/historical-risk

name: historical-risk
description: "Quantify realized risk from historical data using volatility estimators, drawdown analysis, and downside risk metrics. Use when the user asks about historical volatility, maximum drawdown, drawdown duration, historical VaR, downside deviation, semi-variance, or tracking error. Also trigger when users mention 'how risky has this been', 'worst decline', 'Parkinson estimator', 'Yang-Zhang', 'peak-to-trough loss', 'recovery time', 'annualized volatility', or ask how to measure past investment risk."

Historical Risk Analysis

Core Concepts

Close-to-Close Volatility

The simplest and most common volatility estimator. Compute the standard deviation of log returns and annualize.

sigma_annual = sigma_daily * sqrt(N)

where N = number of trading periods per year (typically 252 for daily, 52 for weekly, 12 for monthly).

Log returns are preferred: r_t = ln(P_t / P_{t-1}).

Parkinson (High-Low) Estimator

Uses intraday high and low prices to capture intraday volatility that close-to-close misses. More efficient than close-to-close when the true process is continuous.

sigma^2_Park = (1 / (4 * n * ln(2))) * sum( ln(H_i / L_i)^2 )

This estimator is roughly 5x more efficient than close-to-close for a diffusion process, but is biased downward when there are jumps or when the range is discretized.

Yang-Zhang Estimator

Combines overnight (close-to-open), open-to-close, and Rogers-Satchell components. It is unbiased for processes with both drift and opening jumps.

sigma^2_YZ = sigma^2_overnight + k * sigma^2_open-to-close + (1 - k) * sigma^2_RS

where k is chosen to minimize estimator variance:

k = 0.34 / (1.34 + (n + 1) / (n - 1))

with n the number of observations, and sigma^2_RS is the Rogers-Satchell estimator that uses all four OHLC prices within each period.

Drawdown Analysis

Drawdown at time t measures the decline from the running peak:

DD_t = (Peak_t - Value_t) / Peak_t

where Peak_t = max(Value_s) for all s <= t.

• **Maximum Drawdown (MDD):** MDD = max(DD_t) over the evaluation period.
• **Drawdown Duration:** The number of periods from a peak until a new peak is reached.
• **Recovery Time:** The number of periods from the trough back to the prior peak level.

Historical VaR

The non-parametric (empirical) Value-at-Risk is simply the alpha-percentile of the historical return distribution. No distributional assumptions are made.

VaR_alpha = -Percentile(R, alpha)

For example, 95% VaR uses the 5th percentile of returns. The negative sign is a convention so that VaR is expressed as a positive loss number.

Downside Deviation

Measures dispersion of returns below a Minimum Acceptable Return (MAR):

sigma_d = sqrt( (1/n) * sum( min(R_i - MAR, 0)^2 ) )

Common choices for MAR: 0%, the risk-free rate, or the mean return.

Tracking Error

Standard deviation of the difference between portfolio and benchmark returns, annualized:

TE = std(R_p - R_b) * sqrt(N)

This measures how consistently the portfolio tracks (or deviates from) its benchmark.

Semi-Variance

Variance computed using only returns below the mean (or below a threshold):

SV = (1/n) * sum( min(R_i - mean(R), 0)^2 )

Semi-variance isolates downside risk and is the foundation for the Sortino ratio (see performance-metrics).

Key Formulas

| Formula | Expression | Use Case | |---------|-----------|----------| | Annualized Volatility | sigma_ann = sigma_period * sqrt(N) | Convert period vol to annual vol | | Log Return | r_t = ln(P_t / P_{t-1}) | Compute continuously compounded returns | | Parkinson Variance | sigma^2 = (1 / (4n ln2)) * sum(ln(H/L)^2) | Volatility from high-low data | | Drawdown | DD_t = (Peak_t - Value_t) / Peak_t | Measure peak-to-trough decline | | Max Drawdown | MDD = max(DD_t) | Worst historical decline | | Historical VaR (95%) | 5th percentile of return series | Non-parametric loss estimate | | Downside Deviation | sigma_d = sqrt((1/n) * sum(min(R_i - MAR, 0)^2)) | Asymmetric risk below MAR | | Tracking Error | TE = std(R_p - R_b) * sqrt(N) | Portfolio vs benchmark deviation | | Semi-Variance | (1/n) * sum(min(R_i - mean(R), 0)^2) | Below-mean variance |

Worked Examples

Example 1: Annualized Volatility from Daily Returns

**Given:** A stock has daily log returns with a sample standard deviation of 1.2%. Assume 252 trading days per year.

**Calculate:** Annualized volatility.

**Solution:**

sigma_annual = 0.012 * sqrt(252)
             = 0.012 * 15.875
             = 0.1905
             ~ 19.05%

The stock's annualized volatility is approximately 19%.

Example 2: Maximum Drawdown from a Price Series

**Given:** A fund's NAV follows this path over six months: $120, $135, $150, $130, $105, $125.

**Calculate:** Maximum drawdown and identify the peak and trough.

**Solution:**

Running peaks: $120, $135, $150, $150, $150, $150.

Drawdowns at each point:

• $120: (120-120)/120 = 0%
• $135: (135-135)/135 = 0%
• $150: (150-150)/150 = 0%
• $130: (150-130)/150 = 13.3%
• $105: (150-105)/150 = 30.0%
• $125: (150-125)/150 = 16.7%

**Maximum Drawdown = 30.0%**, occurring from the peak of $150 to the trough of $105. As of the last observation ($125), the drawdown has not yet fully recovered.

Example 3: Historical VaR

**Given:** 500 daily returns sorted from worst to best. The 25th-worst return is -2.8% and the 26th-worst is -2.6%.

**Calculate:** 95% 1-day historical VaR.

**Solution:**

The 5th percentile corresponds to the 25th observation out of 500 (500 * 0.05 = 25).

VaR_95% = -(-2.8%) = 2.8%

Interpretation: On 95% of days, the loss is expected not to exceed 2.8% based on the historical distribution.

Common Pitfalls

• **Not annualizing volatility correctly:** Volatility scales with the square root of time (multiply by sqrt(N)), not linearly. Multiplying daily vol by 252 instead of sqrt(252) produces wildly inflated numbers.
• **Using calen