18.5 Applications of Geometric Sums and Series 581
EXAMPLE 18.12
SOLUTION
The sum of money in the account is
$2001.053 $2001.052 $2001.05$200 $862.03.
Marietta cannot yet take her trip.
On January 1, 2004, before Marietta makes a fth payment,
the rst payment has grown to $2001.054, the second payment has grown to $2001.053, the third payment has grown to $2001.052, the fourth payment has grown to $2001.05.
The sum of money in the account is
$2001.054 $2001.053 $2001.052 $2001.05.
This is just the previous sum multiplied by 1.05 (because the money has collected interest
for another year). $862.03 1.05 $905.12. Marietta can be on her way, with $5.12 in her pocket as a slush fund.
Mass Millions is a state-run lottery encouraging residents to support the states public services by dangling the elusive prize of $1 million for the price of a winning lottery ticket. Instead of a million dollars, however, the winner actually receives 20 annual payments of $50,000. While this is a hefty sum, is this really a prize of $1 million? If you received $1 million today and put it into a bank account paying interest at a rate of 5% per year, just by taking the interest at the end of each year, you could pay yourself $50,000 per year starting one year hence and continuing on forever. The original million would stay in the bank generating interest. The 20 payments of $50,000 spread out over 20 years is not really the same as winning $1 million paid up front now.
Assume the rst payment is made to you today and the 20th payment 19 years later. Lets compute the up-front value of the prize money of 20 annual payments of $50,000. Therstpaymentof$50,000receivedtodayiscertainlyworth$50,000today.Buthowmuch is the second payment of $50,000 worth to you right now? How much is the 20th payment worth right now? Lets rephrase the question. Lets suppose that at the moment you win the prize, the state creates a bank account especially for you. The state puts a certain amount of money, $, into the account today and lets it earn interest. From this account the state dolesoutyour20paymentsof$50,000; thenalpaymentdepletesyouraccount.Howmuch would the state have to deposit today to make all 20 payments, each at the allotted time? This sum depends on the interest rate in the bank account. Lets suppose that the account pays interest at 5% per year compounded annually.15 Find , the total amount of money in the account earmarked for you.
Lets begin by breaking down the problem into manageable pieces.
The state must put away $50,000 for the very rst payment. How much money must the state put away now in order to pay you $50,000 one year from now?
We know that money in this bank account grows according to 01.05. In one year we want $50,000; we must solve for 0.
15 Assuming a xed 5% interest rate over a 20-year period is a rather unrealistic assumption, but we make it to simplify our model. Assume that all payments after the original are made immediately after interest is credited.
582 CHAPTER 18 Geometric Sums, Geometric Series
50,000 01.051 0 50,000 47, 619.05
The state must put $47,619.05 in the account in order to pay you $50,000 in one year. Thequantity$47,619.05iscalledthepresentvalueof$50,000inoneyearataninterest
rate of 5% per year compounded annually.
D e f i n i t i o n
Thepresentvalueof dollarsin yearsatinterestrate peryearcompounded times per year is the amount of money that must be put in an account (paying interest per year compounded times per year) now in order to have dollars at the end of years.
If an account has an interest rate of per year compounded times a year, then
0 1
future present interest per
value value compounding period
where is the amount of money years in the future and 0 is the initial deposit, the present value. In this case the present value of dollars in years is .
1
If an account has interest rate and interest is compounded continuously, then
0 and the present value of dollars in years is 0 .
To continue with our lottery problem, we ask how much money the state must put away in order to pay you $50,000 in two years. (With our new vocabulary, we can rephrase this: What is the present value of $50,000 in two years at an interest rate of 5% per year compounded annually?)
50,000 01.052 so 0 50,000 45,351.47
We continue in this vein.
The present value of the 1st payment is $50,000.
The present value of the 2nd payment is $50,000 $47,619.05. The present value of the 3rd payment is $50,000 $45,351.47.
.
The present value of the 20th payment is $50,000 $19,786.70.
Notice that the amount the state must put away now in order to pay you $50,000 in 19 years is only $50,000 $19,786.70.
The up-front value of the prize can be thought of as the sum of the present values of all of the payments. We can represent this diagramatically. Each of the 20 payments of $50,000 is being pulled to the present.
18.5 Applications of Geometric Sums and Series 583
Now
$50,000 $50,000 $50,000 $50,000
$50,000 $ 1.05 0 1.050 1.05 0
Thisisageometricsumwith 1.05.Wecancomputethesumbyputtingitinclosedform.
$50,000 $50,000 1.050 $50,000
1 $50,000 $50,000 $50,000 $50,000 1.05 1.05 1.052 1.0519 1.0520
1 1.05 $50,000
$50,000 $50,000 $654, 266.04. 1.05
$50,000 1.0520
Work through the following two examples to make sure that the notion of present value makes sense. These examples are designed to emphasize the set-up of the problems, not the summation of a geometric series.
EXAMPLE 18.13 Onnie has just won an award of $1000 per year for four years, with the rst of the four payments being made to him today. Suppose that the money to nance this award is being
kept in a bank account with 5% interest compounded annually. How much must be in the bank right now in order to pay for his award? In other words, what is the present value of his award?
SOLUTION Making Estimates. Although Onnie will receive a total of $4000 dollars, he is not getting it all right now. The bank account earmarked for this award needs less than $4000 in it because the money in the account will earn interest. Our goal is to gure out exactly what sum must be put in the account right now so that after Onnies award has been paid the money in the account is depleted. We expect an answer slightly less than $4000.
Strategy: Treat each of the four payments separately. Calculate how much must be in the account to make the rst payment, the second, the third, and the fourth. Then sum these four gures. In other words, sum the present values of the four payments. We can represent this diagramatically. Well pull each of the payments back to the present.
$1,000 $1,000 $1,000 $1,000
We know that the present value of $1000 in years at an interest rate of 5% per year is given by
present value $1000.
The present value of the rst payment $1000.
584 CHAPTER 18 Geometric Sums, Geometric Series
The present value of the second payment $1000. The present value of the third payment $1000. The present value of the fourth payment $1000.
The present value of the award the sum of the present values of the rst, second, third and fourth payments.
present value of award $1000 1.05 1.050 1.050
There are only four terms here, so well just add them up. This is a geometric sum with
1000, 1.05.
present value of award $1000 $952.38 $907.03 $863.84 $3723.25
The present value of the award is $3723.25.
EXAMPLE 18.14 Julie has just won an award of $1000 per year for four years, with the rst of the four payments being made to her three years from today. Suppose that the money to nance this
award is being kept in a bank account with 5% interest compounded annually. How much must be in the bank right now in order to pay for her award? In other words, what is the present value of her award?
SOLUTION Making Estimates. Although Julie will receive a total of $4000, she is not getting any of it right now. The bank account earmarked for this award needs less than $4000 in it because the money in the account will earn interest. Our goal is to gure out exactly what sum must be put in the account right now so that after Julies award has been paid the money in the account is depleted. We expect an answer substantially less than $3723.25.
Strategy: Treat each of the four payments separately. How much must be in the account to make the rst payment? The second? The third? The fourth? Then sum these four gures.
We can represent this diagramatically. Well pull each of the payments back to the present.
Now 3 years $1,000 $1,000 $1,000 $1,000
We know that the present value of $1000 in years at an interest rate of 5% per year is given by
present value $1000.
The present value of the rst payment $1000. The present value of the second payment $1000. The present value of the third payment $1000. The present value of the fourth payment $1000.
18.5 Applications of Geometric Sums and Series 585
The present value of the award the present values of the rst, second, third, and fourth payments.
present value of award 1.050 $1000 $1000 1.050
present value of award $863.84 $822.70 $783.53 $746.22 $3216.29
The present value of the award is $3216.29.
EXAMPLE 18.15
SOLUTION
Supposeaphilanthropicorganizationwantstostartafundthatwillmakepaymentsof$2000 each year to the American Cancer Society. The payments are to begin in ve years and go onindenitely.(Theyaresettingupwhatisknownasaperpetualannuity.)Thefundwillbe kept in an account with a guaranteed 6% annual interest compounded continuously. How much money should be put in the fund today so the payments can begin ve years from today?
As is the case with many problems, there are several different constructive approaches to obtaining a solution. Well look at two of these.
Approach 1. Our strategy is to gure out how much the organization must put away now in order to make the rst payment of $2000, the second payment, and so on. Well then add up these gures to nd the total amount that must be invested today. In other words, we will lookatthesumofthepresentvaluesofthepayments.Wecanrepresentthisdiagramatically.
Now $2,000 $2,000 $2,000 $2,000
5 years
Interest is 6% compounded continuously, so 0.06. Solving for 0 and denoting the future value by gives 0 .06 .
The present value of the 1st payment $.065 $1481.64. The present value of the 2nd payment $.066 $1395.35. The present value of the 3rd payment $.067 $1314.09.
The amount of money the foundation must put in the account is
$2000 $2000 $2000 .065 .066 .067
This is an innite geometric series with $.065 and .06 . 1, so the series converges to 1 .
$2000 .065
1 1 .06 1 .06 $25,442.17
The organization must use $25,442.17 to set up this perpetuity.
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