Lecture 6 Multiple Cash Flows.pptx

1. Lecture 6: Evaluation of Multiple Cash Flows Prof. Victor Glass
2. Key Concepts and Skills 6F-2 • 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 interest rates are quoted • Understand how loans are amortized or paid off Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
3. Chapter Outline • Future and Present Values of Multiple Cash Flows • Valuing Level Cash Flows: Annuities and Perpetuities • Comparing Rates: The Effect of Compounding • Loan Types and Loan Amortization 6F-3 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
4. Chapter Outline • Derive Simplifying Formulas to calculate present value • Will assume fixed future inflows or inflows that grow at a steady rate • Will assume a fixed required return on investment
5. Problem taxonomy • Typical valuation problems • Initial outlay, generates periodic payments, ends with a lump sum payment • Initial outlay, generates periodic payments, no lump sum ending payment • A fixed payout forever • An initial outlay then periodic payments grow at a fixed percent • Determining a periodic mortgage payment • Other mortgages: interest only, balloon, amortized etc.
6. Problem Taxonomy • Interest rate problems • The effects of faster compounding • Annual Percentage Rate versus Effective Percentage Rate
7. Future Value Problems • Initial Deposit • Periodic Deposits of Equal value for the next several periods • Two approaches • Treat each future cash flow as a single problem and sum results • Use an Excel formula shortcut – if the future deposits are equal and end in the final period
8. Multiple Cash Flows – FV Example 1 • You think you will be able to deposit \$4,000 at the end of each of the next three years in a bank account paying 8 percent interest. • You currently have \$7,000 in the account. • How much will you have in three years? • How much will you have in four years? Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
9. Multiple Cash Flows – FV Example 1 • Find the value at year 3 of each cash flow and add them together  Today (year 0): FV = 7000(1.08)3 = 8,817.98  Year 1: FV = 4,000(1.08)2 = 4,665.60  Year 2: FV = 4,000(1.08) = 4,320  Year 3: value = 4,000  Total value in 3 years = 8,817.98 + 4,665.60 + 4,320 + 4,000 = 21,803.58 • Value at year 4 = 21,803.58(1.08) = 23,547.87 6F-9 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
10. Multiple cash flows: future value using Excel and treating each cash deposit as a separate problem • You think you will be able to deposit \$4,000 at the end of each of the next three years in a bank account paying 8 percent interest. • You currently have \$7,000 in the account. • How much will you have in three years? • How much will you have in four years? • Ans: • Type .08 in a cell, then type 7000, 4000, 4000, and 4000 in rows • Assume .08 is in cell a1. 7000 is in a2, 4000 in a3, 4000 in a4, and 4000 in a5 • =a2*(1+\$a\$1)^3 + a3*(1+\$a\$1)^2 +a4*(1+\$a\$1)^1+ a5 • Note: a1 becomes \$a\$1 by hitting F4 after typing in a1
11. Excel Future Value interest rate = 0.08 Unequal Deposits t Deposits FV 0 7000 8817.98 1 4000 4665.60 2 4000 4320.00 3 4000 4000.00 sum 21803.58 Future Value interest rate = 0.08 Unequal Deposits t Deposits FV 0 7000 =C361*(1+\$F\$358)^3 1 4000 =C362*(1+F358)^2 2 4000 =C363*(1+\$F\$358) 3 4000 =C364 sum =SUM(E361:E364)
12. Using Excel’s FV function • -FV(rate, period, payment, PV) =-FV(.08,3,4000,7000) • Note the negative sign PV 7000 Payment 4000 period 3 r 0.08 FV Formula \$21,803.58
13. Multiple Cash Flows – FV Example 2 – limit of the FV Function • Suppose you invest \$500 in a mutual fund today and \$600 in one year. • If the fund pays 9% annually, how much will you have in two years?  FV = 500(1.09)2 + 600(1.09) = 1,248.05  CAUTION: Can’t use FV formula because you didn’t put in \$600 at the end of year 2. 6F-13 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
14. Multiple Cash Flows – Example 2 Continued • How much will you have in 5 years if you make no further deposits? • First way:  FV = 500(1.09)5 + 600(1.09)4 = 1,616.26 • Second way – use value at year 2:  FV = 1,248.05(1.09)3 = 1,616.26 6F-14 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
15. Multiple Cash Flows – FV Example 3 – a continuation problem • Suppose you plan to deposit \$100 into an account in one year and \$300 into the account in three years. • How much will be in the account in five years if the interest rate is 8%?  FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97 6F-15 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
16. Present Value of Multiple Cash Flows • Two methods • Treat each inflow as a separate problem • Use the special Excel PV formula if the inflows are equal
17. Multiple Cash Flows – PV Example 4 – treat unequal cash flows as separate problems and sum • Receive \$200 at end of year 1, \$400 at end of year 2, \$600 at end of year 3, and \$800 at end of year 4 • Find the PV of each cash flows 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 = 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93 6F-17 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
18. In Excel Unequal Cash flow payments discount rate= 0.12 t Payments 1 200 178.6 2 400 318.9 3 600 427.1 4 800 508.4 1432.9 Unequal Cash flow payments discount rate= 0.12 t Payments 1 200 =C352/(1+\$F\$350)^A352 2 400 =C353/(1+\$F\$350)^A353 3 600 =C354/(1+\$F\$350)^A354 4 800 =C355/(1+\$F\$350)^A355 =SUM(E352:E355)
19. Example 4 Timeline 0 1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93 6F-19 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
21. Multiple Cash Flows –Example 5 • You are considering an investment that will pay you \$1,000 in one year, \$2,000 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 = 1000 / (1.1)1 = 909.09  PV = 2000 / (1.1)2 = 1,652.89  PV = 3000 / (1.1)3 = 2,253.94  PV = 909.09 + 1,652.89 + 2,253.94 = 4,815.92 6F-21 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
22. Proof that PV matches future cash flows Ending t Principal Interest Withrawal s Balance 0 4815.93 1 4815.93 481.59 1000.00 4297.52 2 4297.52 429.75 2000.00 2727.27 3 2727.27 272.73 3000.00 0.00
23. Decisions, Decisions • Your broker calls you and tells you that he has this great investment opportunity. • 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? • PV = 40/(1.15)+ 75/(1.15)^2 = 34.78+56.71= 91.49 • Conclusion: bad deal, losing 8.51 in opportunity cost 6F-23 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
24. Saving For Retirement: Example 6 • You are offered the opportunity to put some money away for retirement. • You will receive five 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%? • Excel • =25000/(1.12)^40 + 25000/(1.12)^41 + …. + 25000/(1.12)^44 = 1084.71 6F-24 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
25. Saving For Retirement Timeline 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 (CF0 = 0) The cash flows in years 1 – 39 are 0 (C01 = 0; F01 = 39) The cash flows in years 40 – 44 are 25,000 (C02 = 25,000; F02 = 5) 6F-25 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
26. Quick Quiz – Part I • 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 cash flows at year 5? • What is the value of the cash flows today? • What is the value of the cash flows at year 3? 6F-26 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
27. Annuities and Perpetuities • These are special cases of PV and FV problems
28. Annuities and Perpetuities Defined • 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 6F-28 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
29. Annuities and Perpetuities Basic Formulas • Perpetuity: PV = C / r • Annuities:                         r r C FV r r C PV t t 1 ) 1 ( ) 1 ( 1 1 6F-29 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
30. Examples 7 and 8 • C= 100 per year • R = .1 per year • T for annuity is 10 years • Perpetuity PV = C/r = 100/.1 =1000 • Annuity PV = C/r*[1-1/(1+r)^t] • =(100/.1)*[1- 1/(1+.1)^10] • =\$614.46 • Annuity chops off all cash flows after the end of year 10; therefore, it has a smaller PV than the perpetuity Could use: =-PV(.1,10,100) = 614.46
31. The PV annuity formula again
32. Annuity (Borrowing) – Example 9 • After carefully going over your budget, you have determined you can afford to pay \$632 per month toward a new sports car. • You call up your local bank and find out that the going rate is 1 percent per month for 48 months. • How much can you borrow? • To determine how much you can borrow, we need to calculate the present value of \$632 per month for 48 months at 1 percent per month. Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
33. Annuity – Example 9 • You borrow money TODAY so you need to compute the present value. • Formula: 54 . 999 , 23 01 . ) 01 . 1 ( 1 1 632 48                PV 6F-33 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved. =-PV(.01,48,632) = 23,999.54
34. Annuity - Sweepstakes Example 10 • Suppose you win the Publishers Clearinghouse \$10 million sweepstakes. • The money is paid in equal annual installments 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 – 1/1.0530] / .05 = 5,124,150.29 6F-34 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
35. Buying a House: Example 11 • 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, and 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? 6F-35 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
36. Buying a House - Continued • Bank loan  Monthly income = 36,000 / 12 = 3,000  Maximum payment = .28(3,000) = 840  PV = 840[1 – 1/1.005360] / .005 = 140,105 • Total Price  Closing costs = .04(140,105) = 5,604  Down payment = 20,000 – 5,604 = 14,396  Total Price = 140,105 + 14,396 = 154,501 6F-36 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
37. Annuities on the Spreadsheet - Example • The present value and future value formulas in a spreadsheet include a place for annuity payments • Click on the Excel icon to see an example 6F-37 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
38. Quick Quiz – Part II • 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? • You want to receive 5,000 per month in retirement. • If you can earn 0.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement? 6F-38 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
39. Finding the Payment; Another Common Problem. Example 12 • 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?  20,000 = C[1 – 1 / 1.006666748] / .0066667  C = 488.26 6F-39 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
40. Finding the Payment on a Spreadsheet • Another special formula in Excel for an equal payments problem is  PMT(rate,nper,pv,fv)  The same sign convention holds as for the PV and FV formulas 6F-40 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
41. Excel Payment Formula • =PMT (rate, nper, pv, [fv], [type]) • Arguments • rate - The interest rate for the loan. • nper - The total number of payments for the loan. • pv - The present value, or total value of all loan payments now. • fv - [optional] The future value, or a cash balance you want after the last payment is made. Defaults to 0 (zero). • type - [optional] When payments are due. 0 = end of period. 1 = beginning of period. Default is 0.
42. Using excel pmt formula: Example 13 PV = 250,000 Cell B3 NPER = 360 (30 years * 12 months per year) RATE = 0.58% (Same as .0058) Monthly Payment = (\$1,656.55) Formula =PMT(B5,B4,B3) = PMT(.0058,360, 250,000) The payment was left as negative to indicate that it is a cash outflow.
44. Finding the Number of Payments – Example 13  t = ln(1/.25) / ln(1.015)  = 93.111 months = 7.76 years 6F-44 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved. 1000= (20/.015)* [1-1/1.015^T] 1000*.015/20 = 1 - 1/1.015^T 1000*.015/20 -1 =-1/1.015^T 1/1.015^T=1-1000*.015/20 = 1-.75 1/1.015^T=.25 1/.25=1.015^T 4=1.1015^T ln4=T*ln(1.015) T= ln(4)/ln(1.015)
45. Special excel formula nper • Another way of solving this problem is to use the NPER formula • NPER(rate, payment, PV) • Make sure you give the payment a negative sign • NPER(.015,-20,1000)= 93.11
46. Finding the Number of Payments – Another Example 14 • Suppose you borrow \$2,000 at 5%, and you are going to make annual payments of \$734.42. • How long before you pay off the loan?  2,000 = 734.42(1 – 1/1.05t) / .05  .136161869 = 1 – 1/1.05t  1/1.05t = .863838131  1.157624287 = 1.05t  t = ln(1.157624287) / ln(1.05) = 3 years 6F-46 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
47. Finding the Rate: Another Basic Problem. Example 15 • 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? • =Rate(nper, pay, PV, guess at rate) • =Rate(60,-207.58,10,000, .05) =..75% = .0075/month 6F-47 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
48. Annuity – Finding the Rate Without a Financial Calculator • Trial and Error Process  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 6F-48 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
49. Quick Quiz – Part III • You want to receive \$5,000 per month for the next 5 years. • How much would you need to deposit today if you can earn 0.75% per month? • What monthly rate would you need to earn if you only have \$200,000 to deposit? • Suppose you have \$200,000 to deposit and can earn 0.75% per month. • How many months could you receive the \$5,000 payment? • How much could you receive every month for 5 years? 6F-49 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
50. Future Values for Annuities: Example 16 • Suppose you begin saving for your retirement by depositing \$2,000 per year in an IRA. • If the interest rate is 7.5%, how much will you have in 40 years?  FV = 2,000(1.07540 – 1)/.075 = 454,513.04  FV =(C/r) * [(1+r)^T-1] 6F-50 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
51. Special formula for FV of an annuity • = -FV(rate,period,payment) • =-FV(.075,40,2000) = \$454,513.04
52. Annuity Due: Example 17 • 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? • FV ={(C/r) * [(1+r)^T-1]]*1.08 shifts the timing of interest payment by one period  FV = 1.08*10,000[(1.083 – 1) / .08] = 35,061.12, then add \$10,000 for last period  Note: since you deposited funds at the beginning of the first period, you need to add that extra interest payment  =FV(.08,3,10000, 10000) “FV(rate,nper,pmt,PV) = 45,061 6F-52 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
54. Perpetuity – Example 18 • Suppose the Fellini Co. wants to sell preferred stock at \$100 per share. • A similar issue of preferred stock already outstanding has a price of \$40 per share and offers a dividend of \$1 every quarter. • What dividend will Fellini have to offer if the preferred stock is going to sell? Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
55. Perpetuity – Example 18 • Perpetuity formula: PV = C / r • Current required return:  C=1  r = .025 or 2.5% per quarter  PV =40 • Dividend for new preferred:  100 = C / .025  C = 2.50 per quarter 6F-55 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
56. Quick Quiz – Part IV • You want to have \$1 million to use for retirement in 35 years. • 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? • What if the first payment is made today? • 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? 6F-56 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
57. Work the Web Example • Another online financial calculator can be found at MoneyChimp. • Click on the web surfer and work the following example:  Choose calculator and then annuity  You just inherited \$5 million. If you can earn 6% on your money, how much can you withdraw each year for the next 40 years?  MoneyChimp assumes annuity due!  Payment = \$313,497.81 6F-57 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
59. Excel Special Functions for constant multiple payments  -FV(rate, nper, payment, PV) -- present value PV is typically 0 in Ch. 5  -PV(rate, nper, payment, FV) -- future value FV is typically 0 in Ch. 5  NPER(rate, payment, PV) payment must be negative Solving for t  -PMT(rate,nper,pv) Solving for C  Rate(nper, pay, PV, guess at rate) pay must be negative Solving for r  Link: https://exceluser.com/1024/excels-five-annuity-functions/
60. Growing Annuity A growing stream of cash flows with a fixed maturity t t r g C r g C r C P V ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 1 2                                   t r g g r C PV ) 1 ( ) 1 ( 1 6F-60 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
61. Growing Annuity: Example 19 A defined-benefit retirement plan offers to pay \$20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent? 57 . 121 , 265 \$ 10 . 1 03 . 1 1 03 . 10 . 000 , 20 \$ 40                   PV 6F-61 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
62. Growing Perpetuity A growing stream of cash flows that lasts forever             3 2 2 ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( r g C r g C r C P V g r C PV   6F-62 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
63. Example 20: Growing Perpetuity The expected dividend next year is \$1.30, and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream? 00 . 26 \$ 05 . 10 . 30 . 1 \$    P V 6F-63 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
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. 6F-64 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
65. Annual Percentage Rate • This is the annual rate that is quoted by law • By definition APR = period rate times the number of periods per year • Consequently, 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 6F-65 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
66. Computing APRs: Example 21 • What is the APR if the monthly rate is .5%?  .5(12) = 6% • What is the APR if the semiannual rate is .5%?  .5(2) = 1% • What is the monthly rate if the APR is 12% with monthly compounding?  12 / 12 = 1% 6F-66 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
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 6F-67 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
68. Computing EARs – Example 22 • 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 if you put it in another account, 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% 6F-68 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
69. EAR - Formula Remember that the APR is the quoted rate, and m is the number of compounding periods per year 6F-69 1 m APR 1 EAR m          Remember that the APR is the quoted rate, and m is the number of compounding periods per year Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved. Special Excel formula: Effect(APR,nper)
70. Decisions, Decisions II • You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?  First account: • EAR = (1 + .0525/365)365 – 1 = 5.39%  Second account: • EAR = (1 + .053/2)2 – 1 = 5.37% • Which account should you choose and why? 6F-70 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved. =Effect(.0525,365) =.0539
71. Decisions, Decisions II, Continued • 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 have more money in the first account. 6F-71 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
72. Computing APRs from EARs • If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:         1 - EA R) (1 m A P R m 1 6F-72 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
73. Example 23: APR • 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?   11.39% or 8655152 113 . 1 ) 12 . 1 ( 12 1 2 / 1     A P R 6F-73 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
74. Computing Payments with APRs: Example 24 • 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 \$3,500. • The loan period is for 2 years, and the interest rate is 16.9% with monthly compounding. • What is your monthly payment?  Monthly rate = .169 / 12 = .01408333333  Number of months = 2(12) = 24  3,500 = C[1 – (1 / 1.01408333333)24] / .01408333333  C = 172.88 6F-74 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
75. Future Values with Monthly Compounding: Example 25 • 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?  Monthly rate = .09 / 12 = .0075  Number of months = 35(12) = 420  FV = 50[1.0075420 – 1] / .0075 = 147,089.22 6F-75 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
76. Present Value with Daily Compounding: Example 26 • 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?  Daily rate = .055 / 365 = .00015068493  Number of days = 3(365) = 1,095  FV = 15,000 / (1.00015068493)1095 = 12,718.56 6F-76 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
77. Continuous Compounding: Example 27 • Sometimes investments or loans are figured based on continuous compounding • EAR = eq – 1  The e is a special function in Excel =exp( ) • Example: What is the effective annual rate of 7% compounded continuously?  EAR = e.07 – 1 = .0725 or 7.25% 6F-77 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
78. Quick Quiz – Part V • 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 in the time value of money calculations? 6F-78 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
79. Pure Discount Loans – Example 6.12 • Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments. • If a T-bill promises to repay \$10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?  PV = 10,000 / 1.07 = 9,345.79 6F-79 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
80. Example: Interest-Only Loan • 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, and we will talk about them in greater detail later. 6F-80 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
81. Example: Amortized Loan with Fixed Principal Payment • Consider a \$50,000, 10-year loan at 8% interest. The loan agreement requires the firm to pay \$5,000 in principal each year plus interest for that year. • Click on the Excel icon to see the amortization table 6F-81 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
82. Example: Amortized Loan with Fixed Payment • Each payment covers the interest expense plus reduces principal • Consider a 4-year loan with annual payments. The interest rate is 8%, and the principal amount is \$5,000.  What is the annual payment? • 4 N • 8 I/Y • 5,000 PV • CPT PMT = -1,509.60 • Click on the Excel icon to see the amortization table 6F-82 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
83. Work the Web Example • There are websites available that can easily prepare amortization tables • Click on the web surfer to check out the Bankrate.com website and work the following example • You have a loan of \$25,000 and will repay the loan over 5 years at 8% interest.  What is your loan payment?  What does the amortization schedule look like? 6F-83 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
84. Quick Quiz – Part VI • 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 amortized loan? • What is a good example of an amortized loan? 6F-84 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
85. Ethics Issues • Suppose you are in a hurry to get your income tax refund. • If you mail your tax return, you will receive your refund in 3 weeks. • If you file the return electronically through a tax service, you can get the estimated refund tomorrow. • The service subtracts a \$50 fee and pays you the remaining expected refund. • The actual refund is then mailed to the preparation service. • Assume you expect to get a refund of \$978. • What is the APR with weekly compounding? • What is the EAR? • How large does the refund have to be for the APR to be 15%? • What is your opinion of this practice? 6F-85 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
86. Comprehensive Problem • An investment will provide you with \$100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually? • What is the present value of the above if the payments are received at the beginning of each year? • If you deposit those payments into an account earning 8%, what will the future value be in 10 years? • What will the future value be if you open the account with \$1,000 today, and then make the \$100 deposits at the end of each year? 6F-86 Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
87. Summary: How to do Valuation Problems • Valuation problems have five elements: PV, FV, C, r, and t. With constant growth add g • PV problem – given FV, C, r, and t. Example: How much would you be willing to pay now for the opportunity to receive \$1000neach year (C) for 3 years (t) and a lump sum payment at the end of three years (FV), if your required return is 8% (r)? • FV problem – You has \$10,00 in the bank now (PV) and will deposit \$1000 each year (C) for three years (t) if you can earn 5% in the bank (r)? • R problem – You borrow \$10,000 (PV) and repay \$2,000 each year(C) for 12 years (t) what interest rate are you being charged? • C problem – You borrow \$10,000 (PV) and must pay it off in four years (t). The interest rate charged is 10%.(r), what is your annual payment? • If the problem is a monthly payment, divide r/12 and multiply 12*t • You can only use the special Excel functions if C is constant.