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:

Value	Reads as
historical_var(...) == 0.028	a 2.8% loss
historical_cvar(...) == 0.042	a 4.2% average loss in the tail
max_drawdown_analysis(...)["max_drawdown"] == 0.325	a 32.5% peak-to-trough decline
analyze_mc_results(...)["var"] == 0.224	a 22.4% loss

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

Method	Formula / Steps	Advantages	Disadvantages
Historical simulation	Sort historical returns and take the quantile	No distribution assumption	Depends on historical samples
Parametric (normal)	VaR = μ - z_α × σ	Easy to compute	Assumes a normal distribution
Monte Carlo	Simulate N paths and take the quantile	Flexible	Computationally intensive

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:

Confidence	Parametric reads above historical
90%	100% of samples
95%	92.7%
97.5%	40.3%
99%	5.3%

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:

Metric	VaR(95%)	CVaR(95%)	Meaning
Typical value	2.1%	3.4%	CVaR is usually 1.3-1.8x VaR
Subadditivity	Not satisfied	Satisfied	CVaR can be used for portfolio risk decomposition
Regulation	Basel II	Basel III	Regulatory trend is shifting toward CVaR

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

Scenario	Period	China A-share Drawdown	US Equity Drawdown	BTC Drawdown	10Y Government Bonds
2008 financial crisis	2008.01-2008.10	-65%	-50%	N/A	yield ↓ 100bp
2015 China equity crash	2015.06-2015.08	-45%	-10%	-20%	yield ↓ 50bp
2018 trade war	2018.01-2018.12	-25%	-20%	-80%	yield ↓ 30bp
2020 COVID shock	2020.01-2020.03	-15%	-35%	-50%	yield ↓ 80bp
2022 hiking cycle	2022.01-2022.10	-20%	-25%	-65%	yield ↑ 200bp

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

Tail-Risk Analysis (Extreme Value Theory, EVT)

POT Method (Peaks Over Threshold)

fit = fit_gpd_tail(returns, threshold_pct=5.0)   # keep the worst 5%
fit["shape_xi"]      # ξ>0 fat tail, ξ=0 exponential tail, ξ<0 bounded tail
fit["shape_stderr"]  # standard error of ξ -- quote ξ with it, never alone
fit["scale_sigma"]   # in units of loss magnitude
fit["tail_type"]     # "fat" | "exponential" | "bounded"
fit["threshold"], fit["n_exceedances"], fit["exceedance_rate"]

Exceedances are non-negative by construction (threshold - return, kept only where the return fell below the threshold), so the GPD location is pinned at zero. Letting loc float instead lets the optimiser absorb tail mass into a shifted origin and biases shape_xi.

tail_type is decided against shape_stderr, not against exact zero. A fitted shape_xi is a float and is never exactly 0.0, so a bare ξ > 0 test would call a genuinely exponential tail "fat" purely on the sign of estimation noise. "fat" therefore means ξ > 2 × shape_stderr, "bounded" means ξ < -2 × shape_stderr, and anything in between is "exponential" — indistinguishable from zero at this sample size. Measured over 200 refits, that 2σ band labels a truly exponential tail "exponential" 96.5% of the time while still catching ξ = +0.40 and ξ = -0.35 100% of the time. A shape_xi of 0.03 with a shape_stderr of 0.02 is not evidence of a fat tail; get more exceedances before you call it one.

Threshold choice is the real judgement call: too high and there is nothing left to fit (fewer than 2 exceedances raises), too low and the EVT limit theorem no longer applies, so the fitted shape stops meaning anything. Check that shape_xi is stable across a few nearby threshold_pct values before quoting it.

Tail-Risk Metrics

Metric	Calculation	Meaning
Kurtosis	returns.kurtosis()	>3 indicates fat tails; China A-shares are often in the 4-8 range
Skewness	returns.skew()	<0 means left-skewed (large drops are more common than large rallies)
Tail ratio	worst 5% / best 5%	>1 means larger downside risk
Hill estimator	Tail index	α<2 implies extremely fat tails

Analysis Framework

Input Requirements

Required:
- Return series (daily or higher frequency) or net-value series
- Portfolio weights (if it is a portfolio)

Optional:
- Benchmark returns (for relative risk analysis)
- Risk budget / constraint settings

Analysis Steps

1. Data preprocessing: compute returns, check missing values, and handle outliers
2. Descriptive statistics: mean / volatility / skewness / kurtosis / maximum drawdown
3. VaR/CVaR calculation: compare three methods at both 95% and 99% confidence levels
4. Monte Carlo simulation: 10,000 paths, output distribution statistics and VaR
5. Stress testing: at least 3 historical scenarios + 2 hypothetical scenarios
6. Tail analysis: fit GPD and determine tail type
7. Risk-control recommendations: provide concrete recommendations based on the results

Output Format

Note the sign flip: the module returns losses as positive numbers, while the report below prints them the way a reader expects to see them (max_drawdown 0.325 → -32.5%). Flip once, here at the presentation layer, and never inside a calculation.

## Risk Analysis Report

### Core Risk Metrics
| Metric | Value |
|------|-----|
| Daily volatility | 1.85% |
| Annualized volatility | 29.3% |
| Maximum drawdown | -32.5% (2024.09.15 → 2024.11.20) |
| VaR(95%, 1D) | -2.8% |
| CVaR(95%, 1D) | -4.2% |
| Skewness | -0.45 |
| Kurtosis | 5.2 (fat tail) |

### Stress-Test Results
| Scenario | Portfolio Loss | Stop Triggered |
|------|---------|----------|
| 2020 COVID replay | -18.5% | No |
| Rates +100bp | -12.3% | No |
| Liquidity dry-up | -28.7% | Yes |

### Monte Carlo Simulation (252 days, 10000 paths)
| Statistic | Value |
|------|-----|
| Expected return | +8.2% |
| Loss probability | 35% |
| Worst 5% scenario | -22.4% |

### Risk-Control Recommendations
1. Recommend setting a portfolio stop-loss at -15%
2. Tail risk is elevated; consider allocating 5% to gold as a hedge
3. Correlations rise in stressed markets, so diversification benefits will be discounted

Notes

1. VaR is not the maximum loss: VaR only says "with 95% probability, losses will not exceed X"; the remaining 5% can be far worse
2. Normality assumption is dangerous: financial returns are almost always fat-tailed, so parametric VaR underestimates risk deep in the tail (99% and beyond). At 90–95% it usually reads higher than the historical figure, because the fat tail inflates the fitted sigma — see the table under "Parametric (normal)". Never cite a 95% parametric VaR as evidence that the normal fit is conservative
3. History does not equal the future: historical simulation fails when structural breaks occur (for example, the first negative oil price)
4. Correlation is unstable: correlation matrices observed in normal markets can collapse in crises (correlations trend toward 1)
5. Monte Carlo seed: always pass seed= to monte_carlo_gbm and quote it in the report, so the numbers can be regenerated; use at least 10,000 paths for stability
6. Holding-period scaling: the square-root-of-time rule only applies under i.i.d. returns; it becomes inaccurate under autocorrelation
7. Risk in backtests: metrics.csv already includes max_drawdown and sharpe; this skill provides deeper analysis
8. Sign discipline: every measure here returns a loss as a positive number. Do not re-derive a measure inline to "get the sign you want" — call the function and flip once when printing