The Mathematics of a Moving Deal

On Monday, the deal works. A buyer is prepared to underwrite an $80 million valuation on an $8 million EBITDA business with roughly 4x leverage, and the model clears the required return with enough margin to support conviction. The lender is comfortable with the structure under base and downside cases, the seller is aligned with the mix of cash at closing and residual exposure, and the process advances with the sense that the essential terms are already in place. By Wednesday, the same deal no longer works. No headline term has been formally changed, no one has walked away, and yet the structure no longer satisfies the conditions required by each participant. Diligence surfaced a concentration risk that increases volatility, the lender recalibrated its tolerance for leverage under stress, and the buyer’s return compressed once both effects were incorporated. The deal changed because it crossed a boundary that was always present and is only visible once the system updates.

A transaction is often treated as a fixed object that is negotiated into agreement. In practice, what participants call “the deal” is a provisional structure that only holds as long as the underlying inputs remain within a narrow range. The seller is not agreeing to a number in isolation, the buyer is not committing to a return in isolation, and the lender is not approving leverage in isolation. Each of those positions depends on a set of assumptions that are continuously revised as new information enters the process. What appears stable early in a transaction is often a temporary alignment of conditions rather than a resolved outcome.

Negotiations are described as converging toward a fixed set of terms, but the reality is that the system itself is moving while that convergence is attempted.

The Feasible Set Moves

In the prior column, I formalized a transaction as a structure x that must satisfy three simultaneous constraints: the seller’s value function V_S(x) must clear a minimum threshold, the buyer’s value function V_B(x) must clear its required threshold, and the lender’s value function V_L(x) must satisfy its downside conditions. The feasible set F is the intersection of all structures where those constraints hold simultaneously. But V_S(x) was written as V_S(P(x), C(x), T(x), −R(x)), treating background conditions as fixed. That formulation is correct when the problem is static. But cash at closing depends on what the financing market will support at the moment of closing, not only on what the structure specifies. Execution certainty depends on what diligence has revealed by the time a commitment is required. Retained risk depends on how the risk profile of the business has been reassessed as information accumulates. Price is the one argument that is directly specified by the structure, while the others are jointly determined by structure and environment.

This requires making the environmental dependence explicit. So define θ(t) as the vector of underlying conditions that participants cannot control through negotiation but that continuously redefine how any structure is evaluated: expected cash flows CF(t), financing market conditions κ(t), business risk assessments ρ(t), and accumulated diligence findings δ(t). The full expression of the seller’s value function then becomes:

V_S(x, θ(t)) = V_S(P(x), C(x, κ(t)), T(x, κ(t), δ(t)), −R(x, ρ(t), δ(t)))

The same logic applies to the buyer and lender. The buyer’s value function, expressed in terms of return, depends on CF(t) and κ(t). The lender’s value function, expressed through coverage constraints, depends on CF(t) under downside scenarios and on ρ(t). The feasible set is therefore time-indexed:

F(t) = {x : V_S(x, θ(t)) ≥ V_S^min, V_B(x, θ(t)) ≥ V_B^min, V_L(x, θ(t)) ≥ V_L^min}

The prior formulation asked whether x satisfies all constraints under current conditions. The dynamic formulation asks whether x(t) ∈ F(t) holds continuously along the entire path from initiation to closing. A structure that cleared every constraint at t = 0 can fail at t = 1 without any term being renegotiated, because θ(t) moved the boundaries of what each participant finds acceptable. The deal that appeared agreed in principle had already crossed a boundary before anyone recognized it.

This is a dynamic system in the mathematical sense: outcomes depend on how variables evolve over time, not only on where they start or end.

Sensitivity Determines What Actually Matters

Not all variables in θ affect the feasible set equally. The boundary of F(t) is defined by constraint surfaces, the hypersurfaces in the space of possible structures where each participant’s value function exactly equals its minimum threshold. The question of which variables matter most is a question about the gradient of those constraint functions with respect to the components of θ.

Leverage is a high-gradient variable because it propagates through multiple constraints simultaneously. Consider how a reduction from 4x to 3.5x leverage affects the system. At 4x, an $8 million EBITDA business supports $32 million of debt, but at 3.5x, debt capacity falls to $28 million. That $4 million reduction increases the required equity at the same purchase price, which compresses the buyer’s IRR. Simultaneously, the reduced debt service load changes the coverage ratio, which shifts the lender’s constraint surface. A single variable moves the boundaries of V_S(x, θ(t)) and V_L(x, θ(t)) at the same time. In gradient terms, leverage has large partial derivatives with respect to multiple constraint functions. It sits near the intersection of constraint surfaces, so small movements in it affect feasibility across the entire system. This is why small adjustments to leverage often feel disproportionately consequential in live processes: they are moving multiple constraint boundaries at once.

Customer concentration behaves differently. It does not primarily shift the base-case value of θ. It widens the distribution of possible θ trajectories. Specifically, it increases the variance of future cash flow paths under stress. A lender’s constraint is evaluated not at the base case but under downside scenarios, so widening the distribution of outcomes tightens V_L(x, θ) even when the expected case is unchanged. The mechanism is not a shift in the gradient but a change in which region of the θ distribution the constraint is evaluated against. These are mathematically distinct failure modes, and conflating them leads to misdiagnosis of why a deal broke.

Rate of Change Breaks Deals

The dynamic problem can be stated precisely. A transaction involves a structure x(t) that is being adjusted by participants and a feasible set F(t) whose boundaries are moving as θ(t) evolves. The condition for a live deal is that x(t) ∈ F(t) holds continuously, not just at signing, but throughout the process. Feasibility is not a test passed once; it is a condition that must hold along the entire trajectory.

What determines whether that condition holds is the relationship between two rates. The first is the rate at which θ(t) moves the feasible boundary over time, which we can represent informally as dF/dt, the speed at which the environment is moving the set of acceptable structures. The second is the rate at which x(t) can be reconfigured in response to that movement, call this dx/dt, the adaptation rate of the structure. The deal survives if dx/dt is sufficient to keep x(t) inside F(t) as the boundary moves. If dF/dt exceeds the system’s ability to adapt, the trajectory exits the feasible region.

This is the precise content of the claim that “timing breaks deals.” It is not that the magnitude of the shock was too large in some absolute sense. It is that the shock arrived faster than the structure could be reconfigured. A cash flow deterioration of a given magnitude, identified at the beginning of a process, allows the buyer to revise underwriting, the lender to recalibrate leverage, and the seller to adjust expectations, all while feasible points still exist. The same deterioration, arriving in the final weeks after financing has been committed and price expectations anchored, moves the feasible boundary faster than the structure can follow. The deal exits F(t) not because conditions were worse, but because the rate exceeded the adaptive capacity of the system.

This also explains path dependence. Two transactions can arrive at identical final conditions and still produce different outcomes. If one path allowed for gradual recalibration while F(t) was moving, the structure tracked the feasible region successfully. If the other path anchored commitments early and encountered the same movement late, the structure could not adapt in time. Final conditions are the same; the trajectory is different; the outcomes diverge.

The deal is path-dependent in the precise sense that the integral of its trajectory, not just its endpoint, determines whether feasibility is maintained.

What Better Dealmakers Track

A deal that appears viable at a point in time may already be on a trajectory that leads outside the feasible region. The structure can hold under current assumptions and still fail because the variables that sustain it are moving in a direction or at a speed that cannot be absorbed. In that sense, the question is not only whether the deal works, but whether it can continue to work as the system evolves.

The practical implication of this framework is a reorientation of attention. Most process management focuses on x(t), the structure, the terms, the negotiated positions. But x(t) can only remain inside F(t) if the boundaries defined by θ(t) are understood and anticipated. A dealmaker who tracks the structure without tracking the environment is navigating by position alone, without reading the movement of the terrain. The high-sensitivity variables identified by the gradient analysis, leverage, cash flow concentration, financing conditions, are the components of θ that move the constraint surfaces most quickly. Addressing them early is the only way to preserve the adaptive capacity that path dependence requires.

The structure that closes is not the one that looked best at any single point in time. It is the one whose trajectory remained inside a feasible region that was moving from the start.