1. Discounted cash flow valuation Chapter 5

2. 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 Understand how interest rates are quoted 5-2 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

3. 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 5-3 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

4. 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 5-4 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. 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 3 years in a bank account paying 8% interest. You currently have $7000 in the account. How much will you have in 3 years? How much in 4 years? 5-5 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

6. Future value: Multiple cash flows Example 5.1—Formulas Find the value at year 3 of each cash flow and add them together – Year 0: FV = $7000(1.08) 3 = $ 8 817.98 – Year 1: FV = $4000(1.08) 2 = $ 4 665.60 – Year 2: FV = $4000(1.08) 1 = $ 4 320.00 – Year 3: value = $ 4 000.00 – Total value in 3 years = $21 803.58 Value at year 4 = $21 803.58(1.08)= $23 547.87 5-6 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

7. Future value: Multiple cash flows Example 5.1—Calculator Calculator keys: – Today (year 0 CF): 3 N; 8 I/Y; -7000 PV; CPT FV = 8817.98 – Year 1 CF: 2 N; 8 I/Y; -4000 PV; CPT FV = 4665.60 – Year 2 CF: 1 N; 8 I/Y; -4000 PV; CPT FV = 4320 – Year 3 CF: value = 4000 – Total value in 3 years = 8817.98 + 4665.60 + 4320 + 4000 = 21 803.58 Value at year 4: 1 N; 8 I/Y; -21 803.58 PV; CPT FV = 23 547.87 5-7 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

8. 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 3 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 = $ 300.00 – Total value in 3 years = $628.49 5-8 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

9. Future value: Multiple cash flows Example 5.2 (cont.) How much in 5 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 5-9 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

10. Future value: Multiple cash flows Example 5.2—Formulas and time line 5-10 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

11. Future value: Multiple cash flows Example 5.2 (cont.) Calculator keys: – Year 1 CF: 2 N; 7 I/Y; -100 PV; CPT FV = 114.49 – Year 2 CF: 1 N; 7 I/Y; -200 PV; CPT FV = 214.00 – Year 3 CF: value = 300 – Total value in 3 years = 114.49+214.00+300.00 = 628.49 – Total FV in 5 years = 628.49*(1.07) 2 = 719.56 5-11 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

12. Future value: Multiple cash flows: Another example Suppose you plan to deposit $100 into an account in one year and $300 into the account in 3 years. How much will be in the account in 5 years if the interest rate is 8%? – FV = 100(1.08) 4 + 300(1.08) 2 = 136.05 + 349.92 = $485.97 Calculator keys: – Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV = 136.05 – Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV = 349.92 – Total FV = 136.05 + 349.92 = 485.97 5-12 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

13. Example—Time line 100 012345 300 136.05 349.92 $485.97 5-13 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

14. Present value: Multiple cash flows Example 5.3 You are offered an investment that will pay: – $200 in year 1; – $400 the next year; – $600 the following year; and – $800 at the end of the 4th year. You can earn 12% on similar investments. What is the most you should pay for this one? 5-14 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

15. Present value: Multiple cash flows Example 5.3—Formula Find the PV of each cash flow and add them: – Year 1 CF: $ 200 / (1.12) 1 = $ 178.57 – Year 2 CF: $ 400 / (1.12) 2 = $ 318.88 – Year 3 CF: $ 600 / (1.12) 3 = $ 427.07 – Year 4 CF: $ 800 / (1.12) 4 = $ 508.41 – Total PV = $ 1432.93 5-15 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

16. Present value: Multiple cash flows Example 5.3—Calculator Calculator: – Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = - 178.57 – Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = - 318.88 – Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV = - 427.07 – Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV = - 508.41 – Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1432.93 5-16 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

17. Present value: Multiple cash flows Example 5.3—Time line 01234 200400600800 178.57 318.88 427.07 508.41 1,432.93 = 1/(1.12) 4 x = 1/(1.12) 3 x Time (years) = 1/(1.12) 2 x 5-17 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

18. Present value: Multiple cash flows Example 5.4 You are offered an investment that will pay: – 3 payments of $5000 the first $5000 in four years from today the second $5000 in five years the third $5000 in six years You can earn 11%. What is the future value of cash flows? 5-18 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

19. Present value: Multiple cash flows Example 5.4 (cont.) First method: – Year 4: FV= 5000(1.11) 2 =6160.50 – Year 5: FV= 5000(1.11)=5550 – Year 6: Value =5000 – Total=16710.50 – PV = 16710.50/(1.11) 6 =8934.12 Second method—using PV one at a time: – Year 6: CF: 5000/(1.11) 6 =2673.20 – Year 5: CF: 5000/(1.11) 5 =2967.26 – Year 4: CF: 5000/(1.11) 4 =3293.65 – Total PV=8934.12 5-19 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

20. Present value: Multiple cash flows Another example—Formula solution You are considering an investment that will pay you $1000 in one year, $2000 in two years and $3000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?  PV = $1,000 / (1.1) 1 = $ 909.09  PV = $2,000 / (1.1) 2 = $1652.89  PV = $3,000 / (1.1) 3 = $2253.94  PV = $4815.92 5-20 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

21. Present value: Multiple cash flows Another example—Calculator solution Calculator : – N = 1; I/Y = 10; FV = 1000; CPT PV = -909.09 – N = 2; I/Y = 10; FV = 2000; CPT PV = -1652.89 – N = 3; I/Y = 10; FV = 3000; CPT PV = -2253.94 – Total PV= $4815.92 5-21 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

22. Calculator hints Use the internal memory of the calculator to store cash flows. Use of cash flow or CF key – Clear all [CF]-[2nd]-[CLR WORK] – Enter period 0 cash flow (use [+/-] to change the sign) – Press [ENTER] to enter the figure in cash flow register 5-22 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

23. Example: Spreadsheet strategies You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows. Setting the data up is half the battle—once it is set up properly, you can simply copy the formulas. Click on the Excel icon for an example. 5-23 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

24. Decisions, decisions… Your broker calls you and tells you that he has a great investment opportunity for you. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment? – Use the CF keys to compute the value of the investment: CF; CF 0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1 NPV; I = 15; CPT NPV = $91.49 – No: the broker is charging more than you would be willing to pay. 5-24 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

25. Saving for retirement You are offered the opportunity to put some money away for retirement. You will receive 5 annual payments of $25 000 each, beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%? – Use cash flow keys: CF; CF 0 = 0; C01 = 0; F01 = 39; C02 = 25000; F02 = 5; NPV; I = 12; CPT NPV = $1084.71 5-25 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

26. Saving for retirement time line 0 1 2 … 39 40 41 42 43 44 0 0 0 … 0 25K 25K 25K 25K 25K Notice that the year 0 cash flow = 0 (CF 0 = 0) The cash flows for years 1 – 39 are 0 (C01 = 0; F01 = 39) The cash flows for years 40 – 44 are 25 000 (C02 = 25 000; F02 = 5) 5-26 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

27. Quick quiz: Part 1 Suppose you are looking at the following possible cash flows: – Year 1: CF = $100 – Years 2 and 3: CFs = $200 – Years 4 and 5: CFs = $300 – The required discount rate is 7%. What is the value of the CFs at year 5? What is the value of the CFs today? 5-27 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

28. Quick quiz: Part 1 Solution: Calculator Use the uneven cash flow keys and find the present value first, then compute the others based on that (easiest solution): – CF 0 = 0; C01 = 100; F01 = 1; C02 = 200; F02 = 2; C03 = 300; F03 = 2; I = 7; CPT NPV = 874.17 – Value in year 5: PV = 874.17; N = 5; I/Y = 7; CPT FV = 1226.07 5-28 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

29. Quick Quiz: Part 1 Solution: Formula Using formulas and one CF at a time: – Year 1 CF: FV 5 = 100(1.07) 4 = 131.08; PV 0 = 100 / 1.07 = 93.46; FV 3 = 100(1.07) 2 = 114.49 – Year 2 CF: FV 5 = 200(1.07) 3 = 245.01; PV 0 = 200 / (1.07) 2 = 174.69; FV 3 = 200(1.07) = 214 – Year 3 CF: FV 5 = 200(1.07) 2 = 228.98; PV 0 = 200 / (1.07) 3 = 163.26; FV 3 = 200 – Year 4 CF: FV 5 = 300(1.07) = 321; PV 0 = 300 / (1.07) 4 = 228.87; PV 3 = 300 / 1.07 = 280.37 – Year 5 CF: FV 5 = 300; PV 0 = 300 / (1.07) 5 = 213.90; PV 3 = 300 / (1.07) 2 = 262.03 – Value at year 5 = 131.08 + 245.01 + 228.98 + 321 + 300 = 1226.07 – Present value today = 93.46 + 174.69 + 163.26 + 228.87 + 213.90 = 874.18 5-29 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

30. Quick quiz: Part 1 Solution: Excel 5-30 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

31. Quick quiz: Part 1 Timeline 5-31 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

32. Valuing level cash flows Annuities and perpetuities Annuity—finite series of equal payments that occur at regular intervals – If the first payment occurs at the end of the period, it is called an ordinary annuity – If the first payment occurs at the beginning of the period, it is called an annuity due Perpetuity—infinite series of equal payments 5-32 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

33. Annuities and perpetuities Basic formulas Perpetuity: PV = C/r Annuities: 5-33 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

34. Annuities and the calculator The [PMT] key on the calculator is used for the equal payment. The sign convention still holds. Ordinary annuity versus annuity due – You can switch your calculator between the two types by using the [2nd] [BGN] [2nd] [SET] on the TI BA-II Plus. – If you see ‘BGN’ or ‘Begin’ in the display of your calculator, you have it set for an annuity due. – Most problems arise with ordinary annuities. 5-34 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

35. Annuities present value Spreadsheet strategy The present value and future value formulas in a spreadsheet include a place for annuity payments. Double-click on the Excel icon to see an example. 5-35 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

36. Annuity Example 5.5 You can afford $632 per month. Going rate = 1%/month for 48 months. How much can you borrow? You borrow money TODAY so you need to compute the present value. 48 [N] 1 [I/Y] 632 [+/-][PMT] 0 [FV] [CPT][PV]= 23,999.54 ($24,000) =PV(0.01,48,-632,0) 5-36 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

37. Annuity—Sweepstakes example Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual instalments of $333 333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? – PV = 333 333.33[1 – 1/1.05 30 ] /.05 = $5 124 150.29 5-37 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

38. Annuity—Sweepstakes example Calculator and Excel solution 5-38 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

39. Buying a house You are ready to buy a house and you have $20 000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36 000. The bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed-rate loan. How much money will the bank loan you? How much can you offer for the house? 5-39 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

40. Buying a house (cont.) Bank loan – Monthly income = 36 000 / 12 = 3000 – Maximum payment =.28(3000) = 840 – Present value= = 840[1 – 1/1.005 360 ] /.005 = $140 105 360 [N] (30*12) 0.5 [I/Y] 840 [+/-][PMT] [CPT][PV]= 140 105 Total Price – Closing costs =.04(140 105) = 5604 – Down payment = 20 000 – 5604 = 14 396 – Total price = 140 105 + 14 396 = 154 501 = PV(.005,360,-840,0) 5-40 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

41. Quick quiz: Part 2 You know the payment amount for a loan and you want to know how much was borrowed. – Do you compute a present value or a future value? 5-41 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

42. Quick quiz: Part 2 (cont.) You want to receive $5000 per month in retirement. If you can earn.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement? – 300 [N]  Months – 0.75 [I/Y]  Monthly rate – 5000 [PMT]  Monthly Payment – 0 [FV] – [CPT][PV] -595 808.11 =PV(0.0075,300,5000,0) 5-42 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

43. Finding the payment Suppose you want to borrow $20 000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 =.66667% per month). If you take a 4-year loan, what is your monthly payment? 4(12) = 48 [N] 0.66667 [I/Y] 20,000 [PV] 0 [FV] [CPT][PMT]= - 488.26 =PMT(0.006667,48,20000,0) 5-43 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

44. Example: Spreadsheet strategies— Annuity payment Another TVM formula that can be found in a spreadsheet is the payment formula: – PMT(rate, nper, pv, fv) – The same sign convention holds as for the PV and FV formulas Click on the Excel icon for an example. 5-44 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

45. Finding the number of payments Example 5.6 $1000 is due on a credit card Payment = $20 month minimum Rate = 1.5% per month – How long would it take to pay off the $1000? – Formula solution: 1000 = 20(1 – 1/1.015 t ) /.015.75 = 1 – 1 / 1.015 t 1 / 1.015 t =.25 1 /.25 = 1.015 t t = ln(1/.25) / ln(1.015) = 93.111 months = 7.75 years – And this is only if you don’t charge anything more on the card! 5-45 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

46. Finding the number of payments Example 5.6 (cont.) Calculator solution: – Sign convention is important Spreadsheet solution 1.5 [I/Y] 1000 [PV] 20 [+/-][PMT] 0 [FV] [CPT][N]= 93.111 months = 7.75 years =NPER(0.015,-20,1000,0) 5-46 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

47. Finding the number of payments— Another example Suppose you borrow $2000 at 5% and you are going to make annual payments of $734.42. How long before you pay off the loan? – 2000 = 734.42(1 – 1/1.05 t ) /.05 –.136161869 = 1 – 1/1.05 t – 1/1.05 t =.863838131 – 1.157624287 = 1.05 t – t = ln(1.157624287)/ln(1.05) = 3 years 5-47 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

48. Finding the number of payments— Another example (cont.) Calculator solution: Spreadsheet solution: 5 [I/Y] 2000 [PV] 734.42[+/-][PMT] 0 [FV] [CPT][N]= 3 years =NPER(0.05,-734.42,2000,0) 5-48 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

49. Finding the rate Suppose you borrow $10 000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate? Calculator keys Spreadsheet function 60 [N] 10000 [PV] 207.58[+/-][PMT] 0 [FV] [CPT][I/Y]=.75% =RATE(60,-207.58,10000,0) 5-49 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

50. Annuity—Finding the rate without a financial calculator Trial and error method: – Choose an interest rate and compute the PV of the payments based on this rate. – Compare the computed PV with the actual loan amount. – If the computed PV > loan amount, then the interest rate is too low. – If the computed PV < loan amount, then the interest rate is too high. – Adjust the rate and repeat the process until the computed PV and the loan amount are equal. 5-50 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

51. Quick quiz: Part 3 You want to receive $5000 per month for the next 5 years. How much would you need to deposit today if you can earn.75% per month? 60 [N] 0.75 [I/Y] 5000 [PMT] 0 [FV] [CPT][PV]= -240866.87 =PV(0.0075,60,5000,0) 5-51 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

52. Quick quiz: Part 3 (cont.) You want to receive $5000 per month for the next 5 years. What monthly rate would you need to earn if you only have $200 000 to deposit? 60 [N] 200000 [+/-][PV] 5000 [PMT] 0 [FV] [CPT][I/Y]= 1.4395% =RATE(60,5000,-200000,0) 5-52 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

53. Quick quiz: Part 3 (cont.) Suppose you have $200 000 to deposit and you can earn.75% per month. – How many months could you receive the $5000 payment? 0.75 [I/Y] 200000 [+/-][PV] 5000 [PMT] 0 [FV] [CPT][N]= 47.73 months ≈ 4 years =NPER(0.0075,5000,-200000,0) 5-53 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

54. Quick quiz: Part 3 (cont.) Suppose you have $200 000 to deposit and you can earn.75% per month. – How much could you receive every month for 5 years? 60 [N] 0.75[I/Y] 200000[+/-][PV] 0 [FV] [CPT][PMT]= 4151.67 =PMT(0.0075,60,-200000,0) 5-54 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

55. Future values for annuities Suppose you begin saving for your retirement by depositing $2000 per year in a superannuation fund. If the interest rate is 7.5%, how much will you have in 40 years? 40 [N] 7.5[I/Y] 0 [PV] 2000[+/-][PMT] [CPT][FV]= 454513.04 =FV(0.075,40,-2000,0) 5-55 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

56. Annuity due An annuity for which the cash flows occur at the beginning of the period. You are saving for a new house and you put $10 000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years? [2nd][BGN][2nd][SET] 3 [N] 8[I/Y] 0 [PV] 10000[+/-][PMT] [CPT][FV]= 35061.12 [2nd][BGN][2nd][SET] =FV(0.08,3,-10000,0,1) 5-56 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

57. Annuity due time line 0 1 2 3 $10 000 $10 000 $10 000 $32 464 $35 061.12 5-57 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

58. Perpetuities— Example 5.7 An annuity in which the cash flows continue forever. Perpetuity formula: PV = PMT / r Current required return: – 40 = 1 / r – r =.025 or 2.5% per quarter Dividend for new preferred: – 100 = PMT /.025 – PMT = 2.50 per quarter 5-58 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

59. Summary of annuity and perpetuity calculations Table 5.2 5-59 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

60. Example: Work the Web is one of the sites that has a financial calculator. <www.moneychimp.com Click on the information icon, and work out the following example using the website calculator. Suppose you retire with $1 000 000. The growth rate is 9%. How much you can withdraw for next 30 years. Do the calculation with a calculator and compare the results. 5-60 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

61. Quick quiz: Part 4 You want to have $1 million to use for retirement in 35 years. Q1: If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month? 420 [N] 1 [I/Y] 0 [PV] 1000000 [FV] [CPT][PMT]= -155.50 =PMT(0.01,420,0,1000000) 5-61 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

62. Quick quiz: Part 4 (cont.) Q2: If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made today? [2nd][BGN][2nd][SET] 420 [N] 1 [I/Y] 0 [PV] 1000000 [FV] [CPT][PMT]= -153.96 [2nd][BGN][2nd][SET] =PMT(0.01,420,0,1000000,1) 5-62 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

63. Quick quiz: Part 4 (cont.) You are considering preferred stock that pays a quarterly dividend of $1.50. If your desired return is 3% per quarter, how much would you be willing to pay? – $1.50/0.03 = $50 5-63 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

64. Effective annual rate (EAR) This is the actual rate paid (or received) after accounting for compounding that occurs during the year. If you want to compare two alternative investments with different compounding periods, you need to compute the EAR and use that for comparison. 5-64 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

65. Annual percentage rate (APR) This is the annual rate that is quoted by law. By definition APR = period rate times the number of periods per year. So, to get the period rate we rearrange the APR equation: – Period rate = APR/number of periods per year You should NEVER divide the effective rate by the number of periods per year—it will NOT give you the period rate. 5-65 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

66. Computing APRs What is the APR if the monthly rate is.5%? –.5(12) = 6% What is the APR if the semi-annual rate is.5%? –.5(2) = 1% What is the monthly rate if the APR is 12%, with monthly compounding? – 12 / 12 = 1% – Can you divide the above APR by 2 to get the semi- annual rate? NO!!! You need an APR based on semi- annual compounding to find the semi-annual rate. 5-66 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

67. Things to remember You ALWAYS need to make sure that the interest rate and the time period match: – If you are looking at annual periods, you need an annual rate. – If you are looking at monthly periods, you need a monthly rate. If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly. 5-67 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

68. EAR formula APR = the quoted rate m = number of compounds per year 5-68 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

69. Computing EARs—Example Suppose you can earn 1% per month on $1 invested today. – What is the APR? 1(12) = 12% – How much are you effectively earning? FV = 1(1.01)12 = 1.1268 Rate = (1.1268 – 1) / 1 =.1268 = 12.68% Suppose you put it in another account, where you earn 3% per quarter. – What is the APR? 3(4) = 12% – How much are you effectively earning? FV = 1(1.03)4 = 1.1255 Rate = (1.1255 – 1) / 1 =.1255 = 12.55% 5-69 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

70. EAR and APR on calculator [2nd][ICONV] [2nd][CLR WORK] 3 fields in worksheet: – NOM (Nominal rate-APR) – EFF (Effective annual rate) – C/Y (Compounding periods/yr) – Enter any 2 values, move to the 3 rd and press [CPT] 5-70 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

71. EAR and NOM (APR) in Excel 2 functions: =EFFECT(Nom, Nper) =NOMINAL(Eff, Nper) All rates entered as decimals Nper = number of compounding periods per year 5-71 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

72. 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 – 1 = 5.39% [2nd][ICONV]:NOM=5.25; C/Y=365 EFF=5.3899 =EFFECT(0.525,365) Second account: EAR = (1 +.053/2) 2 – 1 = 5.37% [2nd][ICONV]: NOM=5.3; C/Y=2EFF=5.3702 =EFFECT(0.53,2) 5-72 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

73. 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 / 2 =.0265 FV = 100(1.0265) 2 = $105.37 You will have more money in the first account. 5-73 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

74. Computing APRs from EARs If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: m = number of compounding periods per year 5-74 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

75. 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? [2nd][ICONV]: EFF = 12 C/Y = 12 NOM[CPT] = 11.3866 =NOMINAL(0.12,12 ) 5-75 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

76. 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 2 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] 0 [FV] [CPT][PMT] = -172.88 Spreadsheet =PMT(0.0140833,24,3500,0) 5-76 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

77. 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) 0 [PV] -50 [PMT] [CPT][FV]= 147,089.22 – Spreadsheet: =FV(0.0075,420,-50,0) 5-77 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

78. Present value with daily compounding You need $15 000 in 3 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) 0 [PMT] 15 000 [FV] [CPT][FV] = -12 718.56 – Spreadsheet: PV(0.00015,1095,0,15000) 5-78 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

79. Quick quiz: Part 5 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 do you need to use for the time value of money calculations? 5-79 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

80. 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: – 1 [N]; 10,000 [FV]; (7*90/365) [I/Y]; [CPT][PV] = -9830.33 – =PV(.07,1,0,10000) (spreadsheet formula) 5-80 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

81. 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

82. 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 / 1.09 5 ] /.09  PMT = 1285.46 =PMT(0.09,5,5000,0) = 1285.46 5 [N]; 9 [I/Y]; 5000 [PV], 0 [FV], [CPT][PMT] = 1285.46 5-82 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

83. Amortised loan with fixed payment Example: Amortisation table Spreadsheet strategies Click on the spreadsheet icon to see the example. 5-83 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

84. Example: Work the Web Several websites have calculators that will prepare amortisation tables quickly. One such website is: http://www.mortgagecalculatorplus.com/ 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? 5-84 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

85. Quick quiz: Part 6 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? 5-85 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

86. Chapter 5 END 5-86