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![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?
[ ]
11.39%or
8655152113.1)12.1(12 12/1
=−+=APR
6C-38](https://image.slidesharecdn.com/chapter6-140702191456-phpapp02/85/Chapter-6-microeconomics-38-320.jpg)
![• For interest calculations, the principal of the
loan is not reduced by the installment payments
• Interest is charged on the full sum of the
principal over 3 yrs.
Monthly payments
= [100,000 + (100,000 X 0.12 X 3)] / 36
= 3777.78
Flat Basis](https://image.slidesharecdn.com/chapter6-140702191456-phpapp02/85/Chapter-6-microeconomics-49-320.jpg)
Uploaded byNoor Asma Zakaria
Chapter 6 microeconomics
This chapter discusses discounted cash flow valuation and concepts related to future and present value of multiple cash flows. It covers annuities, perpetuities, loan amortization, and effective annual rates. Examples are provided to illustrate computing future and present values of cash flows occurring at different times, as well as growing annuities and perpetuities. Key formulas and the distinctions between annual percentage rates, effective annual rates, and stated interest rates are also explained.
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Chapter 6 microeconomics
- 1. Chapter 6 Discounted Cash FlowValuation McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
- 2. Key Concepts andSkills • 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 6C-2
- 3. Chapter Outline • Futureand Present Values of Multiple Cash Flows • Valuing Level Cash Flows: Annuities and Perpetuities • Comparing Rates: The Effect of Compounding • Loan Types and Loan Amortization 6C-3
- 4. Multiple Cash Flows– FV Example 1 • 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? – FV2 = 500(1.09)2 + 600(1.09)1 = 594.05 + 654.00 = 1,248.05 6C-4
- 5. Multiple Cash Flows– FV Example 1 Continued How much will you have in 5 years if you make no further deposits? • First way: – FV5 = 500(1.09)5 + 600(1.09)4 = 769.31 + 846.95 = 1,616.26 • Second way – use value at year 2: – FV5 = FV2 (1.09)3 = 1,248.05(1.09)3 =1,616.26 6C-5
- 6. Multiple Cash Flows– FV Example 2 • 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%? – FV5 = 100 (1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97 6C-6
- 7. Multiple Cash Flows– PV Example • You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay? – PV = 1,000 / (1.10)1 + 2,000 / (1.10)2 +3,000 / (1.10)3 = 4,815.93 6C-7
- 8. Timeline 0 1 23 1,000 2,000 3,000 909.09 1,652.89 2,253.94 4,815.93 6C-8
- 9. Decisions Based onDiscounted Cash Flows • 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 = -100 + 40 / (1.15)1 + 75 / (1.15)2 = -8.51 – No, the investment costs more than what you would be willing to pay. 6C-9
- 10. Saving For Retirement •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%? – PV = 25,000 / (1.12)40 + 25,000 / (1.12)41 +25,000 / (1.12)42 + 25,000 / (1.12)43 +25,000 / (1.12)44 = 1,084.71 6C-10
- 11. Saving For RetirementTimeline 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 (0 CF0) The cash flows in years 1 – 39 are 0 (0 CF1; 39 N) The cash flows in years 40 – 44 are 25,000 (25,000 CF2; 5 N) 6C-11
- 12. Annuities and PerpetuitiesDefined • 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 6C-12
- 13. Annuities and Perpetuities– Basic Formulas Perpetuity with 1st payment at end of the period PV = C / r Annuities with 1st payment at end of the period −+ = + − = r 1r)(1 CFV r r)(1 1 1 CPV t t 6C-13
- 14. Annuities and theCalculator • You can use the PMT key on the calculator 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, it is set for an annuity due – Most problems are ordinary annuities 6C-14
- 15. Annuity – SweepstakesExample • Suppose you win the $10 million sweepstakes. The money is paid in equal annual end-of-year installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? Using financial calculator: 30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29 6C-15 29.150.124,5 0.05 )05.0(1 1 1 333,333.33 r r)(1 1 1 CPV 30t = + − = + − =
- 16. Buying a House •You are ready to buy a house, and you have $20,000 to cover both the down payment and commission costs. Commission 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 lend you? How much can you offer for the house? 6C-16
- 17. Buying a House- Continued • Bank loan – Monthly income = 36,000 / 12 = 3,000 – Maximum mortgage payment = .28(3,000) = 840 • Total Price – Commission costs = .04(140,105) = 5,604 – Down payment = 20,000 – 5,604 = 14,396 – Total Price = 140,105 + 14,396 = 154,501 6C-17 140,105 0.06/12 0.06/12)(1 1 1 840 r r)(1 1 1 CPV 12(30)t = + − = + − =
- 18. 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 wish to make monthly installment payments over a 4- year period, what is your monthly payment? 6C-18
- 19. Finding the Numberof Payments – Another Example • Suppose you borrow $2,000 at 5%, and you are going to make annual payments of $734.42. How long will you take to pay off the loan? 6C-19 3t2,000 0.05 0.05)(1 1 1 734.42 r r)(1 1 1 CPV tt =⇒= + − = + − =
- 20. 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? 6C-20 9%r10,000 r/12 r/12)(1 1 1 207.58 r/12 r/12)(1 1 1 CPV 6012t =⇒= + − = + − =
- 21. Future Values forAnnuities • Suppose you begin saving for your retirement by depositing $2,000 per year in a savings account. If the interest rate is 7.5%, how much will you have in 40 years? 6C-21 454,513.04 0.075 10.075)(1 2,000 r 1r)(1 CFV 40t = −+ = −+ =
- 22. Annuity Due • Youare 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? 6C-22 35,061.12(1.08) 0.08 10.08)(1 10,000r)(1 r 1r)(1 CFV 3t = −+ =+ −+ =
- 23. Annuity Due Timeline 01 2 3 10000 10000 10000 32,464 35,016.12 6C-23
- 24.
- 25. Growing Annuity A growingstream of cash flows with a fixed maturity t t r gC r gC r C PV )1( )1( )1( )1( )1( 1 2 + +× ++ + +× + + = − + + − − = t r g gr C PV )1( )1( 1 6C-25
- 26. Growing Annuity: Example Aretirement 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 6C-26
- 27. Growing Perpetuity A growingstream of cash flows that lasts forever + + +× + + +× + + = 3 2 2 )1( )1( )1( )1( )1( r gC r gC r C PV gr C PV − = 6C-27
- 28. Growing Perpetuity -Example 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$ = − =PV 6C-28
- 29. 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. 6C-29
- 30. 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. • 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. 6C-30
- 31. Computing APRs • Whatis the APR if the monthly rate is 0.5%? – 0.5(12) = 6% • What is the APR if the semiannual rate is 0.5%? – 0.5(2) = 1% • What is the monthly rate if the APR is 12% with monthly compounding? – 12 / 12 = 1% 6C-31
- 32. 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. 6C-32
- 33. 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 and 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% 6C-33
- 34. EAR - Formula 1 m APR 1EAR m − += Rememberthat the APR is the quoted rate, and m is the number of compounding periods per year 6C-34
- 35. Making Decisions usingEAR • 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? 6C-35
- 36. Making Decisions usingEAR 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: FV = 100 (1 + .0525/365)365 = 105.39 – Second Account: FV = 100 (1 + .053/2)2 = 105.37 • You have more money in the first account. 6C-36
- 37. Computing APRs fromEARs If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: += 1-EAR)(1mAPR m 1 6C-37
- 38. APR - Example Supposeyou 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 8655152113.1)12.1(12 12/1 =−+=APR 6C-38
- 39. Computing Payments withAPRs 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? 6C-39 172.88C 0.169/12 0.169/12)(1 1 1 C r r)(1 1 1 C3,500 24t =⇒ + − = + − =
- 40. Future Value withMonthly 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? 6C-40 147,089.22 0.09/12 10.09/12)(1 50 r 1r)(1 CFV 35(12)t = −+ = −+ =
- 41. Present Value withDaily 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? FV = PV(1 + r)t 15,000 = PV (1+0.055/365)3(365) PV = 12,718.56 6C-41
- 42. Continuous Compounding • Sometimesinvestments or loans are figured based on continuous compounding EAR = eq – 1 – The e is a special function on the calculator normally denoted by ex • Example: What is the effective annual rate of 7% compounded continuously? EAR = e.07 – 1 = .0725 or 7.25% 6C-42
- 43. Pure Discount Loans- Example • 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? FV = PV(1 + r)t 10,000 = PV (1+0.07) PV = 9,345.79 6C-43
- 44. 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, and we will talk about them in greater detail later. 6C-44
- 45. 7,451.474C50,000 0.08 0.08)(1 1 1 C r r)(1 1 1 CPV 10t =⇒= + − = + − = Amortized Loan withFixed Payment - Example • Each payment covers the interest expense plus reduces principal • Consider the same 10 year loan with annual payments. The interest rate is 8%, and the principal amount is $50,000. – What is the annual payment? This is an ordinary annuity. . 6C-45 Microsoft Excel 97-2003 Worksheet
- 46. Amortized Loan withFixed Payment – Another Example • 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? This is an ordinary annuity. 6C-46
- 47. APR may differfrom EAR because of the different basis of interest calculation – flat basis – annual rest basis – reducing balance basis Another Reason Why APR May Differ from EAR
- 48. A borrower approacheda bank for a loan. The bank quoted a rate of 12% but did not tell him the basis of interest calculation. Below are the details of the loan. What is the EAR if flat basis applies? If annual rest basis applies? If reducing balance basis applies? Loan=$100,000 Repay over 3 years in 36 equal installments EAR =? Example
- 49. • For interestcalculations, the principal of the loan is not reduced by the installment payments • Interest is charged on the full sum of the principal over 3 yrs. Monthly payments = [100,000 + (100,000 X 0.12 X 3)] / 36 = 3777.78 Flat Basis
- 50. EAR =(1+ APR/12)12 - 1 =23.39% The flat rate of 12% is equivalent to EAR of 23.39%. 21.2%r100,000 r/12 r/12)(1 1 1 3,777.78 r r)(1 1 1 CPV 12(3)t =⇒= + − = + − = Flat Basis Continued
- 51. • For interestcalculation, the principal of the loan will only be reduced by the installment amount at the end of every year. • Interest is charged on balance at beginning of the year. Monthly PMT= 41,634.90 / 12 = 3469.58 41,634.90C100,000 0.12 0.12)(1 1 1 C r r)(1 1 1 CPV 3t =⇒= + − = + − = Annual Rest Basis
- 52. • Based onthe monthly payment of 3,469.58, what is the effective rate charged? APR = 15.06% EAR = (1+APR/12)12 - 1 = 16.1468% 15.06%r100,000 r/12 r/12)(1 1 1 3,469.58 r r)(1 1 1 CPV 3(12)t =⇒= + − = + − = Annual Rest Basis (Continued)
- 53. • For interestrate calculations, the principal of the loan will be reduced as and when the installment payments are made. APR = 12% EAR = (1+APR/12)12 - 1 =12.6825% This is the best basis for the borrower. 3,321.43C100,000 0.12/12 0.12/12)(1 1 1 3,777.78 r r)(1 1 1 CPV 12(3)t =⇒= + − = + − = Reducing Balance Basis
- 54.
Editor's Notes
- #2 The slides present solutions using formulas as well as the financial calculator.
- #8 Formula: PV = 1000 / (1.1)1 = 909.09 PV = 2000 / (1.1)2 = 1,652.89 PV = 3000 / (1.1)3 = 2,253.94
- #10 You can also use this as an introduction to NPV by having the students put –100 in for CF0. When they compute the NPV, they will get –8.51. You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV.
- #14 Lecture Tip: The annuity factor approach is a short-cut approach in the process of calculating the present value of multiple cash flows and that it is only applicable to a finite series of level cash flows. Financial calculators have reduced the need for annuity factors, but it may still be useful from a conceptual standpoint to show that the PVIFA is just the sum of the PVIFs across the same time period.
- #15 Other calculators also have a key that allows you to switch between Beg/End.
- #17 Note that the outstanding balance on the loan at any point in time is simply the present value of the remaining payments.
- #18 The bank will lend you $140,105. At most, you can offer $154,501. Formula PV = 840[1 – 1/1.005360] / .005 = 140,105
- #19 Formula 20,000 = PMT[1 – 1 / 1.006666748] / .0066667 PMT = 488.26
- #20 2000 = 734.42(1 – 1/1.05t) / .05 1/1.05t = .863838131 1.157624287 = 1.05t t = ln(1.157624287) / ln(1.05) = 3 years
- #22 Lecture Tip: It should be emphasized that annuity factor tables (and the annuity factors in the formulas) assumes that the first payment occurs one period from the present, with the final payment at the end of the annuity’s life. If the first payment occurs at the beginning of the period, then FV’s have one additional period for compounding and PV’s have one less period to be discounted. Consequently, you can multiply both the future value and the present value by (1 + r) to account for the change in timing.
- #23 Note that the procedure for changing the calculator to an annuity due is similar on other calculators. What if it were an ordinary annuity? FV = 32,464 (so you receive an additional 2,597.12 by starting to save today.)
- #24 If you use the regular annuity formula, the FV will occur at the same time as the last payment. To get the value at the end of the third period, you have to take it forward one more period.
- #28 Lecture Tip: To prepare students for the chapter on stock valuation, it may be helpful to include a discussion of equity as a growing perpetuity.
- #29 Point out to students that in this example the year 1 cash flow was given as $1.30. If the current dividend were $1.30, then we would need to multiply it by (1+g) to estimate the year 1 cash flow.
- #34 Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.
- #35 Using the calculator: The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates. 2nd I Conv NOM is the quoted rate; down arrow EFF is the effective rate; down arrow C/Y is compounding periods per year. You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.
- #36 Remind students that rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates. Calculator: 2nd I conv 5.25 NOM Enter up arrow 365 C/Y Enter up arrow CPT EFF = 5.39% 5.3 NOM Enter up arrow 2 C/Y Enter up arrow CPT EFF = 5.37%
- #37 It is important to point out that the daily rate is NOT 0.014, it is 0.014383562 Lecture Tip: Here is a way to drive the point of this section home. Ask how many students have taken out a car loan. Now ask one of them what annual interest rate s/he is paying on the loan. Students will typically quote the loan in terms of the APR. Point out that, since payments are made monthly, the effective rate is actually more than the rate s/he just quoted, and demonstrate the calculation of the EAR.
- #39 On the calculator: 2nd I conv down arrow 12 EFF Enter down arrow 12 C/Y Enter down arrow CPT NOM
- #40 See the Lecture Tip in the IM for a discussion of situations that present a mismatch between the payment period and the compounding period.
- #44 Remind students that the value of an investment is the present value of expected future cash flows.
- #46 Note: You may wish to use this opportunity to compare the two types of loan amortizations. We see that the total interest is greater for the fixed payment case. It is equal to (10 x $7451.474)-$50,000=$24,514.74 versus $22,000 for the fixed principal payment case. The reason for this is that the loan is repaid more slowly early on, so the interest is somewhat higher. This doesn’t mean that one loan is better than the other; it simply means that one is effectively paid off faster. For example, the principal reduction in the first year is only $3,451.47 in the fixed payment case as compared to $5,000 in the fixed principal payment case. Lecture Tip: Consider a $200,000, 30-year loan with monthly payments of $1330.60 (7% APR with monthly compounding). You would pay a total of $279,016 in interest over the life of the loan. Suppose instead, you cut the payment in half and pay $665.30 every two weeks (note that this entails paying an extra $1330.60 per year because there are 26 two week periods). You will cut your loan term to just 23.7 years and save almost $69,000 in interest over the life of the loan.Calculations on TI-BAII plusFirst: 200,000 PV; 360 N; 7 I/Y; 12 P/Y; CPT PMT = 1330.60 (interest = 1330.60*360 – 200,000=279,016)Second: 200,000 PV; -665.30 PMT; 7 I/Y; 26 P/Y; CPT N = 616.4 payments / 26 = 23.71 years (interest = 665.30*616.4 – 200,000=210,094.7)