Compounding

With $1,000 and a 10% annual periodic rate, Year 1 growth is $100 and the new balance is $1,100. Year 2 growth is $110 because the rate applies to $1,100, not only to the original $1,000. The two-year total is $1,210:

$1,000 × 1.10 × 1.10 = $1,000 × (1.10)^2

The exponent counts intervals. It is not a date label, a calendar year, or a number copied from the story without a timeline.

The rate must describe the same interval as the exponent. A monthly rate pairs with monthly periods; an annual rate pairs with annual periods. Converting only one side changes the economic assumption while leaving the formula's shape unchanged, so the result can look orderly and still be wrong.

More frequent compounding under the same positive nominal annual quote generally increases effective annual growth because accumulated amounts enter the base sooner. That comparison requires the same principal, horizon, quoted rate convention, and absence of fees or other changes. It does not rank real products whose terms differ.

Compounding moves value toward a later date. Discounting applies the reciprocal factor to move it back. Recomputing the original amount after both operations is a strong internal control.