08 March 2009

The Simpsons and continuous compounding

It appears the Simpsons have a mortgage that has 37% interest compounded every minute.

For the record, if the interest is compounded n times per year, then their interest wud be (1+0.37/n)n-1 compounded annually. If n = 12 (monthly), this is 43.97% per year; if n = 525,600 (every minute), this is 44.7734426% annually; compare 44.7734615% = exp(0.37)-1 for continuous compounding. In other words, compounding every minute might as well be continuous; the difference is one cent per $53,000 or so, per year.

The difference between every-minute and continuous compounding, at an interest rate of r, is the difference

exp(r) - 1 - (1+r/n)n

where n = 525,600; this is asymptotically r2/(2n). (This actually isn't a great approximation here; the next few terms of the series are reasonably large.)

No comments: