Find formula for average compound interest rate given 2 discrete interest rates

Question

For an example, have p=.4 for an interest rate of .1 and $p = .6$ for an interest rate of $-.05$, It seems to me that the average interest rate should be $.1(.6) -.05(.6) = .01$, but when I plug the numbers into excel with $100$ data points I get an average interst rate of about .00737. What am I doing wrong?

Answer 1

Compound interests are not additive.

$$p_1(1+r_1)^n+p_2(1+r_2)^n\ne(1+p_1r_1+p_2r_2)^n.$$

The equivalent rate after $n$ periods would be

$$\sqrt[n]{p_1(1+r_1)^n+p_2(1+r_2)^n}-1.$$

It is not constant, as it depends on $n$.

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With simple interests the weighted average works:

$$p_1(1+nr_1)+p_2(1+nr_2)=1+n(p_1r_1+p_2r_2).$$