risk-analysis

Risk Measurement and Stress Testing

Overview

Systematic risk-measurement methodology covering VaR/CVaR calculation, Monte Carlo simulation, stress-test design, and tail-risk analysis. It provides risk evaluation for backtest results and risk-control constraints for asset allocation.

Risk Measurement Methods

1. VaR (Value at Risk)

Definition: the maximum expected loss over a given horizon at a specified confidence level.

Three Calculation Methods

Historical Simulation Implementation

import numpy as np
import pandas as pd

def historical_var(returns: pd.Series, confidence: float = 0.95, horizon: int = 1) -> float:
    """
    Args:
        returns: Daily return series
        confidence: Confidence level, commonly 0.95 or 0.99
        horizon: Holding period in days, default 1
    Returns:
        VaR value (positive means loss)
    """
    sorted_returns = returns.sort_values()
    index = int((1 - confidence) * len(sorted_returns))
    var_1d = -sorted_returns.iloc[index]
    return var_1d * np.sqrt(horizon)  # square-root-of-time rule

Parametric Implementation

from scipy.stats import norm

def parametric_var(returns: pd.Series, confidence: float = 0.95, horizon: int = 1) -> float:
    mu = returns.mean()
    sigma = returns.std()
    z = norm.ppf(1 - confidence)
    var_1d = -(mu + z * sigma)
    return var_1d * np.sqrt(horizon)

2. CVaR / ES (Conditional VaR / Expected Shortfall)

Definition: the average loss beyond the VaR threshold, more conservative than VaR.

def historical_cvar(returns: pd.Series, confidence: float = 0.95) -> float:
    """CVaR = the mean of all losses beyond VaR."""
    var = historical_var(returns, confidence)
    tail_losses = returns[returns < -var]
    return -tail_losses.mean() if len(tail_losses) > 0 else var