Bitcoin's daily tail is a fifth deeper

A Gaussian crypto risk report misses about a fifth of Bitcoin's 99% daily loss quantile. It gives no measure of how far losses travel after a breach. The paper's practical case rests on that gap, which appears across ten coins.

Osterrieder, Chan, Zhang and Chu estimate seven return distributions by maximum likelihood on daily log-returns: Normal, Student-t, skewed generalized-t, generalised hyperbolic, hyperbolic, normal-inverse Gaussian, variance-gamma. AIC and BIC rank the fits. They then calculate Value-at-Risk and expected shortfall at 95%, 99% and 99.5% using four methods. A generalised Pareto fit to the worst decile of days supplies the extreme-value tail indices. Tail risk means the loss size on bad days, measured unconditionally across the full sample.

The dataset uses CryptoCompare daily USD closes from a frozen snapshot pulled 2026-05-30. It begins no earlier than 23 June 2014 and ends in May 2026. Ten coins are included, with n reaching 4,360 days.

Normality is rejected for all ten assets. Bitcoin's Gaussian 99% VaR is 8.1%, versus 10.5% from the empirical distribution and 10.5% from extreme value theory. Its 99% expected shortfall reaches 14.5%. The median generalised Pareto shape, which controls tail thickness, is 0.19 on the left and 0.24 on the right. No strategy is offered or tested. These estimates matter for capital buffers, margin policy and product disclosure.

We could not run any of this. Our crypto price coverage spans roughly 2018 to 2024, leaving both the 2014-2026 window and the provider-controlled 2014-2016 refit beyond our data. No figure of ours appears anywhere below. The discussion instead examines what their specification establishes for a reader who has the required history.

Put the four Bitcoin estimates together: 8.1% of capital for the Gaussian fit, 10.5% empirical, 11.2% fitted Student-t and 10.5% from peaks-over-threshold. Bitcoin's reported 99% expected shortfall is 14.5% of capital, far above its Value-at-Risk. Relative to the 10.5% extreme-value VaR, the breach carries another 4.0 points of depth.

Solvency turns on that depth. VaR tells a lender where the collateral disappears. Expected shortfall estimates how far losses continue beyond that threshold.

Dogecoin puts the difference in capitals: extreme-value VaR of 23.8% and extreme-value ES of 46.0%, with an average breach nearly twice the threshold already crossed.

Seven densities, ten coins

The authors fit seven densities by maximum likelihood to daily log-returns, then rank them by AIC and BIC. Penalised fit chooses the winner rather than raw likelihood. VaR and ES are estimated at 95%, 99% and 99.5% through four methods. One of the four fits a generalised Pareto to exceedances above the upper tenth percentile. The authors publish an extreme-value index only when at least fifty exceedances exist. Counts in the tail-index table range from 224 for Solana to 436 for Bitcoin, Litecoin, Dogecoin and Dash. The left tail also gets Hill estimates. Jarque-Bera throughout.

Ten assets enter the study. Six cover the full decade: Bitcoin, Ripple, Litecoin, Monero, Dogecoin and Dash, with n ranging between 4,139 and 4,360. Ethereum contributes 3,949 days from 2015-08-08, followed by BNB with 3,204, Cardano with 3,163 and Solana with 2,241 from 2020-04-11. MaidSafeCoin appears solely in the 2014-2016 replication.

The BNC2 volume-weighted composite used by the 2016 predecessor study has been discontinued. The authors therefore refit the early window using the new feed and compare fitted Student-t degrees of freedom with the published values. Median relative change is 9%. Bitcoin remains at 1.82, while Dogecoin shifts from 1.75 to 0.98, a 44% deviation.

All ten assets reject normality. Excess kurtosis stretches from 7.21 for Solana to 46.32 for Dogecoin. For nine of ten assets, the Gaussian produces the lowest of the four 99% VaR estimates. Ripple alone reverses the ordering, at 16.5% Gaussian versus 16.1% historical.

Full-sample Student-t degrees of freedom across the panel range from 1.59 to 3.28. Solana occupies the upper end, though its 2,241 days beginning in April 2020 are left out of the longitudinal comparison. Restrict the analysis to the six-coin cohort carrying the decade claim and the range runs from 1.59 for Dogecoin to 2.57 for Monero. Ten years of liquidity growth brought modest tail moderation at best. The management sections recommend heavy-tailed or extreme-value expected shortfall for capital and limit setting. They also call for retail disclosure of an explicit "worst one percent of days" figure.

Where replication requires judgment

Much of the recipe is precise. The threshold percentile, exceedance minimum, per-asset start dates and snapshot date are printed. The despiking rule for isolated bad prints is also specified: rolling median, window five, ratio 2.5, with 25 prices replaced. Appendix densities are available, alongside fitted Normal and Student-t parameters with standard errors. For Bitcoin, nu is 2.23 with s.e. 0.10, location is 0.126%, and scale is 1.879%. The seven-density log-likelihood table for the cohort is printed as well.

Three decisions remain with the replicator. The printed skewed generalized-t density includes an m and a nu, described as centring and scaling the distribution. We did not find their expressions. Since that family is BIC-best for five of ten assets, reproducing the selected model requires choosing the same convention without guidance from the text. We did not find the Hill tail fraction k either.

Third comes despiking. The paper's data provenance record counts the 25 replaced prints by coin. Data and code can be requested from the corresponding author, which means an outside run cannot identify the replaced observations directly. The mechanism itself is explicit. A price is replaced only if its distance from the local five-day median exceeds the stated ratio. One-day round trips are removed, while sustained moves such as the 2021 Dogecoin rally remain. The authors also name despiking among their limitations because it is a modelling choice.

One conversion matters when turning the 99% VaR table into capital. These estimates are log-return quantiles labelled as percent daily loss. In simple-return terms, Bitcoin's 10.5% becomes a 9.97% capital loss, while Dogecoin's 23.8% becomes a 21.2% loss. That arithmetic is ours. The paper itself flags the log convention when discussing Dogecoin's -125.56% minimum, equal to -71.5% simple.

Two BIC points picked Bitcoin

The abstract says the generalised hyperbolic is selected "for at most one" asset. It also reports that the lighter Akaike penalty shifts four assets toward that family. The model-selection section adds that plain Student-t is AIC-best for none of the ten. The ranking therefore depends on the penalty, as the authors acknowledge.

The fits are close by their own account. Section 3.2 explains the use of a penalty this way: "Because the heavy-tailed families fit almost equally well in likelihood, selection should penalise parameters." The appendix says the ordering "corroborates the main-text finding" even though "the differences among heavy-tailed families are small". Their response is that the family-level conclusion holds. BIC rewards parsimony, and the Gaussian is never selected regardless of which heavy-tailed family leads each row. The per-asset winner table consequently carries less information than its prominence suggests.

BIC margins within the heavy-tailed group are narrow. Bitcoin records -18139 for the skewed generalized-t and -18137 for the generalised hyperbolic. Ethereum gives -13151 against -13146. Ripple has -13338 for Student-t against -13335. Dash prints -13371 for both Student-t and generalised hyperbolic, leaving one of the ten contests unresolved in the published table. The filter for isolated prints affected Dash, Dogecoin, Litecoin and one Ethereum quote.

The family-level separation is far larger. Bitcoin's Normal BIC is -16728, fully 1,411 points behind the best value of -18139. No tie-break among heavy-tailed families brings the Gaussian back into contention.

Use the Student-t or the skewed generalized-t. Both converge for every asset, while two appendix densities require a custom optimiser. The BIC table settles the family question. For Bitcoin, plain Student-t trails the best fit by 84 points, at -18055 versus -18139.

Can an unconditional fit set a margin number?

The analysis is explicitly unconditional, which the authors describe as a deliberate sequencing choice. GARCH, regimes, tail co-movement and "formal backtesting of the risk measures estimated here" are reserved for a companion paper. Their defence cites McNeil and Frey (2000): the marginal law is what the dynamics integrate to, and it provides the benchmark a conditional forecast must beat. That works for model comparison. Limits pose a separate problem.

A 99% ES of 14.5%, fitted across 4,360 days, averages over regimes. We found no exception count in the material we read. The reader therefore cannot tell whether the 8.1% Gaussian estimate is breached on 1% of days or 3%.

Implementers also face an unresolved moment question. Fitted degrees of freedom at or below two imply infinite variance, and three assets meet that condition: Ripple at 1.88, Dogecoin at 1.59 and BNB at 2.00. The paper interprets its generalised Pareto shapes, with medians of 0.19 left and 0.24 right, as consistent with finite variance. Panel-wide Hill estimates for the left tail range from 0.43 to 0.62, compared with generalised Pareto left shapes of 0.11 to 0.44. Bitcoin's pair is 0.52 and 0.14. We did not find this disagreement addressed. Scaling a daily estimate to a ten-day horizon forces the implementer to choose an estimator and accept its implications.

Survivors, by the authors' own account

The conclusion identifies both the original cohort and the 2026 entrants as survivors. It also says that "the disappeared coins had the fattest left tails". Every tail estimate here should therefore be treated as a lower bound for the asset class. On attrition, the authors cite Feder et al. and Grobys and Sapkota for rates on the order of half or more of listed coins. Brown et al. is cited for bias that grows with the lookback. MaidSafeCoin, retained for replication and removed after winding down, supplies one in-sample example. The authors draw the proper conclusion. Their estimates already exceed the Gaussian benchmark, and any limits based on them remain floors that require further stress.

On that reading, the paper earns its place. Bitcoin's 14.5% expected shortfall is a floor. Survivor selection pushes it too low because vanished coins had the heaviest left tails. On a like-for-like basis, Gaussian 99% VaR is 8.1%, compared with 10.5% empirical and 10.5% extreme-value. The understatement is roughly a fifth on the measure that determines whether a levered book lasts through the day.

One promised number would change my view on deployment: a rolling coverage test. If the unconditional 99% extreme-value estimates breach near 1% of days out of sample, I will stop asking for conditional volatility. Until then, they belong in disclosure and capital floors. A margin engine still needs a model with memory.