease. You only care about lowest cost. On to the substance. The 3-year lease costs $2,128.10. The 5-year lease costs Compare 3-year to $3,184.93. Obviously, the 3-year lease is cheaper. Does this mean that the 3-year lease 5-year lease. is better? Obviously not—the 5-year lease gives you 5 years of access, not just 3 years. This is why a 5-year lease is more expensive. So, how can you compare these two leases? You have two methods, which always come to the same answer: 1. Repeated lease: You can repeat both leases until they end up with the same number of years. For example, to compare a 3-year lease with a 5-year lease, you would work out what 15 years worth of leases would cost. That is, you would compare the cost of 5 consecutive 3-year leases with the cost of 3 consecutive 5-year leases. We already worked out that a single 3-year lease beginning now would cost $2,128.10. Thus, the first 3-year lease would cost $2,128.10 in year 0. You would have to repeat it in year 3, when it would cost you another $2,128.10 then. Repeat this in year 6, in year 9, and in year 12. Your present value cost of a 15-year lease is therefore $2, 128.10 $2, 128.10 $2, 128.10 $2, 128.10 $2, 128.10 $6, 509 + 3 + 6 + 9 + 12 1.1 1.1 1.1 1.1 ≈ Your alternative 5-year lease would cost $3,184.93 in year 0, $3,184.93 in year 5, and $3,184.93 in year 10. Therefore, your cost would be 6 Stock and Bond Valuation: Annuities and Perpetuities $3, 184.93 $3, 184.93 $3, 184.93 $6, 390 + 5 + 10 1.1 1.1 ≈ Consequently, the 5-year lease is cheaper. This method works, but it is quite tedious. If you had to compare four different leases, say, a 3-year, 5-year, 7-year, and 11-year lease, you would have to work out what these leases cost over a 1,155-year period. 2. Work out the equivalent annual costs: Instead of comparing leases to one another, work out what their equivalent annual rents would be, and compare these. Well, you have already worked this out for these two leases. The 3-year lease has an EAC of $777.95; the 5-year lease has an EAC of $763.80. Therefore, the 5-year contract is cheaper on a per-annum basis. (If you used the year-end payment EAC, the cost of both would be 10% higher, so the 5-year lease would still be cheaper.) Moreover, you can use this to compare any number of contracts easily.
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