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📊 Volatility

Volatility measures the dispersion of returns — how much an asset's price fluctuates over time. It is the most fundamental risk measure in finance and the building block for nearly all other risk metrics.

🔢 Formula

📐 Standard Deviation of Returns

\[ \sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N} (R_i - \bar{R})^2} \]

where \(R_i\) are individual period returns and \(\bar{R}\) is the mean return.

📈 Annualization

Per-period volatility is annualized by multiplying it by the square root of the number of periods a year contains:

\[ \sigma_{annual} = \sigma_{period} \times \sqrt{f} \]

The factor \(f\) is measured from the observed data, not fixed in advance: it is the number of returns actually used, rescaled to a full calendar year over the span they cover.

\[ f = \frac{N \times 365}{D} \]

where \(N\) is the number of period returns and \(D\) the calendar days they span.

Why a square root?

Returns are assumed to be independent across periods. The variance of a sum of \(f\) independent variables is \(f\) times the individual variance. Therefore:

\[\text{Var}_{annual} = f \times \text{Var}_{period}\]
\[\sigma_{annual} = \sqrt{f} \times \sigma_{period}\]

√252 is a result, not a constant

A daily-priced stock contributes roughly 252 returns over a full calendar year, so \(f = 252 \times 365 / 365 = 252\) and the familiar \(\sqrt{252}\) is recovered — as the outcome of the measurement, not as an assumption written into it. An instrument that trades every calendar day, such as crypto, gives \(f \approx 365\) and therefore \(\approx \sqrt{365}\): a hardcoded \(\sqrt{252}\) would understate its annualized volatility. A weekly-priced fund gives \(f \approx 52\).

→ See Observed Annualization for the derivation, the worked examples and what coverage adds to them.

💡 Interpretation

Annualized Volatility Typical Assets
1-5% Money market, short-term bonds
5-15% Government bonds, investment-grade corporates
15-25% Large-cap stocks, diversified equity ETFs
25-40% Small-cap stocks, single stocks
40-80%+ Crypto, meme stocks, leveraged products

📊 Realized vs Implied Volatility

📈 Realized (Historical) Volatility

Computed from past price data. This is what LibreFolio computes:

\[ \sigma_{realized} = \text{StdDev}(\text{historical returns}) \]

🔮 Implied Volatility

Extracted from options prices using the Black-Scholes model. It represents the market's expectation of future volatility:

\[ C = f(S, K, T, r, \sigma_{implied}) \]

Implied volatility is forward-looking but only available for optionable assets.

🔄 Rolling Window Volatility

Rather than computing a single volatility number for the entire period, rolling window volatility computes \(\sigma\) over a sliding window (e.g., 30 days), producing a time series that shows how volatility evolves:

\[ \sigma_t^{(w)} = \text{StdDev}(R_{t-w+1}, R_{t-w+2}, \ldots, R_t) \]

This is useful for:

  • Identifying volatility regimes (calm vs turbulent periods)
  • Detecting volatility clustering (high-volatility days tend to follow high-volatility days)
  • Setting dynamic position sizes (reduce exposure during high-volatility periods)

📐 Volatility and Portfolio Theory

Volatility plays a central role in Modern Portfolio Theory:

  • It is the denominator of the Sharpe Ratio
  • It determines the width of Bollinger Bands
  • It is the key input for portfolio optimization (minimizing \(\sigma_p\) for a target \(R_p\))
  • Diversification reduces portfolio volatility when asset correlations are less than 1

⚠️ Limitations

Volatility ≠ Risk

Volatility treats upside and downside movements equally. An asset that frequently spikes upward has high volatility but may be very attractive. For a downside-focused measure, use the Sortino Ratio or Max Drawdown.

Non-normality

Financial returns typically have:

  • Fat tails (more extreme events than a normal distribution predicts)
  • Negative skew (large drops more common than large gains)
  • Volatility clustering (calm and turbulent periods)

Standard deviation alone doesn't capture these features.