A few days ago I wrote about now  hyperinflation changes everything. I’d like to follow up with another example of hyperinflation breaking implicit assumptions.

Something I was reading this week gave the approximation for present value

v = 1/(1 + i) ≈ 1 − i + i²

This implicitly assumes that the interest rate i is fairly small. The approximation comes from expanding 1/(1 + i) in a power series. The error in the truncated series is on the order of i³, which is negligible when, say, i = 0.1.

The approximation is absurd under hyperinflation when i > 100. Even for a high rate of inflation less than hyperinflation, the approximation is still bad.

When Javier Milei became president of Argentina in December 2023, the annual rate of inflation that year was 211%.

If you used the approximation above with i = 2.11, you would estimate that that the value of an Argentine Peso (ARS) would triple in value rather than being cut in third. You would predict deflation rather than inflation.

At the time of writing the annual inflation rate in Argentina is 39%. If you used the approximation above, you’d calculate the present value of 1 ARS a year from now to be 0.76 ARS when the actual value is 0.72 ARS. The approximation isn’t very accurate since it implicitly assumes an interest rate less than 39%, but even so it’s not too bad.

My two latest blog posts have been about compound interest. I gave examples of interest rates of 6% up to 18% per year.

Hyperinflation, with rates of inflation in excess of 50% per month, changes everything. Although many economists accept 50% per month as the threshold of hyperinflation, the world has seen worse. Zimbabwe, for example, faced 98% inflation per day. With hyperinflation lots of implicit assumptions and approximations break down.

One take away from the previous posts is that at moderate interest rates, the frequency of compounding doesn’t make that much difference. A nominal interest rate of 12%, compounded continuously, is effectively a 12.75% interest rate. Compound less than continuously, say monthly or daily, and the effective rate will be somewhere between 12% and 12.75%.

But now say interest is 50% per month. Simple interest would be 600% a year, but nobody would accept simple interest. Compounded every month, the effective interest rate would be 12975%. Compounded daily it would be 38433%. And compounded continuously it would be 40343%.

In the numerical post, I said that the difference between continuous interest and interest compounded n times per year was approximately r² / 2n. That works fine when r is say 0.12. When r = 10 it’s off by two orders of magnitude. The implicit assumption that r² < r breaks down when r > 1.

Hyperinflation causes unusual behavior, such as paying for a meal before you eat it rather than afterward, because by the end of the meal the price will be appreciably higher.

What I find hardest to understand about hyperinflation is that people continue to use hyperinflated currency far longer than I would imagine. Once you start using a clock rather than a calendar when doing interest calculations, I would think that people would abandon the inflated currency in favor of something harder, like gold or silver, or even cigarettes. And eventually people do, but “eventually” is further out than I would imagine. It’s absurd to haul paper money in a wheel barrow, and yet people do it.

The previous post looked at the difference between continuously compounded interest and interest compounded a large discrete number of times. This difference was calculated using the following Python function.

    def f(P, n, r) : return P*(exp(r) - (1 + r/n)**n)

where the function arguments are principle, number of compoundings, and interest rate.

When I was playing around with this I noticed

    f(1000000, 365*24*60*60, 0.12)

returned a negative value, −0.00066. If this were correct, it would mean that compounding a 12% loan once a second results in more interest than continuous compounding. But this cannot happen. Compounding more often can only increase the amount of interest.

The problem with the Python function is that when n is very large, exp(r) and (1 + r/n)ⁿ are so nearly equal that their subtraction results in a complete loss of precision, a phenomenon known as catastrophic cancellation. The two terms are indistinguishable within the limits of floating point calculation because the former is the limit of the latter as n goes to infinity.

One way to calculate

exp(r) − (1 + r/n)ⁿ

for moderately small r (such as typical interest rates) and very large n (such as the number of seconds in a year) would be to write out the power series for exp(r), expand (1 + r/n)ⁿ using the binomial theorem, and subtract.

Then we find that

exp(r) − (1 + r/n)ⁿ ≈ r² / 2n

is a good approximation.

When n = 365 × 24 × 60 × 60 and r = 0.12, the approximation gives 2.28 × 10⁻¹⁰ and the correct result is 2.57 × 10⁻¹⁰. The approximation is only good to one significant figure, but it gets the sign and the order of magnitude correct. You could retain more series terms for more accuracy.