🛢️Crude OilCL

Asset thesis · why this asset is different

0.Two-regime by construction

Crude oil is the asset that most cleanly justifies a Markov regime switch. OPEC+ cuts and surplus periods produce visibly different vol clusters, which is why the page's regime panel matters more here than anywhere else on the site. Cornish-Fisher handles the unconditional VaR; the regime probability is the conditional layer that says when to scale exposure.

• EIA crude inventory release (Wednesday 10:30 ET) is a scheduled vol event the page detects without being told.
• WTI–Brent spread proxies US export capacity; a widening spread typically precedes US production growth.
• Negative-price event of April 2020 (CL went to −$37) is the historical reason FHS is included as a method but not the default — that observation distorts the tail when the regime has changed.

Part I · the daily log-return series

1.From prices to returns

Everything that follows works on the daily log return $r_t = \log(P_t / P_{t-1})$. Log returns are time-additive, well-behaved under aggregation, and the natural input for any vol / VaR model. The series below is exactly that, computed over 576 trading days from Supabase data.

The annualised vol used in the EWMA model is the current value from the filter, not a pooled sample estimate. Pooled σ across the entire history would mask the regime structure CQF Module 2.5 cares about.

Part II · stylized facts (CQF Module 2.4)

2.The four moments

Returns are described by their first four sample moments. The first two get most of the airtime; the third and fourth are what break Gaussian VaR.

The Gaussian benchmark has $\gamma_1 = 0$ and $\gamma_2 = 0$. The empirical values on this asset, computed on the live series above, are $\gamma_1 = -0.070$ and $\gamma_2 = 16.03$. Both numbers feed directly into the Cornish-Fisher VaR adjustment in Part III.

3.The empirical density vs normal

Plotting the histogram of $r_t$ against a normal fitted to the same $(\mu, \sigma)$makes the failure of the Gaussian assumption visible. Bars in the < μ − 2σ tail are coloured red; bars in the > μ + 2σ tail are green. A normal would have a fixed 2.5% mass in each tail; if the empirical bars overshoot the curve in those regions, you have fat tails.

4.The Q-Q diagnostic

The Q-Q plot pairs the $i$-th empirical order statistic with the corresponding quantile of $\mathcal{N}(\mu, \sigma^2)$. Under normality every point sits on the 45-degree gold reference line. Departures at the lower-left say the left tail is heavier than normal (extra crash risk); departures at the upper-right say the right tail is heavier (extra rally risk). The two ends are where Gaussian VaR actually fails.

5.Autocorrelation: efficient-market & ARCH

Three autocorrelation panels tell three different stories. The ACF of $r_t$tests whether returns themselves are predictable — efficient-market theory says they shouldn't be, and they typically aren't. The ACF of $|r_t|$ and $r_t^2$ tests for volatility clustering: even though returns are unpredictable, their magnitudeis highly autocorrelated. That's the textbook motivation for any ARCH-family model.

6.Stationarity in the second moment

The rolling 60-day annualised mean and standard deviation make the regime structure visible. A stationary series has both lines hovering around their long-run averages; a non-stationary one has the volatility wandering across regimes. For commodities, the latter is the rule.

7.Findings

Six tests, each with its own per-asset value and verdict. The point isn't to fail facts — most pass on most assets — it's to document that we checked.

Part III · VaR method comparison

8.Gaussian (the baseline)

The textbook VaR at level $\alpha$ under a Gaussian return assumption is $\mathrm{VaR}_\alpha = -(\mu + z_\alpha \sigma)$ where $z_\alpha = \Phi^{-1}(1-\alpha)$. Quick, closed-form, and wrong in either tail of any real-world return series.

9.Cornish-Fisher (Edgeworth correction)

Cornish-Fisher replaces the Gaussian quantile $z_\alpha$ with a polynomial adjustment that absorbs skewness and excess kurtosis.

For a left-skewed, heavy-tailed return series the cubic in $z_\alpha$ pushes the quantile further into the tail than Gaussian, so $\mathrm{VaR}_\alpha^{CF} > \mathrm{VaR}_\alpha^{\text{G}}$. On this asset the empirical $\gamma_1 = -0.070$, $\gamma_2 = 16.03$ drive that correction directly.

10.Filtered Historical Simulation

FHS sits between parametric and non-parametric. The recipe:

1. Estimate a volatility model — here EWMA with λ from MLE.
2. Compute standardised returns $\tilde r_t = r_t / \hat\sigma_t$.
3. Read the empirical $(1-\alpha)$ quantile $q_{1-\alpha}$ from the standardised distribution.
4. Scale back: $\mathrm{VaR}_\alpha^{\text{FHS}} = -q_{1-\alpha} \cdot \hat\sigma_T$.

The non-parametric quantile lets the actual tail shape speak; the EWMA scale lets the vol model react to recent days. Glasserman calls this “the right compromise between Gauss and history.”

11.All three, on the same histogram

The cleanest way to see how the three methods disagree is to draw all three thresholds on the same returns distribution and slide the confidence level around. At low $\alpha$ the three almost coincide. As $\alpha$ approaches 99% the gap widens — and whichever method is most conservative on this asset is the one that was best aware of the tail.

12.Numerical comparison

Part IV · VaR backtest (Kupiec + Christoffersen)

13.Why backtest a VaR at all

Reporting $\mathrm{VaR}_{0.95}$ without checking how often the realised loss exceeded it is the cardinal sin of risk reporting. A correct VaR model breaches at exactly the unconditional rate $p = 1 - \alpha$, and the breaches should be independent in time. Kupiec (1995) and Christoffersen (1998) gave us likelihood-ratio tests for both properties.

14.Kupiec POF (unconditional coverage)

With $x$ breaches in $n$ days, the empirical breach rate is $\hat p = x/n$. The LR statistic is $\chi^2(1)$ under the null that $\hat p = p$.

High p-value means we cannot reject correct coverage; low p-value means the breach rate differs from the target in a way that can't be explained by sampling noise.

15.Christoffersen (independence)

Even when the total breach count is right, breaches that cluster signal a vol model too slow to react. Christoffersen builds a 2×2 transition table over the breach indicator and tests whether $P(\text{breach}_t \mid \text{breach}_{t-1}) = P(\text{breach}_t \mid \text{no breach}_{t-1})$.

with $\pi_{ij}$ the empirical transition rates from state $i$ (no breach / breach) to state $j$. The combined conditional-coverage test adds the two LRs and tests against $\chi^2(2)$ — both at once.

16.Results on this asset

Run on a rolling 60-day window with the recommended VaR method:

17.Breach timeline

Each blue tick is a day with no breach; gold dots trace the negative of the VaR threshold; red ticks are breaches. Clustered red ticks are what Christoffersen catches.

18.Year-by-year breach distribution

Counting breaches per calendar year against the expected rate makes regime episodes obvious. Bars in green sit at or below the gold expected-rate tick; bars in red overshoot.