historical-risk

What this skill does

# Historical Risk Analysis

## Purpose
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.

## Layer
1a — Realized Risk & Performance

## Direction
Retrospective

## When to Use
- 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

## 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, 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, 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.

## Reference Implementation
See `scripts/historical_risk.py` for computational helpers.