300

The Mathematics of the Return from Home OwnershipFloyd Vest Mathematics Department

University of North TexasDenton, Texas 76203

Department of Finance and EconomicsStephen F. Austin State UniversityNacogdoches, Texas 75962

Reynolds Griffith

Mathematics has many important applications, one ofwhichis mathematics offinance with applications to personal finance.Students and teachers should certainly know this most personalarea of application. Mathematics of finance with the aid ofcomputingtechnologyprovides apowerful tool forunderstandinglife’s financial problems. One such problem concerns theeconomic value of home ownership.

Formany people, ahome is theirmostsubstantial investmentof a lifetime. In recent years, many homes have appreciatedsignificantly, resulting for some in a substantial personal asset.Obviously, the decision tobuy a home should be made carefullywith the aid of relevant facts and mathematical calculations.This article constructs a mathematical model or project analysiswhich calculates the financial return from home ownership.This analysis will illustrate an interesting application of suchmathematical topics as compound interest, annuities,amortization schedules, internal rate of return, and otherelementsofschoolandcollegemathematics up through numericalanalysis. First, examine in detail the area of application.

In addition to the personal satisfaction of owning a home,there are the followingpotential financialbenefits (orliabilities):

1. Rentalvalue. Assuming thata person wouldhave to eitherrent or buy, the owning ofa home provides a place to live whichyields a financial benefitwhich is called rental value. This figurecouldbearrivedatby usingtheclassified sectionofthenewspaperto conduct a rent survey for comparable housing.

2. Equity buildup. Homeowners usually accumulate equitybymakingregularmortgagepaymentswhichreducetheprincipalon the loan. An amortization schedule (Bowen, 1976) can bedevelopedwhichgives, foreachpayment, theamountcontributedto interest and principal and gives the balance of the loan afterthe payment.

3. Appreciation. The value of the home may increase withtime, particularly in an era of inflation, thus providing a hedgeagainstinflation. Ithasbeenreported thatfrom 1970to 1980, thenational median value of a home rose by 170% (10.46%compounded annually) (US Department of Commerce, 1987);however, the average rate of increase from 1980 to 1988 wasonly 4.62% (National Association of Realtors, 1989).

4. Tax savings. The homeowner typically receives taxadvantages in the form of an income tax deduction for homemortgage interest and real estate taxes. Also, a tax-free sale ofthe home is allowed for many homeowners.

Analysis of a Property for Purchase

Thepurchasershouldcomplete aprojectedannualoperatingstatement for a property under consideration. The operatingstatement in Table 1 shows how one can analyze a specificproperty for a given year. The hypothetical situation involvesa house which costs $65,000.00. The mortgage is a 95%mortgage at 10.5% interest for 30 years. If the house has arental value of $650.00 per month, it will be credited to theowner, since he would otherwise pay it in rent. Also creditedis $126.00 for renter’s insurance.1 Using the same logic bywhich rental value was credited, it is reasonable to credit thevalueofrenter’sinsurance since theexpensecouldbeconsiderednecessary for renters, and the cost of homeowner’s insuranceprovides benefits included in renter’s insurance (insurance oncontents and personal liability).

Operating expenses of property taxes, homeowner’sinsurance, and maintenance can be estimated as indicated inTable I.2 Crediting the annual rental value and renter’sinsurance to the buyer and subtracting the expenses andmortgage payments leaves a negative cash flow before taxconsiderations of$386.20 per year.3 Assuming a 28% incometax bracket, income tax savings can be estimated at $1,111.36based on deductions for property taxes and mortgage interestresulting in a positive flow of $725.16.4

Recalling the down payment of $3,250.00 and assuming$2,000.00 in closing costs to purchase, the initial equity is$5,250.00. With a$725.16 cash flow in the first year, this givesa rate of return of

72116 =.138 or 13.8%5,250.00

The house would produce a reasonable rate of return.The calculation outlined above is useful but somewhat

simplified. There are other factors that should be taken intoaccount as well. One is simply that the $725.16was not taxableincome. Therefore, it is better than having a 13.8 % return thatwould be taxable. Other simplifications are assuming that thevalue of the house did not change during the year, and notconsidering how long the house will be owned and the prospectof selling expenses (Vest, 1986).

During future years, the assumptions which underlie thecalculations may change. The expenses and implicit income

School Science and Mathematics

Home Ownership 301

Table 1

Example of an Operating Statement for Year One

Table 2

Case Analysis^ Input Data Summary

Price $65,000Credits: Potential Rental Value 12(650) 7800

Renter’s Insurance 126Total Credits 7926Operating ExpensesA. Fixed

Property Taxes (1 to 2 per- 1000cent of purchase price)

Homeowner’s Insurance (ap- 300proximately .5% of pur-chase price)

B. VariableRepairs, maintenance, supplies 234

(2.5 to 14 percent of annual 1534rental value)

Less Total: Operating Expenses 1534Net Operating Value 6392

Less: Debt Service (principal and 6778.20interest on home mortgage)

Cash Flow before Tax �386.20Plus: Estimated Income Tax 1111.36Savings*

Value Flow to Equity after Tax 725.16

^Explanation of "Estimated Income Tax Savings*’:

1000.00 Effective Home-6469,6 owner Ded. 3969.16

Property TaxesMortgage InterestHomeownerDeduction

Assumed OtherDeductions

Total DeductionsLess: StandardDeduction

7469.16 Tax Rate x .28

1500<00 Estimated Income����

8969.16 Tax Savings 1111.36

5000.00

components may change; the value ofthe housemay change. Inorder to take into account possible changes in such relevantfactors, the buyer should do an Input Data Summary whichincludesalternatecases as illustrated in Table2andforeach casea resultant Project Analysis as illustrated in Tables 3 and 4. TheProject Analyses give theexpected return fromhomeownershipresulting from each case.

Selected input data is summarized in Table 2. For example,the ordinary income tax rate is assumed to be 28% with astandard minimum deduction of $5,000.00. Assume that theproperty will be held for seven years and then soldwithoutpaymentofincome taxes on the profit. (Itis reported thatthe average life of a home mortgage is seven years.) Sellingexpenses are assumed to include a 6% brokerage fee, $2,000.00closing costs to sell, and empty house expenses arising from theproperty being unoccupied for the last six years months of the

30 years10.5%

61,750564.85

6778.2028%0%

5,000$1500

3250 down payment2000 closing costs5250

Mortgage InformationMortgage TermMortgage Interest RateInitial MortgageMonthly Payment for

Principal and InterestAnnual Payment

Owner’s Federal Income Tax RateOwner’s State Income Tax RateStandard DeductionOwner’s Income Tax DeductionsOther Than from Homeownership

Initial Equity

Time of sale�at end of seventhyear

Selling ExpensesBrokerage Fee 6%Closing Costs to Sell 2000

Empty House Expenses: Unoccu-pied for the last six months ofthe seventh year

Growth Rates (% compounded annually)Case 1 Case 2Table 3 Table 4

Appreciation 5% 0%Rent Value 5% 0%Insurance 5% 0%Taxes 5% 0%Maintainance 5% 0%

seventh year. This last assumption is an effort to reflect thereality that homes are not always readily sold.

As indicated in Table 2, the rental value is assumed to be$650.00 per month for the first year, and the yearly rates ofappreciation, increase in taxes, insurance, maintenance, andrent value are forCase I assumedtobe5%. Analyses couldbegenerated for other cases of appreciation and inflation such asfor0%.

For the Project Analysis, the rate of return on the initialequity (down paymentplus closing costs) will be calculated asan Internal Rate ofReturn (IRR), which is the rate ofreturn thatequates the sum of the annual discounted after tax cash flowsresulting from home ownership and sale, to the initial equity.The IRR is the i in this equation:Initial Equity = CF, (1 + i)-1 + CE, (1 ^-O^ CF^ (1 + i)-3^

CFJ1 + i)-4-^ CF^(1 + i)-5 + CF^(1 + Q-^CF,(1+Q-7

where CFi is the valueHow for year f. Obviously, this equationis solved with the aid ofan iterative process programmed on acomputer, scientific calculator, or business calculator (Conte

Volume 91(7), Novemberl991

302HomeOwnership

& deBoor. 1980; Hewlett Packard, 1985). Tables 3 and 4present the Project Analyses assuming growth rates of 5% forCase 1 and 0% for Case 2, respectively.

Discussion of Table 3

Table 3 is based upon the assumption of5% yearly increasein the value of the house and other variables as indicated in theCase Analysis Summary in Table 2. The size of the assumedrental value in Column 1 has a significant impact on theestimated return on the investment. An overestimate of therental value ofa certain amount will result in an overestimate ofthe subsequent cash flow (and return on investment) by thesame amount A buyer should be careful and realistic in thisarea. By examining the entries in Column 1, observe thecompounding at5% that is applied. The last entries in Columns1 and 2reflectthe assumptionthatthehouseremainsunoccupiedfor six months while it is being sold.

Column 2 gives credit for the costofrenter’s insurance sincethe cost of the homeowner’s insurance includes coverage forsuch items as furnishingsandpersonal liability normallyincludedin renter’s insurance. Taxes, insurance, and maintenance costsfor home ownership are included in Column 3 (see Table 1).Thiscolumnreflectsa5%annualincreasein operatingexpenses.Annual interest expense and principal amortization for thehome mortgage are presented in Columns 5 and 6 and can bederived from an amortization schedule (Bowen, 1976).5

It is well-known that one of the benefits ofhome ownershiparises from the federal income tax deduction for mortgageinterest and real estate taxes. The value of this deduction isdisplayed inColumn 8; however, aminimum standard deductionis used so that the effective tax savings from home ownershipis based on the excess total deductions above the standarddeduction. To calculate the Effective Tax Deduction fromHome ownership (Column 9) from Column 8:

Let B = Column 8 +deductions other than for home ownership

Then use the logic:

If B < Standard deduction, use 0 in Column 9.If B > Standard deduction, and Other deductions > Standard

deduction, use Column 8 in Column 9.Else use (B - Standard deduction) in Column 9.

The tax savings from home ownership in Column 10 is 28%(the assumed tax rate) of the effective tax deduction for homeownership in Column 9. Since the homeowner’s tax liability isdecreased by the entries in Column 10, this is credited and theafter tax flow is calculated from the before tax flow and the taxeffectas indicated forColumn 11.6 In calculating the tax savings

for Column 10, the investor should include possible stateincome tax savings.

The resale proceeds (Column 14) are calculated fromColumns 12 and 13, assuming the value of the house hasincreased by 5% each year, a 6% brokerage fee, $2,000.00closing costs to sell, and a $58,724.67 mortgage to be paid offleaving $25,249.36 inproceeds from thesale.7 Column 15givesthe net after tax flow for each period of the ownership with aninitial investment of$5,250.00 value flows for years one to sixfrom Column 11, and for year seven a flow equal to the sum ofthe proceeds from the sale (Column 14) and the entry in Column11. Evaluated from Column 15 as the internal rate ofreturn, therate of return on the investment is 38.1 %.8 This rate of returnis equivalent to 52.9% before taxes assuming a 28% bracket.

Summary

There are several classroom applications of the material inthis article:

1. It can be taught as an advanced unit in secondarymathematics courses such as Mathematics for ConsumerEconomics and Advanced Mathematics for Business which areoffered after two years ofalgebra. After students have studiedthe basics of mathematics of finance, multistep problems, andamortization schedules, they areready forthis application. Thismaterial has been taught in a similar course for prospectivemiddle school mathematics teachers simply by lecturing overthe material, deriving formulas, demonstrating calculations,and testing over components of the analysis and computation.The students were not required to use computers, some hadfinancial calculators, and all had been instructed and tested inknowledge of the keyboard functions and speed and accuracyin computations with scientific calculators. Certainlymathematics teachers should be thoroughly familiar withmathematics of finance and its application to personal finance,and be able to pass this very important use of mathematics onto their students. Other topics which have been written for sucha course are: The Mathematics of Financial Planning forRetirement, Examining a Lifetime of Automobile PurchaseExpenses, UsingFinancial Calculators,Computerized BusinessCalculus-Examples from Mathematics of Finance, SolvingAnnuity Formulas for i with a Programmable ScientificCalculator.

2. The material in this article can be used to provide anoutline of an open-ended modeling exercise in modeling thecost of home ownership. This would be similar to modelingactivities suggested by the Consortium for Mathematics and ItsApplications. The teacher would state the problem andprovidehints on spreadsheet analysis, calculations, and input data.Students would research the data and gradually refine variouslevels of solutions to the problem.

3. The material in this article can be used to illustrate

School Science and Mathematics

Home Ownership

Table3

Project Analysis (Case 1, 5%)

Table 4

Project Analysis (Case 2, 0%)

Year

1234567

Year

1234567

Year

1234567

Year

1234567

Year

(1)Rental Value

780081908599.569029.529480.969954.965226.36

(4)«(1)+(2)-(3)Net Operating

Value

63926711.607047.227399.587769.528157.953255.08

(7)"(4)-(5)-(6)Beforc-Tax

Flow

-386.20-66.60269.02621.38991.321379.75

-3523.12

(10)- .28 x (9)Tax Savings

1111.361115.831119.941123.621126.781129.311131.10

(12)Selling PriceLess Expenses

(2)Credit forRentersInsurance

126132.50138.91145.86153.15160.8184.43

(5)Interestexpense

6469.166435.106397.296355.316308.716256.976199.53

(8)-(5)+(propertytaxes)TaxDeductions

7469.167485.107499.797512.947524.227533.257539.63

(11)- (7) + (10)After-Tax

Flow

725.161049.231388.961745.002118.102509.06

-2392.02

(13)MortgageBalance

(3)OperatingExpenses

15341610.701691.251775.801864.591957.822055.71

(6)Principal

Amortization

309.04343.10380.91422.89469.49521.23578.67

(9)Effective

Home-ownerDeduction

3969.163985.103999.794012.944024.224033.254039.63

(14)«(12)-(13)

ResaleProceeds

83,973.8458,724.67 25,249.36

(15)Net After Tax

FlowYear

-5250725.161049.231388.961745.002118.102509.06

22857.15 IRR - 38.1%

Year

1234567

Year

1234567

Year

1234567

Year

1234567

Year

7

Year

01234567

(1)Rental Value

7800780078007800780078003900

(4)»(1)+(2)-(3)Net Operating

Value

6392639263926392639263922429

(7)»(4)-(5)-(7)Before-Tax

Flow

-386.20-386.20-386.20-386.20-386.20-386.20-4349.20

(10)- .28 x (9)Tax Savings

1111.361101.831091.241079.491066.441051.851035.87

(12)Selling PriceLess Expenses

59,100

(15)Net After Tax

Flow

-5250725.16715.63705.04693.29680.24665.65

-2938.00

(2) .

Credit forRentersInsurance

12612612612612612663

(5)InterestExpense6469.166435.106397.296355.316308.716256.976199.53

(8)»(5)+(propertytaxes) TaxDeductions

7469.167435.107397.297355.317308.717256.977199.53

(11)- (7) + (10)After-Tax

Flow

725.16715.63705.04693.29680.24665.65

-3313.33

(13)MortgageBalance

58,724.67

IRR is negative.

(3)OperatingExpenses

1534153415341534153415341534

(6)Principal

Amortization

309.04343.10380.91422,89469.49521.23578.67

(9)Effective

Home-ownerDeduction

3969.163935.103897.293855.313808.713756.973699.53

(14)-(12)- (13)

ResaleProceeds

375.33

Volume 91(7). Novemberl991

304HomeOwnership

important applications ofmathematics presented in the form ofa computer simulation of a real life problem. A Lotus 1-2-3diskette (IBMPC) is available.9 The spreadsheetallows studentsto see a series of calculations all at once and to see patternsdevelop. Students can make and test conjectures and changeinput data and watch immediately the resulting effect, thusproviding active personal involvement, the joy of explorationand discovery, and an experience ofthe powerofmathematics.They can investigate many ofthe mathematical relationships inthe spreadsheet with the aid of a calculator.

Studentswhomaynotleam tousecomputerizedspreadsheetsshould have the opportunity to see spreadsheet analysis appliedas a problem solving tool. Spreadsheet analysis is a veryimportant method of problem solving. Many spreadsheetsincluding the ones in this article can be built withouta computerby simply using calculators and writing tables by hand. Forpeople who are do not frequently use spreadsheet software, it ismore efficient to do the spreadsheet this way. This articleprovides a powerful and practical application of mathematicsand spreadsheet analysis that is of universal interest evenwithout a computer.

There are obviously several levels of mathematical andpedagogical significance of this material. The developmentshould be of mathematical interest since it represents obviousapplication of compound interest, annuities, amortization,internal rate of return, and various mathematical topics upthrough numerical analysis. Teachers are perhaps interested tosee application of this level of mathematics to an area which isofalmost universal interest. For many, it is significant to havea method for evaluating the return from home ownership.

There is an increasing awareness in mathematics educationthat thecomputerand calculator are among the forces ofchangewhich are determining how mathematics is used, learned, andtaught. The above analysis is an example of how fairlysophisticated mathematics and computingare used routinely tosolve a problem that is of interest to many. Such an importantpersonal application ofmathematics canbe used for developinginterest in applications of mathematics and developing a spiritof inquiry for students.

References

Bowen, E. K. (1976). Mathematics with applications inmanagement and economics (4th ed.). Homewood, IL:Richard D. Erwin, Inc.

Conte, S. D., & deBoor, C. (1980). Elementary numericalanalysis (3rd ed.). New York: McGraw-Hill.

Hewlett Packard. (1985). HP-120 Owner’s handbook andproblem solving guide. Corvallis, OR: Hewlett Packard.

National Association ofRealtors. (1989). Home sales (Vol. 3,No, 2). Washington, DC: Author.

US DepartmentofCommerce. (1987). National data bookand

guide to sources, statistical abstract of the United States,1986. Washington, DC: US Government Printing Office.

Vest, F. (1986). Modeling the cost of home ownership.Mathematics Teacher, 79(8), 610-613.

Footnotes

1Information about costs ofreal estate, insurance, mortgagerates, and rental rates can be obtained by consulting suchsources as a local insurance agent or real estate agent orconsulting the real estate section of the newspaper.

information onproperty taxes andinsurancecan beobtainedfrom tax and insurance statements on home mortgages or frominsurance and real estate agents.

^n order to calculate the $6,778.20per year for debt service,calculate the monthly payment for principal and interest usingthe formula

-n wherel-(l+i)A

.105A = 65000 - .05(65000) = 61750, i = 12, n =12(30) = 360, R = monthly payment for principal and interestwhich is calculated to be $564.85. For the year, debt service is12(564.85) = $6,778.20.

Vor further explanation of income tax considerations, seeInternal Revenue Service publications which can be obtainedfrom the public library or by calling IRS.

columns 5 and 6 of Table 3 come from an amortizationschedule. A yearly amortization schedule which approximatesthe values in Columns 5 and 6 can be calculated using ascientific calculator and the formula in Footnote 3. The balanceowed at the end of year 1, B^, is

.105 ,-(360-12)!-(!+12B = (564.85)61440.77

.10512

The principal paid during year 1,P^ = 61,750.00 - 61,440.77= 309.23. The interest paid during year 1,1^ = 12R - P^ =12(564.85) - 309.23 = 6,468.97. Use this pattern to calculateR, P, L for each year needed. The values in Columns 5 and 6were calculated with a computer and are slightly different.

Sincecomputing an amortization scheduleby calculatorcanbe timeconsuming, teachers can usecommonlyavailable menudriven software to generate and supply schedules to students orshow the students how to generate their own (see Footnote 10).

leaders are referred to IRS 1040 publications for furtherexplanation.

"Column 12 = (1.05)7 (65,000) (1 - .06) = 83,973.84. The

School Science and Mathematics

Home Ownership 305

balance of $58,724.67 on the mortgage is derived from anamortization schedule or can be calculated as indicated inFootnote 5 above.

^To check the IRR reported here, the reader can use ascientific calculator to check the following equality:5,250 = 725.16(1 + .381)-1 + 1049.23(1 + .381)-2 + . . . +22,857.15(1 + .381)-7. Atypical scientific calculatorcodewouldbe: 1.381 STOOycl+/-x725.16+RCL0^2+/-x 1049.23+... The teacher can assist students in calculating IRRby usinga financial calculator (for example, a HP12C) or computer (forexample, use a spreadsheet). Certainly all students can use ascientific calculator to check the IRR. This would help them tounderstand the IRR concept.

With the increased use of computing technology, we willfind students and teachers routinely using computer calculation

methods which they don’t fully understand, but we have toadmit that this is already the case when they calculate sine andcosine on the scientific calculator. It may be thatIRR is a moreimportant concept than sine and cosine. There is a growingconsensus thatsuchpowerfulandnearly automatic calculationsshould not be denied students simply because they do notunderstand them. For example, the business and engineeringprofessionsdonotdeny themselvescomputing methods simplybecause they are not fully understood.

^A menu driven spreadsheet which does not requireknowledgeofLotus 1-2-3 andwhich conductsasimilaranalysisofthe financial return ofhome ownership is available from theReal Estate Center, Texas A&M University, College Station,Texas 77843. This diskette (IBM PC, cost about $20) omitsseveral considerations; however, the diskette will doamortization schedules.

Activities for Teaching K-6 Math/Science ConceptsWalter A. Farmer and Margaret A. Farrell

University of Albany, State University of New York

Monograph ^2 in the SSMA Classroom Activities Series

This collection of activities for K-6 classroom use has been developed by the mathematics andscience education faculty at the State University of New York at Albany and representatives ofeight school districts in the Greater Capital District area of New York state.From among several hundred activities prepared, many have been identified and chosen for

inclusion which:

1. combined important mathematics and science learnings in single lessons,2. have been tried by elementary classroom teachers and school children,3. involved "hands-on" activity, and4. use readily available, everyday materials.The authors identify the Procedures, Materials, Key Concepts, Skills and Processes, Background,

Sources of Further Ideas, and Useable Junk for the activities.

Paper, spiral binding, 50 pagesISBN 0-912047-07-0

Copyright 1989Price $7.50 (postage included on prepaid orders)

20 per cent discount to SSMA membersOrder from: School Science and Mathematics Association, 126 Life Science Building, BowlingGreen State University, Bowling Green, OH 43403-0256, telephone (419) 372-7393.

Volume 91(7), Novemberl991