The Essential Fairness of a Mortgage Loan: Developing Financial Literacy

The Essential Fairness of a Mortgage Loan: Developing Financial LiteracyAuthor(s): Wes WhiteSource: The Mathematics Teacher, Vol. 96, No. 7 (OCTOBER 2003), pp. 486-494Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871399 .

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The Essential Fairness of a Mortgage Loan:

Developing Financial Literacy

Financial success

may begin in the

classroom

any people know very little about finance. Few peo

ple, for example, can answer a finance question as natural and useful as, If I set aside $100 a month for 20 years at 5 percent annual interest, how much

money will I save? Some prominent national figures are calling for

financial literacy as a new goal for secondary schools. "In this increasingly complex financial

marketplace, too many people are victims of scam artists and unscrupulous operators, in part because

they lack adequate financial education," said Senator Paul Sarbanes, Chair of the Senate Banking Housing and Urban Affairs Committee (NCTM 2002, p. 1). Federal Reserve Chairman Alan Greenspan and others have challenged mathematics teachers to cover finance by indicating that, "Although your students may not yet be managing a household

budget, saving money to buy a house, starting up a small business, or making investment decisions for a pension plan, their financial success may begin in

your classroom" (NCTM 2002, p. 1). This article attempts to help teachers develop

their students' financial literacy by?

presenting essential finance theory: the relation

ship between mortgages, annuities, and savings accounts;

deriving the annuity and mortgage formulas; and

providing three initial lessons on finance that have succeeded in mathematics classes from ninth grade upward.

During a thirty-year mortgage, a homeowner may

eventually pay back as much as three times the

original loan amount. As we will see, an essential fairness and logic are behind mortgage loans. To

really understand finance, knowing the relationship between a mortgage loan, an annuity account, and a

savings account is essential. Readers can use a graph ing or scientific calculator to follow the examples.

THE FAIRNESS AND LOGIC BEHIND A MORTGAGE LOAN This problem assumes a universal interest rate, that is, all interest in this scenario has the same

annual percentage rate (APR) and is compounded equally often. Suppose that two investors, A and B, each have the same large amount of money?say, $100,000 or more?to invest. A puts her money in a

savings account and simply lets it collect interest for thirty years, whereas takes his $100,000 and lends it to C, a homeowner, on a thirty-year mort

gage loan. As C makes her monthly mortgage pay ments to , deposits each one in an account, which we call B's annuity.

At the end of the thirty-year period, A and will

again have the same amount of money (when rounded). We first verify this result with an example. For

all accounts?A's savings, B's annuity, and C's mort

gage?we assume that the universal interest rate is 6 percent APR and that interest amounts are com

puted monthly The bank's "working" interest rate, w, is therefore

6% W = T2 .06 " 12

= 0.005,

or half of 1 percent per month.

Throughout this article, the working interest rate is the same as the periodic interest rate. I pre fer the term working to periodic, since it is the rate with which the bank actually works to calculate balances and since many finance-related variables

begin with the letter p. We next assume that A and each wish to

invest $100,075 (a number chosen for convenience). The savings account formula is

S = P(l + wT,

where S represents the current balance (sum), is the original principal deposited, w is the bank's

Wes White, [email protected], teaches Advanced Placement statistics and algebra at Alhambra High School, Alhambra, CA 91801. His primary interest is in

reforming the secondary mathematics curriculum so that

it emphasizes interesting explorations and useful applica tions through use of graphing calculators.

486 MATHEMATICS TEACHER



working interest rate used in compounding, and is the number of compoundings over the life of the account. To the nearest cent, the balance of ?s account after 30 years, or 360 payments, is

(1) S = P(l + w)n = $100,075 ? (1.005)360 = $602,709.21.

Meanwhile, according to our scenario, has lent his $100,075?the same amount that A invested? to homeowner C. How much are C's monthly mort

gage payments? The mortgage formula, which we will derive later, is

(2) Pw(l + wf

represents the principal borrowed, is the

monthly payment amount, w is the bank's working interest rate, and is the number of payments over the life of the loan.

The exact payment that C makes every month is

Pw{\ + wf *"

(l + w)n-l

_ $100,075 - 0.005 -

(1.005)360 (1.005)360-1

(3) =$600.000188, or $600.00 when rounded to the nearest cent. This convenient result is the reason for using the figure $100,075. When presenting the forthcoming deriva tions for a class, referring to the "six-hundred dol lar payment" is nice.

immediately deposits each payment into a

monthly annuity account. The annuity formula, which we will also derive, is

(4) s=p[(l + wf-l] w

where S represents the cumulative balance, is the amount of the monthly annuity payment (or deposit), w is the bank's working interest rate, and

is the number of annuity payments (or deposits) over the life of the account.

To the nearest cent, the balance of B's annuity account after 30 years, or 360 payments, is

S = p[(l + w)n-l] w

(5)

$600[(1.005)36Q-1] 0.005

: $602,709.03.

Comparing B's annuity balance (5) with ?s

savings-account balance (1), we see that the differ ence, entirely caused by rounding errors, is only 18 cents.

So ignoring rounding errors in the preceding sce nario about A, B, and C, we see that makes the

same amount of money as A makes from a savings account when lends his money to C and deposits C's payments in an annuity. This outcome is what we mean by the essential fairness of mortgage loans. It also shows the relationship between a

mortgage loan, an annuity, and a savings account. What about the assumption made in the preced

ing scenario about a universal, equal interest rate? Is this rate plausible? No. In real life, mortgage interest rates are higher than savings-account rates. One reason is risk.

Savings accounts are often FDIC-insured up to

$100,000. However, the security offered to mortgage lenders is in their right to foreclose and acquire the

property on which the loan is based. The lender

may incur a loss if the property has lost value. Because depositing a large amount of money in a

savings account is safer than making a mortgage loan, mortgage rates are substantially higher than

savings-account rates to create an incentive. If the two rates were equal, as in our scenario, investor would have no reason to make the mortgage loan.

THE DERIVATION OF THE ANNUITY FORMULA Since we know the logical relationship between sav

ings accounts, annuity deposits, and mortgage pay ments, we can derive the equations of annuity depos its and mortgage payments. To derive the annuity formula, we need to focus, in the preceding scenario, on how much of B's final balance of $602,709.03 is

generated by each $600 mortgage payment deposit ed. Figure 1 shows the total value produced by compound interest for each of the last four deposits.

This outcome is

what we

mean by the essential

fairness of mortgage loans

Deposit number

360 (last payment) 359 (next to last) 358 (third to last) 357 (fourth to last)

Remaining months/ compoundings

-1-1?*

-+?I-1

-1-1

Total value produced

?(1 +*)

?(1 +x)2

?(1 +tf)3

Fig. 1 The later the payment, the less the amount of

compound interest it produces.

The 360th?and last?deposit occurs at the end of the loan, so it earns no interest: its total value is = $600. The 359th, or second-to-last, deposit is

made one month before the end of the thirty-year period, so it compounds a single time. It produces a total value of $600

? (1.005) = $603, or more

Vol. 96, No. 7? October 2003 487



generally, ( + w). Continuing, we note that the 358th deposit, made with two months left, pro duces $600

? (1.005)2 = $606.015, or more generally, p(l + wf. Since a pattern has emerged, we can

expand the table to include all 360 annuity deposits, as shown in table 1.

_TABLE 1_

Total Value Produced by Compounding for Each of B's Deposits Our Tbtal Value

Deposit Example_Produced

360 Oast) 359 (next to last) 358 (third to last) 357 (fourth to last)

1 (first)

Tbtal

$600 $600? (1.005) = $603 $600 ? (1.005P = $606.02 $600 -(1.005)3 = $609.05

p(l + w) ptt+wP p(l + w?

$600 ? (1.005)359 = $3595.57 (1 + w)m

S = $602,709.03* S = 2p-(l + i?y t*0

* Calculated on a TI-82, TI-83, or TI-83 Plus with the command

sum(seq(600 * 1 .005 , X, 0,359,1 ))

Factoring out in table 1 gives 359

S =p?(l+

a geometric series. From the formula for the finite sum of a geometric sequence, we obtain

359

s=pf a + w)\ i=0

_ [(1 + ;)360 -

1] " (l + w)-l

= [(1 + ;)360 - 1] w

This equation is the annuity formula for = 360

monthly deposits. To generalize, we substitute for 360:

pKl + n;)?-!] w

'

the annuity formula (4)

THE DERIVATION OF THE MORTGAGE FORMULA To derive the mortgage formula (2), we begin with the preceding scenario about A, B, and C. Since by the end of the loan period, B's annuity account must equal A's savings account, we simply equate formula (4), the annuity formula, with formula (1), the savings-account formula:

p[(l + wT-l] w

= P(l + w)n

Solving for p, we obtain

Pw(l + wf P

= -

(l + wf-l

which is result (2). We have derived the mortgage formula. In summary, once we understand the original

scenario equating A's savings account, C's mortgage loan from B, and B's annuity account, the deriva tions of the annuity and mortgage formulas follow

directly. Incidentally, we can now answer the finance

question posed at the beginning of this article. Rec

ognizing an annuity, we convert the 5 percent annual interest rate into a monthly rate of

w = W 12

= 0.0041666667.

Then, using the annuity formula (4) with a monthly payment of = $100 over = 240 months, we obtain

S = pKl + Htf1-! w

= $100[(1.0041666667)240-1] 0.0041666667

= $41,103.37.

A CONTEMPORARY APPLICATION OF THE FINANCE FORMULAS The following problem uses the formulas that we

have derived:

In February 2002, a state-lottery winner in California was faced with a difficult decision (poor guy!): He could choose (a) a $33 million lump-sum payment or (?) 26 equal annual payments totaling $64 million, with the first payment occurring immediately. He plans to invest his money in an account that pays a constant rate of interest. (We assume that he will live at least 25 more years, and we neglect tax considerations):

1. At an annual interest rate of 4 percent, which

payoff method would accumulate more money in the long run? How much more?

2. At an annual interest rate of 8 percent, which meth od accumulates more money? How much more?

3. What is the break-even interest rate, that is, the rate at which both payoff methods accumulate

equal amounts? If you were the lottery winner's

accountant, what advice would you give him about the payment method to choose?

Solutions In all three problems, the working interest rate w equals the annual percentage rate, since the

MATHEMATICS TEACHER



interest is compounded annually. The 26 amounts received annually would be $64,000,000/26, or

$2,461,538.46 each. The accumulated money amount for the $33,000,000 lump-sum payoff should be assessed at time t = 25 years, not 26, because with the annual-payments election, option ( ), the first of the 26 payments is received at year 0 and the last occurs at year 25. For a fair compari son, we should assess the accumulated value of the

lump-sum option at year 25, coinciding with the final payment from option (6), the annual-payment option.

Figure 2 diagrams the payouts received over the

twenty-five-year payout period. Twenty-six equal payments are paid by year 25.

CF CF CF CF I-1-1-H

CF CF CF -1-1

o 1 ;

CF = Cash Flow

3 years 23 24 25

Fig. 2 Payments received over the

twenty-five-year payout period

1. The lump-sum accumulation at 4 percent APR is found from formula (1):

s=pa + w)n = $33,000,000 ? (1.04)25 = $87,972,598.94

The equal payments are an annuity whose for mula is (4):

g=p[(i + ̂ r-i] w

$2,461,538.46- [(1.04)26-!] 0.04

= $109,075,063.60

We conclude that the twenty-six annual pay ments at an annual percentage rate of 4 percent yield more than the $33 million lump sum by about $109 million - $88 million, or roughly $21 million.

2. The lump-sum accumulation at an 8 percent APR is again found from formula (1):

S = P(l + wT = $33,000,000

? (1.08)25

= $225,999,681.50

The equal payments are an annuity whose for mula is again (4):

g=p[(l + wyt-l] w

_ $2,461,538.46 -

[(1.08)26- 1] 0.08

= $196,810,867.90

At an APR of 8 percent, we find that the lump sum yields more than the annual payments by roughly $226 million - $197 million, or about $29 million.

We can also illustrate the solution to this

problem on a TI-83 graphing calculator. After the substitution -

w, the input screens are shown in figure 3, where Y1 refers to the equal annual

payments and where Y2 refers to the lump-sum payment.

Pl*ti PlotZ 3 > 4?1538.46*<

a+ > 26 2 33 *a +

> 25 nV? =

nVs =

HINDOU Xnin=0 Xnax=.094 Xscl=.01 Vnin= -85000000 Vmax=330000000 VSC1=33000000 Xres=i

Fig. 3 With

- w, Y1 refers to the equal annual payments

and Y2 refers to the lump-sum payment. The Xmax =.094 allows for clean decimals.

Figure 4 shows the output. The cursors show that at 4 percent, option (6), the monthly pay ment plan Y1, is the better option for the lottery winner; whereas at 8 percent, option (a), the

lump-sum payment plan Y2, is better.

Fig. 4 At 4 percent, the better option is the monthly plan;

at 8 percent, the lump-sum option is better.

The more

of these

comparisons

they make, the greater their intuition about

financial questions becomes

Vol. 96, No. 7? October 2003 489



3. The break-even interest rate occurs where the

graphs cross. Using the same input screens as in

figure 3, we get the screen shown in figure 5.

Intersection K=.0fi3i7B07 V=i??fi3B5Hfi

Fig. 5 The graphs cross at the break-even point,

where the payoff methods are equally lucrative.

The break-even interest rate is - 0.06317807. We next verify this result. According to the sav

ings equation (1), the money accumulated by the

lump-sum payoff at this interest rate is

S = P(l + w)n = $33,000,000 ? (1.06317807)25 = $152,638,534.40.

According to the annuity formula (4), the

money accumulated by the twenty-six equal pay ments at this rate is

5 ̂[(1 + ^-1] w

_ $2,461,538.46 ? [(1.06317807)26 -1] 0.06317807

= $152,638,538.80.

The difference between the amount accumu lated by the lump-sum payout and the twenty-six equal payments, entirely caused by rounding off, is $4.40, a tiny error compared to $153,000,000.

The accountant for the lottery winner might advise him that if he can find an interest rate

higher than 6.3 percent, he should take the lump sum of $33 million. Otherwise, he should choose

option ( ), to be paid in the twenty-six equal pay ments totaling $64 million.

Finally, we note the difficulty of solving prob lem (3) without a graphing calculator or a com

puter. The equation

P(l + w)1 [p(l + ^_l] w

is not easy to solve algebraically for w.

MY THREE INITIAL FINANCE ASSIGNMENTS I use the following three lessons, given on the activ

ity sheets, to help develop students' financial litera

cy. These lessons have been successful for students in grades nine through twelve; I would not discour

age trying them at the middle school level. The teacher should remind students to use parentheses to separate the numerator from the denominator when they enter fractions into the calculator.

Lesson 1, on sheet 1, asks students to fill in every amount in a four-month mortgage table. Some of its finer points (learned by experience) are as follows:

1. The assignment specifies the amount of the

mortgage payment. The reader can verify that the figure came from the mortgage formula dis cussed previously. The students will subsequent ly calculate mortgage payments.

2. The interest is given as a nice round monthly rate, 1 percent. Students will subsequently transform an APR into a monthly interest rate.

3. The final balance is exactly 0, which gives stu dents a sense of satisfaction. More commonly, final mortgage balances have a slight?less than 10 cent?rounding error.

4. Although a four-month mortgage is not common in reality, it is long enough for students to mas ter the arithmetic processes and short enough to avoid monotony.

5. Requiring students to describe in words the

quantitative processes that they have mastered

gives them a communication skill that is highly valued in both academia and the workplace.

Lessons 2 and 3 similarly allow students to mas ter the financial mathematical processes for annu ities and savings accounts, respectively. Most of the

preceding points apply to these lessons, as well. The culmination of the three lessons is when the

students, who have calculated all the figures them

selves, notice that the final balances are the same for the annuity as for the savings account?both are $8324.83.

These three lessons can be covered in a mini mum of one, or a maximum of three, days. In later

lessons, I have students practice calculating with the mortgage, the annuity and the savings formu las. After that, they have fun: they compare two

mortgages or two annuities. I change one value? the interest rate, the principal (for an annuity, I

change the payment), or the length of time?and then ask students to predict the long-range effect.

Which one comes out greater? The more of these

comparisons that students make, the greater their intuition about financial questions becomes. An entire unit such as the one that I have described can provide a strong basis for financial literacy among our students.

SOLUTIONS Sheet 1

1. See table 2. 2. a) Payment made = $2050.25

MATHEMATICS TEACHER



6) Interest paid = working interest rate bal ance owed

c) Principal paid = payment made - interest

paid d) New balance = balance owed - principal paid e) Balance owed = new balance from previous

month

Sheet 2 1. See table 3. 2. a) Deposit made = $2050.25

6) Interest earned = working interest rate balance

c) New balance = balance + interest earned +

deposit made

d) Balance owed = new balance from previous month

Sheet 3 1. See table 4. 2. a) Interest earned = working interest rate

balance

?) New balance = balance + interest earned

c) Balance = new balance from previous month

REFERENCE National Council of Teachers of Mathematics (NCTM).

"Alan Greenspan Suggests Raising Interest in Math ematics." iVeu>s Bulletin 38 (April 2002): 1. Mr

TABLE 2 Sheet 1, Problem 1

Balance Owed

Payment Made

Interest Paid

Principal Paid

New Balance

Feb. 1 $8000.00 Mar. 1 6029.75

Apr. 1 4039.80

May 1 2029.95

$2050.25 $80.00 $1970.25 $6029.75 2050.25 60.30 1989.95 4039.80 2050.25 40.40 2009.85 2029.95 2050.25 20.30 2029.95 0

TABLES Sheet 2, Problemi

Balance Interest Earned

Deposit Made

New. Balance

Feb.l Mar. 1 $2050.25 Apr. 1 4121.00

May 1 6212.46

$2050.25 $2050.25 $20.50 2050.25 4121.00 41.21 2050.25 6212.46 62.12 2050.25 8324.83

TABLE 4 Sheet 3, Problem!

Balance Interest Earned

Deposit Made

New Balance

Jan. 1

Feb.l Mar.l

Apr.l May 1

$8000.00 8080.00 8160.80 8242.41

$80.00 80.80 81.61 82.42

$8000 0 0 0 0

$8000.00 8080.00 8160.80 8242.41 8324.83

(Worksheets begin on page 492)

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Vol. 96, No. 7? October 2003



FINANCE ASSIGNMENT SHEET 1

Investor lends $8,000 to borrower C at a 1.0 percent monthly interest rate for = 4 months.

According to the mortgage formula, the monthly payment is $2050.25. Create the borrower's schedule for this loan. Assume that makes the loan to C on January 1.

C makes her first payment on February 1 and makes her final payment on May 1.

1. Fill in the blanks. Round off to the nearest cent.

_Borrower's Schedule for a Mortgage Loan_ Jan. 1: $8,000 loaned

Balance Payment Interest Principal New

Owed_Made_ Paid_Paid Balance

Feb. 1 $8,000 _ _ _ _

Mar. 1 _ _; _ _ _

Apr. 1_ _ _ _ _

May 1_ _ _

2. Use the finance terms from the column headings to describe in words how you performed the repeating calculations.

a) Payment made =_

b) Interest paid =_

c) Principal paid =_

d) New balance =_

e) Balance owed =_

From the Mathematics Teacher, October 2003




On the first activity sheet, you created the schedule for an $8,000 mortgage loan to be paid back in four equal monthly payments. With a working interest rate, w, of 1 percent, each payment is

$2050.25.

1. Assume that the lender deposits each payment into an annuity account that has the same 1

percent working interest rate. Create a schedule for this annuity account, and determine the final balance.

Since February 1 is the date that the first mortgage payment is received, it is also the date of the first annuity deposit.

_ Lender's Annuity Schedule_ Interest Deposit New

Balance_Earned_ Made _Balance

Feb. 1 ;_

Mar. 1 _ ._ _

Apr. 1 _ _ ? _

May 1 _ _ . _._

2. Describe in words how you did the repeating calculations. Use the finance terms from the col umn headings to describe how you performed the repeating calculations.

a) Deposit made =_

b) Interest earned =_

c) New balance =_

d) Balance =_





Let's turn our attention now to investor A. A deposits her entire amount, in this case $8000, into a

savings account and lets it grow. Since lent his money on January 1, assume that A deposits her

money on January 1, as well.

1. Create a schedule showing the growth of the savings account. Again use w = 1 percent, and allow it to compound monthly. Compounding occurs when the bank computes the interest earned and adds it to the account, just as if it had been deposited.

Savings Account Record_______

Interest Deposit New

Balance_Earned_Made_Balance

Jan. 1 _ _______

Feb. 1 _ _ $0.00_

Mar. 1 _ _ 0.00 _

Apr. 1 _ _ 0.00 _

May 1 _ _ 0.00 _

2. Use the finance terms from the column headings to describe how you performed the repeating calculations.

a) Interest earned =_

b) New balance =_

c) Balance =_




Documents

The Essential Fairness of a Mortgage Loan: Developing Financial Literacy