Historical 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 calendar days vs trading days inconsistently: Use 252 trading days (not 365 calendar days) for equity markets when annualizing. Bond markets and some international markets may differ.
• Survivorship bias in historical data: Data sets that exclude delisted or failed securities understate realized risk.
• Lookback period sensitivity: A 1-year lookback captures different risk regimes than a 5-year lookback. Always state the lookback window and consider whether it spans relevant market conditions.
• Confusing VaR confidence level direction: 95% VaR corresponds to the 5th percentile of returns (the loss tail). The "95%" refers to the confidence level, not the percentile of gains.
• Log returns vs simple returns: For volatility estimation, log returns are preferred because they are additive across time. For reporting cumulative performance, simple returns are more intuitive.

Cross-References

• performance-metrics (wealth-management plugin): Uses volatility, downside deviation, tracking error, and max drawdown as denominators in risk-adjusted ratios (Sharpe, Sortino, Information Ratio, Calmar).
• forward-risk (wealth-management plugin): Historical VaR and historical volatility serve as inputs to forward-looking VaR models and stress tests.
• volatility-modeling (wealth-management plugin): EWMA and GARCH models extend the simple historical volatility estimators covered here into forecasting frameworks.
• retirement-decumulation (wealth-management plugin): drawdown and volatility history drives sequence-of-returns risk in withdrawal-phase portfolios

Running the script

Run with uv run scripts/historical_risk.py (the PEP 723 header resolves numpy automatically) or with python3 scripts/historical_risk.py after pip install numpy scipy. A bare run prints a full risk analysis (annualized and Parkinson volatility, maximum drawdown with timing, 95%/99% historical VaR, downside deviation, semi-variance, tracking error, rolling volatility) on seeded synthetic data with an injected drawdown event. Use --verify to assert outputs match this skill's worked examples and the demo's expected values (exit code 0 on PASS) and --help for an overview of the class. The file is primarily meant to be imported as a module (e.g., from historical_risk import HistoricalRiskAnalyzer).