Transcript Chapter 5
Time Value of Money II: Analyzing Annuity Cash Flows Chapter 5 Fin 325, Section 04 – Spring 2010 Washington State University 1 Introduction The previous chapter involved moving a single cash flow from one point in time to another Many business situations involve multiple cash flows Annuity problems deal with regular, evenlyspaced cash flows Car loans and home mortgage loans Saving for retirement Companies paying interest on debt Companies paying dividends 2 Example Consider the following cash flows: you make a $100 deposit today, followed by a $125 deposit next year and a $150 deposit at the end of the second year. If interest rates are 7%, what is the future value of your account at the end of the 3rd year? -100 -125 -150 ... 0 1 FV3 = $122.50 + $143.11 + $160.50 = $426.11 2 3 FV of Level CFs Suppose that the cash flows are the same each period Level cash flows (annuities) are common in finance (1 i ) 1 FVAN PMT i N For annuities, we can use the PMT key on a financial calculator to input the annuity payment 4 Calculator Solution Example: suppose that $100 deposits are made at the end of each year for five years. If interest rates are 8 percent per year, the future value of the annuity is: INPUT OUTPUT 5 N 8 I/YR 0 PV -100 PMT FV 586.66 5 Another example: Calculate the future value if a $50 deposit is made every year for 20 years at a 6 percent interest rate INPUT OUTPUT 20 N 6 I/YR 0 PV -50 PMT FV 1,839.28 6 What-if Analysis What if the amount deposited doubles to $100 per year? What if the $100 is deposited every year for 40 years rather than 20 years? What if the interest rate is increased from 6 percent to 10 percent? 7 PV of Multiple Cash Flows Consider the example we started with: you make a $100 deposit today, followed by a $125 deposit next year and a $150 deposit at the end of the second year. Interest rates are 7% -100 -125 -150 ... 0 1 2 3 We can find their individual present values and add them up 8 Present Value of Level Cash Flows The present value of an annuity concept has many practical uses: Most loans are set up with even payments throughout the life of the loan The general formula for the present value of an annuity is: 1 1 N (1 i ) PVA N PMT i 10 Example: What is the present value of an annuity consisting of $100 payments made at the end of the next 5 years if interest rates are 8 percent per year? 1 1 (1 i ) N PVAN PMT i 1 1 (1 0.08) 5 100 i $ 399.27 INPUT OUTPUT 5 N 8 I/YR PV 399.27 -100 PMT 0 FV Perpetuities Perpetuities represent a special type of annuity in which the cash flows go on forever The present value of a perpetuity is calculated as: PMT PV of Perpetuity i 12 Example: Find the present value of a perpetuity that pays $100 per year forever if the discount rate is 10 percent. $100 PV 0.10 $1,000 13 Ordinary Annuity vs. Annuity Due ordinary annuity - the payment occurs at the end of each period. annuitiy due - the annuity payments occur at the beginning of each period. 14 FV and PV of Annuity Due FVAN Due FVAN (1 i) (1 i) N 1 (1 i) FVAN Due PMT i PVAN Due PVAN (1 i) 1 (1 i ) (1 i ) N 1 PVA N Due PMT i Set financial calculators to BEGIN mode (BGN) when calculate annuity due problems. 15 Example: Find the present value of an annuity due that pays 100 per year for 5 years if the interest rate is 8 percent. Before you begin: should the PV be larger or smaller than if the payments occur at the end of each period? First Step: Place your calculator in BGN mode INPUT OUTPUT 5 N 8 I/YR PV 431.21 -100 PMT 0 FV 16 Compounding Frequency So far we have assumed that interest is compounded once per year What happens when interest is compounded more frequently? Example: What if 12 percent interest is compounded semiannually? Let’s say that we invest $100. If interest were compounded annually, we would end up with $112. But, semiannual compounding means that our $100 would earn 6 percent halfway through the year and the other 6 percent at the end. We would end up with: FV = $100 x (1+1.06) x (1.06) = $112.36. 17 APR and EAR The quoted, or nominal rate is called the annual percentage rate (APR) The rate that incorporates compounding is called the effective annual rate (EAR) m APR EAR 1 m 1 18 Example: A bank loan has a quoted rate of 12 percent. Calculate the effective annual rate if the interest is compounded monthly 0.12 12 EAR 1 1 12 EAR 12.68% 19 Calculator Solution On the TI BAII Plus calculator, ICONV (interest conversion) function converts nominal rates to effective rates Input two of Nominal rate, Effective rate, and Compounding periods and the calculator solves for the 3rd variable. For the example above: NOM = 12 C/Yr = 12 EFF = 12.68 20 Example: What is the effective rate if the quoted rate is 10 percent compounded daily? ICONV NOM = 10 C/Yr = 365 EFF = 10.5156% 21 Annuity Loans Finding the interest rate Often a business will know the cost of something, as well as the associated cash flows. For example: A piece of equipment costs $100,000 and provides positive cash flows of $25,000 for 6 years. What rate of return does this opportunity offer? INPUT OUTPUT 6 N -100,000 25,000 0 I/YR PV PMT FV 12.98 22 Finding Payments on an Amortized Loan Example: You want a car loan of $10,000. The loan is for 4 years and interest rates are 9 percent per year. Calculate your monthly payment Before we work this problem, we need to discuss how to set our calculator to solve problems that involve payments that are not annual. We typically do this by adjusting the N, I, and PMT to reflect the relevant period 23 For the above problem: Problem Data Calculator input 4 years N = 4 x 12 = 48 months 9% loan I = 9/12 = 0.75% per month Loan amount = $10,000 PV = 10,000 Solution: INPUT OUTPUT 48 N 0.75 10,000 I/YR PV PMT 248.85 0 FV Since N and I are monthly, we know that the PMT is the monthly payment 24 Amortized Loan Schedules • Amortized loans are characterized by level payments, with an increasing portion of the payment consisting of principal, and a decreasing proportion of interest • Example: You have received a $150,00 student loan that is to be repaid in annual payments over 3 years. The interest rate is 10%. Construct an amortization schedule for the loan. 25 The first step is to calculate the payment. INPUT OUTPUT 3 N 10 150,000 0 I/YR PV PMT FV 60,317.22 Beg Bal Payment Interest 1 150,000 60,317.22 15,000 2 104,682.78 60,317.22 10,468.28 49,848.94 54,833.84 3 54,833.84 0.00 60,317.22 5,483.38 Principal End Bal 45,317.22 104,682.78 54,833.84 26 Computing the Time Period How long will it take to pay off a loan? Example: How long will it take to pay off a $5,000 loan with a 19 percent APR which compounds monthly? The payment is $150 per month I = 19/12 = 1.58333 1.58333 5,000 N I/YR PV OUTPUT 47.8 INPUT -150 PMT 0 FV 27