Report

Chapte 8 Slides Developed by: Terry Fegarty Seneca College Time Value of Money Chapter 8 – Outline (1) • The Time Value of Money Time Value Problems Amount Problems—Future Value Other Issues Financial Calculators Spreadsheet Solutions The Present Value of an Amount Finding the Interest Rate Finding the Number of Periods © 2006 by Nelson, a division of Thomson Canada Limited 2 Chapter 8 – Outline (2) • Annuities The Future Value of an Annuity The Future Value of an Annuity—Developing a Formula The Future Value of an Annuity—Solving Problems The Sinking Fund Problem Compound Interest and Non-Annual Compounding The Effective Annual Rate The Present Value of an Annuity—Developing a Formula The Present Value of an Annuity—Solving Problems © 2006 by Nelson, a division of Thomson Canada Limited 3 Chapter 8 – Outline (3) Amortized Loans Loan Amortization Schedules Mortgage Loans The Annuity Due Perpetuities Continuous Compounding • Multipart Problems Uneven Streams Imbedded Annuities © 2006 by Nelson, a division of Thomson Canada Limited 4 The Time Value of Money • $100 in your hand today is worth more than $100 in one year Money earns interest Example • The higher the interest, the faster your money grows Q: How much would $1,000 promised in one year be worth today if the bank paid 5% interest? A: $952.38. If we deposited $952.38 after one year we would have earned $47.62 ($952.38 × .05) in interest. Thus, our future value would be $952.38 + $47.62 = $1,000. © 2006 by Nelson, a division of Thomson Canada Limited 5 The Time Value of Money Example • Present Value The amount that must be deposited today to have a future sum at a certain interest rate The discounted value of a sum is its present value In our example, what is the present value? $952.38 © 2006 by Nelson, a division of Thomson Canada Limited 6 The Time Value of Money • Future Value Example The amount a present sum will grow into at a certain interest rate over a specified period of time In our example, • The • The • The • The present sum (value) is $952.38 interest rate is 5% time is 1 year future value is $1,000 © 2006 by Nelson, a division of Thomson Canada Limited 7 Time Value Problems • Time value deals with four different types of problems Amount— a single amount that grows at interest over time • Future value • Present value Annuity— a stream of equal payments that grow at interest over time • Future value • Present value © 2006 by Nelson, a division of Thomson Canada Limited 8 Time Value Problems • 4 methods to solve time value problems Use Use Use Use formulas financial tables financial calculator financial functions in spreadsheet © 2006 by Nelson, a division of Thomson Canada Limited 9 Amount Problems—Future Value • The future value (FV) of an amount How much a sum of money placed at interest (k) will grow into in some period of time • If the time period is one year • FV1 = PV + kPV or FV1 = PV(1+k) • If the time period is two years • FV2 = FV1 + kFV1 or FV2 = PV(1+k)2 • If the time period is generalized to n years • FVn = PV(1+k)n © 2006 by Nelson, a division of Thomson Canada Limited 10 Amount Problems—Future Value • The (1 + k)n depends on Size of k and n • Can develop a table depicting different values of n and k and the proper value of (1 + k)n Example Example • Can then use a more convenient formula • FVn = PV [FVFk,n] Q: If we deposited $438 at 6% interest for five years, how much would we have? These values can be looked up in an interest factor table. A: FV5 = $438(1.06)5 = $438(1.3382) = $586.13 © 2006 by Nelson, a division of Thomson Canada Limited 11 The Future Value Factor for k and n FVFk,n = (1+k)n Table 8-1: Example 6% 1.3382 1.3382 5 © 2006 by Nelson, a division of Thomson Canada Limited 12 Other Issues • Problem-Solving Techniques Three of four variables are given • We solve for the fourth • The Opportunity Cost Rate The opportunity cost of a resource is the benefit that would have been available from its next best use • Lost investment income is an opportunity cost © 2006 by Nelson, a division of Thomson Canada Limited 13 Financial Calculators • How to use a typical financial calculator in time value Five time value keys • Use either four or five keys Some calculators distinguish between inflows and outflows • If a PV is entered as positive the computed FV is negative © 2006 by Nelson, a division of Thomson Canada Limited 14 Financial Calculators Basic Calculator Keys N Number of time periods I/Y Interest rate (%) PV Present Value FV Future Value PMT Payment © 2006 by Nelson, a division of Thomson Canada Limited 15 Financial Calculators Example Q: What is the present value of $5,000 received in one year if interest rates are 6%? A: Input the following values on the calculator and compute the PV: N 1 I/Y 6 FV 5000 PMT 0 PV 4,716.98 Answer © 2006 by Nelson, a division of Thomson Canada Limited 16 Spreadsheet Solutions • Time value problems can be solved on a spreadsheet such as Microsoft® Excel® • Click on the fx (function) button • Select for the category Financial • Select the function for the unknown variable • For example, to solve for present value: Use PV(k, n, PMT, FV) Interest rate (k) is entered as a decimal, not a percentage © 2006 by Nelson, a division of Thomson Canada Limited 17 Spreadsheet Solutions • To solve for: Select the function for the unknown variable, place the known variables in the proper order within the parentheses and input 0, for the unknown variable. FV use =FV(k, n, PMT, PV) PV use =PV(k, n, PMT, FV) k use =RATE(n, PMT, PV, FV) N use =NPER(k, PMT, PV, FV) PMT use =PMT(k, n, PV, FV) Of the three cash variables (FV, PMT or PV) • One is always zero • The other two must be of the opposite sign • Reflects inflows (+) versus outflows (-) © 2006 by Nelson, a division of Thomson Canada Limited 18 The Present Value of an Amount F Vn P V 1 + k n S o lv e fo r P V 1 P V = F Vn n 1 k In te re s t F a c to r • Either equation can be used to solve any amount problems F V Fk ,n 1 P V Fk ,n Solving for k or n involves searching a table. © 2006 by Nelson, a division of Thomson Canada Limited 19 Example Spreadsheet Solution—PV of Amount Click on the fx (function) button Select for the category Financial In the insert function box, select PV You will see PV(rate,nper,pmt,fv,type) Press OK Select the function for the unknown Fill in rate = B4, nper = C1, variable, place the known variables in pmt = 0, fv = C2 the proper order within the Enter OK parentheses and input 0, for the unknown variable. © 2006 by Nelson, a division of Thomson Canada Limited 20 Example Spreadsheet Solution—PV of Amount © 2006 by Nelson, a division of Thomson Canada Limited 21 Example 8.3: Finding the Interest Rate Q: What interest rate will grow $850 into $983.96 in three years? A: Example Calculator N 3 PV -850 FV 983.96 PMT 0 I/Y 5.0 Financial Table Interest factor is 850.00 / 983.96 = 0.8639 Look up in Table A-2 with n=3 Interest = 5% Answer © 2006 by Nelson, a division of Thomson Canada Limited 22 Spreadsheet Solution Example Example 8.3: © 2006 by Nelson, a division of Thomson Canada Limited 23 Example Example 8.3: Spreadsheet Solution Click on the fx (function) button Select for the category Financial In the insert function box, select RATE You will see RATE(n, PMT, PV, FV) Press OK Fill in n = E1, PMT = 0, PV = -B2, FV = E2 Enter OK © 2006 by Nelson, a division of Thomson Canada Limited 24 Example 8.4: Finding the Number of Periods Q: How long does it take money invested at 14% to double? Example A: The future value is twice the present value. If the present value is $1, the future value is $2 Calculator PV -1 FV 2 Interest factor is 2/1=2 Look up in Table A-2 with k = 14% n = 5 – 6 years PMT 0 I/Y 14 N 5.29 Financial Table Answer © 2006 by Nelson, a division of Thomson Canada Limited 25 Example 8.4: Example Finding the Number of Periods Click on the fx (function) button Select for the category Financial In the insert function box, select NPER You will see NPER(k, PMT, PV, FV) Enter OK Fill in the cell references for • k (= .14) • PMT = 0 • PV (= -1) • FV = 2 Enter OK NPER = 5.29 © 2006 by Nelson, a division of Thomson Canada Limited 26 Spreadsheet Solution Example Example 8.4: © 2006 by Nelson, a division of Thomson Canada Limited 27 Annuities • Annuity A finite series of equal payments separated by equal time intervals • Ordinary annuity • Payments occur at the end of the time periods • Monthly lease, pension, and car payments are annuities • Annuity due • Payments occur at the beginning of the time periods © 2006 by Nelson, a division of Thomson Canada Limited 28 Figures 8.1 and 8.2: Ordinary Annuity and Annuity Due Ordinary Annuity Annuity Due © 2006 by Nelson, a division of Thomson Canada Limited 29 Figure 8.3: Timeline Portrayal of an Ordinary Annuity © 2006 by Nelson, a division of Thomson Canada Limited 30 Future Value of an Annuity • Future value of an annuity The sum, at its end, of all payments and all interest if each payment is deposited when received © 2006 by Nelson, a division of Thomson Canada Limited 31 Figure 8.4: FV of a Three-Year Ordinary Annuity © 2006 by Nelson, a division of Thomson Canada Limited 32 The Future Value of an Annuity— Developing a Formula • Thus, for a 3-year annuity, the formula is F V A = P M T 1 + k P M T 1 + k P M T 1 + k 0 1 2 G e n e ra lizin g th e E xp re ssio n : F V A n = P M T 1 + k P M T 1 + k P M T 1 + k 0 1 2 P M T 1 + k n -1 w h ich ca n b e w ritte n m o re co n v e n ie n tly a s: n FVA n P M T 1 + k ni i= 1 F a cto rin g P M T o u tsid e th e su m m a tio n , w e o b ta in : n FVA n PM T 1 + k i= 1 © 2006 by Nelson, a division of Thomson Canada Limited ni FVFAk,n 33 The Future Value of an Annuity— Solving Problems • There are four variables in the future value of an annuity equation The The The The future value of the annuity itself payment interest rate number of periods © 2006 by Nelson, a division of Thomson Canada Limited 34 Example Example 8.5: The Future Value of an Annuity Q: The Brock Corporation owns the patent to an industrial process and receives license fees of $100,000 a year on a 10-year contract for its use. Management plans to invest each payment until the end of the contract to provide funds for development of a new process at that time. If the invested money is expected to earn 7%, how much will Brock have after the last payment is received? © 2006 by Nelson, a division of Thomson Canada Limited 35 Example Example 8.5: The Future Value of an Annuity A: Use the future value of an annuity equation: FVAn = PMT[FVFAk,n] In Table A-3, look up the interest factor at an n of 10 and a k of 7 Interest factor = 13.8164 Future value = $100,000[13.8164] = $1,381,640 N I/Y PMT PV FV © 2006 by Nelson, a division of Thomson Canada Limited 10 7 100000 0 1,381,645 Answer 36 Example Example 8.5: Spreadsheet Solution Click on the fx (function) button Select for the category Financial In the insert function box, select FV You will see FV(k, n, PMT, PV) Enter OK Fill in the cell references for • • • • k (=.07) n (= 10) PMT (=100000) PV (=0) Enter OK FV = 1,381,644.80 © 2006 by Nelson, a division of Thomson Canada Limited 37 Spreadsheet Solution Example Example 8.5: © 2006 by Nelson, a division of Thomson Canada Limited 38 The Sinking Fund Problem • Companies borrow money by issuing bonds for lengthy time periods No repayment of principal is made during the bonds’ lives • Principal is repaid at maturity in a lump sum • A sinking fund provides cash to pay off a bond’s principal at maturity • Problem is to determine the periodic deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem © 2006 by Nelson, a division of Thomson Canada Limited 39 Example Example 8.6: The Sinking Fund Problem Q: The Greenville Company issued bonds totaling $15 million for 30 years. A sinking fund must be maintained after 10 years, which will retire the bonds at maturity. The estimated yield on deposited funds will be 6%. How much should Greenville deposit each year to be able to retire the bonds? © 2006 by Nelson, a division of Thomson Canada Limited 40 Example 8.6: The Sinking Fund Problem Example A: The time period of the annuity is the last 20 years of the bond issue’s life. On your calculator: N I/Y FV PV 20 6 15000000 0 PMT 407,768.35 Answer © 2006 by Nelson, a division of Thomson Canada Limited 41 Compound Interest and NonAnnual Compounding • Compounding Earning interest on interest • Compounding periods Interest is usually compounded annually, semiannually, quarterly or monthly • Interest rates are quoted by stating the nominal rate followed by the compounding period © 2006 by Nelson, a division of Thomson Canada Limited 42 The Effective Annual Rate • Effective annual rate (EAR) The annually compounded rate that pays the same interest as a lower rate compounded more frequently © 2006 by Nelson, a division of Thomson Canada Limited 43 The Effective Annual Rate— Example Example Q: If 12% is compounded monthly, what annually compounded interest rate will get a depositor the same interest? A: If your initial deposit were $100, you would have $112.68 after one year of 12% interest compounded monthly. Thus, an annually compounded rate of 12.68% [($112.68 $100) – 1] would have to be earned. © 2006 by Nelson, a division of Thomson Canada Limited 44 The Effective Annual Rate • EAR can be calculated for any compounding period using the following formula: EAR k n o m in a l 1 m m - 1 Effect of more frequent compounding is greater at higher interest rates © 2006 by Nelson, a division of Thomson Canada Limited 45 Impact of Compounding Frequency $1,000 Invested at 10% Nominal Rate for One Year $1,106 $1,105 $1,104 $1,103 $1,102 $1,101 $1,100 $1,099 $1,098 $1,097 Annual SemiAnnual Quarterly Monthly © 2006 by Nelson, a division of Thomson Canada Limited Daily 46 The Effective Annual Rate • The APR and EAR Annual percentage rate (APR) • Is actually the nominal rate and is less than the EAR • Compounding Periods and the Time Value Formulas Time periods must be compounding periods Interest rate must be the rate for a single compounding period • For instance, with a quarterly compounding period the knominal must be divided by 4 and the n must be multiplied by 4 © 2006 by Nelson, a division of Thomson Canada Limited 47 Example Example 8.7: The Effective Annual Rate Q: You want to buy a car costing $15,000 in 2½ years. You plan to save the money by making equal monthly deposits in your bank account, which pays 12% compounded monthly. How much must you deposit each month? A: This is a future value of an annuity problem with a 1% monthly interest rate and a 30-month time period. On your calculator: N 30 I/Y 1 FV 15000 PV 0 PMT 431.22 Answer © 2006 by Nelson, a division of Thomson Canada Limited 48 The Present Value of an Annuity— Developing a Formula • Present value of an annuity Sum of all of the annuity’s payments PVA = PMT 1 + k PMT 1 + k 2 PMT 1 + k 3 w h ic h c a n a ls o b e w ritte n a s : P V A = P M T 1 + k 1 P M T 1 + k 2 P M T 1 + k 3 G e n e ra lize d fo r a n y n u m b e r o f p e rio d s : P V A = P M T 1 + k 1 P M T 1 + k 2 P M T 1 + k n F a c to rin g P M T a n d u s in g s u m m a tio n , w e o b ta in : n i P V A P M T 1 + k i= 1 © 2006 by Nelson, a division of Thomson Canada Limited PVFAk,n 49 Figure 8.6: PV of a Three-Payment Ordinary Annuity © 2006 by Nelson, a division of Thomson Canada Limited 50 The Present Value of an Annuity— Solving Problems • There are four variables in the present value of an annuity equation The The The The present value of the annuity itself payment interest rate number of periods • Problem usually presents 3 of the 4 variables © 2006 by Nelson, a division of Thomson Canada Limited 51 Example Example 8.9: The Present Value of an Annuity Q: The Shipson Company has just sold a large machine on an installment contract. The contract calls for payments of $5,000 every six months (semiannually) for 10 years. Shipson would like its cash now and asks its bank to it the present (discounted) value. The bank is willing to discount the contract at 14% compounded semiannually. How much should Shipson receive? © 2006 by Nelson, a division of Thomson Canada Limited 52 Example 8.9: The Present Value of an Annuity Example A: The contract represents an annuity with payments of $5,000. Adjust the interest rate and number of periods for semiannual compounding and solve for the present value of the annuity. N 20 I/Y 7 FV 0 PMT 5000 PV 52,970.07 This can also be calculated using the PVA Table A4. Look up n = 20 and k = 7%. PVA = $5,000[10.594] = $52,970 Answer © 2006 by Nelson, a division of Thomson Canada Limited 53 Amortized Loans • An amortized loan’s principal is paid off regularly over its life Generally structured so that a constant payment is made periodically • Represents the present value of an annuity © 2006 by Nelson, a division of Thomson Canada Limited 54 Example Example 8.10: Amortized Loans Q: Suppose you borrow $10,000 over four years at 18% compounded monthly repayable in monthly installments. How much is your loan payment? A: Adjust your interest rate and number of periods for monthly compounding. On your calculator: N 48 I/Y 1.5 PV 10000 FV 0 PMT 293.75 This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] Answer © 2006 by Nelson, a division of Thomson Canada Limited n = 48 and k = 1.5% $10,000 = PMT[34.0426] = $293.75. 55 Example 8.11: Amortized Loans Example Q: Suppose you want to buy a car and can afford to make payments of $500 a month. The bank makes three-year car loans at 12% compounded monthly. How much can you borrow toward a new car? A: Adjust your k and n for monthly compounding. On your calculator: N 36 I/Y 1 FV 0 PMT 500 PV 15,053.75 This can also be calculated using the PVA Table A4. Look up n = 36 and k = 1%. PVA = $500[30.1075] = $15,053.75 Answer © 2006 by Nelson, a division of Thomson Canada Limited 56 Loan Amortization Schedules • Detail the interest and principal in each loan payment • Show the beginning and ending balances of unpaid principal for each period • Need to know Loan amount (PVA) Payment (PMT) Periodic interest rate (k) © 2006 by Nelson, a division of Thomson Canada Limited 57 Example 8.11: Loan Amortization Schedule Example Q: Develop an amortization schedule for the loan demonstrated in Example 8.11 Note that the Interest portion of the payment is decreasing while the Principal portion is increasing. © 2006 by Nelson, a division of Thomson Canada Limited 58 Mortgage Loans • Mortgage loans (AKA: mortgages) Loans used to buy real estate • Often the largest single financial transaction in a person’s life Typically an amortized loan over 30 years • During the early years of the mortgage nearly all the payment goes toward paying interest • This reverses toward the end of the mortgage Halfway through a mortgage’s life half of the loan has not been paid off © 2006 by Nelson, a division of Thomson Canada Limited 59 Mortgage Loans • Implications of mortgage payment pattern Long-term loans like mortgages result in large total interest amounts over the life of the loan • At 6% interest, compounded monthly, over 25 years, borrower pays almost the amount of the loan just in interest! • Canadian banks compound semi-annually, thus lowering interest charges Early mortgage payments and more frequent mortgage payments provide a large interest saving © 2006 by Nelson, a division of Thomson Canada Limited 60 Mortgage Loans—Example Example Q: Calculate the monthly payment for a 30-year 7.175% mortgage of $150,000. Also calculate the total interest paid over the life of the loan. A: Adjust the n and k for monthly compounding and input the following calculator keystrokes. Monthly payment $1,015.65 X # of payments 360 N 360 I/Y 0.5979 Total payments $365,634 FV 0 - Original Loan $150,000 PV 150000 Total Interest $215,634 Interest / Principal 143.76% PMT 1,015.65 Answer © 2006 by Nelson, a division of Thomson Canada Limited 61 The Annuity Due • In an annuity due payments occur at the beginning of each period • The future value of an annuity due Because each payment is received one period earlier, it spends one period longer in the bank earning interest F V A d n = P M T + P M T 1 + k P M T 1 + k n -1 1 k w h ich w ritte n w ith th e in te re st fa cto r b e co m e s: F V A d n P M T F V F A k ,n 1 k © 2006 by Nelson, a division of Thomson Canada Limited 62 The Future Value of a Three-Period Annuity Due Figure 8.7: © 2006 by Nelson, a division of Thomson Canada Limited 63 Example Example 8.12: The Annuity Due Q: The Baxter Corporation started making sinking fund deposits of $50,000 per quarter today. Baxter’s bank pays 8% compounded quarterly, and the payments will be made for 10 years. What will the fund be worth at the end of that time? A: Adjust the k and n for quarterly compounding and input the following calculator keystrokes. NOTE: Advanced calculators allow you to switch from END (ordinary annuity) to BEGIN (annuity due) mode. N 40 I/Y 2 PMT 50000 PV 0 FV 3,020,099 x 1.02 = 3,080,501 Answer © 2006 by Nelson, a division of Thomson Canada Limited 64 The Annuity Due • The present value of an annuity due Formula P V A d P M T P V F A k ,n 1 k Recognizing types of annuity problems Always represent a stream of equal payments Always involve some kind of a transaction at one end of the stream of payments • • End of stream—future value of an annuity Beginning of stream—present value of an annuity © 2006 by Nelson, a division of Thomson Canada Limited 65 Perpetuities • A perpetuity is a stream of regular payments that goes on forever An infinite annuity • Future value of a perpetuity Makes no sense because there is no end point • Present value of a perpetuity A diminishing series of numbers • Each payment’s present value is smaller than the one before P Vp PMT k © 2006 by Nelson, a division of Thomson Canada Limited 66 Example 8.13: Perpetuities Example Q: The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Investors can earn 8% compounded quarterly on their money. How much can Longhorn sell this security for? A: Convert the k to a quarterly k and plug the values into the equation. P Vp PMT k $5 0 .0 2 $250 You may also work this by inputting a large n into your calculator (to simulate infinity), as shown below. N I/Y PMT FV PV © 2006 by Nelson, a division of Thomson Canada Limited 999 2 5 0 250 Answer 67 Continuous Compounding • Compounding periods can be shorter than a day As the time periods become infinitesimally short, interest is said to be compounded continuously • To determine the future value of a continuously compounded value: F Vn P V e kn Where k = nominal rate, n = number of years, e = 2.71828 © 2006 by Nelson, a division of Thomson Canada Limited 68 Continuous Compounding Example Example 8.15: Q: The First National Bank of Cardston is offering continuously compounded interest on savings deposits. If you deposit $5,000 at 6½% compounded continuously and leave it in the bank for 3½ years, how much will you have? What is the equivalent annual rate (EAR) of 12% compounded continuously? © 2006 by Nelson, a division of Thomson Canada Limited 69 Continuous Compounding Example 8.15: A: To determine the future value of $5,000, plug the appropriate values into the equation FVn = P V e kn = Example FV 3.5= $5,000(2.71828) .065 3.5 = $6.277.29 To determine the EAR of 12% compounded continuously, find the future value of $100 compounded continuously in one year, then calculate the annual return F V 1 = $ 1 0 0 e .1 2 1 EAR= = $ 1 0 0 2 .7 1 8 2 8 .1 2 $112.75-$100 = 12.75% $100 © 2006 by Nelson, a division of Thomson Canada Limited = $112.75 NOTE: Some advanced calculators have a function to solve for the EAR with continuous compounding 70 Table 8.5: Time Value Formulas © 2006 by Nelson, a division of Thomson Canada Limited 71 Multipart Problems • Time value problems are often combined due to complex nature of real situations A time line portrayal can be critical to keeping things straight © 2006 by Nelson, a division of Thomson Canada Limited 72 Example Example 8.16: Multipart Problems Q: Exeter Inc. has $75,000 invested in securities that earn a return of 16% compounded quarterly. The company is developing a new product that it plans to launch in two years at a cost of $500,000. Management would like to bank money from now until the launch to be sure of having the $500,000 in hand at that time. The money currently invested in securities can be used to provide part of the launch fund. Exeter’s bank account will pay 12% compounded monthly. How much should Exeter deposit with the bank each month ? © 2006 by Nelson, a division of Thomson Canada Limited 73 Example Example 8.16: Multipart Problems A: Two things are happening in this problem Exeter is saving money every month (an annuity) and The money invested in securities (an amount) is growing independently at interest We have three steps First, we need to find the future value of $75,000 Then, we subtract that future value from $500,000 to determine how much extra Exeter needs to save via the annuity Then, we solve a future value of an annuity problem for the payment © 2006 by Nelson, a division of Thomson Canada Limited 74 Example 8.16: Multipart Problems Example To find the future value of the $75,000… N I/Y PMT FV PV To find the savings annuity value N I/Y PV FV PMT 24 1 0 8 4 0 75000 102,645 Answer $500,000 - $102,645 = $397,355 397355 14,731 Answer © 2006 by Nelson, a division of Thomson Canada Limited 75 Uneven Streams • Many real world problems have sequences of uneven cash flows These are NOT annuities • For example, if you were asked to determine the present value of the following stream of cash flows $100 $200 $300 Must discount each cash flow individually Not really a problem when attempting to determine either a present or future value Becomes a problem when attempting to determine an interest rate © 2006 by Nelson, a division of Thomson Canada Limited 76 Example 8.18: Uneven Streams Q: Calculate the interest rate at which the present value of the stream of payments shown below is $500. Example $100 $200 $300 A: Start with a guess of 12% and discount each amount separately at that rate. P V = F V1 P V Fk ,1 F V 2 P V Fk ,2 F V 3 P V Fk ,3 $ 1 0 0 P V F1 2 ,1 $ 2 0 0 P V F1 2 ,2 $ 3 0 0 P V F1 2 ,3 $ 1 0 0 .8 9 2 9 $ 2 0 0 .7 9 7 2 $ 3 0 0 .7 1 1 8 $ 4 6 2 .2 7 This value is too low; select a lower interest rate. Using 11% gives us $471.77. The answer is between 8% and 9%. © 2006 by Nelson, a division of Thomson Canada Limited 77 Imbedded Annuities • Sometimes uneven streams of cash flows will have annuities imbedded within them We can use the annuity formula to calculate the present or future value of that portion of the problem © 2006 by Nelson, a division of Thomson Canada Limited 78