Future and Present Values of annuities

Chapter 5

Future and Present Values of Annuities

Chapter Outline

5.1The Time Value Formula for Constant Annuities

5.2 Future Values of Annuities

5.2aEnding Wealth, FV, As the Unknown Variable

5.2bUsing the Annuity and Lump-Sum Formulas Together

5.3 Present Values of Annuities

5.3aBeginning Wealth, PV, As the Unknown Variable

5.3b The Special Case of Perpetuities

5.4Cash Flows Connecting Beginning and Ending Wealth

5.4aCash Flow, CF, As the Unknown Variable

5.4b Other Two-Stage Problems

5.5Amortization Mechanics

5.5aPartitioning the Payment into Principal and Interest

5.5bRe-pricing Loans: Book Versus Market Value

223

224Chapter 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 accommodates 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 periodic 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:

PV

=

N

t =1

CFt (1 + r)t

+

(1

FV + r)N

.

When CF1 = CF2 = ... = CFN the following simplification occurs:

FORMULA 5.1 Constant Annuity Time Value Formula

PV = CF + CF + ... + CF + FV

(1 + r)1 (1 + r)2

(1 + r)N (1 + r)N

{ } = (CF) 1 ? (1 + r)?N + 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 periodic 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 beginning-of-periods. The time line below illustrates essential timing for annuities due:

0

1

N ? 1

N

CF

CF

PV

CF

FV

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 beginning-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 important 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 exactly the same amount, like a 30-year mortgage. Usage of the general time value formula in Equation 4.11 involves summation of 360 different 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 negative. Interpreting the signs on these variables is very important and sometimes complicated. The issue complicates further because different 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 beginning 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 baseline scenario that Formula 5.1 exemplifies. Beginning wealth PV

226Chapter 5

flows into an account, periodic CF flow out 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 flow into the account 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 numerical 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 rearrangements 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 annuity 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. Rearrange 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

{ } FV = PV(1 + r)N ? CF (1 + r)N ? 1 r = PV(1 + r)N ? 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

for an annuity," abbreviated FVIFA. The expression depends only on r and N. The intuitive meaning of FVIFA is simply stated.

Bankers in an earlier era owned "time value books" containing FVIFA tables. The tables list a different N for each row and a different periodic rate for each column. The tables simply compute the value of the expression in curly brackets. Looking at the FVIFA table with a periodic rate equal to 15 percent and N equal to 10, for example, shows a table entry equal to 20.3037.

{ } FVIFArate = 15%, N = 10 =

(1 + .15)10 ? 1 .15

Definition 5.1 Future value interest factor for an annuity (FVIFA)

The future value of one dollar deposits made for N consecutive periods that earn the periodic discount rate r:

FVIFAr, N = (1 + r)N ? 1 r

= 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 principal 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). This lump-sum scenario is shown in the time line below:

0

1

2

PV = $100

CF = $0

CF = $0 FV = $121

Now extend the example. Suppose that $20 is withdrawn from the account at times 1 and 2; that is, CF = $20. This annuity scenario is shown in this time line:

228Chapter 5

0

1

2

PV = $100

CF = $20

CF = $20 FV = ?

Recall that Formula 5.1 (and its rearrangement in 5.2) assumes that when PV, CF, and FV are all positive that CF represents a withdrawal, or return of cash flow. This problem fits that description. On the righthand-side of Formula 5.2 subtract the positive CF from the positive PV(1 + r)N. How much now is the ending balance at time 2? Substitute into Equation 5.2 to find that:

{ } FV = $100(1.10)2 ? $20

(1.10)2 ? 1 0.10

= $121 - $42

= $79

The $42 subtracted-out equals the future value of the withdrawal stream. The withdrawals naturally diminish the ending balance below $121; it falls to $79.

CALCULATOR CLUE 5.1

The figure above also is computable with the time value functions. On the BAII Plus? type 2nd FV to clear the time value memories. Type 2nd I/Y 1 ENTER 2nd CPT to enforce annual compounding. Then solve the preceding problem as follows:

100 PV 20 +/? PMT 2 N 10 I/Y CPT FV

The display shows $-79. The sign on the $100 initial deposit, PV , is positive because cash flows into the account to establish the balance. The sign on the two $20 withdrawals, PMT , is negative because cash flows out of the account. The sign on the ending wealth, FV , is negative because funds are available to flow out of the account.

5.2a Ending Wealth, FV, As the Unknown Variable

Regular savings plans typically involve a series of deposits, a known interest rate, and the unknown variable is ending wealth. Consider the example below.

Future and Present Values of Annuities 229

Example 1 Find Savings Plan Accumulation ?FV10a

You wish to save for the holiday season by starting a regular savings plan at the local bank. Deposits will be made every week, with the first one today. In 40 weeks, at the time you make the final deposit, you will withdraw all accumulated funds. If your deposits are $75 weekly, and the annual interest rate of 6.25 percent compounds weekly, how much will be available for the withdrawal?

Solution

First get the time line right. Deposits commence right now, time zero, and conclude exactly 40 weeks from today.

t = 0

1

2

40 weeks

Deposit #1

Deposit #2

Deposit #3

CF = $75

CF = $75

CF = $75

Deposit #41 CF = $75 FV = ?

Count the number of deposits. The first deposit occurs at time 0, the second deposit at time 1, etc., and finally the forty-first deposit occurs at time 40. Because N is the number of cash flows, N equals 41. This highlights that N is properly thought of as the number of cash flows, not as a time subscript.

Other variable settings include r at 0.0625 ? 52 (the APR, i, is 0.0625 and m is 52). CF definitely is $75, but is it positive or negative? Either approach is correct. Because the deposit flows out of your pocket, set CF to $-75. The beginning wealth, PV, equals zero because there is no account balance preceding the first cash flow. Notice that if a beginning balance were relevant for this problem, the timing requires the beginning wealth exactly one period before the first deposit.

Substitution of all settings into Equation 5.2 shows:

{( ) } FV = $0 ? ($ ?75)

1

+

0.0625 52

41

?

1

.

0.0625

52

= $3,150

The account accumulates $3,150. The total contributed principal equals $3,075 (that is, $3,075 = 41 ? $75). The remainder of the accumulation, $75, is total market interest.

230Chapter 5

CALCULATOR CLUE 5.2

For the algebraic solution to the preceding problem, compute and store in memory the value of the periodic rate. See the discussion in the previous chapter that storing variables in the calculator's memory reduces rounding error. Type

.0625 ? 52 = STO 1

Now compute the present value by typing

RCL 1 + 1 = yx 41 ? 1 = ? RCL 1 X 75 = .

The display shows $3,150.

The remainder of this chapter uses time value functions for solving problems. Solve the preceding problem by typing 2nd FV to clear the time value memories, and 2nd I/Y 52 ENTER 2nd CPT to enforce weekly compounding. Then type:

75 +/? PMT 41 N 6.25 I/Y CPT FV .

The display shows $3,150.

The signs are consistent with the earlier discussion. CF is negative because deposits represent monies flowing out of your pocket. FV takes on an opposite and positive sign, implying that at the end of the investment horizon monies are available to flow into your pocket. Notice, however, that all signs could have been reversed. If CF were positive then FV would be negative, but exactly the same outcome obtains.

Beware! Your calculator gives you wrong answers as well as right ones. It has no conscience! Therefore there is a definite advantage for scenarios when easy approximation of an answer is possible. When the approximation is relatively close to the precise number from your time value calculation then likely the precise answer is correct. Conversely, when the approximation and precise answer are miles apart then this signals a need to double-check. The Rule of 72 from the previous chapter provides approximations within the lump-sum time value framework. The rule modifies for approximating the future value of a constant annuity stream. The modified rule requires some multiplication and is prone to larger approximation errors yet, still, the modified rule may sometimes be useful.

RULE 5.1 The Modified Rule of 72 for Constant Annuities

The Rule of 72 for lump-sums states that the approximate number of years in which a sum of money doubles, D, equals 72 divided by

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