Nonlinear Volatility Transmission with Supply Chain Revenue Exposure

Abstract

Customer shocks can enter supplier valuations through lost sales, order revisions, working-capital exposure, relationship-specific investment, and changes in expected future business. Existing research already establishes that concentrated customer bases are associated with supplier volatility and that shocks propagate through production relationships. The narrower unresolved question is whether the marginal strength of that transmission rises approximately in proportion to customer dependence or accelerates once a customer becomes economically dominant. This paper estimates that exposure-response function using seven directed customer-supplier relationships selected from Altsets historical relationship data, with Supplier Revenue % ranging from 0.10% to 18.68%. Daily equity prices are used to construct customer realized volatility immediately before the June 2026 relationship observation and supplier realized volatility before and after it. A binary-link model, linear concentration interaction, and 10% threshold specification are compared. The threshold specification raises in-sample $R^2$ from 0.089 to 0.686 relative to the linear exposure model, but its estimated curvature is opposite the proposed convex mechanism: transmission increases below the threshold and declines above it. Leave-one-pair-out error is essentially unchanged from the binary baseline, and an exact concentration-label permutation test produces a tail probability of 0.070. A historical Broadcom-Korea Circuit comparison provides an additional within-relationship falsification case. The evidence favors heterogeneous, state-dependent transmission over a mechanically convex response to customer concentration.

Keywords: Realized volatility; Threshold regression; Nonlinear exposure; Volatility transmission; Economic networks

1. Question

Volatility transmission across economically linked firms is already well established enough that another test of whether connected firms share shocks would have little incremental content. Cohen and Frazzini document delayed incorporation of information across principal customer relationships, while Barrot and Sauvagnat show that idiosyncratic supplier disruptions can propagate to customers and that the effect depends on the economic specificity of the affected input. Herskovic, Kelly, Lustig, and Van Nieuwerburgh place related behavior inside a network model of firm volatility in which customer shocks influence suppliers, and their empirical cross section links concentrated customer bases to greater volatility. Most directly, Mihov and Naranjo study customer-base concentration and idiosyncratic volatility and report higher supplier volatility when customers are more concentrated, together with substantial heterogeneity across customer and supplier characteristics. These results make a binary hypothesis such as "customer concentration transmits volatility" an unattractive research target. The remaining object is the shape of the transmission function. If a customer rises from 2% to 4% of supplier revenue, should the change in supplier sensitivity resemble the change from 12% to 14%, or does the second interval alter supplier risk by considerably more because fixed capacity, bargaining exposure, receivables, or replacement frictions begin to bind?

That distinction separates concentration as a level characteristic from concentration as a nonlinear state variable. Mihov and Naranjo measure concentration with customer-base measures such as the sum of squared sales shares and cumulative sales to important customers; their evidence establishes a broad association between concentration and supplier volatility but does not identify a piecewise marginal transmission coefficient for the volatility of one economically weighted customer. Accounting research supplies reasons to avoid assuming a globally linear response. Irvine, Park, and Yildizhan argue that customer-specific investments impose fixed costs and operating leverage early in a relationship, while relationship maturation can produce offsetting efficiencies. Patatoukas likewise reports that concentration can coexist with operating efficiencies rather than behaving purely as a risk penalty. Dhaliwal, Judd, Serfling, and Shaikh find a higher cost of equity for suppliers with concentrated corporate customers, with the association stronger when customer loss is more likely or more damaging. Those findings permit a convex transmission mechanism, but they also permit stabilization, contractual adaptation, or relationship-specific efficiencies to offset it.

The empirical question is therefore falsifiable in a stronger form. Let $c$ denote the percentage of a supplier's revenue associated with a particular customer and let $\lambda(c)$ denote the sensitivity of subsequent supplier volatility to volatility at that customer. The convexity hypothesis requires $d\lambda(c)/dc>0$ and, in its strict form, a larger derivative at high concentration than at low concentration. A linear exposure model instead imposes a constant derivative. A binary relationship model removes economic concentration entirely and gives every linked customer the same transmission coefficient. The analysis treats the relationship as an input to a volatility problem and centers the paper on the marginal transmission function rather than on the existence or ranking of individual links.

2. Mechanism

Consider a supplier whose equity return over a short interval can be locally decomposed as

where $u_{C,t}$ is a customer-specific shock, $\epsilon_{S,t+1}$ contains supplier-specific and unrelated common components, and $\phi(c)$ maps economic dependence into shock exposure. Under a purely proportional sales channel, $\phi(c)=\phi_0+\phi_1 c$. The supplier loses or gains an approximately proportional amount of expected activity as the customer shock changes demand, so each additional percentage point of revenue dependence has the same marginal effect. Such a model can generate a positive concentration-volatility relation without convexity. A nonlinear channel requires some component of exposure whose marginal cost rises with dependence. One representation is

with $h'(c)\geq0$ and $h''(c)>0$ over the region where dedicated capacity, switching difficulty, customer-specific capital, working-capital commitments, or bargaining exposure become increasingly difficult to replace. Conditional on customer shock variance $\sigma_C^2$,

The derivative with respect to customer volatility rises rapidly when $\phi(c)$ is convex. This is a stronger prediction than the ordinary statement that a larger customer produces greater supplier exposure.

The economic distinction is compatible with evidence on both production networks and relationship-specific investment. Barrot and Sauvagnat find larger propagation when affected suppliers produce specific inputs, implying that nominal connectivity is insufficient to describe transmission. Irvine, Park, and Yildizhan attribute part of the economics of concentrated relationships to customer-specific investment, fixed costs, and operating leverage, while also finding that relationship maturity changes the profitability consequences of concentration. A high-revenue customer can therefore create two opposing nonlinearities. Dedicated assets and replacement difficulty can steepen the exposure function, while mature contracting, repeated interaction, inventory planning, and mutual dependence can flatten it. A convexity test identifies which shape is consistent with observed equity volatility rather than assuming that concentration must amplify transmission monotonically.

The threshold specification uses 10% of supplier revenue as an ex ante economic breakpoint rather than choosing the best breakpoint from seven observations. U.S. GAAP segment-reporting taxonomy defines a major customer around the point at which transactions with one external customer amount to 10% or more of entity revenue, with disclosure of the associated revenue and reporting segment. The accounting threshold is not itself a structural law of volatility, particularly for non-U.S. firms. It is useful because it marks a pre-existing boundary at which single-customer dependence is considered sufficiently economically material to warrant separate disclosure. A convex transmission hypothesis should have difficulty surviving if marginal sensitivity fails to strengthen even around a threshold this economically salient.

3. Design

Seven directed relationships are used rather than the full Altsets history. The selection creates an economically broad concentration ladder while keeping the quantitative object interpretable at the relationship level. In Altsets terminology, Supplier Revenue % is the share of the supplier's revenue attributable to the focal customer. Customer Cost % measures the same relationship from the customer's purchasing side, and Relationship Size measures its estimated monetary scale. The June 28, 2026 observations range from Cisco-Samsung Electronics at 0.10% of Samsung revenue to Target-Makalot at 18.68% of Makalot revenue. Three relationships lie above the 10% threshold, one is near 3%, two are near 1% to 2%, and one is close to zero. Several also have multi-year quantified histories: Coca-Cola-Constellium and Intel-SK hynix extend back to 2021, Caterpillar-Teijin has 20 quantified quarterly observations through June 2026, and Broadcom-Korea Circuit has six observations beginning in 2023.

The dependent quantity is deliberately expressed as a change in the supplier's own volatility state rather than as raw post-period volatility. For each customer, $RV_C^{pre}$ is computed from the five close-to-close log returns immediately preceding the relationship observation. Supplier volatility is measured over the corresponding pre-window and over the first five local close-to-close returns after the observation. With $P_0,\ldots,P_n$ denoting the sequence of closing prices,

The modeled supplier response is

A value of zero indicates unchanged realized volatility, a positive value indicates an increase, and a negative value indicates a decline. This transformation removes much of the persistent level difference between a relatively quiet supplier and an intrinsically volatile supplier before asking whether customer volatility interacts with dependence.

The price calculation uses public daily equity closes for both sides of each selected relationship. The June windows incorporate the June 19 U.S. market holiday where relevant and use local-market sessions for Asian suppliers rather than imposing a U.S. calendar on Korean, Japanese, or Taiwanese equities. Public historical series support the Broadcom, Target, Walmart, Coca-Cola, Caterpillar, Intel, and Cisco customer windows and the Korea Circuit, Makalot, McKesson, Constellium, Teijin, SK hynix, and Samsung supplier windows. Supplier-market observations are drawn from the corresponding public historical-price records, including Korea Circuit, Makalot, Constellium, McKesson, Teijin, SK hynix, and Samsung.

Three specifications separate connection from concentration. Since every observation is a validated customer-supplier pair, the binary model permits customer volatility to matter but holds its relationship coefficient constant:

where $x_i=RV_{C,i}^{pre}$. The linear concentration model is

giving a customer-volatility transmission coefficient

The nonlinear specification adds a 10% hinge:

where $(z)_+=\max(z,0)$. Below 10%, $d\lambda/dc=\gamma$. Above 10%, the derivative becomes $\gamma+\psi$. Convex transmission requires the latter to exceed the former and, under the strongest version of the hypothesis, to remain positive. Model comparison uses in-sample $R^2$, leave-one-pair-out prediction error, and an exact permutation test that reassigns the seven observed concentration values across the seven customer-supplier pairs. With only $7!=5{,}040$ assignments, every possible concentration permutation can be evaluated rather than approximating the null with a Monte Carlo sample.

4. Evidence

The realized-volatility calculation does not produce a simple ordering by concentration. Broadcom's pre-window realized volatility is 48.9% annualized while Korea Circuit's supplier volatility falls from 154.6% before the observation to 120.3% afterward. Target is similarly associated with a reduction in Makalot volatility despite the highest Supplier Revenue % in the set. Walmart-McKesson behaves differently: the customer has substantially lower pre-window volatility, 23.8%, but McKesson's supplier volatility rises from 19.1% to 30.0%. Among lower-concentration relationships, Caterpillar-Teijin records the largest positive supplier volatility change even though Caterpillar accounts for only 1.56% of Teijin revenue. The raw observations therefore contain a useful tension: the three relationships above 10% do not form a uniformly high-transmission regime.

A coarse high-versus-low split already shows why a threshold coefficient has to be estimated rather than inferred from the existence of large customers. The mean log supplier-volatility change for the three relationships at or above 10% is $0.008$; for the four relationships below 10% it is $-0.043$. That small difference masks much wider dispersion inside each regime. Walmart-McKesson supplies nearly the entire positive response of the high-concentration group, while Broadcom-Korea Circuit and Target-Makalot move in the opposite direction. Caterpillar-Teijin supplies the strongest positive response in the low-concentration group. Concentration therefore appears capable of distinguishing economically different links, but the unconditional group means are not consistent with a simple high-concentration amplification rule.

The model comparison is sharper. The binary specification produces

with $R^2=0.009$. Allowing transmission to vary linearly with Supplier Revenue % yields

and raises $R^2$ only to 0.089. The fitted marginal customer-volatility coefficient is therefore mildly decreasing with concentration in the globally linear model. The nonlinear specification changes the in-sample fit substantially:

Its $R^2$ is 0.686, and its residual sum of squares falls from 0.4486 under the linear model to 0.1547. A model that permits a bend at economically high concentration therefore describes these seven observations considerably better in sample than one that forces one concentration slope across the full range.

The direction of that bend rejects the paper's proposed convex mechanism within this sample. Below 10%, the fitted transmission function is

so the marginal response rises rapidly as concentration approaches the threshold. Above 10%,

The estimated response function therefore forms an inverted shape rather than a convex one. It rises through the low and intermediate concentration region and then turns downward above the pre-specified threshold. For Broadcom-Korea Circuit, the fitted coefficient remains positive at 16.95%, but it has fallen markedly from its value around 10%. At Target-Makalot's 18.68%, the fitted coefficient is close to zero. The nonlinear model's superior in-sample fit cannot be interpreted as evidence of convex transmission simply because a threshold term is useful. The sign of the curvature is part of the hypothesis.

Cross-validation further reduces the case for treating the threshold shape as a stable empirical law. Leave-one-pair-out root mean squared error is 0.4044 for the binary model, 0.4212 for the linear concentration model, and 0.4040 for the threshold model. The nonlinear specification's large in-sample improvement therefore delivers essentially no out-of-sample gain over a model that knows only customer volatility and the fact that a relationship exists. The linear economic weighting is slightly worse. With seven observations and one high-concentration outcome as distinctive as Walmart-McKesson, that discrepancy between fit and prediction is informative: concentration can organize the observed cross section while still failing to identify a stable marginal response.

An exact permutation test asks whether the reduction in residual sum of squares from the linear to threshold model is unusually large relative to arbitrary assignments of the same seven concentration values. The observed reduction is 0.2939. Across all 5,040 assignments, only 353 generate an improvement at least this large, corresponding to an exact tail frequency of 0.0700 and an add-one adjusted value of 0.0702. The median improvement under permutation is 0.0586; the 90th and 95th percentiles are 0.2525 and 0.3278. The observed nonlinear fit is therefore unusual relative to most shuffled concentration structures, but it does not clear a conventional 5% threshold and its economic shape points in the wrong direction for convexity. The result is better characterized as evidence that the exposure-response relation may contain a breakpoint than as evidence that high customer dependence causes accelerating volatility transmission.

5. Tests

One concern is that modeling a volatility ratio may create nonlinear behavior by normalizing with a noisy pre-period denominator. Re-estimating the comparison in levels gives the supplier's post-period realized volatility as the dependent variable and adds $RV_S^{pre}$ directly as a control. The baseline specification with prior supplier volatility and customer volatility explains 97.5% of the seven-pair cross-sectional variation, with leave-one-pair-out RMSE of 0.118. Adding the linear concentration interaction raises in-sample $R^2$ to 0.984 but worsens cross-validation RMSE to 0.131. The threshold specification raises $R^2$ again to 0.992 while increasing cross-validation RMSE to 0.189. Most of the raw post-window cross section is therefore explained by the supplier's own volatility state. Economic concentration can absorb additional residual variation in sample, but the nonlinear terms become less reliable when asked to predict the omitted relationship.

The other two Altsets economic metrics provide a direct check on whether Supplier Revenue % is merely proxying for transaction scale or reciprocal importance. Relationship Size varies from approximately $120 million for Caterpillar-Teijin to more than $49 billion for Walmart-McKesson, while Customer Cost % ranges from 0.24% for Target-Makalot to 8.20% for Walmart-McKesson. Nominal dollar scale and supplier dependence therefore differ sharply. Walmart-McKesson combines an 11.28% supplier revenue share with an unusually large 8.20% customer cost share, whereas Broadcom-Korea Circuit and Target-Makalot combine even higher supplier dependence with customer cost shares below 1%. A reciprocity statistic

where $q_i$ is Customer Cost %, approaches one when dependence is primarily supplier-side and falls when the customer's purchasing exposure is comparatively important. Adding $A_i x_i$ to the linear transmission regression raises in-sample $R^2$ to 0.267 but produces leave-one-out RMSE of 0.608. Replacing it with a Relationship Size interaction produces RMSE near 0.726. Neither auxiliary metric stabilizes the seven-edge transmission equation.

That failure is economically informative because the three high-concentration observations are structurally different. Target-Makalot has $A\approx0.99$ and Broadcom-Korea Circuit approximately 0.96, describing relationships in which the supplier's dependence is far greater than the customer's measured cost exposure. Walmart-McKesson has $A\approx0.58$, indicating materially greater reciprocity. Yet Walmart-McKesson is the only one of the three in which supplier realized volatility rises in the post-window. A bargaining-power story based solely on one-sided supplier dependence would have anticipated the opposite ordering. Mutual economic importance could instead affect contracting, information flow, order stability, or the market's interpretation of customer shocks. The present calculation provides no stable coefficient for that mechanism, but it does show why Supplier Revenue %, Customer Cost %, and Relationship Size are not interchangeable measures of "relationship strength."

A second failure test separates customer realized variance into downside and upside components,

and

The same concentration interaction can then be estimated using $RV_C^-$ or $RV_C^+$ in place of total realized volatility. The seven relationships provide no stable improvement from this decomposition in leave-one-pair-out prediction. Broadcom's customer window is dominated by negative semivariance, while Target's is dominated by positive semivariance, yet both high-concentration suppliers experience lower post-window volatility. Caterpillar combines substantial positive and negative variation but produces the largest low-concentration increase in supplier volatility. The small panel therefore supplies no evidence that the missing convexity can be recovered simply by restricting the shock measure to customer downside returns.

6. History

Broadcom-Korea Circuit permits a particularly useful within-relationship comparison because its historical Supplier Revenue % changes much more than those of the other long histories. Altsets records Korea Circuit's Broadcom dependence at 0.67% in September 2023, around 0.97% during 2024, and 1.11% in both June and September 2025. By June 2026 the measured share is 16.95%, while Customer Cost % rises from 0.03% in September 2025 to 0.74%. The observed relationship therefore crosses from an economically small supplier revenue exposure to well beyond the 10% major-customer reference level without changing the identity or direction of the pair.

The September 2025 market window provides a falsification case for mechanical convexity. Using Broadcom closes before the September 28 observation and Korea Circuit local-market closes before and after it, Broadcom realized volatility is approximately 15.6% annualized, Korea Circuit pre-window realized volatility is 70.6%, and post-window volatility is 77.4%. The resulting log volatility ratio is $+0.092$. By June 2026, with Supplier Revenue % at 16.95% and Broadcom pre-window volatility much higher at 48.9%, Korea Circuit volatility moves from 154.6% to 120.3%, producing $y=-0.251$. The customer becomes both more volatile and far more economically important to the supplier, yet the subsequent supplier volatility ratio changes in the opposite direction from a monotone convex prediction.

Two observations from one relationship cannot identify a structural elasticity because market regime, supplier news, industry conditions, and the composition of customer shocks also change between dates. Their role here is narrower. A deterministic version of the hypothesis in which greater $c$ mechanically amplifies any customer-volatility state cannot explain the Broadcom-Korea Circuit comparison. The relation requires additional state variables or a weaker probabilistic interpretation. That conclusion also fits the longer low-concentration histories. Coca-Cola's share of Constellium revenue remains near 2% to 3% across much of its quantified history, Caterpillar-Teijin remains around the low single digits before declining to 1.56%, and Intel-SK hynix stays close to the 1% to 2% region. These persistent low-concentration relationships are economically different from a disappearing or newly formed edge, yet their June responses still span positive and negative volatility changes.

The historical evidence changes the interpretation of the cross-sectional threshold coefficient. The inverted bend is less naturally viewed as an anomalous estimate that should be discarded because convexity was expected. It is consistent with a model in which increasing dependence initially strengthens the market relevance of customer shocks, after which high-dependence relationships are selected into different contracts, capacity arrangements, customer-specific investments, or mutual commitments. Irvine, Park, and Yildizhan's relationship-life-cycle evidence is compatible with such an offset because the economics of customer-specific investment vary with relationship maturity rather than remaining a fixed penalty. Herskovic et al. establish that network structure helps explain firm-volatility distributions, but their result similarly leaves room for the mapping from one customer's weight to one supplier's short-horizon equity volatility to vary with relationship state.

7. Close

The seven-relationship calculation rejects the easiest version of the convexity hypothesis. A nonlinear exposure model fits the June 2026 cross section much better than either a binary relationship model or a globally linear concentration interaction, so treating economic concentration as irrelevant would discard structure in the observed data. Yet the estimated structure is not convex. The fitted customer-volatility coefficient rises as Supplier Revenue % approaches 10% and falls above that point. Its large in-sample improvement nearly vanishes under leave-one-pair-out evaluation, while the exact permutation test places the observed threshold improvement at a tail frequency of about 7%. A separate level specification reaches the same practical result: supplier pre-period volatility explains most subsequent volatility, and concentration interactions improve fit while worsening omitted-pair prediction.

The Altsets variables help distinguish what type of null result this is. High Supplier Revenue % is not synonymous with large Relationship Size, and it is not synonymous with reciprocal dependence measured by Customer Cost %. Broadcom-Korea Circuit, Target-Makalot, and Walmart-McKesson all exceed 10% of supplier revenue but occupy very different positions on those other dimensions. The within-relationship Broadcom-Korea Circuit history is more damaging to a mechanical concentration story: customer dependence moves from 1.11% to 16.95% and customer realized volatility rises sharply, while the supplier's post-to-pre volatility response moves from positive to negative. The observable relationship weight therefore acts as a state variable whose effect is conditional rather than as a monotone volatility multiplier.

The resulting quantitative object is the function $\lambda(c)$, not a classification of suppliers as concentrated or unconcentrated. Existing literature supports transmission, concentration risk, customer-specific investment, and production-network propagation. The present calculation isolates a narrower proposition that those results do not settle: whether the marginal volatility response itself accelerates at economically high dependence. In this seven-edge historical application, it does not. The evidence is more consistent with a response surface that bends, changes sign in its concentration derivative, and depends on features of the relationship beyond the supplier's revenue share. That distinction is material for volatility modeling because a risk model that converts a 15% customer share into five times the transmission coefficient of a 3% customer share imposes an economic shape that the observed relationships do not support.

References

Altsets supply chain dataset.

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