> ## Documentation Index
> Fetch the complete documentation index at: https://docs.chicago.global/llms.txt
> Use this file to discover all available pages before exploring further.

# Risk-Adjusted Returns

> Understanding risk-adjusted returns - evaluating performance relative to risk taken

Risk-adjusted returns measure investment performance relative to the risk taken to achieve those returns. A 15% return with 20% volatility may be worse than 10% return with 5% volatility.

## Beginner

### What It Means

Raw returns don't tell the whole story. Risk-adjusted returns answer the question: "How much return did I get for each unit of risk I took?" Higher risk-adjusted returns mean more efficient use of risk.

### Portfolio Example

| Portfolio   | Return | Volatility | Risk-Adjusted? |
| ----------- | ------ | ---------- | -------------- |
| Portfolio A | 15%    | 20%        | Lower          |
| Portfolio B | 12%    | 8%         | Higher         |

Portfolio B is actually better - it generated nearly as much return with far less risk. You could leverage Portfolio B to match Portfolio A's risk and get higher returns.

### Why It Matters

Comparing raw returns is misleading. A hedge fund returning 20% with 40% volatility isn't necessarily better than an index fund returning 10% with 15% volatility. Risk-adjusted metrics reveal true performance quality.

***

## Advanced

### Key Risk-Adjusted Metrics

| Metric                | Formula                      | Best For               |
| --------------------- | ---------------------------- | ---------------------- |
| **Sharpe Ratio**      | (Return - Rf) / Volatility   | General comparison     |
| **Sortino Ratio**     | (Return - Rf) / Downside Dev | Downside-focused       |
| **Treynor Ratio**     | (Return - Rf) / Beta         | Diversified portfolios |
| **Information Ratio** | Alpha / Tracking Error       | Active managers        |
| **Calmar Ratio**      | Return / Max Drawdown        | Drawdown-sensitive     |

### Sharpe Ratio

The most common risk-adjusted measure:

```
Sharpe Ratio = (Rp - Rf) / σp

Where:
- Rp = Portfolio return
- Rf = Risk-free rate
- σp = Portfolio volatility
```

| Sharpe    | Interpretation   |
| --------- | ---------------- |
| Below 0.5 | Below average    |
| 0.5 - 1.0 | Good             |
| 1.0 - 1.5 | Very good        |
| Above 1.5 | Excellent (rare) |

### Sortino Ratio

Focuses only on downside risk:

```
Sortino Ratio = (Rp - Rf) / Downside Deviation

Downside Deviation = Only negative returns count
```

<Note>
  Sortino is often preferred because investors don't mind upside volatility - only downside hurts. A stock that jumps 10% isn't "risky" in any meaningful sense.
</Note>

### Calmar Ratio

Uses maximum drawdown as the risk measure:

```
Calmar Ratio = Annual Return / |Maximum Drawdown|
```

| Calmar    | Interpretation |
| --------- | -------------- |
| Below 0.5 | Poor           |
| 0.5 - 1.0 | Acceptable     |
| 1.0 - 2.0 | Good           |
| Above 2.0 | Excellent      |

### Comparing Metrics

| Metric          | Risk Measure     | Strengths               | Weaknesses          |
| --------------- | ---------------- | ----------------------- | ------------------- |
| **Sharpe**      | Total volatility | Simple, universal       | Penalizes upside    |
| **Sortino**     | Downside only    | More intuitive          | Fewer data points   |
| **Calmar**      | Max drawdown     | Captures tail risk      | Single worst event  |
| **Information** | Tracking error   | Active management focus | Benchmark dependent |

### Practical Application

**When to use each metric:**

* **Sharpe**: Comparing any strategies against each other
* **Sortino**: When you care specifically about losses
* **Calmar**: When drawdowns are your primary concern
* **Information Ratio**: Evaluating active managers vs. benchmark
* **Treynor**: Comparing well-diversified portfolios

### Limitations

* **Backward-Looking**: Past risk-adjusted returns don't guarantee future performance
* **Distribution Assumptions**: Most assume normal returns (reality has fat tails)
* **Time Period Sensitive**: Results vary significantly by measurement period
* **Gaming**: Some strategies artificially inflate ratios (selling options, smoothing)

<Warning>
  A high Sharpe Ratio from selling options or illiquid strategies may mask hidden tail risks. Always understand the source of risk-adjusted returns.
</Warning>

### Related Terms

<CardGroup cols={3}>
  <Card title="Sharpe Ratio" href="/glossary/sharpe-ratio">
    Most common measure
  </Card>

  <Card title="Volatility" href="/glossary/volatility">
    The denominator in most metrics
  </Card>

  <Card title="Drawdown" href="/glossary/drawdown">
    Used in Calmar Ratio
  </Card>
</CardGroup>