How this calculator works
The balance ratio determines an annual growth factor. The selected compounding frequency translates that factor into a nominal quoted rate; changing the frequency leaves the effective annual return unchanged. A surplus can be positive or negative provided the final balance stays positive. Deposits and withdrawals would distort this two-balance rate.
Formula
j = m[(F/P)^(1/(mt)) - 1]; effective rate = (F/P)^(1/t) - 1. Under continuous compounding, j = ln(F/P)/t.
Worked example
$10,000 growing to $12,100 over two years implies 10% nominal with annual compounding. With $100 becoming $144 in two years, the annual effective rate is 20%.
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Frequently asked questions
Why does the nominal rate depend on frequency?
More compounding periods need a lower nominal quoted rate to create the same effective annual growth.
Can the surplus be negative?
Yes. A final balance below the initial balance implies a negative compound rate.
What if the final balance is zero?
There is no finite continuously compounded rate for a total loss. This solver requires a positive final balance for consistent frequency comparisons.