# Arithmetic versus Geometric Average Return

`D00-F02-A09` · Financial Mathematics, Statistics, and Data Foundations → Financial Arithmetic, Time Value, and Returns · archetype `record-transform` · difficulty 1/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/foundations/financial-arithmetic-time-value-and-returns/arithmetic-geometric-return/
Agent skill: `npx skills add IslamBaraka90/Fintech-Algorithms-Library` — https://docs.thefintechbuilder.com/guides/agent-skill/

## Install and import

```bash
npm install fintech-algorithms
```

```ts
import { arithmeticVersusGeometricAverageReturn } from "fintech-algorithms/foundations/financial-arithmetic-time-value-and-returns/arithmetic-geometric-return";
```

## Signature

```ts
arithmeticVersusGeometricAverageReturn(input)
```

Averages the same return series two ways, by adding and by compounding, so the gap between the plain mean and the rate actually realised is visible.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | A plain object. Every F02 topic first reads `principal`, `rate` and `periods` and validates them before any per-topic branch runs. Before reaching this branch the engine also rejects a `startValue` or `endValue` that is zero or negative (omitting them entirely is not caught, because `Number(undefined)` is `NaN`), and `returns` to be a non-empty array of finite numbers none of which is at or below -1. Only `returns` feeds the result; the rest are validated but not used. |

## Returns

`{ arithmeticAverage: number; geometricAverage: number }`

`arithmeticAverage` is the plain mean of the series. `geometricAverage` is the compound growth factor taken to the power `1 / n` and reduced by 1, where n is the number of returns.

## Errors

- When input is not a plain object — throws TypeError
- When principal is negative, periods is negative, or rate is at or below -1 — throws RangeError
- When startValue or endValue is zero or negative — throws RangeError
- When returns is missing, empty, not an array, or contains a non-finite number — throws RangeError
- When any entry of returns is at or below -1, a loss of 100% or worse — throws RangeError

## Complexity

Time `O(n)`, space `O(n)`.

## Worked example

This is the worked example published in the article, replayed by the test suite on every run. The output cannot drift.

### Input

`input`:

```json
{
  "principal": 1000,
  "rate": 0.05,
  "periods": 3,
  "compoundsPerPeriod": 12,
  "futureValue": 1200,
  "cashFlows": [-1000, 400, 400, 400],
  "startValue": 100,
  "endValue": 110,
  "returns": [0.1, -0.05, 0.08],
  "frequency": 12,
  "periodicReturn": 0.01
}
```

### Call

```ts
arithmeticVersusGeometricAverageReturn(input)
```

### Returns

object with 2 fields: arithmeticAverage, geometricAverage

```json
{
  "arithmeticAverage": 0.043333333333333335,
  "geometricAverage": 0.04115010832757027
}
```

## Verification and provenance

Tier: **verified** (via input-expected).

The worked example below is the figure published in this algorithm's article, replayed and asserted by the test suite on every build. The arithmetic cannot drift without the build failing.

Both tiers guarantee the signature. Full explanation: https://docs.thefintechbuilder.com/guides/verification/

Generated from the docs.json payload shipped inside fintech-algorithms@0.13.0.
The signature and parameter list are checked against the compiled implementation at build time,
so a description that contradicts the code fails the build rather than reaching this file.

## Links

- Article (how it works, step by step): https://thefintechbuilder.com/foundations/financial-arithmetic-time-value-and-returns/arithmetic-geometric-return/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/financial-arithmetic-time-value-and-returns/arithmetic-geometric-return/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt