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Copyright © 2002, Joel Anderson

How to tell if your economics professor knows anything about interest:

If he thinks I=PRT is sound, he doesn't know anything.[1]

If, using the same rate of interest, he calculates different amounts by ert, and (1+r)t, he doesn't know anything.[2]


Understanding interest rate accounting

Logarithms

Logarithms and log tables are merely aids to computation for people lacking computers.

Interest calculations use powers and roots. In the absence of computers, powers and roots are easily handled by multiplying and dividing logarithms.

By either method, logs or integers, the results are identical. If different, one of the methods is wrong. Invariably, it is the method using logarithms. The common mistake is that the log-user neglected to convert the stated rate into a log-rate suitable for use with the log-base in question.

Logarithms to base-e or base-10, or any other log-base the user chooses (there are an infinite number of log bases), can be used to make the calculation. Done correctly, the log calculation will always agree with the integer answer to (1+r)time.

This is interest:

Interest is real money paid into, or out of, an account. Interest is either paid and exists, or isn't paid and doesn't exist. There is no intermediate state.

This is the simple rate of interest:

The rate of simple interest is calculated by dividing the amount of interest paid, by the amount of principal on which interest is being paid. The interest can be paid into the account, added to principal, in which case it earns further interest; or paid out of the account to the account owner as cash in hand.

Note: the simple interest rate has nothing to do with time (i. e., When interest is paid).

Rate conventions. Here's were the confusion starts:

The rate of interest, as we commonly know it, is the conventional restatement of periodic simple interest to a timebase.

The problem with rate conventions is that they mire the borrower or saver who wants to understand interest in the minutiae of the conventional restatements. In fact, most bankers do not know how their bank calculates interest.

Yes, bankers know in principle how interest is calculated, they can do it on their pocket financial calculators. But they don't know how their bank calculates interest on your account. The only person who knows how the bank calculates interest is the computer programmer somewhere in the bank's fifth sub-basement working on the bank's interest rate algorithm.


Reverse engineering

The bank starts out knowing where the Aces are, it's up to you to find them.

On savings, the bank works forward from the periodic simple interest rate to a statement of APY on the conventional (rubber) timebase.

Though the APY could be stated to any degree of accuracy, the rate is after all the result of a mathematical calculation, the Federal Reserve's standard for accuracy is less accurate than gas at the pump.

The saver works backward from the conventional statement of APY on the (rubber) timebase to the periodic simple interest, the actual interest paid per period. Reverse engineering the amount paid in any given period from the stated rate is a non-trivial task.

You could ask your bank how they calculate interest on your account. Legally, they don't have to tell you. They're only required to disclose a rate according to the applicable law. Most of them couldn't tell you even if they wanted to. They simply don't know.


The USA has two rate conventions: APY, a true rate embodied in Truth In Savings (Federal Reserve Regulation DD) and, APR, a nominal rate embodied in Truth In Lending (Reg Z). APR is a rate in name only, that is, it's called a rate but its not really a rate. APR is what the banks could agree upon when the enabling legislation was being written in Congress. It results in borrowers being charged more interest than they would under APY, a true rate. Were banks 100% efficient, had they no overhead and deployed all their savings as loans, the difference between the APR convention and the APY convention would permit banks to borrow money from savers and lend money to borrowers at the same rate - and still make money. Such is the magic of a powerful lobby.

A problem with both APR and APY is that both can (legally) have a "rubber" timebase, that is, the timebase can be 365 days, or 366 days in a leap year. Many banks adhere to a 365-day timebase, leap year or not. It saves arguing with customers who notice that they're receiving less interest on the same stated rate APY. When $billions are on deposit, there is a financial inducement to change to a 366-day timebase during a leap year.

A "rubber" timebase. Could you build a house not knowing if a foot was 12 or 13 inches? If you don't know the duration of the timebase, then you don't know what the stated rate means in practice, in either APY or APR.

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Notes:

1. I=PRT, too simple and WRONG

I=PRT, Interest =
Principal * Rate * Time

I=PRT has a logic flaw. The formula requires a limbo state where interest accumulates while time marches on, an intermediate state between interest in the account earning interest, and interest paid out as cash in hand.

Under I=PRT, the interest resulting from P*R is neither paid into the account, in which case it would be compounded; nor is it paid out of the account, in which case it wouldn't be there at the end as part of a lump payout - which is what the formula predicts. I=PRT is bogus.

The simple interest formula is I=PR,

Interestperiodic =

      Principal * Rateperiodic

No timeperiodic enters in.

The Rateperiodic is for timeperiodic

On $100 principal, if $100 interest is paid at the end of 10 years, that's simple interest at 100% on a 10-year period coincident with a 10-year timebase.

The 100% 10-year rateperiodic cannot be restated as an "annual" rate. It is not a 10% rateperiodic for 10 years as I=PRT would have it; not $100*.1*10=$100

When interest is paid determines both Rateperiodic and timeperodic

Timeperiodic is used in conjuction with a timebase to restate the rateperiodic on the timebase, (1+Rperiodic)(timebase/timeperiodic)-1.
In the I=PRT example, interest is paid at 10 years. And 10 years is the timebase. The correct way to state an interest payment of $100 on a 10-year period and a 10-year timebase is I=PR, $100*100%=$100, the formula for interest at the Rateperiodic is I=PR.

In the I=PRT example, there is no "annual" interest paid, there is only a single payment at 10 years. This is simple interest. The simple interest period begins when the account is initialized and ends when interest is paid into, or out of, the account - here, out, at 10 years.

Wrap your head around this: simple interest can only be stated on a timebase that is equal to or longer than the simple interest period. Until interest is paid, the simple interest rate, R=I/P, is zero. Whomever invented I=PRT didn't understand this.

See: The Limits of Analysis



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2. Continuous compounding,    ert    vs    (1+r)t

Logarithms are an easier way to do interest-rate math using look-up tables when you don't have access to a computer or pocket calculator. Using logs on a computer or pocket calculator is like installing a fireplace in Hell. If you have the computational power, why not keep things simple and use it?

But if you do use logs on a pocket or desktop computer, and you get an answer that is different from an integer calculation using powers and roots, it's your log calculation that's wrong. Logs base-e, base-10 (and base-3, base-8, and base-42.6 just for fun), all provide identical results.

What's usually forgotten in interest computations using logs is that the stated rate of real world (integer world) interest must be converted to a log-rate matching the base of the system of logs in use.
E. g: an APY interest rate calculation solved using both integers and logs:


What is the end balance for this account after 500 days?

 Given:
      Initial balance $,  100
       APY (APY is a %),    6
        timebase (days),  365
                   time,  500

Solutions (in spreadsheet notation):

1. Integer method, powers and roots

      Use .06 decimal yield "as is"

      endBal = initBal * (1+r)^(time/timebase)
             = 100*(1+.06)^(500/365)
      endBal = 108.309255071

2. Logarithms, base "e" (convert rate to an eRate)

     Convert .06 decimal yield to an equivalent eRate:
      eRate = @LN(1+0.06) = 0.0582689081

     endBal = initBal * e^(eRate/time*timebase)
            = 100 * 2.7182818285^(.0582689081/365*500)
            = 100*@EXP(1)^(@LN(1+0.06)/365*500)
     endBal = 108.309255071

3. Logarithms, base 10 (convert rate to an base10Rate)

     Convert .06 yield to an equivalent base10Rate:
     base10Rate = @Log(1+0.06) = 0.0253058653

     endBal = initBal * 10^(base10Rate/time*timebase)
            = 100 * 10^(.0253058653/365*500)
            = 100*10^(@LOG(1+0.06)/365*500)
     endBal = 108.309255071

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Understanding Compound Interest

Timebases

Measurement of time-dependent variables, of which interest rates is one, are only as knowable as the timebase is stable.

A periodic rate can be restated to any timebase equal to or longer than periodic time. Prior to Truth in Lending, and Truth in Savings, many timebases were in use, commonly 360-days, 364-days, 365-days, 366- days. Prior to the passage of Truth in Lending there were over 6000 ways used to calculate interest rate disclosures. The measurement of interest in the USA is still legally measurable on a rubber timebase, 365 or 366 days.

E. g: RPM, Revolutions Per Minute. The "minute" is the timebase. Would you know how fast a shaft was turning if the minute was defined as 59 or 61 seconds? Rotational speed on different timebases, revolutions per second, revolutions per week, are readily converted to RPM. Interest rates can be converted from one timebase to another by time-exponential.

Understanding compound interest is predicated on knowing the periodic rate, and the timebase, and the mathematics of the conventional restatement(s) of the periodic rate to the timebase.

The "Ohm's law " of interest: I=PR. If you know any two variables, you know the third.
Simple interest:
    I=PR
    Interest = Principal * Rate

Compounding, periodic rate:
    R=I/P
    Rateperiodic = Interestpaid/Principal.

When interest is paid, and the amount of interest paid, determines both the Rateperiodic and the Timeperiodic

Interest can be paid into an account, or out of an account.

A Rateperiodic can be adjusted to any timebase equal to or longer than the timeperiodic by time-exponential.
See: The Limits of Analysis

(1+rperiodic)(timebase/timeperiodic)-1, will adjust the rateperiodic to the timebase.

A rate stated one one timebase can be moved to another timebase by time-exponential.

How much time in a timebase?. Timebases are user defined. When users are lenders bent on deception? that's part of the problem. A timebase can be any length of time the user chooses; 1 second? 365 days? 366 days? 1 million years?