# Volatility and Annualized Volatility

`D00-F11-A01` · 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/volatility-and-annualized-volatility/
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 { volatilityAndAnnualizedVolatility } from "fintech-algorithms/foundations/financial-risk-and-performance-statistics/volatility-and-annualized-volatility";
```

## Signature

```ts
volatilityAndAnnualizedVolatility(input)
```

Takes the sample standard deviation of a return series and rescales it to an annual horizon by the square root of the observations per year.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | Reads `returns` and `benchmark`, two aligned non-empty lists of finite periodic returns, and `frequency`, the number of periods per year. Only `returns` and `frequency` enter this calculation; `benchmark` is still validated. |

## Returns

`D00Output`

`periodicVolatility` is the per-period sample standard deviation and `annualizedVolatility` is that figure 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

## 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
{
  "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
volatilityAndAnnualizedVolatility(input)
```

### Returns

object with 2 fields: periodicVolatility, annualizedVolatility

```json
{
  "periodicVolatility": 0.02,
  "annualizedVolatility": 0.3174901573277509
}
```

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