Amortization with daily periodic rate - Loan_365.WK1

How lenders exploit Truth-in-Lending's fundamental weakness^[1]

Lenders regularly maximize the division of APR (Annual Percentage Rate) to the periodic rate.

A larger APR divisor means more interest income for lenders while preserving the same nominal APR, the rate they quote to you.

Why stop at 365? An APR divisor of 10,000 adds almost nothing to the interest paid on a typical loan Legally, a lender could observe the number of hours (minutes or seconds) between payment dates and base a loan's periodic rate on the number of hours (minutes or seconds) between dates. The advantage of more frequent division of APR (to the perPeriodRate, pPR) is that it permits a lender to receive more interest on the same nominal APR than the loan would yield with fewer APR divisions. Since the return from large divisors is small, no lender yet finds it worthwhile to antagonize regulators and borrowers with a scheme to lend and charge interest by the hour. Though you can be sure, were the return significant, lenders would would find a way to overcome objections.

Loan payments are calculated using the periodic rate, APR divided by the number of periods. Lenders are not required to disclose the periodic rate.

APR fall short of being entirely useful to borrowers. APR is only sort of an 'executive summary' of the periodic rate. It isn't really useful because it doesn't disclose the periodic rate, the rate on which the loan is based.

Absent an amortization table, a list of all the loan's dates and payments, it's difficult to determine the true rate of interest on a loan. Ask for an amortization table for any loan you're considering. If your lender is reluctant to provide an amortization table?

As a practical matter, given the number of possible variations, absent a loan's amortization table it's impossible to reverse engineer a loan to its rate in finite time (say 8 hours) from APR. It may be possible to come close.

This is legal

If a lender observes the number of days (i. e., the number of days in a month), they can base a loan's periodic rate on days (rather than the 'month' which could be 28, 29, 30, or 31 days long), and the loan's payments based on days between monthly payments. Loans payments slated to be paid on a particular day of the month are regularly adjusted for a shorter (or longer) time to the first payment.

365 (366 in leap years) is about the largest practical APR divisor. Any divisor larger than 365 (366) would be difficult to justify. See sidebar.

---

NOTE: Truth-in-Lending's APR is a nominal rate.
    ''Nominal'' means ''in name only,'' that is, not a true rate.
      APR is the calculation lenders could agree upon at the time Truth-in-Lending was legislated.
      APR is adjusted by time-geometric, (division and multiplication), and by time-exponential (powers and roots). True rates are adjusted only by time-exponential (powers and roots). APY (Federal Reserve Reg. DD, Truth-in-Savings) is a true rate.
      The difference between rate definitions; a nominal rate (APR on loans), and a true rate (APY on savings); makes it possible were there no overhead for lenders to borrow savings and lend them out at the same stated rate and make money. «back»

---

Spreadsheet

Download spreadsheet  

A lender-optimized 365-divisor days-between-dates loan

Initialize loan:
Amount	10000.00
APR, nominal, decimal	0.13	<- "13.000% Annually"
Term (months)	48
Pmt, monthly [a]	268.84	<- from lender's table
First Period's interest	110.41	<- calculated
Last Pmt [b]	235.40	<- from lender's table

a] The monthly Pmt amount is arbitrary. It is designed to over repay (on a straight amortization basis), then close-out the loan with a smaller last payment.

Calculate the periodic rate (perPeriodRate, pPR):
    NOTE: This loan uses a daily perPeriod Rate (pPR).

    pPR(daily) = APR/100/365 = 13/100/365 = 0.000356

        From the pPR, it follows:
    APR = pPR*365*100 = (.000356*365*100) = 13
    Yield, %      =      100*((1+0.000356)^365-1) = 13.88020

This amortization table uses days-between-dates:
na=not applicable

Pmt_#	Date	Days	Pmt	IntPd	PrinPd	PrinBal
Init	10/15/1997	na	na	na	na	10000.00
1	11/15/1997	31	268.84	110.41	158.43	9841.57
2	12/15/1997	30	268.84	105.16	163.68	9677.89
3	1/15/1998	31	268.84	106.85	161.99	9515.90
4	2/15/1998	31	268.84	105.07	163.77	9352.13
[. . . table is truncated for online viewing . . .]
46	8/15/2001	31	268.84	8.36	260.48	496.26
47	9/15/2001	31	268.84	5.48	263.36	232.90
48	10/15/2001	30	235.40	2.49	232.91	-0.01 [b]

b] The balance doesn't end in zero. It should. But it doesn't. The spreadsheet accurately reproduces an amortization table furnished by a lender for a real loan. Why make things up when reality is more fun?