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

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.