Geometric return chain

Calculation contract

W = product(1 + r_t); R = W - 1; R_ann = W^(K/n) - 1

Method and interpretation

A 10% gain followed by a 10% loss leaves wealth at 0.99, not 1.00. Adding a 5% gain takes the example to 1.0395. The ordering of these returns does not change terminal compounded wealth, although it can change path-dependent drawdown.

Annualization uses the supplied observation count and frequency. It is a mathematical equivalent rate, not a prediction. With a complete loss, wealth remains zero under the no-external-flow convention; later positive percentages cannot replenish an extinguished balance.

Where the reference stops

The reference is not money-weighted performance, a cash-flow accounting engine or a broker statement reconstruction. Returns below -100% are outside its nonnegative-wealth model. Extreme short-sample annualization can overflow and is rejected instead of producing an infinite result.

Published synthetic inputs

{
  "returns": [
    0.1,
    -0.1,
    0.05
  ],
  "periods_per_year": 12
}

Frozen calculation output

{
  "ending_wealth": 1.0395,
  "compounded_return": 0.0395,
  "annualized_return": 0.16761045388006252,
  "periods": 3
}

Equal-length periods with no external cash flows. Annualized short samples are arithmetic demonstrations, not expected returns.

Verification coverage

• Multiplicative chain, total loss, order-invariant terminal wealth and overflow rejection.
• No summation substituted for compounding.

Unit and independent-reference checks test arithmetic and rejection rules, not future investment performance.

Method references

• PerformanceAnalytics: cumulative return

Read the assumptions in context