Financial Risk Management: VaR and Beyond

Module: Fintech AI | Difficulty: Advanced

Value at Risk (VaR)

Expected Shortfall (CVaR)

Cornish-Fisher Expansion

where = skewness, = excess kurtosis.

Backtesting

| Method | Unconditional Coverage | Independence | |--------|----------------------|--------------| | Kupiec | LR test | No | | Christoffersen | LR test | LR test | | Basel Traffic Light | Green/Amber/Red | No |

import numpy as np
from scipy import stats

class RiskMetrics:
    def __init__(self, confidence=0.99):
        self.confidence = confidence
    def historical_var(self, returns):
        return -np.percentile(returns, (1-self.confidence)*100)
    def historical_es(self, returns):
        var = self.historical_var(returns)
        return -returns[returns <= -var].mean()
    def parametric_var(self, returns, mu=None, sigma=None):
        if mu is None: mu = returns.mean()
        if sigma is None: sigma = returns.std()
        z = stats.norm.ppf(1-self.confidence)
        return -(mu + sigma * z)
    def cornish_fisher_var(self, returns):
        mu, sigma = returns.mean(), returns.std()
        s = stats.skew(returns)
        k = stats.kurtosis(returns)
        z = stats.norm.ppf(1-self.confidence)
        z_cf = z + (z**2 - 1)*s/6 + (z**3 - 3*z)*k/24
        return -(mu + sigma * z_cf)
    def kupiec_test(self, returns, var_estimate):
        n = len(returns)
        failures = np.sum(returns <= -var_estimate)
        p_hat = failures / n
        p = 1 - self.confidence
        lr = -2 * (np.log((1-p)**(n-failures) * p**failures) -
                   np.log((1-p_hat)**(n-failures) * p_hat**failures))
        return 1 - stats.chi2.cdf(lr, 1)

Research Insight: VaR has well-known flaws: it is not subadditive and ignores tail risk. Expected Shortfall (ES) is a coherent risk measure that addresses these issues. Basel III/IV is transitioning from VaR to ES for market risk capital requirements.