WEB APPENDIX 5B

GROWING ANNUITIES

Normally, an annuity is defined as a series of constant payments to be receivedover a specified number of periods. However, the term growing annuity is used todescribe a series of payments that are growing at a constant rate. For example, apayment of $100 that is growing at a rate of 5% per year for 20 years is a growingannuity. As we shall see, growing annuities are quite important in the real world.

Example 1: Finding a Constant Real Income, FirstWithdrawal Made Immediately

Growing annuities are often used in the area of financial planning. Suppose aprospective retiree wants to determine the constant real, or inflation-adjusted,withdrawals he or she can make from a given amount of money over a specifiednumber of years. For example, suppose your uncle, who is 65 years old, is con-templating retirement. He expects to live for another 20 years, has a $1 million nestegg, expects to earn 8% on his or her investments, expects inflation to average 3%per year, and wants to withdraw a constant real amount annually over the next20 years. If the first withdrawal is to be made today, what is the amount of theinitial withdrawal?

This problem can be solved two ways:1 (1) Use a financial calculator, wherewe first calculate the real rate of return, which is the nominal rate adjusted forinflation, and then use it for I to find the initial withdrawal. (2) Set up a spread-sheet model that is similar to an amortization table, where the account earns 8%per year and withdrawals rise at the 3% inflation rate. We then use Excel’s GoalSeek function to find the initial inflation-adjusted withdrawal that produces a zerobalance at the end of the 20th year. We illustrate these procedures in the GrowingAnnuity tab of the chapter spreadsheet model. The calculator approach is easier touse, but the spreadsheet model shows the value of the retirement portfolio,earnings, and each withdrawal over the 20-year planning horizon. Also, thespreadsheet model can be used to create a graph that makes it easy to see what ishappening.

To implement the calculator approach, we first use this formula to find theexpected real, or inflation-adjusted, rate of return, where rr is the real rate andrNOM is the nominal rate of return:2

Real rate ¼ rr ¼ ½ð1þ rNOMÞ=ð1þ InflationÞ� � 1:0

¼ ½1:08=1:03� � 1:0 ¼ 0:048543689 ¼ 4:8543689%

Next, we set the calculator to Begin Mode, after which we input N ¼ 20, I/YR ¼ real rate ¼ 4.8543689, PV ¼ –1,000,000, and FV ¼ 0. We then pressPMT to get $75,585.53, which is the initial withdrawal, at Time 0, today, andthat will increase at the inflation rate, 3%, for 20 years. Thus, the retiree will have aconstant real income over the next 20 years.

Growing AnnuityA series ofpayments thatgrow at a constantrate.

1A relatively complex formula can also be used to find the payment, but the two methods discussed here areeasier to use.2The real rate is discussed in detail in Chapter 6.

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5B-1

Example 2: Constant Real Income, End-of-YearWithdrawals

In the preceding example, we assumed that the first withdrawal would be madetoday. If withdrawals are to be made at the end of each year, the procedure will beslightly different. We would find the real rate the same way and enter the sameinputs into the calculator, but with the calculator set to End Mode. The calculatedPMT would be $79,254.73. However, that value is in beginning-of-year dollars.Since 3% inflation is expected during the year, we must make the followingadjustment to find the inflation-adjusted initial payment:

Initial withdrawal ¼ $79,254:73ð1þ InflationÞ ¼ $79,254:73ð1:03Þ ¼ $81,632:38

Thus, the first withdrawal at the end of the year would be $81,632.38; it wouldgrow by 3% per year; and after the 20th withdrawal, the balance in the retirementfund would be zero.

The end-of-year payment is also analyzed in the chapter spreadsheet model.There we set up an “amortization table” that shows the beginning balance, theannual earnings, the annual withdrawals, and the ending balance for each of the20 years. This analysis confirms that the $81,632.38 initial withdrawal foundpreviously is correct.3

Example 3: Initial Deposit to Accumulate a Future SumAs another example of a growing annuity, suppose you need to accumulate$100,000 in 10 years. You plan to make a deposit now, at Time 0, and then to make9 more deposits at the beginning of the following 9 years, for a total of 10 deposits.The bank pays 6% interest, and you expect to increase your initial deposit amountby the 2% inflation rate each year. How much would you need to deposit initially?First, we calculate the real rate:

Real rate ¼ rr ¼ ½1:06=1:02� � 1:0 ¼ 0:0392157 ¼ 3:92157%

Next, since inflation is expected to be 2% per year, in 10 years, the target $100,000will have a purchasing power as follows:

$100,000=ð1þ 0:02Þ10 ¼ $82,034:83

Now we can find the size of the required initial payment by setting a financialcalculator to the “BEG” mode and then inputting N ¼ 10, I/YR ¼ 3.92157, PV ¼ 0,and FV ¼ 82,034.83. Then when we press the PMT key, we get PMT ¼ –6,598.87.Thus, a deposit of $6,598.87 made at time zero and growing by 2% per year willaccumulate to $100,000 by Year 10 if the interest rate is 6%. Again, this result isconfirmed in the chapter model. The key to this analysis is to express I, PV, andPMT in real, inflation-adjusted terms.

3It is tempting to just input the values in the calculator set to END MODE to find the initial payment. However, thatprocedure is incorrect because you would calculate a payment amount that didn’t include the 3% inflation forthat year.

5B-2 Web Appendix 5B

QUESTIONS

5B-1 Differentiate between a regular annuity and a growing annuity.

5B-2 What two methods are generally used to deal with growing annuities?

PROBLEMS

5B-1 REAL RATE OF RETURN If the nominal interest rate is 10% and the expected inflation rate is5%, what is the expected real rate of return?

5B-2 REAL RATE OF RETURN You plan to make annual deposits into a bank account thatpays a 5.0% nominal annual rate. You think inflation will amount to 2.5% per year. Whatis the expected annual real rate at which your money will grow?

5B-3 BEGINNING-OF-YEAR REAL WITHDRAWAL Your father now has $1,000,000 invested inan account that pays 9%. He expects inflation to average 3%, and he wants to make annualconstant dollar (real) beginning-of-year withdrawals over each of the next 20 years and end upwith a zero balance after the 20th year. How large will his initial withdrawal be (andthus constant dollar (real) withdrawals)?

5B-4 END-OF-YEAR REAL WITHDRAWAL Your father now has $1,000,000 invested in an accountthat pays 9%. He expects inflation to average 3%, and he wants to make annual constantdollar (real) end-of-year withdrawals over each of the next 20 years and end up with azero balance after the 20th year. How large will his initial withdrawal be (and thus constantdollar (real) withdrawals)?

5B-5 BEGINNING-OF-YEAR REAL WITHDRAWAL You anticipate that you will need $1,500,000when you retire 30 years from now. You plan to make 30 deposits, beginning today, ina bank account that will pay 6% interest, compounded annually. You expect to receiveannual raises of 4%, so you will increase the amount you deposit each year by 4%. (That is,your second deposit will be 4% greater than your first, the third will be 4% greater thanthe second, and so forth.) How much must your first deposit be if you are to meet your goal?

5B-6 BEGINNING-OF-YEAR REAL WITHDRAWAL You want to accumulate $2,500,000 in your401(k) plan by your retirement date, which is 35 years from now. You will make 35 depositsinto your plan, with the first deposit occurring today. The plan’s rate of return typicallyaverages 9%. You expect to increase each deposit by 2% as your income grows withinflation. (That is, your second deposit will be 2% greater than your first, the third will be2% greater than the second, and so forth) How much must your first deposit at t ¼ 0 beto enable you to meet your goal?

Web Appendix 5B 5B-3