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- 1. Chapter Five – Time Value of Money
- 2. Learning Goals LG1 Understand the concepts of future value and present value, their calculation for single amounts, and the relationship between them. LG2 Find the future value and the present value of both an ordinary annuity and an annuity due, and find the present value of a perpetuity. LG3 Calculate both the future value and the present value of a mixed stream of cash flows. LG4 Understand the effect that compounding interest more frequently than annually has on future value and the effective annual rate of interest. LG5 Describe the procedures involved in (1) determining deposits needed to accumulate a future sum, (2) loan amortization, (3) finding interest or growth rates, and (4) finding an unknown number of periods. © 2012 PEARSON EDUCATION 5-2
- 3. 3 Based on positive time preference ~ a ringgit today is worth more than a ringgit expected in the future TVM tools are used to; Calculate deposits required to accumulate a future sum Amortize loans by calculating loan payments schedules Determine interest or growth rates of money streams Evaluate perpetuities Find the required rate of return
- 4. The Role of Time Value in Finance •Most financial decisions involve costs & benefits that are spread out over time. •Time value of money allows comparison of cash flows from different periods. •Question: Your father has offered to give you some money and asks that you choose one of the following two alternatives: ◦ RM1,000 today, or ◦ RM1,100 one year from now. •What do you do?
- 5. The Role of Time Value in Finance (cont.) •The answer depends on what rate of interest you could earn on any money you receive today. •For example, if you could deposit the RM1,000 today at 12% per year, you would prefer to be paid today. •Alternatively, if you could only earn 5% on deposited funds, you would be better off if you chose the RM1,100 in one year. 5-5
- 6. Future Value versus Present Value •Suppose a firm has an opportunity to spend RM15,000 today on some investment that will produce RM17,000 spread out over the next five years as follows: •Is this a wise investment? •To make the right investment decision, managers need to compare the cash flows at a single point in time. 5-6 Year Cash flow 1 $3,000 2 $5,000 3 $4,000 4 $3,000 5 $2,000
- 7. 7 Basic Concepts
- 8. 8 TMV Solution Methods 1. Numerical – using regular calculator w/o financial functions 2. Interest Tables - given with the text book a. Present Value Interest Factor b. Present Value Interest Factor for Annuity c. Future Value Interest Factor d. Future Value Interest Factor for Annuity 3. Financial Calculator 4. Worksheet
- 9. 9 Time Lines Show the timing of cash flows. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period. CF0 CF1 CF3 CF2 Time 0 1 2 3 i% End of period 2 & beginning of period 3
- 10. 10 Time line illustrations 100 100 100 0 1 2 3 i% 100 0 1 2 5% RM100 lump sum due in 2 years End of period 2 PV FV 3 year RM100 ordinary annuity (fixed periodic payment) 10%
- 11. 11 100 50 75 0 1 2 3 i% -50 Uneven cash flow stream CF0 CF1 CF2 CF3
- 12. 12 12 Key Description Clear all data No of payment per year No of year Annual interest rate Present value Future value (No keyed in) x (P/YR) Begin End Calculates amortization table C ALL P/YR PV FV I/YR N xP/YR BEG/END AMORT
- 13. 13 Future and Present Values
- 14. 14 Future Value Future Value – the value at a given future date of an amount placed on deposit today and earning interest at a specified rate. Found by applying compound interest over a specified period of time. Ending amount of your account at the end of n periods In determining the final value of a cash flow or series of cash flows, compound interest will be applied. What is compound interest? Is it the same thing as simple interest? The process of going from today’s values, or PV to future value is called compounding.
- 15. 15 Present Value •It is based on the idea that a dollar today is worth more than a dollar tomorrow. •Discounting cash flows is the process of finding present values; the inverse of compounding interest. Present Value is the current dollar value of a future amount—the amount of money that would have to be invested today at a given interest rate over a specified period to equal the future amount. Beginning amount in your account
- 16. 16 Future Value & Present Value BASIS FOR COMPARISON COMPOUNDING DISCOUNTING Meaning The method used to determine the future value of present investment The method used to determine the present value of future cash flows Concept If we invest some money today, what will be the amount we get at a future date What should be the amount we need to invest today, t get a specific amount in the future Use of Compound Interest Rate Discount Rate Known Present Value Future Value Factor Future Value Factor or Compounding Factor Present Value Factor or Discounting Factor Formula FV=PV (1 + r)^n PV = FV / (1 + r)^n
- 17. 17 Simple interest; 105 110 2nd period; (Principal) [(2 x 0.05) + 1 ] = RM110 or, [100 (1.05) - 100] + 100(1.05) = RM110 0 1 2 5% 100 0 1 2 5% Compounding interest; 100 105 110.25 2nd period; (Principal + interest)(1 + i) (RM100 + RM5) (1 + 0.05) = RM110.25 Future Value
- 18. 18 Mimi has decided to invest RM100 in a savings account that earns 12% interest. She invested for two years. How much would she earn at the end of year 2? Bill Smith deposited RM80 in a savings account for 4 years at annual interest rate of 8%. What is Bill’s interest and compounded amount? Bill Smith deposited RM80 in a savings account for 4 years at annual interest rate of 8%. What is Bill’s simple interest and maturity value? Activity 1
- 19. 19 Future Value : One period case Case 1: Given that a car dealer offers a car for RM20,000 in cash or RM25,000 on credit for one year. Given an annual i.r. of 10%, which payment is better for you & which for the car dealer? 0 1 i% = ? 20,000 25,000 Given RM20,000 now, in 1 year at ir of 10%, the money deposited will be ? 20,000 0 1 10% FV = ? Future Value
- 20. 20 FV1 = PV + INT = PV + PV (i) = PV ( 1 + i ) Numerical Solution (N/S) ; FV1 = PV ( 1 + i ) = 20,000 ( 1 + 0.1) = RM22,000 Future Value Interest Factor for i & n (FVIFi,n) ~ the future value of RM1 left on deposit for n periods at a rate of i percent per period ~ where ( 1 + I )n = FVIFi,n Tabular Solution (T/B) ; FV1 = PV (FVIFi,n) = 20,000 ( 1.1000) = RM22,000 Future Value FV1 = PV + (1 + i) n
- 21. 21 21 Find the FV of RM20,000 given an interest rate of 10% in one year. Data Key Clear all data 1 No of payment per year 1 No of year 10 Annual interest rate 20,000 Present value 22,000.00 C All P/YR PV FV I/YR N + / -
- 22. 22 Present Value : One period case FV1 = PV ( 1 + i )n , so PV = FV1 (1 + i) n C2 : Given the annual ir of 10%, at what amount of cash would the car dealer be indifferent to receiving RM25,000 at time 1? 0 1 PV = ? 10% 25,000
- 23. 23 n n i i PVIF ) 1 ( 1 ) ( , Numerical solution ; PV = FV1 / (1 + i) = 25,000 = RM22,727.27 1 + 0.1 Tabular Solution ; Present Value Interest Factor for i & n (PVIFi,n) ~ the present value of RM1 due n periods in the future discounted at i percent per period PV = FV (PVIFi,n) = 25,000 (0.9091) = RM 22,727.50 ~ where;
- 24. 24 Find the PV of RM25,000 given an interest rate of 10% in one year. Data Key Clear all data 1 Payment per year 1 10 Annual interest rate 25,000 -22727.27273 C ALL P/YR FV PV I/YR N
- 25. 25 1. What is the future value of RM10,000 twenty years from now given interest rate of 6%? 2. What is the present value of RM100,000 ten years from now given the same annual rate of 6% 3. What will the present value be RM1,000 to be received eight years from today if the discount rate is 5%? 4. Mandy plans to invest RM50,000 in a fixed deposit that pays 4% interest annually for five years. Later, she plans to invest in another marketable securities that can bring 6% interest per annum for another two years. Advise Mandy how much her money would grown in seven years Activity 2
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- 28. 28 Future Value & Present Value : Multi – period case Important terms; Compound interest – interest earned on the principal & on the accumulated interest Discount interest rate – the rate that will make the future value equivalent to the present value Fair (Equilibrium) Value – the price at which investors are indifferent btw buying or selling a security 0 1 2 discounting 5% compounding
- 29. 29 The discount rate is often also referred to as the opportunity cost, the required return, and the cost of capital.
- 30. 30 C3 : Find the FVo RM100 left for 3 years in an account paying 10 percent, annual compounding; FV = ? 0 1 2 3 10% 100 FV1 = PV + INT = PV + PV (i) = PV ( 1 + i ) FV2 = FV1 (1 + i) = PV ( 1 + i ) (1 + i) = PV (1 + i)2 FV3 = PV (1 + i)3 N/S; FV3 = 100 (1.10)3 = 133.10 FVn = PV (1 + i)n = PV (FVIFi,n) = 100 (1.10)3 = 100 (1.3310) = RM133.10 T/S;
- 31. 31 31 Data Key 1 3 10 -100 133.10 C ALL P/YR PV FV I/YR N +/-
- 32. 32 C4 : Find the PV of RM100 to be received in 3 years if the appropriate ir is 10 percent, compounded annually: 100 0 1 2 3 10% PV = ? PVn = FV (1 + i)n PV3 = FV (1 + i)3 N/S; PVn = FVn/ (1 + i)n = FVn 1 n = FVn(PVIFi,n) 1 + i = 100 (1/1.10)3 = 100 (0.7513) = RM75.13 T/S;
- 33. 33 Data Key 1 3 10 100 -75.13 C ALL P/YR FV PV I/YR N
- 34. 34 n (periods) and i (interest rate)
- 35. 35 Solving for n in TVM problems C5: How long will it take a firm’s sales to double, if sales are growing at a 15% rate? 0 15% n = ? n - 1 RM1 RM2 FVn = PV (1 + i)n 2 = 1 (1.15)n 2 = (1.15)n FVn = PV (FVIFi,n) (FVIFi,n) = FV / PV = 2 / 1 = 2 T/S; Look in FVIF Table for (FVIF15%,n) = 2 n 5 periods N/S;
- 36. 36 0 15% n = ? n - 1 RM1 RM2 Financial Calculator Solution ; Data Key 1 1 15 2 4.96 C ALL P/YR FV PV I/YR N +/-
- 37. 37 Solving for interest rate C6: What annual ir will cause RM100 to grow to RM125.97 in 3 years? 125.97 0 1 2 3 i = ? 100 T/S; 100 (1 + i) 100 (1 + i)2 100 (1 + i)3 100 (1 + i)3 = 125.97 100 (FVIFi,3) = 125.97 FVIFi,3 = 1.2597 Look at Row 3 of FVIF Table. 1.2597 is in the 8% column
- 38. 38 Tutorial Activity • Differentiate between compounding and discounting • What is future value and present value
- 39. 39 Activity 1 • You are offered RM2,500 today, RM10,000 in 12 years or RM25,000 in 25 years. Assuming you can earn 10% on your money, which should you choose? • Azrul has decided to place RM500, which he received as a birthday gift, in a saving account paying 4% interest. How much will accrue to Azrul’s account in six years time. • Faizal intends to buy a new car, the MyVi for RM57,650, but only has RM20,000 in cash. How many years will it take for RM20,000 to grow to RM 57,650 if it is invested at 10% interest compounded annually? FVn = PV (FVIFi,, n) • Let us say Faizal intends to buy the Myvi in five years’ time. At what rate must his RM20,000 be compounded annually for it to grow to RM57,650 in five years?
- 40. 40 1. What is the future value of RM10,000 twenty years from now given interest rate of 6%? 2. What is the present value of RM100,000 ten years from now given the same annual rate of 6% 3. What will the present value be RM1,000 to be received eight years from today if the discount rate is 5%? 4. Mandy plans to invest RM50,000 in a fixed deposit that pays 4% interest annually for five years. Later, she plans to invest in another marketable securities that can bring 6% interest per annum for another two years. Advise Mandy how much her money would grown in seven years Activity 2
- 41. 41 Annuities & Perpetuities
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- 46. 46 Annuities An annuity is a series of equal payments made at fixed intervals for a specific number of periods Ordinary annuity - payments occur at the end of each period - eg. Students loan Annuity due – payments are made at the beginning of each period - eg. Monthly rentals, insurance premiums
- 47. 47 What is the difference between an ordinary annuity and an annuity due? Both are 3-yr annuities ( 3 pmts) Ordinary Annuity PMT PMT PMT 0 1 2 3 i% PMT PMT 0 1 2 3 i% PMT Annuity Due
- 48. 48 Future Value of an Ordinary Annuity C7: What is the future value of an ordinary annuity of RM100 per period for 3 yrs if the ir is 10 percent, compounded annually? Time line approach; 0 1 2 3 10% 100 100 100 110 121 331 100(1 + i)2 100(1 + i) + Twice compounding
- 49. 49 N/S; FVA3 = PMT (1 + i) + PMT (1 + i)1 + PMT (1 + i)2 = 100 (1) + 100 (1.10) + 100 (1.21) = RM331 T/S; FVAn = PMT (FVIFAi,n) FVA3 = 100 (FVIFA10%,3) = 100 (3.3100) = RM331 FC; = 331.00 10 3 -100 Make sure no BGN sign PMT I/YR N FV P/YR 1
- 50. 50 Future Value of an Annuity Due C8: What is the future value of RM100 payments made at beginning of each year for 3 yrs in a saving account that pays 10 percent, compounded annually? Time line approach; 0 1 2 3 10% 100 100 0 110 121 133.10 100(1 + i)2 100(1 + i) + Triple compounding 100 100 (1 + i)3 364.10
- 51. 51 N/S; FVAD3 = PMT (1 + i)3 + PMT (1 + i)2 + PMT (1 + i)1 = 100 (1.331) + 100 (1.21) + 100 (1.10) = RM364.10 T/S; FVADn = FVA3 (1 + i) or = PMT (FVIFA10%,3) (1 + i) = 331 (1.10) = 100 (3.3100) (1.10) = RM364.10 = 364.10 FC; = 364.10 10 3 -100 Make sure BGN sign P/YR PMT I/YR N 1 FV BEG/END
- 52. 52 Present Value of an Ordinary Annuity C9: What is the PV of an annuity of RM100 per period for 3 years if the ir is 10 percent annually? Time line approach; 0 1 2 3 10% 100 100 100 /(1 + i)2 100 / (1 + i) Triple discounting 0 100 / (1 + i)3 90.91 82.64 75.13 248.68 + 100
- 53. Activity Three 1. What is the present value of a 5 year RM 1,000 ordinary annuity discounted back to the present at 10% 2. How much must Sharifah deposit at the end of each year in a savings account earning 10% annual interest to accumulate RM10,000 at the end of six years? 3. What is the present value of a 4year RM1,000 annuity discounted back to the present of 9%? 4. What is the present value of a RM10,000 perpetuity discounted at 8%? 53
- 54. Activity Four 1. Danny deposits RM 1200 at the end of each year into an ordinary annuity that pays 6% annual interest at the beginning of each year. What is the future value at 5 years? 2. Kamal deposit RM100 each month into an annuity pays of 6% annual interest at the end of each year. How much money she have at this account at the end of 5 years? 3. Rose wants to have RM800,000 in an annuity by the time he retires 30 years from now. If the annuity pays a fixed interest of 5% at the end of each year, how much money should she deposit into his account for the next 30 years? 54
- 55. Activity Five 1. Mazni has decided to invest RM100 in a savings account paying 12% interest for two year. How much will she earn? 2. What is the present value of a RM600 payment you expect to receive in three years if the rate of interest is 12% compounded annually? 3. Liza is certain of receiving RM50,000 10 years from now. She wishes to sell this future amount to Amrul. What maximum price can Amrul afford to pay if he intends to earn 7% per year over the next 10 years? RM 4. In order for Lai Ming to enjoy a holiday in Japan, she will need RM 7,480. She is able to save RM 729.04 at the end of each year for this purpose. She has discovered that a mutual fund pays 7% interest compounded annually. How long will it be before Lai Ming can embark on her holiday? 5. What is the present value of a 5year ordinary annuity with annual payment of RM200 and discounted at 15% interest? 6. What is the present value of 16 year annuity if the payments are RM2,000 per year and the rate of return is 7%? 55
- 56. 56 Present Value of an Annuity Due C10: How much lump sum today to make it equivalent with a 3 year annuity paying RM100 at beginning of each year? Time line approach; 0 1 2 3 10% 100 100 100 /(1 + i)2 100 / (1 + i) Double discounting 100 90.91 82.64 273.55 + Make sure BGN sign
- 57. 57 An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.
- 58. 58 Perpetuities - is a stream of equal payments expected to continue forever - a type of annuity PV (Perpetuity) = payment = PMT interest rate i the current price
- 59. 59 C11: A perpetual bond promised to pay RM100 per year in perpetuity. What would the bond’s worth today if the opportunity cost, or discount rate was 5 percent PV (Perpetuity) = RM100 = RM2000 0.05 As the interest rate increases, the perpetuity’s value will drop. When ir = 10%; PVp = 100 = RM1000 0.1
- 60. 60 Uneven Cash Flow Stream Payment (PMT) - equal cash flows at regular intervals Cash flow (CF) - uneven cash flows Examples of uneven cash flows; - common stock’s dividend - returns from fixed asset investments ~ production income ~ rentals
- 61. 61 Present Value of an Uneven Cash Flow Stream C12: Find PV of the following cash flows stream, discounted at 10% 0 1 2 3 4 10% 0 100 300 300 -50 PV = CF0 1 0 + CF1 1 1 + CF2 1 2 1 + i 1 + i 1 + i + CF3 1 3 + CF4 1 4 1 + i 1 + i CF0 CF1 CF2 CF3 CF4
- 62. 62 For cash flow calculation ; Key Clear all No of periods per year Cash flow j No of consecutive times CFj occurs Internal rate of return per year Net present value C ALL P/YR Nj CFj IRR/YR NPV
- 63. 63 Key 0 100 300 300 50 10 530.09 C ALL I/YR CFj NPV CFj CFj CFj +/- Key 0 100 300 2 50 10 530.09 C ALL I/YR CFj NPV CFj CFj Nj +/- CFj CFj
- 64. 64 0 1 2 3 4 5 10% 0 100 50 200 200 200 C13: Find the PV of the following c/f discounted at 10% 497.38 200 (PVIFA10%,3) = 200 (2.4869) 100(1/1 + i)1 50(1/1 + i)2 497.38(1/1 + i)2 or 200 (PVIFA10%,3) or 200 (1/1 + i)3 200 (PVIFA10%,4) or 200 (1/1 + i)4 200 (PVIFA10%,5) or 200 (1/1 + i)5 90.91 41.32 411.03 543.26 150.26 136.60 124.18
- 65. 65 Future Value of an Uneven Cash Flow Stream C14: Find FV of the following cash flows stream, compounded at 10% 0 1 2 3 4 5 10% 0 100 50 200 200 200 FV = CF5 (1 + i)0 + CF4 (1 + i)1 + CF3 (1 + i)2 + CF2 (1 + i)3 + CF1 (1 + i) 4 + CF0 (1 + i)5 CF0 CF1 CF2 CF3 CF4 CF5
- 66. 66 0 1 2 3 4 5 10% 0 100 50 200 200 200 420 ( 1 + i) 200 (FVIFA10%,2) = 200 (2.1) 200 (1 + i)2 200 (1 + i) 50 (1 + i)3 = 50 (1.331) 100 (1 + i)4 = 100 (1.4641) 220 242 66.55 146.41 NFV = 874.96 462 462
- 67. 67 0 1 2 3 4 5 10% 0 100 50 200 200 200 Data Key 1 0 100 50 200 3 C ALL P/YR I/YR CFj CF0 CF1 CF2 CF3 CF4 CF5 NFV=? Data Key 10 543.28 5 0 847.93 CFj Nj CFj NPV N PMT FV CFj Using formula; NFV = NPV ( 1 + i)n = 543.26 (1 + 0.1)5 = 874.93
- 68. Different Compounding Periods - annually / semiannually / quarterly / monthly / daily compounding - the quoted interest rate is normally the annual one. - If bank promised 10% annual interest rate semiannually what does it means ? ~ interest will be added each 6 months but will the interest be 10% as quoted or more/less?
- 69. Say that you want to deposit your money in the bank which offer you the highest return. As you shopped around, you come up with these rates: Bank A : 15 percent, compounded daily Bank B : 15.5 percent, compounded quarterly Bank C : 16 percent, compounded annually Which one has the best rates for deposits?
- 70. C15: A bank declares that it pays a 6% annual ir semiannually & you want to deposit RM 100. What is FV at the end of 3rd year? P/YR 1 PV (-)100 n 3 I% 6 FV 119.1 P/YR 1 2 PV (-)100 (-)100 N 6 6 I% 6/2 6 FV 119.41 119.41 0 1 2 3 6% Annual compounding Semiannual compounding 0 1 2 3 4 5 6 i% FV3 = PV (1 + i)3 = 100 (1 + 0.06)3 = RM119.10 -100 FV = ? -100 FV = ? FV6 = PV (1 + i)6 = 100 (1 + 0.03)6 = RM119.41 i% = 6% /2 = 3% n = 3 x 2 = 6
- 71. Different compounding periods are used for different types of investment In order to compare securities with different compounding periods, need to put them on a common basis. Types of interest rates; Nominal or quoted interest rates Annual percentage rates (APR) Effective annual rates (EAR)
- 72. 1. Nominal or Quoted interest rates , inom Is the contracted, or stated, or declared ir. The rate which is given by the bank or issuer. Annual Percentage Rate (APR) The interest rate charged per period multiplied by the number of periods per year. C16: If a bank is charging 1.2% per month on car loans, what is the APR? APR = 1.2% x 12 = 14.4% It is the nominal rates for loan that some government requires the bank to display to customers.
- 73. Periodic rate is the nominal rate at each period; where m is the no of compounding periods per year Eg: 6% compounded quarterly. periodic rate, iper = inom / m = 6% / 4 = 1.5% Periodic Rate m i i nom per
- 74. So periodic rate is the rate charged by a lender or paid by a borrower in each period. C17: A bank charges 18% annual interest rates monthly on credit card loans, what is the periodic rate? iper = inom / m = 18% / 12 = 1.5% Bank will charge 1.5% of interest monthly or per month So if we delayed paying our credit card debt for a year, will the debt be the same as we take a loan of the same amount at 18% annual interest?
- 75. Notice that iper is the rate that is shown on time lines and used in certain calculations, not the annual rate. C18: How much would you have at the end of the 2nd year when you make RM100 deposit in an account that pays 12% interest rate semiannually. 0 1/2 1 1/2 2 (x2) = N 6% -100 FV = ? For calculation of FV given only PV, must use this rate not the annual rate given in this case. AND maintain P/YR = 1.
- 76. 2. Effective Annual Rates (EAR) or (EFF) The rate which would produce the same ending (future) value if annual compounding has been used. ~ (the interest rate expressed as if it were compounded once per year) 0 1 2 3 6% Annual compounding Semiannual compounding 0 1 2 3 4 5 6 3% -100 119.41 -100 119.10 0 3 i = ? -100 119.41 EAR ; The annual rate that produces the same FV as if we had compounded at a given periodic rate m times per year
- 77. An investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually. Why? C19: EFF% or EAR for 10% semiannual investment. % EFF 10.25% 1 2 0.10 1 1 m i 1 EAR or EFF% 2 m nom
- 78. C20: What is the FV of RM100 compounded semiannually for 3 years if inom = 10%? Would it be different if it were compounded quarterly? Quarterly compounding Semiannual compounding 0 1 2 3 5% -100 FV = ? 0 1 2 3 -100 FV = ? 2.5% 134.01 134.49
- 79. FVn = PV 1 + inom mn m FV3 = 100 1 + 0.1 2(3) 2 = 100(1.05)6 = 134.01 EAR = ( 1 + inom / m )m - 1 = ( 1 + 0.10 / 2 )2 – 1 = 10.25% Semiannual compounding Quarterly compounding FVn = PV 1 + inom mn m FV3 = 100 1 + 0.1 4(3) 4 = 100(1.025)12 = 134.49 EAR = ( 1 + inom / m )m - 1 = ( 1 + 0.10 / 4 )4 – 1 = 10.38%
- 80. F/C ; Given annual i.r. of 10%, compounded semiannually; 10.25 To find APR, given the EAR of 10.25%; 10.00 The APR formula; APR = 1 + EFF 1/m - 1 x m x 100 100 NOM% 10 2 P/YR EFF% EAR NOM% P/YR EFF% 10.25 2
- 81. Compounding More Frequently than Annually • Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently. • As a result, the effective interest rate is greater than the nominal (annual) interest rate. • Furthermore, the effective rate of interest will increase the more frequently interest is compounded.
- 82. Nominal & Effective Rates The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower. The effective interest rate is the rate actually paid or earned. The effective rate > nominal rate whenever compounding occurs more than once per year EAR > inom EAR = inom = iper If compounding occurs only once a year, then;
- 83. Nominal & Effective Rates C21: What is the effective rate of interest on your credit card if the nominal rate is 18% per year, compounded monthly? EAR = (1 + .18/12) 12 -1 EAR = 19.56%
- 84. Why is it important to consider effective rates of return? An investment with monthly payments is different from one with quarterly payments. Must put each return on an EFF% basis to compare rates of return. Must use EFF% for comparisons. See the following values of EFF% rates at various compounding levels. EARANNUAL 10.00% EARQUARTERLY 10.38% EARMONTHLY 10.47% EARDAILY (365) 10.52%
- 85. Fractional Time Periods Before, payments only occur at beginning or end of periods. What if, payments occur at some date within a period? 0 1st 2nd month 20th day month 1% -100 FV = ?
- 86. C22: Deposits RM100 in a bank that pays 12% ir compounded monthly. How much the amount will be then in 1 month and 20 days? N/S; FVn = PV (1 + i)n = PV (1 + i)1+ 20/30 = 100(1 + 0.1)1.67 = RM101.68 F/C ; N 20÷30+1 I/YR 1 PV (-)100 P/YR 1 FV 101.67 If use 360-day year; iper = 0.12 /360 = 0.00033333 per day No of days deposited; = 50 days FVn = PV (1 + iper)n = 100 ( 1.0003333)50 = RM 101.68
- 87. Note of Caution; Rule 1: Money saved/deposited/invested should be in negative sign. Money withdrawn/received should be in positive sign. C23: RM1000 is deposited today for a semiannual payment of RM300 for 3 years. Given an interest rate of 10% semiannually, how much would be left in the account in 3 years time? 0 1 2 3 5% -1000 300 300 300 300 300 300 FV = ?
- 88. Rule 2 : If there is only PV & no PMT, either; a. If use periodic ir, keep P/YR = 1. b. If use nominal rate, change P/YR accordingly. C24: RM1000 is deposited today. Given an interest rate of 10% semiannually, how much would be in the account in 3 years time? 0 1 2 3 5% -100 FV = ? N 3x2 = 6 I/YR 10÷2 = 5 PV (-)100 PMT 0 P/YR 1 FV 134.01 N 3x2 = 6 I/YR 10 PV (-)100 PMT 0 P/YR 2 FV 134.01
- 89. Rule 3: For case with PMT or PMT and PV, N = no of payment made, I/YR = annual interest rate, P/YR = no of payments made per year. 0 1 2 3 8% -100 -150 -150 -150 -150 -150 -150 FV = ? C25: Interest is 8% compounded quarterly. Initial deposit is RM100, and regular payments of RM150 will be made every semiannually. 2 P/YR 3x2 = 6 N 8.08 I/YR (-)100 PV (-)150 PMT 1122.77 FV 8 NOM% 4 P/YR 8.24 I/YR 8.24 EFF% 2 P/YR 8.08 NOM%
- 90. C26: Someone offers to sell you a note calling for the pmt of RM1000, 15 months from today for RM850. You have RM850 in the bank, which pays a 7% nominal rate with daily compounding. Should you buy the note or leave your money in the bank. An Example of Everything 0 456 days -850 1,000 iper = 7%/365 = 0.0192% How to solve this? Have to compare both investments on similar grounds; Fvnote vs. FVbank PVnote vs. PVbank EARnote vs. EARbank
- 91. Fvnote vs. Fvbank Bank : FV = 850 (1.000192)456 = 927.67 Note : FV = 1,000 Buy note (more value in future) PVnote vs. Pvbank Bank : PV = RM850 Note : PV = 1,000 (1.000192)-456 = 916.27 Buy note (more value now) EARnote vs. EARbank Bank : iper = 0.0192% Note : 1,000 = 850 (1 + i)456, solving i = 0.0356% Buy note (higher iper means higher EAR)
- 92. C27: Cost of note = RM850 PMT = RM190 quarterly for 5 quarters inom = 7% compounded daily Is this a good investment? 0 91 182 274 366 456 days -850 190 190 190 190 190 iper for bank = 7% / 365 = 0.0192%
- 93. PVAnote = 190 (1.000192)-91 + 190 (1.000192)-182 + … + 190 (1.000192)-456 = 901.68 PVApocket = 850 Buy note (more value now) EARnote ; finding iper; inom = (iper) (m) = (3.83) (4) = 15.3% So for daily rate = inom / 365 = 0.0419% Buy note coz iper,note > iper, bank = 0.0192% -850 CFj 190 CFj 5 Nj IRR 3.82586 Quarterly iper
- 94. Loan Types 1. Pure Discount Loans - the borrower receives money today & repays a single sum at some time in the future - eg. A 1-year, 10% RM100 pure discount loan, would require the borrower to repay RM110 in one year. 2. Interest-Only Loans - a loan that has a repayment plan that calls for the borrower to pay interest each period & repay the entire principal (original loan amount) at some point in the future - eg. With a 3-year, 10%, interest-only loan of RM1000, the borrower would pay RM1000(0.1) = 100 in interest at the end of 1st & 2nd years. At the end of 3rd year, the borrower would return the principal along with RM100 in interest for that year.
- 95. 3. Amortized Loans - a loan that is repaid in equal payments over its life. - eg. Car & home loans C28: Say, borrow RM1,000 at 10% interest and have to pay equally at the end of each of the next 3 years. 0 1 2 3 -1,000 PMT PMT PMT 10% T/S ; PVAn = PMT (PVIFAi,n) 1,000 = PMT(PVIFA10%,3) PMT = 1,000 / (2.4869) = RM402.11 F/C ; 3 N 10 I/YR -1000 PV 0 FV PMT 402.11
- 96. Constructing an amortization table: Repeat steps 1 – 4 until end of loan Interest paid declines with each payment as the balance declines. Year Beginning Balance (1) PMT (2) Interest (3) Princip Repmt (4) End Balance 1 RM1,000 RM402 RM100 RM302 RM698 2 698 402 70 332 366 3 366 402 37 366 0 Total 1,206.34 206.34 1,000 - PVn(1 + i) (2) – (3) (1) – (4)
- 97. 97 1 P/YR 3 N 10 I/YR -1000 PV 0 FV PMT 402.11 1 INPUT AMORT 1-1 = 302.11 PRIN = 100.00 INT = -697.89 BAL 1 INPUT 2 AMORT 1-2 = 634.43 PRIN = 169.79 INT = -365.57 BAL