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# Interest and stability (II)

## Introduction

## Suitable model?

### Using the model for something else

### Using the model for a loan with interest

#### Inputs

#### Outputs

### Yes, instability can occur

### Examples of possibly unstable situations

#### Not paying the interest

#### Paying the interest, but no redemptions

#### Paying interest and redemptions

## Stable after all

## More

Text 24–25 August 2012, expressing earlier ideas

In part 12 I discussed the document by Sergio Domínguez and Marc Gauvin called “Formal Stability Analysis of Common Lending Practices and Consequences of Chronic Currency Devaluation”. This Analysis states that interest-bearing loans, and therefore a financial system using interest, is inherently unstable.

I don’t think so and tried to show why not. In my previous article I focussed on the mathematics, the meaning of variables and graphs. Now in part 13, I will look at the first model that is used. Is it valid and usable for the topic at hand?

The model that the authors are using (see their introduction and chapters 1 and 2) is suitable, for example, for a (digital or analogue) sound filter and for the shock absorber of a car. Stability is indeed important there.

A sound filter has a electric sound signal as its input, and a modified version of that sound signal as its output. The modification is usually in the frequency response. The filter can be low-pass, high-pass or band-pass, so it favours certain sound frequencies while suppressing others.

You certainly want stability there: zero input (barring some inevitable noise) should produce zero output. The filter’s output should not suddenly go to plus or minus supply voltage for no reason. It shouldn’t generate any frequencies itself, i.e., a filter is not supposed to be an oscillator.

The modelling mathematics that the authors of the Analysis use, are suitable to guarantee the filter’s stability already in the design phase, before it is built.

A shock absorber of a car should leave the wheel in fixed position while the car is standing still. When the car is riding on a smooth surface, the wheel also shouldn’t move. (The wheels are turning, of course, but that is not the kind of motion that a shock absorber is all about.)

On a bumpy road, the shock absorbers, in cooperation with the springs, may let the wheels move, but in such a way that the people in the car don’t notice much of it. The shock absorber should smooth out the bumps and not get out of hand, not continue to let the wheel move after the car is past the bumps.

Here too, the control system theory model can help design a good shock absorber and spring system, already before it is built. Stability is an important factor.

Now back to the subject of the Analysis: interest-bearing loans. There isn’t just one input now, but several: the principal (sometimes misspelled ‘principle’), which is the initial loan amount before any redemption has taken place. There are interest payments, there may be penalty interest if such a thing has been agreed upon, and there are redemption instalments.

Having many inputs need not be a problem, the model can still be valid and usable.

The output is what the authors call “debt”. But they define it inconsistently: see my description here versus the one here.

The authors sometimes look at debt being the resulting principal after any redemption instalments. At other times they include any interest in their notion of ‘debt’, regardless of whether that interest was already paid as agreed, or whether the borrower is in arrears.

I think that’s incorrect and confusing. For clarity, I’d rather consider several different outputs:

Principal, taking into account any redemption that has taken place.

Interest due, that is, interest that the borrower should pay but hasn’t already.

Interest already paid is not an important output of the model, where stability is concerned. Interest paid is over and done with.

Those two outputs, remaining principal, and interest yet to be paid, together constitute the liability of the borrower vis-à-vis the bank. That is the important point: how much do you owe them and when do you have to pay those amounts? (The latter has to do with the maturity.)

And here I agree with the authors of the Analysis: an interest-bearing loan can indeed be unstable and under many conditions will be unstable. However, what matters is: how often do these conditions occur and what can be done to avoid them?

By the way, personally I don’t need maths to see that instability. I can also understand it intuitively. But it’s good to know that doing the maths confirms the possibility of instability.

If you take out a loan and never pay the interest, the total amount of unpaid interest will keep rising. Depending on what has been contractually agreed, and is legally permitted to be agreed, the bank may also charge penalty interest or apply compound interest (also known as anatocism), which makes the instability worse if you don’t pay that either.

Simple solution: do pay the interest.

The bank will encourage that by contractually agreeing that you should pay the interest. If you don’t, the bank will send reminders, more reminders, then dunning letters. Failing that, the bank will take debt collection measures.

In case of a mortgage loan, or a loan with some other collateral, the bank will eventually force you to sell that collateral in an auction (foreclosure) and use the proceeds to redeem part or all of what you owe the bank.

So what we see here is:

Yes, the system (here: loan) is inherently unstable, but:

Mechanisms are built-in (in control system theory terms: feedbacks) in order to promote the odds that the system will actually be stable.

Of course, these stability measures will not be able to avoid any and all problems. But they will be effective in a large number of cases.

To say that interest-bearing loans are inherently unstable because in some cases they can be, and sometimes are, is a wrong and misleading representation of the situation.

(Addition 18 September 2012: Well, loans ARE inherently unstable under a strictly mathematical definition of ‘stability’ in control systems engineering. It’s just that I don’t think such a definition is meaningful in the real world of finance. More on that here.)

This scenario is stable in the sense that the principal remains constant. Of course, the borrower will have to keep paying interest for ever if nothing changes. But if he does, the debt, that is, what the borrower owes the bank, does not increase.

Quite a few people in the Netherlands, where I live, have such a mortgage. The reason is that paid interest is tax-deductible here. At the same time, tax facilities exist for building up capital based on savings, bonds and/or shares. This was often combined (when such mortgages were still popular; they aren’t any more) with a life insurance policy.

If all goes well, at the end of such a mortgage loan’s maturity, the situation becomes even more stable, in that the principal is reduced to zero, which is reason for erasure of the inscription of the mortgage.

Of course, where bonds and shares didn’t perform as well as expected, problems can arise. But even then it is not the interest that has made the situation unstable.

If interest is paid and regular redemption payments are made, the principal will eventually become zero and no more interest needs to be paid. The mortgage inscription will be erased.

Several schemes exist, like annuity mortages, in which total periodic payments (consisting of interest and redemption) are constant, and linear mortgages, in which periodic redemptions are constant and the interest gradually decreases with the diminishing principal.

Interestingly, in the Analysis, on pages 6, 7, 8, 11 and 12, we see those alarming graphs with lines that keep going up and up, illustrating the alleged inherent instability of all interest-bearing loans.

But the calculated table on page 20, showing a typical annuity mortgage, nicely ends with “Balance Loan 1” at zero.

So it has become stable after all.

This article is continued in part 14.

Copyright © 2012 R. Harmsen. All rights reserved.