Risk Measurement and Stress Testing

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.

The measures below are implemented once, with tests, in src/quantlib/risk.py. Call them; do not retype the formulas, because a hand-retyped VaR is where the sign convention silently flips.

from src.quantlib.risk import (
    historical_var, parametric_var, historical_cvar,
    max_drawdown_analysis, monte_carlo_gbm, analyze_mc_results, fit_gpd_tail,
)

Sign convention

A loss is a positive number, uniformly, across every function in the module:

Quantities that are returns rather than losses keep their natural sign and are named *_return (mean_return, worst_5pct_return, best_5pct_return), so a bad outcome there is negative. Report VaR to the user with the sign the user expects, but never re-derive it — flip it at the presentation layer only.

cvar >= var holds by construction whenever both come from the same sample at the same confidence level. If you ever compute a CVaR below its VaR, the tail mask is wrong.

This is not cvar >= var >= 0. The magnitudes are never clipped, so a sample whose tail contains no actual loss reports a negative loss — a gain. That is deliberate and informative; do not assert non-negativity on a VaR and do not clip it, or you destroy the distinction between "small loss" and "no loss at all".

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

Reads the loss straight off the sorted sample, so it inherits whatever fat tails the history actually had. horizon scales by the square-root-of-time rule, which is only valid under i.i.d. returns.

historical_var(returns, confidence=0.95)              # 1-day 95% VaR
historical_var(returns, confidence=0.99, horizon=10)  # 10-day 99% VaR

The quantile is a non-interpolating lower order statistic: element ceil((1 - confidence) * n) - 1 of the ascending-sorted returns, negated. The result is therefore always a return that was actually observed, never a blend of two neighbours.

Parametric (normal)

parametric_var(returns, confidence=0.95)

Fits mu and the sample sigma (ddof=1) and returns -(mu + z*sigma) with z = norm.ppf(1 - confidence). Needs at least 2 observations.

Do not assume the parametric figure is the lower one. The direction of the gap depends on the confidence level. A fat tail inflates the fitted sigma, which pushes the normal quantile outward at moderate confidence, where the empirical quantile is still sitting in the well-behaved body. Measured over 300 t(4) samples of 750 daily returns:

So the familiar "parametric understates risk" result only appears at 99% and deeper. At the 95% default it is normally the higher of the two, and that is not a sign your code is wrong. Quote both at 99% when the point is to expose the tail.

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

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

historical_cvar(returns, confidence=0.95)
historical_cvar(returns, confidence=0.99, horizon=10)

Averages the VaR order statistic together with everything worse than it (inclusive), which is the standard expected shortfall and is what makes cvar >= var structural rather than incidental.

VaR vs CVaR comparison:

3. Maximum Drawdown Analysis

dd = max_drawdown_analysis(equity)   # equity = a strictly positive net-value Series
dd["max_drawdown"]        # 0.325 -> fell 32.5% below its running peak (POSITIVE)
dd["peak_date"], dd["trough_date"], dd["recovery_date"]
dd["recovered"]           # False when the series ends still underwater

Full return keys: max_drawdown, peak_date, trough_date, recovery_date, recovered, peak_to_trough_periods, trough_to_recovery_periods, underwater_days, recovery_days.

• Recovery means reaching the peak value again, not merely bouncing off the trough; recovery_date is None and recovered is False if it never happens.
• underwater_days / recovery_days are calendar days and require a DatetimeIndex; on any other index they come back None and you should use the *_periods counts, which are always populated.
• Non-positive equity raises — a drawdown ratio is undefined at or below zero. Rebase a signed PnL series to a positive net value first.

4. Monte Carlo Simulation

Geometric Brownian Motion (GBM)

paths = monte_carlo_gbm(
    s0=100.0, mu=0.10, sigma=0.20,   # mu/sigma are ANNUALISED
    n_steps=252, n_paths=10_000,
    seed=42,                          # keyword-only; required for a reproducible run
)
paths.shape        # (10000, 253) -- n_steps + 1 columns
paths[:, 0]        # exactly s0 on every path

Always pass seed. It is keyword-only so it cannot be supplied by accident, and leaving it None draws fresh OS entropy — the run is then unreproducible and the numbers in your report cannot be regenerated. Use steps_per_year if the step is not a 252-day trading day.

Column 0 is the starting price, so paths[:, -1] / paths[:, 0] - 1 is the total return over the whole simulation. Terminal expectation is s0 * exp(mu * n_steps / steps_per_year); the median sits lower, at s0 * exp((mu - 0.5*sigma**2) * T), and that gap is the volatility drag, not a bug.

Simulation Result Analysis

summary = analyze_mc_results(paths, confidence=0.95)
summary["var"], summary["cvar"]                 # positive loss magnitudes
summary["mean_return"], summary["prob_loss"]
summary["worst_5pct_return"], summary["best_5pct_return"]   # signed returns

var / cvar are computed with exactly the same order-statistic convention as historical_var / historical_cvar, so a simulated VaR and a historical VaR are directly comparable.

Stress-Testing Framework

Historical Scenario Stress Tests

Hypothetical Scenario Design

STRESS_SCENARIOS = {
    'rate_shock_up_100bp': {
        'equity': -0.10,    # equities down 10%
        'bond_10y': -0.08,  # 10-year bonds down 8%
        'bond_2y': -0.02,   # short bonds down 2%
        'gold': +0.05,      # gold up 5%
        'btc': -0.15,       # BTC down 15%
    },
    'credit_crisis': {
        'equity': -0.25,
        'bond_10y': +0.05,  # government bonds act as a safe haven
        'credit_bond': -0.15,
        'gold': +0.10,
        'btc': -0.30,
    },
    'liquidity_dry_up': {
        'equity': -0.20,
        'bond_10y': -0.05,  # when liquidity is poor, everything falls
        'gold': -0.05,
        'btc': -0.40,
        'cash': 0.0,
    },
    'geopolitical_conflict': {
        'equity': -0.15,
        'bond_10y': +0.03,
        'gold': +0.15,
        'oil': +0.30,
        'btc': -0.20,
    },
}

Stress-Test Implementation Steps

1. Select a scenario: either historical or hypothetical
2. Apply shocks: multiply scenario shocks by the current positions
3. Compute portfolio loss: portfolio_loss = Σ(weight_i × shock_i × position_i)
4. Assess adequacy: compare loss vs risk budget and whether stop-loss thresholds are triggered