Compounding works both ways: returns, losses and the path between

The base changes after each period

Start with a hypothetical $1,000 and apply two annual gains of 10%. The first year ends at $1,100 and the second at $1,210 because the second gain applies to the larger balance. Adding the percentages to get $1,200 misses the return earned on the first year’s gain.

With no contributions or withdrawals and a constant annual rate r for n years, the simplified formula is principal multiplied by (1+r) to the power n. Actual investment returns usually vary. The constant-rate formula is useful for exploring scenarios, but a smooth line from a calculator should not be presented as a realistic year-by-year market forecast.

A loss needs a larger percentage recovery

A 50% decline takes $1,000 to $500. Returning to $1,000 then requires a 100% gain on the smaller balance. More generally, a loss fraction L requires a recovery of L divided by (1−L), provided the loss is less than 100%. The asymmetry becomes more severe as losses deepen.

Equal gains and losses therefore do not cancel. A 20% increase followed by a 20% decrease produces 1.20 times 0.80, or 96% of the starting value. Reversing that order produces the same ending value when there are no intervening cash flows. With contributions or withdrawals between periods, however, the sequence can change the investor’s dollar outcome.

Contributions are not investment profits

When regular deposits are included, separate total money contributed from growth. A portfolio can finish above its starting value even if investments lost money, simply because the owner added cash. Conversely, withdrawals can reduce the ending balance even while the investments earned a positive return.

The compound calculator on this site assumes deposits at the end of each month and converts an effective annual scenario return into an equivalent monthly rate. Those assumptions are stated beside the controls. Beginning-of-month deposits would have more time to compound, so another calculator using a different timing convention may correctly show a different answer.

Fees and inflation belong in the interpretation

A positive nominal balance does not establish a positive real return. Inflation changes what the balance can buy, while fees reduce what remains invested. For a simple no-cash-flow example, divide the nominal growth factor by the price-level growth factor to calculate the change in purchasing power.

Try a range of outcomes rather than one attractive rate. Include a low or negative scenario and inspect how much of the ending value came from your own deposits. For a real portfolio with irregular cash flows, a money-weighted return calculation may be needed to describe the investor’s experience. The elementary formula remains valuable because it makes the changing base visible—and prevents the common mistake of treating percentages as dollars that can simply be added together.

I prefer to begin with the contribution plan because that is the part a saver may be able to influence directly. A return assumption is useful for exploring possibilities, but making the spreadsheet more optimistic does not make those possibilities more likely.