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Copyright © 2002, Joel Anderson
The Limits of Analysis
- or -
Some fictions are more ridiculous than others

Men in general are more affected by what a thing appears to be
than by what it is, and are frequently influenced
more by appearances than by the reality.

- Machiavelli

An account is best described by the amount and timing of its flows.

Only accounts with flows at equal periods, at a constant rate, are well described by a single number - typically, savings and loan accounts. All other accounts, including simple interest accounts with long periods (simple interest is the basic building block of compound interest accounts), are not well described by a single number.

I=PRT's logically dubious method, using time-geometric, can extract an annual rate from a simple interest rate with a period longer than one year.

Discounted cash flow [DCF], using time-exponential, can extract an annual rate from a simple interest rate that has a period longer than one year. Do it at your own risk: a simple interest account's flows and timing better describe the account.

DCF treats simple interest as a two-flow problem, an in-flow and an out-flow. DCF can provide an annual rate (though a rate can be calculated on any timebase), for theoretical flows, at timebase-time, compounded annually. The problem with a DCF analysis of simple interest is that simple interest has no flows at timebase-time. You can't dine out on theoretical flows.
A discounted cash flow two-flow annual-rate extraction:
Given:
I=PR, a simple interest account (i. e., one period)
$100 principal
10-year period
100% rate


Discounted Cash Flow, 2 flows (spreadsheet notation)
APY = 100*((outFlow/inFlow)^(timebase/timeBetweenFlows)-1)
APY = 100*((200/100)^(365/3650)-1) = 7.177346


Reverse-engineer the DCF (compound interest):
inFlow*(1+%RateAnnual/100)^(time/timebase) = outFlow
  $100*(1+7.177346/100)^(3650/365) = $200


Notes:
1. A rate can be stated on any timebase.
2. Time and timebase must be stated in the same units.
Just because you can reverse-engineer a simple interest rate to an annualized compound rate, that doesn't mean that you have a useful description of the simple interest rate account. The real flows for $100 at 100% interest, 10-years, are very different from the hypothetical flows for $100 at 7.177346% interest compounded annually. There is no compounding in a simple interest account. The 7.177346% rate is a simulation, not a representation, of reality.

Men have become tools of their tools.   - Thoreau