Taylor Series and the Argentine Peso

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 + i^2

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 ofi^3, 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.

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