Presentation on theme: "The Time Value of Money (Part Two)"— Presentation transcript:
1 The Time Value of Money (Part Two) Chapter 4The Time Value of Money(Part Two)
2 1. Explain and illustrate an annuity. LEARNING OBJECTIVES1. Explain and illustrate an annuity.2. Determine the future value of an annuity.3. Determine the present value of an annuity.4. Adjust the annuity equation for present value and future value for an annuity due.5. Distinguish between the different types of loan repayments.6. Build and analyze amortization schedules.7. Calculate waiting time and interest rates for an annuity.
3 4.1 Future Value of Multiple Payment Streams With unequal periodic cash flows, treat each of the cash flows as a lump sum and calculate its future value over the relevant number of periods.Sum up the individual future values to get the future value of the multiple payment streams.
5 4.2 Future Value of an Annuity Stream Annuities are equal, periodic outflows/inflows at regular intervals, e.g. rent, lease, mortgage, car loan, and retirement annuity payments.An annuity stream can begin at the start of each period (annuity due) as is true of rent and insurance payments or at the end of each period, (ordinary annuity) as in the case of mortgage and loan payments.The formula for calculating the future value of an ordinary annuity stream is as follows:FV = PMT x (1+r)n -1rwhere PMT is the term used for the equal periodic cash flow, r is the rate of interest, and n is the number of payments, one at the end of each period (ordinary annuity).
6 4.2 Future Value of an Annuity Stream Example: Future Value of an Ordinary Annuity Stream Jill has been faithfully depositing $2,000 at the end of each year over the past 10 years into an account that pays a guaranteed 8% per year. How much money has she have accumulated in the account?
7 4.2 Future Value of an Annuity Stream Example Answer (via the long way)Future Value of Payment One = $2,000 x = $3,998.01Future Value of Payment Two = $2,000 x = $3,701.86Future Value of Payment Three = $2,000 x = $3,427.65Future Value of Payment Four = $2,000 x = $3,173.75Future Value of Payment Five = $2,000 x = $2,938.66Future Value of Payment Six = $2,000 x = $2,720.98Future Value of Payment Seven = $2,000 x = $2,519.42Future Value of Payment Eight = $2,000 x = $2,332.80Future Value of Payment Nine = $2,000 x = $2,160.00Future Value of Payment Ten = $2,000 x = $2,000.00Total Value of Account at the end of 10 years $28,973.13
8 4.2 Future Value of an Annuity Stream Example Answer (short way) FORMULA METHOD FV = PMT x (1+r)n -1 r where, PMT = $2,000; r = 8%; and n=10. FVIFA =[((1.08)10 - 1)/.08] = , FV = $2000 x = $28, USING A FINANCIAL CALCULATOR N= 10; PMT = -2,000; I/Y = 8; PV=0; CPT FV = $28,973.134.2 Future Value of an Annuity Stream
9 4.2 Future Value of an Annuity Stream USING AN EXCEL SPREADSHEET Enter =FV(8%, 10, -2000, 0, 0); Output = $28, Rate = 0.08, Nper = 10, Pmt = -2,000, PV =0, and Type is 0, for ordinary annuities USING FVIFA TABLE (A-3), page 575 Find the FVIFA in the 8% column and the 10 period row; FVIFA = FV = 2000 x = $ (off by 7 cents)
10 FIGURE 4.3 Interest and principal growth with different interest rates for $100 annual payments.
11 4.3 Present Value of an Annuity To calculate the value of a series of equal periodic cash flows at the current point in time, we can use the following simplified formula:The last portion of the equation is thePresent Value Interest Factor of an Annuity (PVIFA).Practical applications include figuring out the nest egg neededprior to retirement or lump sum needed for college expenses.
12 FIGURE 4.4 Time line of present value of annuity stream.
13 4.3 Present Value of an Annuity Example problem for the four solution methodsYou are now holding the winning lottery ticket that will pay the holder of the ticket $10,000 per year for the next 20 years. A friend has offered to buy the winning ticket from you. What should you sell the ticket for assuming you have a discount rate of 6% on future dollars (this is your opportunity rate for the future cash flow)?Four ways to solveFormulaCalculatorSpreadsheetTable
14 4.3 Present Value of an Annuity FormulaInputs? N = 20, r = 0.06, PMT = $10,000 andCompute PV,PV = $10,000 x [1 – 1/(1.06)20] / 0.06 = $114,699.21CalculatorInputs? N = 20, I/Y = 6.0, PMT = 10,000, FV = 0Compute PVPV = -$114,699.21
15 4.3 Present Value of an Annuity Spreadsheet, use PV functionInputs? Rate = 0.06, Nper = 20, Pmt = 10,000, Fv = 0PV = -$114,699.21TableFirst find the PVIFA with n = 20 and r = 6.0% on page 576, PVIFA =Calculate PV = $10,000 x = $114, (off by 21 cents)
16 4.4 Annuity Due and Perpetuity A cash flow stream such as rent, lease, and insurance payments, which involves equal periodic cash flows that begin right away or at the beginning of each time interval is known as an annuity due.
17 4.4 Annuity Due and Perpetuity Formula AdjustmentPV annuity due = PV ordinary annuity x (1+r)FV annuity due = FV ordinary annuity x (1+r)PV annuity due > PV ordinary annuityFV annuity due > FV ordinary annuityCan you see why?Financial calculatorMode set to BGN for annuity dueMode set to END for an ordinary annuitySpreadsheetType = 0 or omitted for an ordinary annuityType = 1 for an annuity due.
18 4.4 Annuity Due and Perpetuity Example: Annuity Due versus Ordinary Annuity Let’s say that you are saving up for retirement and decide to deposit $3,000 each year for the next 20 years into an account which pays a rate of interest of 8% per year. By how much will your accumulated nest egg vary if you make each of the 20 deposits at the beginning of the year, starting right away, rather than at the end of each of the next twenty years?
19 4.4 Annuity Due and Perpetuity FV ordinary annuity = $3,000 x [((1.08)20 - 1)/.08]= $3,000 x= $137,285.89FV of annuity due = FV of ordinary annuity x (1+r)FV of annuity due = $137, x (1.08) = $148,268.76Difference is $10,982.87Example Answer Given information: PMT = -$3,000; n=20; i= 8%; PV=0;
20 4.4 Annuity Due and Perpetuity Perpetuity A Perpetuity is an equal periodic cash flow stream that will never cease. The PV of a perpetuity is calculated by using the following equation:
21 4.4 Annuity Due and Perpetuity Example: PV of a perpetuityIf you are considering the purchase of a consol that pays $60 per year forever, and the rate of interest you want to earn is 10% per year, how much money should you pay for the consol? Answer:r=10%, PMT = $60; and PV = ($60/0.10) = $600$600 is the most you should pay for the consol.You can think of it this way, if you put $600 in the bank earning 10% you can withdraw the annual interest of $60 every year forever without touching the principal.
22 4.5 Three Payment Methods Discount loan Interest-only loan Loan payments can be structured in one of 3 ways:Discount loanPrincipal and interest is paid in lump sum at endInterest-only loanPeriodic interest-only payments, principal due at end.Amortized loanEqual periodic payments of principal and interest
23 4.5 Three Payment MethodsExample: Discount versus Interest-only versus Amortized loans Roseanne wants to borrow $40,000 for a period of 5 years. The lenders offers her a choice of three payment structures: 1) Pay all of the interest (10% per year) and principal in one lump sum at the end of 5 years; 2) Pay interest at the rate of 10% per year for 4 years and then a final payment of interest and principal at the end of the 5th year; 3) Pay 5 equal payments at the end of each year inclusive of interest and part of the principal. Under which of the three options will Roseanne pay the least interest and why? Calculate the total amount of the payments and the amount of interest paid under each alternative.
24 4.5 Three Payment MethodsMethod 1: Discount Loan. Since all the interest and the principal is paid at the end of 5 years we can use the FV of a lump sum equation to calculate the payment required, i.e. FV = PV x (1 + r)n FV5 = $40,000 x (1+0.10)5 = $40,000 x = $64, Interest paid = Total payment - Loan amount Interest paid = $64, $40,000 = $24,420.40
27 4.5 Three Payment Methods Loan Type Total Payment Interest Paid Discount Loan $64, $24,420.40Interest-only Loan $60, $20,000.00Amortized Loan $52, $12,759.31Why does the equal annual payments of principal and interest each period have the lowest total interest?
28 4.6 Amortization Schedules Tabular listing of the allocation of each loan payment towards interest and principal reduction Helps borrowers and lenders figure out the payoff balance on an outstanding loan. Procedure: 1) Compute the amount of each equal periodic payment (PMT) using the ordinary annuity formula. 2) Calculate interest on unpaid balance at the end of each period, minus it from the PMT, reduce the loan balance by the remaining amount, 3) Continue the process for each payment period, until we get a zero loan balance.
29 4.6 Amortization Schedules Example: Loan amortization schedule. Prepare a loan amortization schedule for the amortized loan option given in the previous Example with the five annual payments for the $40,000 at 10% annual interest rate. What is the loan payoff amount at the end of 2 years? Step One, determine the annual payment: PV = $40,000; n=5; I/Y=10.0; FV=0; CPT PMT = $10, (rounded to nearest whole cent)
30 Amortization TableYearBeg. BalPaymentInterestPrin. RedEnd. Bal140,000.0010,551.904,000.006,551.9033,448.1023,344.817,207.0926,241.0132,264.107,927.8018,313.2141,831.328,720.589,592.645959.260.00The loan payoff amount at the end of 2 years is $26,241.01
31 4.7 Waiting Time and Interest Rates for Annuities Problems involving annuities typically have 4 variables, i.e. PV or FV, PMT, r, nIf any 3 of the 4 variables are given, we can easily solve for the fourth one.This section deals with the procedure of solving problems where either n or r is not given.For example:Finding out how many deposits (n) it would take to reach a retirement or investment goal;Figuring out the rate of return (r) required to reach a retirement goal given fixed monthly deposits,
32 4.7 Waiting Time and Interest Rates for Annuities Example: Solving for the number of annuities involved Martha wants to save up $100,000 as soon as possible so that she can use it as a down payment on her dream house. She figures that she can easily set aside $8,000 per year and earn 8% annually on her deposits. How many years will Martha have to wait before she can buy that dream house?
33 4.7 Waiting Time for Annuities Method 2: Using a financial calculator INPUT ? TVM KEYS N I/Y PV PMT FV Compute Method 3: Using an Excel spreadsheet Using the “NPER” function we enter the following: Rate = 8%; Pmt = -8000; PV = 0; FV = ; Type = 0 or omitted; display in excel = NPER(8%,-8000,0,100000,0) The cell displays
34 4.7 Waiting Time for Annuities Method 1: Formula (uses natural logs) N = ln ([FV x r]/PMT + 1) / ln (1+r) N = ln ([100,000 x 0.08]/8, ) / ln 1.08 N = ln 2 / ln 1.08 = / = Method 4: Tables You need to interpret from the tables… Take FV / PMT to find the FVIFA, Look for under the 8% column, find its close to n = 9 (its between 9 and 10 but very close to 9)
35 4.8 Finding the interest rate Solving a Lottery Problem In the case of lottery winnings, 2 choices1) Annual lottery payment for fixednumber of years, OR2) Lump sum payout.How do we make an informed judgment?Need to figure out the implied rate of return of both options using TVM functions.
36 4.8 Finding the interest rate Example: Calculating an implied rate of return given an annuity Let’s say that you have just won the state lottery. The authorities have given you a choice of either taking a lump sum of $26,000,000 or a 30-year annuity of $1,500,000. Both payments are assumed to be after-tax. What will you do? The missing variable is the implied interest rate on the two payment choices.
37 4.8 Finding the interest rate Using the TVM keys of a financial calculator, enter: PV=26,000,000; FV=0; N=30; PMT = -1,500,000; CPT I/Y = 3.98% 3.98% = rate of interest used to determine the 30-year annuity of $1,500,000 versus the $26,000,000 lump sum pay out. Choice: If you can earn an annual after-tax rate of return higher than 4.0% over the next 30 years, go with the lump sum. Otherwise, take the annuity option.
38 4.8 Finding the interest rate We could use the Spreadsheet functions (Rate function) to find the 3.98%.We could use the Tables to estimate the interest rate by looking at the PVIFA at 30 years with the PVIFA calculated as PV / PMT but again we will need to estimate between to interest rates although in this case it will be very close to 4.0% (PVIFA is )We can not use the formula to solve for interest rate, it is an iterative process (trial and error)
39 4.9 Ten Important Points about the TVM Equation Amounts of money can be added or subtracted only if they are at the same point in time.The timing and the amount of the cash flow are what matters.It is very helpful to lay out the timing and amount of the cash flow with a timeline.Present value calculations discount all future cash flow back to current time.Future value calculations value cash flows at a single point in time in the future
40 4.9 Ten Important Points about the TVM Equation An annuity is a series of equal cash payments at regular intervals across time.The time value of money equation has four variables but only one basic equation, and so you must know three of the four variables before you can solve for the missing or unknown variable.There are three basic methods to solve for an unknown time value of money variable:(1) Using equations and calculating the answer;(2) Using the TVM keys on a calculator;(3) Using financial functions from a spreadsheet.
41 4.9 Ten Important Points about the TVM Equation There are 3 basic ways to repay a loan:Discount loans,Interest-only loans, andAmortized loans.Despite the seemingly accurate answers from the time value of money equation, in many situations not all the important data can be classified into the variables of present value, i.e., time, interest rate, payment, or future value.