How to tell if your economics professor knows anything about interest:
If he thinks I=PRT is sound, he doesn't know
If, using the same rate of interest, he calculates different amounts by
ert, and (1+r)t, he doesn't know
Understanding interest rate accounting
Logarithms and log tables are merely
aids to computation for people lacking
Interest calculations use powers and roots. In
the absence of computers, powers and roots are
easily handled by multiplying and dividing
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
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.
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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1. I=PRT, too simple and WRONG
I=PRT, Interest =
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.
Principal * Rate * Time
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,
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
Principal * Rateperiodic
No timeperiodic enters in.
The Rateperiodic is for timeperiodic
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
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.
Timeperiodic is used in conjuction with a timebase to restate the
rateperiodic on the timebase,
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
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?
Initial balance $, 100
APY (APY is a %), 6
timebase (days), 365
Solutions (in spreadsheet notation):
1. Integer method, powers and roots
Use .06 decimal yield "as is"
endBal = initBal * (1+r)^(time/timebase)
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)
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)
endBal = 108.309255071
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Understanding Compound Interest
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
Measurement of time-dependent
variables, of which interest rates is one,
are only as knowable as the timebase is
A periodic rate can be restated to any
timebase equal to or longer than periodic time.
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.
The "Ohm's law " of interest: I=PR. If you know any
two variables, you know the third.
Interest = Principal * Rate
Compounding, periodic rate:
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
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