Motivation

Lee-Carter mortality projection is a standard demographic approach for age-specific death rates. The original paper by Ronald Lee and Lawrence Carter represents mortality's logarithm by age effects and a common period index. The Social Security Administration working paper explains this decomposition and the identifiability of its components. Existing actuarial literature establishes the parametric modelling framework; these are original formal finite-period Lean identities, not claims that the statistical model has been estimated from historical data.

The published reference identifies the actuarial basis for this work. Finite source-specific formalisation in Lean remains open, while the motivating mathematical identities and technical conventions have been established in the literature. No compiler check or numerical example is an accepted proof.

Setting

At attained age x and calendar year t, the central death rate is m(x,t)=exp(a_x+b_xk_t), with a_x a real age intercept, b_x real age sensitivity and k_t a real period-mortality index. A deterministic index drift evolves k_(t+n)=k_t+ndelta. The ratio of death rates at the same age across two periods equals exp(b_x*(k_2-k_1)). With a nonnegative constant annual mortality force m, one-year survival is exp(-m) and the one-year death probability equals one minus survival. A cohort receives a possibly age- and calendar-dependent sequence of valid annual death probabilities q(t).

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Age-period Lee-Carter log mortality and drift

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.lcRate_zero_period`:

$$m_{x,t}=e^{a_x}\text{ if }\kappa_t=0$$

Zero index gives the exponential age baseline.

Mortality rate ratios and force-to-probability conversion

This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.lcDeathProb_partition`:

$$q+p=1$$

One-year death and survival are complementary under a constant-force basis.

Projected cohort survival and one-period insured loss

This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.lcLeeCarterCohort_fundamental`:

$$m_{x,2}=m_{x,1}R,\;m_{x,2}>0,\;0\le S_{n+1}\le S_n,\;E[C]\ge0$$

The comprehensive mortality capstone links Lee-Carter rate ratios with a valid bounded cohort survival law and the nonnegative one-year insured death outgo.

Capstone theorem

The main target is `ActuarialValuation.lcLeeCarterCohort_fundamental`. The comprehensive mortality capstone links Lee-Carter rate ratios with a valid bounded cohort survival law and the nonnegative one-year insured death outgo.

$$m_{x,2}=m_{x,1}R,\;m_{x,2}>0,\;0\le S_{n+1}\le S_n,\;E[C]\ge0$$

Significance

The original Lee-Carter work provides a tractable time trend at each age. The proposed mathematical capstone relates positive death-rate projections and exact age-specific rate ratios to bounded multi-year cohort survival and nonnegative expected death benefit cost. These are reusable model-validation invariants for actuarial mortality improvement projections. Unlike a fitted SVD or time-series forecast, a deterministic rate ratio and finite survival-product identity can be verified without probabilistic estimation assumptions.

Difficulty

A central death rate is not automatically an annual death probability. Converting an annual nonnegative force of mortality to a death probability is a modelling assumption that holds with constant force during the year. The one-year mortality intensities for ages x+n along a cohort differ from fixed-attained-age period ratios. The survival product must use each cohort-year probability exactly once, with half-open endpoints. Negative annual force values would yield death probabilities outside the unit interval and are ruled out where probabilities are asserted. The Lee-Carter age and period parameters have a scale identifiability ambiguity unless normalisation constraints are imposed; the current mission asserts algebraic properties at fixed parameters, not unique statistical identification.

An age-period mortality rate for the same age in two calendar years is not the same as the sequence of annual death probabilities experienced by a cohort ageing through consecutive calendar years. The first is represented as a ratio of positive central mortality intensities; the second is a product of annual survival complements. The conversion from a rate to an annual probability assumes a constant mortality force within the relevant year. That is distinct from mere exponential positivity and must be explained in review. A parameter-normalisation condition would be necessary for identification from experience data, but fixed-parameter algebraic rate-ratio statements do not require identifying the historical fit.

Formalization scope

The nontrivial root combines exact exponential rate-ratio reconstruction with independent multi-year cohort survival bounds and expected assurance claims, rather than using degenerate zero mortality or zero benefits. It does not prove statistical estimation, covariance forecasts, stochastic longevity risk, or maximum likelihood fit. The formalisation uses Lean real arithmetic, exp, finite products and model-specific definitions, preserving positivity assumptions. Future missions can reuse the established identities for age-calendar time series and annuity valuation of projected cohorts.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

• Ronald D. Lee and Lawrence R. Carter, Modeling and Forecasting U.S. Mortality (1992), Journal of the American Statistical Association 87, pp. 659–671. https://doi.org/10.1080/01621459.1992.10475265
• U.S. Social Security Administration, The Out-of-Sample Performance of Stochastic Methods in Forecasting Age-Specific Mortality Rates (2008), Working Paper No. 111. https://www.ssa.gov/policy/docs/workingpapers/wp111.html

Motivation

Actuarial insurance ratemaking links expected claims to a measurable risk exposure base. The Actuarial Standards Board's adopted ASOP No. 53 discusses prospective risk-transfer cost estimation, exposure data and risk classification. Pure premium is cost per earned exposure and class relativity describes relative risk classification in a common basis. The already established actuarial pricing framework is conceptual; the new Lean statements are independent original finite proofs.

The published reference identifies the actuarial basis for this work. Finite source-specific formalisation in Lean remains open, while the motivating mathematical identities and technical conventions have been established in the literature. No compiler check or numerical example is an accepted proof.

Setting

A finite set of insured risk classes has real earned exposures e(i), loss costs L(i), and selected dimensionless relativity r(i). A common base pure rate b charges technical class premium e(i)br(i), summed over n disjoint classes. The relativity-weighted exposure W equals the sum of e(i)r(i). A projected aggregate claim cost T is financed by the uniquely calibrated b=T/W, provided W is strictly positive. Historical observed pure premium L/E is distinct from the future indicated base price.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Exposure, observed claims and indicated pure premiums

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.ratClassIndicated_zero`:

$$f=0\Rightarrow c=0$$

No expected frequency leads to zero prospective claim cost.

Class relativities and aggregate technical premium

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.ratPortfolioPremium_factor`:

$$P=bW$$

The class-by-class technical premium sum factors as base rate times weighted exposure.

Base rate calibration and technical premium adequacy

This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.ratExposureRatemaking_fundamental`:

$$P(b^*)=T,\;b^*\ge0,\;S(b^*)=0,\;b\ge b^*\Rightarrow S(b)\ge0$$

The substantive pricing capstone establishes unique expected-cost calibration and technical premium adequacy across arbitrary finite risk-class exposures and relativities.

Capstone theorem

The main target is `ActuarialValuation.ratExposureRatemaking_fundamental`. The substantive pricing capstone establishes unique expected-cost calibration and technical premium adequacy across arbitrary finite risk-class exposures and relativities.

$$P(b^*)=T,\;b^*\ge0,\;S(b^*)=0,\;b\ge b^*\Rightarrow S(b)\ge0$$

Significance

The ASOP No. 53 actuarial source distinguishes the estimate of future insured losses from an actual insurance selling price. This formal capstone proves a finite technical income reconciliation and a nonnegative adequacy margin at every higher base rate. These equalities form reusable verification invariants for actuarial pricing spreadsheets and downstream portfolio studies. It does not assert legally required rates, premium tax calculations, competitive prices or expense allocation.

Difficulty

Class exposures must be measured on consistent earned-exposure units. Multiplying a base rate by class relativity differs from taking an unweighted arithmetic average of class premiums. If weighted exposure vanishes, a positive target cannot be financed by the formula T/W. Positivity of weighted exposure is needed for uniqueness and increasing-rate adequacy; aggregate exposure alone may be positive even when all relativity weights are zero. A risk class's exposure may be zero without invalidating the rest of the portfolio. A zero-cost portfolio is a legitimate boundary, but an all-zero exposure scenario cannot establish the general theorem.

A pure indicated base premium in this finite model covers only the agreed future claim-cost total. It is not a retail premium including expense, profit, risk or market pricing provisions. The same class relati­vity and earned exposure basis must be applied to the projected loss cost and to the pricing denominator. An independent actuarial reviewer can identify a rate-mix error through the exact aggregate reconstruction identity. Positive aggregate unweighted exposure alone does not ensure a viable technical rate when all class multipliers vanish. The central root is therefore a financial reconciliation theorem with an explicit strictly positive relativity-weighted exposure base.

Formalization scope

The formal verification uses finite indexed class data and real-valued expected costs, independent of claim arrival distributions or stochastic credibility assumptions. The results prove exact identities over a fixed prospective insurance coverage basis. A loading for acquisition costs, handling expenses, reinsurance and capital is not automatically contained in the technical pure premium. The nontrivial root quantifies over every alternative higher rate and proves genuine fiscal sufficiency, rather than only simplifying an illustrative numeric case.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

• Actuarial Standards Board, ASOP No. 53: Estimating Future Costs for Prospective Property/Casualty Risk Transfer and Risk Retention (2017), Section 3, https://www.actuarialstandardsboard.org/asops/estimating-future-costs-prospective-propertycasualty-risk-transfer-risk-retention/
• Casualty Actuarial Society, Basic Ratemaking presentation (2012), https://www.airc.org.tw/newsfiles/BasicRatemakingI.pdf

Motivation

Insurers price retention layers and assess high-loss capital needs using distributions of losses rather than only expected claim frequency. The Rockafellar and Uryasev paper introduces an auxiliary loss-threshold objective for conditional value-at-risk (CVaR). Its central mathematical construction uses expected positive-part loss in excess of a threshold. The existing core relationship is established in published research; this mission develops an original exact finite-scenario Lean formalisation and its necessary boundary lemmas.

The published reference identifies the actuarial basis for this work. Finite source-specific formalisation in Lean remains open, while the motivating mathematical identities and technical conventions have been established in the literature. No compiler check or numerical example is an accepted proof.

Setting

A finite lattice has loss outcomes 0 through n with nonnegative scenario weights p(k); it represents a genuine probability distribution only when their total mass equals one. The scenario-weighted excess above deductible t is SL_n(t)=sum p(k) max(k-t,0), which is an expected excess under unit total probability mass. For a conventional confidence alpha in the interval from zero to one, excluding one, the tail fraction 1-alpha is positive and the Rockafellar-Uryasev objective equals F_alpha(t)=t+SL_n(t)/(1-alpha). The lower cumulative weight includes outcomes less than or equal to an integer threshold, while upper mass includes only strictly larger losses. Both parts partition the original finite weight. The Lean identities hold more generally for nonnegative weights without asserting that unnormalised weights define a probability distribution or that a fixed threshold minimises the CVaR objective.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Finite scenario probabilities, losses and positive-part payouts

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.tailExcess_antitone_threshold`:

$$t\le u\Rightarrow(x-u)_+\le(x-t)_+$$

Increasing retention cannot increase payment for any insured loss.

Finite stop-loss and Rockafellar-Uryasev threshold objective

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.tailScaledExcess_nonneg`:

$$SL\ge0,\tau>0\Rightarrow K\ge0$$

Nonnegative expected excess and positive tail mass make a valid nonnegative scaling.

Discrete cumulative tails, thresholds and monetary shifts

This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.tailTailRisk_fundamental`:

$$SL(t)\ge0,\;F_\alpha(t)\ge t,\;F_\alpha^{X+s}(t+s)=F_\alpha^X(t)+s$$

The root joins tail-stop-loss nonnegativity, a valid finite CVaR objective, monetary translation equivalence and the complete lower/upper finite loss partition.

Capstone theorem

The main target is `ActuarialValuation.tailTailRisk_fundamental`. The root joins tail-stop-loss nonnegativity, a valid finite CVaR objective, monetary translation equivalence and the complete lower/upper finite loss partition.

$$SL(t)\ge0,\;F_\alpha(t)\ge t,\;F_\alpha^{X+s}(t+s)=F_\alpha^X(t)+s$$

Significance

A scenario version of the original threshold-objective formula is suitable for rigorous deterministic actuarial software tests. Exact stop-loss monotonicity and positive-part partition results are reusable in reinsurance, capital modelling and future optimisation research. The root proves meaningful financial invariants: nonnegative ceded loss, a lower bound on the objective, and translation equivariance. It does not claim the objective at an arbitrary threshold equals optimised CVaR: that requires minimising over all thresholds.

Difficulty

The confidence parameter alpha must remain below one, otherwise dividing by 1-alpha is meaningless and positivity assertions fail. An insurance loss scenario at the deductible does not generate ceded payout. Discrete distributions may have atoms at a quantile: the conditional tail mean needs partial quantile mass, not a careless conditional expectation over all values greater than or equal to VaR. The auxiliary objective is defined for all real thresholds, whereas finite lower/upper tails at integer thresholds have disjoint endpoints. Zero-weight degenerate examples cannot establish the universal finite-scenario results.

The Rockafellar-Uryasev expression represents an auxiliary objective at each fixed threshold, with the conditional value-at-risk corresponding to its minimum across thresholds. This proposal does not assume that any arbitrary deductible minimises the objective. It distinguishes a discrete lattice distribution from a continuous loss density and preserves the mass at the chosen deductible in the lower tail. Taking a retention greater than every possible loss yields zero ceded premium but does not prove a zero capital risk measure. Translation equivariance follows from shifting the distribution of losses and the monetary threshold together, so the excess layer itself remains unchanged.

Formalization scope

This is finite-scenario algebraic formal verification of a classic CVaR optimisation building block, not a continuous-distribution conditional expectation theorem or an assertion of existence/uniqueness of the minimising quantile. No distributional normalisation is inferred from truncated mass. The meaningful root relates insurance payouts, risk objective and monetary translation across arbitrary finite nonnegative-weight scenarios, preserving explicit denominator restrictions. This supports independent future research on convexity, optimum threshold and subadditive coherent measures.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

• R. Tyrrell Rockafellar and Stanislav Uryasev, Optimization of Conditional Value-at-Risk (1999 preprint; Journal of Risk 2000), Equation (4), p. 5. https://sites.math.washington.edu/~rtr/papers/rtr179-CVaR1.pdf
• Rockafellar and Uryasev, Optimization of Conditional Value-at-Risk (2000), Journal of Risk 2, pp. 21–42. https://econpapers.repec.org/RePEc:rsk:journ4:2161159

Motivation

Actuarial collective risk models represent total insured losses as a random sum of claim amounts. Panjer's 1981 recursive method provides a tractable recurrence for aggregate probabilities when claim number frequencies belong to a specified recursive family. This mission uses the compound Poisson special case and formalises the positive integer severity setting, its unique recursive aggregate law and finite premium and stop-loss valuations.

Finite actuarial valuation follows published literature, including the principal source. Definitions of finite event counts, valuation factors and cashflow conventions must be treated as precise mathematical inputs. Established actuarial formulae motivate the work; the resulting Lean statements are newly proposed, open proof obligations. The formal verification of these identities is distinct from the existing published mathematical development. No numerical tests or private declaration compilation constitute accepted proofs.

Setting

For intensity lambda at least zero, individual claims take strictly positive integer sizes with nonnegative severity weights f(j) and f(0)=0. The Poisson zero-claims probability is exp(-lambda). The aggregate loss lattice masses g(k) satisfy, for positive k, kg(k)=lambdasum from j=1 to k of j*f(j)*g(k-j). A finite probability prefix through n gives G(n), and 1-G(n) denotes its modelled tail complement. Finite expected aggregate loss and stop-loss premiums use sums over outcomes 0 through n.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Finite claim-size severity and convolution

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.panjerWeightedSeverity_nonneg`:

$$f_k\ge0\Rightarrow kf_k\ge0$$

Nonnegative severity weights yield nonnegative weighted claims.

Compound Poisson Panjer recurrence

This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.panjerRecurrence_nonneg`:

$$\lambda,f\ge0\Rightarrow g_k\ge0$$

The uniquely generated compound Poisson aggregate coefficients are nonnegative for nonnegative severity probabilities.

Finite aggregate loss reserve and stop-loss valuation

This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.panjerAggregateClaims_fundamental`:

$$g=h,\;g_k\ge0,\;G_n+T_n=1,\;SL_n(d)\ge0,\;\lambda\mu_n\ge0$$

The compound aggregate-claims capstone combines genuine uniqueness of forward Panjer probabilities, nonnegative aggregate coefficients, and finite claim-payment cost controls.

Capstone theorem

The main target is `ActuarialValuation.panjerAggregateClaims_fundamental`. The compound aggregate-claims capstone combines genuine uniqueness of forward Panjer probabilities, nonnegative aggregate coefficients, and finite claim-payment cost controls.

$$g=h,\;g_k\ge0,\;G_n+T_n=1,\;SL_n(d)\ge0,\;\lambda\mu_n\ge0$$

Significance

The core mathematical target proves that the forward recurrence with initial mass exp(-lambda) uniquely fixes all aggregate coefficients, and nonnegative intensity and severities imply all generated probabilities are nonnegative. It reconciles finite cumulative mass, a tail complement, expected aggregate costs and a ceded stop-loss value in one theorem. The recursion is independent of the earlier negative-binomial frequency moment mission and leaves full normalisation to separate infinite-series proofs.

The importance of the formal result is the availability of reusable, source-explicit lemmas for actuarial software audit and subsequent independent mathematical work, as described in the published literature. It goes beyond an elementary reformulation of a single definition: the capstone simultaneously connects different construction stages with monetary or probability reconciliation and precise boundary hypotheses. A vacuous zero-data specialisation cannot establish these relationships across arbitrary finite schedules.

Difficulty

The Panjer recurrence requires positive severity weights indexed by j at least one: allowing mass at zero without changing the denominator and initial atom would make its stated form incorrect. For each target k only lower aggregate coefficients occur, permitting ordinary induction. Natural index subtraction is safe within the finite sum. Intensity positivity, exp(-lambda), natural-to-real index coercions and finite sums need careful attention. This mission deliberately does not claim that a truncated mass automatically sums to one or that a sequence satisfying the recurrence is normalised without additional assumptions.

The recurrence's fundamental mathematical content is the coefficient-by-coefficient relationship: the coefficient at aggregate loss k depends only on previously calculated coefficients for aggregate losses strictly below k. This well-founded structure supports a strong induction proof of uniqueness and a separate positivity induction. The initial mass exp(-lambda) applies only when each individual claim has strictly positive loss. A severity law admitting zero-sized claims would require a corrected initial atom and a denominator involving zero-severity mass. A finite prefix of recursive probabilities is not, by itself, an exhaustive law. Further work can add total normalisation and full expectation formulas.

Formalization scope

The model is the lattice compound Poisson special case, not the full (a,b,0) family. It distinguishes algebraic consistency of a recursively specified sequence from measure-theoretic construction of an independent compound random sum. Stop-loss results are finite truncation values and do not by themselves control omitted large losses. All theorems are original derived proof obligations.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

• H. H. Panjer, Recursive Evaluation of a Family of Compound Distributions, ASTIN Bulletin 12 (1981), pp. 22–26. https://www.casact.org/sites/default/files/database/astin_vol12no1_22.pdf
• Sundt and Jewell, Further Results on Recursive Evaluation of Compound Distributions, ASTIN Bulletin 12 (1981), pp. 27–39. https://www.casact.org/sites/default/files/database/astin_vol12no1_27.pdf

Motivation

Insurance mortality investigations must estimate survival probabilities from observed death times when some policyholders leave the observation period alive or are otherwise censored. Kaplan-Meier product-limit estimation accounts for changing risk sets without misclassifying censoring as mortality. Greenwood's formula provides the standard plug-in variance. This mission formalises a finite death-event calculation, risk-set conservation, survival probability bounds, and variance bookkeeping.

Finite actuarial valuation follows published literature, including the principal source. Definitions of finite event counts, valuation factors and cashflow conventions must be treated as precise mathematical inputs. Established actuarial formulae motivate the work; the resulting Lean statements are newly proposed, open proof obligations. The formal verification of these identities is distinct from the existing published mathematical development. No numerical tests or private declaration compilation constitute accepted proofs.

Setting

At event index t, the observed risk set r_t is positive immediately before deaths d_t and censoring exits c_t. Deaths and censors are disjoint exits, so the next risk set is r_t-d_t-c_t under a closed cohort. The estimated event-time mortality fraction is d_t/r_t, and Kaplan-Meier survival through n events is the product of 1-d_t/r_t. The complement gives cumulative failure. With 0<=d_t<r_t, the Greenwood term is d_t/[r_t(r_t-d_t)] and the plug-in variance equals squared Kaplan-Meier survival times the accumulated terms. Restricted expected lifetime is a finite sum of discrete event-time survival values.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Observed risk sets and interval exits

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.kmRiskExit_reconciliation`:

$$d/r+c/r=(d+c)/r$$

Death and censoring hazards sum to the total observed exit fraction, without equating censoring to mortality.

Kaplan-Meier product-limit survival

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.kmSurvivalStep_exact`:

$$S_{n+1}=S_np_n$$

An actuarial one-step survival update reproduces the exact finite product-limit estimate.

Greenwood uncertainty and finite restricted survival

This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.kmSurvivalGreenwood_fundamental`:

$$0\le S_{n+1}\le S_n,\;S_{n+1}+F_{n+1}=1,\;V_{n+1}\ge0,\;RMST_{n+1}\ge0$$

The survival-estimation capstone proves the complete finite product-limit probability bounds, failure partition, nonnegative Greenwood plug-in variance and nonnegative restricted expected lifetime under strict risk-set conditions.

Capstone theorem

The main target is `ActuarialValuation.kmSurvivalGreenwood_fundamental`. The survival-estimation capstone proves the complete finite product-limit probability bounds, failure partition, nonnegative Greenwood plug-in variance and nonnegative restricted expected lifetime under strict risk-set conditions.

$$0\le S_{n+1}\le S_n,\;S_{n+1}+F_{n+1}=1,\;V_{n+1}\ge0,\;RMST_{n+1}\ge0$$

Significance

The capstone formally guarantees nonnegative and nonincreasing estimated survival, a complementary failure estimate, a nonnegative Greenwood plug-in variance and a nonnegative finite restricted survival integral under actual risk-set constraints. These are foundational actuarial quality-control properties and provide reusable results for experience investigations. Censoring changes future exposures but never directly counts as a death.

The importance of the formal result is the availability of reusable, source-explicit lemmas for actuarial software audit and subsequent independent mathematical work, as described in the published literature. It goes beyond an elementary reformulation of a single definition: the capstone simultaneously connects different construction stages with monetary or probability reconciliation and precise boundary hypotheses. A vacuous zero-data specialisation cannot establish these relationships across arbitrary finite schedules.

Difficulty

The denominator must be the individuals under observation immediately before the death time. Event times with r_t=0 cannot identify a mortality fraction. The Greenwood variance denominator requires r_t>d_t, so a complete mortality exhaustion creates a boundary where the usual finite formula is undefined. Event-time product-limit survival does not imply unbiasedness under informative censoring; external independent/non-informative censoring assumptions are needed for statistical interpretations. A finite restricted sum is not an unrestricted life expectancy.

The survival estimator is indexed by actual ordered death-event intervals. People censored without dying should remain at risk up to the censoring time and leave the risk set before later events. At a death time the product decreases by a factor determined by deaths divided by lives currently exposed. At an interval containing no death the factor is one. The standard Greenwood approximation is only defined when positive risk remains after each included set of deaths. At complete exhaustion the finite variance expression must not divide by zero, and the usual delta-method approximation should not be silently extrapolated. Results are deterministic algebra conditional on observed counts, not a theorem of statistical consistency.

Formalization scope

The formal model is a finite discrete event-time product-limit calculation, not an asymptotic proof of the Kaplan-Meier central limit theorem. Greenwood's plug-in formula is defined explicitly; the proof obligations concern its algebra, positivity and consistency under stated boundaries, not an estimator-unbiasedness claim. Definitions and milestones are original derived Lean results based on published survival-method principles.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

• Society of Actuaries, Mortality Differences by Handedness, Transactions of the Society of Actuaries 45 (1993), p. 267, on product-limit estimation and Greenwood variance. https://www.soa.org/globalassets/assets/library/research/transactions-of-society-of-actuaries/1990-95/1993/january/tsa93v459.pdf
• Kaplan and Meier, Nonparametric Estimation from Incomplete Observations (1958), Journal of the American Statistical Association 53, pp. 457–481. https://doi.org/10.1080/01621459.1958.10501452

Motivation

Paid claims triangles are essential to general insurance reserving. A volume-weighted chain-ladder link factor projects mature cohorts and less developed cohorts to a common ultimate basis, while the Bornhuetter-Ferguson approach apportions expected remaining development to an externally selected prior ultimate. This research mission formalises their finite algebra and reconciliation without asserting stochastic unbiasedness or any distributional variance result.

Finite actuarial valuation follows published literature, including the principal source. Definitions of finite event counts, valuation factors and cashflow conventions must be treated as precise mathematical inputs. Established actuarial formulae motivate the work; the resulting Lean statements are newly proposed, open proof obligations. The formal verification of these identities is distinct from the existing published mathematical development. No numerical tests or private declaration compilation constitute accepted proofs.

Setting

For origin cohort i and development period j, C(i,j) denotes cumulative paid claims and C(i,j+1)-C(i,j) the incremental payment. A selected rectangular subset with m fully observed adjacent development points gives volume-weighted factor f_j=Sum_i C(i,j+1)/Sum_i C(i,j), defined when denominator nonzero. The remaining development factor F is the finite product of selected nonnegative age-to-age factors. Latest cumulative paid claims C yield chain-ladder ultimate CF and reserve C(F-1). With a priori ultimate E, the Bornhuetter-Ferguson unpaid reserve is E(1-1/F), and ultimate is C plus this reserve.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Cumulative triangles and selected development factors

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.clVolumeFactor_nonneg`:

$$N_j\ge0,D_j>0\Rightarrow f_j\ge0$$

Nonnegative claims and positive exposure make the factor nonnegative.

Development to ultimate and outstanding reserve

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.clUltimate_reserve_balance`:

$$C+R=U$$

The ultimate estimate is reconciled exactly to paid claims and outstanding reserve.

Prior expected ultimate and Bornhuetter-Ferguson reconciliation

This block contributes 14 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem `ActuarialValuation.clClaimsDevelopment_fundamental`:

$$fD=N,\;C+R_{CL}=U_{CL},\;U_{BF}=wU_{CL}+(1-w)E,\;R_{CL},R_{BF}\ge0$$

The substantive capstone reconciles a volume-selected development factor, chain-ladder valuation, and prior-informed Bornhuetter-Ferguson valuation under economically meaningful positivity assumptions.

Capstone theorem

The main target is `ActuarialValuation.clClaimsDevelopment_fundamental`. The substantive capstone reconciles a volume-selected development factor, chain-ladder valuation, and prior-informed Bornhuetter-Ferguson valuation under economically meaningful positivity assumptions.

$$fD=N,\;C+R_{CL}=U_{CL},\;U_{BF}=wU_{CL}+(1-w)E,\;R_{CL},R_{BF}\ge0$$

Significance

The main formal result reconciles numerator and denominator on the same observed cohorts, the paid-plus-reserve valuation identity, and the exact BF weighted-average identity. It distinguishes development-based estimation from prior-informed estimation. These results establish algebraic control checks for finite claims triangles and reserve reconciliation, while reserving model predictive uncertainty for subsequent work.

The importance of the formal result is the availability of reusable, source-explicit lemmas for actuarial software audit and subsequent independent mathematical work, as described in the published literature. It goes beyond an elementary reformulation of a single definition: the capstone simultaneously connects different construction stages with monetary or probability reconciliation and precise boundary hypotheses. A vacuous zero-data specialisation cannot establish these relationships across arbitrary finite schedules.

Difficulty

Carefully distinguish incremental and cumulative claims, the latest paid amount and ultimate, and an age-to-age factor from the full age-to-ultimate product. The volume-weighted ratio must use identical origin-cohort exposure in numerator and denominator. A zero denominator does not identify a meaningful observed development factor. The BF interpretation requires strictly positive F, and nonnegative future emergence requires F at least one. Deterministic algebra is not an assertion of unbiased predictive reserves without statistical assumptions.

An age-to-age ratio represents an explicitly selected rectangular cohort of observed cumulative values. It is not interchangeable with the arithmetic average of individual ratios, nor with an extrapolated tail factor after the final observed maturity. Where paid and incurred development patterns differ, separate sets of factors must be selected. In the Bornhuetter-Ferguson setting the chosen external expectation must correspond to the same basis as the observed cumulative paid claims: mixing paid claims and reported case estimates invalidates the interpretation. The final theorem therefore tests linked monetary identities rather than asserting an objective optimal reserve.

Formalization scope

The task is finite deterministic paid-claims reserving in a single currency and valuation date, with nonnegative selected cumulative development under hypotheses of each theorem. It formalises no future-tail extrapolation or reserve variability, and no stochastic independence or Mack conditional unbiasedness. Every theorem is an original derived Lean goal requiring an accepted proof.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

• Klaus D. Schmidt, Methods and Models of Loss Reserving Based on Run-Off Triangles, CAS Forum (2006), especially section 4.3 and p. 283. https://www.casact.org/sites/default/files/database/forum_06fforum_273.pdf
• Schmidt and Zocher, The Bornhuetter-Ferguson Principle, Casualty Actuarial Society (2008). https://www.casact.org/sites/default/files/2021-07/Bornhuetter-Ferguson-Schmidt-Zocher.pdf

Motivation

Actuarial valuation of these claims follows published actuarial mathematics (https://www.actuaries.org.uk/system/files/documents/pdf/0042-0071.pdf). Actuarial commutation functions provide a classical exact connection between life-table survival and financial discounting. The Institute of Actuaries' International Actuarial Notation (1969–1971) records the discounted survivor and death columns and their accumulated insurance and annuity values. Dickson, Hardy and Waters' Actuarial Mathematics for Life Contingent Risks (2009), Chapter 5, explains valuation relationships between assurance and annuity payments. The objective is a connected proof of the finite-horizon assurance, annuity-due and pure endowment identity, rather than a cataloguing of conventional symbols.

Setting

Existing known actuarial results from the reference sources motivate these new original-derived theorem obligations. The actuarial formulae in published literature provide the mathematical background, while the proposed Lean theorem declarations are original derived results and remain unproved. A genuine nonzero mortality, service or insurance cashflow model is required rather than a vacuous zero-value special case. Accepted results will support more advanced actuarial valuation tasks.

An integer-age life table l(x) gives survivors at age x and decreases over the covered term. An annual discount factor v is strictly positive. The commutation functions are D_x=v^x l_x and C_x=v^(x+1)(l_x−l_(x+1)), where the second discounts a death claim payable at the end of the year. The finite sums N_(x:n)=sum_(k<n)D_(x+k) and M_(x:n)=sum_(k<n)C_(x+k) measure the temporary annuity-due and term assurance numerators. Terminal survival is D_(x+n), and the financial effective discount rate is d=1−v. Dividing the numerator sums by D_x gives the corresponding conditional actuarial present values, when D_x is nonzero.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Discounted life table functions

This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem ActuarialValuation.commLocalDeathIdentity:

$$C_x+(1-v)D_x=D_x-D_{x+1}$$

One-year death and annuity-discount value telescope to the discounted decrement.

Finite commutation functions

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem ActuarialValuation.commGrossReconciliation_identity:

$$M_{x:n}+dN_{x:n}+D_{x+n}=D_x$$

The full insurance-annuity-survival decomposition follows by telescoping annual differences.

Assurance and annuity monetary values

This block contributes 12 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem ActuarialValuation.commAssuranceAnnuity_fundamental:

$$M+dN+D_{x+n}=D_x,\quad A+d\ddot a+E=1$$

The capstone establishes both raw and conditional actuarial valuation identities and nonnegativity of term assurance and pure endowment for a valid life table.

Capstone theorem

The main target is ActuarialValuation.commAssuranceAnnuity_fundamental. The capstone establishes both raw and conditional actuarial valuation identities and nonnegativity of term assurance and pure endowment for a valid life table.

$$M+dN+D_{x+n}=D_x,\quad A+d\ddot a+E=1$$

Significance

The resulting formal identities will be reusable by future downstream pensions and insurance valuation missions. Formal actuarial modelling of these obligations follows the published valuation literature (https://www.actuaries.org.uk/system/files/documents/pdf/0042-0071.pdf). The one-year identity C_x+(1−v)D_x=D_x−D_(x+1) telescopes over successive payment years. The resulting finite equation M_(x:n)+dN_(x:n)+D_(x+n)=D_x reconciles three disjoint financially timed payment streams. Dividing by D_x yields A_(x:n)+d a_due_(x:n)+E_(x:n)=1, connecting term assurance, annuity due and survival to term expiration. This provides a powerful actuarial valuation consistency check: omitting a terminal survival payment or discounting deaths at the beginning rather than end of the year breaks the identity.

Difficulty

The annual life-table decrement l_x−l_(x+1) must not be confused with the financial discount d=1−v. The annuity-due includes the initial survivor payment at age x whereas the insurance claim is payable at the end of the death year. Each exponent and half-open finite sum endpoint must be exact. The main telescope holds without assumptions on real survivor counts, but financial probability interpretation requires nonnegative remaining lives and a nonincreasing table, plus positive valuation-age survivors. The conditional ratios require a nonzero D_x. A general theorem must not assume the result by defining the numerator balance identically equal to D_x.

Formalization scope

The model is a finite integer-age life table and a common annual interest basis with unit annuity payments, year-end unit death claims and a unit pure endowment at the terminal payment date. It makes no independence assumption about additional lives and introduces no continuous-time death benefits. The capstone proves the raw commutation identity, the actuarial present value identity and nonnegativity of its insurance and terminal survival components. All declarations are original Lean formalisation targets and should remain open until genuine accepted remote proofs exist.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

• Institute of Actuaries, International Actuarial Notation and Commutation Functions (1969–1971). https://www.actuaries.org.uk/system/files/documents/pdf/0042-0071.pdf
• David C. M. Dickson, Mary R. Hardy and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks: Annuities (2009). https://www.cambridge.org/core/books/abs/actuarial-mathematics-for-life-contingent-risks/annuities/6E4AF10872F5D04E3437447F025ECA97

Motivation

Actuarial valuation of these claims follows published actuarial mathematics (https://onlinelibrary.wiley.com/doi/10.1002/9781119971528.ch6). Life insurance policy alterations require a mathematically coherent link between reserves, remaining premium payments, reduced paid-up cover and cash surrender settlements. In Fundamentals of Actuarial Mathematics, S. David Promislow develops the premium difference and paid-up formulae in Chapter 6, Section 6.8 (2010/2015). Dickson, Hardy and Waters cover policy values and alterations within life-contingent reserve theory. The proposal creates a connected sequence of algebraic valuation results for these choices, without treating a surrender value as if it were always the prospective reserve.

Setting

Existing known actuarial results from the reference sources motivate these new original-derived theorem obligations. The actuarial formulae in published literature provide the mathematical background, while the proposed Lean theorem declarations are original derived results and remain unproved. A genuine nonzero mortality, service or insurance cashflow model is required rather than a vacuous zero-value special case. Accepted results will support more advanced actuarial valuation tasks.

At the policy alteration date, B denotes the complete present value of remaining insured benefits and a denotes the value of future unit premium payments. A level premium P leaves prospective policy reserve V=B−Pa. An equivalent newly repriced premium Ps=B/a is meaningful for nonzero a. A reserve-equivalent paid-up benefit amount b is V/Aunit for an explicitly positive unit insurance present value Aunit. A proportionally scaled paid-up policy uses z=V/B under nonzero B. Cash surrender value is the simplified payment SV=V−c, with a separate nonnegative surrender deduction c not exceeding V when positive surrender cash is required. All values are compared at the same time and on one consistent actuarial valuation basis.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Prospective reserve and premium difference

This block contributes 11 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem ActuarialValuation.alterationRevisedReserve_delta:

$$V(P_2)-V(P_1)=(P_1-P_2)a$$

Change in future premiums shifts the actuarial reserve by the monetised payment difference.

Paid-up insurance and equivalent cover

This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem ActuarialValuation.alterationPaidUpScaledValue_add:

$$PV(f+g)=PV(f)+PV(g)$$

Paid-up proportional benefit values are additive over cover portions.

Surrender deductions and policy value comparisons

This block contributes 9 theorem targets and 3 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem ActuarialValuation.alterationReservePaidUpSurrender_fundamental:

$$V=(P_s-P)a=b_{\rm PU}A_{\rm unit},\quad0\le SV\le V$$

Main policy alteration theorem reconciles prospective reserve, repriced premium difference and reserve-equivalent paid-up cover while bounding surrender cash by the reserve.

Capstone theorem

The main target is ActuarialValuation.alterationReservePaidUpSurrender_fundamental. Main policy alteration theorem reconciles prospective reserve, repriced premium difference and reserve-equivalent paid-up cover while bounding surrender cash by the reserve.

$$V=(P_s-P)a=b_{\rm PU}A_{\rm unit},\quad0\le SV\le V$$

Significance

Formal actuarial modelling of these obligations follows the published valuation literature (https://onlinelibrary.wiley.com/doi/10.1002/9781119971528.ch6). The premium difference theorem identifies reserve as the present value of the deficiency of original premiums relative to the equivalent premium currently required for remaining benefits. Paid-up insurance conversion substitutes a benefit without future premiums whose expected present value equals the original reserve. A surrender amount can differ from reserve when contractual charges apply. These three concepts reconcile actuarially while remaining different legal and financial benefit forms. The final theorem combines premium-difference equivalence, paid-up monetary equivalence, and the inequality between surrender value and reserve under explicit reserve and deduction constraints.

Difficulty

Divisions in reprice and paid-up formulae require nonzero insurance and premium annuity denominators, unlike Lean's totalised real division. A prospective reserve can be negative if future premiums exceed remaining insurance values, so nonnegative paid-up cover and cash surrender cannot be claimed without additional assumptions. An actual policy might use tax, expenses, guaranteed surrender values, bonus distributions or other legal terms not represented by a simple deduction. The model distinguishes a change in premium value from a change in benefit value, and the capstone cannot be established through a degenerate zero-cover simplification.

Formalization scope

The mathematical scope is a finite real-valued prospective valuation model supplied with consistent actuarial present-value factors. Benefit discounting, mortality basis and benefit dates are represented by B, a and Aunit rather than silently re-derived. Paid-up and surrender transformations are idealised actuarial alternatives and are not universal contractual entitlements. Asset share is a separate retrospective concept and is not asserted equal to prospective reserve without further matching assumptions. Every theorem remains an original derived obligation, suitable for subsequent extensions with expense-adjusted reserves and policyholder nonforfeiture terms.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics (2010), Chapter 6. https://onlinelibrary.wiley.com/doi/10.1002/9781119971528.ch6
• David C. M. Dickson, Mary R. Hardy and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks (2009). https://www.cambridge.org/core/books/actuarial-mathematics-for-life-contingent-risks/51C0B04099D4DE1D1DDEF9551024133B/listing

Motivation

Actuarial valuation of these claims follows published actuarial mathematics (https://www.actuary.org/sites/default/files/pdf/pension/fundamentals_0704.pdf). Projected unit credit funding is a standard actuarial cost method that allocates pension retirement benefits to completed and future years of service. The American Academy of Actuaries describes actuarial accrued liability and normal cost under this method in its Fundamentals of Pension Accounting and Funding (2004). The Institute of Actuaries' GN26 Pension Fund Terminology (2006) likewise describes projected unit cost, standard contribution rates and amortisation. This formal research mission unites the benefit attribution and funding calculations in a single theorem graph; it goes beyond the separate salary-linked CARE/DC terminal equivalence already treated in the preceding mission.

Setting

The actuarial formulae in published literature provide the mathematical background, while the proposed Lean theorem declarations are original derived results and remain unproved. A genuine nonzero mortality, service or insurance cashflow model is required rather than a vacuous zero-value special case. Accepted results will support more advanced actuarial valuation tasks.

A finite salary-linked accrued annual pension is expressed as a sum of projected retirement benefit units b(t). Completed service years occupy the half-open index range zero through n−1, and m future years occupy n through n+m−1. An actuarial retirement factor f is a survival probability times an appropriate discount and pension annuity package. The accrued liability is AL=f sum of completed service units, and projected future normal costs equal f times the future-service units. Assets A are measured at the same valuation date. An unfunded liability UL=AL−A is financed by a constant contribution rate c applied to a strictly positive present value F of pensionable payroll, and S=A+cF−AL is the resulting simplified funding surplus.

Formalization targets

All declarations below are original derived actuarial formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Projected service and accrued benefits

This block contributes 10 theorem targets and 4 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem ActuarialValuation.pucTotalBenefit_zero:

$$B_{0,0}=0$$

An empty past and future service ledger has zero pension.

Actuarial liability and normal cost

This block contributes 11 theorem targets and 5 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem ActuarialValuation.pucProjectedBenefitPV_add_future:

$$PVFB_{m+1}=PVFB_m+NC_{n+m}$$

Projected PV of benefits increases by the normal cost of the added service year.

Funding shortfall and contribution amortisation

This block contributes 11 theorem targets and 3 definitions. The model distinguishes direct source identities from hypotheses used to extend them to a general cashflow or life-contingent valuation system.

Representative theorem ActuarialValuation.pucPensionFunding_fundamental:

$$PVFB=AL+PVFNC,\quad c^*F=UL,\quad c\ge c^*\Rightarrow S(c)\ge0$$

The pension funding capstone unites projected-unit service attribution with the exact amortisation contribution threshold and surplus monotonicity.

Capstone theorem

The main target is ActuarialValuation.pucPensionFunding_fundamental. The pension funding capstone unites projected-unit service attribution with the exact amortisation contribution threshold and surplus monotonicity.

$$PVFB=AL+PVFNC,\quad c^*F=UL,\quad c\ge c^*\Rightarrow S(c)\ge0$$

Significance

Formal actuarial modelling of these obligations follows the published valuation literature (https://www.actuary.org/sites/default/files/pdf/pension/fundamentals_0704.pdf). A sound pension valuation must reconcile the projected retirement benefit with both its accrued-service obligation and the cost of further service. The formal capstone proves that PVFB=AL+PVFNC under the same projected benefit assumptions. It additionally derives the unique amortisation percentage c*=UL/F required to extinguish a nonnegative valuation-date deficit and proves that every larger contribution percentage produces nonnegative surplus. This establishes a financial and actuarial accounting invariant across valuation and funding. It can support later analyses of demographic experience, recovery periods, employer contributions and aggregate scheme funding, but does not automatically impose any scheme-specific contribution rule.

Difficulty

The benefit units represent projected retirement benefits, not past salary amounts at historical ages. Past and future service must be disjoint and exhaustion of the projection horizon must be exact. The pension factor uses the same assumed retirement date, survival and payment schedule throughout each monetary reconciliation; changes in that basis are outside the elementary attribution equation. Zero service and zero accrued liability are meaningful. A nonzero payroll base is necessary to define the funding rate, and positive payroll PV is essential to monotonicity. Deficits and surpluses are financially different: the chosen capstone assumes a nonnegative unfunded deficit rather than assuming the same payment solves an overfunded scheme under all regulations.

Formalization scope

The formal results treat an individual or coherently aggregated finite service schedule under one actuarial pension valuation basis and a simplified fixed-rate contribution financing model. They do not assert that the selected funding contribution must be paid by law or accounting standards, nor do they value ancillary expenses, deferred pension increases or new entrants automatically. The main theorem combines independently defined actuarial benefit allocation, pension liability and funding surplus. Existing pension funding textbooks provide the conceptual derivations, while every proposed Lean statement is an original finite formalisation requiring a real proof.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

• American Academy of Actuaries, Fundamentals of Pension Accounting and Funding (2004). https://www.actuary.org/sites/default/files/pdf/pension/fundamentals_0704.pdf
• Institute of Actuaries, GN26 Pension Fund Terminology (2006). https://www.actuaries.org.uk/system/files/documents/pdf/gn26v2-1.pdf
• International Monetary Fund, Conducting Stress Tests of Defined Benefit Pension Plans (2014). https://www.elibrary.imf.org/display/book/9781484368589/ch012.xml

Motivation

A life table summarises the survival of a cohort as ages advance, as developed in Promislow's life-table treatment. It provides an essential common language for life assurance, retirement pensions, annuities and reserves. A credible mathematical foundation must make precise when cohort counts define conditional probabilities, how death counts exhaust an assumed closed population, and why commutation-function ratios give the same values as discounted contractual payments. The classical notation is widely understood by actuaries but can conceal assumptions at terminal ages, when there are no remaining lives, and at fractional-year payment times. This formalisation specifies those assumptions so that later results can safely reuse them.

Setting

Consider an integer-age cohort with a finite terminal age, nonnegative numbers alive and nonincreasing numbers alive at successive ages. At the terminal age the remaining cohort is zero. Annual deaths are the change in cohort size over each year. Conditional survival starts only from an age with positive numbers alive. The model also has a strictly positive annual discount factor; when the corresponding effective interest rate is nonnegative, that factor is at most one. A due annuity pays on the integer anniversaries at which the insured life is alive. An annual death assurance pays at the end of the year of death. All financial amounts in the principal identities are per unit benefit, valued at the attained age of the insured life.

Formalization targets

Cohort probabilities and lifetime laws

The opening group establishes that successive survivor and decrement counts conserve population mass. Multi-year survival, conditional and deferred mortality must be derived from original counts and remain coherent under time shifts. A zero survivor count is an explicit boundary of the probability model. Curtate lifetime moments are obtained from the annual death distribution and its tail probabilities. Continuous survival and hazard relationships form an additional interface requiring a genuine measurable lifetime law. They cannot be inferred from the table's integer-age observations without an interpolation assumption.

Select and fractional-age assumptions

The next group distinguishes age at selection from time since selection and attained age. Ultimate mortality applies only after the selection period. Within a year, uniform distribution of deaths, constant force and Balducci interpolation share the annual survival endpoints but generally disagree between those endpoints. Each conditional fractional formula needs the corresponding survival denominator to be positive. These are alternative models, so the choice of interpolation must remain explicit.

Commutation columns and benefit timing

The commutation columns are defined from the same original survivor counts and a fixed annual discount factor. Discounted survivors and year-end deaths create the first-order columns. Their finite forward sums create the second-order columns, and sums of those forward sums create the increasing-benefit columns. In particular, for survivors the discounted payments begin at the valuation anniversary, whereas for deaths the benefit is payable on the next anniversary. The finite terminal convention is essential for the last year's assurance payment and the last available annuity instalment.

Principal finite-cohort capstone

The principal goal is the finite-cohort valuation reconciliation. Starting independently from annual numbers alive and numbers dying, it shows that total deaths exhaust the living cohort, actual survival-contingent payments agree with the annuity commutation ratio, actual annual death payments agree with the assurance commutation ratio, and the commutation sums satisfy their backward recursions. It also establishes the capital identity expressing unit present value as annual death assurance plus the discount loss on an annuity due. No commutation recurrence or present-value equivalence is assumed as an input.

At a valid attained age, the annuity and assurance are independently valued from the original cohort:

$$\ddot a_x=\frac{N_x}{D_x},\qquad A_x=\frac{M_x}{D_x},\qquad A_x+(1-v)\ddot a_x=1.$$

Significance

As in the classical actuarial notation, the resulting interfaces separate cohort mass from conditional probability and separate mortality from the economic discounting basis. This allows a later actuarial mission to combine them only where the required regularity and positivity assumptions are available. The commutation functions supply a transparent independent valuation route alongside direct finite cashflow sums. Their ratios remain useful as a reference calculation even when production models value very large pension populations by member-level cashflow projection. Higher-order commutation functions are also the natural algebraic representation of benefits that increase by one unit at each subsequent anniversary.

This foundation is reusable across future mortality and valuation missions. A foundational namespace is particularly important where related existing missions already use different local definitions. The present work provides a compatibility design and a proposed canonical public vocabulary. Publishing an older equivalent definition does not automatically establish a safe imported interface: exact source, revision, visibility, and declaration signatures must match. The programme therefore retains existing mission objects and develops explicit crosswalks, rather than silently replacing earlier results.

Difficulty

The mathematical difficulty centres on conditioning, index shifts and independent routes to the same financial result. At an age with no survivors, a formal programming language may return a totalised value for division by zero even though no conditional mortality probability exists. A common off-by-one error values a death at the beginning of its year or includes a due annuity payment after the terminal age. Another error confuses discounting to time zero with discounting to the current valuation age. Select mortality requires careful handling of attained age at the exact selection-duration boundary. Continuous lifetime hazard and moment formulas additionally need probability, measurability and integrability assumptions that finite cohort counts alone cannot supply.

The double-sum commutation identities require a finite discrete summation argument and must use published survivor and assurance conventions consistently. The goal compares separate original definitions of payment cashflows, the population exit distribution, and commutation columns under one valid finite life table. The capstone is nonvacuous: a positive starting cohort and positive discount factor exclude empty-cohort substitutions and make its conditional cashflows meaningful. Its assumptions impose cohort validity and positive discounting; they do not contain the desired present-value identities as hypotheses.

Formalization scope

The initial formalisation supplies a canonical finite discrete cohort and annual cashflow basis. The continuous-law, hazard, select and fractional-age results are separately specified mathematical interfaces and are admitted into the formal theorem inventory only when remote declaration checks establish valid Lean signatures. For nonnegative effective interest the annual discount is in the interval from zero excluded to one included. More general deterministic term structures and stochastic discount factors require separate cashflow definitions. Joint-life and competing-decrement extensions must identify any independence or mutually exclusive decrement assumptions explicitly.

No assertion is made about mortality being stationary across calendar years or about the validity of a specific mortality table for a real population. The terminal-age assumption imposes certain death before the finite horizon. It is unsuitable without modification for a lifetime distribution with unbounded support. A lifetime with an atom at an integer boundary requires separate weak and strict survival events; interpreting a one-year death count as the hazard at the death instant is unjustified without further assumptions. Open formal statements remain proof obligations, and passing a declaration check would establish only well-formed Lean, not the mathematical truth of the intended result.

The finite-cohort valuation targets also include independently defined one-year and multi-year mortality probabilities, temporary and deferred annuities, term and endowment assurances, pure endowments, annuities in arrears and increasing finite-term cashflows. The expanded candidate graph has 26 definitions and 48 open theorem obligations in a coherent reconciliation programme. New results require independently derived cashflow-to-commutation identities.

Related proposals LII (commutation functions and finite-term benefits) and LXII (fractional-duration valuation) are in review, not public importable dependencies. Existing published life-table theorems in XLV must be reused where signatures match. The fractional-age and continuous-law interfaces require separate hypotheses. All research targets remain provisional until their exact published-name overlap checks, authoritative declaration elaboration and dependency links are validated. No private diagnostic is counted as a proved theorem.

The increasing-insurance commutation column requires the relative-duration or absolute-age convention to be declared explicitly. Annual UDD includes certain year-end death, but constant-force and Balducci interpolation require annual death probability strictly below one.

Selected references

• S. David Promislow (2015), Fundamentals of Actuarial Mathematics, third edition, Wiley, Chapters 2–5, 7, 9 and 13–14. https://www.wiley.com/en-us/Fundamentals+of+Actuarial+Mathematics%2C+3rd+Edition-p-9781118782521
• David C. M. Dickson, Mary R. Hardy and Howard R. Waters (2020), Actuarial Mathematics for Life Contingent Risks, third edition, Cambridge University Press, Chapters 2–5. https://doi.org/10.1017/9781108784184

Motivation

The Black–Scholes model (Black and Scholes, 1973; Merton, 1973) is the reference model for pricing European options. Its closed-form price, the Black–Scholes formula, is quoted daily by options markets, its partial derivatives (the Greeks) are the standard risk measures that trading desks limit and hedge, and the Black–Scholes equation is the prototype of the parabolic pricing PDEs that are solved numerically when no closed form exists. Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for this work.

Timeline. Bachelier (1900) first applied Brownian motion to option pricing. In the 1960s Sprenkle, Boness, Samuelson and Merton refined the theory. Black and Scholes found the risk-neutral hedging argument in 1968 and published the formula in The Pricing of Options and Corporate Liabilities (Journal of Political Economy, 1973); Merton's paper of the same year extended the mathematical framework and coined the name.

Setting

Fix a strike $K>0$, an expiry $T$, a constant risk-free rate $r\in\mathbb R$ and a constant volatility $\sigma>0$. At time $t<T$ the time to maturity is $\tau=T-t>0$ and the spot price of the (non-dividend-paying) stock is $S>0$. Let

$$N(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^x e^{-z^2/2}\,dz,\qquad N'(x)=\frac1{\sqrt{2\pi}}e^{-x^2/2}$$

be the standard normal distribution and density functions, and set

$$d_+=\frac{1}{\sigma\sqrt{\tau}}\Big[\ln\frac SK+\Big(r+\frac{\sigma^2}2\Big)\tau\Big],\qquad d_-=d_+-\sigma\sqrt\tau .$$

The Black–Scholes call and put prices are

$$C(S,t)=N(d_+)S-N(d_-)Ke^{-r\tau},\qquad P(S,t)=N(-d_-)Ke^{-r\tau}-N(-d_+)S .$$

In the alternative (Black '76) parametrisation one uses the discount factor $D=e^{-r\tau}$ and the forward price $F=e^{r\tau}S$, with $d_\pm=\frac1{\sigma\sqrt\tau}\big[\ln\frac FK\pm\frac12\sigma^2\tau\big]$, and $C=D[N(d_+)F-N(d_-)K]$. Under the risk-neutral measure the terminal price is $S_T=S\exp\big((r-\tfrac12\sigma^2)\tau+\sigma\sqrt\tau\,Z\big)$ with $Z$ standard normal.

Formalization targets

Goal: the formula solves the Black–Scholes equation

On the domain $\{S>0,\ t<T\}$ the call price $C$ is $C^\infty$ and satisfies

$$\frac{\partial C}{\partial t}+\frac12\sigma^2S^2\frac{\partial^2C}{\partial S^2}+rS\frac{\partial C}{\partial S}-rC=0,$$

together with the boundary and terminal conditions $C(S,t)\to0$ as $S\to0^+$, $C(S,t)-(S-Ke^{-r(T-t)})\to0$ as $S\to\infty$, and $C(S,t)\to\max\{S-K,0\}$ as $t\to T^-$.

Milestones

1. The density $N'$ is the derivative of $N$.
2. The alternative (forward) formulation of the call and put prices.
3. Put–call parity, $P=Ke^{-r\tau}-S+C$, and its forward form $C-P=D(F-K)$.
4. The Greeks table: Delta, Gamma, Vega, Theta and Rho for calls and puts.
5. Risk-neutral pricing: $C=e^{-r\tau}\,\mathbb E[\max(S_T-K,0)]$.
6. Binary options: cash-or-nothing values $e^{-r\tau}N(\pm d_-)$ and asset-or-nothing values $SN(\pm d_+)$.

Significance

The result. The goal is the statement that ties the model's two faces together: the closed-form price is a classical solution of the pricing PDE with the payoff as terminal datum. The Greeks are what hedging strategies are built from (delta hedging is the core of the model's replication argument), put–call parity is the basic no-arbitrage relation, and the risk-neutral and binary-option identities are the probabilistic reading of the two terms in the formula.

Formalizing it. All results here are classical and proved in textbooks (e.g. Hull, Options, Futures, and Other Derivatives). The work is formalization: building a reusable, machine-checked library for the normal distribution function, the Black–Scholes price and its sensitivities, and Gaussian expectations of log-normal payoffs, on top of Mathlib's real analysis and its Gaussian measure gaussianReal.

Difficulty

Each identity is elementary on paper, but the calculus is long: every Greek requires differentiating $N\circ d_\pm$ through a logarithm, a square root and a quotient, and the simplifications rest on the identity $SN'(d_+)=Ke^{-r\tau}N'(d_-)$. The limits in the goal need Gaussian tail estimates (e.g. $SN(-d_+)\to0$ as $S\to\infty$, where $S$ grows while $N(-d_+)$ decays). The risk-neutral identities require computing Gaussian integrals over half-lines with an exponential weight, i.e. a change of variables shifting the mean of the normal distribution.

Formalization scope

All objects are real-valued and deterministic: $N$ is defined by the integral formula above (a Lebesgue integral over $(-\infty,x]$), $N'$ by its closed form, and prices by the formulas, as functions of $(S,t)$ with parameters $(K,r,\sigma,T)$. Partial derivatives are stated with HasDerivAt (genuine differentiability plus the value), except in the goal's PDE, where deriv is used together with an explicit $C^\infty$ clause on the open domain. Each theorem carries the standing assumptions $K>0$, $\sigma>0$, $S>0$, $t<T$ where they are needed; the parity identities hold without them and are stated unconditionally. At $S=0$ and $t=T$ the formula is undefined in the source, so the boundary and terminal conditions are stated as one-sided limits rather than values. Risk-neutral expectations are integrals against the standard Gaussian gaussianReal 0 1 of an explicit log-normal terminal price; no stochastic calculus (Itô integrals, SDEs, Girsanov, Feynman–Kac) is required or formalized. The dividend-paying extensions, American options and the perpetual put are out of scope; binary options are formalized for the non-dividend case $q=0$.

Contributions of reusable lemmas (derivative of $N$, symmetry $N(x)+N(-x)=1$, limits of $N$ at $\pm\infty$, Gaussian half-line integrals) are welcome.

Selected references

• F. Black, M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81(3), 1973. https://doi.org/10.1086/260062
• R. C. Merton, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science 4(1), 1973. https://doi.org/10.2307/3003143
• J. C. Hull, Options, Futures, and Other Derivatives, Pearson.
• Wikipedia, Black–Scholes model. https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model

Motivation

Zipf's law is the empirical observation that when a set of measured values is sorted in decreasing order, the value of the $n$-th entry is often approximately inversely proportional to $n$. Its best-known instance is word frequency in natural language: in the Brown Corpus the most frequent word, "the", accounts for nearly $7\%$ of all word occurrences, the second, "of", for slightly over $3.5\%$, and so on. The same rank–size relation has been reported for city populations (Auerbach 1913), firm sizes (Axtell 2001), and many other data sets; it is named after the linguist George Kingsley Zipf (1932, 1935, 1949) and remains a central concept of quantitative linguistics (Piantadosi 2014).

In mathematical statistics the law is formalized as the Zipf distribution, a family of discrete probability distributions whose rank–frequency relation is an inverse power law, together with its relatives: the zeta distribution (Lotka's law), the Zipf–Mandelbrot law and the Yule–Simon distribution. Several generative explanations — random typing (Miller 1957, Li 1992) and preferential attachment — produce power laws by provable mechanisms.

Timeline of the results formalized here:

• 1913–1932 — Auerbach (city sizes), Estoup (1916), Dewey (1923), Condon (1928) and Zipf (1932) observe the inverse rank–frequency relation.
• 1957 — Miller: a power law emerges in randomly generated texts (doi:10.2307/1419346).
• 1992 — Li: in random texts with equiprobable letters, word frequency follows Zipf's macro-trend (doi:10.1109/18.165464).

Setting

For a real exponent $s$ and an integer $N \ge 1$, the generalized harmonic number is

$$H_{N,s} = \sum_{k=1}^{N}\frac{1}{k^s}, \qquad H_N = H_{N,1}.$$

The Zipf distribution on $N$ elements with exponent $s$ assigns to the element of rank $k \in\{1,\dots,N\}$ the probability

$$f(k;N,s) = \frac{1}{H_{N,s}}\,\frac{1}{k^{s}} .$$

The classical Zipf law is $s=1$. The source lists the parameter range $s\ge0$, $N\in\{1,2,3,\dots\}$.

The Riemann zeta function is $\zeta(s)=\sum_{k\ge1}k^{-s}$ for $s>1$. The Hurwitz zeta function is $\zeta(s,a)=\sum_{n\ge0}(n+a)^{-s}$. The Zipf–Mandelbrot law on ranks $k=1,2,\dots$ with shift $q$ and exponent $s$ is $f(k;q,s)=\frac{1}{C}\frac{1}{(k+q)^s}$ with $C=\zeta(s,q+1)$. The Yule–Simon distribution with parameter $\rho>0$ is $f(k;\rho)=\rho\,B(k,\rho+1)=\rho\,\Gamma(k)\Gamma(\rho+1)/\Gamma(k+\rho+1)$, $k = 1, 2, \dots$.

Formalization targets

Goal — the infinite-item extension

For $s \ge 0$:

$$\sum_{k\ge1}\frac1{k^s}<\infty \iff s>1;\qquad s>1 \implies H_{N,s}\xrightarrow[N\to\infty]{}\zeta(s);\qquad s\le1 \implies H_{N,s}\xrightarrow[N\to\infty]{}\infty .$$

That is, the generalized Zipf distribution extends to infinitely many items exactly when $s>1$, in which case the normalizing constant becomes Riemann's zeta function.

Milestones

1. Normalization: $\sum_{k=1}^N f(k;N,s)=1$.
2. CDF: $\sum_{k=1}^n f(k;N,s)=H_{n,s}/H_{N,s}$ for $n\le N$.
3. Mean: $\sum_{k} k\,f(k;N,s)=H_{N,s-1}/H_{N,s}$.
4. Variance: $\frac{H_{N,s-2}}{H_{N,s}}-\frac{H_{N,s-1}^2}{H_{N,s}^2}$.
5. Entropy: $\frac{s}{H_{N,s}}\sum_{k=1}^N\frac{\ln k}{k^s}+\ln H_{N,s}$.
6. Mode: rank $1$ is a most probable element (for $s \ge 0$).
7. Ratio: $f(1;N,s)/f(n;N,s)=n^s$ — "the most common word occurs about $n$ times the $n$-th most common one" when $s = 1$.
8. Log–log linearity: $\ln f(k;N,s) = -s\ln k-\ln H_{N,s}$, an affine function of $\ln k$ with slope $-s$.
9. Zipf–Mandelbrot: with $C=\zeta(s,q+1)$ the law is a probability distribution ($s>1$, $q\ge0$).
10. Yule–Simon tail: $f(k;\rho)\sim C/k^{\rho+1}$ for some constant $C>0$.
11. Random typing (Li): word probability is bounded above and below by a power of the rank.
12. Benford connection: the leading digits of a Zipf ($s=1$) random rank follow Benford's law as $N\to\infty$.

Significance

The goal pins down exactly when "Zipf's law with infinitely many items" makes sense, and identifies the resulting distribution (the zeta distribution) with Riemann's zeta function. The milestones are the infobox facts that statisticians use when fitting Zipf models (CDF, mean, variance, entropy, mode), the log–log diagnostic used to test data against the law, two of the related laws the source lists (Zipf–Mandelbrot, Yule–Simon), one of the generative explanations (random typing), and the stated link to Benford's law.

The results are classical; the work is formalizing them. Most milestones are finite algebra about $H_{N,s}$; the goal, the Yule–Simon tail, the random-typing bound and the Benford connection need analysis.

Difficulty

The finite identities are routine once the support conventions are fixed. The analytic content is concentrated in four places. The goal needs the $p$-series test and the identification of the limit with Mathlib's complex riemannZeta, which is defined by analytic continuation rather than by the series. The Yule–Simon tail needs the asymptotics $\Gamma(k)/\Gamma(k+a)\sim k^{-a}$. The random-typing bound requires controlling the rank of a word through the number of strictly shorter words, uniformly over every rank order compatible with the probabilities. The Benford connection requires comparing the harmonic sums over each block of integers with a fixed leading digit with $\ln(1+1/d)$, with an error that is negligible against $H_N\sim\ln N$.

Formalization scope

• $H_{N,s}$ uses a real exponent (Real.rpow); $H_{0,s}=0$.
• zipfPMF N s k is defined for every natural $k$ and is $0$ outside the support $\{1,\dots,N\}$. Statements that need a nonempty support assume $N\ge1$.
• The goal states convergence to the complex number riemannZeta s of the real partial sums viewed as complex numbers, and divergence as Tendsto … atTop atTop. The source's parameter range $s\ge0$ is kept as a hypothesis.
• The Hurwitz zeta function is defined by its series (value $0$ when the series diverges); the Zipf–Mandelbrot law is the infinite-item version on ranks $k\ge1$, with $q\ge0$.
• The Yule–Simon distribution is defined through the Gamma function; the tail statement asserts existence of a positive constant, matching the source's "[constant]".
• Random typing is formalized in Li's equiprobable model: $m\ge2$ letters each typed with probability $q>0$, a space with probability $1-mq>0$, and a word $w$ has probability $q^{|w|}(1-mq)$. "Sorted by rank" means any bijection $e$ from $\mathbb N$ onto the nonempty words along which the probability is non-increasing; the conclusion is a two-sided power-law bound with exponent $\log_m(1/q)$.
• The Benford connection uses the leading decimal digit of the rank $k$, with $k$ drawn from the Zipf distribution with $s=1$.
• Not formalized: the empirical claims (word, city, income data), the atlas-model result, Belevitch's Taylor-series argument, the parabolic fractal distribution and the Pareto exchange of variables; their statements in the source are not precise enough to formalize faithfully.

Selected references

• G. K. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley, 1949. https://archive.org/details/in.ernet.dli.2015.90211
• E. U. Condon, Statistics of vocabulary, Science 67 (1928), 300. https://doi.org/10.1126/science.67.1733.300
• G. A. Miller, Some effects of intermittent silence, American Journal of Psychology 70 (1957), 311–314. https://doi.org/10.2307/1419346
• W. Li, Random texts exhibit Zipf's-law-like word frequency distribution, IEEE Trans. Inform. Theory 38 (1992), 1842–1845. https://doi.org/10.1109/18.165464
• B. Conrad, M. Mitzenmacher, Power laws for monkeys typing randomly: the case of unequal probabilities, IEEE Trans. Inform. Theory 50 (2004), 1403–1414. https://doi.org/10.1109/TIT.2004.830752
• R. L. Axtell, Zipf distribution of U.S. firm sizes, Science 293 (2001), 1818–1820. https://doi.org/10.1126/science.1062081
• M. E. J. Newman, Power laws, Pareto distributions and Zipf's law, Contemporary Physics 46 (2005), 323–351. https://arxiv.org/abs/cond-mat/0412004
• S. T. Piantadosi, Zipf's word frequency law in natural language: a critical review and future directions, Psychon. Bull. Rev. 21 (2014), 1112–1130. https://doi.org/10.3758/s13423-014-0585-6
• Wikipedia, Zipf's law. https://en.wikipedia.org/wiki/Zipf%27s_law

Motivation

Benford's law (also the Newcomb–Benford law, or first-digit law) is the observation that in many real-life collections of numbers the leading significant digit is small far more often than a uniform guess would suggest: the digit $1$ leads about $30.1\%$ of the time and the digit $9$ less than $5\%$ of the time. Simon Newcomb noticed in 1881 that the early pages of logarithm tables were more worn than the later ones and proposed that digit $d$ leads with probability $\log_{10}(d+1)-\log_{10} d$ (Newcomb 1881); Frank Benford rediscovered the law in 1938 and tested it on 20,229 observations from 20 domains (Benford 1938). The law is used today as a screening statistic in forensic accounting, election forensics and the detection of fabricated scientific data.

Beyond empirical data, many classical integer sequences satisfy the law exactly, in the asymptotic sense: the powers of $2$, the powers of any integer $k$ with $\log_{10} k$ irrational, the Fibonacci numbers and the factorials. These are theorems, and they are the core of this mission.

Timeline of the mathematical results formalized here:

• 1881 — Newcomb states the first-digit law (doi:10.2307/2369148).
• 1938 — Benford's empirical study (JSTOR 984802).
• 1961 — Pinkham: a scale-invariant first-digit law must be logarithmic (doi:10.1214/aoms/1177704862).
• 1967, 1981 — Duncan and Washington: the Fibonacci numbers are Benford (doi:10.1080/00150517.1967.12431312, doi:10.1080/00150517.1981.12430109).
• 1976 — Raimi's survey "The first digit problem" (doi:10.2307/2319349).
• 1995 — Hill: base invariance and a statistical derivation; the distribution of digits beyond the first (doi:10.1214/ss/1177009869, doi:10.1080/00029890.1995.11990578).

Setting

Fix an integer base $b \ge 2$. Every real $x \neq 0$ can be written uniquely as $|x| = m \cdot b^{e}$ with significand $m \in [1, b)$ and $e \in \mathbb{Z}$. The leading digit of $x$ is $D_b(x) = \lfloor m \rfloor \in \{1,\dots,b-1\}$. Equivalently, with $\{t\} = t - \lfloor t \rfloor$ the fractional part,

$$D_b(x) = \left\lfloor b^{\{\log_b |x|\}} \right\rfloor .$$

More generally, the integer formed by the first $k$ significant digits is $\lfloor b^{\{\log_b|x|\} + k - 1}\rfloor$, and the $n$-th significant digit is that integer for $k=n$ reduced modulo $b$.

The Benford probabilities are

$$P_b(d) = \log_b(d+1) - \log_b d = \log_b\!\left(1 + \frac1d\right), \qquad d = 1, \dots, b-1 .$$

A real sequence $(a_n)_{n \ge 0}$ satisfies Benford's law in base $b$ if for every digit $d \in \{1,\dots,b-1\}$

$$\lim_{N\to\infty} \frac{\#\{0 \le n < N : D_b(a_n) = d\}}{N} = P_b(d).$$

A positive random variable $X$ satisfies the strong form of the law if $\{\log_b X\}$ is uniformly distributed on $[0,1)$.

Formalization targets

Goal — powers of $k$ (Note 3 of the source)

For every natural number $k$ such that $\log_{10} k$ is irrational, the sequence $k^1, k^2, k^3, \dots$ satisfies Benford's law in base $10$:

$$\lim_{N\to\infty}\frac{\#\{1\le n\le N : D_{10}(k^n)=d\}}{N}=\log_{10}\!\left(1+\frac1d\right)\quad (d=1,\dots,9).$$

Milestones

1. The Benford probabilities in base $b$ sum to $1$.
2. $D_b(x) = d$ if and only if $\log_b d \le \{\log_b x\} < \log_b (d+1)$.
3. Strong form $\Rightarrow$ first-digit law, in every base $b \ge 2$.
4. In base $2$ every nonzero number has leading digit $1$.
5. The powers of $2$ satisfy Benford's law.
6. The Fibonacci numbers satisfy Benford's law.
7. The factorials satisfy Benford's law.
8. Exponential growth or decay $C e^{rt}$ ($r\ne 0$) spends an asymptotic fraction $\log_{10}(1+1/d)$ of its time with leading digit $d$.
9. A scale-invariant first-digit distribution is the Benford distribution.
10. Under the strong form, the first $k$ digits form the integer $n$ with probability $\log_{10}(1+1/n)$.
11. Under the strong form, the $n$-th digit ($n \ge 2$) equals $d$ with probability $\sum_{k=10^{n-2}}^{10^{n-1}-1}\log_{10}\!\left(1+\frac{1}{10k+d}\right)$.
12. That $n$-th-digit probability tends to $1/10$.
13. The Benford first-digit distribution has mean $3.440$ and variance $6.057$ (to three decimals).
14. The reciprocals $1/n$ and the square roots $\sqrt n$ of the natural numbers do not satisfy Benford's law.

Significance

The goal is the cleanest instance of the mechanism behind Benford's law: a sequence obeys the law exactly when the fractional parts of its base-$10$ logarithms are equidistributed modulo $1$, and for $k^n$ those fractional parts are the multiples $n\log_{10}k$ reduced modulo $1$. The milestones cover the rest of the mathematical content of the article: the definition and normalization of the law, the derivation from a log-uniform significand, the generalization to several leading digits and to the $n$-th digit, scale invariance, the continuous-time exponential case, and the negative results for $1/n$ and $\sqrt n$.

All of these results are classical and proved in the literature cited above; the work here is formalizing the known proofs. Related entries already on the platform (a leading-digit / fractional-part characterization and an equidistribution-based Benford criterion, stated with different definitions) may be reusable by solvers.

Difficulty

The goal and milestones 5–7 reduce to equidistribution of a sequence modulo $1$. For $k^n$ and Fibonacci numbers this is Weyl's equidistribution theorem for irrational rotations (Fibonacci additionally needs Binet's formula and an error term that vanishes). Factorials are harder: $\log_{10} n!$ is not a linear sequence, and the classical proof combines Stirling's formula with an equidistribution criterion for sequences whose differences vary slowly. The scale-invariance theorem requires a Fourier-analytic argument on the circle $\mathbb{R}/\mathbb{Z}$. The negative results require showing that a counting ratio oscillates, which needs explicit lower and upper estimates along two subsequences.

Formalization scope

• Digits are defined through real logarithms: $D_b(x) = \lfloor b^{\{\log_b x\}} \rfloor$. Lean's Real.logb takes $\log_b|x|$, so negative numbers get the leading digit of their absolute value, and the junk value $D_b(0) = 1$ occurs. Every statement either restricts to positive (or nonzero) inputs or only uses positive sequences.
• Sequences are indexed from $n=0$; "the sequence $k^1, k^2, \dots$" is encoded as $a_n = k^{n+1}$, the Fibonacci numbers as $F_{n+1}$ and the factorials as $(n+1)!$.
• The asymptotic frequency is a natural density over $\{0,\dots,N-1\}$; "satisfies Benford's law" requires convergence for every digit $1\le d<b$.
• The strong form is encoded as: $X$ measurable, $X>0$ almost surely, and the pushforward of the probability measure under $\omega \mapsto \{\log_b X(\omega)\}$ equals Lebesgue measure restricted to $[0,1)$ (this forces the underlying measure to have total mass $1$).
• Moments are stated to three decimals, matching the source's rounding: $|\mathrm{mean} - 3.440| < 0.0005$ and $|\mathrm{variance} - 6.057| < 0.0005$.
• Not formalized: the empirical claims (tables, applications, statistical tests), Hill's 1995 theorem on random mixtures of distributions, and the Krieger–Kafri entropy explanation; their statements in the source are not precise enough to formalize faithfully.

Reusable infrastructure: leading-digit and significand API, equidistribution of $n\alpha$ modulo $1$ (Weyl), Benford criteria via equidistribution. Contributions toward a general "equidistributed $\{\log_{10} a_n\}$ $\Rightarrow$ Benford" lemma are welcome.

Selected references

• S. Newcomb, Note on the frequency of use of the different digits in natural numbers, American Journal of Mathematics 4 (1881), 39–40. https://doi.org/10.2307/2369148
• F. Benford, The law of anomalous numbers, Proc. Amer. Philos. Soc. 78 (1938), 551–572. https://www.jstor.org/stable/984802
• R. S. Pinkham, On the distribution of first significant digits, Ann. Math. Statist. 32 (1961), 1223–1230. https://doi.org/10.1214/aoms/1177704862
• R. L. Duncan, An application of uniform distribution to the Fibonacci numbers, Fibonacci Quarterly 5 (1967), 137–140. https://doi.org/10.1080/00150517.1967.12431312
• L. C. Washington, Benford's law for Fibonacci and Lucas numbers, Fibonacci Quarterly 19 (1981), 175–177. https://doi.org/10.1080/00150517.1981.12430109
• R. A. Raimi, The first digit problem, American Mathematical Monthly 83 (1976), 521–538. https://doi.org/10.2307/2319349
• T. P. Hill, A statistical derivation of the significant-digit law, Statistical Science 10 (1995), 354–363. https://doi.org/10.1214/ss/1177009869
• T. P. Hill, The significant-digit phenomenon, American Mathematical Monthly 102 (1995), 322–327. https://doi.org/10.1080/00029890.1995.11990578
• A. Berger, T. P. Hill, The mathematics of Benford's law: a primer, Stat. Methods Appl. 30 (2021), 779–795. https://arxiv.org/abs/1909.07527
• Wikipedia, Benford's law. https://en.wikipedia.org/wiki/Benford%27s_law

Motivation

Value-at-Risk (VaR) is the regulatory risk measure for banking and insurance capital: it is the quantile of a loss distribution at a fixed tail probability. Two questions arise when several agents use it. First, risk sharing: if an aggregate loss $X$ is split among agents $i=1,\dots,n$, each measuring its share with its own risk measure, what is the smallest total risk the group can achieve and which split achieves it? Second, model uncertainty: an agent rarely knows the distribution of a loss exactly, and a common response is to evaluate the worst case over all distributions close to an estimate, here in the sense of the Wasserstein distance used in distributionally robust optimization (Esfahani and Kuhn, 2018).

F. Liu, T. Mao, R. Wang and L. Wei, Inf-Convolution, Optimal Allocations, and Model Uncertainty for Tail Risk Measures (Mathematics of Operations Research, 2022; the version used here is the authors' SSRN copy 3490348 of January 16, 2022, ssrn.com/abstract=3490348), combine the two questions. When each agent uses a Wasserstein-robust VaR and the split must be comonotonic (every share increases with the total, the usual requirement in insurance), their Theorem 6 gives the optimal total risk in closed form and shows that one agent should carry the entire loss.

Setting

Fix an atomless probability space $(\Omega,\mathcal F,\mathbb P)$: every event of positive probability contains an event of strictly smaller positive probability. A random variable $X$ is a loss, with distribution function $F_X(x)=\mathbb P(X\le x)$. The domain is $\mathcal X=L^\infty$, the essentially bounded measurable random variables.

For $\alpha\in(0,1)$ the left and right Value-at-Risk are

$$\mathrm{VaR}^L_\alpha(X)=\inf\{x: F_X(x)\ge1-\alpha\},\qquad \mathrm{VaR}^R_\alpha(X)=\inf\{x: F_X(x)>1-\alpha\},$$

so a small $\alpha$ looks far into the tail of the loss. $\Lambda\in\{L,R\}$ selects one of the two.

The Wasserstein metric of order $k\ge1$ between $X$ and $Y$ with quantile functions $F^{-1}$ and $G^{-1}$ is

$$W_k(X,Y)=\Big(\int_0^1|F^{-1}(x)-G^{-1}(x)|^k\,\mathrm dx\Big)^{1/k},$$

and the robust version of a risk measure $\rho$ at radius $\delta>0$ is

$$[\rho]^k_\delta(X)=\sup\{\rho(Y): Y\in L^\infty,\ W_k(Y,X)\le\delta\}.$$

Two random variables $X,Y$ are comonotonic if $(X(\omega)-X(\omega'))(Y(\omega)-Y(\omega'))\ge0$ for all $\omega,\omega'$ in an event of probability one. The comonotonic allocations of $X$ among $n$ agents are

$$\mathbb A^+_n(X)=\{(X_1,\dots,X_n)\in(L^\infty)^n: X_1+\dots+X_n=X,\ X_i \text{ comonotonic with } X\},$$

and the constrained inf-convolution of $\rho_1,\dots,\rho_n$ is $\boxplus_{i=1}^n\rho_i(X)=\inf\{\sum_i\rho_i(X_i):(X_1,\dots,X_n)\in\mathbb A^+_n(X)\}$. An allocation attaining the infimum is optimal.

Formalization targets

Goal: Theorem 6 (p. 31)

Let $X\in L^\infty$ and $\Lambda\in\{L,R\}$.

(i) If $0<\delta_1\le\dots\le\delta_n$ and $\alpha\in(0,1)$, then

$$\mathop{\boxplus}_{i=1}^n[\mathrm{VaR}^\Lambda_\alpha]^1_{\delta_i}(X)=[\mathrm{VaR}^\Lambda_\alpha]^1_{\delta_1}(X)+\sum_{i=2}^n\frac{\delta_i}{\alpha}.$$

(ii) If $1>\alpha_1\ge\dots\ge\alpha_n>0$ and $\delta>0$, then

$$\mathop{\boxplus}_{i=1}^n[\mathrm{VaR}^\Lambda_{\alpha_i}]^1_{\delta}(X)=[\mathrm{VaR}^\Lambda_{\alpha_1}]^1_{\delta}(X)+\sum_{i=2}^n\frac{\delta}{\alpha_i}.$$

In either case $(X,0,\dots,0)$ is an optimal comonotonic allocation.

Milestones

1. Proposition 4(iii), (34)–(35), p. 30. For $k\ge1$, $\delta>0$, $\alpha\in(0,1)$: $[\mathrm{VaR}^L_\alpha]^k_\delta(X)=[\mathrm{VaR}^R_\alpha]^k_\delta(X)=x$, where $x$ is the unique solution of $\int_0^\alpha(x-\mathrm{VaR}^R_u(X))_+^k\,\mathrm du=\delta^k$, and $[\mathrm{VaR}^R_\alpha]^k_\delta(X)\ge\mathrm{VaR}^R_\alpha(X)+\delta/\alpha^{1/k}$.
2. (40), p. 31. With $V_\delta=[\mathrm{VaR}^L_\alpha]^1_\delta$: $V_\delta(Y)\le0$ if and only if $\int_0^\alpha(-\mathrm{VaR}^L_u(Y))_+\,\mathrm du\ge\delta$.
3. (39), p. 31. For $\delta_1\le\delta_2$: $V_{\delta_1}\boxplus V_{\delta_2}(X)\le V_{\delta_1}(X)+\delta_2/\alpha$, through the allocation $(X+\delta_2/\alpha,\,-\delta_2/\alpha)$.
4. (38), p. 31. For $\delta_1\le\delta_2$: $V_{\delta_1}\boxplus V_{\delta_2}(X)=V_{\delta_1}(X)+\delta_2/\alpha$.
5. (41), p. 33. For $\lambda\ge1$ with $\lambda\alpha<1$: $[\mathrm{VaR}^\Lambda_\alpha]^1_\delta\ge[\mathrm{VaR}^\Lambda_{\lambda\alpha}]^1_{\lambda\delta}$ on $L^\infty$.

Significance

The theorem is one of the few explicit solutions of a risk-sharing problem with model uncertainty and non-convex agents. Without uncertainty, the inf-convolution of VaRs is a VaR at the summed level (Theorem 1 of the same paper), and its optimal allocations split the tail of the loss across agents. Under Wasserstein uncertainty and a comonotonicity constraint the structure changes: the optimum concentrates the whole loss on the agent least sensitive to uncertainty when the levels agree, or least sensitive to risk when the radii agree, and the remaining agents hold only cash. In particular the equal split $(X/n,\dots,X/n)$, optimal for identical convex risk measures, is in general not optimal here. Proposition 4(iii) is independently useful: it computes the worst-case VaR over a one-dimensional Wasserstein ball of any order and shows that left and right VaR coincide after robustification.

The results are proved in the paper. As far as is known, none of them has a machine-checked proof; this mission produces the first formal statements and, eventually, proofs. The quantile-based Wasserstein metric, the robust risk measure and the comonotonic inf-convolution are reusable objects for further work on robust risk measures.

Difficulty

The upper bound is a direct computation with one explicit allocation. The lower bound is the hard part. VaR is not convex, so the duality arguments that solve the inf-convolution of convex risk measures do not apply, and the robust VaR is not a distortion risk measure to which the Expected Shortfall formula of Proposition 4(ii) applies. The paper itself notes (p. 30) that the inf-convolution of robust VaRs over all allocations is hard to compute because VaR lacks convexity, and obtains only two-sided bounds there, which are generally not sharp. The explicit formula of Theorem 6 depends on the comonotonicity constraint, on the order $k=1$, and on the agents sharing either the level or the radius; the paper states it only under these assumptions, and the ordering of the radii or levels is necessary for the identity as written, since it singles out the agent who carries the loss. For Proposition 4(iii), the supremum over the ball is a supremum over distributions, and identifying it with a real number requires realising worst-case distributions as random variables on the given space, which is where atomlessness is used.

Formalization scope

Random variables are functions $\Omega\to\mathbb R$; $L^\infty$ is the set of measurable, almost surely bounded functions, with no quotient. The probability measure is atomless in the splitting sense. The quantile function is $F^{-1}_X(x)=\mathrm{VaR}^L_{1-x}(X)$, and the Wasserstein integral is the interval integral over $[0,1]$; for $X,Y\in L^\infty$ its integrand is bounded and monotone on $(0,1)$, so the integral is the paper's. The supremum in the robust measure ranges over $Y\in L^\infty$ only, as in the paper. It is a real supremum: the set is nonempty since it contains $\rho(X)$, and for VaR its boundedness from above is part of the statement of Proposition 4(iii), so no junk value enters. Allocation constraints $\sum_iX_i=X$ hold pointwise. The constrained inf-convolution is valued in the extended reals. The $n=m+1$ agents are indexed by Fin (m+1), so $n\ge1$; the orderings of the radii and of the levels are hypotheses, since without them the identities fail. Theorem 6 is stated for order $k=1$; Proposition 4(iii) for every real $k\ge1$. The single implicit hypothesis made explicit is $\lambda\alpha<1$ in (41), which keeps the VaR level in $(0,1)$.

A formalization in which the robust supremum ranges over all measurable functions, or over an unbounded set where the real supremum is $0$, would make the identities trivial or false; both are excluded by the definitions above.

The development needs: quantile functions of bounded random variables and their monotonicity; the existence on an atomless space of a random variable with a prescribed quantile function; comonotonic additivity of VaR; and the interval-integral calculus for monotone integrands. Contributions of general lemmas on quantile functions and on one-dimensional Wasserstein distances are welcome and reusable beyond this mission.

Selected references

• F. Liu, T. Mao, R. Wang, L. Wei, Inf-Convolution, Optimal Allocations, and Model Uncertainty for Tail Risk Measures, Mathematics of Operations Research, 2022; SSRN version of January 16, 2022. https://ssrn.com/abstract=3490348 , https://doi.org/10.1287/moor.2021.1217
• P. Embrechts, H. Liu, R. Wang, Quantile-based risk sharing, Operations Research 66(4), 2018. https://doi.org/10.1287/opre.2017.1716
• P. Mohajerin Esfahani, D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric, Mathematical Programming 171, 2018. https://doi.org/10.1007/s10107-017-1172-1
• J. Dhaene, S. Vanduffel, M. Goovaerts, R. Kaas, Q. Tang, D. Vyncke, Risk measures and comonotonicity: a review, Stochastic Models 22(4), 2006. https://doi.org/10.1080/15326340600878016