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

| Method | Description | Pros | Cons | |--------|-------------|------|------| | Historical simulation | Replay past returns | No distributional assumption | Assumes past repeats | | Variance-covariance | Assume normal returns | Fast, analytical | Underestimates tail risk | | Monte Carlo simulation | Simulate from fitted model | Flexible distributions | Computationally expensive | | Filtered historical simulation | GARCH + historical innovations | Captures volatility clustering | More complex |

Implementation

python
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),
    }

Monte Carlo VaR

python
def monte_carlo_var(returns: np.ndarray, n_simulations: int = 100000,
                     confidence: float = 0.99,
                     horizon_days: int = 10) -> dict:
    """
    Monte Carlo VaR using GBM (Geometric Brownian Motion).
    """
    mu = np.mean(returns)
    sigma = np.std(returns, ddof=1)

    # Simulate daily returns for the horizon
    rng = np.random.default_rng(42)
    simulated = rng.normal(
        mu * horizon_days,
        sigma * np.sqrt(horizon_days),
        size=n_simulations,
    )

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

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

Credit Risk Modeling

Probability of Default Estimation

python
from sklearn.linear_model import LogisticRegression

def build_pd_model(features: pd.DataFrame,
                    default_flag: pd.Series) -> dict:
    """
    Build a Probability of Default (PD) model using logistic regression.
    Common features: debt-to-income, credit utilization, payment history,
    employment length, loan amount.
    """
    model = LogisticRegression(max_iter=1000, class_weight="balanced")
    model.fit(features, default_flag)

    # Coefficient interpretation
    coef_df = pd.DataFrame({
        "feature": features.columns,
        "coefficient": model.coef_[0],
        "odds_ratio": np.exp(model.coef_[0]),
    }).sort_values("coefficient", ascending=False)

    # Model discrimination
    from sklearn.metrics import roc_auc_score
    pred_proba = model.predict_proba(features)[:, 1]
    auc = roc_auc_score(default_flag, pred_proba)

    return {
        "auc": round(auc, 4),
        "coefficients": coef_df.to_dict("records"),
        "intercept": round(model.intercept_[0], 4),
    }

Loss Given Default and EAD

python
def compute_expected_loss(pd_score: float, lgd: float,
                           ead: float) -> dict:
    """
    Compute Expected Loss = PD x LGD x EAD.
    pd_score: probability of default (0-1)
    lgd: loss given default (0-1, fraction of exposure lost)
    ead: exposure at default (dollar amount)
    """
    el = pd_score * lgd * ead
    return {
        "PD": pd_score,
        "LGD": lgd,
        "EAD": ead,
        "Expected_Loss": round(el, 2),
        "Unexpected_Loss_99": round(el * 2.33 * np.sqrt(pd_score * (1 - pd_score)), 2),
    }

Stress Testing

Scenario-Based Stress Tests

python
def run_stress_test(portfolio_returns: pd.DataFrame,
                     scenarios: dict[str, dict]) -> pd.DataFrame:
    """
    Apply macroeconomic stress scenarios to a portfolio.
    scenarios: {name: {factor: shock_value}} where factors are
    macroeconomic variables (interest_rate, gdp_growth, unemployment, etc.)
    """
    # Factor sensitivities (betas from regression)
    # In practice, estimated via historical regression
    factor_betas = {
        "interest_rate": -0.15,    # portfolio loses 15bp per 1% rate increase
        "gdp_growth": 0.08,        # gains 8bp per 1% GDP growth
        "unemployment": -0.12,     # loses 12bp per 1% unemployment increase
        "equity_market": 0.45,     # 45bp per 1% equity market move
        "credit_spread": -0.25,    # loses 25bp per 1% spread widening
    }

    results = []
    for name, shocks in scenarios.items():
        portfolio_impact = 0
        for factor, shock in shocks.items():
            beta = factor_betas.get(factor, 0)
            portfolio_impact += beta * shock

        results.append({
            "scenario": name,
            "portfolio_impact_pct": round(portfolio_impact * 100, 2),
            "shocks": shocks,
        })

    return pd.DataFrame(results)

# Example scenarios
scenarios = {
    "Mild Recession": {
        "interest_rate": -0.5, "gdp_growth": -2.0,
        "unemployment": 2.0, "equity_market": -15.0,
        "credit_spread": 1.5,
    },
    "Severe Recession": {
        "interest_rate": -1.0, "gdp_growth": -5.0,
        "unemployment": 5.0, "equity_market": -40.0,
        "credit_spread": 4.0,
    },
    "Rate Shock": {
        "interest_rate": 3.0, "gdp_growth": -1.0,
        "unemployment": 1.0, "equity_market": -10.0,
        "credit_spread": 1.0,
    },
}

Regulatory Framework

Basel III Capital Requirements

| Risk Type | Measurement | Capital Charge | |-----------|-------------|---------------| | Market risk | FRTB (Fundamental Review of the Trading Book) | ES at 97.5%, stressed calibration | | Credit risk | SA or IRB approach | PD, LGD, EAD based risk weights | | Operational risk | Basic Indicator / Standardized | Business indicator x ILM | | Liquidity risk | LCR and NSFR ratios | High-quality liquid assets buffer |

Tools and Libraries

• QuantLib (Python/C++): Derivatives pricing and risk analytics
• riskfolio-lib: Portfolio risk and optimization in Python
• arch (Python): GARCH models for volatility estimation
• pyfolio: Portfolio performance and risk analysis
• OpenGamma Strata: Open-source market risk analytics (Java)
• Moody's Analytics / Bloomberg PORT: Commercial risk platforms