Volume Shock Therapy: Why Markowitz Risk Might Be Lying to You

TL;DR for operators

Markowitz variance is usually treated as the clean mathematical backbone of portfolio risk. Olkhov’s paper asks a narrower and more awkward question: what if that familiar covariance formula is only what remains after trade-volume randomness has been quietly set to zero?¹

The paper’s answer is mechanism-first. It constructs a buy-and-hold portfolio as if it were a synthetic single traded security. To do that, it rescales the observed market trades of each constituent so their normalised volumes match the investor’s actual holdings, then aggregates those normalised trade values and volumes into portfolio-level trade series. Once the portfolio has its own synthetic trade values $Q(t_i)$, volumes $W(t_i)$, and implied prices $s(t_i)$, its variance can be computed in the same market-based way as the variance of any single security.

That move changes the risk object. The variance is no longer driven only by return covariance. It also depends on the coefficient of variation of trade values, the coefficient of variation of trade volumes, and the normalised covariance between trade values and trade volumes. In the paper’s notation, the market-based return variance takes the form:

where $\psi$ captures variation in trade values, $\chi$ captures variation in trade volumes, and $\varphi$ captures value-volume covariance. When $\chi=0$, the expression collapses into the Markowitz-style variance approximation. That is the knife twist. Not dramatic. Just inconvenient.

For portfolio managers, the implication is not “throw out Markowitz”. That would be the usual LinkedIn-grade overreaction. The better interpretation is: keep Markowitz as the baseline covariance estimate, but add a market-based variance overlay when trade-volume fluctuations are large, unstable, or strategically relevant. This is most relevant for risk desks, execution-sensitive portfolios, concentrated portfolios, smaller or fragmented markets, and any setting where liquidity conditions are part of the risk, not background scenery.

The boundary is important. The paper is theoretical. It provides derivations and limiting cases, not empirical backtests. It does not prove that a specific live portfolio would have suffered a specific loss because of volume-adjusted variance. It does show that, within its market-based framework, ignoring random trade volumes can make Markowitz variance either too high or too low. The direction depends on the return fluctuation regime and the sign and size of value-volume covariance.

A risk number can be wrong before optimisation even starts

Portfolio risk often enters the room wearing a covariance matrix. It looks serious, speaks in Greek letters, and reassures everyone that optimisation is now under adult supervision.

The standard setup is familiar. Estimate expected returns, estimate covariances, choose weights, optimise. If the final allocation looks odd, blame the inputs, regularise the covariance matrix, add constraints, or gently remind the committee that models are not crystal balls. All fair. But Olkhov’s paper targets a step before that familiar machinery begins: the definition of the variance being fed into the machinery.

The paper does not argue that diversification is fake. It does not say Markowitz was “wrong” in the tabloid sense. It makes a more specific claim: the Markowitz variance formula can be recovered as an approximation under the assumption that consecutive trade volumes are constant. Real markets, tragically, did not sign that assumption.

That matters because trade volume is not a decorative market statistic. Consecutive trades have values, volumes, and prices. A price is the ratio of value to volume. If the volumes fluctuate, then the distribution of observed prices and returns is not only a story about price movement. It is also a story about how much traded when those prices appeared.

Most practitioners know this informally. A thin print at a strange price is not the same as a deep market move. VWAP exists for a reason. Liquidity risk exists for a reason. Execution desks do not stare only at last price and hope for spiritual enlightenment. What the paper tries to do is push that intuition into the variance formula itself.

The portfolio becomes a synthetic traded security

The paper begins with a buy-and-hold investor. At past time $t_0$, the investor bought a portfolio of $J$ securities. Since then, the investor does not trade. At current time $t$, the investor observes market trades in those same securities over an averaging interval $\Delta$.

This is already a useful framing. The investor’s portfolio itself is not traded as a single instrument. But the underlying securities are traded. The question is whether those observed constituent trades can be transformed into a time series that behaves like trades of the portfolio itself.

Olkhov’s construction says yes.

For each security $j$, the market provides consecutive trade values $C_j(t_i)$, trade volumes $U_j(t_i)$, and prices:

The investor owns $U_j(t_0)$ shares of security $j$. The paper rescales the observed trade series by a factor:

where $U_{\Sigma j}(t)$ is the total traded volume of security $j$ during the averaging interval. This scale transformation produces normalised trade values and volumes:

The key property is simple: the total normalised volume for security $j$ over the interval equals the number of shares the investor actually holds. In plain English, the observed market trades are resized so that, across the interval, they represent the investor’s holding in that security.

Then the portfolio-level synthetic trade series is built by summing across securities:

The synthetic portfolio price at each trade time is then:

This is the first real contribution of the paper. It gives the buy-and-hold portfolio a constructed market-trade representation. The investor is not pretending the portfolio ETF exists. The investor is creating a portfolio-level value-volume-price sequence from constituent market trades.

That mechanism is more interesting than the slogan. The slogan says “Markowitz ignores volume”. The mechanism says: once a portfolio has a synthetic trade series, its market-based variance can be computed using the same value-volume logic as a single security. That is where the Markowitz formula becomes a special case rather than the whole game.

VWAP is not a side note; it is the bridge

The average price of the synthetic portfolio is not a simple arithmetic average of prices. It is volume-weighted:

The same logic applies to the component securities. Their average prices are also VWAP-style averages over the interval.

This matters because it clarifies the paper’s target. It is not trying to bolt a volume signal onto a return model after the fact. It is redefining the measurement of current average price, return, and variance from the underlying trade values and volumes.

The average portfolio return still decomposes into security returns weighted by initial relative investment:

So the paper does not break the familiar linear return decomposition. The disruption arrives at the variance layer.

Once the portfolio is represented as a synthetic traded security, the market-based variance of its price is written in terms of the dispersion of synthetic prices around the average synthetic price. The return variance is the price variance scaled by the initial portfolio price:

The algebra then expresses that variance through the coefficient of variation of trade values, the coefficient of variation of trade volumes, and their covariance.

The result is:

and therefore:

The important variables are:

The common misconception is that return covariance already absorbs everything important about volume because volume shows up indirectly through traded prices. The paper’s answer is no, not under this market-based construction. Prices are ratios of values to volumes; when the denominator is random, the variance of the ratio cannot be reduced to price covariance without losing information about volume behaviour.

Financial models often hide their assumptions politely. This one leaves the fingerprints.

Markowitz appears when trade-volume fluctuation disappears

The paper’s most commercially useful idea is not the full formula. It is the nesting result.

When trade volumes are constant, the coefficient of variation of portfolio trade volumes is zero:

With constant volumes, the value-volume covariance term also drops out in the relevant approximation. The market-based variance reduces to:

The paper then shows that this equals the familiar Markowitz quadratic form:

This is the paper’s hinge. Markowitz variance is not presented as nonsense. It is presented as the zero-$\chi$ case.

That distinction is important for practitioners. The useful question is not “is Markowitz dead?” It is: “when is $\chi$ large enough that the zero-volume-fluctuation approximation becomes a bad measurement of market-based variance?”

That is a better question because it can be operationalised. Compute the baseline covariance estimate. Compute the market-based variance from trade values and volumes. Track the gap. Stress the gap. Ask whether the gap is stable, concentrated, or regime-dependent.

The paper does not provide a production recipe for all of that. But it gives the theoretical reason such a diagnostic is not merely an execution-desk superstition wearing a spreadsheet.

The Taylor correction tells you when the lie points up or down

The paper next derives a Taylor expansion of the market-based variance with respect to $\chi$, the coefficient of variation of trade volumes. The covariance term is represented using the Cauchy-Schwarz-Bunyakovskii inequality:

Here $a$ captures the sign and relative strength of value-volume covariance. Positive $a$ means trade values and trade volumes tend to move together; negative $a$ means they tend to move against each other.

The second-order Taylor approximation is:

Equivalently, using the Markowitz baseline:

This is where the paper avoids a simplistic conclusion. Volume fluctuation does not always increase risk. It can increase or decrease the market-based variance depending on the regime.

That is not a bug. It is the entire point.

The limiting cases are not empirical tests. They are mathematical stress cases. Their job is to show that the sign and magnitude of the error are not fixed. Markowitz can be too pessimistic or too complacent. Risk models, being models, enjoy finding ways to be wrong in both directions.

When Markowitz can overstate risk

In the first limiting case, the paper considers very high fluctuations of portfolio returns. The zero-volume-fluctuation coefficient $\psi_0$ is near one, so the Markowitz approximation is close to its maximum form:

When value-volume covariance is positive, $a>0$, the Taylor expression includes a negative first-order term:

In the high-return-fluctuation case, the paper approximates:

The interpretation is straightforward. If large trade values tend to arrive with large trade volumes, then volume weighting can dampen the market-based variance relative to the simple Markowitz approximation. Under the paper’s limiting assumptions, Markowitz may overvalue variance and therefore overstate risk.

For operators, this is the less intuitive side of the argument. Most discussions of ignored liquidity risk imply “hidden danger”. Here, the hidden volume term can also make the standard estimate too conservative. That matters in capital allocation. Overstated variance can push a portfolio away from exposures that are less dangerous under the market-based value-volume measure than the covariance matrix suggests.

This does not mean the portfolio is safe. It means the measurement changes once volume is allowed to move.

When Markowitz can understate risk

The opposite case is more familiar and more dangerous. Suppose return fluctuations are low, so $\psi_0\ll 1$. The Markowitz baseline is then small:

If volume fluctuation is large relative to return fluctuation, the $\chi^2$ term can dominate. The paper writes the approximation as:

In this regime, the portfolio looks calm through the return-covariance lens, but market-based variance can rise because the synthetic trade volumes are unstable. The portfolio’s quiet price path may not be telling the whole story.

This is the case that risk teams should recognise immediately. Low realised return variance often seduces risk models into reducing capital charges, increasing leverage, or relaxing limits. If volume instability is growing underneath that calm surface, the paper’s framework says the Markowitz baseline can understate risk.

The zero-covariance case is even more pointed. If $\varphi=0$, then $a=0$, and the approximation becomes:

If $\psi_0^2$ is small and $\chi^2$ approaches one, the market-based variance can approach $R^2(t,t_0)$ while the Markowitz baseline remains much smaller. In other words, even without value-volume covariance doing anything exotic, volume fluctuation alone can create a large gap.

That is the paper’s strongest practical warning. A portfolio can look low-risk under constant-volume thinking because the return series is stable. But if the trading volume process is unstable, the market-based variance may be larger than the covariance estimate implies.

The business use is a diagnostic overlay, not a replacement religion

The sensible business interpretation is not to replace every covariance matrix with Olkhov’s formula by Monday morning. That would be heroic, which in risk management is often a synonym for expensive and poorly documented.

A more useful path is to treat the paper as a diagnostic framework.

A practical implementation would start modestly.

First, compute the familiar Markowitz variance from returns and portfolio weights. This remains the baseline.

Second, construct synthetic portfolio trade series using constituent trade values and volumes. That requires deciding the averaging interval, dealing with asynchronous trades, handling missing prints, and cleaning obvious bad ticks. None of this is glamorous. All of it matters.

Third, estimate $\psi$, $\chi$, and $\varphi$ over the interval. Track their history. Ask when $\chi$ moves from background noise to an active driver.

Fourth, compare the market-based variance estimate against $\Theta_M$. The useful signal is the spread, not the ideological victory of one formula over another.

Fifth, attribute the spread. Appendix C sketches how the portfolio-level volume coefficient can be decomposed through constituent volume variation and cross-volume covariances. That points toward an attribution layer: which assets are contributing most to the volume shock embedded in portfolio variance?

For a large asset manager, this could become a risk-control panel. For a market maker or execution desk, it could connect risk measurement with liquidity state. For a wealth platform, it might be overkill unless the platform handles instruments where volume instability is material. Not every model improvement deserves a product roadmap. Some deserve a warning light.

The paper is theoretical, and that boundary matters

The paper’s limitation is not that it lacks another paragraph saying “future research is needed”. Everyone has that paragraph. The important limitation is that the paper proves a mathematical relationship inside a specific market-based variance framework. It does not provide empirical backtests, out-of-sample comparisons, portfolio case studies, or transaction-cost-aware implementation results.

That affects how the result should be used.

The derivations show that random trade volumes can change the variance estimate. They do not tell us how often the effect dominates in liquid large-cap equity portfolios, corporate bond portfolios, crypto portfolios, emerging-market baskets, or intraday multi-asset strategies. Those are empirical questions.

The construction also assumes the investor can observe and process the relevant trade sequences. In real markets, trades are asynchronous, venues fragment liquidity, reported volumes can require filtering, and the definition of a useful averaging interval is not obvious. The paper simplifies by considering finite sequences of trades over an interval. Production systems would have to make many additional choices.

The forecasting boundary is even sharper. Olkhov notes that predicting future market-based variance would require predicting future time series of trade values and trade volumes for the relevant securities over the future averaging interval. That is a much harder object than forecasting a covariance matrix from historical returns. The paper therefore fits most naturally as a current-risk diagnostic and stress-testing lens, not as a turnkey long-horizon optimisation engine.

There is also a conceptual boundary. The paper deliberately says it is not part of the standard price-volume relation literature. It is not claiming that volume predicts returns in the usual alpha sense. The argument is about variance measurement when the market-based price and return are derived from value-volume trade series. Confusing those two topics would be a quick way to build an impressive-looking model with the wrong label on the box.

What a risk team should actually take from this

The most useful takeaway is a change in model governance language.

Instead of saying, “Our portfolio variance is $X$,” a risk team might say:

“Our Markowitz baseline variance is $X$. The market-based value-volume variance estimate is $Y$. The difference is currently driven by volume coefficient $\chi$ and value-volume covariance $\varphi$ in these constituents.”

That is a better sentence. It separates baseline theory from market-state adjustment. It also gives operators something to investigate.

A portfolio whose Markowitz variance and market-based variance usually move together is probably not the immediate problem. A portfolio where they diverge during liquidity shocks deserves attention. A portfolio where Markowitz variance falls while $\chi$ rises deserves even more attention, because that is the exact regime where low return volatility may be buying false comfort at a discount.

The paper also suggests a useful mental model for managers: volume is not merely an execution variable after the portfolio is chosen. Under this framework, volume fluctuation is part of the portfolio’s measured variance itself. That does not collapse risk management into execution management. It does make the boundary less clean than many dashboards pretend.

Conclusion: the covariance matrix is not the whole witness statement

Markowitz variance remains one of the most durable ideas in finance because it is simple, elegant, and operationally useful. Those are rare virtues. But elegant formulas often survive by leaving something outside the frame.

Olkhov’s paper identifies one such outside object: the randomness of consecutive trade volumes. By constructing a buy-and-hold portfolio as a synthetic traded security, the paper shows how portfolio variance can be written in market-based value-volume terms. In that construction, the familiar Markowitz expression emerges when trade volumes are constant. Once volume fluctuation enters, the variance can move above or below the Markowitz estimate.

The business lesson is measured rather than revolutionary. Do not throw away the covariance matrix. Interrogate it. Ask whether the portfolio’s risk estimate is stable once trade-volume variation is admitted into the calculation. Build the diagnostic before building the sermon.

Finance does not need another grand theory declaring the old tools dead. It needs better instruments for knowing when the old tools are being asked to operate outside their assumptions. Volume shock therapy is not pleasant. That is probably why it is useful.

References

Cognaptus: Automate the Present, Incubate the Future.