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Please explain the apr formula used by the Fed. Reserve -...

<< Back to: sci.math FAQ: Formula for Compound Interest

Question by rick
Submitted on 4/13/2004
Related FAQ: sci.math FAQ: Formula for Compound Interest
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Please explain the apr formula used by the Fed.
Reserve - regulation Z

Answer by jawinterink@juno.com
Submitted on 5/2/2004
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Dear Rick,
This is an attempt to explain the APR formula used by the Federal Reserve - Regulation Z:

Amount financed equals the sum of present value of the scheduled payments.

Formula involves the use of arithmetic and algebra.

Johan de Witt wrote “Waerdije van lyfrenten naar proportie van losrenten” ‘s Gravenhage 1671.
This is the first paper on annuities. An English translation of de Witt’s paper exists in PDF format and can be obtained: 1. Type http://www.google.com 2. in the search block type Treatise of life annuities

John D. Smart, in 1726, calculates interest for fractional periods, using the compound interest method.

Regulation Z, Appendix J,  http://www.federalreserve.gov/regulations/#z

Note, interest for a fractional period is calculated using the simple interest method.
Now, "Regulation Z Appendix J (6) Complex single-advance transaction. Example (ii) Skipped-payment loan plus single payments.
A loan of $7350 on 3-3-78 ...", requires using a formula which yields an accurate periodic interest rate i=0.008512869954918.
Substitute this value for i in the four rational expressions on the right side of the equation and show that their sum is approximately 7350.0000000000.

Note, the Truth in Lending Act states: accurate value i "will yield a sum equal to the amount of the finance charge when it is applied to the unpaid balances of the amount financed, calculated according to the actuarial method of allocating payments made on a debt between the amount financed and the amount of the finance charge, pursuant to which a payment is applied first to the accumulated finance charge and the balance is applied to the unpaid balance financed." However,
the periodic interest rate i=0.008512869954918 does not permit the amortization of $7350 loan mentioned above. Hence, Regulation Z’s  prescribed formula violates the law.

A United States Military education manual issued to me during the Koren War uses the formula required by TILA. Also the European Union requires the use of the formula required by The Truth in Lending Act to calculate the annual percentage rate.


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