Negative interest rates

Echonomists have been very busy lately talking about how somehow, somebody now has negative interest rates and how OMFGWTFBBQ!!!111!!! this means something or other and has totally surprised someone or other.  If you are confused by any of this, allow me to spell it out for you.

In point of fact, all fiat currency is issued at an effective interest rate of negative infinity.  What someone charges you to borrow currency from them is another matter.  When you borrow fiat, typically it is from somebody who borrowed from somebody else who also borrowed.  How much you pay for the new units roughly defines your status in a hierarchy of recipients of these units.  This hierarchy has also been described as "interest rate apartheid".

An example or two might as usual help elucidate things further.

If we browse the darknet for newly printed euros, shipped directly to your home, we find several purveyors.  Because we expect low quality (if anything at all) from these folks, they offer us the notes at a discount.  While a standard euro might go for four or five millies, these euros sell for one or two millies each.  And why not?  After all, these notes cost their manufacturers (a.k.a. counterfeiters) almost nothing to issue.  It makes sense they can still make some profit even passing discount on to us.  Thus for a shipment of 100 euros to our house, we repay with 50 euros.  The repayment in full, for an amount less than the original amount of the loan, represents a payment with an effective negative interest rate.  The business described here is exactly what is happening on a larger scale with all fiat currency purveyors.  They can still make really good money with a negative interest rate, because they are paying an original rate of negative infinity percent.  Why would anyone be surprised that they can charge a negative interest rate?

Recall the formula for the amount owed on a loan of principal P compounded continually:

 A=Pe^{rt}

Where A is the amount owed, P is the principal, r is the interest rate, and t is the time passed.  So for A to be zero, that is for you to not owe anything (at any time t and for any principal P), r must be equal to negative infinity.

So we can see, a negative interest rate is not so strange after all.  It's what the sun charges you for UV light.  An infinitely negative interest rate is how we say gratis in the language of usury.  And so we see clearly that all fiat currency is in fact issued at an infinitely negative interest rate.

Unfortunately this post is rather useless in the department of actionable advice.  If somebody is charging you 20% or 200% interest, and you are under contract, you are still going to be under that contract.  You will pay up or go bankrupt.  If somebody is charging you -10% and you are under contract, you will also have to pay up.  How much are people issuing to themselves at r=-\infty?  I don't have a clue.  However it looks like we will find out eventually.  That is another tale for another day.