govtwork / loans / APR, Truth in Lending's fundamental weakness

Copyright 2001, 2002 Joel Anderson

Amortization with daily periodic rate - Loan_365.WK1

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

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 hours (minutes or seconds) between dates.

The advantage of more frequent division of APR (to pPR, the perPeriodRate) is that it permits a lender to receive more interest on the same nominal APR than the loan would yield with fewer APR divisions.

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

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

This is legal

If a lender observes the number of days (e.g: in a month), they can base a loan's periodic rate on days (rather than months), and payments on days between monthly payment dates.

365 is about the largest practical divisor. Any divisor larger than 365 would be difficult to justify - no one borrows by the hour. Since the return from large divisors is small, no lender finds it worthwhile to antagonize regulators and borrowers with a scheme to charge interest by the hour. Though you can be sure, were the return significant, lenders would would find a way to overcome objections.


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

Initialize loan:
  Amount                      10000.00
  APR, nominal dec.               0.13   <- "13.000% Annually"
  Term (mo)                         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 perPeriodRate (pPR):
  NOTE: This loan uses a daily pPR, perPeriod Rate.

  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    Day     Pmt    IntPd   PrinPd    PrinBal
           Init 10/15/97            na       na       na   10000.00
              1 11/15/97   31   268.84   110.41   158.43    9841.57
              2 12/15/97   30   268.84   105.16   163.68    9677.89
              3 1/15/98    31   268.84   106.85   161.99    9515.90
              4 2/15/98    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?


  1. Truth-in-Lending's APR is a nominal rate. "Nominal" means "in name only." 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 by time-exponential, powers and roots, only. APY (Federal Reserve Reg. DD, Truth-in-Savings) is a true rate. The difference between 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 at the same stated rate and make money. «back»