# Active Return and Tracking Error

`D00-F11-A09` · Financial Mathematics, Statistics, and Data Foundations → Financial Risk and Performance Statistics · archetype `record-transform` · difficulty 1/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error/
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 { activeReturnAndTrackingError } from "fintech-algorithms/foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error";
```

## Signature

```ts
activeReturnAndTrackingError(input)
```

Subtracts the benchmark from the portfolio period by period, then reports the average and the annualized volatility of that difference.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | Reads `returns` and `benchmark`, two aligned non-empty lists of finite periodic returns, `frequency`, the number of periods per year, and `confidence`, used when the loss series is prepared. |

## Returns

`D00Output`

`activeReturns` holds the per-period differences, `averageActiveReturn` their mean, and `trackingError` their sample standard deviation times the square root of `frequency`.

## Errors

- When `returns` or `benchmark` is absent, empty, or holds a non-finite number — throws RangeError
- When `returns` and `benchmark` differ in length, or hold fewer than two observations — throws RangeError
- When `frequency` is zero or negative — throws RangeError
- When `confidence` is absent, or outside the range zero to one — the engine reads it for every topic from A04 onward, including those that never use it — throws RangeError

## Complexity

Time `O(n log 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
{
  "returns": [0.01, -0.02, 0.015, -0.01, 0.03],
  "benchmark": [0.008, -0.01, 0.012, -0.006, 0.02],
  "frequency": 252,
  "target": 0,
  "confidence": 0.8,
  "riskFree": 0.0001,
  "weights": [0.6, 0.4],
  "covarianceMatrix": [
    [0.04, 0.01],
    [0.01, 0.09]
  ]
}
```

### Call

```ts
activeReturnAndTrackingError(input)
```

### Returns

object with 2 fields: activeReturns, averageActiveReturn

```json
{
  "activeReturns": [0.002, -0.01, 0.002999999999999999, -0.004, 0.009999999999999998],
  "averageActiveReturn": 0.0001999999999999995
}
```

## 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-risk-and-performance-statistics/active-return-and-tracking-error/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt