Every damages model, and when each one applies.

The mathematicsA difference of two streams, brought to one date. D = sum over the damages period of [R*(t) - R(t) - dC(t)] x (1 + r)^-t, where R*(t) is but-for revenue, R(t) is actual revenue, and dC(t) is the incremental cost that would have been incurred to earn the missing revenue. The cost term is what makes this a profit measure rather than a revenue measure, and it is a slope, not a share: only costs that vary with output belong in it. Past losses are carried forward with interest and future losses discounted back, so a single rate r appears on both sides of the valuation date with opposite effect.

The mathematicsA difference of two valuations at a single date: V* - V, the enterprise value in the but-for world less its value as it is. Each V is a discounted sum, V = sum of FCF(t) x (1 + WACC)^-t over the forecast, plus a terminal value TV = FCF(n+1) / (WACC - g) that capitalizes everything beyond it. Because the terminal term has the growth rate g in its denominator, a small change in the spread between WACC and g moves the whole answer by a multiple, which is why the discount rate decides more than the cash flows do. Lost profits over the interim and a value differential at the end describe the same stream twice, so the two measures cannot simply be added.

The mathematicsA sum over the defendant's books rather than the plaintiff's: G = a x sum of [Rd(t) - Cd(t)], where Rd is the defendant's revenue from the conduct, Cd is the cost the law permits it to deduct, and a is the apportionment fraction, the share of the gain attributable to the conduct rather than to the defendant's own contributions. The fight is entirely in Cd and a. Cd ranges from incremental cost to fully absorbed cost, and a from one to a small fraction, so the same revenue can support gains that differ by an order of magnitude with no disagreement about the sales.

The mathematicsThe value of performance minus the value received, plus consequential loss, minus cost avoided: E = V(performed) - V(actual) + consequentials - costs saved by not performing. V(performed) is what the promised bundle was worth to the plaintiff, usually the market value of the promised goods or the profit the contract would have generated. The identity is simple; the economics is in V(performed), which for a long contract is a discounted stream and inherits every property of present value.

The mathematicsAn accounting sum with no counterfactual in it: L = sum of expenditures made in reliance on the promise that are now wasted, less any salvage value they retain. Because it adds up what was spent rather than what would have been earned, it needs no projection, only an interest adjustment to bring dated outlays to a common date. The mathematical ceiling is the expectation measure: reliance cannot exceed what full performance would have produced, so a losing contract caps the recovery at the contract's own value.

The mathematicsThe market value of what was conferred: Q = units of benefit delivered x the reasonable rate per unit, where the rate comes from the market for that kind of work rather than from the failed contract. It is a pricing problem rather than a loss problem. When the benefit was hours, Q is hours x a defensible hourly rate; when it was goods or a completed structure, Q is what the recipient would have had to pay a stranger for the same thing. The contract price can serve as evidence of the rate, or as a cap, depending on the jurisdiction, which is where the mathematics hands off to law.

The mathematicsA price paid minus a true value at the same moment: O = P - V(true), both measured at the date of the transaction. V(true) is what the asset was actually worth given the facts that were concealed: a valuation as of a single historical date with the fraud removed from the price and nothing else changed. Later movements in value are excluded by construction, which is the property that distinguishes this from expectation and from rescission: the measure freezes at the moment of the exchange.

The mathematicsA pre-agreed function of the breach rather than a measurement of loss: LD = f(breach), typically a fixed sum, or a per-day or per-unit rate multiplied by a count of days or units. The arithmetic is trivial; the economics is a test of whether f was a reasonable forecast of the loss at the time of contracting, which compares the stipulated rate against the expected loss then, E[loss], and against how hard that loss would have been to prove. A rate far above any plausible expected loss reads as a penalty, and the comparison is made ex ante, not against the loss that actually occurred.

The mathematicsDamages scaled by a probability. Award = P(the chance) x V(the outcome), where P is the probability the plaintiff would have obtained the benefit but for the conduct (a cure, a contract, a promotion) and V the value of the benefit had it been obtained, itself an ordinary damages measure. The structure is the expected value of a single-branch decision tree, and it differs from case-within-a-case in what is compensated: the chance itself, at its probability, rather than the outcome conditional on winning. P comes from statistical evidence (survival rates, award rates, promotion rates) and the argument is over whether it is the right population's rate for this plaintiff.

The mathematicsA forecast of the plaintiff's own series with the conduct removed: R*(t) = f(t, X(t)) for each period of the damages window, where f is fitted to pre-conduct history and X(t) holds the drivers that would have applied anyway (market growth, seasonality, capacity). The projection is the counterfactual, and everything downstream inherits its uncertainty, which widens with the horizon: a forecast error of a few percent a year compounds into a wide band by year ten. The discipline is that f must be estimated from data that predate the conduct, or it is fitting the answer.

The mathematicsThe plaintiff as its own control. Loss per period L(t) = B - A(t), where B is the pre-conduct benchmark (a level, a trend, or a seasonal profile) and A(t) is the actual result. The identifying assumption is that nothing else changed between the before period and the after period, which is a statement about the other causes, not about the plaintiff. In its regression form it is a single indicator for the damages period with time and market covariates alongside, and the coefficient on the indicator is the estimate. A clean before period is the whole method; a contaminated one leaves nothing to compare against.

The mathematicsA comparator as the control. R*(t) = R(comparator, t) x k, where k is a scaling factor that maps the comparator onto the plaintiff (relative size, share, or the pre-conduct ratio between them). In its regression form the comparator's series enters as a covariate and the plaintiff's shortfall against the fitted relationship is the loss. The whole inference rests on k being stable, that is, on the two having moved together before the conduct; if the pre-period ratio wandered, the yardstick measures the wandering. Averaging several comparators dilutes idiosyncrasies, which is the road toward synthetic control.

The mathematicsOrdinary least squares on the plaintiff's history, y(t) = b0 + b1 x1(t) + ... + bk xk(t) + e(t), fitted on pre-conduct data and projected through the damages window with the actual values of the x's. The gap between projection and actual is the estimate, with a confidence band from the residual variance that widens as the forecast reaches further from the sample. The coefficients are conditional means, so the model captures what usually accompanied the drivers, not causation; time series fitted to trending data can also produce spurious fits, which is why differencing and stationarity get raised on cross-examination.

The mathematicsTwo subtractions. D = [Y(treated, after) - Y(treated, before)] - [Y(control, after) - Y(control, before)]. The first bracket is the plaintiff's change, the second removes whatever the control changed by over the same period, and the residual is attributed to the conduct. The identifying assumption is parallel trends: absent the conduct, the two would have moved alike, which can be checked in the pre-period but never verified in the post-period. In regression form it is the interaction of a treated indicator with an after indicator; with staggered timing across many units the two-way fixed effects estimator can weight comparisons perversely, which is the Goodman-Bacon result.

The mathematicsA decomposition of the observed shortfall into named causes: total loss = L(conduct) + L(recession) + L(management) + L(other), where only the first term is claimed. Mechanically it is a regression with a covariate for each competing cause, or a sequence of adjustments to the counterfactual that strip out each non-conduct effect before the residual is attributed. The mathematics forces a statement of what every competing cause contributed; an analysis that attributes the whole gap to the conduct has implicitly set the other terms to zero, and that is the assumption the other side attacks.

The mathematicsA price difference times a quantity: D = [P(cover) - P(contract)] x Q, or [P(market) - P(contract)] x Q where no substitute was bought, plus incidental costs, less expenses saved. The economics is in P(market): the price at the time and place performance was due, for goods of the same grade, which is an index or a set of contemporaneous quotes rather than a projection. There is no discounting and no counterfactual beyond the price, which is why this is the cleanest measure in the field and why disputes reduce to which price series is the right one.

The mathematicsA capacity question dressed as a profit question. The seller recovers the margin on the lost sale, M = P - VC, only if the resale did not use capacity the breached sale would have used, that is, only if supply was not the binding constraint. Formally the seller is a lost volume seller when its profit-maximizing output exceeded the level actually sold, so that both sales would have been made. The proof is about the marginal cost curve: constant or falling marginal cost at the relevant output supports the claim, rising marginal cost defeats it, because the second sale would then have cost more than the first.

The mathematicsA downstream sum: C = sum over affected activities of the loss each suffered because the primary breach occurred, each activity's loss being its own lost profits, extra expense or diminution calculation with its own counterfactual. The structure is a chain: breach, then operational consequence, then economic consequence. The mathematics of each link is ordinary; the difficulty is that the chain multiplies uncertainty, since each link carries a probability that it would have unfolded as claimed. Foreseeability and the contract's limitation clauses then cap or cut the chain by law rather than by calculation.

The mathematicsThe value of a terminated relationship as a present value of a stream whose length is itself uncertain. E[V] = sum over t of S(t) x profit(t) x (1 + r)^-t, where S(t) is the probability the relationship survives to period t, estimated from renewal history and the attrition of comparable relationships, and profit(t) is the dealer's projected margin. The survival function is where the dispute lives: assuming perpetual renewal sets S(t) to one everywhere and produces the largest possible answer, while the contract's nominal term sets it to zero at expiry and produces the smallest.

The mathematicsA comparison of two financing paths. Loss = V(business with the promised credit) - V(business as it actually financed), where the first term projects the firm with the facility in place and the second traces what happened: replacement credit at a higher rate, assets liquidated, opportunities forgone. When the business failed, the measure collapses into lost business value at the date the credit was withdrawn. The cost-of-replacement-credit component is simpler, sum over t of [r(replacement) - r(promised)] x balance(t), a spread times a balance over time.

The mathematicsRent on an asset for the period it could not be used. LOU = sum over the deprivation period of rental value(t), where the rental value is the market rate for a substitute of like kind, whether or not one was actually hired; or, for an income-producing asset, the net income it would have produced over the period. It is a stream bounded by the asset's own value: a loss of use that exceeds what the asset was worth has priced the period wrong or run it too long, and the period is bounded by the reasonable time to repair or replace. The rate per period is documentary; the length of the period is the counterfactual.

The mathematicsAn audit difference. Underpayment = sum over periods of [royalty due under the contract's formula on the correct base - royalty actually paid], where the correct base is the contract's definition of net sales applied to the licensee's records, with the deductions it allows and no others. The mathematics is the contract's own formula run twice; the work is reconstructing the base from the licensee's sales data, which may need sampling when the records are large, and the disputes are definitional: what counts as a sale, which deductions apply, how bundled products are allocated. Interest on each period's shortfall runs from when it was due.

The mathematicsA productivity ratio from the contractor's own job. Productivity p = output / labor hours in an unimpacted stretch of the work; in the impacted stretch it is p'. Lost hours = actual hours in the impacted period - (impacted output / p), and damages are those hours times the loaded labor rate. The identification is that the unimpacted stretch is the same work, the same crew and the same conditions, so the fall in p is the disruption. It is a before-and-after design inside a single project, and it stands or falls on whether the unimpacted period is genuinely unimpacted.

The mathematicsThe simplest identity and the least trusted: D = actual cost - bid cost, sometimes adjusted for the contractor's own errors (the modified total cost method). It assumes the bid was accurate, the actual costs reasonable, and every dollar of overrun caused by the owner, three assumptions the arithmetic cannot check. It is treated as a method of last resort precisely because the formula has no term that separates the owner's causes from the contractor's; the measured mile exists to supply that term.

The mathematicsA pro rata allocation of home office overhead to a delay, in three steps. Contract's share of overhead = (contract billings / total firm billings during the contract period) x total home office overhead for that period. Daily overhead rate = that share / days of contract performance. Damages = daily rate x days of compensable delay. Its logic is that overhead continued to accrue while the delayed contract could not absorb it; the attack is on the premise that the contractor was on standby and could not take replacement work, since if it could, the overhead was absorbed elsewhere and the formula counts it twice.

The mathematicsA time-and-rate structure on top of a schedule model. Delay damages = compensable delay days x daily extended cost (site overhead, equipment, escalation), where the compensable days come from a critical path analysis comparing as-planned and as-built schedules, since only delay on the critical path extends completion. The schedule is a network: the longest path through activities with durations and dependencies. Acceleration damages are the premium cost of compressing that path: overtime premiums, added crews, productivity lost to stacked trades. Concurrent delay, where both parties delayed the same critical path, is the term that zeros out the claim.

The mathematicsA policy formula rather than a tort measure. Loss = sum over the period of restoration of [net income that would have been earned + continuing normal operating expenses] - [net income actually earned], the period of restoration being the time reasonably required to repair, not the time actually taken. The but-for net income is projected from pre-loss results and trend, so the counterfactual mathematics is the same as lost profits; what differs is that the policy fixes the period, the triggers and the treatment of continuing expenses, so the formula's terms are defined by contract rather than by economics.

The mathematicsA pure cost sum: E = sum of costs incurred to continue operating that would not have been incurred but for the loss, less any reduction in the business interruption loss those costs achieved. It is an incremental cost identity with an offset, and the offset is the point: an expense that avoided a larger income loss is recoverable, an expense that avoided nothing is contested. No discounting and no projection, only the classification of each cost as extra or normal.

The mathematicsBusiness interruption with the counterfactual running through someone else's loss. L = the insured's but-for income minus actual over the period the dependent property was down, multiplied by a dependency fraction: how much of the insured's throughput actually ran through the damaged supplier or customer and could not be sourced elsewhere. If substitute supply existed, the fraction falls, and the measure shifts toward the extra cost of the substitute rather than lost income.

The mathematicsTwo valuations of the same property. Replacement cost RC is the current cost of replacing with like kind and quality; actual cash value ACV = RC - depreciation, where depreciation is a function of age, condition and useful life, either straight-line (RC x age / useful life) or market-based (what a buyer would pay for the used item). The dispute is the depreciation term: whether it covers physical wear only or also obsolescence, and whether labor depreciates. Under a replacement cost policy the holdback RC - ACV is paid only on completion of the repair.

The mathematicsLosses caused by the delay or denial, over and above the policy benefit. C = sum over consequences of the cost of each: financing obtained because the claim was not paid (a spread times a balance over time), assets sold under pressure (sale price against value), business losses from an interruption that was not funded (the lost profits structure), and distress where the law allows. Each is a counterfactual against timely payment, and the date the claim should have been paid sets the start of the interest and the beginning of every consequential stream, so it is the one input every term shares.

The mathematicsA rate times a base: R = royalty rate x royalty base x units, with the rate the outcome of a hypothetical negotiation at the date infringement began and the base the smallest saleable unit that practices the invention. The negotiation is a bargaining problem: each side's outside option (the infringer's best non-infringing alternative, the patentee's licensing history) sets the range, and the split within the range is where the Georgia-Pacific factors and, in economic form, Nash bargaining operate. The result is sensitive to the base in a way a percentage disguises: a modest rate on the whole product can exceed a large rate on the component.

The mathematicsA fraction applied to a value: damages attributable = total value x a, where a is the share of the value driven by the infringing feature rather than by everything else in the product. Methods for a include conjoint-derived willingness to pay for the feature relative to the whole, regression of price on features, comparable licenses that priced the feature alone, and the incremental profit the feature adds over the next best alternative. Every method produces a fraction between zero and one, and it multiplies the entire answer, so the choice of method matters as much as the choice of base.

The mathematicsLost profits with four gates: demand for the patented product, absence of acceptable non-infringing substitutes, capacity to make the additional sales, and the profit that would have been made on them. The calculation is then sum of [infringer's sales x capture share] x the patentee's incremental margin, where the capture share is the fraction of the infringer's sales that would have gone to the patentee in a market without the infringer, estimated from market shares or from a model of consumer substitution. Sales not captured fall back to a reasonable royalty, so the answer is a mixture of two measures.

The mathematicsA price difference on the sales the patentee did make: E = [P* - P] x Q, where P* is the but-for price without the infringer's competition and Q the units sold at the depressed price. P* comes from a demand model: with the infringer removed, the residual demand facing the patentee shifts out and the profit-maximizing price rises. Consistency then bites: at the higher P* the patentee would have sold fewer units, so Q* < Q, and an erosion claim that keeps Q at its actual level while raising P to P* has ignored its own demand curve. Elasticity is the parameter the whole claim turns on.

The mathematicsLost profits on the unpatented items sold alongside the patented one: C = lost patented units x attach rate x margin on the convoyed goods, where the attach rate is the fraction of patented sales that carried the convoyed item, drawn from the patentee's own sales records. The mathematics is a multiplication; the fight is a functional-unit test, whether the convoyed goods were sold with the patented item for reasons of function rather than convenience, which caps the set of items whose margin enters the product.

The mathematicsA time advantage converted to money. The head start h is the time the defendant would have needed to develop the secret lawfully; damages are the plaintiff's lost profits over h, the defendant's gain over h, or the development cost avoided, depending on the theory. Under the gain version, G = sum over t up to h of the defendant's profits from the secret, discounted; under the loss version, L = sum over t up to h of the plaintiff's shortfall. Everything is proportional to h, so the estimate of the lawful development timeline is the single input the answer is linear in.

The mathematicsA cost the defendant did not bear: A = [research, engineering, testing and time costs of independent development] - what the defendant actually spent, sometimes with a probability-of-success adjustment, since an independent effort might have failed. The engineering estimate is a project costing exercise, hours x rates plus materials plus the cost of the elapsed time, and the optional term E[A] = A / p(success) inflates the figure for the risk that the lawful path would not have worked. It is the measure when neither side can show profits attributable to the secret.

The mathematicsThe defendant's profit on the infringing goods, with the burden shifted on costs: G = a x [revenue from the infringing articles - costs the defendant proves are attributable to them], with a the apportionment fraction where the law allows one. Revenue is a sum from the defendant's records; the deduction term ranges from direct cost to fully absorbed cost with overhead allocated. Because the burden on cost lies with the defendant, an incomplete cost record raises the answer by default, which is the reverse of every plaintiff-side measure.

The mathematicsThe cost of repairing a belief. C = the expenditure required to return consumer perception to its pre-conduct state, estimated either as a fraction of the defendant's advertising spend that created the false impression (the historical rule of thumb) or bottom-up as reach x frequency x cost per impression needed to move measured awareness. The bottom-up version is a media planning model with a survey measurement at each end, belief before and belief after, and the spend needed to close the gap. The fraction-of-spend version has no behavioral model in it, which is its weakness.

The mathematicsA rate bounded by a commitment. The FRAND rate is capped by the value the patented technology adds over the next best alternative the standard could have adopted, with hold-up value (the value that comes from being in the standard rather than from the invention) excluded by construction. Two structures are used: top-down, where an aggregate royalty for the whole standard is set and divided by the share of essential patents held, R = aggregate x [portfolio's essential patents / all essential patents]; and comparable licenses, with observed rates adjusted for scope and timing. Royalty stacking, the sum over all holders, is the consistency check on any single rate.

The mathematicsA price difference times a quantity: O = [P(actual) - P(but-for)] x Q purchased over the conduct period. P(but-for) is estimated by before-and-after (prices outside the conspiracy regressed on cost and demand covariates), by yardstick (an unaffected market), or by a structural model of competitive pricing (price as a mark-up over marginal cost under the counterfactual market structure). In the reduced-form regression the coefficient on a conduct-period indicator is the overcharge; the structural version simulates equilibrium prices. Both must control for the costs that moved prices for innocent reasons.

The mathematicsThe share of an overcharge a buyer passed to its own customers, which reduces the buyer's loss and creates the indirect purchaser's. Pass-through rate p = the change in downstream price per unit change in input cost, estimated by regressing downstream price on input cost with controls; the direct purchaser's net loss is O x (1 - p) plus the lost sales from the higher downstream price. Theory gives the range: under perfect competition with constant marginal cost p = 1, under linear demand and monopoly p = 1/2, and under other demand curvatures anything from below one half to above one. That theoretical spread is why the empirical estimate is fought so hard.

The mathematicsOvercharge on purchases from non-conspirators. Non-cartel sellers price under the cartel's umbrella, so their price also rises: U = [P(fringe, actual) - P(fringe, but-for)] x Q bought from the fringe. The but-for fringe price comes from an oligopoly model in which the fringe's best response to the cartel price is estimated, or empirically from the same before-and-after regression applied to fringe prices. The claim is that the cartel moved the whole market's price, and it requires showing that the fringe's price tracked the cartel's rather than the fringe's own costs.

The mathematicsLost profits from a market the plaintiff was kept out of. L = sum over t of [share*(t) x market size(t) x margin*] - actual profits, where share* is the market share the plaintiff would have held absent the exclusion, estimated from its share in unaffected markets or periods, from a model of entry and competition, or from what the excluding conduct itself displaced. Margin and market size are ordinary projections; share* is a counterfactual about competitive dynamics and carries most of the uncertainty and most of the cross-examination.

The mathematicsAn overcharge on the buying side. Underpayment = [W(competitive) - W(actual)] x Q supplied, where W is the wage or input price and the competitive W is the level at which supply equals demand without the buyer's power. The model mirrors monopoly with the supply elasticity in place of the demand elasticity: at the buyer's optimum, the proportional mark-down of the wage below the worker's marginal revenue product equals one divided by the elasticity of labor supply, so that elasticity sets the size of the harm. Reduced-form estimation regresses wages on a measure of buyer concentration with controls.

The mathematicsA regression and a residual. The market model fits R(i,t) = alpha + beta x R(market,t) + e over an estimation window before the event; the abnormal return on the event day is AR = R(i,event) - [alpha + beta x R(market,event)], the part of the stock's move the market cannot explain. A t-statistic on AR against the residual standard deviation says whether the move is distinguishable from noise; cumulative abnormal return sums AR over a window. The design assumes the market impounds information promptly and that nothing else firm-specific happened in the window, and confounding news on the event day is the standard rebuttal.

The mathematicsThe per-share inflation as a time series: I(t) = P(actual, t) - P(true, t) for every day of the class period, where P(true) is the price with the misstatement removed. A constant-dollar ribbon sets I(t) equal to the corrective-disclosure price drop until the disclosure, then zero; a constant-percentage ribbon scales with price; backcasting builds I(t) from the sequence of disclosures, working backward from each drop. A claimant's damages are I(purchase date) - I(sale date), so the ribbon's shape, not only its height, decides who recovers what, and the ribbon is only as sound as the event study that measured each drop.

The mathematicsA decomposition of a price decline into disclosure and everything else. Recoverable decline = the abnormal return on the corrective disclosure date, the event-study residual, not the raw drop; market and industry movements that day are removed by the regression, and confounding company news is separated where it can be. Formally, loss = AR(disclosure) x shares held. What the law adds is that only the portion attributable to the truth coming out counts, so where several facts were disclosed at once the residual has to be decomposed further, and that second decomposition is where most of the argument sits.

The mathematicsThe event study run where no price moved. The question is whether a confirmatory false statement kept the price from falling, so the counterfactual is the abnormal return that would have followed a truthful statement, estimated from the abnormal return on the later corrective disclosure, or from analyst expectations and the reaction of comparable firms. The test is a null result followed by a drop: no abnormal return on the date of the false statement, and a significant negative one when the truth emerged.

The mathematicsA statutory difference: D = price paid (capped at the offering price) - value at the time of suit, or the sale price if sold before suit, or the sale price if sold after suit for a smaller loss. The formula fixes the measure and shifts the burden: the defendant may reduce it by proving that part of the decline came from causes other than the misstatement, which is a negative-causation event study decomposing the decline into disclosure-related and unrelated components. The plaintiff's arithmetic is subtraction; the defendant's defense is the regression.

The mathematicsUnwinding rather than compensating. R = consideration paid, with interest, - value of what was received, brought to the date of judgment or to the date it could reasonably have been sold, rather than frozen at the transaction. The difference from out-of-pocket is the date: rescission gives the plaintiff the benefit or burden of every subsequent movement in value, which is why it is available only where the plaintiff can show it would have held or sold on those terms, and why the choice of date can move the answer more than the fraud did.

The mathematicsA valuation at the merger date with the deal's own synergies excluded: FV = value of the company as a going concern on the plan in place before the merger, typically by DCF, V = sum of FCF(t) x (1 + WACC)^-t + TV, cross-checked against comparable companies and the unaffected market price. Courts have increasingly weighted deal price minus synergies, FV ~ deal price - the synergy value the buyer paid to capture, and the unaffected trading price, treating the DCF as one input among several. Disagreement concentrates in the terminal value and the discount rate, where a one-point change moves fair value by a large fraction.

The mathematicsThree tests, each a comparison at a historical date. Balance sheet: fair value of assets against face value of liabilities, with contingent liabilities probability-weighted. Cash flow: projected cash flows against debts as they come due, period by period, under a stress on the assumptions. Adequate capital: the firm's capital against the volatility of its cash flows, whether a reasonable shock would leave it unable to pay. The third is explicitly a distribution question, so it is often run as a Monte Carlo over cash flow scenarios, asking what fraction of paths leave the firm unable to service its debt.

The mathematicsA contingent payment as a function of a metric: E = f(metric over the earnout period), where f is the contract's formula (a threshold, a percentage above a hurdle, a tiered schedule) and the metric is defined by the contract's accounting terms. Damages for breach of an operating covenant are E(metric as it would have been with the covenant honored) - E(actual metric), which requires a counterfactual for the business under the buyer's promised conduct. Because f is often kinked at a threshold, a small change in the counterfactual metric can move the payment from zero to the full amount, so the sensitivity is discontinuous rather than smooth.

The mathematicsPricing a security by its cash flows and its contingencies. Straight debt: the present value of coupons and principal at a credit-adjusted yield. Options and option-like features: Black-Scholes-Merton or a lattice, with value a function of the underlying price, strike, volatility, time and rates, where volatility is the input the model cannot observe. Structured products: a Monte Carlo over paths of the underlying with the payoff rule applied to each path, averaged and discounted. The model choice sets what can be argued: in a lattice the dispute is the volatility, in a simulation it is the distribution and the correlations.

The mathematicsThe price with the manipulation removed, as a series. Artificiality A(t) = P(actual, t) - P(but-for, t) for each day of the manipulation, where the but-for price comes from a model of the price's normal relationship to its fundamentals (related contracts, the spot market, the cost of carry) fitted outside the manipulation window, or from the abnormal return around the manipulative trades in an event-study frame. Damages for a trader are the sum over their trades of A(t) x quantity x direction. The but-for model's specification is the dispute, and a manipulation that lasted long enough to change the fundamentals themselves is the hard case.

The mathematicsA value at a date, where the date is the argument. For property with a stable value, D = V at the date of conversion, with interest to judgment. For property whose value fluctuates (securities, commodities) the jurisdictions diverge: value at conversion, the highest value between conversion and a reasonable time after the plaintiff learned of it, or value at judgment. The mathematics is a lookup on a price series at a chosen date, or the maximum over a window, and the choice of rule can move the answer by the full range of the asset's movement over the period, which is why the rule is argued before the valuation is.

The mathematicsThe growth of the firm's deficit during a period it should have been wound up. DI = [liabilities - asset value] at the actual failure date - the same deficit at the date the firm should have ceased trading. It requires a solvency determination at the earlier date, a valuation of assets at both dates, and a showing that continued operation rather than market movement produced the growth. The measure has been rejected as an independent theory in several jurisdictions and survives as a way of measuring damages for a recognized wrong, which is the mathematics outliving the cause of action.

The mathematicsA deterministic re-run of the waterfall, period by period, under two sets of inputs. In each payment period the cash available from the collateral pool is applied in the indenture's priority: fees, then interest by seniority, then coverage tests that divert cash to principal when a ratio falls below its trigger, then principal by seniority, then the residual to equity. The distribution to tranche k in period t is a function of available cash, the pool's state and the test outcomes, D(k, t) = W(cash(t), state(t), tests(t)). The loss is the sum over t of [D*(k, t) - D(k, t)] compounded to the judgment date, where D* uses the correct classification of collateral and the correct test arithmetic and D is what was paid. Because the coverage tests are step functions, a small misclassification of collateral can flip a test and move a whole period's cash between tranches, so the sensitivity is discontinuous rather than smooth. Where the counterfactual pool performance is itself uncertain, the re-run becomes a Monte Carlo over default and recovery paths, and the loss is reported as a distribution rather than a point.

The mathematicsA present value of an earnings stream with a life cycle. L = sum over t from injury to the end of worklife of [E*(t) - E(t)] x P(alive and working at t) x (1 + r)^-t, where E*(t) is but-for earnings on a growth path (age-earnings profile plus wage growth), E(t) is residual earning capacity, and the survival and participation probabilities come from life and worklife tables. Fringe benefits add a fraction; taxes and personal consumption are subtracted where the law directs. Growth and discounting enter as a spread, and the net discount rate, r less g, is the single parameter the present value is most sensitive to.

The mathematicsTwo sums either side of the judgment date. Back pay = sum from termination to judgment of [W*(t) - W(t)] with interest, where W* is the wage in the lost position with the raises it would have carried and W is actual earnings, including what mitigation should have produced. Front pay = sum from judgment forward of the same gap, discounted, over a horizon bounded by the expected time to find comparable employment or by expected remaining tenure, whichever the theory calls for. The front pay horizon is a survival question about re-employment, and it is where the disagreement usually sits.

The mathematicsExpected years of working life from a Markov model of labor force states. A person is active or inactive at each age; transition probabilities (active to inactive, inactive to active, and mortality, each by age) are estimated from longitudinal survey data; worklife expectancy is the expected number of future years spent active, WLE = sum over future ages of P(active at that age, given the current state). The increment-decrement model allows re-entry, which the older static life table method did not. The output is an expectation; the distribution around it is wide, and a claim that uses the expectation as if it were certain has discarded the variance.

The mathematicsHours times a replacement wage, over time and survival. H = sum over t of [hours*(t) - hours(t)] x w(t) x P(alive at t) x (1 + r)^-t, where hours*(t) is the time the person would have spent on household production (from time-use survey data by age, sex and household composition), hours(t) is what they can still do, and w(t) is the market wage for the replacement services. Every term is a lookup or a projection; the fight is over which services count and whether the replacement wage is the specialist's rate or the generalist's.

The mathematicsA deduction, in a wrongful death claim, for what the decedent would have consumed. Net loss = gross earnings stream x (1 - c), where c is the personal consumption fraction, estimated from consumer expenditure survey data as a function of household income and the number of household members. c is larger for one-person households and smaller as income rises; the choice between consumption (what the decedent spent on themself) and maintenance (the minimum needed to keep themself working) moves c by several points, and every point is a percent of the whole award.

The mathematicsA present value of a schedule of future costs, each item with its own quantity, price, growth and horizon: LC = sum over items i and years t of q(i,t) x p(i) x (1 + g(i))^t x P(alive at t) x (1 + r)^-t. Medical cost growth g typically exceeds general inflation, and the spread between g and r is the driver: when g is close to r, discounting barely reduces the sum and the award approaches the undiscounted total; when r exceeds g by several points, the far years fade. Life expectancy is not a cut-off but a declining weight. The demonstration at /the-rate-decides/ shows exactly this stream.

The mathematicsA fraction of wages, or an itemized cost. Simple form: benefits = wages x f, where f is the employer's benefit cost as a share of wages from employer compensation surveys, covering legally required contributions, retirement and health insurance. Itemized form: each benefit on its own basis, health insurance at the employer's premium share, a defined benefit pension as the present value of the lost accrual, a defined contribution plan as the employer match plus growth. The itemized version is more defensible and more work; the fraction version is standard and attacked when the person's actual benefits diverged from the average.

The mathematicsA price on the enjoyment of life, derived from the value of a statistical life. VSL is estimated from wage-risk regressions, how much extra wage workers accept per unit of fatality risk, w = f(risk, X), with the risk coefficient scaled up to a whole life, or from what people pay for safety. Hedonic loss is then VSL x the fraction of life enjoyment lost x years remaining, discounted. The mathematics is a compensating-differential regression; the leap courts reject is from a marginal willingness to pay for small changes in risk to the value of an entire life, since the trade-off is not linear over that range.

The mathematicsWhat savings would have grown to. LA = sum over t of s(t) x (1 + r(investment))^(T - t), the future value at retirement or death of the savings that would have been made from lost income, discounted back to the judgment date at the discount rate; the two rates work against each other, and if they are equal the accumulation equals the present value of the savings themselves. It is a claim about a portion of lost earnings and their investment return, so it can double count with the lost earnings that funded it; the model has to subtract the savings from the earnings claim or present one measure, not both.

The mathematicsA single formula applied to every member: damages(i) = f(x(i)), where f is estimated once from common evidence and x(i) is each member's own data (units, dates, prices). The requirement that impact be shown with common proof is a requirement on f: the same regression, with the same coefficient, must plausibly describe every member, or at least a definable subclass. Where the estimated overcharge is positive for the average member but not for identifiable subgroups, the model is said to prove impact for people it cannot show were harmed, and that is the objection its structure has to answer.

The mathematicsInference from a sample to a population. Estimate = sample mean x N, with a standard error of (s / square root of n) x N and a confidence interval built from it; a stratified design reduces the variance by sampling within groups that differ, and a probability sample is what licenses the extrapolation at all. The mathematics is elementary and the disputes are about design: whether the draw was random, whether n supports the claimed precision, whether the population sampled is the population extrapolated to, and whether the interval's width was disclosed next to the point estimate.

The mathematicsThe value of a feature from choices between bundles. Respondents choose among product profiles that vary attributes and price by design; a discrete choice model, usually a logit with random coefficients, estimates a utility for each attribute level, U = sum of b(attribute) + b(price) x price, and willingness to pay for a feature is its utility divided by the price coefficient, WTP = b(feature) / -b(price). The design is orthogonal so the attributes can be separated. Market value needs a supply side too: WTP is a demand-side quantity, and the price that would have prevailed is an equilibrium, which is the supply-side critique.

The mathematicsThe part of the price paid for a claim or feature that was false or absent. Premium = P(with the claim) - P(without), estimated by hedonic regression of price on characteristics across products and time, by conjoint-derived willingness to pay, or by the price drop when the claim was withdrawn; damages are premium x units sold. In the hedonic version the coefficient on the claim indicator is the premium and the identification is that the other characteristics are fully controlled for; the conjoint version carries the supply-side gap; the event version needs a clean withdrawal with nothing else changing.

The mathematicsThe lesser of two measures. Repair cost C(repair) is a sum of contractor estimates; diminution DV = V(before) - V(after), two appraisals as of the same date, or a paired-sales or hedonic analysis of what damaged properties sell for against undamaged ones. The rule in most jurisdictions awards the smaller, subject to exceptions when repair is economically wasteful or the property has personal value. The comparison is where the economics lives: when C(repair) exceeds DV, the mathematics says restore the value, not the thing.

The mathematicsThe residual value loss after remediation. Stigma = V(remediated, formerly contaminated) - V(never contaminated), estimated by hedonic regression on sales of properties with and without a remediation history, controlling for characteristics, or by paired sales and appraiser studies; the coefficient on a remediation-history indicator is the stigma. Its decay is part of the model: an interaction with years since cleanup tests whether the market forgets, and a persistent coefficient is the claim, a decaying one the defense.

The mathematicsThe value of lost services from the resource over the injury period, brought to a common date. The service-to-service approach (habitat equivalency analysis) equates the discounted service-years lost, sum of services lost(t) x (1 + r)^-t, with the discounted service-years a restoration project would provide, and solves for the scale of restoration. The value-to-value approach prices the services, by revealed preference (travel cost, hedonic) or stated preference (contingent valuation), and sums the lost value over time. In either structure the discount rate applies to service units as well as dollars, and the recovery curve of the injured resource is the counterfactual.

The mathematicsThe value of reputation and relationships, measured as an intangible residual. Goodwill = business value - value of identifiable net assets, so lost goodwill is the change in that residual, estimated from the change in expected excess earnings capitalized: dG = sum over t of d(excess earnings)(t) x (1 + r)^-t, where excess earnings are what the business earns above a normal return on its tangible assets. In a defamation or disparagement claim the change in earnings must be tied to the statement, a before-and-after or event-study design on revenue; the capitalization then converts the earnings effect into a value effect.

The mathematicsDividing a cleanup cost among responsible parties by equitable factors. Share(i) = w(i) / sum over j of w(j), where w(i) is a weight built from the volume each party contributed, the toxicity of what it contributed, its degree of involvement, its cooperation, and whatever other factors the court adopts; the total cost divides in those proportions. Volumetric allocation sets w(i) to volume alone; a toxicity-weighted version multiplies volume by a hazard factor. Orphan shares (insolvent or unknown parties) are reallocated among the rest, a renormalization of the weights. It is apportionment by formula, and the weights are the argument.

The mathematicsYield shortfall times price, less costs not incurred. L = [Y*(acres) - Y(actual)] x P - C(saved), where Y* is the expected yield from the grower's own history, county averages or trial plots, P the price at the harvest that was lost (market or contract), and C(saved) the harvest and marketing costs avoided on the crop not produced. Multi-year damage to perennial crops or to soil becomes a present value of shortfalls over the recovery period. The yardstick is usually the neighboring field or the grower's other fields, so this is a before-and-after or yardstick design on a biological process with weather as the confounder.

The mathematicsA sum of future amounts weighted by a discount factor: PV = sum of CF(t) x (1 + r)^-t. A level stream of n payments has a closed form, PV = CF x [1 - (1 + r)^-n] / r; a stream growing at g has PV = CF x [1 - ((1 + g) / (1 + r))^n] / (r - g), which shows that only the spread r - g matters for a growing stream. The function is convex in r: the first point of rate moves the answer more than the fifth, and long horizons amplify the effect, since the factor at year t is (1 + r) to the power minus t. Real versus nominal is a consistency condition: real cash flows take a real rate, nominal take nominal, and mixing them is an error of a few points that reads as a valuation.

The mathematicsCompounding a past loss forward to the judgment date. Simple interest: L x (1 + i x T). Annual compounding: L x (1 + i)^T. Compounding m times a year: L x (1 + i/m)^(mT). For a stream of losses at different dates, each is compounded from its own date, the mirror image of discounting. The rate and the compounding convention are set by statute or by the court's discretion, and over a long pre-judgment period the choice between simple and compound, or between a statutory and a market rate, can change the award by a large fraction: ten years at seven percent multiplies a loss by about 1.97 compounded and 1.70 simple.

The mathematicsA cap defined by an optimization. The recoverable loss is the loss the plaintiff would have had if it had taken the reasonable steps, plus the cost of those steps, so Loss = L(with reasonable mitigation) + C(mitigation). Reasonable mitigation is the action that minimized the loss at reasonable cost given what the plaintiff knew at the time, so the counterfactual is a decision problem solved under the plaintiff's information set, not with hindsight. Where the plaintiff mitigated well, the offset is its actual mitigation income; where it did not, the model substitutes what it should have earned.

The mathematicsSubtractions from the gross loss. Net loss = gross loss - sum of offsets, the candidates being payments the plaintiff received because of the injury (insurance, benefits, mitigating earnings) and costs it avoided by not performing. Which offsets count is legal, and the collateral source rule excludes many; the mathematics is timing and tax: an offset received at a different date from the loss must be brought to the same date, and an offset taxed differently from the loss must be compared after tax, or the subtraction mixes bases.

The mathematicsPutting the award on the same tax footing as the lost stream. If the lost earnings would have been taxed and the award is not, the after-tax equivalent is L x (1 - t), with t the effective rate on that income; if the award will be taxed at a different rate or all at once, a gross-up L / (1 - t') restores the after-tax position. The term is multiplicative, so the effective rate matters directly, and a progressive schedule makes a lump sum taxed in one year cost more than the same income taxed across many, which is the gross-up argument. The direction of the adjustment depends on the tax treatment of the claim type.

The mathematicsNot a damages measure but a valuation offered for a different purpose: the defendant's net worth, as evidence for calibrating a punitive award. Net worth = assets at fair value - liabilities, as of a date near trial, from financial statements adjusted for items not carried at fair value (real estate at cost, unrecorded intangibles, contingent liabilities). Whose net worth counts, for a subsidiary or a parent, is legal. The mathematics is a balance sheet at fair value; the discipline is that a punitive award expressed as a fraction of net worth is a ratio, and numerator and denominator must be measured on the same date and basis.

The mathematicsRent on the value the regulation withheld, for the time it withheld it. T = sum over the taking period of V(unrestricted) x k(t), where V(unrestricted) is the property's value without the restriction and k is a periodic return, the rate a fair rental of that value would have earned; or, as a difference of use values, T = sum over t of [use value(unrestricted, t) - use value(restricted, t)]. It is a stream, not a lump sum, and it is bounded by the value of the property itself, so a taking measured by the development profit a project would have earned exceeds what the mathematics of use value supports.

The mathematicsThe value taken, as of the date of taking. Total taking: J = fair market value on that date, by comparable sales, income capitalization or cost. Partial taking: J = V(whole, before) - V(remainder, after), which captures severance damage to what is left. Temporary taking: the rental value over the period. Every version is an appraisal at a date, and the date matters because the government's own project can move the market: value is measured with the project's influence removed, a counterfactual on the market rather than on the property.

The mathematicsThe cost of holding an asset through a delay. CC = sum over the delay period of [interest on the capital tied up + taxes + insurance + other holding costs], each period's charge being the capital balance times the rate plus the fixed holding costs, with the balance growing as carrying costs are themselves financed. It is a compounding sum on a balance, and it double counts with a diminution measure taken at the end of the delay, since the diminished value already reflects the delayed cash flows. One of the two, not both.

The mathematicsA cost index applied over time. E = sum over cost components i of C(i) x [I(i, actual date) / I(i, planned date) - 1], where I is a construction cost index for that component (labor, materials, equipment) and the ratio is its escalation between the date the work was planned and the date it was done. It is an index-number calculation; the disputes are about which index (national or regional, composite or component-specific), and whether the component would have been bought at the planned date at all, since a contract not yet let has no locked price to escalate from.

The mathematicsThe profit a project would have earned, as a present value. LDP = [V(completed project) - sum of development costs brought to the same date] discounted from completion to the valuation date, with a probability-of-completion factor where the project had not cleared its risks. V(completed) is a valuation of a building that does not exist, by capitalizing projected income or by comparable sales; the cost side is a construction budget; and the profit is the residual between two large numbers, so a small error in either moves the residual by a large fraction. That is the general property of residual measures: they amplify the uncertainty of their inputs.

The mathematicsThe cost of getting the same profit later. If a project of value V would have completed at date t1 and instead completes at t2, the loss is V x (1 + r)^-t1 - V x (1 + r)^-t2, plus the carrying costs over the interval and any change in V itself across it (market movement, escalation). It is the time value of a deferred lump sum: the profit is not lost, only delayed, and the loss is the discounting of the delay plus what the delay cost to carry. This is the measure that fits when the project will still be built, and it is a fraction of the lost development profit that fits only when it will not.

The mathematicsA proportion and its interval. From a probability sample of the relevant consumers, the share who say the claim mattered to their purchase, or who took the message from it, is the sample proportion p, with a standard error of the square root of p(1 - p)/n; a control group shown a version without the claim gives a baseline, and net materiality is p(test) - p(control), the difference removing yea-saying and guessing. The design elements (universe, question order, the control stimulus) determine whether the proportion measures what it claims to; the arithmetic is a difference of two proportions with a test on it.

The mathematicsA demand model closed with a supply model. Conjoint or other choice data give each consumer's utility for attributes; a product's market share is the sum of choice probabilities over consumers, s(j) = sum over i of P(i chooses j, given prices and attributes); firms set prices to maximize profit given rivals' prices, which yields an equilibrium (typically Bertrand-Nash) where each firm's first-order condition holds. The but-for price of a product with a feature removed is the new equilibrium, not the shift in willingness to pay, and the gap between the two is the supply-side correction: competition absorbs part of what consumers would have paid.

The mathematicsA change in the premium or volume the brand commands, tied to the conduct. Dilution = d(brand premium) x volume, where the premium is estimated from conjoint (brand as an attribute, WTP for it relative to an unbranded equivalent) or from hedonic regression of price on brand and other characteristics, and the change is measured between exposed and unexposed groups. The experimental version compares two randomized groups' brand associations and willingness to pay; the loss is the difference times the market, and the leap from a measured shift in associations to a shift in purchase behavior is where the design is tested.

The mathematicsDiverted sales times margin. L = [sales the defendant made because of the false claim] x [fraction that would have gone to the plaintiff] x [plaintiff's incremental margin]. The first term comes from a survey or experiment measuring how many buyers the claim moved (the difference in purchase intent between exposed and control groups, applied to the defendant's volume), or from a before-and-after regression on the defendant's sales around the campaign; the diversion fraction from market shares or a substitution study; the margin from the plaintiff's cost records. Three multiplied estimates, each with its own interval, so the product's uncertainty is wider than any one of them.

The mathematicsNot a damages formula but the audit of the instrument the others depend on. A survey estimate is a proportion with a margin; the audit asks whether it measures the right population (universe), whether the sample was drawn so the interval is meaningful (probability sampling, response rates), whether the question elicited the answer or led to it (order, wording, demand effects), whether a control condition removed background noise (p(test) - p(control) rather than p alone), and whether coding of open-ended responses was blind. Each failure biases the proportion in a known direction, which is why a rebuttal expert can often say not only that the number is wrong but which way.

The mathematicsHours times rate, summed over a class, from records or a sample. Unpaid wages = sum over employees and pay periods of [hours worked - hours paid] x the applicable rate, with overtime at the premium multiple for hours over the threshold. Where time records are missing, hours are estimated from a representative sample (time studies, surveys, badge data) and extrapolated with a confidence interval; where the class is heterogeneous, stratification by job and site keeps the extrapolation honest. The rate arithmetic is fixed by statute; the sample design is the fight.

The mathematicsA regression with a protected-class coefficient. log(pay) = b0 + b(group) x group + sum of b(k) x controls(k) + e, where the controls are the legitimate determinants (tenure, level, location, performance) and b(group) is the residual gap after them. Its t-statistic against zero is the statistical significance; the two-or-three standard deviations convention comes from here. The choice of controls is the model: including a variable that is itself the product of discrimination (a grade assigned unequally) explains the gap away, omitting a legitimate one inflates it. The remedy is the sum over affected employees of the gap x pay, over the period.

The mathematicsValuing options and shares that were forfeited. Vested options: Black-Scholes-Merton or a lattice, value = f(stock price, strike, volatility, time to expiry, rate, dividends), at the forfeiture date. Unvested awards: the same valuation times the probability the award would have vested, P(still employed at vesting) x P(performance condition met), with the value at the expected vesting date discounted back. Restricted stock: shares x price at the vesting date, probability-weighted. Volatility is unobservable and the vesting probabilities are survival estimates, so the answer carries both an option-pricing and a survival-analysis dispute.

The mathematicsThe value of the accounts and people that left. Diversion = sum over diverted clients of [expected profit from the client over its expected remaining life] discounted, where remaining life comes from the firm's own retention data (a survival curve fitted to its attrition history), profit is revenue less cost to serve, and the sum is reduced by the probability the client would have left anyway. The people cost separately: recruiting and training replacements, and productivity lost during the ramp. It is customer lifetime value applied to a book of business whose hazard of loss the raid raised.

The mathematicsAn expected value of the underlying case. L = P(would have won) x E[award, given a win] x P(collectible) - what was actually recovered. Each term is estimated separately: the probability from the merits of the underlying claim as a fact-finder would have seen it, the award from the damages that claim would have supported (an ordinary damages model nested inside this one), and collectibility from the original defendant's assets and insurance. It is a decision tree with one branch, and the nesting means the malpractice case carries the full uncertainty of the case it replaced.

The mathematicsThe tax cost of the wrong advice. L = tax paid under the advice actually given - tax that would have been paid under competent advice, plus penalties and interest attributable to the error, less any tax benefit the plaintiff obtained from the erroneous position in other years. Each term is a tax computation on the same facts under a different position, so the model is a pair of returns, with the timing of the two liabilities reconciled to a common date. Where competent advice would have led to a different transaction altogether, the counterfactual expands into that transaction's economics.

The mathematicsRestoring the trust to the position proper administration would have produced. Surcharge = V(trust under prudent management) - V(trust as managed), as of the date of judgment, with the prudent counterfactual built from the portfolio the fiduciary should have held; or, where the fiduciary profited, disgorgement of the profit. The prudent portfolio is a benchmark return applied to the corpus over the period, compounded, so the measure is the difference of two compounded values, and the choice of benchmark (a balanced index, the investment policy statement, a peer group) sets the answer. Interest on the shortfall runs from each breach.

The mathematicsThe benchmark comparison made explicit. Loss = the value the portfolio would have reached at the benchmark return - the value it did reach, that is, corpus compounded at r(benchmark, t) less corpus compounded at r(actual, t), period by period. The benchmark is a portfolio at the risk level the beneficiary's objectives called for, built from index returns in the policy's asset allocation; where the fiduciary took more risk than that, the loss is not only the shortfall but the value of the extra risk taken, sometimes measured as a difference in risk-adjusted return. The measure is sensitive to the start date and the weights, and a benchmark chosen after seeing the results is the standard rebuttal.

The mathematicsA fee gap times assets, compounded. Excess fees = sum over t of assets(t) x [f(actual) - f(benchmark)], where f is the annual fee rate and the benchmark is the fee available for the same service at comparable scale (a peer group, or the same manager's institutional rate). Because fees reduce the base that compounds, the loss to a participant exceeds the sum of the fees: it is the terminal shortfall, corpus x [(1 + r - f(benchmark))^T - (1 + r - f(actual))^T], which grows with the horizon. Peer-group construction is the dispute, since the benchmark rate is a distribution and the choice of percentile is the answer.

The mathematicsA measure that inherits the deal's own pricing. If the buyer priced the target at a multiple M of a metric (EBITDA, revenue, ARR) and the breach reduced the metric permanently by dE, then loss = M x dE, the price the buyer would have paid with the truth known; if the reduction was one-time, loss = dE, the cash effect only. The whole dispute is whether dE is recurring, which decides whether it is multiplied, and the multiplier turns a modest accounting error into a large claim. The evidence for M is the deal model and the letter of intent; the evidence for recurrence is the nature of the item.

The mathematicsAn accounting difference defined by contract. Adjustment = [working capital at closing - target working capital], or [closing net debt - estimated net debt], computed under the accounting principles the agreement specifies, in the order it specifies (typically the seller's historical practices, then GAAP). The arithmetic is a subtraction; the dispute is which principles govern each line and whether an item is a debt-like item or a working capital item, because moving a liability between categories changes the adjustment dollar for dollar. An earnout on the same metric raises the same classification questions with a formula on top.

The mathematicsTwo tests and a valuation, in sequence. The disproportionality test is a yardstick: the target's change in the metric (revenue or EBITDA) over the signing-to-closing window, less the change in a matched peer set over the same window, d(target) - d(peers); in regression form it is the coefficient on a target indicator with time and industry effects, so the carve-outs for industry-wide and general economic conditions are literally the controls. The durational test asks whether the decline persists in the projections a reasonable buyer would make at the closing date, a comparison of the pre-signing forecast with a post-decline forecast over a horizon of years rather than quarters. The valuation converts the surviving decline into value, V(at signing) - V(at closing), using the same discounted cash flow structure as lost business value, and how that compares with the deal value is where the materiality argument is made; what fraction is enough is a question of law. Everything turns on the peer set and the window: a peer group that fell as far shows no disproportion, and a window that ends before recovery shows duration that a longer window would not.

The mathematicsA sum of separable harms, each with its own model. Response costs: an accounting sum. Business interruption: the lost profits structure over the outage. Customer loss: churn attributable to the breach, from a difference-in-differences on retention against a control cohort, times customer lifetime value. Individuals' losses: the expected cost of fraud, P(misuse) x E[loss, given misuse], plus mitigation such as monitoring, which is the medical monitoring structure applied to identity. Regulatory and contractual: fixed or formulaic. The total is additive, and the double-count risk is between customer loss and interruption, which can describe the same lost revenue.

The mathematicsTwo counterfactuals, one for cost and one for benefit. Cost side: amounts paid and internal costs incurred on the failed project, less the salvage value of anything usable, plus the cost of the replacement system where that is the theory. Benefit side: the benefits the system would have delivered (savings, revenue) from the promised go-live date, discounted, less the benefits the replacement delivers from its actual go-live, a deferred-benefit measure with the same structure as deferred development profit. Limitation clauses then cap or exclude the benefit side by law; the mathematics of both sides is ordinary and the contract decides which survives.

The mathematicsThe present value of a customer's expected margin over an uncertain life. CLV = sum over t of m(t) x S(t) x (1 + r)^-t, where m(t) is the margin per period and S(t) the probability the customer is still active at t, from a retention model. With a constant retention rate p and a stream that begins next period the sum has a closed form, CLV = m x p / (1 + r - p), which shows the answer is driven by how close p is to one. Damages for lost customers are the sum of their CLV at the date of loss, less the CLV they would have had anyway given normal churn, so the claim is really about the excess churn attributable to the conduct, estimated by comparing the affected cohort's retention curve with a control cohort's.

The mathematicsThe government's loss from the false claims, multiplied by statute, with penalties per claim added. Single damages = sum over false claims of [amount paid - value of what was received]; where the goods or services had no value, the whole amount, otherwise the difference. Trebling and per-claim penalties are then arithmetic. When the claims are too numerous to review, single damages come from a probability sample: the error rate and the mean overpayment are extrapolated to the population with a confidence interval, and the sampling design (frame, stratification, size) is the expert issue, while the penalty count turns on how a claim is defined.

The mathematicsA cost-based settlement, not a profit measure. Recovery = costs incurred on the terminated work + a reasonable profit on that work + settlement expenses - amounts already paid; anticipatory profit on the unperformed portion is excluded, and if the contract would have lost money, a loss adjustment reduces the recovery proportionally. The cost terms are an allocation problem: what was incurred for this contract, what overhead is allocable, what is unabsorbed. The profit term is a rate applied to costs incurred, and the loss ratio, (total estimated cost - contract price) / total estimated cost, scales the whole settlement down.

The mathematicsA position on a distribution, then a ratio, then a conditional. Fair market value is tested by placing the compensation C against a benchmark distribution of pay for the specialty and market, usually as the percentile it occupies, and comparing that with the percentile of the physician's productivity measured in work relative value units or collections: pay at the ninetieth percentile against productivity at the fiftieth is the pattern the test exists to catch. The ratio C / wRVU against its own benchmark distribution controls for the fact that a productive physician legitimately earns more. Commercial reasonableness is a separate question with no benchmark: whether a reasonable entity would enter the arrangement absent referrals, tested by removing the referral revenue from the arrangement's economics and asking whether it still makes sense. If either test fails, the claims submitted for services referred by that physician during the non-compliant period become the base for False Claims Act damages, which is where this model hands off. Benchmarks are survey data with their own sampling error, and the choice of survey, year, region and specialty definition moves the percentile, which is the usual fight.

The mathematicsThe cost of a testing program, priced as an expected present value. MM = sum over t of [P(an exposed person needs the test at t) x cost of the test at t x number exposed] x (1 + r)^-t, over the latency period the science supports, with the program's content set by what a reasonable physician would prescribe for the exposure over and above ordinary care. The incremental term is the discipline: only monitoring the exposure makes necessary counts, so the model subtracts the baseline screening the population would have had anyway. Class size, latency and test cost growth are the inputs it is most sensitive to.

The mathematicsAn accounting sum with a lost-profits tail. Direct cost = notification + retrieval + logistics + refund or repair + disposal, each a count of units times a unit cost. Indirect cost = lost sales during the recall period, projected against the pre-recall trend, plus the longer-run share loss where the brand was damaged, a before-and-after on sales with a market control. Where the recall was caused by a supplier's component, the allocation to that supplier is the fraction of units affected and the contract's indemnity terms. The direct sum is documentary; the indirect tail is where the counterfactual and the disagreement live.

The mathematicsAn overcharge measured against a counterfactual entry date. Purchasers' loss = sum over t from the but-for entry date to actual entry of [P(brand, t) - P(but-for blended, t)] x Q(t), where the blended price weights brand and generic prices by the generic substitution rate that would have applied, itself a curve rising over the months after entry, estimated from other molecules' launches. The generic's own lost profits, in the mirror image, are its but-for share times the market times its margin over the delay. The entry date and the erosion curve are the two counterfactual inputs, and both are estimated from comparable launches.

The mathematicsThe value of the investment on the day before the taking became known, by discounted cash flow where there is a track record and by cost or comparables where there is not. FMV = sum of FCF(t) x (1 + r)^-t + TV, with r built up from a risk-free rate, an equity premium and a country risk premium reflecting the host state's risk as it stood before the measure. The valuation date is chosen to exclude the effect of the expropriation itself on value. Because r carries the country premium and the cash flows carry the project's specific risk, the tribunal's task is to avoid counting the same risk twice, and the premium's size is where most awards are contested. Interest to the award compounds at a rate the tribunal sets.