The Mathematics of Money

452 Chapter 10 Consumer Mathematics

Installment Plan Interest Rates: Tables and Spreadsheets

As we noted early in this section, the realistic interest rate actually being paid on one of

these loans may be easily misunderstood. The Truth in Lending Act requires lenders to

disclose the “actual” interest rate for a loan, generally speaking meaning the interest rate

that would result in the plans’ payments if the interest were calculated in the amortized way

that we have previously seen. This “actual rate” is generally referred to as the APR.

The regulations surrounding calculation of the APR are complex, but we can get a gen-

eral idea from some of the mathematical tools we have already seen.

We noted that the actual rate for Tonya’s piano loan must be much higher than the 8%

nominal simple interest rate. We know the term (2 years), the payments ($87/month) and

the initial balance ($1,800), but calculating the APR for this poses a bit of a challenge.

As we saw in Chapter 5, a problem of this type can be solved by setting up an amorti-

zation table, and then using guess and check methods to find the interest rate that works.

Realistically speaking this is probably the most efficient method to use today. We can find

the rate she is actually paying by setting up an amortization table, using $1,800 as the pres-

ent value and $87.00 as the payment, and then using either guess and check or goal seek to

find the rate that would have the payment killing off an $1,800 debt in 24 months.

Example 10.3.5 Calculate the actual interest rate Tonya is paying on her piano

installment plan, using a spreadsheet amortization table like the one here.

Rows Omitted

26 24 $87.00 $1.04 $85.96 -$0.66

25 25 $87.00 $2.08 $84.92 $85.30

1 Rate: 14.65% Initial Balance: $1,800

2

3

A B C D

Month Payment To Principal Ending Balance

1 $87.00 $65.02 $1,734.98

To Interest

$21.98

4 2 $87.00 $21.18 $65.82 $1,669.16

5 3 $87 00 $38 38 $66 60 $1 603 54

E

When the Truth in Lending law was passed, though, personal computer technology was not

what it is today, and that sort of solution was not a realistic option. One option is to find

the rate using tables. The idea behind this table approach rate is to find the present value

annuity factor implied by the present value and payments. For example, for Tonya’s loan,

we would start with the formula:

PV
PMT a _n (^) |i
Then, we would plug in the known values of PV and PMT, and solve for a _n (^) | (^) i.
$1,800
$87a _n (^) |i
a _n (^) |i
20.68965517
The next step would be to go to a table of annuity factors, looking at the annuity factors for
various different interest rates with n
24. The interest rate that gives the closest match
to the annuity factor we calculated is then determined to be the actuarial rate, since using
that rate with its annuity factor would produce the same payments.
While tables are not much used today, an excerpt from such a table is shown below as
an illustration.
ANNUITY FACTORS BY RATE AND NUMBER OF PAYMENTS
14.50% 14.55% 14.60% 14.65% 14.70% 14.75%
n
24 20.72563428 20.71546249 20.70529779 20.69514016 20.68498961 20.67484613
Since the factor shown for the 14.65% rate is the closest we can get to the annuity factor
we are looking for, we would conclude that this is the APR.