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Chapter 5

Chapter 5. Time Value of Money 2: Analyzing Annuity Cash Flows. 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, evenly-spaced cash flows

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Chapter 5

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  1. Chapter 5 Time Value of Money 2: Analyzing Annuity Cash Flows ACT3211 FINANCIAL MANAGEMENT

  2. ACT3211 FINANCIAL MANAGEMENT

  3. 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, evenly-spaced cash flows • Car loans and home mortgage loans • Saving for retirement • Companies paying interest on debt • Companies paying dividends ACT3211 FINANCIAL MANAGEMENT

  4. -100 -125 -150 ... 0 1 2 3 Consider the following cash flows: you make a RM100 deposit today, followed by a RM125 deposit next year and a RM150 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? ACT3211 FINANCIAL MANAGEMENT

  5. -100 -125 -150 ... 0 1 2 3 • Notice that the first deposit will compound for 3 years, the second deposit will compound for 2 years, and the last deposit will compound for 1 year. • We can calculate the future value of each deposit individually and add them up to get the total FV3 = RM122.50 + RM143.11 + RM160.50 = RM426.11 ACT3211 FINANCIAL MANAGEMENT

  6. (1+i) - 1 i FVAN = PMT( ) • Now, suppose that the cash flows are the same each period • Level cash flows are common in finance. These problems are known as annuities ACT3211 FINANCIAL MANAGEMENT

  7. Annuities and the Financial Calculator In the previous chapter, the level payment button PMT was always set to zero Now, for annuities, we can use the PMT key to input the annuity payment 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: ACT3211 FINANCIAL MANAGEMENT

  8. Calculator Solution -100 INPUT 5 8 0 N I/YR PV PMT FV OUTPUT 586.66 ACT3211 FINANCIAL MANAGEMENT

  9. -50 INPUT 20 6 0 N I/YR PV PMT FV OUTPUT 1,839.28 Another example: Calculate the future value if a RM50 deposit is made every year for 20 years at a 6 percent interest rate ACT3211 FINANCIAL MANAGEMENT

  10. What if the amount deposited doubles to RM100 per year? • The future value doubles to $RM,678.56 • What if the RM100 is deposited every year for 40 years rather than 20 years? Does the future value double as well? • No – remember that the time and interest rate variables are exponentially related to value ACT3211 FINANCIAL MANAGEMENT

  11. -100 INPUT 40 6 0 N I/YR PV PMT FV OUTPUT 15,476.20 The future value more than quadruples when the time is doubled in this example ACT3211 FINANCIAL MANAGEMENT

  12. -100 INPUT 40 10 0 N I/YR PV PMT FV OUTPUT 44,259.26 What if the interest rate is increased from 6 percent to 10 percent? The size of the periodic payments, the number of years invested, and the interest rate significantly impact the future value of an annuity ACT3211 FINANCIAL MANAGEMENT

  13. Present Value of Multiple Cash Flows -100 -125 -150 ... 0 1 2 3 Consider the example we started with: you make a RM100 deposit today, followed by a RM125 deposit next year and a RM150 deposit at the end of the second year. Interest rates are 7% To find the present value of these cash flows we recognize that we can find their individual present values and add them up ACT3211 FINANCIAL MANAGEMENT

  14. Present Value of Level Cash Flows 1 i 1 i(1+i)n PVA = PMT( - ) • 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: ACT3211 FINANCIAL MANAGEMENT

  15. 1 .08 1 .08(1+.08)5 PV = 100 ( ) PV = 100(3.9927) PV = 399.27 Example: What is the present value of an annuity consisting of RM100 payments made at the end of the next 5 years if interest rates are 8 percent per year? ACT3211 FINANCIAL MANAGEMENT

  16. -100 0 INPUT 5 8 N I/YR PV PMT FV OUTPUT 399.27 Calculator solution: ACT3211 FINANCIAL MANAGEMENT

  17. Perpetuities PMT i PV of perpetuity = • Perpetuities represent a special type of annuity in which the cash flows go on forever • Real-life applications of perpetuities • Preferred stock • British 2 ½ % Consolidated Stock (a debt known as consols) • The present value of a perpetuity is calculated using a simple equation: ACT3211 FINANCIAL MANAGEMENT

  18. Example: Find the present value of a perpetuity that pays RM100 per year forever if the discount rate is 10 percent. PV = 100/.10 = $1000 ACT3211 FINANCIAL MANAGEMENT

  19. Ordinary Annuities vs. Annuities Due So far we have worked problems where the payment occurs at the end of each period. This is called an ordinary annuity Sometimes, however, the annuity payments occur at the beginning of each period. These are called annuities due ACT3211 FINANCIAL MANAGEMENT

  20. In calculating the future value of an annuity due, we recognize that the payments all occur one period sooner than for an ordinary annuity, and therefore earn an extra period of interest. We can adjust the FV as follows: FVAN due = FVAN x (1+i) ACT3211 FINANCIAL MANAGEMENT

  21. Likewise, in calculating the present value of an annuity due, we discount each cash flow one less period. We can adjust the PV using the following equation: PVAN due = PVAN x (1+i) • In our financial calculators, we need to tell the calculator that the payments occur at the beginning of each period. We do this by putting the calculator in BEGIN mode (represented by BGN) ACT3211 FINANCIAL MANAGEMENT

  22. -100 0 INPUT 5 8 N I/YR PV PMT FV OUTPUT 431.21 • 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 ACT3211 FINANCIAL MANAGEMENT

  23. 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 RM100. If interest were compounded annually, we would end up with RM112. But, semiannual compounding means that our RM100 would earn 6 percent halfway through the year and the other 6 percent at the end. We would end up with: FV = RM100 x (1+1.06) x (1.06) = RM112.36. ACT3211 FINANCIAL MANAGEMENT

  24. We end up with more than RM112 due to compounding. The RM6 interest we earned in the first half earns RM0.36 in interest in the second half. The quoted, or nominal rate is called the annual percentage rate (APR) The rate that incorporates compounding is called the effective annual rate (EAR) ACT3211 FINANCIAL MANAGEMENT

  25. The relationship between APR and EAR is as follows: ACT3211 FINANCIAL MANAGEMENT

  26. Example: A bank loan has a quoted rate of 12 percent. Calculate the effective annual rate if the interest is compounded monthly EAR = 12.68% ACT3211 FINANCIAL MANAGEMENT

  27. Calculator solution: • Financial calculators have a function that converts nominal rates to effective rates • These functions have 3 variables: Nominal rate, Effective rate, and Compounding periods. The user inputs two of them and the calculator solves for the 3rd. • On the TI BAII Plus calculator the function is ICONV (interest conversion) • For the example above: • NOM = 12 • C/Yr = 12 • EFF = 12.68 ACT3211 FINANCIAL MANAGEMENT

  28. Example: What is the effective rate if the quoted rate is 10 percent compounded daily? • ICONV • NOM = 10 • C/Yr = 365 • EFF = 10.5156% ACT3211 FINANCIAL MANAGEMENT

  29. Annuity Loans -100,000 25,000 0 INPUT 6 N I/YR PV PMT FV 12.98 OUTPUT • 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? ACT3211 FINANCIAL MANAGEMENT

  30. Finding Payments on an Amortized Loan • Example: You want a car loan of RM10,000. 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 (we will assume we leave the calculator set to 1 payment per year, i.e. P/YR=1) ACT3211 FINANCIAL MANAGEMENT

  31. 48 0.75 10,000 0 INPUT N I/YR PV PMT FV 248.85 OUTPUT For the above problem: Solution: Since N and I are monthly, we know that the PMT is the monthly payment ACT3211 FINANCIAL MANAGEMENT

  32. 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: Your business has received a RM150,00 loan that is to be repaid in annual payments over 3 years. The interest rate is 10%. Construct an amortization schedule for the loan. ACT3211 FINANCIAL MANAGEMENT

  33. 3 10 150,000 0 INPUT N I/YR PV PMT FV 60,317.22 OUTPUT The first step is to calculate the payment. ACT3211 FINANCIAL MANAGEMENT

  34. 1.58333 5,000 -150 0 INPUT N I/YR PV PMT FV 47.8 OUTPUT • Computing the Time Period • How long will it take to pay off a loan? • Example: How long will it take to pay off a RM5,000 loan with a 19 percent APR which compounds monthly? The payment is RM150 per month • I = 19/12 = 1.58333 ACT3211 FINANCIAL MANAGEMENT

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