How do you read a QQ plot to assess whether financial return data follows a normal distribution?

Question

AcadiFi · Accepted Answer

A **Quantile-Quantile (QQ) plot** compares the quantiles of your observed data against the quantiles of a theoretical distribution (usually the normal). If the data is perfectly normal, all points fall on a straight 45-degree line. Deviations reveal departures from normality.

**Reading the Plot**

| Pattern | Interpretation | Risk Implication |
|---|---|---|
| Points on the line | Data matches normal distribution | Parametric VaR is appropriate |
| Upper-right curves above line | Right tail heavier than normal (positive excess kurtosis) | More extreme gains than normal predicts |
| Lower-left curves below line | Left tail heavier than normal | **More extreme losses** — VaR underestimates risk |
| S-shaped deviation | Both tails heavier (leptokurtic) | Normal VaR severely underestimates tail risk |
| Systematic curve above/below | Skewness present | Asymmetric risk not captured by symmetric VaR |

```mermaid
flowchart TD
    A["Collect daily returns for Fenwick Global Equity Fund"] --> B["Sort returns, compute empirical quantiles"]
    B --> C["Compute theoretical normal quantiles"]
    C --> D["Plot empirical vs theoretical"]
    D --> E{"Points on diagonal?"}
    E -->|Yes| F["Normal assumption OK for VaR"]
    E -->|No: S-shape| G["Fat tails — use t-distribution or historical VaR"]
    E -->|No: Curve| H["Skewness — use Cornish-Fisher expansion"]
```

**Worked Example**

Fenwick Global Equity Fund has 1,260 daily returns (5 years). On the QQ plot:

- The middle 90% of data points hug the diagonal closely
- The lowest 2% of returns bend sharply **below** the line (left-tail fat)
- The highest 2% bend moderately **above** the line (right-tail fat, but less extreme)

This tells us the fund's returns exhibit **excess kurtosis** (fat tails) and mild **negative skew**. A normal-distribution VaR at 99% confidence would underestimate the actual 1st percentile loss.

**Quantitative Check**

If the 1st percentile under a normal distribution with mean 0.04% and standard deviation 1.2% is:

VaR_normal = 0.04% - 2.326 x 1.2% = **-2.75%**

But the actual 1st percentile from the empirical distribution is **-3.85%**, the QQ plot's left-tail departure of roughly 1.1% translates into a 40% underestimation of tail risk.

**Alternatives When QQ Plot Shows Non-Normality:**
- Use a **Student's t-distribution** with low degrees of freedom (4-6)
- Apply the **Cornish-Fisher expansion** to adjust percentiles for skew and kurtosis
- Switch to **historical simulation** or **Monte Carlo** methods

For more on distribution testing, visit our FRM Part I question bank.

How do you read a QQ plot to assess whether financial return data follows a normal distribution?

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