What is the Definition of APR?

As I understand it, it depends on the compounding rate. Investments are generally express as APR, but actually mean R_yearly/12 and are compounded monthly. For example that $100 would go to 101.67, 103.36,

105.08... 121.94.

I think some loans are compounded daily which would be very close to R_continuous.

[...] [...]

There is no clear answer.

APR is the annualized rate without compounding, so it depends on the period of the interest, and the calculation method! If you invest $100 at

20% APR where the interest is credited once per year, you'll have $120 at the end of the year. If you invest $100 at 20% on a 365/365 calendar (where interest is calculated daily, based on 1/365 of the APR), then you'll have (assuming I used the calculator correctly :-) $122.13. Less than continuous compounding, but close.

Another common calculation is 365/360 compounding. Divide the APR by 360 (12 "standardized" 30-day months), but compound daily. Back when savings banks' interest was regulated (before the 1980s), they offered APRs of

5.25%. But you would usually get 5.39% yield (365/365 calendar) or 5.47% (365/360 calendar).

This is why banks are required to advertise annual percentage yield (APY) - the equivalent of interest credited once per year.

(APR vs. APY) Mark Freeland snipped-for-privacy@sbcglobal.net

It's worth pointing out that APR for mortages is a different beast from APR/APY for bank deposits. This is where finance *really* diverges from math.

APR for mortages *is* a standardized number. But it includes so much that you may not recognize it. APR is for mortgages what APY is for banks (the rate that you would be charged if you paid your mortgage once per year, e.g. a $100K mortgage at 6% would result in $6K of interest if you paid at the end of the year).

But ... The APY attempts to incorporate interest that goes beyond the basic interest due from the outstanding loan. It incorporates points (which the IRS considers interest, just paid up front), some loan fees, and mortgage insurance.

HUD's definition: A measure of the cost of credit, expressed as a yearly rate. It includes interest as well as other charges. Because all lenders follow the same rule to ensure the accuracy of the annual percentage rate, it provides consumers with a good basis for comparing the cost of loans, including mortgage plans.

If you want the full and gory details (I've never looked through them): "Appendix J of Reg. Z contains the formulas and instructions. You cannot do these calculations manually, a computer is required." (It has a link to Reg Z, containing Appendix J: ) Mark Freeland snipped-for-privacy@sbcglobal.net

As wikipedia points out there is another important facter than just the compounding period. One time fees such as points are taken into consideration to figure out your equivalent cost of borrowing (APR). A "no points" loan of 6% is quite different than a "2 points" loan at

6%. APR is a better "apples to apples" comparison than just rate.

Actually, for a given year, the APR is not equivalent to the ln R_(t))/ ln R(t-1) -as you imply- . APR is the effective interest rate, adding fees and expenses (in the case of loans). The effective interest rate depends on when the interest accrues, and it is closer but not equal to the interest rate found by applying logarithms. When you apply the exponential function to a given amount (and the logarithm to calculate the implicit APR) you are assuming that the interest accrues continuously. That's not the case of the APR.

The general formula to calculate the effective interest rate is r (1+i/n)^n - 1, where i is the nominal rate, and n is the number of periods in which the interest accrues. In your example, assuming that the interest accrues every 12 months, the nominal interest rate of 20% implies a nominal interest rate of 1.666667 % per month (20%/12), and the APR would be r = (1+1.666667%)^12 -1 = 21.93%, which differs from the exp(0.2) = 22.14% that you calculated.

The way I think of it is this: APR is the number which, in combination with the compounding rules (schedule), is sufficient information to define the interest that will actually be paid/owed. Neither APR nor compounding rules by themselves are sufficient information to define the amount of interest that will be paid. However, the two together are sufficient.

APR is definitely not the same thing as R_continuous. R_continuous implies a set of compounding rules (specifically, that compounding is done continuously). APR implies no compounding rules.

The APR is expressed in annual terms. It is pro-rated across the compounding interval(s). For example, if the compounding rules specify that interest is compounded every 30 days, then the interest paid for a 30-day interval is APR*30/365. If the compounding rules specify that interest is paid on the 1st of every month, the interest paid could in theory be APR*28/365 one month and APR*31/365 another month. In the case of continuously-compounding interest, it's probably easier not to think in terms of an infinite number of prorated interest amounts over and infinite number of infinitely small intervals, although there probably is a mathematically valid way to do it. :-)

The big gray area in my mind is really the definition of "annual". As far as I know, there is no fixed mathematical definition of "year". Some years are 365 days and some are 366. The length of an average year is a physical phenomenon and can have no precise mathematical foundation. Maybe the law has solved this problem by making a fixed definition of "year" for certain purposes and in certain jurisdictions.

The caveat with all of the above is that I'm neither an expert on math or financial matters. :-) So, if I have any wrong assumptions, it would actually be nice to hear about that.

- Logan

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