Compounding and Geometric vs Arithmetic Returns

Compounding is what happens when returns are reinvested and start earning returns of their own. Over long periods, it is the most powerful force in investing: modest returns, compounded for decades, produce large results. But compounding works in both directions. Losses shrink the base that future gains build on, which is why a 50% loss needs a 100% gain to recover and why volatility quietly lowers long term growth. Understanding compounding helps traders set realistic expectations and avoid sizing that looks fine on average but destroys capital over time.

The basic formulas#

future value = present value × (1 + r)^n
continuous compounding: future value = present value × e^(r × n)

The rule of 72#

A quick way to estimate doubling time:

years to double ≈ 72 / annual return in percent

At 8%, money doubles in about 9 years; at 12%, about 6 years.

Losses hurt more than gains help#

gain needed to recover = 1 / (1 - loss) - 1

This asymmetry is why protecting capital matters so much. Large drawdowns take a long time to recover from. See Maximum Drawdown and the Drawdown Recovery Calculator.

Volatility drag#

Two return streams with the same average return can compound very differently.

Compounding and position sizing#

Because losses compound against you, betting too much can reduce long term growth even with a positive edge. The Kelly criterion finds the bet size that maximises compound growth; betting more than Kelly lowers growth and greatly increases drawdowns. Many traders use a fraction of Kelly for this reason. See Kelly Criterion and Fractional Kelly.

Leveraged ETFs and daily rebalancing#

Leveraged ETFs reset their leverage daily. In choppy markets, the daily compounding causes them to lose value even if the underlying index ends flat, because of volatility drag. Over long periods, their returns can differ greatly from simple multiples of the index's return. They are designed mainly for short term use.

Compounding costs#

Fees compound too. A 1% annual fee on a portfolio earning 7% reduces the ending value after 30 years by roughly a quarter compared with no fee. Trading costs on active strategies compound in the same way. See All-In Trading Cost.

Compounding in a trading account#

Compounding in a trading account means sizing positions as a percentage of current equity. After gains, positions grow; after losses, they shrink. This automatically slows losses during drawdowns and accelerates growth during good periods. Fixed dollar sizing does not compound and can leave traders overexposed after losses. See Fixed Percentage vs Fixed Dollar Risk.

Frequently asked questions#

What is compounding?#

Earning returns on previous returns by reinvesting gains, so growth accelerates over time.

Why does a 50% loss need a 100% gain to recover?#

Because after losing half, the remaining capital is smaller, so it must double to return to the original amount.

What is volatility drag?#

The reduction in compound returns caused by volatility; higher volatility lowers long term growth even if the average return is the same.

Next, learn how money's value depends on time in Time Value of Money.