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historical-risk

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Historical Risk Analysis

历史风险分析

Purpose

用途

Quantify how risky an investment or portfolio has been using historical return and price data. This skill covers volatility estimation (close-to-close, Parkinson, Yang-Zhang), drawdown analysis, historical Value-at-Risk, downside deviation, tracking error, and semi-variance. All measures are backward-looking and computed from observed data.
利用历史收益率和价格数据量化投资或投资组合的过往风险。本技能涵盖波动率估计（收盘价-收盘价、Parkinson、Yang-Zhang）、回撤分析、历史VaR、下行偏差、跟踪误差和半方差。所有指标均为回溯性指标，基于观测数据计算得出。

Layer

层级

1a — Realized Risk & Performance
1a — 已实现风险与表现

Direction

方向

Retrospective
回顾性

When to Use

适用场景

  • Understanding how risky an investment has been over a past period
  • Computing historical (realized) volatility using various estimators
  • Measuring drawdowns: maximum drawdown, drawdown duration, and recovery time
  • Calculating historical VaR (non-parametric, directly from the return distribution)
  • Computing downside deviation or semi-variance for asymmetric risk assessment
  • Measuring tracking error of a portfolio relative to a benchmark
  • 了解某一投资在过去一段时间内的风险水平
  • 使用多种估计器计算历史（已实现）波动率
  • 衡量回撤：最大回撤、回撤持续时间和恢复时间
  • 计算历史VaR（非参数法，直接基于收益率分布）
  • 计算下行偏差或半方差以进行非对称风险评估
  • 衡量投资组合相对于基准的跟踪误差

Core Concepts

核心概念

Close-to-Close Volatility

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}).
最简单且最常用的波动率估计器。计算对数收益率的标准差并年化。
sigma_annual = sigma_daily * sqrt(N)
其中N = 每年的交易周期数（通常日度数据为252，周度为52，月度为12）。
优先使用对数收益率：r_t = ln(P_t / P_{t-1})。

Parkinson (High-Low) Estimator

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.
利用日内最高价和最低价来捕捉收盘价-收盘价估计器所遗漏的日内波动率。当真实过程为连续过程时，其效率高于收盘价-收盘价估计器。
sigma^2_Park = (1 / (4 * n * ln(2))) * sum( ln(H_i / L_i)^2 )
对于扩散过程，该估计器的效率约为收盘价-收盘价估计器的5倍，但当存在跳空或区间离散化时，会产生向下偏差。

Yang-Zhang Estimator

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, and sigma^2_RS is the Rogers-Satchell estimator that uses all four OHLC prices within each period.
结合了隔夜（收盘价-开盘价）、开盘价-收盘价以及Rogers-Satchell组件。对于同时存在漂移和开盘跳空的过程，该估计器是无偏的。
sigma^2_YZ = sigma^2_overnight + k * sigma^2_open-to-close + (1 - k) * sigma^2_RS
其中k的选择旨在最小化估计器方差，sigma^2_RS是使用每个周期内所有四个OHLC价格的Rogers-Satchell估计器。

Drawdown Analysis

回撤分析

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.
时间t的回撤衡量从运行峰值到当前的跌幅：
DD_t = (Peak_t - Value_t) / Peak_t
其中Peak_t = 所有s <= t时的max(Value_s)。
  • Maximum Drawdown (MDD)： MDD = 评估期内的max(DD_t)。
  • Drawdown Duration： 从峰值到达到新峰值的周期数。
  • Recovery Time： 从谷底回到之前峰值水平的周期数。

Historical VaR

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.
非参数（经验）风险价值（VaR）就是历史收益率分布的α分位数。无需做分布假设。
VaR_alpha = -Percentile(R, alpha)
例如，95% VaR使用收益率的第5分位数。负号是惯例，以便VaR表示为正的损失数值。

Downside Deviation

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.
衡量收益率低于最低可接受收益率（MAR）的离散程度：
sigma_d = sqrt( (1/n) * sum( min(R_i - MAR, 0)^2 ) )
MAR的常见选择：0%、无风险利率或平均收益率。

Tracking Error

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.
投资组合与基准收益率差值的标准差，年化后的值：
TE = std(R_p - R_b) * sqrt(N)
该指标衡量投资组合与基准的跟踪（或偏离）一致性。

Semi-Variance

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).
仅使用低于均值（或低于阈值）的收益率计算的方差：
SV = (1/n) * sum( min(R_i - mean(R), 0)^2 )
半方差专注于下行风险，是Sortino比率的基础（参见performance-metrics）。

Key Formulas

关键公式

FormulaExpressionUse Case
Annualized Volatilitysigma_ann = sigma_period * sqrt(N)Convert period vol to annual vol
Log Returnr_t = ln(P_t / P_{t-1})Compute continuously compounded returns
Parkinson Variancesigma^2 = (1 / (4n ln2)) * sum(ln(H/L)^2)Volatility from high-low data
DrawdownDD_t = (Peak_t - Value_t) / Peak_tMeasure peak-to-trough decline
Max DrawdownMDD = max(DD_t)Worst historical decline
Historical VaR (95%)5th percentile of return seriesNon-parametric loss estimate
Downside Deviationsigma_d = sqrt((1/n) * sum(min(R_i - MAR, 0)^2))Asymmetric risk below MAR
Tracking ErrorTE = 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
公式表达式适用场景
年化波动率sigma_ann = sigma_period * sqrt(N)将周期波动率转换为年化波动率
对数收益率r_t = ln(P_t / P_{t-1})计算连续复利收益率
Parkinson方差sigma^2 = (1 / (4n ln2)) * sum(ln(H/L)^2)基于高低价数据计算波动率
回撤DD_t = (Peak_t - Value_t) / Peak_t衡量峰谷跌幅
最大回撤MDD = max(DD_t)历史最大跌幅
历史VaR（95%）收益率序列的第5分位数非参数损失估计
下行偏差sigma_d = sqrt((1/n) * sum(min(R_i - MAR, 0)^2))MAR以下的非对称风险
跟踪误差TE = std(R_p - R_b) * sqrt(N)投资组合与基准的偏差
半方差(1/n) * sum(min(R_i - mean(R), 0)^2)低于均值的方差

Worked Examples

示例演练

Example 1: Annualized Volatility from Daily Returns

示例1：基于日度收益率计算年化波动率

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%.
已知： 某股票的日度对数收益率样本标准差为1.2%。假设每年有252个交易日。
计算： 年化波动率。
解答：
sigma_annual = 0.012 * sqrt(252)
             = 0.012 * 15.875
             = 0.1905
             ~ 19.05%
该股票的年化波动率约为19%。

Example 2: Maximum Drawdown from a Price Series

示例2：基于价格序列计算最大回撤

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.
已知： 某基金的资产净值（NAV）在六个月内的走势为：$120, $135, $150, $130, $105, $125。
计算： 最大回撤并确定峰值和谷底。
解答：
运行峰值：$120, $135, $150, $150, $150, $150。
各时点的回撤：
  • $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%
最大回撤 = 30.0%，发生在峰值$150到谷底$105期间。截至最后一个观测值（$125），回撤尚未完全恢复。

Example 3: Historical VaR

示例3：历史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.
已知： 500个日度收益率从最差到最好排序。第25差的收益率为-2.8%，第26差的为-2.6%。
计算： 95%的1日历史VaR。
解答：
第5分位数对应500个观测值中的第25个（500 * 0.05 = 25）。
VaR_95% = -(-2.8%) = 2.8%
解释：根据历史分布，95%的交易日中，损失预计不会超过2.8%。

Common Pitfalls

常见误区

  • 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.
  • 波动率年化错误： 波动率随时间的平方根缩放（乘以sqrt(N)），而非线性缩放。将日度波动率乘以252而非sqrt(252)会产生严重夸大的数值。
  • 日历日与交易日混用： 年化股票市场数据时使用252个交易日（而非365个日历日）。债券市场和部分国际市场可能有所不同。
  • 历史数据中的生存偏差： 排除已退市或破产证券的数据集会低估已实现风险。
  • 回溯期敏感性： 1年回溯期与5年回溯期捕捉的风险状态不同。始终明确回溯窗口，并考虑其是否涵盖相关市场环境。
  • 混淆VaR置信水平方向： 95% VaR对应收益率的第5分位数（损失尾部）。"95%"指的是置信水平，而非收益的分位数。
  • 对数收益率与简单收益率： 波动率估计优先使用对数收益率，因为其具有时间可加性。报告累计表现时，简单收益率更直观。

Cross-References

交叉引用

  • performance-metrics (wealth-management plugin, Layer 1a): 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, Layer 1b): Historical VaR and historical volatility serve as inputs to forward-looking VaR models and stress tests.
  • volatility-modeling (wealth-management plugin, Layer 1b): EWMA and GARCH models extend the simple historical volatility estimators covered here into forecasting frameworks.
  • performance-metrics（财富管理插件，层级1a）：在风险调整比率（Sharpe、Sortino、信息比率、Calmar）中使用波动率、下行偏差、跟踪误差和最大回撤作为分母。
  • forward-risk（财富管理插件，层级1b）：历史VaR和历史波动率作为前瞻性VaR模型和压力测试的输入。
  • volatility-modeling（财富管理插件，层级1b）：EWMA和GARCH模型将此处介绍的简单历史波动率估计器扩展为预测框架。

Reference Implementation

参考实现

See 
scripts/historical_risk.py
for computational helpers.
计算辅助工具请参见
scripts/historical_risk.py
。