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# Business Mathematics Jerome Chapter 08

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### Business Mathematics Jerome Chapter 08

1. 1. 8-1 Compound Compound 8 Interest Interest Compound Chapter 8 McGraw-Hill Ryerson©
2. 2. Compound Compound 8 Interest Interest 8-2 Learning Objectives After completing this chapter, you will be able to: Calculate the… LO-1 …Maturity Value(MV), Future Value (FV), and Present Value(PV) in compound interest applications, by both the algebraic method and the pre-programmed financial calculator …Maturity Value of compound interest for method Guaranteed Investment Certificates (GICs) …Price of "strip" bonds McGraw-Hill Ryerson©
3. 3. Compound Compound 8 Interest Interest 8-3 Learning Objectives Calculate the… LO-2 … Redemption Value of a compound interest bearing Canada Savings Bond …Payment on any date that is equivalent to one or more payments on other dates …Economic Value of a payment stream And be able to… …Adapt the concepts and equations of compound interest to cases of compound growth McGraw-Hill Ryerson©
4. 4. Compound Compound 8 Interest Interest LO-1 McGraw-Hill Ryerson© 8-4
5. 5. Compound Compound 8 Interest Interest 8-5 To better understand how Compound Interest is calculated, let’s review how we calculate Simple Interest! The formula on which we base our calculation is… Formula Formula I = Prt Here we have an amount, the Principal, which is multiplied by the Interest Rate and the Time over which the Interest is earned! As we will now see, Compound Interest uses the Sum of P & I as a base on which to calculate new Interest! McGraw-Hill Ryerson©
6. 6. Compound Compound 8 Interest Compound Interest 8-6 - Future Value Interest …the interest on the principal plus the interest of prior periods e.g. Principal + prior period interest = \$1100.00 \$1000.00 \$100.00 Interest for the next period is calculated on \$1100.00. This method will continue over the life of the loan or investment. (See later example) McGraw-Hill Ryerson©
7. 7. Compound Compound 8 Interest Interest Compound Interest 8-7 - Future Value …is the compounded amount and is the FINAL amount of the loan or investment at the end of the last period! Contrast this with… ...is the value of a loan or investment TODAY! McGraw-Hill Ryerson©
8. 8. Compound Compound 8 Interest Interest Compound Interest 8-8 - Future Value …the calculation of interest over the life of the loan or investment Let’s assume that the interest rate is 10% pa. Example: Principal + prior period interest = \$1100.00 Interest is now calculated on \$1100.00 Principal(Compounded) * 0.10 = \$110.00 New P \$1210.00 to start next period McGraw-Hill Ryerson© Graphically…
9. 9. Compound Interest Compound Compound 8 Interest 8-9 - Future Value Interest Interest Interest Interest Interest Interest Interest Interest Interest Amount \$1000 Amount \$1000 133.1 1331 1210 1100 1000 121 100 110 100 121 110 100 110 100 Compounding Compounding Compounding Period Period Period 0 McGraw-Hill Ryerson© 1 2 Time(Years) 3 Compounding Period 4
10. 10. Compound Compound 8 Interest Interest Compound Interest 8 - 10 - Future Value What happens if the interest rate changes during the life of an investment? Example… Example… McGraw-Hill Ryerson©
11. 11. Compound Compound 8 Interest Interest Compound Interest 8 - 11 - Future Value You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested \$1000 at the beginning of the term, how much will you have at the end of the last year? McGraw-Hill Ryerson©
12. 12. Compound Compound 8 Interest Interest Compound Interest 8 - 12 - Future Value You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested \$1000 at the beginning of the term, how much will you have at the end of the last year? Year 1 \$1000 \$1000 * (1 + .04) = \$1040 McGraw-Hill Ryerson© Year 2 \$1040 \$1040 * (1 + .08) Year 3 \$1123.20 \$1123.20 * (1 - .10) = \$1123.20 = \$1010.88 Year 4 \$1010.88 \$1010.88 * (1 +.09) = \$1101.86 …Alternative …Alternative
13. 13. Compound Compound 8 Interest Interest Compound Interest 8 - 13 - Future Value You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested \$1000 at the beginning of the term, how much will you have at the end of the last year? 1000(1.04)(1.08)(.90)(1.09) = \$1101.86 Solving Solving Alternative Alternative 1 -10% Solve for all for all Solve 4 years at 4 years at It is rare for interest to be It is rare for interest to be once! once! compounded only once per year! compounded only once per year! McGraw-Hill Ryerson©
14. 14. 8 - 14 Compound Compound 8 Interest Interest Compounding Frequencies and Periods Frequency Frequency No. per Year No. per Year Period Period Annually 1 1 year Semiannually 2 6 months Quarterly 4 3 months Monthly 12 1 month Daily 365 1 day McGraw-Hill Ryerson©
15. 15. 8 - 15 Compound Compound 8 Interest Interest Development of a Formula Formula Nominal or Annual Rate Number of compoundings per year Periodic Rate per period Total Number of Periods Periods (j) m (i) n Determining values for n and i McGraw-Hill Ryerson©
16. 16. Compound Compound 8 Interest Interest 8 - 16 Formulae Formulae To Determine To Determine n n Time(Years) * # of Compounding Frequencies p.a.(m) To Determine To Determine ii Annual Interest Rate(j) # of Compounding Frequencies p.a. (m) McGraw-Hill Ryerson©
17. 17. Compound Compound Determining values for 8 Interest 8 - 17 n Interest If you compounded \$100 for 3 years at 6% annually, semiannually, or quarterly, what are the values for n and i ? Formula Time(Years) * # of Compounding Frequencies per year (m) Annually 3* Semiannually 3 * Quarterly 3 * McGraw-Hill Ryerson© No. n 1 = 3 = 6 = 12 2 4
18. 18. 8 - 18 Compound Compound 8 Interest Interest Determining values for i If you compounded \$100 for 3 years at 6% annually, semiannually, or quarterly, what are the values for n and i ? Formula Formula Annual Interest Rate (j) # of Compounding Frequencies per year (m) 6% / Annually Semiannually 6% / Quarterly 6% / McGraw-Hill Ryerson© No. 1 2 4 Rate Rate -- ii = 6% = 3% = 1.5%
19. 19. 8 - 19 Compound Compound 8 Interest Interest Development of a Formula for Future Value Formula FV = PV(1 + i)n Where… PV= Present Value(Principal) i = rate per period n = number of periods McGraw-Hill Ryerson©
20. 20. Compound Compound 8 Interest Interest Compound Interest 8 - 20 - Future Value FV = PV(1 + i)n Formula Formula Steve Smith deposited \$1,000 in a savings account for 4 years at a rate of 8% compounded semiannually. What is Steve’s interest and compounded amount? Extract necessary data... PV = \$1000 n = 4X2=8 i = .08/2 = .04 McGraw-Hill Ryerson© Solve…
21. 21. Compound Compound 8 Interest Interest Compound Interest 8 - 21 - Future Value FV = PV(1 + i)n Formula Formula Solve… Using PV = \$1000 n = 8 i= .04 FV = \$1000(1 + .04)8 = \$1000(1.368569) = \$1,368.57 Principal \$1,000.00 + Interest 368.57 Compounded \$1,368.57 McGraw-Hill Ryerson©
22. 22. Compound Compound 8 Interest Interest BOTH BOTH ways will ways will be be shown! shown! McGraw-Hill Ryerson© There are two methods that can be used to 8 - 22 calculate compound interest: Use a calculator and algebraic sequencing Use the TI BAII Plus financial calculator!
23. 23. 8 - 23 Compound Compound 8 Interest Interest Use a calculator and algebraic sequencing Use a calculator and algebraic sequencing Solve… \$1000(1 + .04)8 1368.57 .04 1 \$1,368.57 \$1,368.57 8 1000 McGraw-Hill Ryerson©
24. 24. Compound Compound 8 Interest Interest 8 - 24 Use a calculator and algebraic sequencing Use a calculator and algebraic sequencing Find the following KEYS: Find the following KEYS: The Power function Key. Used to calculate the value of exponents. Used to access symbols located “above” another key, i.e. its acts like the SHIFT key on a computer keyboard. Changes the sign of the data value of the number being displayed. McGraw-Hill Ryerson©
25. 25. 8 - 25 Compound Compound 8 Interest Interest Use a calculator and algebraic sequencing Use a calculator and algebraic sequencing Find the following KEYS: Find the following KEYS: Some calculators have the yx symbol above the calculator key. The key stroke sequence to evaluate an EXPONENT that is… 8 Positive (1.04) is…1.04 Positive 8 1.368569 Negative Negative (1.04)-8 is…1.04 0.73069 McGraw-Hill Ryerson© 8
26. 26. Compound Compound 8 Interest Interest 8 - 26 Use a calculator and algebraic sequencing Use a calculator and algebraic sequencing Find the following KEYS: Find the following KEYS: Used to Store or save displayed values. Used to Recall the saved values. This calculator can store up to McGraw-Hill Ryerson© 10 values. Therefore, the calculator must be informed as to where the values are to be stored. Let’s Practise Let’s Practise
27. 27. 8 - 27 Compound Compound 8 Interest Interest Use a calculator and algebraic sequencing Use a calculator and algebraic sequencing Using the Using the key key e.g. you want to store the value ’45’. The key stroke sequence ‘to store’ is: 45 ..choose from 0 - 9 …’clear’ display The key stroke sequence ‘to recall’ is: …where you stored the value! McGraw-Hill Ryerson©
28. 28. Compound Compound 8 Interest 8 - 28 Interest Some key Keys! Some key Keys! McGraw-Hill Ryerson©
29. 29. Compound Compound 8 Interest Interest 8 - 29 Find the following KEYS: Find the following KEYS: 1. Number of compoundings (for lump payments) 2. Number of payments (for annuities) The nominal interest rate (Interest/Year) Present Value or initial(first) lump sum Represents the Periodic Annuity Payment (used in chapter 10) Future Value or terminal(last) lump sum McGraw-Hill Ryerson© Tells the calculator to compute (CPT)
30. 30. Compound Compound 8 Interest 8 - 30 Find the following KEYS: Find the following KEYS: Interest Previously, it was noted that …it is rare for interest to be …it is rare for interest to be compounded only once per year! compounded only once per year! However, we can now 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©
31. 31. 8 - 31 Compound Compound 8 Interest Interest The 12 The 12 is a is a default default setting setting 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! …more …more
32. 32. 8 - 32 Compound Compound 8 Interest Interest If the calculation does not involve more than one payment must be given the same value as Illustration Illustration McGraw-Hill Ryerson©
33. 33. 8 - 33 Compound Compound 8 Interest Interest Illustration Illustration … represents the number of Compoundings per Year In Compound Interest, P/Y must be In Compound Interest, P/Y must be given the same value as C/Y. given the same value as C/Y. Setting a new value for P/Y Setting a new value for P/Y will will automatically change the entry for C/Y automatically change the entry for C/Y to the same value to the same value We must key in this default, i.e. P/Y We must key in this default, i.e. P/Y …to scroll as the as the sequence sequence to close any worksheet to close any worksheet you have opened. you have opened. McGraw-Hill Ryerson©
34. 34. Compound Compound 8 Interest Interest There are two methods that can be used to 8 - 34 calculate compound interest: Using the TI BAII Plus financial calculator! McGraw-Hill Ryerson©
35. 35. 8 - 35 Compound Compound 8 Interest Interest Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator Steve Smith deposited \$1,000 in a savings account for 4 years at a rate of 8% compounded semiannually. What is Steve’s interest and compounded amount? Step 1 Step 1 Set the Set the frequency frequency of of interest interest compounding compounding Step 2 Step 2 Input values Input values into the into the financial keys financial keys Using McGraw-Hill Ryerson©
36. 36. 8 - 36 Compound Compound 8 Interest Interest Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator Step 1 Set the Step 1 Set the frequency frequency of of interest interest compounding compounding Steve Smith Steve Smith deposited \$1,000 deposited \$1,000 in a savings in a savings account for account for 4 years at a 4 years at a rate of 8% rate of 8% compounded compounded semiannually. semiannually. 2 What is Steve’s What is Steve’s interest and interest and compounded compounded amount? amount? McGraw-Hill Ryerson© Step 2 Input values Step 2 Input values into the into the 8.0 financial FV= 1368.57 1000 financial keys keys 4*2 0 \$1,368.57 \$1,368.57
37. 37. 8 - 37 Compound Compound 8 Interest Interest …there is no need to keep inputting 0 each time! The calculator remembers this step! You only need to input the values that have changed! McGraw-Hill Ryerson©
38. 38. 8 - 38 Compound Compound 8 Interest Interest 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 --
39. 39. Compound Compound 8 Interest Interest What is the effect on the Future Value of different Compounding Periods of Interest? McGraw-Hill Ryerson© 8 - 39
40. 40. Compound Compound 8 Interest Compound Interest 8 - 40 - Future Value Interest If you compounded \$100 for 3 years at 6% annually, semiannually, or quarterly, what are the final amounts that you would have at the end of the three (3) years ? Annual Annual FVA = 100(1.06)3 SemiSemi- FVS = 100(1.03)6 \$119.10 \$119.10 \$119.41 \$119.41 Semi = 6%/2 Quarterly Quarterly FVQ = 100(1.015)12 Quarterly = 6%/4 McGraw-Hill Ryerson© \$119.56 \$119.56
41. 41. Compound Interest Compound Compound 8 Interest 8 - 41 - Future Value Interest The Components of the Future Value of \$100 S or FV S or FV Future Value Future Value 250 Interest on FV=PV(1+i)n Interest 200 Interest on Original Principal 150 S=P(1+rt) 100 Original Principal 50 0 McGraw-Hill Ryerson© Time(Years) 1 2 3 4 5 6 7 8 9 10 11
42. 42. Compound Compound 8 Interest 8 - 42 Interest Comparisons McGraw-Hill Ryerson©
43. 43. Compound Compound 8 Interest 8 - 43 Simple Vs Compound Interest Interest Al Jones deposited \$1,000 in a savings account Al Jones deposited \$1,000 in a savings account for 5 years at 10% p.a.. for 5 years at 10% p.a.. Annual Simple nterest Annual Simple IInterest Rate of 10% Rate of 10% Annual Compound Annual Compound Rate of 10% Rate of 10% What is Al’s What is Al’s Simple nterest and Simple IInterestand What is Al’s What is Al’s IInterestand nterest and Maturity Value? Maturity Value? McGraw-Hill Ryerson© Compounded Value? Compounded Value?
44. 44. Compound Compound 8 Interest 8 - 44 Simple Vs Compound Interest Interest Al Jones deposited \$1,000 in a savings account for 5 years at 10% Al Jones deposited \$1,000 in a savings account for 5 years at 10% Simple Simple n = 5 * 1 = 5 i = .10 I = Prt I = \$1,000 * .10 * 5 = \$500 FV = \$1,000 + \$500 = \$1,500 McGraw-Hill Ryerson© Formulae Formulae Compare Compare Compound Compound FV = PV(1 + i)n I = FV – PV = \$1610.51 - \$1000 = \$610.51 FV = \$1000(1.1)5 = \$1,000 *1.6105 = \$1,610.51
45. 45. 8 - 45 Compound Compound 8 Future Values of \$100 at Interest Interest Various Rates of Interest Compounded Annually 12% Future Value FV Future Value FV 1800 1600 1400 1200 10% 1000 800 8% 600 6% 400 200 100 0 1 McGraw-Hill Ryerson© 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Years to Maturity, 16 17 n 18 19 20 21 22 23 24 25 26
46. 46. Compound Compound 8 Interest Interest Beginning Beginning Nominal Rates of Interest Compared McGraw-Hill Ryerson© + 6% + 6% Ending Ending Balance Balance Semiannual \$1,060.00 \$1,060.90 Quarterly \$1,061.36 Daily \$1,000 \$1,000 Compounding Compounding Period Period Annual Balance Balance Nominal Nominal Rate Rate 8 - 46 \$1,061.83
47. 47. 8 - 47 Compound Compound 8 Interest Future Values of \$100 at the same Nominal Rate but Future Value FV Future Value FV Interest 2500 Different Compounding Frequencies 2000 12% Compounded 1500 monthly 1000 ed nd u po y om all C 2% Annu 1 500 100 0 McGraw-Hill Ryerson© 5 10 15 Time 20 25
48. 48. Compound Compound 8 Interest 8 - 48 Compounding Daily Interest Interest Calculate the Future Value of \$2,000 compounded daily for 4 years at 4.5%. FV = PV(1 + i)n Formula Formula n = 4 * 365 = 1460 i = .045 /365 = 0.0001232 FV = \$2000(1+ .045/365)1460 = \$2,000 * 1.1972 = \$2,394.41 = \$2,000 * 1.1972 = \$2,394.41 McGraw-Hill Ryerson©
49. 49. Compound Compound 8 Interest Interest 8 - 49 Compounding Daily Interest Solve FV = 2394.41 \$2000(1+ .045/365)1460 .045 365 1 1460 2000 McGraw-Hill Ryerson© = \$2,394.41 = \$2,394.41
50. 50. Compound Compound 8 Interest 8 - 50 Compounding Daily Interest Interest Step 1 Set the Step 1 Set the frequency frequency of of interest interest compounding compounding Calculatethe the Calculate Future Value Future Value of \$2,000 of \$2,000 compounded compounded daily for 4 daily for 4 years at 4.5%. years at 4.5%. 365 FV= - 2394.41 Step 2 Input values Step 2 Input values into the into the 4.5 financial keys financial keys 2000 4 * 365 0 \$2,394.41 \$2,394.41 McGraw-Hill Ryerson©
51. 51. Compound Compound 8 Interest 8 - 51 Interest You invested \$6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2% compounded monthly. What amount will you have 41/2 years after the initial investment? Prepare a ‘time-line’ as part of the solution McGraw-Hill Ryerson©
52. 52. 8 - 52 Compound Compound 8 Interest Interest You invested \$6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2% compounded monthly. What amount will you have 41/2 years after the initial investment? 0 \$6000 2 years FV1 = PV2 i = .045/4 n = (2*4) = 8 FV1 = 6000(1+.045/4)8 = 6000(1.0936) = 6561.75 McGraw-Hill Ryerson© 4.5 years FV2 i = .052/12 n = 2.5*12 = 30 FV2 = 6561.75(1+.052/12)30 = 6561.75(1.1385) = \$7470.61
53. 53. 8 - 53 Compound Compound 8 Interest Interest Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator Step 1 Set the Step 1 Set the frequency frequency of of interest interest compounding compounding You invested \$6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 4 5.2% compounded monthly. What amount will you have 41/2 years after the initial investment? McGraw-Hill Ryerson© FV = PV FV11= PV22 Step 2 Input values Step 2 Input values into the into the 6000 financial financial keys keys 6,571.75 4*2 4.5 \$6,561.75 \$6,561.75 FV2
54. 54. 8 - 54 Compound Compound 8 Interest Interest Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator Step 1 Set the Step 1 Set the frequency frequency of of interest interest You invested compounding compounding \$6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 12 5.2% compounded monthly. What amount will you have 41/2 years after the initial investment? McGraw-Hill Ryerson© FV FV22 Step 2 Input values Step 2 Input values into the into the financial financial keys keys 7470.61 2.5*12 5.2 \$7,470.61 \$7,470.61
55. 55. Compound Compound 8 Interest 8 - 55 Interest You borrowed \$5000 at 7% compounded monthly. On the first and second anniversaries of the loan, you made payments of \$2500. What is the balance outstanding immediately following the second payment? Prepare a ‘time-line’ as part of the solution McGraw-Hill Ryerson©
56. 56. Compound Compound 8 Interest 8 - 56 Interest You borrowed \$5000 at 7% compounded monthly. On the first and second anniversaries of the loan, you made payments of \$2500. What is the balance outstanding immediately following the second payment? 1 year 2 years 0 \$5000 FV1 - \$2500 = PV2 FV2 i = .07/12 n = 12 i = .07/12 n = 12 12 FV1 = 5000(1+.07/12) FV2 = 2861.45 (1+.07/12)12 = 5000(1.072290) = 2861.45(1.072290) = 5361.45 = \$3068.30 PV2 = 5361.45 – 2500.00 = \$3068.30 – 2500.00 = 2861.45 New Balance = \$568.30 New Balance McGraw-Hill Ryerson©
57. 57. 8 - 57 Compound Compound 8 Interest Interest Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator FV – 2500 = PV FV11– 2500 = PV22 Step 1 Step 1 You borrowed \$5000 at 7% compounded monthly. On the 1st. and 2nd anniversaries of the loan, you made payments of \$2500. What is the balance outstanding immediately after the 2nd payment? McGraw-Hill Ryerson© 5000 FV= -5361.45 -2861.45 7.0 12 12 2500 \$2,861.45 \$2,861.45 FV2
58. 58. 8 - 58 Compound Compound 8 Interest Interest Using the TI BAII Plus financial calculator Using the TI BAII Plus financial calculator Step 2 Step 2 You borrowed \$5000 at 7% compounded monthly. On the 1st. and 2nd anniversaries of the loan, you made payments of \$2500. What is the balance outstanding immediately after the 2nd payment? McGraw-Hill Ryerson© FV FV22 -2861.45 - 2,861.45 3068.30 568.30 2500 \$568.30 \$568.30
59. 59. Compound Compound 8 Interest Interest McGraw-Hill Ryerson© 8 - 59
60. 60. Compound Compound 8 Interest 8 - 60 Formula for Present Value Interest Formula PV = FV(1 + i)-n Formula i Keys 1 This is the only change to the usual sequence! \$PV \$PV McGraw-Hill Ryerson©
61. 61. Compound Compound 8 Interest Interest 8 - 61 Calculating Present Value Calculating Present Value You expect to need \$1,500 in 3 years. Your bank offers 4% interest compounded semiannually. How much money must you put in the bank today (PV) to reach your goal in 3 years? Prepare the solution…(a) algebraically, and (b) by financial calculator McGraw-Hill Ryerson©
62. 62. 8 - 62 Compound Compound 8 Interest Calculating Present Value Calculating Present Value Interest Formula PV = FV(1 + i)-n Formula You expect to need \$1,500 in 3 years. Your bank offers 4% interest compounded semiannually. How much money must you put in the bank today (PV) to reach your goal in 3 years? n=3*2=6 i = .04/2 = .02 (a) PV = \$1500(1+.02)-6 = \$1500 * .8880 1,331.96 0.88797 1.02 6 = \$1,331.96 1500 McGraw-Hill Ryerson©
63. 63. Compound Compound 8 Interest Interest 8 - 63 Calculating Present Value Calculating Present Value You expect to need \$1,500 in 3 years. Your bank offers 4% interest compounded semiannually. How much money must you put in the bank today (PV) to reach your goal in 3 years? 3*2 (b) 4 1500 PV= -1,331.96 0 2 \$1331.96 \$1331.96 McGraw-Hill Ryerson©
64. 64. 8 - 64 Compound Compound 8 Interest Calculating Present Value Calculating Present Value Interest Formula PV = FV(1 + i)-n Formula What amount must you invest now at 5% compounded daily to accumulate to \$6000 after 1 year? j = 5% m = 365 PV = \$6000(1+.05/365)-365 = \$6000 * .9512 i = .05/365 n = 1*365 = 365 FV = \$6000 = \$5,707.40 5,707.40 0.9512 0.0001 1.001 .05 365 1 365 McGraw-Hill Ryerson© 6000
65. 65. 8 - 65 Compound Compound 8 Interest Calculating Present Value Calculating Present Value Interest What amount must you invest now at 5% compounded daily to accumulate to \$6000 after 1 year? 1 * 365 5 6000 PV= - 5,707.40 0 365 \$5707.40 \$5707.40 McGraw-Hill Ryerson©
66. 66. Compound Compound 8 Interest Interest 8 - 66 Equivalent Payments Equivalent Payments Two payments of \$2200 each must be made 1 and 4 years from now. If money can earn 5% compounded monthly, what single payment 3 years from now would be equivalent to the two scheduled payments? Step 1 Draw a Time-line Draw a Time-line Step 1 Step 2 Find the FV of the payment that Step 2 Find the FV of the payment that is moved from Year 1 to Year 3 is moved from Year 1 to Year 3 Step 3 Find the PV of the payment that Step 3 Find the PV of the payment that is moved from Year 4 to Year 3 is moved from Year 4 to Year 3 Prepare the solution…(a) algebraically, and (b) by financial calculator McGraw-Hill Ryerson©
67. 67. Compound Compound Equivalent Payments Equivalent Payments 8 Interest 8 - 67 Interest Step 1 Draw a Time-line Draw a Time-line Step 1 Two payments of \$2200 each must be made 1 and 4 years from now. If money can earn 5% compounded monthly, what single payment 3 years from now would be equivalent to the two3 years scheduled payments? 0 1 year 2 years 4 years PV1 \$2200 \$2200 FV2 FV i = .05/12 Step 2 Step 2 Find the FV of Find the FV of the payment the payment that is moved that is moved from Year to from Year 11to Year Year 33 McGraw-Hill Ryerson© 1 n = 2*12 = 24 PV 2 2430.87 (a) FV1 = 2200(1+.05/12)24 = 2200(1.1049) = 2430.87 Now 0
68. 68. Compound Compound 8 - 68 (b) 8 Interest Interest 0 1 year 2 years 3 years PV1 \$2200 FV1 i = .05/12 n = 2*12 = 24 PV2 2*12 2200 Step 2 Step 2 Find the FV of Find the FV of the payment the payment that is moved that is moved from Year to from Year 11to Year Year 33 4 years \$2200 FV2 5 2430.87 0 12 Now McGraw-Hill Ryerson© 0
69. 69. Compound Compound Equivalent Payments Equivalent Payments 8 Interest 8 - 69 Interest 0 1 year 2 years PV1\$2200 3 years FV1 4 years \$2200 PV2 Step 3 Step 3 i = .05/12 n =1*12=12 Find the PV of Find the PV of (a) PV2 = 2200(1+.06/12)-12 the payment the payment that is moved that is moved from Year to = 2200(0.9513) from Year 44to Year Year 33 = 2092.92 Finally, this PV amount can be added to that put into memory… \$4523.79 0 2430.87 McGraw-Hill Ryerson©
70. 70. Compound Compound 8 - 70 (b) 8 Interest Interest 0 1 year 2 years PV1\$2200 2,092.92 4,523.79 1*12 2200 McGraw-Hill Ryerson© 3 years FV1 4 years \$2200 PV2 n =1*12=12 Finally, this PV amount can be added to that put into memory… Some of the Some of the values values have not have not changed so changed so there is no there is no need to need to enter them enter them again! again! 0 2430.87 \$4523.79
71. 71. 8 - 71 Compound Compound 8 Interest Interest What regular payment will an investor receive from a \$10,000, 3 year, monthly payment GIC earning a nominal rate of 4.8% compounded monthly? Interest rate per payment interval is: i = j/m = . = 0.0040 048/12 …the monthly PV * I = \$10000 * 0.0040 …the monthly payment will be: payment will be: = \$40.00 Making a choice!… Making a choice!… McGraw-Hill Ryerson©
72. 72. Compound Compound 8 Interest Interest 8 - 72 Making a choice!… Making a choice!… Suppose a bank quotes nominal annual interest rates of 6.6% compounded annually, 6.5% compounded semi-annually, and 6.4% compounded monthly on five-year GICs. Which rate should an investor choose for an investment of \$1,000? McGraw-Hill Ryerson©
73. 73. Compound Compound 8 Interest Interest j = 6.6% compounded annually Suppose bank Suppose aabank quotes nominal quotes nominal annual interest annual interest rates of rates of 6.6% 6.6% compounded compounded annually, annually, 6.5% compounded 6.5% compounded semi-annually, semi-annually, and and 6.4% compounded 6.4% compounded monthly monthly on fiveon fiveyear GICs. year GICs. McGraw-Hill Ryerson© Which Which 5*1 j = 6.5% 6.6 compounded semi-annually compounded monthly 6.5 6.4 2 12 1376.89 1375.96 5*2 1000 j = 6.4% 8 - 73 5 * 12 0 1 1376.53 Comparisons
74. 74. Compound Compound Comparisons 8 Interest Interest j = 6.6% 8 - 74 Results Results compounded annually j = 6.5% 1376.53 1376.53 compounded semi-annually j = 6.4% 1376.89 1376.89 compounded monthly 1375.96 1375.96 the 6.5% compounded semi-annually the 6.5% compounded semi-annually provides for the best provides for the best rate of return on investment! rate of return on investment! McGraw-Hill Ryerson©
75. 75. Compound Compound 8 Interest 8 - 75 Interest of Interest Rates McGraw-Hill Ryerson©
76. 76. 8 - 76 Compound Compound 8 Interest Interest of Interest Rates An investment in a GIC might have a… Fixed Rate Fixed Rate Step-up Rate Step-up Rate Variable Rate Variable Rate …the interest …the interest rate does not rate does not change over the change over the term of the GIC. term of the GIC. …the interest rate is …the interest rate is increased every 6 increased every 6 months or every year months or every year according to a preaccording to a predetermined schedule. determined schedule. ... is adjusted every ... is adjusted every year or every 6 year or every 6 months to reflect months to reflect market rates… market rates… may be a minimum may be a minimum “floor” “floor” below which rates below which rates cannot drop cannot drop McGraw-Hill Ryerson©
77. 77. Compound Compound 8 Interest Interest Payment of Payment of Interest Interest Regular Interest Regular Interest version version Interest is paid Interest is paid to the investor to the investor every year or every 66 every year or every months months McGraw-Hill Ryerson© 8 - 77 Compound Interest Compound Interest version version Interest is periodically Interest is periodically converted to principal converted to principal and and paid at maturity paid at maturity
78. 78. 8 - 78 Compound Compound 8 Interest Interest C anadian S avings B onds McGraw-Hill Ryerson©
79. 79. Compound Compound 8 Interest Interest C anadian S avings B onds 8 - 79 - Can be purchased from financial institutions but funds go to federal government to help finance its debt - usual term is 10 or 12 years - variable interest rates - interest rate is changed on each anniversary, with minimum rates for subsequent 2 years To view current rates of interest and redemption values Go to http://www.cis-pec.gc.ca/ McGraw-Hill Ryerson©
80. 80. Compound Compound 8 Interest Interest McGraw-Hill Ryerson© 8 - 80
81. 81. Compound Compound 8 Interest Interest C anadian S avings B onds 8 - 81 All CSBs issued up to 1988 (Series 1 to 43) have matured and are no All CSBs issued up to 1988 (Series 1 to 43) have matured and are no longer earning interest. longer earning interest. The rates of interest for Series 45 to 70 for subsequent years to maturity The rates of interest for Series 45 to 70 for subsequent years to maturity will be announced at future dates. will be announced at future dates. McGraw-Hill Ryerson©
82. 82. Compound Compound 8 Interest Interest McGraw-Hill Ryerson© 8 - 82
83. 83. 8 - 83 Compound Compound 8 Interest Interest Concepts The fair market value of an investment is the sum of the Present Values of the expected cash flows. McGraw-Hill Ryerson© The discount rate used should be the prevailing market determined rate of return required on this type of investment.
84. 84. Compound Compound 8 Interest Interest McGraw-Hill Ryerson© 8 - 84
85. 85. Compound Compound 8 Interest 8 - 85 Interest … owner will receive single … owner will receive aasingle payment (called the face value payment (called the face value of the bond) on the of the bond) on the bond’s maturity date bond’s maturity date McGraw-Hill Ryerson© … the maturity date … the maturity date could be as much as 30 could be as much as 30 years in the future. years in the future. No interest will be received No interest will be received in the interim! in the interim!
86. 86. 8 - 86 Compound Compound 8 Interest Interest Suppose a \$10,000 face value strip bond matures 18 years from now. The owner of this bond will receive a payment of \$10,000 in 18 years. What is the appropriate price to pay for the bond today if the prevailing rate of return is 5.75%, compounded semi-annually? FV = \$10000 i = .0575/2 n = 18 * 2 = 36 McGraw-Hill Ryerson© PV = 10000(1+.0575/2)-36 = 10000(0.3605) = \$3604.50
87. 87. Compound Compound 8 Interest Interest j = 5.75% m=2 n = 18*2 = 36 FV = \$10000 8 - 87 Suppose a \$10,000 face value strip bond matures 18 years from now. The owner of this bond will receive a payment of \$10,000 in 18 years.What is the appropriate price to pay for the bond today if the prevailing rate of return is 5.75%, compounded semi-annually? 18 * 2 5.75 10000 PV = -3,604.50 0 2 \$3604.50 \$3604.50 McGraw-Hill Ryerson©
88. 88. 8 - 88 Compound Compound 8 Interest Interest This completes Chapter 8 McGraw-Hill Ryerson©