A Downside-Risk Framework for the Index Book

Date: 2026-06-07 Author: OLTA Research Abstract: Sharpe is a symmetric, Gaussian-world summary, and crypto returns are neither symmetric nor Gaussian. This paper documents the downside-risk framework OLTA now computes for every index in the book: historical-simulation Value-at-Risk at the 7-day and 30-day holding horizons, sitting alongside the Sortino ratio, conditional Value-at-Risk, tail beta to Bitcoin, and drawdown-recovery statistics already in the engine. We set out the method, the order-statistic convention, the horizon logic, and what the cross-book VaR distribution reveals about which families carry structural downside and which do not. The headline reading is consistent with the Sharpe leaderboard but adds information the Sharpe number cannot carry: the cross-asset and tokenised-equity families clear a 7-day VaR band of roughly five to nine percent, the crypto-pure families a band of roughly twelve to twenty percent, and the gap is a property of construction rather than of the window.

1. Why Sharpe is not enough

The Sharpe ratio answers one question well: how much excess return did a basket earn per unit of total volatility. It is the right first number and OLTA reports it on every surface. But it is a second-moment statistic. It treats an upside surprise and a downside surprise as equally costly, and it assumes the return distribution is well-behaved enough that mean and variance describe it. Crypto violates both assumptions.

The violation is measurable. Empirical kurtosis on a daily Bitcoin return series over the trailing two years runs well above the Gaussian benchmark of three; on a concentrated altcoin basket it runs several multiples higher again. Heavy kurtosis means the worst day in a window is not two or three standard deviations from the mean, as a normal distribution would predict, but four or five. A parametric Value-at-Risk that scales a daily standard deviation by a normal quantile will systematically understate the loss an allocator actually faces, because the parametric model has no term for the fat left tail that crypto reliably produces.

The institutional consequence is direct. A basket can post an attractive Sharpe and still carry a left-tail signature that the Sharpe number does not surface. Two baskets with identical Sharpe ratios can have very different answers to the only question a risk committee asks before sizing a position: how much can this lose over the holding period I actually care about. The downside-risk framework exists to answer that question natively, on the same surface as the Sharpe, for every index in the book.

2. What the framework measures

OLTA computes a downside-risk stack rather than a single number. Each layer answers a different version of the loss question.

Value-at-Risk (VaR). The maximum expected loss over a holding horizon at a stated confidence level. OLTA reports historical-simulation VaR at 95 percent confidence over a 7-day and a 30-day horizon. Read as: with 95 percent confidence, the basket does not lose more than the VaR figure over the horizon. The fifth-percentile reading is the threshold of the worst one-in-twenty outcomes.

Conditional Value-at-Risk (CVaR), also expected shortfall. The average loss in the tail beyond the VaR threshold. Where VaR is the edge of the worst five percent, CVaR is the mean of that worst five percent. CVaR is the more honest tail number because it is sensitive to how bad the tail is, not just where it begins. A basket whose losses pile up just past the VaR line and a basket whose losses run far past it can share a VaR but differ sharply in CVaR. The ratio of CVaR to VaR is itself a fat-tail diagnostic: near 1.25 is the Gaussian expectation, materially above that is a fat-tail signature.

Sortino ratio. Return per unit of downside deviation, where the deviation counts only returns below a minimum acceptable return. OLTA uses a minimum acceptable return of zero, so the Sortino asks the loss-only version of the Sharpe question and rewards baskets whose volatility is mostly to the upside. The convention is documented in the methodology paper.

Tail beta to Bitcoin. The sensitivity of basket returns to Bitcoin returns conditional on Bitcoin's worst days, estimated on the bottom-decile Bitcoin sessions rather than the full sample. Ordinary beta averages calm and crisis together; tail beta isolates how the basket behaves precisely when the market is breaking, which is the only regime in which the number matters.

Drawdown and recovery statistics. Maximum peak-to-trough drawdown, and separately the recovery profile from a deep drawdown. The two are distinct: a basket can have a moderate maximum drawdown that takes a long time to heal, or a deep one that recovers quickly, and an allocator sizing a position needs both the depth and the time-underwater.

This paper foregrounds the VaR layer because it is the most recently computed across the entire book and because the holding-horizon framing maps directly onto how allocators actually think about sizing. The other layers are documented for completeness and were established in prior work.

3. Historical simulation, not parametric

OLTA computes VaR by historical simulation: it reads the empirical distribution of the basket's own realised returns and takes the relevant percentile directly. It does not fit a normal distribution and read a quantile off the fitted curve. This choice is deliberate and it is the institutional standard for assets with fat tails.

The mechanics are simple and reproducible. For a holding horizon of h days, the engine forms the distribution of overlapping h-day returns from the basket NAV series: each observation is the return from day t minus h to day t. It sorts that distribution ascending and reads the fifth-percentile order statistic. The result is reported as a negative fraction, the loss. The same order-statistic convention is used at every horizon, so the figures reconcile against each other and against the one-day tail-risk export the engine already produced; at a one-day horizon the method reduces exactly to that one-day historical VaR.

Two design points are worth stating because they distinguish an honest VaR from a convenient one.

Overlapping multi-day returns, not square-root-of-time scaling. The textbook shortcut is to compute a one-day VaR and scale it by the square root of the horizon. That shortcut assumes returns are independent and identically distributed across days, which crypto returns are not: they cluster, they autocorrelate in stress, and the fat tail at a multi-day horizon is worse than the one-day tail scaled up. OLTA instead reads the empirical distribution of actual h-day returns, so the figure captures the basket's own multi-day path dependence rather than importing an independence assumption the data rejects.

A statistical floor, and honest unavailability. A percentile is only meaningful with enough observations behind it. The engine requires a minimum sample of overlapping observations before it will report a horizon, which in practice means a 30-day VaR needs roughly two months of history and longer is better. A basket too young for a horizon has that horizon reported as unavailable rather than estimated from too little data. Likewise, if a window is so short and so strongly upward that the fifth percentile is not actually a loss, the engine returns unavailable rather than advertising a misleading zero-risk reading. The framework would rather show nothing than show a false precision.

The cost of historical simulation is that it can only see losses the basket has actually lived through. A window that did not contain a particular kind of shock cannot price that shock. This is the central limitation and Section 7 treats it directly.

4. The cross-book VaR distribution

VaR is now computed for the full index book over each basket's longest available window. Reported here in banded form by family, because the per-basket figures and the construction inputs behind them sit in the methodology brief, not on the public surface.

The pattern is unambiguous and it lines up with everything the Sharpe and stress-test work has shown. Lower-correlation construction produces a materially lighter left tail.

Three readings follow.

First, the Diversified and tokenised-Equities families clear a 7-day VaR band of roughly five to nine percent, while the crypto-pure families sit at roughly twelve to twenty percent and beyond. That is a two-to-threefold difference in the worst-one-in-twenty weekly loss, between families holding the same calendar. The cross-asset and equity baskets are not merely higher-Sharpe; they are structurally less able to lose a large amount in a week.

Second, the ordering at 7 days and at 30 days is the same. The families with the lightest weekly tail also have the lightest monthly tail. The downside advantage does not wash out as the horizon lengthens, which is what one would expect if it were a path-smoothing artifact rather than a structural property.

Third, VaR severity tracks tail beta to Bitcoin almost mechanically. The families with heavy VaR are the families whose tail beta to Bitcoin sits near or above one; the families with light VaR are the families whose tail beta sits well below one. The driver of the downside difference is the same driver as the Sharpe difference: correlation to the single asset that dominates crypto's left tail. A basket that does not move with Bitcoin on Bitcoin's worst days does not inherit Bitcoin's worst days.

5. VaR and Sharpe tell the same story, with one exception class

For most of the book the downside stack and the Sharpe leaderboard agree. The Diversified family is high-Sharpe and light-tailed. The Sector and Ecosystem families are low-Sharpe and heavy-tailed. An allocator screening on either number reaches the same shortlist.

The framework earns its keep on the exceptions, where the two numbers diverge and the Sharpe alone would mislead a sizing decision. There are two recognisable patterns.

The concentrated high-Sharpe basket with a fat single-day tail. A small basket of highly intra-correlated names can post an excellent Sharpe over a favourable window and still carry a single-day loss far outside what the Sharpe implies, because the basket is functionally one position with several tickers. The Sharpe rewards the long run of positive sessions; the tail records the one session where the correlated names all fell together. Here the CVaR and the worst-day statistic, not the Sharpe, are the correct sizing inputs, and the right response is an explicit position cap rather than a Sharpe-proportional weight.

The label-versus-behaviour mismatch. A basket can carry a family label implying a hedged, differentiated risk character while its constituents trade with the high-beta crypto complex. The tail beta and VaR expose the mismatch directly: the basket behaves like the cluster it actually trades with, not the cluster its name implies. An allocator relying on the family label to estimate risk would be miscalibrated, sometimes by a factor of two, and the downside stack is what catches it.

Both patterns share a structure: a Sharpe-only screen sizes the position, and the tail the Sharpe could not see determines whether that size survives a bad week. This is the precise gap the framework closes. The institutional discipline is to read VaR and CVaR alongside Sharpe and to let the binding constraint set the size.

6. Using the framework at order time

A risk number is only useful if it reaches the decision. OLTA surfaces the horizoned VaR on the trade surface so the loss figure sits next to the size the allocator is choosing, in the holding-horizon framing the allocator already uses. The intent is to make the downside the allocator is accepting legible at the moment of the trade, rather than buried in a methodology appendix.

The framework supports three institutional sizing disciplines without prescribing any of them.

Horizon-matched reading. An allocator with a one-week tactical horizon reads the 7-day VaR; an allocator rebalancing monthly reads the 30-day VaR. Matching the VaR horizon to the actual holding period is the single most important discipline, and it is the reason the framework reports two horizons rather than one annualised abstraction.

Tail-budget sizing. A position can be sized so that its contribution to a portfolio-level tail budget stays within a stated limit. Given a target tail loss at the portfolio level, the maximum allocation to a basket is the size at which the basket's CVaR contribution consumes the remaining budget and no more. This is the conversation institutional risk teams already have internally; the framework supplies the per-basket inputs that conversation needs.

Cap-and-disclose for the exception class. For the concentrated high-tail baskets identified in Section 5, the discipline is an explicit position cap independent of the Sharpe-implied weight, with the worst-day and CVaR figures shown so the allocator sees why the cap exists. The constraint is informational and not a block: the framework's posture is to give allocators more context, not more guardrails.

None of these disciplines requires the allocator to recompute anything. The figures are pre-computed for every basket and bound to the surface directly.

7. Limitations and caveats

The framework is honest about what it can and cannot see. The limitations below are not incidental; they bound the correct interpretation of every number above.

Historical simulation prices only what happened. A VaR read from a window can only reflect the losses that window contained. A window that did not include a particular shock cannot price that shock, and a basket whose worst historical day was moderate will show a moderate tail even if its construction could in principle produce a worse one. The figures are descriptive of a realised distribution, not predictive of an unrealised one.

One day dominates the recent crypto tail. A single market-wide risk-off session in the recent window was the worst day for the large majority of crypto-correlated baskets simultaneously. The two-year tail picture is therefore conditioned heavily on one event. The correct forward-looking stress is a recurrence of that kind of day, not an average across calm and crisis treated as independent draws. An allocator should weight the conditional tail, given that such a day occurs in a forward window, above the unconditional VaR.

The tail-correlation assumption can break. The cross-asset families are light-tailed because their legs have historically not crashed together. That benefit depends on the historical cross-asset correlation structure persisting. A macro regime in which the hedge legs and the crypto leg all sell off at once, the everything-correlated-to-one scenario, would compress the diversification benefit precisely when it is most needed. The framework reports the historical tail; it cannot guarantee the correlation regime that produced it.

Window dependence and horizon overlap. As with every figure in this collection, the VaR is conditioned on a specific window and a specific regime, and a different window would produce different numbers. Overlapping h-day returns are also serially dependent by construction, so the effective sample behind a multi-day percentile is smaller than the raw observation count; the minimum-sample floor mitigates but does not eliminate this.

No transaction costs, no risk-free rate, survivorship. Consistent with the rest of the collection, the underlying NAV series carry no transaction-cost or slippage model, the Sortino uses a zero minimum acceptable return, and the constituent set is today's set. These conventions are documented in the methodology paper and apply unchanged here.

VaR is a threshold, not a worst case. A 95 percent VaR says nothing about the worst five percent except where it begins. The maximum loss can exceed the VaR by a wide margin; that is what CVaR is for, and why the framework reports both rather than VaR alone.

8. What is disclosed and what is not

This paper publishes the framework: the metrics, the historical-simulation method, the order-statistic and horizon conventions, the statistical floor, the cross-book reading in banded form, and the order-time disciplines the framework supports. A desk that wants to build the same downside stack can do so from this material and the public methodology paper.

The methodology brief reserves the execution-relevant detail: the per-basket VaR, CVaR, and tail-beta figures; the full per-basket correlation matrices; the exact constituent weights behind each basket's NAV series; and the per-basket sizing-cap parameters. This follows the institutional convention used across this collection and by established index providers, which is to publish the rules and methodology in full while reserving the per-constituent recipe and any front-runnable specifics. The framework is credible because it is independently modelable; it is protected because the per-basket inputs are not disclosed.

The brief is available to institutional counterparties and grant reviewers under standard research-distribution terms.

9. References

• Engine source: lib/backtest.js
• VaR export and method: scripts/export-var.mjs; consolidated figures lib/var-results.js (auto-generated)
• Tail-risk metrics (CVaR, tail beta, drawdown clusters, recovery half-life): one-day tail-risk export and methodology brief
• Sharpe, Sortino, drawdown, and beta conventions: 01-methodology.md
• Crisis-window behaviour by family: 04-stress-test-report.md
• Sharpe driver attribution at the framework level: 05-sharpe-decomposition.md
• Academic basis for crypto tail behaviour: Borri (2019), "Conditional Tail-Risk in Cryptocurrency Markets," Journal of Empirical Finance; and the crypto risk-return literature review in research/literature/06-crypto-risk-and-return.md

Methodology version 2026.06. Per-basket VaR, CVaR, tail-beta, and the sizing-cap parameters are documented in a private appendix issued to institutional counterparties and grant reviewers under NDA on request. Every number traces back to the engine in lib/backtest.js and the export in scripts/export-var.mjs.

OLTA Research Desk · 2026-06-07