risk-modeling-guide

Risk Modeling Guide

A skill for quantitative financial risk modeling, covering Value at Risk, Expected Shortfall, credit risk, stress testing, and Monte Carlo simulation methods. Essential for financial engineering research and regulatory risk analysis.

Market Risk: Value at Risk

VaR Methodologies

Implementation

import numpy as np
import pandas as pd
from scipy.stats import norm, t as t_dist

def historical_var(returns: np.ndarray, confidence: float = 0.99,
                    horizon_days: int = 1) -> dict:
    """
    Compute Value at Risk using historical simulation.
    returns: array of daily log returns
    confidence: confidence level (e.g., 0.99 for 99% VaR)
    horizon_days: risk horizon in days
    """
    # Scale returns to horizon
    if horizon_days > 1:
        # Rolling sum for overlapping returns
        scaled_returns = pd.Series(returns).rolling(horizon_days).sum().dropna().values
    else:
        scaled_returns = returns

    alpha = 1 - confidence
    var = -np.percentile(scaled_returns, alpha * 100)
    es = -np.mean(scaled_returns[scaled_returns <= -var])

    return {
        "VaR": round(var, 6),
        "Expected_Shortfall": round(es, 6),
        "confidence": confidence,
        "horizon_days": horizon_days,
        "n_observations": len(scaled_returns),
    }

def parametric_var(returns: np.ndarray, confidence: float = 0.99,
                    distribution: str = "normal") -> dict:
    """
    Parametric VaR assuming normal or Student-t distribution.
    """
    mu = np.mean(returns)
    sigma = np.std(returns, ddof=1)

    if distribution == "normal":
        z = norm.ppf(1 - confidence)
        var = -(mu + sigma * z)
        # Analytical ES for normal
        es = -mu + sigma * norm.pdf(norm.ppf(1 - confidence)) / (1 - confidence)
    elif distribution == "student-t":
        # Fit Student-t
        df, loc, scale = t_dist.fit(returns)
        z = t_dist.ppf(1 - confidence, df)
        var = -(loc + scale * z)
        # ES for Student-t
        t_pdf = t_dist.pdf(t_dist.ppf(1 - confidence, df), df)
        es = -loc + scale * (t_pdf / (1 - confidence)) * ((df + z**2) / (df - 1))
    else:
        raise ValueError(f"Unknown distribution: {distribution}")

    return {
        "VaR": round(var, 6),
        "Expected_Shortfall": round(es, 6),
        "distribution": distribution,
        "mean": round(mu, 6),
        "std": round(sigma, 6),
    }