I NTERDISCIPLINARY L IVELY A PPLICATIONS P … · Thomas L. Fitzkee (Mathematics)...

I N T E R D I S C I P L I N A R Y L I V E L Y A P P L I C A T I O N S P R O J E C T

AUTHOR:Thomas L. Fitzkee(Mathematics)[email protected] Marion UniversityFlorence, SC

EDITORS:Richard West andNeil RileyFrancis Marion UniversityFlorence, SC

FINANCIAL ADVISOR:George E. Love, Jr.Lincoln Financial AdvisorsCamden, SC

The UMAP Journal 24 (1) (2003) 53–78. © Copyright 2003 by COMAP, Inc. All rightsreserved. Permission to make digital or hard copies of part or all of this work for per-sonal or classroom use is granted without fee provided that copies are not made ordistributed for profit or commercial advantage and that copies bear this notice.Abstracting with credit is permitted, but copyrights for components of this workowned by others than COMAP must be honored. To copy otherwise, to republish, topost on servers, or to redistribute to lists requires prior permission from COMAP.

Tuition Prepayment Plan 53

Tuition PrepaymentPlanMATHEMATICS CLASSIFICATIONS:

Discrete mathematics, difference equations, discrete dynamical systems, sequences.

DISCIPLINARY CLASSIFICATIONS: Economics, Finance, Business

PREREQUISITE SKILLS:1. Modeling with difference equations or sequences.2. Performing iteration of difference equations using a

computational tool (spreadsheets on computers or calculators in sequence mode).

PHYSICAL CONCEPTS EXAMINED: Classical mechanics, kinetics, and kinematics .

COMPUTING REQUIREMENT: Either a spreadsheet program on a computer or else a calculator in sequence mode.

54 The UMAP Journal 24.1 (2003)

Contents1. Setting the Scene2. Lump Sum Payment3. 48 Monthly Payments4. Monthly Payments until College5. Comparison of OptionsSample SolutionReferences, About the Author

1. Setting the SceneCongratulations, you earned your bachelor’s degree last summer from Fran-

cis Marion University in Florence, SC. But you wonder about your daughtergoing to college. Right now, in late November 2002, she is in the 5th grade, butyou are already worried about how to finance her college tuition.

She expects to be in college from September 2010 till June 2014. The tu-ition for each year must be paid in advance, on July 1. (We set this date forconvenience in calculation and so that the funds will be on hand.)

You have heard about the South Carolina Tuition Prepayment Program(SCTPP) (http://www.scgrad.org), but you wonder if it is a good deal. Itallows you to prepay your daughter’s college tuition and guarantee paymentto attend any South Carolina public four-year institution for four years. Thecost of college tuition for many years has risen at a rate higher than inflation, andthe future cost is not easily predictable; so a prepayment plan may be desirable.(However, other investment alternatives to prepayment may be more attractivebecause of their interest rates or tax considerations, and changes in interest ratesand in the family’s economic situation might provoke an alteration in strategy.)1

The options available under the plan (see Table 1) are:

• you can pay a one-time lump sum right now,

• you can pay 48 fixed monthly payments starting now, or

• you can pay fixed monthly payments starting now until your daughter at-tends college.

Any one of these options will cover her tuition, no matter how high thetuition might rise before your daughter attends college. In each case, youbegin the investment on Jan. 1, 2003.

The SCTPP covers tuition only, so a student could still apply for financial aidfor housing. For ending the plan due to stopping payments, death, and other

1This project is created solely as an example to illustrate how mathematics—specifically, differ-ence equations—can be used to analyze a real-world problem. It is not to be construed or used asfinancial advice. The author cannot be held responsible for any financial decisions made due tothis analysis.


Table 1.

Current rates and conditions for available investments.

Institution Rate Compounding Otherperiod

SC State Credit Union—Money Market Account < $1,000: 0.60% monthly

≥ $1,000: 1.85%

—Certificates of Deposit 3.85% quarterly Term of 60 mos. or more

U.S. Savings Bonds 2.57% semiannually Denominations: $50, $75,$100, $500, $1,000, $5,000,$10,000

Maximum $30,000 per yearForfeit 3 mos. interest if re-

deemed before 5 yrs. old

reasons, the student/parent would receive the amount invested plus someinterest. For students attending a private school, SCTPP will pay the amountof tuition equal to the average of the SC public institutions’ tuition.

Table 2 shows tuition fees for the last 10 years of the South Carolina publiccolleges and universities.

2. Lump Sum PaymentAccording to the SCTPP, the lump sum payment required for your 5th-grade

daughter is $16,989. You wish to determine if this is a bargain or if you can beatit with another investment.

Table 2.

10-year summary of tuition fees for full-time in-state undergraduates at South Carolina publiccolleges and universities, for academic year beginning in year shown.http://www.che400.state.sc.us/web/Chemis/abstract%202001.pdf

Institution 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

The Citadel 2,949 3,080 3,176 3,275 3,297 3,498 3,631 3,396 3,404 3,727Clemson Univ. 2,762 2,954 3,036 3,112 3,112 3,252 3,344 3,470 3,590 5,090Coastal Carolina U. 2,170 2,470 2,710 2,800 2,910 3,100 3,220 3,340 3,500 3,770Coll. of Charleston 2,650 2,950 3,060 3,090 3,190 3,290 3,390 3,520 3,630 3,780Francis Marion U. 2,440 2,800 2,920 3,010 3,010 3,270 3,350 3,350 3,360 3,790Lander Univ. 2,920 3,220 3,340 3,400 3,550 3,600 3,700 3,700 3,888 4,152SC State Univ. 2,200 2,500 2,500 2,550 2,730 2,974 3,184 3,410 3,724 4,096USC at Columbia 2,818 3,090 3,196 3,280 3,363 3,534 3,530 3,740 3,868 4,064USC at Aiken 2,120 2,320 2,500 2,578 2,708 2,974 3,018 3,318 3,558 3,738USC at Spartanburg 2,120 2,320 2,500 2,578 2,708 2,974 3,018 3,360 3,624 3,868Winthrop Univ. 3,112 3,470 3,620 3,716 3,818 3,918 4,032 4,126 4,262 4,668Medical U. of SC 2,212 2,560 2,819 2,910 3,202 3,648 4,034 4,626 5,180 5,824


2.1 Money Market AccountRequirement 1

Suppose that you deposit the $16,989 lump sum payment into a moneymarket account at the South Carolina State Credit Union (http://www.scscu.com/savings.html) on Jan. 1, 2003. Currently, a money market account pays1.85% interest compounded monthly. Make and use a mathematical model(for instance, a discrete dynamical system) to determine how much money youwould have in the account when your daughter goes to college in 7.5 years.

Requirement 2When you read the fine print, you realize that you get the interest rate of

1.85% compounded monthly only if you have at least $1,000 in the account.Otherwise, you only get 0.60%. How will this change your account amount?

Requirement 3Is this enough money for her college tuition? To answer this question, you

need to estimate the cost of four years of your daughter’s tuition. Consider thetuition costs per year of SC public four-year institutions as listed in Table 2.

a) For each year, which tuition should you consider? Remember, the SCTPPallows your daughter to attend any one of these institutions. What is thepercentage increase per year of these tuitions?

b) Your child is in 5th grade, so she will enter college in 2010. Using thepercent increase from a), what tuition amount should you be ready to pay forher first year? What about her second year? What about her third year? Whatabout her fourth year? What would be the total cost of tuition for her fouryears?

Requirement 4Judging from your calculations, would you have enough money in your

money market account to cover this total cost of tuition?

Requirement 5You can stretch your money by realizing that you do not have to pay the

tuition for the upcoming year until the previous year has ended. For instance,you do not have to pay the 2nd-year tuition until after the 1st year finishes.This schedule allows you to leave your money in your account longer, whichgains more interest. Revise your mathematical model from Requirement 1 (or2) so that the yearly tuition amounts are deducted at the appropriate times.How does this investment regimen compare with the previous model?


2.2 Savings BondsAnother investment for college is Series I Savings Bonds offered by the U.S.

Treasury Department (http://www.savingsbonds.com), currently offering aninterest rate of 2.57% compounded semiannually. (Moreover, when used fortuition, interest on the bonds can be excluded from taxes.) The Series I bondscan be purchased in denominations of $50, $75, $100, $500, $1,000, $5,000, and$10,000 and can earn interest for up to 30 years.

Requirement 6Suppose that you buy savings bonds with the $16,989 lump sum. How

many and what denominations of savings bonds would you buy?

Requirement 7How much money would you have in savings bonds when your daughter

enters college in 7.5 years? Make and use a mathematical model to determinehow much money you would have in your savings bonds when your daughtergoes to college in 7.5 years.

Requirement 8.a) Once again, you read the fine print and realize that you can buy only

$30,000 worth of savings bonds each year. How does this change your situation?b) Another requirement is that you can only redeem the bonds once they

are 6 months old. How does this change your total amount?c) Finally, you must forfeit the last three months of interest on bonds that

are redeemed before they are 5 years old. How does this change your totalamount?

Requirement 9Judging from your calculations in Requirements 2 and 7, would you have

enough money in savings bonds to cover the total cost of tuition? How doesthis investment compare with the total amounts of other investments?

Requirement 10As in Requirement 5, you can stretch your money by realizing that you do

not have to pay the tuition for the upcoming year until the previous year hasended. Revise your mathematical model from Requirement 7 (or 8) so that theyearly tuition amounts are deducted at the appropriate times. How does thisinvestment regimen compare with the previous models?


2.3 Certificates of DepositA third investment is certificates of deposits (CDs) offered by the South

Carolina State Credit Union. The 60+ month CD has an interest rate of 3.85%compounded quarterly.

Requirement 11Suppose you purchase a CD using the $16,989 lump sum. Make and use a

mathematical model to determine how much money you would have in yourCD when your daughter goes to college in 7.5 years.

Requirement 12Judging from your calculations in Requirement 2 and Requirement 11,

would you have enough money in your CD to cover the total cost of tuition?How does this investment compare with the total amounts of other invest-ments?

Requirement 13Similar to Requirement 5, you can stretch your money by realizing that you

do not have to pay the tuition for the upcoming year until the previous yearhas ended. Revise your mathematical model from Requirement 11 so that theyearly tuition amounts are deducted at the appropriate times. How does thisinvestment regimen compare with the other model?

3. 48 Monthly PaymentsAnother option of the SCTPP for your 5th-grade daughter is starting now

(November 2002) make 48 monthly payments of $413 each. You wish to deter-mine if you can beat it with another investment.


Suppose that you deposit the 48 monthly payments of $413 into your moneymarket account at the SC State Credit Union, currently paying 1.85% interestcompounded monthly. Make and use a mathematical model (for instance adiscrete dynamical system) to determine how much money you would have inthe account in 7.5 years.


Requirements 2–5Complete Requirements 2–5 of the Lump Sum Payment situation using

this second option of 48 monthly payments of $413 each.

3.2 Savings BondsRequirement 6

Suppose you buy savings bonds every six months with the six accumulated$413 payments. How many and what denominations of savings bonds wouldyou buy? (See Table 1.)


this second option of 48 monthly payments of $413 each.

4. Monthly Payments until CollegeYet another option of the SCTPP for your 5th-grade daughter is to make

monthly payments until your daughter enters college. The SCTPP determinesthat this would require 89 monthly payments of $253. You wish to determineif you can beat it with another investment.


Suppose that you deposit the 89 monthly payments $253 into your moneymarket account at the SC State Credit Union, currently paying 1.85% interestcompounded monthly. Make and use a mathematical model to determine howmuch money you would have in the account in 7.5 years.


the option of 89 monthly payments of $253 each.

4.2 Savings BondsRequirement 6

Suppose that you buy savings bonds every six months with the six accu-mulated $253 payments. How many bond purchases will you make? How


many and what denominations of bonds would you buy for each savings bondpurchase? (See Table 1.).

Requirements 7–10Complete Requirements 7-10 of the Lump Sum Payment situation using

the option of 89 monthly payments of $253 each.

5. Comparison of OptionsSuppose that you decide to enroll in the SCTPP. With its three options, you

are unsure which option is the better choice:

• a lump sum payment of $16,989,

• 48 monthly payments of $413, or

• monthly payments of $253 until your daughter enrolls in college (89 pay-ments).

Requirement 1Compare the lump sum option to the 48 monthly payment option using

your money market account at the SC State Credit Union. Make and use amathematical model to determine how many monthly withdrawals of $413you can make from an account that starts with an initial balance of $16,989.Which option is better?

Requirement 2Compare the lump sum option to the monthly-payment-until-college option

using your money market account at the SC State Credit Union. Make and usea mathematical model to determine how many monthly withdrawals of $253you can make from an account that starts with an initial balance of $16,989.Which option is better?

Requirement 3Compare the 48 monthly payments option to the monthly-payment-until-

college option using your money market account at the SC State Credit Union.Make and use a mathematical model to determine the account balances after7.5 years resulting from 48 monthly payments of $413 and resulting from 89monthly payments of $253. Which option is better?


Title: Tuition Prepayment Plan: Sample Solutions

Sample Solutions

Lump Sum PaymentRequirement 1

You are depositing the lump sum payment of $16,989 into an account thatpays 1.85% interest compounded monthly for 7.5 years (90 months), from Jan-uary 1, 2003 through June 30, 2010. Since you are concerned with the amountin the account each month, let a(n) = amount in the account after n months,where n = 0, 1, 2, . . . , 90. You must also define an initial starting point (ini-tial condition), which is a(0) = 16,989. Since the interest rate is compoundedmonthly, the annual interest rate of 0.0185 is equivalent to an interest rate of0.0185/12 per month.

Discrete dynamical systems define the future as the present plus any changeduring the current period. So, in our example:

a(n + 1) = a(n) + interest gained on a(n)a(n + 1) = a(n) + a(n) × 0.0185/12.

With this discrete dynamical system, you can use a spreadsheet to calculatea(n) for n = 0, 1, . . . , 90, as in Table S1. You will have $19,515.50 in the accounton July 1, 2010, just months before your daughter enters college.

Table S1.

Spreadsheet for solution to Requirement 1.

month previous + interest = newamount change amount

0 16,989.001 16,989.00 + 26.19 = 17,015.192 17,015.19 + 26.23 = 17,041.423 17,041.42 + 26.27 = 17,067.70...

88 19,425.52 + 29.95 = 19,455.4789 19,455.47 + 29.99 = 19,485.4690 19,485.46 + 30.04 = 19,515.50

Requirement 2The $1,000 minimum balance condition will not affect your account, since

your lump sum of $16,989 is greater than $1,000.


Requirement 3To determine if your account balance of a(90) = $19,515.50 is enough to

pay for your daughter’s tuition, you need to analyze the tuitions for the past10 years listed in Table 1 and predict future tuition costs for 2010, 2011, 2012,and 2013.

• Since your daughter might attend any of the public institutions, you couldchoose the highest tuition for each year and calculate its average percentageincrease, based on the annual percentage increases.

• You can obtain another conservative estimate by using just the tuition atMUSC (Medical University of South Carolina), since it will most likely bethe highest tuition in the future.

• You could use the Annualized Holding Period Yield (AHPY), which is usedin finance to calculate annualized returns and growth rates. The formula is

AHPY =(

Ending amountBeginning amount

) 1# of periods − 1.

In our case,

AHPY =(

1992–93 tuition2001–02 tuition

) 19 − 1.

To no surprise, the institution with the highest AHPY is MUSC, at 11.36%.

We use the first method to estimate the percentage increase from the datashown in Table S2. The average rate of increase per year is 7.32%.

Table S2.

Percentage increases in Requirement 3.

Year 92 93 94 95 96 97 98 99 00 01

Max tuition 3,112 3,470 3,620 3,716 3,818 3,918 4,034 4,626 5,824 5,824% Increase 11.50 4.32 2.65 2.74 2.62 2.96 14.68 11.98 12.43

You can use a discrete dynamical system to calculate future tuitions basedon an annual increase of 7.32%. You start from $5,824 (the highest tuition in2001–02) and add an annual increase of 7.32% per year for 12 years. Since youare concerned with the tuition amount for each year, let

t(n) = tuition amount n years after 2001,

where n = 0, 1, 2, . . . , 12. The initial condition is t(0) = 5,824. Again, thediscrete dynamical system assumes that future equals the present plus andchange:

t(n + 1) = t(n) + percent increase of t(n)


t(n + 1) = t(n) + t(n) × 0.0732.

You can use a spreadsheet to calculate t(n) forn = 0, 1, . . . , 12, as in Table S3.The estimated total tuition for all four years is $49,069.81.

Table S3.


year previous + tuition = estimatedtuition change new tuition

2001 5,824.002002 5,824.00 + 426.36 = 6,250.362003 6,250.36 + 457.57 = 6,707.93

...2010 10,249.23 + 750.32 = 10,999.552011 10,999.55 + 805.25 = 11,804.802012 11,804.80 + 864.20 = 12,669.002013 12,669.00 + 927.47 = 13,596.46

Requirement 4You would not have enough money in the money market account to pay for

her tuition, since the account balance would be $19,515.50 and the estimatedtotal tuition would be $49,069.81.

Requirement 5Because discrete dynamical systems are very flexible, you do not have to

redo the whole system—just revise it. First, continue the money market accountsystem in Requirement 1 until the July 1, 2013 (n = 126), when the last tuitionamount is paid. Then subtract each year’s tuition from the amount accumulatedin the account on July 1 of that year. The spreadsheet in Table S4 starts whenthe first year’s tuition is withdrawn.

Again, the amount in the account in June (month 91 for 2010, month 103 for2011, and month 115 for 2012) is the tuition subtracted from the May amount.Stretching your money still will not cover the estimated cost of tuition since thetuition for the 2nd year ($11,804.80) is more than the account balance ($8,674.84)which gives a negative balance.

Requirement 6You are purchasing savings bonds using the lump sum payment of $16,989.

Due to the denominations, you can purchase only $16,975 in savings bonds.


Table S4

Spreadsheet for Requirement 5.

year month previous + interest = new tuitiondue amount change amount

1st 90 19,485.46 + 30.04 = 19,515.50 10,999.5591 8,515.96 + 13.13 = 8,529.09

...101 8,648.16 + 13.33 = 8,661.49

2nd 102 8,661.49 + 13.35 = 8,674.84 11,804.80103 −3,129.95 + −4.83 = −3,134.78

...113 −3,178.54 + −4.90 = −3,183.44

3rd 114 −3,183.44 + −4.91 = −3,188.35 12,669.00115 −15,857.35 + −24.4 = −15,881.80

...125 −16,103.52 + −24.83 = −16,128.35

4th 126 −16,128.35 + −24.86 = −16,153.21 13,596.46−29,749.67

Requirement 7The savings bonds pay 2.57% interest compounded semiannually for 7.5

years. Since you are concerned with the amount in the savings bonds each6-month period, let

b(n) = amount in the savings bonds after n 6-month periods,

where n = 0, 1, 2, . . . , 15 with an initial condition of b(0) = 16,975. Since theinterest rate is compounded semiannually, the annual interest rate of 0.0257 isequivalent to an interest rate of 0.0257/2 per 6-month period.

You have

b(n + 1) = b(n) + interest gained on b(n)b(n + 1) = b(n) + b(n) × 0.0257/2.

You can use a spreadsheet to calculate b(n) for n = 0, 1, . . . , 15, as in Table S5.You will have $20,558.28 in savings bonds on July 1, 2010.

Requirement 8None of the stipulations apply since:

• you purchased only $16,975 worth of savings bonds, which is less than the$30,000 limit per year;

• you let them mature for 7.5 years, which is more than the 6-month require-ment; and

• you let them mature for 7.5 years, which is more than the 5-year requirement.


Table S5.


period previous + interest = newamount change amount

0 16,975.001 16,975.00 + 218.13 = 17,193.132 17,193.13 + 220.93 = 17,414.06...

13 19,785.70 + 254.25 = 20,039.9414 20,039.94 + 257.51 = 20,297.4615 20,297.46 + 260.82 = 20,558.28

Requirement 9Since the savings bonds investment totals $20,558.28, you would have more

money than in a money market account ($19,515.50); but the estimated totaltuition ($49,069.81) would be even greater.

Requirement 10The spreadsheet in Table S6 starts when the first year’s tuition is withdrawn.

The amount in the account at the end of the day on July 1 of the years whenshe is attending college will be the amount accumulated by July 1 minus thatyear’s tuition.

Table S6.


year month previous + interest = new tuitiondue amount change amount

1st 15 20,297.46 + 260.82 = 20,558.28 10,999.5516 9,558.73 + 122.83 = 9,681.56

2nd 17 9,681.56 + 124.41 = 9,805.97 11,804.8018 −1,998.83 + −25.68 = −2,024.51

3rd 19 −2,024.51 + −26.01 = −2,050.53 12,669.0020 −14,719.53 + −189.15 = −14,908.67

4th 21 −14,908.67 + −191.58 = −15,100.25 13,596.46−28,696.71

Stretching your money still will not cover the estimated cost of tuition,since the tuition for the 2nd year ($11,804.80) is more than the account balance($9,805.97).


Requirement 11The CD pays 3.85% interest compounded quarterly for 7.5 years. Since you

are concerned with the amount in the account each quarter, let

c(n) = amount in the account after n quarters,

where n = 0, 1, 2, . . . , 30 with an initial condition of c(0) = 16,989. Since theinterest rate is compounded semiannually, the annual interest rate of 0.0385 isequivalent to an interest rate of 0.0385/4 per quarter.

You have

c(n + 1) = c(n) + interest gained on c(n)c(n + 1) = c(n) + c(n) × 0.0385/4.

You can use a spreadsheet to calculate c(n) for n = 0, 1, . . . , 30, as in Ta-ble S7. You will have $22,644.91 in your CD on July 1, 2010.

Table S7.


month previous + interest = newamount change amount

0 16,989.001 16,989.00 + 163.52 = 17,152.522 17,152.52 + 165.09 = 17,317.61...

28 22,003.43 + 211.78 = 22,215.2129 22,215.21 + 213.82 = 22,429.0430 22,429.04 + 215.88 = 22,644.91

Requirement 12The CD at 7.5 years of maturity is $22,644.91, which is better than the money

market account ($19,515.50) and the savings bonds ($20,558.28) but still notenough for the estimated total tuition ($49,069.81).

Requirement 13The spreadsheet in Table S8 starts when the first year’s tuition is withdrawn.Stretching your money still will not cover the estimated cost of tuition,

since the tuition for the 3rd year ($12,669.00) is more than the account balance($306.97).


Table S8.


month previous + interest = new tuitionamount change amount

1st year 30 22,429.04 + 215.88 = 22,644.91 10,999.5531 11,645.37 + 112.09 = 11,757.4532 11,757.45 + 113.17 = 11,870.6233 11,870.62 + 114.25 = 11,984.87

2nd year 34 11,984.87 + 115.35 = 12,100.23 11,804.8035 295.43 + 2.84 = 298.2736 298.27 + 2.87 = 301.1537 301.15 + 2.90 = 304.04

3rd year 38 304.04 + 2.93 = 306.97 12,669.0039 −12,362.03 + −118.98 = −12,481.0140 −12,481.01 + −120.13 = −12,601.1441 −12,601.14 + −121.29 = −12,722.43

4th year 42 −12,722.43 + −122.45 = −12,844.88 13,596.46−26,441.35

48 Monthly PaymentsRequirement 1

You deposit 48 monthly payments of $413 into an account that pays 1.85%interest compounded monthly for 7.5 years (90 months). Since you are con-cerned with the amount in the account each month, let

d(n) = amount in the account after n months,

where n = 0, 1, 2, . . . , 90 with an initial condition of d(0) = 413. Since the inter-est rate is compounded monthly, the annual interest rate of 0.0185 is equivalentto an interest rate of 0.0185/12 per month.

You have, until 47th month,

d(n + 1) = d(n) + interest gained on d(n) + next paymentd(n + 1) = d(n) + d(n) × 0.0185/12.

You can use a spreadsheet to calculate d(n) for n = 0, 1, . . . , 90, as in Table S9.You will have $21,967.47 in your account on July 1, 2010.

Requirement 2The $1,000 minimum balance condition needed to receive 1.85% interest

will affect your account, since your starting balance is only $413. Starting withthe 1st month, change each month’s interest rate from 1.85% to 0.60% until youraccount reaches (or exceeds) $1,000. This will occur after the third payment, atthe beginning of the 3rd month. So, starting with the 3rd month, the interestrate becomes 1.85%. This lower interest rate reduces your total amount to$21,966.00.


Table S9.


month previous + interest + monthly = newamount change payment amount

0 413.00 413.001 413.00 + 0.64 + 413.00 = 826.642 826.64 + 1.27 + 413.00 = 1,240.91...

46 19,672.14 + 30.33 + 413.00 = 20,115.4747 20,115.47 + 31.01 + 413.00 = 20,559.4848 20,559.48 + 31.70 + 0.00 = 20,591.1849 20,591.18 + 31.74 + 0.00 = 20,622.92

...88 21,866.19 + 33.71 + 0.00 = 21,899.9089 21,899.90 + 33.76 + 0.00 = 21,933.6690 21,933.66 + 33.81 + 0.00 = 21,967.47

Requirement 3See Requirement 2 of the Lump Sum Payment situation.

Requirement 4You would not have enough money in your money market account to pay for

her tuition, since your account balance would be $21,966.00 and the estimatedtotal tuition would be $49,069.81.

Requirement 5The spreadsheet in Table S10 starts when the first year’s tuition is with-

drawn. Stretching your money still will not cover the estimated cost of tuition,since the tuition for the 2nd year ($11,804.80) is more than the account balance($11,171.06).

Requirement 6You purchase savings bonds every six months with the six accumulated

$413 payments which total 6 × $413 = $2,478. Due to the denominations, youcan purchase only $2,475 in savings bonds.



e(n) = amount in the savings bonds after n 6-month periods,


Table S10.


year month previous + interest + monthly = new tuitiondue amount change payment amount

1st 90 21,932.18 + 33.81 + 0.00 = 21,966.00 10,999.5591 10,966.45 + 16.91 + 0.00 = 10,983.35

...101 11,136.69 + 17.17 + 0.00 = 11,153.86

2nd 102 11,153.86 + 17.20 + 0.00 = 11,171.06 11,804.80103 −633.74 + −0.98 + 0.00 = −634.72

...113 −643.58 + −0.99 + 0.00 = −644.57

3rd 114 −644.57 + −0.99 + 0.00 = −645.57 12,669.00115 −13,314.56 + −20.53 + 0.00 = −13,335.09

...125 −13,521.26 + −20.85 + 0.00 = −13,542.10

4th 126 −13,542.10 + −20.88 + 0.00 = −13,562.98 13,596.46−27,159.45

where n = 0, 1, 2, . . . , 15. Since you need to accumulate the six payments,you have e(0) = 0 and e(1) = 2, 475. Since the interest rate is compoundedsemiannually, the annual interest rate of 0.0257 is equivalent to an interest rateof 0.0257/2 per 6-month period.

You have, until the 8th 6-month period,

e(n + 1) = e(n) + interest gained one(n) + next bond purchasee(n + 1) = e(n) + e(n) × 0.0659/2 + 2475.

You can use a spreadsheet to calculate e(n) for n = 0, 1, . . . , 15, as in Table S11.You will have $22,650.35 in savings bonds on July 1, 2010.

Requirement 8• Since you purchased only $2,475 in savings bonds twice a year, you did not

reach the $30,000 annual limit.

• Since even your most recently purchased bonds matured for 3.5 years, the6-month redemption condition does not apply.

• Since you purchased bonds 4.5, 4.0, and 3.5 years ago, the 5-year redemptioncondition does apply and thus you will lose the last three months of intereston each of those bonds. So you need to calculate the interest gained each6-month period on $2,475 in bonds. (Three months of interest is consideredhalf of the interest during the 6-month period.) See Table S12.


Table S11.


period previous + interest + new bonds = newamount change bought amount

0 0.001 0.00 + 0.00 + 2,475.00 = 2,475.002 2,475.00 + 31.80 + 2,475.00 = 4,981.80...8 18,007.37 + 231.39 + 2,475.00 = 20,713.769 20,713.76 + 266.17 + 0.00 = 20,979.93...

14 22,079.27 + 283.72 + 0.00 = 22,362.9915 22,362.99 + 287.36 + 0.00 = 22,650.35

Table S12.


period previous + interest = new 3 monthsamount change amount of interest

1 2,475.00 + 31.80 = 2,506.80 15.902 2,506.80 + 32.21 = 2,539.02 16.113 2,539.02 + 32.63 = 2,571.64 16.314 2,571.64 + 33.05 = 2,604.69 16.525 2,604.69 + 33.47 = 2,638.16 16.746 2,638.16 + 33.90 = 2,672.06 16.957 2,672.06 + 34.34 = 2,706.39 17.178 2,706.39 + 34.78 = 2,741.17 17.399 2,741.17 + 35.22 = 2,776.40 17.61

• For the bonds purchased 4.5 years ago (matured for nine 6-month periods),you lose $17.17.

• For the bonds purchased 4.0 years ago (matured for eight 6-month periods),you lose $17.39.

• For the bonds purchased 3.5 years ago (matured for seven 6-month periods),you lose $17.61.

Therefore, you lose a total of $52.17 in interest due to this redemption condition,so the new adjusted total amount is $22,598.18.

Requirement 9Since your savings bonds investment totals $22,598.18, you would have

more money than your money market account ($21,966.00), but the estimatedtotal tuition ($49,069.81) is greater.



drawn. Note that we can use the total from Requirement 7 ($22,650.35) with-out the redemption loss in Requirement 8c, since the bonds will mature longer(i.e., more than 5 years).

Table S13.


year period previous + interest + bonds = new tuitiondue amount change bought amount

1st 15 22,137.10 + 284.46 + 0.00 = 22,650.35 10,999.5516 11,422.01 + 146.77 + 0.00 = 11,568.78

2nd 17 11,568.78 + 148.66 + 0.00 = 11,717.44 11,804.8018 −87.35 + −1.12 + 0.00 = −88.48

3rd 19 −88.48 + −1.14 + 0.00 = −89.61 12,669.0020 −12,758.61 + −163.95 + 0.00 = −12,922.56

4th 21 −12,922.56 + −166.05 + 0.00 = −13,088.62 13,596.46−26,685.08

Stretching your money still will not cover the estimated cost of tuition,since the tuition for the 2nd year ($11,804.80) is more than the account balance($11,717.44).

Monthly Payments until CollegeRequirement 1

You deposit 89 monthly payments of $253 into an account that pays 1.85%interest compounded monthly for 7.5 years (90 months). Since you are con-cerned with the amount in the account each month, let

f(n) = amount in the account after n months,

where n = 0, 1, 2, . . . , 90 with an initial condition of f(0) = 253. Since the inter-est rate is compounded monthly, the annual interest rate of 0.0185 is equivalentto an interest rate of 0.0185/12 per month. You have, until the 88th month,

f(n + 1) = f(n) + interest gained on f(n) + next paymentf(n + 1) = f(n) + f(n) × 0.0185/12 + 253.

You can use a spreadsheet to calculate f(n) for n = 0, 1, . . . , 90, as in Table S14.You will have $24,189.43 in your account on July 1, 2010.

Requirement 2The $1,000 minimum balance condition needed to receive 1.85% interest

will affect your account, since your starting balance is only $253. Starting with


Table S14.


month previous + interest + monthly = newamount change payment amount

0 253.00 253.001 253.00 + 0.39 + 253.00 = 506.392 506.39 + 0.78 + 253.00 = 760.17...

87 23,536.00 + 36.28 + 253.00 = 23,825.2888 23,825.28 + 36.73 + 253.00 = 24,115.0189 24,115.01 + 37.18 + 0.00 = 24,152.1990 24,152.19 + 37.23 + 0.00 = 24,189.43

the 1st month, change each month’s interest rate from 1.85% to 0.60% until youraccount reaches (or exceeds) $1,000. This will occur at the beginning of the 4thmonth. So, starting with the 4th month, the interest rate becomes 1.85%. Thelower interest rate reduces your total amount to $24,187.62.

Requirement 3See Requirement 2 of the Lump Sum Payment situation.

Requirement 4You would not have enough money in your money market account to pay for

her tuition, since your account balance would be $24,187.62 and the estimatedtotal tuition is $49,069.81.


drawn. Stretching your money still will not cover the estimated cost of tuition,since the tuition for the 3rd year ($12,669.00) is more than the account balance($1,659.73).

Requirement 6You purchase savings bonds every six months with the six accumulated $253

payments, which total 6 × $253 = $1,518. Due to the denominations, you canpurchase only $1,500 in savings bonds. Since the option requires 89 monthlypayments of $253, you can make 14 purchases of $1,500 plus 1 last purchaseof $1,500 using the remaining 5 monthly payments and the left-over amountsfrom the first 14 purchases.


Table S15.


year month previous + interest + monthly = new tuitiondue amount change payment amount

1st 90 24,150.38 + 37.23 + 0.00 = 24,187.62 10,999.5591 13,188.07 + 20.33 + 0.00 = 13,208.4092 13,208.40 + 20.36 + 0.00 = 13,228.76

...101 13,392.80 + 20.65 + 0.00 = 13,413.45

2nd 102 13,413.45 + 20.68 + 0.00 = 13,434.13 11,804.80103 1,629.33 + 2.51 + 0.00 = 1,631.84

...113 1,654.62 + 2.55 + 0.00 = 1,657.17

3rd 114 1,657.17 + 2.55 + 0.00 = 1,659.73 12,669.00115 −11,009.27 + −16.97 + 0.00 = −11,026.24

...125 −11,180.18 + −17.24 + 0.00 = −11,197.41

4th 126 −11,197.41 + −17.26 + 0.00 = −11,214.68 13,596.46−24,811.14



g(n) = amount in the savings bonds after n 6-month periods,

where n = 0, 1, 2, . . . , 15. Since you need to accumulate the six payments,you have g(0) = 0 and g(1) = 1,500. Since the interest rate is compoundedsemiannually, the annual interest rate of 0.0257 is equivalent to an interest rateof 0.0257/2 per 6-month period.

You have, until the 15th 6-month period:

g(n + 1) = g(n) + interest gained on g(n) + next bond purchaseg(n + 1) = g(n) + g(n) × 0.0659/2 + 1500.

You can use a spreadsheet to calculate g(n) for n = 0, 1, . . . , 15, as in Table S16.You will have $24,641.04 in savings bonds on July 1, 2010.

Requirement 8a) Your total bond purchases are 15 × $1,500 = $22,500, which is less than

the $30,000 limit per year.b) The 6-month redemption condition applies to your last bond purchase of

$1,500. So you would have to find some other investment (such as the money


Table S16.


period previous + interest + new bonds = newamount change bought amount

0 0.001 0.00 + 0.00 + 1,500.00 = 1,500.002 1,500.00 + 19.28 + 1,500.00 = 3,019.28...

14 21,076.62 + 270.83 + 1,500.00 = 22,847.4515 22,847.45 + 293.59 + 1,500.00 = 24,641.04

market account) for the $1,500. But the calculations did not include any intereston the $1,500, since it was purchased bonds during the 15th period; so yourtotal amount remains the same.

c) The 5-year redemption condition does apply to all bonds redeemed lessthan five years after purchases, so you will lose the last three months of intereston each of them. You need to calculate the interest gained each 6-month periodon $1,500 in bonds. (Three months of interest is considered half of the interestduring the 6-month period.) See Table S17.

Table 18.

Spreadsheet for solution to Requirement S17.

period previous + interest = new 3 monthsamount change amount of interest

1 1,500.00 + 19.28 = 1,519.28 9.642 1,519.28 + 19.52 = 1,538.80 9.763 1,538.80 + 19.77 = 1,558.57 9.894 1,558.57 + 20.03 = 1,578.60 10.015 1,578.60 + 20.28 = 1,598.88 10.146 1,598.88 + 20.55 = 1,619.43 10.277 1,619.43 + 20.81 = 1,640.24 10.408 1,640.24 + 21.08 = 1,661.32 10.549 1,661.32 + 21.35 = 1,682.66 10.67

• For the bonds purchased 4.5 years ago (matured for nine 6-month periods),you lose $10.67.

• For the bonds purchased 4.0 years ago (matured for eight 6-month periods),you lose $10.54.

• For the bonds purchased 3.5 years ago (matured for seven 6-month periods),you lose $10.40.

• . . .


• For the bonds purchased 0.5 years ago (matured for one 6-month period),you lose $9.64.

Therefore, you lose a total of $91.33 in interest due to this redemption condition,so the new adjusted total amount is $24,549.71.

Requirement 9Since the savings bonds investment totals $24,549.71, you would have more

money than in a money market account ($24,187.62); but your estimated totaltuition ($49,069.81) would be still greater.


drawn. Note that you can use the total from Requirement 7 ($24,641.04) with-out the redemption loss in Requirement 8c, since almost all the bonds willmature long enough (i.e., more than 5 years). Stretching your money stillwould not cover the estimated cost of tuition, since the tuition for the thirdyear ($12,669.00) is more than the account balance ($2,246.71).

Table S18.


year period previous + interest + new bonds = new tuitionamount change bought amount

1st 15 22,847.45+ 293.59 + 1,500.00 = 24,641.04 10,999.5516 13,641.49 + 175.29 + 0.00 = 13,816.79

2nd 17 13,816.79 + 177.55 + 0.00 = 13,994.33 11,804.8018 2,189.53 + 28.14 + 0.00 = 2,217.67

3rd 19 2,217.67 + 28.50 + 0.00 = 2,246.17 12,669.0020 −10,422.83 + −133.93 + 0.00 = −10,556.76

4th 21 −10,556.76 + −135.65 + 0.00 = −10,692.42 13,596.46−24,288.88

Comparison of SCTPP OptionsRequirement 1

You are withdrawing $413 from an account with an initial balance of $16,989that pays 1.85% interest compounded monthly. Since you are concerned withthe amount in the account each month, let

h(n) = amount in the account after n months,

where n = 0, 1, 2, . . . , 48 with an initial condition of h(0) = 16,989 − 413 =16,576. Since the interest rate is compounded monthly, the annual interest rateof 0.0185 is equivalent to an interest rate of 0.0185/12 per month.


You have:

h(n + 1) = h(n) + interest gained on h(n) − withdrawalh(n + 1) = h(n) + h(n) × 0.0185/12 − 413.

You can use a spreadsheet to calculate h(n) for n = 0, 1, . . . , 48, as in Table S19.

Table S19.


month previous + interest + monthly = new amountamount change payment amount

0 16,576.001 16,576.00 + 25.55 − 413.00 = 16,188.552 16,188.55 + 24.96 − 413.00 = 15,800.51...

40 1,014.49 + 1.56 − 413.00 = 603.0541 603.05 + 0.93 − 413.00 = 190.9842 190.98 + 0.29 − 413.00 = −221.72

The balance becomes negative after the 42nd month. Thus, if someonewere to offer you $16,989 instantly or 48 monthly payments of $413, you wouldchoose the 48 monthly payments of $413 because it is “more money.” Con-versely, in this tuition scenario, if you were given the choice to pay eitherof these options, you would choose the $16,989 lump sum because it is “lessmoney”—provided you had $16,989 on hand.

You might have guessed that the lump sum is “less money” because its bal-ance after 90 months in Requirement 1 of the Lump Sum Payment situation($19,515.50) is less than the balance for the 48 monthly payments in Require-ment 1 of the 48 Monthly Payments situation ($21,967.47).

Requirement 2You are withdrawing $253 from an account with an initial balance of $16,989

that pays 1.85% interest compounded monthly. Since you are concerned withthe amount in the account each month, let

i(n) = amount in the account after n months,

where n = 0, 1, 2, . . . , 89 with an initial condition of i(0) = 16,989 − 253 =16,739. Since the interest rate is compounded monthly, the annual interest rateof 0.0185 is equivalent to an interest rate of 0.0185/12 per month.

You have:

i(n + 1) = i(n) + interest gained on i(n) − withdrawali(n + 1) = i(n) + i(n) × 0.0185/12 − 253.

You can use a spreadsheet to calculate i(n) for n = 0, 1, . . . , 89, as in Table S20.


Table S20.

Spreadsheet for Requirement 2.

month previous + interest − monthly = newamount change payment amount

0 16,736.001 16,736.00 + 25.80 − 253.00 = 16,508.802 16,508.80 + 25.45 − 253.00 = 16,281.25...

68 712.74 + 1.10 − 253.00 = 460.8369 460.83 + 0.71 − 253.00 = 208.5470 208.54 + 0.32 − 253.00 = −44.13

The balance becomes negative after the 70th month. Thus, if someone wereto offer you $16,989 instantly or 89 monthly payments of $253, you wouldchoose the 89 monthly payments of $253 because it is “more money.” Con-versely, in this tuition scenario, if you were given the choice to pay eitherof these options, you would choose the $16,989 lump sum because it is “lessmoney” (if you had the $16,989 on hand).

You might have guessed that the lump sum is “less money” because its bal-ance after 90 months in Requirement 1 of the Lump Sum Payment situation($19,515.50) is less than the balance for the 48 monthly payments in Require-ment 1 of the Monthly Payments until College situation ($24,189.43).

Requirement 3This requirement is the most interesting of the three for this situation, since

Requirements 1 and 2 assume that you have $16,989 for a lump sum—whichmost parents don’t. You make the comparison by evaluating the calculationsin Requirement 1 of the 48 Monthly Payments situation ($21,967.47) and Re-quirement 1 of Monthly Payments until College situation ($24,189.43). Sincethe 48 monthly payments is “less money” than monthly payments until college,

you should choose 48 monthly payments.

ReferencesBeezer, Robert A. 1996. Closing in on the internal rate of return. UMAP Mod-

ules in Undergraduate Mathematics and Its Applications: Module 750.Reprinted in UMAP Modules: Tools for Teaching 1996, edited by Paul J. Camp-bell, 47-–78. Arlington, MA: COMAP, 1997.

The internal rate of return (IRR) is a natural measure of the performance ofan investment. This Module describes three iterative methods for findingthe IRR of an investment; two of these methods are general and wellknown (bisection, Newton’s method), and the third is peculiar to the


computation of the IRR. Each method is illustrated by an application toan actual investment in stocks, bonds, or a mutual fund.

Lindstrom, Peter A. 1983. Nominal vs. effective rates of interest. UMAP Mod-ules in Undergraduate Mathematics and Its Applications: Module 464.Reprinted in UMAP Modules: Tools for Teaching 1987, edited by Paul J. Camp-bell, 21-–53. Arlington, MA: COMAP, 1988.

Paley, Hiram, Peter F. Colwell, and Roger E. Cannady. 1984. Internal rates ofreturn. UMAP Modules in Undergraduate Mathematics and Its Applica-tions: Module 640. In UMAP Modules: Tools for Teaching 1983, 493—548.Lexington, MA: COMAP, 1984.

This Module is concerned with situations where an investment has severalvalues for the IRR; it gives a criterion for determining which is the mostappropriate.

About the Author

Tom Fitzkee is an assistant professor of mathematicsat Francis Marion University in Florence, SC. Tom re-ceived his B.S. from Salisbury (MD) University in 1989,his M.S. from Virginia Tech in 1993, and his Ph.D. fromGeorge Washington University in tiling dynamical sys-tems under E. Arthur Robinson, Jr., in 1998. He has beenteaching at FMU for five years and teaches a wide varietyof mathematics courses.

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