16 structures, 115 models, 29 verified sources
Everything on this page is written with symbols and without values. It explains how a damages figure is built and produces none: no formula here is evaluated, no matter is valued, and no opinion is offered on what any claim is worth or what is legally recoverable. Those are the work of a retained expert and of counsel.
Each structure is set out four ways: what it is, in plain language; its general form, as an expert would write it; what drives the answer, meaning the term or assumption the result is most sensitive to; and how it is attacked, the standard line of cross-examination. Under each is the list of damages models that run on it, linked to their entries in the Index, and the foundational literature where a verified source exists.
Discounting, compounding and the time value of money
A dollar in ten years is worth less than a dollar today, and a dollar lost ten years ago is worth more. Discounting converts future amounts to a present equivalent; compounding does the reverse for past ones. Nearly every damages stream passes through one or the other, and a long-horizon claim passes through both, on either side of the valuation date.
General formPV = sum over t of CF(t) x (1 + r)^-t
Level stream of n payments: PV = CF x [1 - (1 + r)^-n] / r
Stream growing at g: PV = CF x [1 - ((1 + g) / (1 + r))^n] / (r - g)
Past loss carried forward: FV = L x (1 + i)^T, or L x (1 + i x T) if simple
- What drives the answer
- The rate, and more precisely the spread between the discount rate and the growth rate of the stream. The present value is convex in r, so the first point of rate moves the answer more than the fifth, and the effect grows with the horizon: at forty years a few points of rate is a multiple, not a percentage. Real versus nominal is a consistency condition: real flows take a real rate, nominal flows a nominal one.
- How it is attacked
- That the rate is unsupported, that it is inconsistent with the growth assumption (a nominal growth rate against a real discount rate, or the reverse), or that risk has been counted twice, once in the cash flows and once in the rate. The demonstration at /the-rate-decides/ shows the sensitivity on fixed streams.
38 models in the Index run on it
No single foundational paper is cited for this structure; it is textbook material, and the Institute lists only sources it has verified.
The counterfactual: potential outcomes and the but-for world
Almost every damages measure is a difference between two states of the world, one observed and one not. The formal version is the potential outcomes framework: each unit has an outcome under the conduct and an outcome without it, only one of which can ever be seen. Damages is the gap between them, and every method for building the counterfactual is a way of estimating the outcome that was not observed.
General formD = Y(0) - Y(1), where Y(1) is the observed outcome under the conduct and Y(0) the outcome that would have obtained without it
Y(0) is never observed and must be estimated: from the plaintiff's own past (before-and-after), from a comparator (yardstick), from a model (regression, difference-in-differences), or from a market equilibrium (structural simulation)
- What drives the answer
- The identifying assumption: the statement about what would have happened that licenses the estimate of Y(0). Before-and-after assumes nothing else changed; yardstick assumes the comparator would have moved with the plaintiff; difference-in-differences assumes parallel trends. The assumption is not tested by the data in the damages period, because that is exactly where the counterfactual is unobservable.
- How it is attacked
- That the identifying assumption fails on the facts: the before period was contaminated, the comparator is not comparable, the trends were not parallel, or the projection was fitted to the answer. Also that the counterfactual attributes to the conduct what other causes produced, which is the disaggregation problem.
42 models in the Index run on it
Foundational literature
- Rubin, Donald B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66(5), 688-701. https://doi.org/10.1037/h0037350 The potential outcomes framework that makes the but-for world a formal object.
- Holland, Paul W. (1986). Statistics and Causal Inference. Journal of the American Statistical Association, 81(396), 945-960. https://doi.org/10.1080/01621459.1986.10478354 States the fundamental problem: only one potential outcome is ever observed.
Regression and econometric control
Regression estimates how an outcome moves with its drivers, holding the others fixed. In damages it does two jobs: it builds the counterfactual, by fitting the plaintiff's history to its drivers and projecting, and it isolates an effect, by putting the conduct in as one variable among many and reading its coefficient. Difference-in-differences, synthetic control and hedonic pricing are all regression with a particular design.
General formy(t) = b0 + b1 x1(t) + ... + bk xk(t) + e(t), fitted by least squares
Effect of the conduct: the coefficient on an indicator for the conduct period or the treated group
Difference-in-differences: D = [Y(treated, after) - Y(treated, before)] - [Y(control, after) - Y(control, before)]
Forecast: y-hat(t) for t in the damages window, with a band that widens with distance from the sample
- What drives the answer
- Specification: which drivers are in the model and which are left out. An omitted driver that moved with the conduct loads its effect onto the conduct's coefficient. In a pay-equity model, the choice of controls is the entire case; in an overcharge model, the cost variables decide how much of the price rise is innocent.
- How it is attacked
- Omitted variables, a control that is itself a product of the conduct, spurious fits to trending series, serial correlation that understates the standard errors, and a treated indicator that captures a period effect rather than the conduct. With staggered treatment across many units, the two-way fixed effects estimator can weight comparisons perversely.
18 models in the Index run on it
Foundational literature
- Bertrand, Marianne, Duflo, Esther & Mullainathan, Sendhil (2004). How Much Should We Trust Differences-in-Differences Estimates? The Quarterly Journal of Economics, 119(1), 249-275. https://doi.org/10.1162/003355304772839588 Serial correlation and the overstatement of significance in difference-in-differences.
- Goodman-Bacon, Andrew (2021). Difference-in-differences with variation in treatment timing. Journal of Econometrics, 225(2), 254-277. https://doi.org/10.1016/j.jeconom.2021.03.014 The decomposition showing how staggered timing can distort the two-way fixed effects estimate.
- Abadie, Alberto, Diamond, Alexis & Hainmueller, Jens (2010). Synthetic Control Methods for Comparative Case Studies. Journal of the American Statistical Association, 105(490), 493-505. https://doi.org/10.1198/jasa.2009.ap08746 The weighted-comparator method that formalises the yardstick.
- Heckman, James J. (1979). Sample Selection Bias as a Specification Error. Econometrica, 47(1), 153-161. https://doi.org/10.2307/1912352 Why a sample that selected itself biases the coefficients, and the correction.
The market model and the event study
In a liquid market the price is the measuring instrument. The event study fits how a security normally moves with the market, then asks whether its move on a particular day was larger than that relationship predicts. The excess is the abnormal return, and it is the building block of securities damages, loss causation, and price artificiality.
General formMarket model: R(i,t) = alpha + beta x R(market,t) + e(t), fitted over an estimation window before the event
Abnormal return: AR(event) = R(i,event) - [alpha + beta x R(market,event)]
Significance: t = AR / standard deviation of the residuals
Cumulative abnormal return: CAR = sum of AR over the event window
- What drives the answer
- Whether the event day was clean. The residual is everything the market did not explain, so any other firm-specific news that day is inside it. The estimation window, the choice of market or industry index, and the length of the event window each move the residual, and the significance test decides whether the residual is distinguishable from noise at all.
- How it is attacked
- Confounding news on the event day, an estimation window contaminated by the fraud itself, a thin market in which prices do not impound information promptly, and a disclosure that revealed several things at once so the residual cannot be assigned to one of them.
6 models in the Index run on it
Foundational literature
- Fama, Eugene F., Fisher, Lawrence, Jensen, Michael C. & Roll, Richard (1969). The Adjustment of Stock Prices to New Information. International Economic Review, 10(1), 1-21. https://doi.org/10.2307/2525569 The original event study.
- Brown, Stephen J. & Warner, Jerold B. (1985). Using daily stock returns: The case of event studies. Journal of Financial Economics, 14(1), 3-31. https://doi.org/10.1016/0304-405X(85)90042-X The properties of the method on daily data, which is how it is used in litigation.
Survival analysis, hazards and retention curves
Many damages streams do not end on a known date. A customer churns, an employee finds new work, a franchise is not renewed, a patient dies. Survival analysis models the probability that something is still going at each future time, and the hazard is the chance it stops in the next period given that it has lasted this long. A damages stream is then weighted by survival, period by period, rather than cut off at an assumed date.
General formS(t) = probability the stream is still running at time t
Hazard h(t) = probability it ends in period t, given it survived to t
Expected value of a stream: E[V] = sum over t of S(t) x CF(t) x (1 + r)^-t
With a constant retention rate p and a stream beginning next period: E[V] = CF x p / (1 + r - p)
- What drives the answer
- The shape of S(t), and above all how close the retention rate is to one. In the closed form, the denominator is 1 + r - p, so a retention rate that rises from ninety to ninety-five percent can nearly double the value. The horizon is not a cut-off but a declining weight, which is why a reduced life expectancy or a high churn rate changes the answer gradually rather than at a point.
- How it is attacked
- That the survival curve was fitted to a period that is not representative, that the conduct-period churn is compared with the wrong baseline, that the curve is extrapolated beyond the data, and that the horizon assumed (perpetual renewal, lifetime retention) sets S(t) to one where the evidence does not support it.
11 models in the Index run on it
Foundational literature
- Kaplan, E. L. & Meier, Paul (1958). Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53(282), 457-481. https://doi.org/10.1080/01621459.1958.10501452 The estimator of the survival curve from data in which some observations are still running.
- Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B, 34(2), 187-202. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x The proportional hazards model: how covariates shift the hazard.
Markov chains and labour force transitions
A Markov chain describes a system that moves between states with probabilities that depend only on the state it is in. In forensic economics the states are active and inactive in the labour force, and alive or dead, and the transition probabilities by age come from longitudinal survey data. Worklife expectancy is the expected number of future years spent in the active state, and it is the horizon of every lost earnings claim.
General formStates: active, inactive, dead. Transition probabilities by age: p(active to inactive), p(inactive to active), p(to dead)
P(active at age a + k, given the state at age a) follows by multiplying the transition matrices
Worklife expectancy: WLE = sum over future ages of P(active at that age, given the current state)
- What drives the answer
- The transition probabilities for the person's cohort and, more subtly, the starting state: an initially active person and an initially inactive person of the same age have different expectancies, and the increment-decrement model's allowance for re-entry is what separates it from the older static life table. The output is an expectation with a wide distribution around it.
- How it is attacked
- Departures from the published tables asserted rather than argued, a cohort mismatched to the person (education, sex, attachment history), and using the expectation as if it were a certainty, which discards the variance that a jury is entitled to know about.
2 models in the Index run on it
Foundational literature
- Skoog, Gary R. & Ciecka, James E. (2006). Worklife Expectancy via Competing Risks/Multiple Decrement Theory with an Application to Railroad Workers. Journal of Forensic Economics, 19(3), 243-260. https://doi.org/10.5085/0898-5510-19.3.243 Applies competing-risks theory to worklife estimation.
- Ciecka, James E. & Skoog, Gary R. (2017). Expected Labor Force Activity and Retirement Behavior by Age, Gender, and Labor Force History. Statistics and Public Policy, 4(1), 1-8. https://doi.org/10.1080/2330443x.2017.1358125 Extends worklife to second-order models using labour force history.
Expected value, probability weighting and decision trees
When an outcome was uncertain, its value is the probability of it times what it would have been worth. Loss of chance, case-within-a-case, unvested equity, a project not yet through its permits: each is a value scaled by a probability, and several of them are nested, a probability applied to a value that is itself a damages model. The decision tree is the picture of the nesting, and the value of information is the same tree read backward: how much would it be worth to know a branch's outcome before choosing.
General formE[V] = sum over outcomes of P(outcome) x V(outcome)
Single branch: Award = P(the chance) x V(the outcome)
Nested: L = P(win) x E[award, given a win] x P(collectible) - actual recovery
Value of resolving an uncertainty = E[V with the information] - E[V without it]
- What drives the answer
- The probability, because it multiplies everything, and where it comes from: a population rate applied to an individual is the usual source and the usual argument. In nested structures the probabilities multiply, so three plausible seventy percent chances together are about a third.
- How it is attacked
- That the probability is the wrong population's rate for this plaintiff, that it was assigned rather than estimated, that the tree omits a branch (the deal that would have failed anyway), and that the expected value is presented as if it were the outcome, when the law in many settings compensates the outcome and not the chance.
11 models in the Index run on it
Foundational literature
- Howard, Ronald A. (1966). Information Value Theory. IEEE Transactions on Systems Science and Cybernetics, 2(1), 22-26. https://doi.org/10.1109/TSSC.1966.300074 The value of information as the difference in expected value with and without it.
Optimisation, equilibrium and bargaining
A but-for price is not a guess; it is the price a market would have settled at. Firms are modelled as choosing prices or quantities to maximise profit given what rivals do, and the equilibrium is the set of choices from which no one would deviate. Cournot wrote the first version in 1838. Reasonable royalties add a second idea: a hypothetical negotiation, modelled as a bargaining problem in which each side's outside option sets the range and a rule sets the split.
General formEach firm chooses price or quantity to maximise profit given rivals' choices; the equilibrium is where every first-order condition holds at once
Monopoly mark-up at the optimum: (P - MC) / P = 1 / (elasticity of demand); the monopsony mark-down mirrors it with the supply elasticity
Nash bargaining: the split that maximises the product of each side's gain over its outside option
Lost volume: the seller recovers the margin only if its profit-maximising output exceeded the units actually sold
- What drives the answer
- Elasticity. The demand elasticity sets the monopoly mark-up and the price erosion a competitor caused; the supply elasticity sets the monopsony underpayment; the cross-elasticities set how many of an infringer's sales the patentee would have captured. In bargaining, the outside options set the range and the rule sets the split, and the non-infringing alternative is the outside option that decides most royalty disputes.
- How it is attacked
- That the model's assumptions about conduct (Bertrand versus Cournot, static versus dynamic) drive the answer more than the data does, that elasticities were assumed rather than estimated, that the counterfactual holds one side's quantity fixed while moving the price (the price erosion inconsistency), and that the bargaining range ignores a real alternative the infringer had.
13 models in the Index run on it
Foundational literature
- Nash, John F. (1950). The Bargaining Problem. Econometrica, 18(2), 155-162. https://doi.org/10.2307/1907266 The axiomatic solution to a two-party bargain, the model behind the hypothetical negotiation.
- Nash, John (1951). Non-Cooperative Games. The Annals of Mathematics, 54(2), 286-295. https://doi.org/10.2307/1969529 The equilibrium concept every but-for market simulation solves for.
- Berry, Steven, Levinsohn, James & Pakes, Ariel (1995). Automobile Prices in Market Equilibrium. Econometrica, 63(4), 841-890. https://doi.org/10.2307/2171802 Demand estimation closed with a supply-side equilibrium, the structure behind market simulation.
Sampling and statistical inference
When there are too many claims, employees, transactions or consumers to examine each one, a sample stands in for the population and the result is extrapolated with a stated margin of error. The mathematics is elementary; what is not elementary is the design that makes the extrapolation legitimate: a defined population, a probability draw, adequate size, and stratification where the population is heterogeneous. Surveys are the same structure applied to what people believe.
General formEstimate = sample mean x N
Standard error = (s / square root of n) x N; the interval is the estimate plus or minus a multiple of it
Proportion: p with standard error square root of p(1 - p)/n
Net of a control: p(test) - p(control)
- What drives the answer
- The design, not the arithmetic. A probability sample from a defined frame licenses the inference; a convenience sample does not, whatever its size. Stratification reduces variance where the strata differ; the size sets the width of the interval; and the control condition, in a survey, is what separates the effect of the stimulus from what respondents would have said anyway.
- How it is attacked
- A frame that does not match the population extrapolated to, a non-random draw, a size too small for the claimed precision, a point estimate presented without its interval, and in surveys the whole apparatus of leading questions, order effects, missing controls and unblinded coding.
14 models in the Index run on it
Foundational literature
- Neyman, Jerzy (1934). On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection. Journal of the Royal Statistical Society, 97(4), 558-625. https://doi.org/10.2307/2342192 The foundation of probability sampling and the case against purposive selection.
- National Research Council (2011). Reference Manual on Scientific Evidence, Third Edition. The National Academies Press. https://doi.org/10.17226/13163 The reference guides on statistics, multiple regression and survey research that federal courts use.
Discrete choice and demand estimation
How much was a feature, a claim, or a brand worth to the people who bought it? Discrete choice models answer by observing choices, in the market or in a designed experiment, and estimating the utility each attribute contributes. Willingness to pay for an attribute is its utility measured in the currency of the price coefficient. Conjoint analysis is the experimental version and the workhorse of apportionment and price premium claims.
General formUtility of option j for person i: U(i,j) = sum of b(attribute) x attribute(j) + b(price) x price(j) + e
Choice probability (logit): P(i chooses j) = exp(U(i,j)) / sum over k of exp(U(i,k))
Willingness to pay for a feature: WTP = b(feature) / -b(price)
Market share: s(j) = sum over i of P(i chooses j)
- What drives the answer
- The price coefficient, because every willingness to pay is divided by it, and the design, because attributes can only be separated if they were varied independently. Then the supply side: willingness to pay is what consumers would give, not what they would have paid, and the market price is an equilibrium in which competition absorbs part of the difference.
- How it is attacked
- Attribute levels outside the market's range, a design that confounds attributes, respondents who are not the buyers, the leap from willingness to pay to market price without a supply side, and aggregating heterogeneous consumers into a single premium.
6 models in the Index run on it
Foundational literature
- McFadden, Daniel (1974). The measurement of urban travel demand. Journal of Public Economics, 3(4), 303-328. https://doi.org/10.1016/0047-2727(74)90003-6 The conditional logit model of choice among discrete alternatives.
- Green, Paul E. & Srinivasan, V. (1978). Conjoint Analysis in Consumer Research: Issues and Outlook. Journal of Consumer Research, 5(2), 103-123. https://doi.org/10.1086/208721 The survey of conjoint methods that fixed the field's vocabulary.
- Allenby, Greg M., Brazell, Jeff D., Howell, John R. & Rossi, Peter E. (2014). Economic valuation of product features. Quantitative Marketing and Economics, 12(4), 421-456. https://doi.org/10.1007/s11129-014-9150-x Why willingness to pay is not market value without a supply side.
Valuation identities: DCF, multiples and capitalisation
A business, a property or an investment is worth the present value of what it will produce. The discounted cash flow model states that directly; a capitalisation rate collapses it to a single year divided by a rate; a multiple borrows the market's rate from comparable transactions. All three are the same identity at different levels of detail, and the terminal value, which capitalises everything beyond the forecast, usually carries most of the answer.
General formV = sum over the forecast of FCF(t) x (1 + WACC)^-t + TV x (1 + WACC)^-n
Terminal value: TV = FCF(n + 1) / (WACC - g)
Capitalisation: V = income / cap rate
Multiple: V = metric x M, with M from comparable companies or transactions
Cost of equity under the CAPM: r = risk-free rate + beta x equity premium
- What drives the answer
- The spread between the discount rate and the terminal growth rate, because it is the denominator of the term that dominates the sum, and the choice of comparables, because a multiple is only as good as the set it was taken from. A valuation date matters as much as a rate: the same company on two dates a year apart is two different answers, and which date the law fixes is often the first argument.
- How it is attacked
- An unsupported discount rate, comparables that are not comparable, a terminal growth rate that outruns the economy, risk counted in both the cash flows and the rate, and double counting between a value differential and a lost profits claim over the same stream.
24 models in the Index run on it
Foundational literature
- Gordon, Myron J. & Shapiro, Eli (1956). Capital Equipment Analysis: The Required Rate of Profit. Management Science, 3(1), 102-110. https://doi.org/10.1287/mnsc.3.1.102 The growing perpetuity formula behind the terminal value.
- Sharpe, William F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3), 425-442. https://doi.org/10.1111/j.1540-6261.1964.tb02865.x The capital asset pricing model, the usual source of the cost of equity.
Contingent claims and option pricing
Some things are worth something only if a condition is met: an option is worth exercising only above the strike, an earnout pays only above a threshold, an unvested award pays only if the employee stays. Contingent claims are valued by modelling the distribution of the underlying and pricing the payoff over it. The models are exact in their assumptions and the assumptions are where the dispute lives, above all the volatility, which cannot be observed.
General formValue = f(underlying price, strike, volatility, time to expiry, risk-free rate, dividends)
Black-Scholes-Merton gives f in closed form for a European option; a binomial lattice steps the underlying up or down each period and works back from the payoff
Kinked payoff (an earnout): E = f(metric), zero below the threshold and rising above it
Unvested award: value x P(still employed at vesting) x P(condition met), discounted
- What drives the answer
- Volatility, because it is the one input no record supplies, and the kink, because a payoff that is zero below a threshold has a sensitivity that is discontinuous: a small change in the counterfactual metric can move the payment from nothing to everything.
- How it is attacked
- The volatility assumption, the use of a closed-form model on an instrument whose features it does not fit (early exercise, path dependence), and a counterfactual metric placed just above a threshold by an assumption rather than by evidence.
3 models in the Index run on it
Foundational literature
- Black, Fischer & Scholes, Myron (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654. https://doi.org/10.1086/260062 The closed-form option pricing model.
- Merton, Robert C. (1973). Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4(1), 141-183. https://doi.org/10.2307/3003143 The general theory, and the treatment of dividends and early exercise.
- Cox, John C., Ross, Stephen A. & Rubinstein, Mark (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229-263. https://doi.org/10.1016/0304-405X(79)90015-1 The binomial lattice, the workhorse for instruments the closed form does not fit.
Apportionment and allocation
A gain, a cost or a value often has several causes and the law compensates one of them. Apportionment is the fraction assigned to the conduct; allocation is the same idea across several parties or several products. Whatever the method, the output is a set of weights that sum to one, and because the weight multiplies the whole answer, a dispute over apportionment is a dispute over the entire award in disguise.
General formAttributable amount = total x a, with a between zero and one
Across parties: share(i) = w(i) / sum over j of w(j)
Pro rata by billings (Eichleay): contract share = contract billings / total billings
Orphan shares reallocated: the remaining weights renormalised to sum to one
- What drives the answer
- How a is derived. Conjoint gives a demand-side share of value; incremental profit over the next best alternative gives an economic share; comparable licences give a market share; pro rata gives an accounting share. These are different numbers for the same question, and the method chosen is the answer chosen.
- How it is attacked
- No apportionment at all (the whole gain attributed to the conduct), a method that assigns the feature's value by counting features rather than by measuring value, weights asserted rather than estimated, and an allocation that ignores contributions the defendant or the other parties made.
16 models in the Index run on it
Foundational literature
- Shapley, L. S. (1953). A Value for n-Person Games. In Contributions to the Theory of Games (AM-28), Volume II, 307-318. Princeton University Press. https://doi.org/10.1515/9781400881970-018 The axiomatic attribution of a joint outcome to its contributors.
Cost behaviour and accounting identities
Revenue is not profit, and the difference is the part of the field most often got wrong. Costs that vary with output are avoided when the output is not produced and must be deducted from lost revenue; costs that do not vary continue regardless and must not be. Overhead absorption, incremental cost, extra expense, and the cost side of every gain-based measure are all statements about which costs move with what.
General formIncremental profit on lost sales = lost revenue - variable cost of those sales
Variable cost is a slope, not a share: dC / dQ, estimated from the cost accounts or by regressing cost on output
Extra expense = costs incurred to continue operating - costs that would have been incurred anyway
Unabsorbed overhead: fixed cost that continued while the activity that carried it stopped
- What drives the answer
- The fixed versus variable classification, over the relevant range and horizon. A cost that is fixed for a month is variable over a year; a cost that is variable per unit may be step-fixed per plant. The classification decides the margin, and the margin multiplies the lost volume.
- How it is attacked
- A margin that deducts too little (fixed costs treated as if they would have been avoided, or the reverse on the defendant's side), overhead allocated by a convenient rule, costs classified by their accounting label rather than their behaviour, and a total cost figure that includes the claimant's own inefficiency.
28 models in the Index run on it
No single foundational paper is cited for this structure; it is textbook material, and the Institute lists only sources it has verified.
Monte Carlo simulation and sensitivity
When several inputs are uncertain and the output is a non-linear function of them, no single-point calculation shows what the answer could be. Monte Carlo draws each input from a distribution, computes the output, and repeats until the distribution of outputs is known. In damages it appears in solvency analysis, structured instrument valuation, and any expert report honest enough to show a range rather than a point.
General formFor each of many trials: draw each input from its distribution, compute the output; the collection of outputs is its distribution
Sensitivity: change one input across its range with the others held at base, and record the swing in the output
Probability of a threshold: the fraction of trials in which the output crosses it
- What drives the answer
- The input distributions and their correlations. Independent draws understate the spread when the inputs move together (revenue and margin in a downturn), and the tails of the output depend on the tails assumed for the inputs, which the data rarely pins down.
- How it is attacked
- Distributions assumed without support, correlations ignored, a simulation that dresses a point estimate in a spurious interval, and a range so wide that it says nothing, which is sometimes the truthful answer and sometimes a way of avoiding one.
2 models in the Index run on it
Foundational literature
- Metropolis, Nicholas & Ulam, S. (1949). The Monte Carlo Method. Journal of the American Statistical Association, 44(247), 335-341. https://doi.org/10.1080/01621459.1949.10483310 The original statement of the method.
Index numbers, escalation and real versus nominal
Prices at two dates are not comparable until one is restated in the other's dollars. An index number is the ratio that does the restating: a construction cost index escalates a budget from the planned date to the actual one, a consumer price index converts nominal earnings to real, and the same ratio in reverse deflates a projection. Which index, at what geography and granularity, is the whole question.
General formEscalation = C x [I(actual date) / I(planned date) - 1]
Real amount = nominal amount / (I(t) / I(base))
Composite index: a weighted average of component indices, weighted by the cost shares of the components
- What drives the answer
- The match between the index and the thing being escalated. A national composite applied to a regional, component-specific cost can miss the actual movement by a wide margin, and the base date fixes what counts as escalation at all: a contract not yet let has no locked price to escalate from.
- How it is attacked
- The wrong index (national for regional, composite for component), a base date chosen to maximise the movement, and inconsistency between the index used to escalate and the rate used to discount, which double counts inflation.
1 model in the Index run on it
Foundational literature
- Diewert, W. E. (1976). Exact and superlative index numbers. Journal of Econometrics, 4(2), 115-145. https://doi.org/10.1016/0304-4076(76)90009-9 The theory of which index formulas measure a price level exactly.
A note from the Executive Director
I studied mathematical methods in the social sciences and economics at Northwestern, then quantitative finance at the University of Chicago, and then spent thirty-two years placing the economists who use these structures for a living without ever becoming one of them. This page is the part of the field I have always found most beautiful: that underneath every damages number a court has ever seen there is the same short list of ideas, recombined. If you can see which structure a claim runs on, you can usually see where it will be attacked before the other side does, and that is the reason the Institute publishes this rather than keeping it in the back room.
The same formal tradition, applied to research decisions rather than to disputes, is the method behind Vista Research. Vista Research →