Risk Measures Var Cvar

Risk measures: VaR and CVaR

"The 99 % VaR is 2.3 %" is not a fact until three things are stated: the estimator, the horizon, and how the number was scaled to that horizon. All three are choices, and on the same data they move the answer by more than the risk does.

Every number below is printed by scripts/var_cvar.py (numpy / scipy, seed 0, about 6 s). The DGP is a GARCH(1,1) with alpha 0.08, beta 0.90 and unit-variance Student-t(5) innovations at 1 %/day unconditional volatility, so the truth is known: the unconditional quantile from a 1,000,000-day path, and the conditional quantile in closed form.

🔑 Sign convention here: every VaR and ES is a POSITIVE LOSS, losses = -returns.

1. Four estimators, one tail, and they do not agree

✅ Measured on a 2,500-day sample, 99 % 1-day, against the 1,000,000-day truth:

• The normal estimator is not slightly wrong, it is wrong in the direction that matters: it understates ES by 35 % on fat-tailed data. ✅ The true Student-t(5) conditional quantile at 99 % is 2.606 sigma against the normal's 2.326, and the conditional ES is 3.449 sigma against 2.665.
• 🚨 Cornish-Fisher stops being a quantile function. ✅ At skew 0 the expansion's derivative is 1 + (3z^2 - 3)K/24, minimised at z = 0 where it equals 1 - K/8: the break is at excess kurtosis 8 exactly (a 0.05-step search reports 8.05). This sample has skew -0.75 and excess kurtosis 12.59, and ✅ its expansion is decreasing over z in [-0.50, +0.67], which is to say the "74.9 % VaR" it returns (-0.114 sigma) is smaller than the "30.9 % VaR" (+0.280 sigma). A function that goes down as the confidence level goes up is not a quantile function. The +56 % at 99 % is not conservatism, it is a series evaluated outside its domain. Call cornish_fisher_is_monotone(skew, exkurt) before using it - a daily equity index is routinely past the threshold.
• EVT is the best of the four and still 7 % low, because 2,500 days contains 25 tail observations. ✅ The fit: tail index xi = 0.115 (a genuinely heavy tail), scale 0.00586, 250 exceedances over the 90th percentile. Its closed-form ES agrees with numerical integration of the fitted Pareto tail to four decimals: 3.3428 % both ways.
• ✅ Source-verified numerically in scipy 1.13.0: genpareto.pdf(x, c) equals (1 + c x)^(-1 - 1/c) for c != 0 and exp(-x) at c = 0, so scipy's shape parameter c is the tail index xi with no reparameterization. Fit the exceedances with floc=0; leaving loc free re-estimates the threshold you already chose.

⚠️ Threshold choice is a researcher degree of freedom: the 90th percentile is a convention, not a result. Report the threshold, the exceedance count and xi alongside the number, and record the choice as a trial in ../../../fin-core/skills/backtest-validation/SKILL.md.

2. The rolling backtest: count and timing are different failures

✅ Measured: 500-day rolling window, 2,000 test days, 99 % 1-day VaR. pi11 is P(exception | exception yesterday); the unconditional rate is 1 %.

The two halves of the problem are separable. The EWMA filter supplies the right scale day by day but a normal quantile still misses the shape, so EWMA + normal fails the count at p = 0.013 while carrying the lowest mean VaR of the lot (2.019 % - it is cheap and wrong). Filtered historical simulation keeps the EWMA scale and takes the empirical quantile of the standardized residuals, and it passes everything with 21 exceptions against 20 expected. Cornish-Fisher errs the other way: 15 exceptions, and ✅ 36 % more capital than the oracle's mean VaR (3.180 % against 2.346 %) for the privilege.

🚨 The independence test rejects nothing here - and that is not evidence of independence. pi11 runs 4 to 7 times the unconditional 1 % in every unconditional method, yet no p-value is below 0.10, because 20 exceptions in 2,000 days produce one or two consecutive pairs to test on. Section 3 measures what that is worth.

3. The backtests themselves, and their real size

✅ The statistic, computed two ways: kupiec_pof(28, 2000, 0.01) returns 2.8748121042 and -2 * (binom.logpmf(28, 2000, 0.01) - binom.logpmf(28, 2000, 28/2000)) returns 2.8748121042 (difference 2.1e-14). The binomial coefficient cancels between the two hypotheses, which is why the likelihood ratio may be written either way.

• Kupiec proportion-of-failures: LR_uc = -2 ln[(1-p)^(n-x) p^x / ((1-x/n)^(n-x) (x/n)^x)], chi2(1). ✅ It is exactly 0 when x = p n.
• Christoffersen independence: a first-order Markov chain on the hit sequence against a constant hit probability, LR_ind, chi2(1). ✅ It is 0 when pi01 = pi11.
• Conditional coverage: LR_cc = LR_uc + LR_ind, chi2(2).

🚨 Their nominal size is not their real size. ✅ Measured on 10,000 iid Bernoulli(0.01) hit sequences of 500 days, nominal 5 %:

The hit count is a discrete random variable with about 5 expected exceptions in 500 days, so the chi2 approximation is coarse in both directions. A passed independence test at the 99 % level is close to no evidence at all; at the 95 % level there are five times as many exceptions and the test has something to work with.

🚨 And "independence rejected" does not mean clustering. ✅ Measured: a perfectly regular pattern - exactly one exception every fifth day - gives LR_ind = 50.1, p = 0.000. The test compares pi01 with pi11 and fires just as loudly on under-dispersion as on clustering.

🚨 Whenever pi01 = pi11 it sees nothing, however predictable the sequence. ✅ Measured: the block pattern 1,1,0,0 repeated has pi01 = pi11 = 0.50, so LR_ind = 2.5e-03, p = 0.960 - a sequence you could forecast perfectly passes the independence test. The Markov alternative only looks one day back; anything structured at a longer lag is invisible to it by construction.

4. 🚨 sqrt(h) scaling is wrong in both directions

✅ Source-verified in the RiskMetrics Technical Document, Fourth Edition (December 1996), Sec. 5.2, Eq. [5.21]: the square-root-of-time relation is derived inside the EWMA/IGARCH model, and the document itself warns that the rule normally "results from the assumption that variances are constant" while in its own derivation they vary. It names three cases where scaling up is problematic: mean-reverting rates or prices, boundaries that limit movements, and using a volatility estimate optimised for one horizon at another.

✅ Measured, 99 %, loss = the sum of h daily returns:

The sign of the error is not fixed. Fat tails alone make sqrt(h) overstate, because a sum of h draws is closer to normal than one draw is. Starting from a quiet day makes it understate, because volatility mean-reverts upward over the horizon. "sqrt(h) is conservative" is false as often as it is true, and the direction depends on the state you are in when you compute it - which is exactly when you care.

✅ And the error grows with the horizon:

A quarterly VaR scaled from a quiet day understates the truth by 29 %. The guard. scale_var_sqrt_time(var, h) refuses unless you pass iid_normal_zero_mean=True, so the assumption appears in the code that relies on it. Simulate the horizon from the current conditional variance instead (horizon_var_from_state).

5. VaR is not subadditive; ES is

✅ Measured, exactly: two independent positions, each losing 1.0 with probability 2 %, at the 97.5 % level.

VaR(A+B) > VaR(A) + VaR(B): diversification makes the VaR go up, which is not a paradox to be explained away - it is why VaR is not a coherent risk measure and why an optimizer that minimises VaR can be told to concentrate. ES obeys the inequality. Optimize and report ES; quote VaR when a regulator asks for it. ../portfolio-optimizers/SKILL.md has the linear program that minimises ES directly.

6. Traps

• 🚨 A VaR without its estimator, horizon and scaling stated. The four estimators here span 2.14 % to 4.16 % on one sample.
• 🚨 Cornish-Fisher past excess kurtosis 8. It is no longer a quantile function.
• 🚨 A normal quantile on a fat-tailed series. -35 % on ES here.
• 🚨 Passing Kupiec and declaring victory. Count and timing are different failures, and at 99 % the timing test has 1.4 % power against a nominal 5 %.
• 🚨 sqrt(h) scaling. Section 4.
• 🚨 Minimising VaR. Section 5.
• 🚨 Sign and definition mismatches across libraries. ✅ Source-verified in quantstats 0.0.81: qs.stats.value_at_risk(returns, sigma=1, confidence=0.95) returns norm.ppf(1 - confidence, mu, sigma * std) - a parametric normal VaR as a negative return, not an empirical quantile and not a positive loss (measured: -0.016236 on a seeded normal series). conditional_value_at_risk is the mean of returns[returns < var], also negative (-0.020824), and it uses a strict <. Its confidence is a confidence level, not a tail probability, and a value above 1 is silently divided by 100. See ../../../fin-libraries/skills/lib-quantstats/SKILL.md.
• ⚠️ Two historical-ES conventions. The mean of losses at or beyond the interpolated quantile, and the Rockafellar-Uryasev form with the atom at VaR split fractionally. ✅ On this data they agree to four decimals at n = 2,500, 500 and 100 and the VaR differs only in the fourth decimal - but two libraries reporting the two conventions will not tie out to the digit, so say which you used.
• ⚠️ The EVT threshold, the window length, the decay factor and the refit cadence are all trials.

7. Scripts

• scripts/var_cvar.py - all five sections. var_es_historical, var_es_normal, var_es_cornish_fisher, cornish_fisher_is_monotone, var_es_evt with gpd_es_numeric as its independent check, kupiec_pof with kupiec_via_binomial, christoffersen_independence, conditional_coverage, backtest_size_under_null, simulate_garch_t, rolling_var (seven methods including an oracle), horizon_var_from_state, scale_var_sqrt_time (the guard) and discrete_var_es. numpy and scipy only; no optional dependency. About 6 s.

8. Where this sits

../covariance-and-risk-models/SKILL.md supplies the covariance a parametric or portfolio VaR needs, and owns the EWMA recursion used here; ../portfolio-optimizers/SKILL.md minimises CVaR as a linear program; ../../../fin-core/skills/portfolio-and-risk/SKILL.md owns the Sharpe, drawdown and annualization conventions and the wider risk-metric audit; ../../../fin-core/skills/backtest-validation/SKILL.md counts the estimator, threshold and window choices as trials; ../../../fin-libraries/skills/lib-arch/SKILL.md is the deep dive on fitting the GARCH model whose conditional variance the good rows above depend on; and ../../../fin-libraries/skills/lib-quantstats/SKILL.md owns the reporting library whose sign convention is the opposite of this skill's.