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# 4.86 Crashes Travel on a Different Network Than Returns
- URL: https://aligrithm.com/crashes-travel-on-a-different-network-than-returns/
- Published: 2026-11-04T05:11:00.000Z
- Updated: 2026-11-04T05:10:59.000Z
- Description: Downside lead-lag: estimate bad COVOL and lead-lag strength on negative-return days, then hedge the high out-strength source before followers reprice.
- Author: ali askar
- Tags: 4. Market Structure Notes

An average correlation puts a quiet day and a selloff on the same edge, and so does the symmetric graph in the old article ["Network Momentum as a Cross-Asset Factor."](https://aligrithm.com/network-momentum-as-a-cross-asset-factor/) Pham, Han, Nguyen, Pham and Do estimate common volatility, the shared multiplier on residual variance they call COVOL, separately on positive returns and on negative returns. The largest negative-return shock in their crypto sample is 98.49 on 3 April 2017\. The largest positive-return shock is 58.93, and it falls on 15 September 2016\. You hedge the wrong name if you rank nodes on the blended graph. Treat the two signs as two lead-lag networks, the way a commute map and an evacuation map are two drawings of the same streets, and rank who to short by out-strength on the evacuation map.

## Bad COVOL is the common shock on down days

Each coin's excess return is an AR(1), a regression on its own previous return, plus the first principal component of the 25 excess-return series, with GARCH(1,1) variance (today's variance depends on yesterday's squared residual and yesterday's variance). That block is their equations (1) through (3). What is left is a standardised residual with unit variance on an ordinary day. COVOL is a latent factor, mean one, that rescales that residual when many coins are surprised together. The same fit, run on the positive part of the return and again on the negative part, is good COVOL and bad COVOL.

$$ r\_{i,t}^{+} = \\begin{cases} r\_{i,t} & \\text{if } r\_{i,t} > 0 \\\\ 0 & \\text{otherwise} \\end{cases} \\qquad r\_{i,t}^{-} = \\begin{cases} r\_{i,t} & \\text{if } r\_{i,t} \\le 0 \\\\ 0 & \\text{otherwise} \\end{cases} $$

A positive day is copied into the plus series and zeroed in the minus series, and a negative day does the opposite. Their equation (9) prints a hat on the kept negative return and a tilde in the condition; the two symbols are the same return, and the split above is the rule the paragraph describes. Worked day: a coin return of −3% contributes −3% to the minus series and 0 to the plus series. A +2% day contributes nothing to bad COVOL. You then run the COVOL fit on each series alone.

$$ e\_{i,t} = \\sqrt{g(s\_i\\, x\_t)}\\; \\epsilon\_{i,t}, \\qquad g(s\_i\\, x\_t) = s\_i\\,(x\_t - 1) + 1 $$

Here x is the COVOL factor on that series and s is the coin's loading, how much of the common shock it absorbs. On an average day x equals 1, g equals 1, and the residual stays at unit variance. Worked on their bad-COVOL pair, which belongs to the same fit: Bitcoin's bad loading is 0.243 and bad COVOL on 3 April 2017 is 98.4933\. Then g equals 0.243 times 97.4933, plus 1, which is 24.69\. The volatility multiplier is the square root, 4.97\. That day's common negative shock made Bitcoin's residual about five times as wide as on a day when bad COVOL sits at its mean of 1\. The same arithmetic on 12 March 2020, bad COVOL 43.5893, gives g of 11.35 and a multiplier of 3.37.

The two fits produce two rankings. Solana's bad-COVOL loading is 0.2148, ninth of the 25 coins. Its good-COVOL loading is 0.16, twenty-second, with only LEO, MNT and TON printed lower. Binance Coin is seventh on bad COVOL at 0.2195 and nineteenth on good COVOL at 0.18\. The good-side column is printed to two decimals, so trust the rank more than the hundredths. Bitcoin is first on bad COVOL at 0.243\. The prose says it ranks seventh on good COVOL; the printed good-COVOL column places it eighth, after Litecoin, at 0.22\. Keep the table. Bitcoin's full-sample loading in their Table 2 is 0.2311, between the bad-list value and the good-list value, and that single number describes neither list.

Good COVOL exceeds bad COVOL on 1,754 days in the sample, and bad exceeds good on 1,588 days. The gap is their equation (10).

$$ d\_t = \\hat\\chi\_t^{\\sigma,+} - \\hat\\chi\_t^{\\sigma,-} $$

Positive d means the common shock that day was larger on up moves. Negative d means the common shock was larger on down moves. A single day with good COVOL at 4 and bad COVOL at 9 has d equal to −5, and that is the day you read the downside graph. The old article ["Good vs Bad COVOL in Crypto: A Common-Volatility Tilt"](https://aligrithm.com/good-vs-bad-covol-in-crypto-a-common-volatility-tilt/) turns a 20-day average of d into a 0–100 index and tilts a long crypto book. That tilt is a position in the whole market. Which coin leads the others when d is negative is a separate graph.

## Downside semivariance is how hard the follower falls

Lead-lag strength, below, is a pure number. Downside semivariance is what tells you whether the follower's down move is a scrape or a hole. Pu, Roberts, Dong and Zohren report a Sortino ratio as expected return over downside deviation, and they never print the deviation. The deviation their ratio needs, with a target of zero, is the square root of the average squared shortfall:

$$ \\sigma\_{i,-} = \\sqrt{\\frac{1}{T}\\sum\_{t=1}^{T}\\min(r\_{i,t},\\, 0)^{2}} $$

Positive days contribute zero inside the sum. The semivariance is the square of this deviation. Worked path, four days, +2%, +1%, −3%, −4%. The squared shortfalls are 0, 0, 0.0009 and 0.0016\. The average is 0.000625, so the downside deviation is 2.50%. The average of all four squared returns is 0.00075, and the down days are five sixths of that. A full-sample variance, or a full-sample correlation, spends most of its weight on a path whose squared movement is already a down-day fact, and still lets the two up days vote.

You keep a follower in the crash graph when its downside deviation is large enough to be worth hedging, and you size that hedge with the deviation. A tail day, for the sample you estimate on, is a day in that asset's own bottom decile of returns, or a day when bad COVOL itself is in the extreme upper tail. Both 3 April 2017 and 12 March 2020 sit on the short list of largest bad-COVOL dates in their Table 5\. The decile cut is arbitrary. None of the four papers estimated a lead-lag network inside it.

## The edge has a direction, and you estimate it on down days

Bender, Cestonaro and Schmidt measure lead-lag on irregular trade times with the Hayashi-Yoshida correlation, shifted by a lag. The old article ["Lead-Lag Is Multidimensional: Trade Price and OBI Predict Others' Midpoints"](https://aligrithm.com/lead-lag-is-multidimensional-trade-price-and-obi-predict-others-midpoints/) is the long version. The estimator you need is the overlap of increments after you move one series:

$$ \\hat\\rho(\\ell) = \\frac{\\sum\_{i,j}\\Delta X(I\_i)\\,\\Delta Y(I\_j)\\,1\_{\\{I\_i \\cap (I\_j+\\ell)\\ne\\emptyset\\}}} {\\sqrt{\\sum\_i\[\\Delta X(I\_i)\]^{2}\\;\\sum\_j\[\\Delta Y(I\_j)\]^{2}}} $$

The denominator is fixed for a given pair of series. The lag changes which increments overlap. Worked pair of increments: X moves +0.04 then −0.01, Y moves +0.02 then +0.03\. The denominator is the square root of 0.0017 times 0.0013, which is 0.001487\. If both pairs overlap, the numerator is 0.0005 and ρ is 0.34\. If the lag leaves only the first pair overlapping, the numerator is 0.0008 and ρ is 0.54\. Same variances, different lead.

Their lead-lag strength, LLS, compresses the whole curve of shifted correlations into one signed number. The displayed equation in the paper is missing a parenthesis. The prose is the reliable reading: take the ratio of summed squared correlations on positive lags to summed squared correlations on negative lags, subtract one, and scale by 100 times the average squared correlation on the grid. If the ratio is below one, swap the ratio and attach a minus sign, so the sign flips cleanly when you swap which series you call Y.

$$ R = \\frac{\\sum\_{\\ell>0}\\hat\\rho(\\ell)^{2}}{\\sum\_{\\ell<0}\\hat\\rho(\\ell)^{2}}, \\qquad LLS = \\begin{cases} (R-1)\\,\\dfrac{100}{n\_L}\\sum\_{\\ell}\\hat\\rho(\\ell)^{2} & \\text{if } R \\ge 1 \\\\\[6pt\] -\\left(\\dfrac{1}{R}-1\\right)\\,\\dfrac{100}{n\_L}\\sum\_{\\ell}\\hat\\rho(\\ell)^{2} & \\text{otherwise} \\end{cases} $$

n\_L is the number of lags on the grid, including the contemporaneous point. Worked curve on lags −2, −1, 0, +1, +2, so n\_L is 5, with correlations 0.10, 0.20, 0.50, 0.40, 0.30\. Squared positive lags sum to 0.25\. Squared negative lags sum to 0.05\. R is 5\. All five squared correlations sum to 0.55\. Then 100/5 times 0.55 is 11, and LLS is 4 times 11, which is 44\. That curve is a cartoon: short, and badly skewed. On their real grid of 35 lags, the 95th percentile of LLS across instrument pairs, for transaction price leading midpoint, is 2.29\. A value of 44 would be an absurd outlier on that grid. Use 44 to check the arithmetic, and use 2.29 as the empirical bar for a strong pair.

The Euro Stoxx 50 future's transaction price leads the DAX future's midpoint, and the paper marks that lead with three stars: the same sign of LLS on 99% of days in the January–June 2021 sample. The lag that maximises the curve is 100 microseconds. Euro Bobl and Euro Schatz transaction prices lead Euro Bund midpoints, and the reverse lead is absent. Bobl is the less liquid contract, and it is the one that leads. Prediction accuracy of the next midpoint change correlates 0.67 with LLS, but the average accuracy is 51.8% against a 50.5% base rate that always guesses the more common sign. Restricting the horizon to a window around the estimated lag lifts the average to 53.3%. The association is real. The average hit rate is a poor trade, and they assume transaction costs away.

Estimate that LLS twice. Once on positive-return days, once on negative-return days, or on the tail days defined above. You get two directed graphs. Pu's graph learner cannot store either of them. The optimisation in their Section 3.1 forces the adjacency to be symmetric and nonnegative: A\_ij equals A\_ji, and A\_ij is at least zero. An edge that runs from Bobl to Bund on down days, and a different edge that runs the other way on up days, is two matrices. One symmetric matrix trained on all days is the commute map with the arrows rubbed out.

## Out-strength is who you hedge

Downside source centrality is the sum of the outgoing downside edges. Keep only positive LLS. A lagging name contributes zero.

$$ S\_i^{-} = \\sum\_{j \\ne i} \\max(\\mathrm{LLS}\_{i \\to j}^{-},\\, 0) $$

LLS here is the downside estimate, the one computed on negative-return or tail days. S is a pure number, a sum of lead-lag strengths. You rank sources by S. You size the hedge in each follower with that follower's downside deviation, not by stuffing the deviation into S, because the deviation is in return units and LLS is not.

Worked book, three assets. These LLS figures are arithmetic for the example. On up days, Q leads P with LLS 1.50 and leads M with LLS 0.80, and P leads M with LLS 0.10\. Upside out-strength is 0.10 for P, 0 for M, and 2.30 for Q. On down days, P leads M with LLS 2.00 and leads Q with LLS 1.20, and M leads Q with LLS 0.30\. Downside out-strength is 3.20 for P, 0.30 for M, and 0 for Q. Look at Asset Q in the chart: the upside bar is 2.30 and the downside bar is 0\. Asset P is the reverse, 0.10 against 3.20.

![Worked example of out-strength: Asset Q leads on up days and leads nobody on down days; Asset P does the reverse](https://storage.ghost.io/c/27/cb/27cb0fc8-2c77-4434-af9e-d5d32a916994/content/images/2026/10/article_541-out_strength.png)

Average the two out-strengths, which is what a single graph trained on all days approximates, and you get 1.65 for P, 0.15 for M, and 1.15 for Q. The blend still ranks Q as a hub. Q leads nobody on the evacuation map. If you short the top two names of the blend going into a selloff, you short P, the source, and Q, a name with downside out-strength of zero. The short in Q is the hedge the commute map talked you into.

That is the same pattern as the loadings, in a different object. Solana is peripheral on good COVOL and near the top of the bad-COVOL list. Bobl leads Bund while Bund is the more liquid contract. On 22 December 2006, Pu's learned graph still ties equities to fixed income. On 5 April 2022 those edges are gone and equities sit in a tighter cluster. The topology moves when the regime moves. Their graph is still symmetric, so the picture shows a new neighbourhood, and direction still has to come from LLS.

## The unsigned graph still posts the pretty Sharpe

Look at the two equity bars. They are an equity sleeve of the futures book, volatility-scaled to a 15% annual target. The other three bars are the full 64-future book, or a restricted edge set of that book, on the same scaling, from Pu, Roberts, Dong and Zohren, Table 2.

![Volatility-scaled Sharpe ratios: single-class equity 0.30, cross-class equity 0.95, inter-class edges only 0.91, intra-class edges only 1.21, full network momentum 1.51](https://storage.ghost.io/c/27/cb/27cb0fc8-2c77-4434-af9e-d5d32a916994/content/images/2026/10/article_541-graph_sharpe.png)

Single-class network momentum on equities has a Sharpe of 0.30\. Letting that sleeve hear the other asset classes lifts it to 0.95\. The full book, all four asset classes, has a Sharpe of 1.51\. The rest of that table: intra-class edges alone earn 1.21, inter-class edges alone earn 0.91, and the no-graph regression on the same eight momentum features, volatility-scaled, earns 1.13\. Every row in that table is an all-days graph. The 1.51 is the commute map, carefully estimated. A crash hedge reads the downside graph.

Two neighbouring comparisons sit beside that table. The horse race against them is still unrun. Pham and coauthors' 30–70 tilt of the crypto index has a Sharpe of 0.56 against 0.40 for buy-and-hold, which matches their claim that the Sharpe outperforms by 42% (0.5648 divided by 0.3971 is 1.42). Bender and coauthors' 0.67 correlation between LLS and hit rate sits on top of a 51.8% average accuracy. The old article ["Stock-Bond Correlation: The Sign Flips and So Do Its Drivers"](https://aligrithm.com/stock-bond-correlation-the-sign-flips-and-so-do-its-drivers/) covers McMillan's G7 sample from 1980 to 2023: the correlation is positive into the late 1990s and negative after about 2000 for six of the seven markets. To stay on the pre-2000 trend of a 50/50 US book, his text says you needed 65% in stocks from 2000 through 2008 and 80% from 2009 on. Bonds had changed from diversifier to hedge. A full-sample correlation near zero is those two regimes averaged, which is the same mistake as averaging S+ and S−.

## Charge the hedge, and demand the horse race

At the desk, before any order, read the sign of d, or of bad COVOL against its own trailing upper tail, and pick the graph. Rank names by downside out-strength. Size each follower with its downside deviation, and haircut the package by cost.

The cost number you already have is from the unsigned strategy, and it attaches to the raw-signal Sharpe of 1.36\. The volatility-scaled Sharpe of 1.51 is a different row, and it is gross of this cost schedule. Pu and coauthors say that raw-signal network momentum stays positive through 3 basis points of cost and prints a cost-adjusted Sharpe of −0.67 at 5 basis points. The linear regression without the graph is at −1.05 at that same 5 basis points. A crash hedge that rebuilds a directed graph will turn over at least as fast. I am against putting the downside-source rank on a live book until it beats three baselines, after that 5 basis point charge, on the thing it claims to forecast: the next tail day, and the hedge P&L through it. The baselines are full-sample correlation, beta (the slope of the name on a market factor), and ordinary network momentum, the symmetric GMOM graph. None of the four papers ran that horse race. The pieces show the graphs differ. The hedge P&L is unmeasured.

The FESX lead of 100 microseconds is a six-month microstructure fact on 19 Xetra and Eurex names. A daily futures hedge needs LLS recomputed on daily down days. If the lead dies when you leave the microsecond grid, you have the sniping fact the old article ["Lead-Lag Is Multidimensional: Trade Price and OBI Predict Others' Midpoints"](https://aligrithm.com/lead-lag-is-multidimensional-trade-price-and-obi-predict-others-midpoints/) already priced, and the crash network is still unestimated.

The portfolio check, for a two-asset hedge, is the usual variance. McMillan's displayed decomposition uses 50% of each variance, the raw weight, where the expansion below uses the squared weight, 25%, and it adds the correlation in the slot where the expansion multiplies. Plug numbers into his line and the shares fail to add to the portfolio variance. The source may have typeset the usual formula wrong. Keep his 65% and 80% as the statements in the text. For the variance you compute, use the expansion:

$$ \\sigma\_p^{2} = w\_s^{2}\\sigma\_s^{2} + w\_b^{2}\\sigma\_b^{2} + 2\\, w\_s w\_b\\, \\sigma\_s \\sigma\_b\\, \\rho $$

Worked book with round volatilities: weights 50/50, stock volatility 15%, bond volatility 6%. At a correlation of +0.30 the variance is 0.007875 and the volatility is 8.87%. At −0.30 the variance is 0.005175 and the volatility is 7.19%. The covariance term changed sign and took 1.68 percentage points of volatility with it. A hedge ratio fit on the full sample uses a ρ near the average of those two regimes, and that ratio is off in each regime separately.

Two source numbers do not reconcile, and I am leaving them as printed. The introduction says the Sortino ratio outperforms buy-and-hold by 47%. Table 6 prints 0.3584 against 0.2695, and 0.3584 divided by 0.2695 is 1.33, a 33% gap. The same table prints an accumulated return of 141.98% beside an average monthly return of 2.48% from March 2017 to April 2024\. Over those 86 months the monthly mean sums to 213% and compounds to about 722%. Neither is 141.98%. The Sharpe pair is the cell that agrees with the paper's own 42% sentence. Treat the accumulated return, and the 47% Sortino claim, as possibly wrong.

Their relative-index formula, equation (12), rescales a 20-day average by the minimum and maximum of that average. If you take the min and the max once over the whole sample, later crashes set the scale of earlier days. Use an expanding window, or a long trailing window. The old article ["Stock-Bond Correlation: The Sign Flips and So Do Its Drivers"](https://aligrithm.com/stock-bond-correlation-the-sign-flips-and-so-do-its-drivers/) is where the macro drivers of the sign flip are worked. The real rate is the coefficient that keeps a positive sign most often after the late 1990s. You still build the downside graph.

![](https://storage.ghost.io/c/27/cb/27cb0fc8-2c77-4434-af9e-d5d32a916994/content/images/2026/10/article_541-visual-0.png)

## KEY POINTS

- Split the return by sign before you estimate common volatility. Bad COVOL peaks at 98.49 on 3 April 2017; good COVOL peaks at 58.93 on 15 September 2016.
- Read the bad-COVOL loading list. Solana is ninth there, at 0.2148, and twenty-second on the good-COVOL list.
- Estimate lead-lag strength on down days. On their 35-lag grid the 95th percentile of that strength is 2.29, and that is the bar for a strong pair.
- Rank the hedge by downside out-strength. In the worked book the blended rank still scores Asset Q at 1.15 when its downside out-strength is 0.
- Treat the symmetric network-momentum Sharpe of 1.51 as the all-days graph. Race downside out-strength against full-sample correlation, beta, and that graph before you hedge.

## References

- [Network Momentum as a Cross-Asset Factor](https://aligrithm.com/network-momentum-as-a-cross-asset-factor/)
- [Good vs Bad COVOL in Crypto: A Common-Volatility Tilt](https://aligrithm.com/good-vs-bad-covol-in-crypto-a-common-volatility-tilt/)
- [Lead-Lag Is Multidimensional: Trade Price and OBI Predict Others' Midpoints](https://aligrithm.com/lead-lag-is-multidimensional-trade-price-and-obi-predict-others-midpoints/)
- [Stock-Bond Correlation: The Sign Flips and So Do Its Drivers](https://aligrithm.com/stock-bond-correlation-the-sign-flips-and-so-do-its-drivers/)
- [Good versus Bad COVOL in Cryptocurrency Markets: A Measure of Asymmetric Common Volatility](https://papers.ssrn.com/sol3/papers.cfm?abstract%5Fid=6038623&ref=aligrithm.com)
- [Network Momentum across Asset Classes](https://arxiv.org/abs/2308.11294?ref=aligrithm.com)
- [Lead-Lag Relationships in Market Microstructure](https://papers.ssrn.com/sol3/papers.cfm?abstract%5Fid=5043260&ref=aligrithm.com)
- [Stock-Bond Return Correlation: Understanding the Changing Behaviour](https://papers.ssrn.com/sol3/papers.cfm?abstract%5Fid=5193234&ref=aligrithm.com)

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**A note on AI.* The ideas, research, analysis, and conclusions in this article are my own. I use AI tools to help with editing and wordsmithing, because English is not my first language, and I am not shy about that. AI-generated ideas and AI-assisted writing are not the same thing: the first is empty slop from a generic prompt, the second is a tool for communicating years of real research more clearly. Judge the work by its substance, not by whether software helped polish the prose.*