Chapter 5 Future and Present Values of Annuities Chapter Outline 5.1 The Time Value Formula for Constant 5.4 Cash Flows Connecting Beginning and Annuities Ending Wealth 5.4a Cash Flow, CF, As the Unknown 5.2 Future Values of Annuities Variable 5.2a Ending Wealth, FV, As the 5.4b Other Two-Stage Problems Unknown Variable 5.2b Using the Annuity and 5.5 Amortization Mechanics Lump-Sum Formulas Together 5.5a Partitioning the Payment into Principal and Interest 5.3 Present Values of Annuities 5.5b Re-pricing Loans: Book Versus 5.3a Beginning Wealth, PV, As the Market Value Unknown Variable 5.3b The Special Case of Perpetuities 223 224 Chapter 5 Combining cash flows at different points in time requires accounting for differences in time value. The general time value formula for mixed cash flows from the previous chapter (Formula 4.11) properly handles all situations. That approach, however, is very general because it ac- commodates situations where the cash flow each period is possibly a different amount. For some financial situations the cash flow each period is exactly the same amount. Consumer and mortgage loans, for example, generally have a fixed payment that is exactly the same every month. Many investment or savings plans, too, stipulate a constant pe- riodic cash flow. Procedures simplify when the cash flows are all the same amount. In this chapter we examine cash flow streams in which the cash flow each period is exactly the same. 5.1 The Time Value Formula for Constant Annuities Recall the previous chapter’s general time value formula for mixed cash flow streams from Formula 4.11: N CF FV PV = t + . ∑ t N t =1 (1 + r) (1 + r) When CF1 = CF2 = … = CFN the following simplification occurs: FORMULA 5.1 Constant Annuity Time Value Formula CF CF CF FV PV = + + … + + (1 + r)1 (1 + r)2 (1 + r)N (1 + r)N 1 – (1 + r)–N = (CF) + FV(1 + r)–N { r } . Equation 5.1 is the constant annuity time value formula. Variable definitions and cash flow timing are the same as before. CF is the pe- riodic cash flow that occurs at times 1 through N. Each period CF is the same amount. There are N unique cash flows of amount CF. PV equals the beginning wealth one period before the first periodic cash flow. The ending wealth N periods later is FV. The last CF occurs at the same time as FV. The periodic interest rate r equals APR ÷ m, where APR is the annual percentage interest rate and m is the number of compounding periods per year. The time line below illustrates the essential timing of cash flows. 0 1 2 N PV CF CF CF FV Future and Present Values of Annuities 225 Some textbooks refer to cash flows consistent with the preceding time line as ordinary annuities. That perspective maintains that cash flows occur at the end-of-periods. An alternative scenario pertains to annuities due in which case the cash flows are said to occur at the be- ginning-of-periods. The time line below illustrates essential timing for annuities due: 0 1 N – 1 N CF CF CF FV PV With an annuity due the first CF is concurrent with PV, the last CF occurs one period before FV, and still there are N occurrences of CF. Most calculators allow setting whether cash flows occur at end or be- ginning-of-periods. Practically speaking, however, as far as a time line goes the end of one period is the beginning of the next and so this distinction is a little arbitrary and potentially confusing. The im- portant fact is occurrence of the first and last CF! All lessons in this book avoid potential confusion by eliminating labels ordinary annuities and annuities due. Instead, the lessons explicitly specify timing of cash flows—all Calculator Clues assume that you keep your calculator set to end-of-period! The most significant simplification inherent with Formula 5.1 is elimination of the summation expression. For example, suppose a cash flow stream contains 360 monthly cash flows (N = 360) and all are ex- actly the same amount, like a 30-year mortgage. Usage of the general time value formula in Equation 4.11 involves summation of 360 dif- ferent terms. The constant annuity time value formula in Equation 5.1 does not involve that summation. Instead, an exponent in one of the terms takes on the value 360. Five variables appear in Formula 5.1: FV, PV, N, CF, and r. When any four of the variables are set to numerical values, the fifth becomes an unknown that takes on a value satisfying the equation. Almost always the signs on N and r are positive and easy to interpret. The signs for FV, PV, and CF, however, may sometimes be positive and other times neg- ative. Interpreting the signs on these variables is very important and sometimes complicated. The issue complicates further because dif- ferent calculators sometimes adopt different rules regarding signage. Here are three short lessons about variable signs for FV, PV, and CF in Formula 5.1 (or any of its rearrangements shown in this chapter). 1. Signage is simple to interpret when one of the three variables is zero. For example, if PV equals zero because there is no begin- ning wealth but simply there are deposits CF and ending wealth FV then signage is simple. Likewise in the lump-sum relation when CF is zero then the signs on FV and PV are easy to interpret. 2. When FV, PV, and CF are all non-zero then remember the base- line scenario that Formula 5.1 exemplifies. Beginning wealth PV 226 Chapter 5 flows into an account, periodic CF flowout of the account (like withdrawals), and ending wealth FV is the balance immediately after the last CF. For the preceding scenario all variables are positive. For scenarios that reverse the flow then reverse the sign. For example, when periodic deposits CF flowinto the ac- count assign in Formula 5.1 a negative sign to CF. 3. Usually there are two approaches for signing all variables. Whatever is positive in approach 1 is negative in approach 2, and vice versa. Both approaches lead to the same correct nu- merical answer. For example, the previous paragraph states that when PV and FV are positive then periodic deposits have negative signs. An alternative approach reverses signs: when PV and FV are both negative then assign a positive sign to periodic deposits. The choice of signs in a problem is a relative issue. The preceding paragraphs apply to Formula 5.1 or any of its rear- rangements shown throughout this chapter. Calculators adopt their own unique rules. On the BAII Plus© financial calculator variable signs are easier to interpret by taking the perspective of one of the problem participants. Assign a positive sign to money flowing into your pocket such as withdrawals or stock dividends. Deposits, however, flow out of your pocket and into the asset account. They are leaving your pocket so give them a negative sign. The sections below discuss scenarios that rely on the constant an- nuity time value formula. Exercises 5.1 CONCEPT QUIZ 1. Explain how inflation integrates into the constant annuity time value formula. 5.2 Future Values of Annuities Suppose you make a series of identical deposits and want to know the ending balance. For this scenario, FV is the unknown variable. Rear- range and isolate FV on the left-hand-side: Future and Present Values of Annuities 227 FORMULA 5.2 Future Value of a Constant Annuity Stream (1 + r)N – 1 FV = PV(1 + r)N – CF { r } N = PV(1 + r) – CF{FVIFArate = r, periods = N}. Solving for FV requires assigning numerical values for PV, N, CF, and r. The expression in curly brackets is the “future value interest factor Definition 5.1 Future for an annuity," abbreviated FVIFA. The expression depends only on r value interest factor and N. The intuitive meaning of FVIFA is simply stated. for an annuity (FVIFA) Bankers in an earlier era owned “time value books” containing The future value of one FVIFA tables. The tables list a different N for each row and a different dollar deposits made for N periodic rate for each column. The tables simply compute the value of consecutive periods that the expression in curly brackets. Looking at the FVIFA table with a pe- earn the periodic discount riodic rate equal to 15 percent and N equal to 10, for example, shows a rate r: table entry equal to 20.3037. (1 + r)N – 1 FVIFAr, N = r (1 + .15)10 – 1 FVIFA = rate = 15%, N = 10 { .15 } = 20.3037 This means that if one dollar per year is deposited for ten years, and interest of 15 percent per year accrues, the account balance equals $20.30 immediately after the last deposit. Because contributed prin- cipal equals $10, the total market interest equals $10.30. FVIFA tables enable easy computation of future sums even though the deposit is different than one dollar. With a $500 deposit, and the same rate of 15 percent for 10 years, the future value equals $500 × 20.3037, or $10,152. The tables are easy, but financial calculators and spreadsheets pretty much make the tables obsolete. The variable signs in Equation 5.2 deserve discussion. Begin with an example in which 10 percent interest compounds annually in a savings account for 2 years. With a beginning wealth PV of $100, and CF of $0, the ending FV wealth two periods later is $121 (that is, $121 = $100 × 1.102).