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# Interest and stability (I)

## Summary

## Challenge

## Sound maths

## Stability?

### Meaning of symbols Y

#### Debt is what you owe the bank

#### Debt is principal PLUS paid interest?

#### Text and graphs

### A wrong notion of ‘debt’

Ideas 28 July and 8 August, thinking and writing 17−20 and 22 August 2012

If the mathematics are right, but the definitions, concepts and reasoning are wrong, you will arrive at the wrong conclusions.

The maths can make the conclusions seem convincing, especially to those who are not well versed in mathematics. But then those conclusions can still be wrong.

In a comment on an article by Anthony Migchels, Larry from Pittsburgh pointed me to a mathematical analysis of the stability of a financial system with interest-bearing loans. Its conclusion is that interest makes such systems inherently unstable.

In Larry’s words:

“As long as the amount of new debt
adequately grows, it is possible for the debts to be repaid.
Perpetual growth is impossible to sustain in a finite world
as eventually the amount of new debt will be inadequate to
satisfy the interest claims.

The interest bearing debt money system is nothing more than
a pyramid scheme. There will be defaults as a mathematical
certainty and there will chronic shortages of money.

If you doubt the math,
I suggest that you read Marc Gauvin’s
excellent thesis entitled *“Formal Stability Analysis
of Common Lending Practices and Consequences of Chronic
Currency Devaluation”* –
http://bibocurrency.org/English/Formal Stability Analysis and experiment (final) rev 3.4.pdf
”

(Addition 7 October 2017: in the meantime the document was moved elsewhere (‘/English/’ became ‘/images/pdfdownloads/’). My hyperlink now points to a working location again.)

My intuition tells me that that “Formal Stability Analysis” (hereafter called: Analysis) must be wrong somehow. One reason is having seen the three spreadsheets, mentioned in an earlier discussion (links here), which show small but realistic examples of economies, with interest-bearing loans, and no growth. These examples are fully stable!

Larry’s challenge (repeated by webmaster Anthony Migchels in a later discussion) was that I should show how and why Marc Gauvin’s Analysis is wrong.

In this article and the next ones, I will try to do so.

When I first looked at the Analysis, I was deterred by the maths. I couldn’t quickly grasp what was going on. Being tired and sleepy, I put it aside until a next occasion.

On 8 August I tried again and noticed this:

“The analysis tool that has been used to arrive at the conclusions has been the Z transform.”

I had never heard of Z transform, but Wikipedia told me it is to discrete values, what Laplace transform is to continuous signals and systems.

That was a relief, because Laplace transform is very well known to me! I learnt about that around 1975 when studying to become an electrical engineer.

Laplace transform is very useful in enabling people to interpret capacitors and coils as complex impedances, and then analyse the frequency response and pulse response of a circuit (e.g. a sound filter) by simply using Ohm’s law and Kirchhoff’s circuit laws, instead of differential and integral calculus. So much easier once the transforms have been done.

So I can follow the maths in the Analysis by Sergio Domínguez and Marc Gauvin. And I think their maths are sound. Mathematics is not where they went wrong. But as I said in my summary, if the reasoning is wrong, you can arrive at incorrect conclusions, even with the correct mathematical derivations.

On page 3 of the Analysis, Y is called “Total debt in each period”. Page 4 gives the formula:

*Y _{k} =
P_{k} + R_{1 k} + R_{2 k} =
P_{k} +
(I_{k} − X_{k}) +
(D_{k} − W_{k})
*

That means that if the interest is paid as agreed
(*I _{k} − X_{k}*
equals zero) and therefore the penalty interest
is zero
(

In other words, what is called “Total debt” in that case is always the same amount as the “Principal”.

In my view, that is correct, because the “Total debt” represents what at any moment, the borrower owes the bank. It is the liability of the borrower towards the bank (credit in the borrower’s books, if he has them), and also the claim that the bank has on the borrower (debit in the bank’s books).

Pages 9 and 10 provide a similar description and formulas for a slightly different situation, as explained in the Analysis.

So far, so good. But now comes the tricky part.

On page 5 we are represented a different variable,
namely *y _{k}*.
Note that the variables discussed
previously were all uppercase

This lowercase *y* is also called
“debt”.
It is calculated as:

*y _{k}* =

Similar formulas for slightly different situations (as described in the text) appear on pages 6 and 11.

All those formulas for lowercase *y* clearly
show that **this** interpretation of debt,
contrary to that for uppercase *Y*,
**includes** any interest (regular
interest and penalty or compound interest), even if
the borrower has already paid that interest to the
bank – so he won’t have to pay it again.

In my view, that is an incorrect and misleading way
to use the notion of
“debt”. Also
it is inconsistent with Domínguez and Gauvin’s
previous definitions of uppercase *Y*, also called
“debt”.

Note that the graphs on pages 6 (Figure 1), 7, 8,
11 and 12 (Figures 5 and 6) are all based on lowercase
*y* (which includes interest, whether paid
or unpaid), not on uppercase *Y*, which
includes only **unpaid** interest.

Therefore, the lines in the graphs are always rising, due to the interest. Whether the interest is paid or not, makes no difference.

The text of the Analysis in several places adds to the inconsistency about whether the interest is paid or not.

Page 5, and I quote:

“The hypothesis for this simulation is a loan where only the regular interest is paid at the end of each period. This means that no reduction on the principal of the loan is provided.”

OK, if they want to consider that scenario and it is clearly defined, why not? It’s fine with me. The interest is paid but no redemption takes place. So the principal remains constant. Yet the graph (Figure 1 on page 6) shows a rising line, because the interest, although already paid, is still counted as debt.

The authors don’t hide that, but clearly state it themselves, at the bottom of page 5:

“This means that both principal and interests (already paid) are summed up to give the total value of the debt.”

Then on page 6:

“The hypothesis for this simulation is a loan where no funds are paid at any time, i.e. no principal, regular or penalty interests are paid.”

So here, contrary to the previous situation, the
borrower **does not** pay the interest
due. The line in the graph (Figure 2 on page 7) also
goes up (slightly faster due to the penalty interest),
and this time that is correct, because the unpaid
interest adds to what the borrower owes the bank.

Page 10:

“The hypothesis for this
simulation is a loan where no interest is paid and
no reduction on the principal of the loan is
provided^{11}.”

No interest and no redemption. Same as before, except that the calculation now works with compound interest instead of penalty interest. The line goes up due to the accumulating unpaid interest.

There’s a footnote 11 to the quoted sentence, which reads:

“It is straight forwardly proved that if the interest is regularly paid, principal doesn’t grow and the behavior of the loan is the same as a standard loan (linear growth of the debt)”

All of this reminds me of the famous line from the famous song “Hotel California” by The Eagles:

“You can check out any time you like, but you can never leave!”

Here: “You can pay interest as much as you like, it makes no difference anyway, it is still debt and will continue to grow, no matter what you do or do not do.”

That is simply wrong. An incorrect notion of ‘debt’. This alone makes the whole Analysis invalid.

But there’s more. That’ll appear in a follow-up article.

Copyright © 2012 R. Harmsen. All rights reserved.