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# Business Mathematics Jerome Chapter 10

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### Business Mathematics Jerome Chapter 10

1. 1. 10-1 Ordinary Ordinary 10 O rdinary A nnuities O Annuities Annuities Chapter 10 McGraw-Hill Ryerson© McGraw-Hill Ryerson©
2. 2. Ordinary Ordinary 10 Annuities Annuities 10-2 Learning Objectives After completing this chapter, you will be able to: LO-1 Define and distinguish between… … ordinary simple annuities and ordinary general annuities Calculate the… LO-2 … Future Value and Present Value of ordinary simple annuities LO-3 McGraw-Hill Ryerson© … fair market value of a cash flow stream that includes an annuity
3. 3. 10-3 Ordinary Ordinary Learning Objectives 10 Annuities Annuities Calculate the… LO-4 … principal LO-5 … LO-6 McGraw-Hill Ryerson© balance owed on a loan immediately after any payment … Present Value of and period of deferral of a deferred annuity Future Value and Present Value of ordinary general annuities
4. 4. Ordinary Ordinary Terminology 10 Annuities Annuities 10-4 Annuity LO-1 - A series of equal payments at regular intervals Term of the Annuity - the time from the beginning of the first payment period to the end of the last payment period Present Value the amount of money needed to invest today in order to receive a series of payments for a given number of years in the future McGraw-Hill Ryerson© Future Value the future dollar amount of a series of payments plus interest
5. 5. Ordinary Ordinary 10 Terminology 10-5 Annuities Annuities PMT PMT … is the amount of each payment in an annuity n … is the number of payments in the annuity payment interval … is the time between successive payments in an annuity ordinary annuities … are ones in which payments are made at the end of each payment interval McGraw-Hill Ryerson©
6. 6. Ordinary Ordinary 10 Terminology 10-6 Annuities Annuities Suppose you obtain a personal loan to be repaid by 48 equal monthly payments McGraw-Hill Ryerson© Term 48 months or 4years. payment interval 1 month ordinary annuities first payment will be due 1 month after you receive the loan, i.e. at the end of the first payment interval
7. 7. Ordinary Ordinary Terminology 10 10-7 Annuities Annuities … for an n-payment Ordinary Annuity Payment interval 0 1 2 PMT PMT 3 n-1 Interval number PMT PMT PMT Term of the annuity McGraw-Hill Ryerson© n
8. 8. Ordinary Ordinary 10 Annuities Annuities Ordinary Annuity Ordinary Ordinary Simple Annuities Simple Annuities The payment interval The payment interval = = 10-8 Ordinary Ordinary General Annuities General Annuities The payment interval The payment interval differs from differs from the compounding the compounding interval interval Monthly payments, Monthly payments, and interest is and interest is compounded monthly compounded monthly McGraw-Hill Ryerson© the compounding interval the compounding interval Monthly payments, Monthly payments, but interest is but interest is compounded semi-annually compounded semi-annually
9. 9. Ordinary Ordinary 10 Annuities Annuities Future Value 10-9 Future Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity LO-2 Assume that there are four(4) annual \$1000 payments with interest at 4% 0 1 2 \$1000 3 \$1000 \$1000 n=1 n=2 n=3 4 Interval number \$1000 \$1000 (1.04)1 \$1000 (1.04)2 \$1000 (1.04)3 Sum = FV of annuity …the sum of the future values of all the payments McGraw-Hill Ryerson©
10. 10. Ordinary Ordinary 10 Annuities Annuities Future Value 10-10 Future Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity Assume that there are four(4) annual \$1000 payments with interest at 4% 0 1 \$1000 4 Interval number \$1000 \$1000 \$1000 n = 1 \$1000 (1.04)1 n=2 \$1000 (1.04)2 n=3 \$1000 (1.04)3 Sum = FV of annuity 2 3 FV of annuity = \$1000 + \$1000(1.04) + \$1000(1.04)2 + \$1000(1.04)3 = \$1000 + \$1040+ \$1081.60 +\$1124.86 = \$4246.46 McGraw-Hill Ryerson©
11. 11. Future Value 10-11 Future Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity Ordinary Ordinary 10 Annuities Annuities Suppose that you vow to save \$500 a month for the next four months, with your first deposit one month from today. If your savings can earn 3% converted monthly, determine the total in your account four months from now. 0 1 \$500 2 \$500 4 Month 3 \$500 \$500 \$500(1+.03/12) \$500(1+.03/12)2 \$500(1+.03/12)3 Sum = FV of annuity McGraw-Hill Ryerson© Result Result \$ 500.00 501.25 502.50 503.76 \$2,007.51
12. 12. Ordinary Ordinary 10 Annuities Annuities Future Value 10-12 Future Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity Now imagine that you save \$500 every month for the next three years. Although the same logic applies, I certainly don’t want to do it this way! Since your ‘account’ was empty when you began… PV = 0 n = 3 yrs * 12 payments per year = 36 payments Using the McGraw-Hill Ryerson© …
13. 13. Ordinary Ordinary 10 Annuities Annuities Future Value 10-13 Future Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity You save \$500 every month for the next three years. Assume your savings can earn 3% converted monthly. Determine the total in your account three years from now. 12 Note Note Keys direction FV = P/Y= 3 18810.280 12 36 0 500 McGraw-Hill Ryerson© Using the formula Using the formula
14. 14. Ordinary Ordinary 10 Annuities Annuities …the Future Value 10-14 Future Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity sum of the future values of all the payments Formula Formula McGraw-Hill Ryerson© FV = PMT [ (1+ i)n - 1 i ]
15. 15. Ordinary Ordinary 10 Annuities Annuities Future Value 10-15 Future Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity You save \$500 every month for the next three years. Assume your savings can earn 3% converted monthly. Determine the total in your account three years from now. FV = PMT [ (1+ i)n - 1 i 18810.28 37.6206 0.0941 1.0025 1.0941 0.0025 12 .03 1 1 500 McGraw-Hill Ryerson© ] 36
16. 16. Ordinary Ordinary 10 Solving earlier Question Solving earlier Question using Annuities using Annuities 10-16 Annuities Annuities You vow to save \$500/month for the next four months, with your first deposit one month from today. If your savings can earn 3% converted monthly, determine the total in your account four months from now. Since your ‘account’ was empty when you began… PV = 0 n = 4 payments PMT = -500 McGraw-Hill Ryerson©
17. 17. 10-17 Ordinary Ordinary 10 Annuities Annuities Cash Flows Cash Flows ..a term that refers to payments ..a term that refers to payments that can be either … that can be either … … payments received e.g. receipts Positives Positives + + McGraw-Hill Ryerson© Treated as: Treated as: … payments made e.g. cheques Negatives Negatives -- Therefore… Therefore…
18. 18. 10-18 Ordinary Ordinary Cash Flow Sign Convention 10 Annuities Annuities Therefore… Therefore… …when you are making payments, or even making deposits to savings, these are cash outflows, Really Really payments to payments to the bank! the bank! and therefore the values must be negative! Using the McGraw-Hill Ryerson© …
19. 19. Ordinary Ordinary 10 Annuities Annuities Future Value 10-19 Future Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity PV = 0 n = 4 payments PMT -500 3 You vow to save You vow to save 0 \$500/monthfor the for the 12 \$500/month FV = 2007.51 next four months, next four months, 500 with your first with your first deposit one month deposit one month from today. from today. 4 If your If your savings can earn savings can earn We already have We already have 3% converted 3% converted these from before, so monthly,determine we don’t have to enter determine these from before, so monthly, thetotal in your we don’t have to enter the total in your them again! them again! account four account four months from now. months from now. Formula solution Formula solution McGraw-Hill Ryerson©
20. 20. Ordinary Ordinary 10 Annuities Annuities 10-20 You vow to save \$500/month for the next You vow to save \$500/month for the next four months, with your first deposit four months, with your first deposit one month from today. If your savings can one month from today. If your savings can earn 3% converted monthly, determine the earn 3% converted monthly, determine the total in your account four months from now. total in your account four months from now. Formula Formula FV = PMT PMT = \$500 n= 4 i = .03/12 = 0.0025 2007.51 4.0150 0.0100 1.0100 1.0025 0.0025 .03 12 1 1 500 McGraw-Hill Ryerson© [ (1+ i)n - 1 i 4 ]
21. 21. Ordinary Ordinary 10 10-21 Not seeing the total picture! Annuities Annuities When you use formula or a calculator’s financial functions to calculate an annuity’s Future Value, the amount each payment contributes to the future value is NOT apparent! McGraw-Hill Ryerson©
22. 22. Ordinary Ordinary 10-22 Contribution \$ FV Contributions FV Contributions 10 10% Compounded Annually 10% Compounded Annually FV FV \$10.00 14.64 \$10.00 Annuities Annuities \$10.00 \$10.00 \$10.00 \$10.00 13.31 \$10.00 \$10.00 12.10 \$10.00 \$10.00 11.00 10.00 0 McGraw-Hill Ryerson© 1 2 3 Years Years 4 5 \$61.05 \$61.05
23. 23. Ordinary Ordinary 10 Annuities Annuities Future Value 10-23 Future Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity You decide to save \$75/month for the next four years. If you invest all of these savings in an account which will pay you 7% compounded monthly, determine: a) the total in the account after 4 years b) the amount you deposited c) the amount of interest earned Extract necessary data... PMT = - \$75 =7 = 12 n = 4 * 12 = 48 PV = 0 FV = ? Total Deposits = \$75* 48 = \$3,600 Solve… McGraw-Hill Ryerson©
24. 24. 10-24 Ordinary Ordinary 10 Annuities Annuities You decide to save You decide to save \$75/month for the \$75/month for the next four years. next four years. If you invest If you invest all of these savings all of these savings in an account which in an account which will pay you 7% will pay you 7% compounded compounded monthly, determine: monthly, determine: a) the total in the a) the total in the account after 4 account after 4 years b) the amount years b) the amount you deposited you deposited c) the c) the amount of interest amount of interest McGraw-Hill Ryerson© FV == P/Y 7 12 0 4140.69 12 48 75 FV……….. \$4,140.69 Deposits…... 3,600.00 Interest Earned = \$ 540.69 Formula solution Formula solution
25. 25. Ordinary Ordinary 10 Formula Formula FV = PMT Annuities Annuities You decide to save You decide to save \$75/month for the \$75/month for the next four years. next four years. If you invest If you invest all of these savings all of these savings in an account which in an account which will pay you 7% will pay you 7% compounded compounded monthly, determine: monthly, determine: a) the total in the a) the total in the account after 4 account after 4 years b) the amount years b) the amount you deposited you deposited c) the c) the amount of interest amount of interest earned McGraw-Hill Ryerson© [ 10-25 (1+ i) - 1 i n ] 55.20924 1.005833 0.005833 0.32205 1.32205 4140.6927 .07 12 1 48 1 75 FV \$4,140.69 - Deposits 3,600.00 = Interest Earned \$540.69
26. 26. Ordinary Ordinary 10 Annuities Annuities …the PresentValue 10-26 PresentValue of an of an Ordinary Simple Annuity Ordinary Simple Annuity sum of the present values of all the payments Formula Formula McGraw-Hill Ryerson© PV = PMT [ 1-(1+ i)-n i ]
27. 27. Present Value 10-27 Present Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity Assume that there are four(4) annual \$1000 payments with interest at 4% Ordinary Ordinary 10 Annuities Annuities 0 \$1000 (1.04)-1 \$1000 (1.04) -2 \$1000 (1.04) -3 \$1000 (1.04) Sum = PV of annuity -4 McGraw-Hill Ryerson© 1 \$1000 n=1 2 \$1000 3 \$1000 4 Interval Number \$1000 n=2 n=3 n=4 …the sum of the present values of all the payments
28. 28. Ordinary Ordinary 10 Annuities Annuities Present Value 10-28 Present Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity Assume that there are four(4) annual \$1000 payments with interest at 4% 0 1 \$1000 n=1 2 \$1000 3 \$1000 4 Interval Number \$1000 \$1000 (1.04)-1 n=2 \$1000 (1.04)-2 n=3 \$1000 (1.04)-3 n=4 \$1000 (1.04)-4 PV of annuity Sum = PV of annuity = \$1000(1.04)-1 + \$1000(1.04)-2 + \$1000(1.04)-3 + \$1000 (1.04)-4 = \$961.54 + \$924.56 + \$889.00 + \$854.80 = \$3629.90 McGraw-Hill Ryerson©
29. 29. Ordinary Ordinary 10 Annuities Annuities Present Value 10-29 Present Value of an of an Ordinary Simple Annuity Ordinary Simple Annuity You overhear your friend saying the he is repaying a loan at \$450 every month for the next nine months. The interest rate he has been charged is 12% compounded monthly. Calculate the amount of the loan, and the amount of interest involved. …Since you are making payments, not receiving them, PMT = 450 …Since you are making payments, not receiving them, PMT = -450 … n = 9 payments … Repaid 9 payments at \$450 = \$4,050 … Interest - use 12, not .12 when using financial calculator … Interest - use 12, not .12 when using financial calculator … At the end of the loan, you don’t owe any money, so FV = 0 McGraw-Hill Ryerson© Solve…
30. 30. 10-30 Ordinary Ordinary 10 Annuities Annuities You overhear your You overhear your friend saying the friend saying the he is repaying a he is repaying a loan at \$450 every loan at \$450 every month for the next month for the next nine months. nine months. The interest rate he The interest rate he has been charged is has been charged is 8% compounded 8% compounded monthly. Calculate monthly. Calculate the amount of the the amount of the loan, and the loan, and the amount of interest amount of interest involved. involved. McGraw-Hill Ryerson© PV = 3,918.24 12 8 9 0 450 Amount Borrowed (PV) \$ 3,918.24 Repaid.…………………. 4,050.00 Interest Paid = \$ 131.76 Formula solution Formula solution
31. 31. Ordinary Ordinary 10 PV = PMT Formula Formula Annuities Annuities You overhear your You overhear your friend saying the friend saying the he is repaying a he is repaying a loan at \$450 every loan at \$450 every month for the next month for the next nine months. nine months. The interest rate he The interest rate he has been charged is has been charged is 8% compounded 8% compounded monthly. Calculate monthly. Calculate the amount of the the amount of the loan, and the loan, and the amount of interest amount of interest involved. involved. McGraw-Hill Ryerson© 10-31 -n [ 1-(1+ i) ] i -0.0580479 1.006667 0.006667 3,918.24 0.94195 .08 12 1 9 1 450 Repaid \$4,050.00 - Borrowed \$3,918.24 = Interest Charged \$131.76
32. 32. 10-32 Ordinary Ordinary 10 Annuities Annuities Contribution of Each Payment to an Annuity’s Present Value McGraw-Hill Ryerson©
33. 33. Ordinary Ordinary 10 Annuities Annuities PV PV PV Contributions PV Contributions \$10.00 \$10.00 9.09 8.20 \$10.00 \$10.00 \$10.00 \$10.00 7.51 \$10.00 \$10.00 \$10.00 \$10.00 6.83 \$10.00 \$10.00 0 McGraw-Hill Ryerson© 10-33 Contribution \$ 1 2 3 Years Years 4 6.21 5 \$37.91 \$37.91
34. 34. Ordinary Ordinary 10 10-34 Annuities Annuities LO-3 …of a cash flow McGraw-Hill Ryerson© stream that includes an annuity
35. 35. Ordinary Ordinary 10-35 You have received two offers on a building lot that you want to sell. Annuities Annuities 10 LO-3 Ms. Armstrong’s offer is \$25,000 down plus a \$100,000 lump sum payment five years from now. Mr. Belcher has offered \$20,000 down plus \$5000 every quarter for five years. Compare the economic values of the two offers if money can earn 5% compounded annually. McGraw-Hill Ryerson©
36. 36. Ordinary Ordinary 10 Annuities Annuities 10-36 On what information On what information ocu should we should we The economic value of a payment stream ffocus? ocus? on a particular date (focal date) refers to a single amount that is an economic substitute for the payment stream WE need to choose a focal date, and determine the values of the two offers at that focal date. (Obvious choices would be today, the date of the offers, or the end of the term i.e. 5 years from now.) McGraw-Hill Ryerson© Back to Offer Comparison Back to Offer Comparison
37. 37. Ordinary Ordinary 10 Annuities Annuities 10-37 You have received two offers on a building lot that you want to sell. Ms. Armstrong’s offer is \$25,000 down plus a \$100,000 lump sum payment five years from now. Mr. Belcher has offered \$20,000 down plus \$5000 every quarter for five years. Compare the economic values of the two offers if money can earn 5% compounded annually. Ms. Armstrong \$25,000 down Mr. Belcher \$20,000 down plus a \$100,000 lump plus \$5000 every quarter sum payment for five years five years from now Focal Date: Today Focal Date: Today Preparing Time Lines McGraw-Hill Ryerson©
38. 38. Ordinary Ordinary 10 Annuities Annuities Time Lines 10-38 A \$25,000 down plus a \$100,000 lump sum payment A five years from now B B \$20,000 down plus \$5,000 every quarter for five years 0 1 2 Years 3 \$25,000 Ms. Armstrong \$20,000 Mr.Belcher \$5000 every quarter \$20,000 \$20,000 \$20,000 \$20,000 \$20,000 McGraw-Hill Ryerson© 4 5 \$100,000
39. 39. 10-39 Ordinary Ordinary 10 Annuities Annuities Step 1–Determine today’s value of Ms. Armstrong’s offer You have received two offers on a building lot that you want to sell. Ms. Armstrong’s offer is \$25,000 down plus a \$100,000 lump sum payment five years from now. Mr. Belcher has offered \$20,000 down plus \$5000 every quarter for five years. Compare the economic values of the two offers if money can earn 5% compounded annually. McGraw-Hill Ryerson© PV= 103,352.62 78352.692 100,000 1 5 today’s today’s value today’s today’s value value of of value of Ms. A’s of Ms. A’s lumpsum sum lump total total offer offer 5 0 25,000 Step 2… Step 2…
40. 40. 10-40 Ordinary Ordinary 10 Annuities Annuities Step 2 – Determine today’s value of Mr. Belcher’s offer. You have received two offers on a building lot that you want to sell. Ms. Armstrong’s offer is \$25,000 down plus a \$100,000 lump sum payment five years from now. Mr. Belcher has offered \$20,000 down plus \$5000 every quarter for five years. Compare the economic values of the two offers if money can earn 5% compounded annually. McGraw-Hill Ryerson© C/Y = 99,376.93 P/Y = 79,376.93 PV 40 1 4 5 4500 0 20 1 20000 today’s value today’s today’s value today’s ofMr. B’s Mr. of valueof of valueB’s lump total sum total lump sum offer offer
41. 41. 10-41 Ordinary Ordinary 10 Annuities Annuities Total Value Total Value of each offer of each offer Ms. Armstrong Mr.Belcher \$103,352.62 99,376.93 Difference in Offers \$ 3,975.69 Better off accepting Ms. Armstrong’s offer! McGraw-Hill Ryerson©
42. 42. Ordinary Ordinary 10 Annuities Annuities LO-4 Calculating the Calculating the 10-42 Original Loan Original Loan and a Subsequent and a Subsequent Balance Balance payment on The required monthly a five-year loan, bearing 8% interest, compounded monthly, is \$249.10. a) What was the original principal amount of the loan? a) What was the original principal amount of the loan? b) What is the balance owed just after the twentieth payment? b) What is the balance owed just after the twentieth payment? Since you are “borrowing” money, you are looking for PV Since you are “borrowing” money, you are looking for PV … and FV = 0 once you have repaid the loan! … and FV = 0 once you have repaid the loan! n = 5 yrs * 12 payments per year = 60 payments n = 5 yrs * 12 payments per year = 60 payments McGraw-Hill Ryerson©
43. 43. 10-43 Ordinary Ordinary 10 Annuities Annuities Original Principal = PV of all 60 payments PMT = 249.10 FV = 0 The required The required monthly payment monthly payment on a five-year loan, on a five-year loan, bearing 8% bearing 8% interest, interest, compounded compounded monthly, is \$249.10. monthly, is \$249.10. a) What was the a) What was the original principal original principal amount of the loan? amount of the loan? b) What is the b) What is the balance owed just balance owed just after the twentieth after the twentieth payment? payment? McGraw-Hill Ryerson© n = 5*12 = 60 i = .08/12 Original loan Original loan value value PV = 12,285.22 0 0 12 249.10 60 c= 8 1
44. 44. Ordinary Ordinary 10 Annuities Annuities Balance after 20 payments 10-44 = PV of 40 payments left PMT = 249.10 FV = 0 n = 60 - 20 = 40 i = .08 The required The required monthly payment monthly payment on a five-year loan, on a five-year loan, bearing 8% bearing 8% interest, interest, compounded compounded monthly, is \$249.10. monthly, is \$249.10. a) What was the a) What was the original principal original principal amount of the loan? amount of the loan? b) What is the b) What is the balance owed just balance owed just after the twentieth after the twentieth payment? payment? McGraw-Hill Ryerson© PV = 8,720.75 New loan New loan balance balance 40 We will leave it to you to do We will leave it to you to do the algebraic solution…! the algebraic solution…!
45. 45. 10-45 Ordinary Ordinary 10 Annuities Annuities LO-5 A Deferred Annuity may be viewed as an ordinary annuity that does not begin until a time interval (named the period of deferral) has passed McGraw-Hill Ryerson©
46. 46. Ordinary Ordinary 10 Annuities Annuities A Deferred A Deferred Annuity Annuity may be viewed as may be viewed as an an ordinary annuity ordinary annuity that does not begin that does not begin until time until aatime interval interval (named the period (named the period of deferral) of deferral) has passed has passed McGraw-Hill Ryerson© 10-46 Deferred Annuities d = Number of payment intervals in the period of deferral Two-step procedure to find PV: Two-step procedure to find PV: Calculate the present value, PV1, of the payments at the end of the period of deferral — this is just the PV of an ordinary annuity Calculate the present value, PV2, of the STEP 1 amount at the beginning of the period of deferral
47. 47. Ordinary Ordinary 10 10-47 Annuities Annuities … your friend saying the he is repaying a loan at \$450 every … your friend saying the he is repaying a loan at \$450 every month for four months. The interest rate he has been charged is month for four months. The interest rate he has been charged is 8% compounded monthly. Calculate the amount of the loan, and 8% compounded monthly. Calculate the amount of the loan, and the amount of interest involved. the amount of interest involved. …this same friend doesn’t begin to repay his loan …this same friend doesn’t begin to repay his loan for another 11 months, at rate \$500 every for another 11 months, at aarate \$500 every month for four months. The interest month for four months. The interest rate is still 8% compounded monthly. rate is still 8% compounded monthly. Determine the size of the loan Determine the size of the loan. . McGraw-Hill Ryerson© Solve…
48. 48. Ordinary Ordinary 10 Annuities Annuities Present Value Present Value of of aa Deferred Annuity Deferred Annuity 10-48 Step 1 – Determine PV of Annuity 10 months from now 10 0 11 \$500 12 \$500 13 \$500 14 Months \$500 PV PV …of the Annuity Step 2 Discount for 10 months to get today’s Loan Value Step 2 -- Discount for 10 months to get today’s Loan Value Hint: (Use Compound Discount) McGraw-Hill Ryerson©
49. 49. 10-49 Ordinary Ordinary 10 Annuities Annuities …this same friend …this same friend doesn’t begin to doesn’t begin to repay his loan repay his loan for for another 11 months, another 11 months, at rate \$500 at aarate \$500 every month every month for four for four months. The months. The interest rate is still interest rate is still 8% 8% compounded compounded monthly. monthly. Determine the size Determine the size McGraw-Hill Ryerson© PV = FV PV - 1967.11 1840.65 0 12 loanvalue value value10 months loan 10 months value today now from today now from 8 500 4 0 10
50. 50. 10-50 Ordinary Ordinary 10 Annuities Annuities LO-6 The payment interval The payment interval differs from differs from the compounding interval the compounding interval e.g. A typical Canadian mortgage has Monthly payments, but the interest is compounded semi-annually McGraw-Hill Ryerson© Using calculators… Using calculators…
51. 51. 10-51 Ordinary Ordinary 10 Annuities Annuities For those who are using For those who are using this type of calculator, this type of calculator, the C/Y the C/Y See following REVIEW worksheet worksheet will now be used will now be used For those who are using For those who are using aa non-financial calculator, non-financial calculator, new formulae new formulae will be added to find will be added to find the solution the solution McGraw-Hill Ryerson© See following
52. 52. Ordinary Ordinary 10 10-52 Annuities Annuities We can input the number of compoundings per year into the financial calculator. This can be performed by using the symbol To access this symbol use: …and you will see McGraw-Hill Ryerson©
53. 53. 10-53 Ordinary Ordinary 10 The 12 The 12 is a is a default default setting setting Annuities Annuities This display is referred to as “the worksheet”. … represents the number of Payments per Year … represents the number of Compoundings per Year To access use: Appears Appears automatically automatically Note: You can override these values by entering McGraw-Hill Ryerson© new ones! …Example …Example
54. 54. 10-54 Ordinary Ordinary 10 Annuities Annuities Typical Typical Canadian Canadian mortgage mortgage Interest is Interest is compounded compounded semi-annually semi-annually C/Y = P/Y = Using 12.00 12.00 2.00 12 2 and and payments are payments are each month. each month. Adding New Formulae McGraw-Hill Ryerson©
55. 55. 10-55 Ordinary Ordinary 10 Annuities Annuities Step Step 11 C= Step 2 Step 2 Adding New Formulae Adding New Formulae Determine the number of Interest periods per compounding interval number of interest compoundings per year number of payments per year Use c to determine i2 Use i2 = (1+i)c - 1 to calculate the equivalent periodic rate that matches the payment interval Step Step 33 McGraw-Hill Ryerson© Use this equivalent periodic rate as the value for “i” in the appropriate simple annuity formula …Example …Example
56. 56. Ordinary Ordinary 10 Step Step 11 Annuities Annuities Typical Typical Canadian Canadian mortgage mortgage 6% Interest is 6% Interest is compounded compounded semi-annually semi-annually and and payments are payments are each month. each month. Find C and i . Find C and i 2. 2 McGraw-Hill Ryerson© 10-56 To determine the number of Interest periods per ompounding interval c number of interest compoundings per year C= number of payments per year 0.166666 = C 2 12 Step 2 Step 2 Use c to determine i 2
57. 57. Ordinary Ordinary Step 2 Step 2 Use 10 Annuities Annuities 6% Interest is 6% Interest is compounded compounded semi-annually semi-annually 2 McGraw-Hill Ryerson© c to determine i 2 i2 = (1+i)c - 1 i2 = (1+ .06/2) Typical Typical Canadian Canadian mortgage mortgage and and payments are payments are each month. each month. Find C and i . Find C and i 2. 10-57 .16666 -1 i 0.166666 = 0.0049 1.0049 2 1.03 1 …another example …another example
58. 58. 10-58 Ordinary Ordinary 10 Annuities Annuities Mortgage Mortgage 5% interest 5% interest is is compounded compounded monthly monthly and and payments payments are each are each week week McGraw-Hill Ryerson© Step11 To determine the number of Step compoundings number of interest compoundings per year C= number of payments per year 0.23076 = C 12 52 Step 2 Step 2 Use c to determine i 2
59. 59. Ordinary Ordinary 10 Annuities Annuities Step 2 Step 2 Use 5% interest 5% interest is is compounded compounded monthly monthly and and payments payments are each are each week week McGraw-Hill Ryerson© c to determine i i2 = (1+i)c - 1 i2 = (1+ .05/12) Mortgage Mortgage 10-59 2 .2308 -1 1.0041667 = i2 0.0041667 0.230769 0.00096 1.00096 0.05 12 1 1 …another example …another example
60. 60. Ordinary Ordinary Is the following a 10 10-60 General Annuity? Annuities Annuities You decide to save \$50/month for the next three years. If you invest all of these savings in an account which will pay you 7% compounded semi-annually, determine the total in the account after 3 years. Criteria Criteria The payment interval The payment interval differs from differs from the compounding interval the compounding interval As the Criteria have been met, therefore, As the Criteria have been met, therefore, we need to determine C we need to determine C McGraw-Hill Ryerson©
61. 61. Ordinary Ordinary 10 Step Step 11 Find 10-61 c Annuities Annuities 0.1666 0.00575 1.00575 You decide to save \$50/month 12 2 for the next three years. Step 2 If you Step 2 Find i2 invest all of these savings in an account which will pay you 7% 1.035 compounded semi-annually, 1 determine the total in the account after 3 years. McGraw-Hill Ryerson© i2 = (1+i) - 1 i2 = (1+ .07/2).1666-1 i2 = 0.00575 c Step33 Use Step i2
62. 62. Ordinary Ordinary 10 Use Step Step 33 Annuities Annuities You decide to save \$50/month for the next three years. If you invest all of these savings in an account which will pay you 7% compounded semi-annually, determine the total in the account after 3 years. McGraw-Hill Ryerson© 10-62 i2 in the appropriate formula Formula Formula FV = PMT PMT = 50 i = .07/2 [ (1+ i)n - 1 i ] PV = 0 n = 3*12 = 36 c = 2/12 = .16666 i2 = 0.00575 0.229255 1.229255 1993.51 39.8702 0.00575 1.00575 1 36 1 50 Solve…
63. 63. 10-63 Ordinary Ordinary 10 Annuities Annuities You decide to save \$50/month for the next three years. If you invest all of these savings in an account which will pay you 7% compounded semi-annually, determine the total in the account after 3 years. McGraw-Hill Ryerson© FV C/Y = P/Y== 1993.51 120 12 2 12 50 2 7 36 0
64. 64. 10-64 Ordinary Ordinary 10 number of interest compoundings per year C= number of payments per year Annuities Annuities the value for c can be a repeating decimal SAVE c in memory… Improving the Accuracy of Calculated Results …your calculator retains at least two more digits than you see displayed! when you need the exponent for Simply the c value from memory! The value for i2 should be saved in memory as soon you calculate it! McGraw-Hill Ryerson© it later!
65. 65. Ordinary Ordinary 10 10-65 Annuities Annuities Reid David made annual deposits of \$1,000 to Fleet Bank, which pays 6% interest compounded annually. After 4 years, Reid makes no more deposits. What will be the balance in the account 10 years after the last deposit? McGraw-Hill Ryerson©
66. 66. 10-66 Ordinary Ordinary Reid David made annual deposits of \$1,000 to Fleet Bank, which pays 6% interest compounded annually. After 4 years, Reid makes no more deposits. What will be the balance in the account 10 years after the last deposit? 10 Annuities Annuities Step 1 – Determine FV1 of Annuity 10 years from now 0 1 2 3 \$1000 \$1000 \$1000 4 14 Years \$1000 FV FV11 …of the Annuity Step 2 – Determine FV using compound interest FV FV22 McGraw-Hill Ryerson©
67. 67. 10-67 Ordinary Ordinary 10 Annuities Annuities Step 1 – Determine FV1 of Annuity 10 years from now Reid David Reid David made annual made annual deposits of \$1,000 to deposits of \$1,000 to Fleet Bank, Fleet Bank, that pays that pays 6% 6% interest interest compounded compounded annually. annually. After years, Reid After 44years, Reid makes no more makes no more deposits. deposits. What will be the What will be the balance in the balance in the account account McGraw-Hill Ryerson© value at value at end of end of 4 years 4 years C/Y = P/Y = 4374.62 1.000 1.00 FV = 1 6 1 1000 0 4 Step 2… Step 2…
68. 68. 10-68 Ordinary Ordinary 10 Annuities Annuities Step 2 – Determine FV2 using compound interest Reid David Reid David made annual made annual deposits of \$1,000 to deposits of \$1,000 to Fleet Bank, Fleet Bank, that pays that pays 6% 6% interest interest compounded compounded annually. annually. After years, Reid After 44years, Reid makes no more makes no more deposits. deposits. What will be the What will be the balance in the balance in the account account McGraw-Hill Ryerson© FV = = 7834.27 4374.62 value 14 years value 14 years from now from now 0 10 Formula solution Formula solution
69. 69. Ordinary Ordinary 10 10-69 Step 1 – Determine FV of Annuity 4 years from now Annuities Annuities Reid David Reid David made annual made annual deposits of \$1,000 to deposits of \$1,000 to Fleet Bank, Fleet Bank, that pays that pays 6% 6% interest interest compounded compounded annually. annually. After years, Reid After 44years, Reid makes no more makes no more deposits. deposits. What will be the What will be the balance in the balance in the account account McGraw-Hill Ryerson© ] [ (1+ i)n - 1 Formula Formula FV = PMT i PMT = 1000 n = 4 i = 0.06 c = 1 0.262477 1.262477 4374.62 value at end value at end of 4 years of 4 years 4 1.06 1 0.06 1000 Step 2… Step 2…
70. 70. Ordinary Ordinary 10 10-70 Step 2 – Determine FV using compound interest Annuities Annuities Reid David made annual deposits of \$1,000 to Fleet Bank, which pays 6% interest compounded annually. After 4 years, Reid makes no more deposits. What will be the balance in the account 10 years McGraw-Hill Ryerson© Formula FV = PV(1 + i)n Formula PV =4374.62 n = 10 i = 0.06 11.262477 0.262477 .1708477 7834.27 4374.62 1.06 10 value 14 years value 14 years from now from now
71. 71. 10-71 Ordinary Ordinary 10 Annuities Annuities Step 1 – Determine FV of Annuity 4 years from now How much more interest will Reid David accumulate over the 14 years if his account earns 6% compounded daily? McGraw-Hill Ryerson© 1000 1 365 0 C/Y = P/Y = 4386.52 C/Y = FV 365 1 1 value at end value at end of 4 years of 4 years 4 6 0
72. 72. 10-72 Ordinary Ordinary 10 Annuities Annuities How much more interest will Reid David accumulate over the 14 years if his account earns 6% compounded daily? McGraw-Hill Ryerson© Step 2 – Determine FV in 10 years using compound interest P/Y = 7992.37 FV = 4386.52 FV = 365 1 0 365 0 3650 value 14 years value 14 years from now from now
73. 73. 10-73 Ordinary Ordinary 10 Annuities Annuities Interest Interest \$7,992.37 McGraw-Hill Ryerson© \$7,834.27
74. 74. Ordinary Ordinary 10 10-74 Annuities Annuities This completes Chapter 10 McGraw-Hill Ryerson©