In this chapter, students will be able to understand: Be able to compute the future value of multiple cash flows, be able to compute the present value of multiple cash flows, be able to compute loan payments, be able to find the interest rate on a loan, understand how loans are amortised or paid off, understand how interest rates are quoted.
Discounted cash flow valuation Chapter Key concepts and skills • Be able to compute the future value of multiple cash flows • Be able to compute the present value of multiple cash flows • Be able to compute loan payments • Be able to find the interest rate on a loan • Understand how loans are amortised or paid off ã Understand how interest rates are quoted Copyright â 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-2 Chapter outline • Future and present values of multiple cash flows • Valuing level cash flows: Annuities and perpetuities • Comparing rates: The effect of compounding periods • Loan types and loan amortisation Copyright ©2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-3 Future value with multiple cash flows—Drawing and using a time line • Suppose you deposit $100 today in an account paying 8% In one year, you will deposit another $100 How much will you have in two years? – At the end of first year = 100* (1.08)+100=208 – At the end of second year = 208*(1.08)=224.64 Copyright ©2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-4 Future value: Multiple cash flows Example 5.1 • You think you will be able to deposit $4000 at the end of each of the next years in a bank account paying 8% interest • You currently have $7000 in the account • How much will you have in years? • How much in years? Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-5 Future value: Multiple cash flows Example 5.1—Formulas • Find the value at year of each cash flow and add them together – – – – – Year 0: FV = $7000(1.08)3 Year 1: FV = $4000(1.08)2 Year 2: FV = $4000(1.08)1 Year 3: value Total value in years = $ 817.98 = $ 665.60 = $ 320.00 = $ 000.00 = $21 803.58 • Value at year = $21 803.58(1.08)= $23 547.87 Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-6 Future value: Multiple cash flows Example 5.1—Calculator • Calculator keys: – Today (year CF): N; I/Y; -7000 PV; CPT FV = 8817.98 – Year CF: N; I/Y; -4000 PV; CPT FV = 4665.60 – Year CF: N; I/Y; -4000 PV; CPT FV = 4320 – Year CF: value = 4000 – Total value in years = 8817.98 + 4665.60 + 4320 + 4000 = 21 803.58 • Value at year 4: N; I/Y; -21 803.58 PV; 5-7 CPT FV = 23 547.87 Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh Future value: Multiple cash flows Example 5.2 • You deposit $100 in one year, $200 in two years and $300 in three years • How much will you have in years at 7% interest? – Year 1: FV = $100(1.07)2 = $ 114.49 – Year 2: FV = $200(1.07) = $ 214.00 – Year 3: FV = $300 – Total value in years Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh = $ 300.00 = $628.49 5-8 Future value: Multiple cash flows Example 5.2 (cont.) • How much in years if you don’t add additional amounts? – Amount in three years = 628.49 – Year 5: FV=628.49(1.07)2 = $719.56 – This can also be calculated by calculating the future value of each amount separately – Year 1: FV= 100(1.07)4 = 131.08 – Year 2: FV= 200(1.07)3 = 245.01 – Year 3: FV=300(1.07)2 = 343.47 – Total = 719.56 Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-9 Future value: Multiple cash flows Example 5.2—Formulas and time line TIMELINE -$100.00 -$200.00 -$300.00 7% $300.00 200*(1.07) = $214.00 100*(1.07)^2 = $114.49 $628.49 Total interest = $628.49-600=28.49 * (1.07)^2 = Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh $719.56 5-10 Decisions, decisions… II • Which savings accounts should you choose: – 5.25%, with daily compounding – 5.30%, with semiannual compounding • First account: • EAR = (1 + 0525/365)365 – = 5.39% • [2nd][ICONV]:NOM=5.25; C/Y=365 EFF=5.3899 • =EFFECT(0.525,365) • Second account: • EAR = (1 + 053/2)2 Copyrightã â[2nd][ICONV]: 2011 McGraw-Hill Australia Pty Ltd NOM=5.3; C/Y=2 PPTs t/a Essentials of Corporate Finance 2e by Ross et al • =EFFECT(0.53,2) Slides prepared by David E Allen and Abhay K Singh = 5.37% EFF=5.3702 5-72 Decisions, decisions… II (cont.) • Let’s verify the choice Suppose you invest $100 in each account How much will you have in each account in one year? – First account: • Daily rate = 0525 / 365 = 00014383562 • FV = 100(1.00014383562)365 = $105.39 – Second account: • Semiannual rate = 0539 / = 0265 • FV = 100(1.0265)2 = $105.37 • You will have more money in the first account Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-73 Computing APRs from EARs • If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: APR m (1 EAR) 1 m m = number of compounding periods per year Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-74 APR—Example • Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis What APR must they pay? APR 12 (1 12)1/ 12 1138655 or 11.39% [2nd][ICONV]: EFF = 12 C/Y = 12 NOM[CPT] = 11.3866 =NOMINAL(0.12,12) Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-75 Computing payments with APRs • Suppose you want to buy a new computer system and the store is willing to allow you to make monthly payments The entire computer system costs $3500 The loan period is for years and the interest rate is 16.9%, with monthly compounding What is your monthly payment? • Calculator • • • • • 2(12) = 24[N] 16.9 / 12 = 1.40833 [I/Y] 3500 [PV] [FV] [CPT][PMT] = -172.88 • Spreadsheet =PMT(0.0140833,24,3500,0) Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-76 Future values with monthly compounding • Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding How much will you have in the account in 35 years? – Calculator: • • • • • 420 [N] (35*12) 0.75 [I/Y] (9/12) [PV] -50 [PMT] [CPT][FV]= 147,089.22 – Spreadsheet: =FV(0.0075,420,-50,0) Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-77 Present value with daily compounding • You need $15 000 in years for a new car If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit? – Calculator: • 1095 [N] (3*365) • 015068493[I/Y] (5.5/365) • [PMT] • 15 000 [FV] • [CPT][FV] = -12 718.56 – Spreadsheet: PV(0.00015,1095,0,15000) Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-78 Quick quiz: Part • What is the definition of an APR? • What is the effective annual rate? • Which rate should you use to compare alternative investments or loans? • Which rate you need to use for the time value of money calculations? Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-79 Loan types and loan amortisation Pure discount loans—Example 5.11 • Bank bills are excellent examples of pure discount loans – Principal amount is repaid at some future date – No periodic interest payments are paid • If a promissory note promises to repay $10 000 in 90 days and the market interest rate is 7%, how much will the bill sell for in the market? – Calculator: – [N]; 10,000 [FV]; (7*90/365) [I/Y]; [CPT][PV] = -9830.33 – =PV(.07,1,0,10000) (spreadsheet formula) Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-80 Interest only loan—Example • Consider a 5-year, interest only loan with a 7% interest rate The principal amount is $10 000 Interest is paid annually – What would the stream of cash flows be? • Years 1–4: Interest payments of 07(10 000) = $700 • Year 5: Interest + principal = $10 700 • This cash flow stream is similar to the cash flows on corporate bonds; we will talk about them in greater detail later 5-81 Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh Amortised loan with fixed payment— Example • Each payment covers the interest expense plus reduces principal • Consider a 5-year loan with annual payments The interest rate is 9% and the principal amount is $5000 – What is the annual payment? • 5000 = PMT[1 – / 1.095] / 09 PMT = 1285.46 • =PMT(0.09,5,5000,0) = 1285.46 • [N]; [I/Y]; 5000 [PV], [FV] , [CPT][PMT] 1285.46 Copyright= © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-82 Amortised loan with fixed payment Example: Amortisation table • Spreadsheet strategies • Click on the spreadsheet icon to see the example Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-83 Example: Work the Web • Several websites have calculators that will prepare amortisation tables quickly • One such website is: • Try the following example: The amount of the loan is $250 000 You will repay the loan over the next 30 years at 6.5% What are your monthly payments? Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-84 Quick quiz: Part • What is a pure discount loan? What is a good example of a pure discount loan? • What is an interest only loan? What is a good example of an interest only loan? • What is an amortised loan? What is a good example of an amortised loan? Copyright © 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-85 Chapter END 5-86 ... t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-18 Present value: Multiple cash flows Example 5.4 (cont.) • First method: – – – – – Year... t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 5-14 Present value: Multiple cash flows Example 5.3—Formula Find the PV of each cash flow. .. make annual payments of $734.42 How long before you pay off the loan? – 2000 = 734.42(1 – 1/1.05t) / 05 – 136161869 = – 1/1.05t – 1/1.05t = 863838131 – 1.157624287 = 1.05t – t = ln(1.157624287)/ln(1.05)