{
  "slug": "compound-return",
  "title": "Geometric return chain",
  "topic": "validation",
  "summary": "Reconcile sequential decimal returns to terminal wealth and an explicitly annualized equivalent.",
  "version": "1.0.0",
  "published": "2026-10-16",
  "status": "Educational reference",
  "formula": "W = product(1 + r_t); R = W - 1; R_ann = W^(K/n) - 1",
  "inputs": [
    [
      "returns",
      "1-250 decimals, -1 to 10",
      "Chronological equal-length returns without external flows"
    ],
    [
      "periods_per_year",
      "Number, 1 to 366",
      "Periods corresponding to one year"
    ]
  ],
  "method": [
    "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."
  ],
  "limits": [
    "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."
  ],
  "checks": [
    "Multiplicative chain, total loss, order-invariant terminal wealth and overflow rejection.",
    "No summation substituted for compounding."
  ],
  "related": [
    "compound-return-and-cash-flows",
    "drawdown-path-and-sampling"
  ],
  "sources": [
    {
      "title": "PerformanceAnalytics: cumulative return",
      "url": "https://search.r-project.org/CRAN/refmans/PerformanceAnalytics/html/Return.cumulative.html"
    }
  ],
  "example": {
    "returns": [
      0.1,
      -0.1,
      0.05
    ],
    "periods_per_year": 12
  },
  "frozen_result": {
    "schema_version": 1,
    "model_id": "compound-return",
    "model_version": "1.0.0",
    "engine_version": "1.0.0",
    "published_at": "2026-10-16",
    "scope": "Synthetic educational reference calculation. Not a fitted investment strategy, forecast, provider model or client return.",
    "inputs": {
      "returns": [
        0.1,
        -0.1,
        0.05
      ],
      "periods_per_year": 12
    },
    "outputs": {
      "ending_wealth": 1.0395,
      "compounded_return": 0.0395,
      "annualized_return": 0.16761045388006252,
      "periods": 3
    },
    "notes": [
      "Equal-length periods with no external cash flows. Annualized short samples are arithmetic demonstrations, not expected returns."
    ]
  }
}