> ## 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.

# Return

> Understanding investment returns - measuring gains and losses

Return measures the percentage change in investment value over a period, including any income received. It's the most fundamental measure of investment performance.

## Beginner

### What It Means

Return tells you how much you made (or lost) on an investment, expressed as a percentage of your starting value.

### Basic Calculation

```
Return = (Ending Value - Beginning Value + Income) / Beginning Value

Example:
- Buy stock at $100
- Receive $3 dividend
- Sell at $110
- Return = ($110 - $100 + $3) / $100 = 13%
```

### Types of Returns

| Type                  | What It Includes                |
| --------------------- | ------------------------------- |
| **Price Return**      | Price change only               |
| **Total Return**      | Price change + dividends/income |
| **Annualized Return** | Return expressed as yearly rate |
| **Cumulative Return** | Total return over entire period |

### Why It Matters

Return is the bottom line - did your investment make money? Everything else (risk, volatility, Sharpe ratio) provides context, but return is what you actually earn.

***

## Advanced

### Return Calculation Methods

| Method             | Formula                       | Use Case               |
| ------------------ | ----------------------------- | ---------------------- |
| **Simple**         | (End - Start) / Start         | Single period          |
| **Logarithmic**    | ln(End / Start)               | Continuous compounding |
| **Time-Weighted**  | Geometric link of sub-periods | Manager evaluation     |
| **Money-Weighted** | IRR of cash flows             | Investor experience    |

### Time-Weighted vs. Money-Weighted

| Type                     | Considers Cash Flows? | Best For            |
| ------------------------ | --------------------- | ------------------- |
| **Time-Weighted (TWR)**  | No                    | Manager performance |
| **Money-Weighted (MWR)** | Yes                   | Investor experience |

<Note>
  Time-weighted return removes the effect of when money was added/withdrawn, isolating manager skill. Money-weighted return shows what the investor actually earned.
</Note>

### Annualization

Converting returns to annual rates:

```
Annualized Return = (1 + Total Return)^(1/Years) - 1

Example:
- 50% cumulative return over 3 years
- Annualized = (1.50)^(1/3) - 1 = 14.5% per year
```

### Geometric vs. Arithmetic Mean

| Type           | Formula             | When to Use                   |
| -------------- | ------------------- | ----------------------------- |
| **Arithmetic** | Sum / N             | Expected single-period return |
| **Geometric**  | (Product)^(1/N) - 1 | Actual compound growth        |

```
Example: Years with +50% then -50%
- Arithmetic mean: (50 + -50) / 2 = 0%
- Geometric mean: √(1.5 × 0.5) - 1 = -13.4%
- Reality: $100 → $150 → $75 (you lost money!)
```

<Warning>
  Arithmetic mean overstates compound returns. Always use geometric mean for multi-period performance.
</Warning>

### Real vs. Nominal Returns

| Type        | Definition                         |
| ----------- | ---------------------------------- |
| **Nominal** | Return before inflation adjustment |
| **Real**    | Return after subtracting inflation |

```
Real Return ≈ Nominal Return - Inflation

Example:
- Nominal return: 8%
- Inflation: 3%
- Real return: ~5%
```

### After-Tax Returns

Different return measures for tax impact:

| Measure                          | Description                     |
| -------------------------------- | ------------------------------- |
| **Pre-Tax**                      | Return before taxes             |
| **After-Tax (Pre-Liquidation)**  | After taxes paid during holding |
| **After-Tax (Post-Liquidation)** | After all taxes including sale  |

### Return Attribution

Breaking down where returns came from:

| Component             | Description              |
| --------------------- | ------------------------ |
| **Market Return**     | Broad market movement    |
| **Sector Allocation** | Sector over/underweights |
| **Stock Selection**   | Individual stock picks   |
| **Interaction**       | Combined effects         |

### Historical Context

Long-term annualized returns (US):

| Asset     | Return | Period       |
| --------- | ------ | ------------ |
| Stocks    | 10%    | 1926-present |
| Bonds     | 5%     | 1926-present |
| T-Bills   | 3%     | 1926-present |
| Inflation | 3%     | 1926-present |

### Return Expectations

<Note>
  Historical returns don't guarantee future returns. Current valuations, interest rates, and economic conditions affect forward expectations.
</Note>

### Related Terms

<CardGroup cols={3}>
  <Card title="Alpha" href="/glossary/alpha">
    Excess return vs. benchmark
  </Card>

  <Card title="Risk-Adjusted Returns" href="/glossary/risk-adjusted-returns">
    Return relative to risk
  </Card>

  <Card title="Benchmark" href="/glossary/benchmark">
    Standard for comparison
  </Card>
</CardGroup>