Probability

Why short-horizon outcome windows

Randomized experiments (A/B tests) on ride-sharing marketplaces, recommendation systems and digital-health services assign each arriving user to treatment or control. The assignment changes the system state seen by later users, so observations are not independent: this is temporal interference, or carryover. The difference-in-means estimator ignores it, and in Markovian models its bias can be as large as the treatment effect itself (Farias, Li, Peng, Zheng 2022; Hu, Wager 2022). Corrections based on off-policy evaluation or on differences of $Q$-values assume a stationary environment, while real systems have time-varying dynamics and the experimenter observes a single trajectory.

Johari, Peng and Xing propose the truncated policy gradient (TPG) estimator: each instantaneous outcome in the difference-in-means estimator is replaced by the sum of the next $k+1$ outcomes. The paper shows that this estimates, without bias, the derivative of a truncated policy value, and derives bias and variance bounds, a central limit theorem and a consistent variance estimator in nonstationary Markovian environments. This mission formalizes those statements.

Setting

Time is $t=0,1,\dots,T-1$ (the paper's time $t+1$). The state space $S$ is finite and the action $Z_t\in\{0,1\}$ is control or treatment. For each time $t$ and action $z$ there is a transition matrix $P_z^t$ with $\mathbb P(X_{t+1}=x'\mid X_t=x,Z_t=z)=P_z^t(x,x')$; the matrices may change arbitrarily with $t$, which is the nonstationarity. The initial state $X_0$ has distribution $\rho$. The reward $Y_t$ is drawn from a distribution $R(\cdot\mid X_t,Z_t)$ on $\mathbb R$, independently of everything else given $(X_t,Z_t)$, and $R([-M,M]\mid x,z)=1$ for all $x,z$. Write $r(x,z)=\int y\,R(dy\mid x,z)$.

The policy $\pi_\theta$ assigns treatment with probability $\theta$ at every time, independently of everything else. $\pi_1$ is all-treatment, $\pi_0$ is all-control, and the experiment runs $\pi_{1/2}$. Write $P_\theta^t=\theta P_1^t+(1-\theta)P_0^t$ and $\mathbb E_\theta$ for expectation under $\pi_\theta$. The value function and the global average treatment effect (GATE) are

$$J(\theta)=\frac1T\sum_{t=0}^{T-1}\mathbb E_\theta[Y_t],\qquad \tau=J(1)-J(0).$$

The TPG estimator with truncation size $k$ is

$$\widehat\tau_k=\frac1T\sum_{u=0}^{T-1}\Bigl(\frac{\mathbf 1\{Z_u=1\}}{1/2}-\frac{\mathbf 1\{Z_u=0\}}{1/2}\Bigr)\sum_{t=u}^{\min(u+k,\,T-1)}Y_t ,$$

so $\widehat\tau_0$ is the difference-in-means estimator.

Two quantities control the bounds. The mixing assumption (Assumption 1) asks for $\gamma\in(0,1)$ with $\|f'P_z^t-fP_z^t\|_{\mathrm{TV}}\le\gamma\,\|f'-f\|_{\mathrm{TV}}$ for all $t<T$, all $z$ and all probability vectors $f,f'$ on $S$, where $\|f-f'\|_{\mathrm{TV}}=\tfrac12\|f-f'\|_1$. The kernel deviation is $\delta=\max_{t<T}\max_{x}\|P_1^t(x,\cdot)-P_0^t(x,\cdot)\|_{\mathrm{TV}}$.

Formalization targets

Goal: Theorem 1, bias and variance of the TPG estimator

There are absolute constants $C_1,C_2>0$ such that for every environment, every $T\ge1$, every $0\le k\le T$ and every $\gamma\in(0,1)$ and $M$ as above,

$$\bigl|\mathbb E_{1/2}[\widehat\tau_k]-\tau\bigr|\le C_1\Bigl(\frac{1-\gamma^k}{(1-\gamma)^2}\,\delta^2M+\frac{\gamma^k}{1-\gamma}\,\delta M\Bigr),\qquad \operatorname{Var}_{1/2}(\widehat\tau_k)\le C_2\Bigl(\frac{(k+1)^2M^2}{T}+\frac{\gamma\,(k+1)M^2}{T(1-\gamma)}\Bigr).$$

The constants are left unspecified, as in the paper's $O(\cdot)$ statement, so an improvement of the constants does not invalidate the goal.

Explicit constants: Propositions 5 and 6

For data collected under $\pi_\theta$ with $\theta\in(0,1)$, the paper gives both bounds with explicit constants, for the TPG estimator and for a reward-corrected variant. At $\theta=1/2$, Proposition 5 reads

$$\bigl|\mathbb E[\widehat\tau_k]-\tau\bigr|\le\frac{2\delta M\gamma^k}{1-\gamma}+2\delta^2M\,\Psi_k(\gamma),\qquad \operatorname{Var}(\widehat\tau_k)\le\frac{4M^2(k+1)}{T}\Bigl(2k+3+\frac{4\gamma}{1-\gamma}\Bigr),$$

with $\Psi_k(\gamma)=\sum_{j=0}^{k-2}(j+1)\gamma^j$.

Policy-gradient identities: Propositions 1–4

$J$ and its truncated analogue $J_k$ are differentiable at $1/2$, with derivative equal to the horizon average of differences of ($k$-truncated) $Q$-values, and $\mathbb E_\theta[\widehat\tau^{\mathrm{TPG}}_{\theta,k}]$ equals the corresponding derivative for every $\theta\in(0,1)$. Without the mixing assumption, the untruncated estimator satisfies $|\mathbb E[\widehat\tau_T]-\tau|\le C\,T^2\delta^2M$ and $\operatorname{Var}(\widehat\tau_T)\le2M^2(T+1)$.

Asymptotics: Theorem 2, Lemmas 1–5 and Corollary 1

For fixed $k$, under mixing at all times, a positive limit $\sigma_k^2$ of the normalized variance (Assumption 2) and a Cesàro mean-stability condition (Assumption 3),

$$\sqrt T\,\bigl(\widehat\tau_k-\mathbb E[\widehat\tau_k]\bigr)\Rightarrow\mathcal N(0,\sigma_k^2),\qquad \widehat\Omega_T\xrightarrow{p}\sigma_k^2,$$

where $\widehat\Omega_T$ is a Newey–West estimator with bandwidth $L_T\to\infty$, $L_T=O(T^{1/3})$. The supporting statements are a covariance inequality under strong mixing, the bound $\alpha(h)\le\gamma^{h-1}$ on the mixing coefficients of the experimental process and its consequence for the summands of the estimator, the central limit theorem alone, and consistency of the ideal and of the feasible variance estimators.

What the results give

Theorem 1 separates two error sources: a term of second order in $\delta$ that does not grow with $T$, and a term that decays geometrically in $k$. Together with Proposition 2 it shows what truncation buys: a variance bound of order $(k+1)^2/T$ instead of order $T$, at the price of a bias term of order $\gamma^k$. Theorem 2 supplies standard errors for the estimator from a single nonstationary trajectory.

All results of the paper are proved on paper; none has a machine-checked proof. The work of this mission is the formal proofs of the stated results. The definition layer (a finite-state nonstationary Markov decision process with real rewards, its trajectory law under time-varying randomization, total-variation contraction, strong-mixing windows, a Newey–West estimator) can be reused for other results on experiments with Markovian interference.

Where the difficulty lies

The first idea is to bound $|J'(1/2)-(J(1)-J(0))|$ by the second derivative of $J$ on $[0,1]$. Every pair of times contributes to that second derivative, so this route gives the $T^2\delta^2M$ of Proposition 2 and nothing uniform in the horizon. Theorem 1 is uniform in $T$, and its bound has no term of first order in $\delta$ beyond the truncation term; Proposition 5 shows that such a term is present at every allocation $\theta\ne1/2$. Formal proofs have to handle ordered products of time-varying stochastic matrices and signed measures of total mass zero, with a contraction that is assumed one step at a time.

The variance bound and Theorem 2 concern a dependent, non-identically distributed sequence observed once. Mathlib has no central limit theorem for nonhomogeneous Markov chains and no covariance inequality for strongly mixing sequences; the paper relies on Arlotto and Steele (2016) for the former and on an inequality from Doukhan (1994) for the latter.

Formalization scope

TPG.Env S packages $(P,\rho,R)$ for a finite type S. Kernels are real matrices with nonnegative entries and rows summing to one, indexed by all $t\in\mathbb N$; a statement with horizon $T$ uses only the times below $T$. A trajectory is a function Fin T → S × Bool × ℝ, and Env.trajLaw is its law for an arbitrary sequence of treatment probabilities, built as a finite mixture over state and action paths of product reward laws. $J$, $\tau$, $J_k$ and the $Q$-values are defined from this law, and expectations given $\{Z_u=z\}$ are taken under the conditional measure. Rewards are bounded through the hypothesis E.RewardBound M; the standing assumptions of the paper enter each theorem as hypotheses, and S carries the discrete measurable structure.

Statements require $T\ge1$ and $0\le k\le T$, as in the paper. Asymptotic statements compute the law of each statistic under the horizon-$T$ measure: convergence in distribution is tested against bounded continuous functions, and convergence in probability is stated for each $\varepsilon>0$. The limit $\sigma_k^2$ is a parameter constrained by Assumption 2. For $\theta\notin[0,1]$ the law of $\pi_\theta$ is not a probability law and no statement depends on it. The hypotheses are satisfiable for every $T$, $k$ and $\gamma$: an environment with one state and a nonzero constant reward satisfies all of them, including Assumptions 2 and 3, so no statement is vacuous.

Not included: Definition 1 and Lemma 6 of the paper (Dobrushin coefficients and the Arlotto–Steele theorem on general Borel state spaces), the stationary difference-in-$Q$ discussion of Appendix C, the constant-reward example showing that the variance order is attained, and the numerical studies. Formal statements of those, and auxiliary lemmas about Env.trajLaw, ordered matrix products and total-variation contraction, are welcome contributions.

Selected references

• R. Johari, T. Peng, W. Xing, Estimation of Treatment Effects Under Nonstationarity via the Truncated Policy Gradient Estimator, arXiv preprint, 2025–2026. https://arxiv.org/abs/2506.05308
• V. F. Farias, A. A. Li, T. Peng, A. Zheng, Markovian Interference in Experiments, Advances in Neural Information Processing Systems 35, 2022. https://arxiv.org/abs/2206.02371
• Y. Hu, S. Wager, Switchback Experiments under Geometric Mixing, arXiv preprint, 2022. https://arxiv.org/abs/2209.00197
• A. Arlotto, J. M. Steele, A Central Limit Theorem for Temporally Nonhomogeneous Markov Chains with Applications to Dynamic Programming, Mathematics of Operations Research 41(4), 2016. https://doi.org/10.1287/moor.2016.0784
• P. Doukhan, Mixing: Properties and Examples, Lecture Notes in Statistics 85, Springer, 1994. https://doi.org/10.1007/978-1-4612-2642-0

Motivation

The established IFoA actuarial mathematics syllabus discusses the relevant mathematics (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). CM1 actuarial foundations form the mathematical background. Continuous-time reserves connect mortality and financial discounting to the dynamics of insurance liabilities. The standard Thiele differential equation is a recognised result in actuarial mathematics and describes the effect of mortality death strain, premium income and interest force on prospective policy reserves. A significant missing foundation of the preceding primarily discrete-time missions is the exact analytic relationship between a closed-form constant-force reserve and its differential equation. The Oxford actuarial reserve lectures and the Encyclopedia of Mathematics give independent descriptions of the equation.

Setting

Exact actuarial equivalence is conditional on the stated model assumptions. The previously known actuarial mathematics in the published textbooks motivates these original-derived Lean targets. They remain unproved until real theorem solutions are accepted. Nontrivial cashflow and probability examples exclude vacuous zero-benefit substitutes, and the resulting identities can support future actuarial missions.

An insured is alive at valuation time t, with term insurance expiring at time T. A death benefit B is payable immediately upon death within the contract term. Continuous premiums arrive at annual rate P while the insured is alive. The mortality force μ and financial interest force δ are constant. Survival over u years is exp(-μu), financial discount is exp(-δu), and their product discounts by exp(-(δ+μ)u). The combined force k=δ+μ must be strictly positive for the capstone's conventional closed-form quotient. The prospective reserve at time t is V(t)=(μB-P)(1-exp(-k(T-t)))/k for 0≤t≤T.

Formalization targets

All declarations below are original derived CM1-style formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Continuous survival and discount

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.cm1ForceDiscount_le_one:

$$\delta,t\ge0\Rightarrow v(t)\le1$$

Nonnegative interest force implies ordinary discount factors no greater than one.

Insurance net outgo and prospective reserves

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.cm1ThieleRHS_zero_benefit:

$$B=0\Rightarrow V'=(\delta+\mu)V+P$$

Without insurance benefit, reserve drift includes only premium and liability unwind.

Valuation of continuous benefits

This block contributes 11 theorem targets and 2 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.cm1ThieleContinuousValuation_fundamental:

$$V(T)=0,\quad V(0)=PV_B-PV_P,\quad V(t)\ge0,\quad V'=P+\delta V-\mu(B-V)$$

The capstone proves the exact continuous term reserve issue value, terminal zero, nonnegative on the term and the actual Thiele ODE, not merely a definitional rewriting.

Capstone theorem

The main target is ActuarialValuation.cm1ThieleContinuousValuation_fundamental. The capstone proves the exact continuous term reserve issue value, terminal zero, nonnegative on the term and the actual Thiele ODE, not merely a definitional rewriting.

$$V(T)=0,\quad V(0)=PV_B-PV_P,\quad V(t)\ge0,\quad V'=P+\delta V-\mu(B-V)$$

Significance

The professional actuarial syllabus provides the background to this reusable work (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). The closed form reserve formula can be verified independently through expected benefit and premium present values, and must satisfy the continuous-time insurance reserve balance at every intermediate time. Thiele's equation states V'(t)=P+δV(t)-μ(B-V(t)), where B−V is the death strain at risk. The terminal reserve vanishes at expiry of the pure term insurance. Proving all conditions together guards against wrong signs in mortality costs, incorrect interest accumulation, and omitting the premium stream. This formal special case may be reusable for continuous-time multi-state reserves or variable force extensions, provided their assumptions are stated explicitly.

Difficulty

The formula is valid under a constant mortality force and a constant financial force and is not a universal solution of a time-varying Thiele equation. Real exp differentiation and finite-term time inversion require correct signs. The quotient involves k=δ+μ, so algebraic cancellation requires k nonzero. For nonnegative reserves one additionally needs μB≥P and t≤T. At t=T the closed form vanishes, but no positive term must be inferred after expiry. The ODE must be established by differentiating the actual prospective reserve, not inserting it as an extra axiom. Zero mortality and zero premiums provide legitimate boundary cases.

Formalization scope

The model describes a pure continuous term insurance with immediate death benefit and continuous premium income. Expenses, profit margins, maturity endowments, lapses and stochastic changing forces are excluded. Financial force δ need not individually be nonnegative if the combined k is positive, but for a mortality probability interpretation μ must be nonnegative. The closed-form expected claim and premium PV definitions use the same integrated combined discount factor. The capstone is a nontrivial analytic theorem joining boundary values, PV valuation, reserve nonnegativity and Thiele's linear ODE and is suitable for reuse in richer insurance reserve models.

These are original derived finite/analytic CM1 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

• Encyclopedia of Mathematics, Thiele Differential Equation (2012). https://encyclopediaofmath.org/wiki/Thiele_differential_equation
• University of Oxford Department of Statistics, Actuarial Reserves and Thiele's Differential Equation (2020), lecture notes. https://www.stats.ox.ac.uk/~winkel/bs4b11.pdf
• David C. M. Dickson, Mary R. Hardy and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks (2020), Chapter 7. https://www.kriso.ee/actuarial-mathematics-life-contingent-risks-3rd-db-9781108478083.html
• Institute and Faculty of Actuaries, CM1 Actuarial Mathematics Syllabus (2026). https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf

Motivation

The established IFoA actuarial mathematics syllabus discusses the relevant mathematics (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). CM1 actuarial foundations form the mathematical background. Salary growth, defined contribution funding and defined benefit pension accrual are usually treated in separate actuarial calculations, even though they can be joined through a coherent finite cashflow model. The standard pension formula B=alpha times years of service times pensionable salary and the distinction between final salary and career-average revalued earnings are explained by Dickson, Hardy and Waters. A larger formal mission can prove the exact deterministic contribution rate that finances a salary-linked CARE pension, rather than stopping at a definition of accrued pension. This fills a gap not addressed by earlier tranche factor and commutation missions.

Setting

Exact actuarial equivalence is conditional on the stated model assumptions. The previously known actuarial mathematics in the published textbooks motivates these original-derived Lean targets. They remain unproved until real theorem solutions are accepted. Nontrivial cashflow and probability examples exclude vacuous zero-benefit substitutes, and the resulting identities can support future actuarial missions.

At service year t, salary S(t) is annual pensionable pay. Annual benefit accrual at rate alpha is alpha S(t). The CARE retirement benefit revalues each year's accrual with a tranche-specific real factor R(t), giving B=alpha sum S(t)R(t) over the service period. A pension capitalisation factor a at retirement converts this annual pension to liability L=aB. A DC member or sponsor contributes a fixed fraction c of salary each year, with terminal growth multiplier G(t), producing assets A(c)=c sum S(t)G(t). The relevant salary contribution base F is the last sum. All monetary quantities and multipliers must use the same common terminal valuation date.

Formalization targets

All declarations below are original derived CM1-style formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Salary scales and cumulative service

This block contributes 9 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.cm1FinalSalaryPension_nonneg:

$$B_{\rm FS}\ge0$$

Nonnegative service, salary and accrual rate produce a nonnegative final-salary pension.

Pension accrual and valuation

This block contributes 10 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.cm1CARELiability_add_last:

$$L_{n+1}=L_n+\alpha S_n R_n a_R$$

An additional year of pension accrual adds its own capitalised liability slice.

DC contribution accumulation and matching

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.cm1SalaryPensionFunding_fundamental:

$$c^*\ge0,\quad A_n(c^*)=L,\quad c\ge c^*\Rightarrow A_n(c)\ge L$$

Capstone proves the unique actuarially consistent contribution threshold funding a CARE pension, with nonnegative surplus for rates above the threshold.

Capstone theorem

The main target is ActuarialValuation.cm1SalaryPensionFunding_fundamental. Capstone proves the unique actuarially consistent contribution threshold funding a CARE pension, with nonnegative surplus for rates above the threshold.

$$c^*\ge0,\quad A_n(c^*)=L,\quad c\ge c^*\Rightarrow A_n(c)\ge L$$

Significance

The professional actuarial syllabus provides the background to this reusable work (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). The formal result makes the relation between salary-linked pension promises and their financing explicit. An increasing final-salary pension need not equal a career-average pension, and a DC accumulated account value is not directly an annual pension. CARE revaluation and contribution fund growth can differ significantly, so the unique equivalent rate c=L/F must use the separately defined factors rather than an arbitrary average. The equations can support pension valuation controls, funding reconciliations and salary projection sensitivities. They can become reusable infrastructure for subsequent UK defined benefit funding and DC pension options.

Difficulty

The common time origin for earnings, CARE revaluation and contribution accumulation must be maintained. Each cashflow slice is counted exactly once and the result is not a closed-form salary sum unless growth is genuinely constant. The rate c=L/F is only actuarially meaningful if the accumulation base F is positive; division by zero must not make a false equivalence claim. Zero service, negative growth, increasing salary and nonnegative annuity factors are genuine boundary cases. The capstone must prove monetary matching and the threshold property, not assume either as an axiom.

Formalization scope

The model is finite and deterministic. Salary projection uses a constant growth special case, while salary and revaluation cashflow schedules can vary arbitrarily. It does not pretend that a matching terminal DC fund guarantees an identical DB pension after retirement: the annuity capitalisation factor must be agreed and both schemes may have different risk allocation. The final salary formula is a separate valuation component. The nontrivial root proves a contribution-rate funding threshold and is reusable in future stochastic contribution and employer covenant extensions.

These are original derived finite/analytic CM1 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

• David C. M. Dickson, Mary R. Hardy and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks (2009), Chapter 9. https://doi.org/10.1017/CBO9780511800146
• Cambridge University Press, Actuarial Mathematics for Life Contingent Risks: Pensions (2020), Chapter 1. https://api.pageplace.de/preview/DT0400.9781108787406_A49239377/preview-9781108787406_A49239377.pdf
• Institute and Faculty of Actuaries, CM1 Actuarial Mathematics Syllabus (2026). https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf

Motivation

The established IFoA actuarial mathematics syllabus discusses the relevant mathematics (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). CM1 actuarial foundations form the mathematical background. A pension service table projects active members through competing causes of leaving employment, including retirement, death, disability and withdrawal. A comprehensive valuation must account for those mutually exclusive events without creating or destroying pension scheme members. Dickson, Hardy and Waters Chapter 9 discusses pension service tables and Chapter 8 introduces multiple decrement models. Earlier missions consider a single-year decrement law and mortality variance; this mission develops multi-year cohort conservation and a complete reconciliation between two implementations of cause-specific pension benefit valuation.

Setting

Exact actuarial equivalence is conditional on the stated model assumptions. The previously known actuarial mathematics in the published textbooks motivates these original-derived Lean targets. They remain unproved until real theorem solutions are accepted. Nontrivial cashflow and probability examples exclude vacuous zero-benefit substitutes, and the resulting identities can support future actuarial missions.

An initial active-service cohort l(0) is projected to year N. For each year t, d(t,j) records the nonnegative number of scheme members leaving active service due to cause j during that year, where causes are mutually exclusive. The service-table accounting law is l(t+1)+sum_j d(t,j)=l(t). Individual cause probabilities conditional on active membership are q(t,j)=d(t,j)/l(t) when l(t)>0. The probability of remaining active at year t is l(t)/l(0). The unconditional probability of a specified exit is d(t,j)/l(0). A benefit amount b(t,j) and a discount factor v(t) attach to that exit event. The expected present value can be constructed either directly from unconditional cause exit mass or by multiplying active survival and conditional cause probability.

Formalization targets

All declarations below are original derived CM1-style formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Pension cohort accounting

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.cm1ServiceValid_telescoping:

$$\sum_{t<N}\sum_jd_{t,j}+l_N=l_0$$

All exits during the horizon plus active lives at its end exactly reconcile to the initial cohort.

Cause probabilities and transitions

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.cm1ServiceMassPartition:

$$P(\mathrm{active\ at\ }N)+P(\mathrm{exit\ before\ }N)=1$$

The full pension cohort partitions into all competing causes of exit and eventual continued active service.

Benefit cashflows and service value

This block contributes 10 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.cm1ServiceCohortValuation_fundamental:

$$P(\mathrm{stay})+\sum_{t,j}P(\mathrm{exit}_{t,j})=1,\quad V_{\rm unconditional}=V_{\rm conditional}\ge0$$

The nontrivial capstone simultaneously proves full multi-year service-table probability conservation, consistent cause-specific pension benefit PV and nonnegative liability.

Capstone theorem

The main target is ActuarialValuation.cm1ServiceCohortValuation_fundamental. The nontrivial capstone simultaneously proves full multi-year service-table probability conservation, consistent cause-specific pension benefit PV and nonnegative liability.

$$P(\mathrm{stay})+\sum_{t,j}P(\mathrm{exit}_{t,j})=1,\quad V_{\rm unconditional}=V_{\rm conditional}\ge0$$

Significance

The professional actuarial syllabus provides the background to this reusable work (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). A service table must reconcile deaths, retirements, withdrawals and remaining active membership throughout the projection. The multi-period accounting result says cumulative decrement counts plus terminal active count equal the original cohort; after normalisation, the corresponding probabilities sum to one. This provides an important actuarial QC invariant. The same model supplies a direct payment valuation and a survival-conditioned valuation, which must match if probability denominators and calendar payment dates are consistently defined. Both the conservation invariant and cashflow equality can be reused for scheme funding, actuarial gain analysis and member-level pension cost models.

Difficulty

Each decrement cause must form a disjoint partition and the yearly cohort equation has no additional inflow from new entrants. To divide by remaining active service, survivor counts must be nonzero in each year where conditional rates are used. At the terminal horizon no more cause payment years are included. The exact finite-sum telescope must account for every annual exit once and only once. Negative claim sizes, negative exit counts or zero initial cohorts cannot support the probability interpretation. The capstone deliberately validates both valuation routes rather than obtaining equality by defining the two formulas identically.

Formalization scope

The results model a closed, finite cohort with given competing decrement counts. They do not infer cause-specific hazards from observed data or assume independent causes. Causes are disjoint observed outcomes; the conditional exit probabilities sum with the surviving-in-service probability to one. The service table can later be enhanced by new entrants, state transitions and pension commencement options, but these are outside this precise closed-cohort theorem. The calculations retain common valuation dates and financial units for all benefits. The root establishes probability conservation, PV equivalence and nonnegativity, and is reusable in other pension missions.

These are original derived finite/analytic CM1 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

• David C. M. Dickson, Mary R. Hardy and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks (2009), Chapters 8 and 9. https://www.cambridge.org/core/books/abs/actuarial-mathematics-for-life-contingent-risks/multiple-state-models/BFEB2CD04A3EA012FCB1C7E3A326E9A0
• Institute and Faculty of Actuaries, CM1 Actuarial Mathematics Syllabus (2026). https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf
• Institute and Faculty of Actuaries, CM1 Actuarial Mathematics Syllabus (2026). https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf

Motivation

This work is motivated by the IFoA CM1 financial and life-contingent mathematics syllabus (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). A life-table survival model underpins almost every actuarial life contingent assurance, annuity and pensions result. The earlier actuarial missions concentrate on valuation of given death and survival probabilities. CM1_new_formula.pdf sections on life-table identities, select mortality, fractional-year assumptions, joint lives and contingent annuities motivate a larger proof mission establishing those foundations rather than restating assurance formulas already formalised. The principal application is the correct value of a spouse's reversionary annuity when the original member dies while the beneficiary survives.

Setting

The established CM1 modelling conventions and previously known actuarial identities provide the conceptual background, while the following larger original-derived Lean results still need accepted proofs. The stated nontrivial capstone excludes vacuous equal cashflow or zero-duration substitutes. Each accepted module could become reusable infrastructure for future actuarial formalisation missions.

A life table l(x) records survivor counts at integer ages. Conditional survival and death probabilities divide later survivor counts by a nonzero starting cohort. Select rates use within an explicit initial select period, with ultimate rates from the boundary onwards. Within an integer year UDD fractional survival is linear in elapsed fraction, whereas constant force of mortality gives an exponential survival model; neither assumption can be inferred from the other. For two lifetimes, pX(t) and pY(t) are marginal survival probabilities and pBoth(t) is the joint probability of both lives surviving. A valid joint model obeys 0 ≤ pBoth ≤ each marginal probability, but does not need independence. The difference pY(t)−pBoth(t) is the probability of y surviving while x has died. Finite annuity present values sum positive discount factors multiplied by appropriate survival-event indicators.

Formalization targets

All declarations below are original derived CM1-style formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Life tables and survival composition

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.cm1DeferredDeath_nonneg:

$${}_{k|n}q_x\ge0$$

Valid cohort life counts yield a nonnegative deferred mortality probability.

Select, UDD and constant force

This block contributes 12 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.cm1SelectUltimateRate_zero_select:

$$k=0\Longrightarrow q=q^U$$

A select period of zero years uses ultimate rates immediately.

Joint, last-survivor and reversionary benefits

This block contributes 18 theorem targets and 7 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.cm1ReversionaryValue_fundamental:

$$a_{x|y}\ge0,\ a_{x|y}+a_{xy}=a_y,\ a_{\rm last}+a_{xy}=a_x+a_y$$

The genuine contingent annuity capstone combines positivity, reversionary partition and inclusion-exclusion for any valid dependent joint survival model, not just independent lives.

Capstone theorem

The main target is ActuarialValuation.cm1ReversionaryValue_fundamental. The genuine contingent annuity capstone combines positivity, reversionary partition and inclusion-exclusion for any valid dependent joint survival model, not just independent lives.

$$a_{x|y}\ge0,\ a_{x|y}+a_{xy}=a_y,\ a_{\rm last}+a_{xy}=a_x+a_y$$

Significance

The established actuarial syllabus identifies the relevant relationships and applications (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). Pension spouse benefits, contingent pensions, first death and last survivor assurance contracts all depend on correctly partitioning joint and marginal survival events. Using pX pY for all joint probabilities would silently impose independence, which is neither universally correct nor necessary for the most important reversionary identity. The capstone shows a contingent pension to y can be computed as the y-only annuity minus the joint-life annuity, with correct survival-event assumptions and finite payment dates. It also proves the separate last-survivor inclusion-exclusion relation. These foundational results can serve as accepted dependencies for detailed UK DB spouse entitlement and guarantee missions.

Difficulty

Conditional life-table quotients require a positive or nonzero cohort count at the reference age and intermediate age. A monotone life table is needed for probabilities to lie within zero and one. A survival decomposition cannot assume independence across adjoining conditional periods without precise population conditioning, and multiplicativity of constant-force survival cannot be falsely attributed to UDD within the year. Select and ultimate mortality require disjoint endpoints. The reversionary decomposition must keep both-alive and y-alone events distinct and retain the correct payment dates, while allowing arbitrary joint lifetime dependence. The full capstone is the exact conjunction of substantive annuity identities and a meaningful positivity inequality.

Formalization scope

The model is a finite discrete collection of integer-age and payment-date statements augmented by separate algebraic fractional-age rules. The UDD and constant-force models are explicitly assumed and are not asserted as general mortality laws. Annuities are generic finite discounted expected payments; contract guaranteed payments or calendar pension increases would enter through schedule-specific weights in a later mission. The theorem capstone does not impose independent lifetimes, and the independent product model is kept as an optional separate special case. There is no double-counting of survivor states or need to invent an average life expectancy.

These are original derived finite/analytic CM1 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 and Faculty of Actuaries, CM1 Actuarial Mathematics Syllabus (2026). https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf
• David C. M. Dickson, Mary R. Hardy and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks (2009), chapters on mortality and joint lives. https://www.cambridge.org/core/books/actuarial-mathematics-for-life-contingent-risks/51C0B04099D4DE1D1DDEF9551024133B/listing

Motivation

This work is motivated by the IFoA CM1 financial and life-contingent mathematics syllabus (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). Actuarial life insurance pricing is incomplete without the link between gross premiums, future reserve liabilities, emerging cashflow profits and risk capital requirements. The IFoA CM1 syllabus explicitly includes gross prospective and retrospective reserves, death strain, mortality profit, profit signatures, profit testing for unit-linked business and elimination of negative non-unit profit cashflows by reserve zeroisation. Dickson, Hardy and Waters (2009), Chapter 11 (Emerging costs for traditional life insurance) supplies the broader profit testing theory. We combine these elements into one connected mission with an optimisation property as its capstone.

Setting

The established CM1 modelling conventions and previously known actuarial identities provide the conceptual background, while the following larger original-derived Lean results still need accepted proofs. The stated nontrivial capstone excludes vacuous equal cashflow or zero-duration substitutes. Each accepted module could become reusable infrastructure for future actuarial formalisation missions.

A contract has dated benefit payments, expenses, premiums and probability/discounter weights over a finite horizon N. Gross level premium divides the total expected benefit and expense PV by a nonzero premium annuity factor. A prospective reserve equals expected future outgo less future premiums, and a retrospective reserve uses prior accumulated premiums less outgo on the same valuation basis. Death strain at risk measures the claim amount net of the next reserve release. For unit-linked products the unit fund develops through allocation, charges and investment returns. Non-unit projected cashflow c(t) combines premium and charges less expenses and claims. The non-unit reserve R(t), growth factor g(t)>0 and survival/continuation weight s(t)≥0 determine projected profit Π(t)=c(t)+g(t)R(t)-s(t)R(t+1), with terminal reserve R(N)=0.

Formalization targets

All declarations below are original derived CM1-style formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Gross premium equivalence

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.cm1ExpectedBenefitPV_add_last:

$$B_{N+1}=B_N+b_Nv_Np_N$$

Finite expected benefit PV recurses correctly when one extra payment term is appended.

Prospective reserve and mortality profit

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.cm1MortalityProfit_favourable:

$$a\le e,\ DSAR\ge0\Longrightarrow M\ge0$$

With positive death strain, fewer actual deaths than expected produce favourable mortality profit.

Projected profits and unit-linked funds

This block contributes 10 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.cm1ProfitMargin_nonneg:

$$NPV\ge0,\ PV(P)>0\Longrightarrow m\ge0$$

Profit margin is nonnegative if projected profit is nonnegative and the premium base positive.

Backwards non-unit zeroisation

This block contributes 12 theorem targets and 2 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.cm1ZeroisationOptimality_fundamental:

$$(\forall t<N:\ \Pi_t(R)\ge0)\ \land\ (\forall t\le N:\ R_t\le Q_t)$$

The finite-horizon capstone proves that a backwards-zeroised reserve produces nonnegative projected profits and is pointwise minimal among all nonnegative feasible reserve paths with the same terminal boundary.

Capstone theorem

The main target is ActuarialValuation.cm1ZeroisationOptimality_fundamental. The finite-horizon capstone proves that a backwards-zeroised reserve produces nonnegative projected profits and is pointwise minimal among all nonnegative feasible reserve paths with the same terminal boundary.

$$(\forall t<N:\ \Pi_t(R)\ge0)\ \land\ (\forall t\le N:\ R_t\le Q_t)$$

Significance

The established actuarial syllabus identifies the relevant relationships and applications (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). The zeroisation recursion can be used to eliminate projected negative future non-unit profits while avoiding excessive reserve financing. It is valuable to show that the backwards reserve is not only feasible but pointwise minimal among all competing nonnegative reserve profiles satisfying the same profit and terminal constraints. Such a theorem is an auditable dynamic optimisation guarantee, not merely a spreadsheet check. It also provides a bridge from earlier finite cashflow PV and insurance reserve missions to modern policy profit testing. The intermediate theorem groups separate gross premium equivalence, death strain, reserve valuation, unit funds, discounted profit signatures and zeroisation.

Difficulty

The formal model carefully distinguishes net benefit cashflows and reserve-adjusted profit, and never calls a gross premium formula universally correct without its equivalence assumptions. In zeroisation, g(t) must be strictly positive before division and multiplication inequalities can be reversed, while s(t)≥0 is necessary to compare different next-year reserve requirements. The floor max(0,(sRnext-c)/g) is a genuine feasibility threshold. The minimality result needs backward induction from the same terminal reserve, not a false forward monotonicity argument. Negative non-unit cashflows and zero survival are meaningful cases. The result does not equate the risk discount rate used for net present value with the investment growth rate g(t).

Formalization scope

This is a finite-horizon deterministic expected-cashflow model rather than a fully stochastic regulatory reserve basis. Probability, benefit and expenses schedules enter as given finite arrays. Gross premiums use the simplified actuarial equivalence principle and not arbitrary commission or investment-profit target structures. The non-unit zeroisation capstone is conditional on a reserve path satisfying the specified backward recursion; its existence can be constructed by a later recurrence theorem and should not be replaced by a hidden axiom. All explicitly stated profit constraints and nonnegative reserve bounds are maintained for each pre-terminal valuation date. The theorem is meaningful even when cashflow c(t) is negative.

These are original derived finite/analytic CM1 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 and Faculty of Actuaries, CM1 Actuarial Mathematics Syllabus (2026). https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf
• David C. M. Dickson, Mary R. Hardy and Howard R. Waters, Actuarial Mathematics for Life Contingent Risks: Emerging Costs for Traditional Life Insurance (2009), Chapter 11. https://doi.org/10.1017/CBO9780511800146.012

Motivation

This work is motivated by the IFoA CM1 financial and life-contingent mathematics syllabus (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). Financial mathematics provides the time-value backbone of actuarial pricing and pension liability management. The CM1 formula sheet records effective interest and discount rates, annuity-certain relationships, loan amortisation and bond duration. These should be proved as one connected mathematical construction culminating in the local Redington immunisation conditions, rather than as separate missions of two-line rearrangements. A correct proof handles actual discounted dated cashflows and the consistency of their rate derivatives. The IFoA 2026 CM1 financial mathematics syllabus provides the subject boundaries.

Setting

The established CM1 modelling conventions and previously known actuarial identities provide the conceptual background, while the following larger original-derived Lean results still need accepted proofs. The stated nontrivial capstone excludes vacuous equal cashflow or zero-duration substitutes. Each accepted module could become reusable infrastructure for future actuarial formalisation missions.

Real-valued effective annual interest i is restricted to i greater than minus one where cashflow discount factors need to remain positive. A finite dated cashflow c(k) is payable at time k+1. The annual discount factor is (1+i) raised to the negative integer horizon, implemented as a reciprocal of an accumulated factor. Unit annuities in advance and arrears use dates k and k+1 respectively. The cashflow valuation V(i) is a finite weighted sum. A loan balance at t is the present value of remaining loan installments. Duration and convexity numerators aggregate discounted times and squared-time coefficients of cashflows. Asset and liability streams are compared at a single flat effective rate, so rate derivatives are well-defined on the positive accumulation domain.

Formalization targets

All declarations below are original derived CM1-style formal results. The supporting milestone order builds an intentional mathematical chain to a single root theorem.

Interest rate algebra

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.cm1NominalInterest_scale:

$$j^{(m)}=mj$$

Nominal rate scales the periodic effective interest input.

Annuity certain identities

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.cm1IncreasingImmediate_succ:

$$(Ia)_{n+1}=(Ia)_n+(n+1)v^{n+1}$$

Increasing annuity recurrence uses the next arithmetically increasing payment.

General cashflows and loan balances

This block contributes 10 theorem targets and 2 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.cm1LoanBalance_nonneg:

$$B_t\ge0$$

Remaining positive scheduled loan payments have nonnegative prospective value.

Bond pricing and immunisation

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.cm1RedingtonImmunisation_fundamental:

$$S(i_0)=S'(i_0)=0,\ S''(i_0)>0 \Longrightarrow S(i)\ge0\text{ locally}$$

The genuine Redington second-derivative test gives a local surplus minimum at the matching effective yield; it does not assert global or arbitrary curve immunisation.

Capstone theorem

The main target is ActuarialValuation.cm1RedingtonImmunisation_fundamental. The genuine Redington second-derivative test gives a local surplus minimum at the matching effective yield; it does not assert global or arbitrary curve immunisation.

$$S(i_0)=S'(i_0)=0,\ S''(i_0)>0 \Longrightarrow S(i)\ge0\text{ locally}$$

Significance

The established actuarial syllabus identifies the relevant relationships and applications (https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf). Matching current assets and liabilities without checking interest-rate sensitivity is inadequate. A valuation system should reconcile cashflow present values, annuities, prospective loan balances, duration and convexity, and verify that asset surplus cannot turn negative under sufficiently small yield movements when Redington's strictly positive second-derivative condition holds. These equations are reused in actuarial annuity and pension schemes. The mission formalises genuine financial relationships as a graph: compound accumulation leads to discount factors, then discounted cashflow valuation and rate derivatives, ending with a strict local minimum argument. The latter is more substantial than algebraic definition expansion.

Difficulty

Theorems must preserve distinctions between interest and discount rates and between annuities in advance and arrears. Positive accumulation is essential where inverse factors are cancelled or differentiated. Loan prospective value must include only payments remaining after the valuation date. Duration is a ratio whose denominator must be nonzero. Redington immunisation is a genuinely local twice-differentiable result requiring matched PV, first derivative and strictly positive second derivative at an interior yield. It does not provide protection from arbitrary nonparallel yield-curve shifts or sufficiently large interest-rate movements. A proof must establish smoothness of the finite rational cashflow surplus near i₀, rather than assuming local minimisation as a hidden premise.

Formalization scope

The model is deterministic finite cashflow financial mathematics. Mortality-weighted uncertain payments can later be inserted as expected cashflow amounts. Explicit coupon, redemption, loan and annuity terms share the same discounting convention. The capstone uses the actual finite asset-minus-liability discounted cashflows. This is a flat-yield Redington local test only. It is mathematically strong and may be difficult to complete because the second derivative theorem requires calculus and finite-sum differentiability, but no artificial axiom or proof placeholder may count as a solved result.

These are original derived finite/analytic CM1 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 and Faculty of Actuaries, CM1 Actuarial Mathematics Syllabus (2026). https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf
• 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
• F. M. Redington, Review of the Principles of Life-Office Valuations (1952); contextual discussion and institutional course treatment in CM1 Actuarial Mathematics Syllabus (2026). https://actuaries.org.uk/media/yfnkmkbq/cm1_syllabus-2026-_final-proof.pdf

Motivation

A defined benefit pension commencement lump sum cannot generally be determined by multiplying the pre-commutation pension by a cash conversion factor without considering the remaining pension. HMRC's Pensions Tax Manual PTM063240 (https://www.gov.uk/hmrc-internal-manuals/pensions-tax-manual/ptm063240) establishes the applicable amount for a DB arrangement and gives a simplified exact solution 20fg/(20+3f) when the pension is partially surrendered at commutation factor f. PTM063230 (https://www.gov.uk/hmrc-internal-manuals/pensions-tax-manual/ptm063230) identifies other statutory allowance restrictions. This mission formalises the HMRC arithmetic under explicit assumptions, and then extends the model to separate pension tranche reductions rather than pretending all pensions must use a single commutation factor.

Setting

Let g denote the annual pension before commutation, L the pension commencement cash and f the cash paid per pound of annual pension surrendered. The remaining annual pension is P_B=g−L/f. For the simplified DB arrangement with no disqualifying pension credit and no other relevant linked lump sum, the statutory test becomes 3L≤20P_B. Solving this with f>0 gives the maximum applicable cash L_max=20fg/(20+3f). A real value A can represent an externally determined available allowance, giving the capped amount min(L_max,A). For a collection of tranches, annual reduction x(i) and individual cash factor f(i) generate cash contributions f(i)x(i). The residual annual pension is the sum of P(i)−x(i).

Formalization targets

The following declarations are original derived results, not direct quotations of numbered textbook or statutory theorems. Exact hypotheses and corner cases are fixed by the formal Lean statements.

Fundamental boundary

The named result ActuarialValuation.dbCommutedPension_zero_cash captures the relevant actuarial relationship. Original derived theorem for UK DB pension valuation. If no pension is commuted, the original annual DB pension remains unchanged. In the published mission the exact Lean binders and all positivity conditions determine the actuarial model. This proposition is a new derived finite or real-algebraic statement and is not asserted to be a numbered theorem in a pension statute.

$$L=0\Longrightarrow P_B=g$$

Intermediate valuation relation

The named result ActuarialValuation.dbCappedPCLS_le_max captures the relevant actuarial relationship. Original derived theorem for UK DB pension valuation. The allowance cap cannot increase cash above the base applicable amount. In the published mission the exact Lean binders and all positivity conditions determine the actuarial model. This proposition is a new derived finite or real-algebraic statement and is not asserted to be a numbered theorem in a pension statute.

$$L_{\rm cap}\le L_{\max}$$

Main actuarial capstone

The named result ActuarialValuation.dbPCLSMaximum_fundamental captures the relevant actuarial relationship. Original derived theorem for UK DB pension valuation. The capstone proves the attained basic HMRC cash limit, verifies the allowance-limited cash remains admissible, and retains a nonnegative pension tax headroom. In the published mission the exact Lean binders and all positivity conditions determine the actuarial model. This proposition is a new derived finite or real-algebraic statement and is not asserted to be a numbered theorem in a pension statute.

$$H(L_{\max})=0,\quad0\le L_{\rm cap}\le\min(L_{\max},A),\quad H(L_{\rm cap})\ge0$$

Additional supporting theorems establish cancellations, finite-sum identities, positivity and uniqueness. In all cases the generated formal theorem type rather than an informal diagram is authoritative.

Significance

The equations are financially useful because the HMRC basic limit depends on the annual pension remaining after cash has been commuted. Using the original pension instead of P_B would overstate the permitted amount. HMRC PTM063240 (https://www.gov.uk/hmrc-internal-manuals/pensions-tax-manual/ptm063240) gives the explicit formula and examples, while PTM063230 (https://www.gov.uk/hmrc-internal-manuals/pensions-tax-manual/ptm063230) explains the allowance components of the permitted maximum. The formal result provides a check of cash amount, pension reduction, binding constraint and residual HMRC headroom. Separate tranche cash and residual pension definitions are appropriate when benefit tranches have different commutation factors and noncommutable components. They will support more advanced finite optimisation and pension rights constraints.

Difficulty

Cross multiplication and the residual pension test require f strictly positive. Without this, division by f does not correspond to a legal pension surrender and reversed inequality steps are possible. The formula L_max is required to attain zero HMRC headroom in the simplified case, and every lower nonnegative cash amount must satisfy the same basic maximum relation. Allowance caps must themselves be nonnegative to preserve a nonnegative paid amount. An actual arrangement can also have disqualifying pension credits, other linked lump sums, protected rights, scheme minimum pensions or a finite available lump sum and death benefit allowance. The reduced mathematics must not be misrepresented as covering these cases automatically.

Formalization scope

The Lean model uses exact real arithmetic for the single-factor formula and finite sums for the separate-tranche amounts. It proves basic nonnegativity, monetary identities, headroom monotonicity, the cash upper bound and an allowance-capped admissible solution. Noncommutable GMP and scheme-specific rules are deliberately outside this first formal capstone, with reductions of zero available as an abstract representation of such tranches. The external allowance input does not calculate a member's statutory tax allowances. This limits the formal claims to the HMRC simplified applicable-amount formula, while providing separate tranche arithmetic that a subsequent mission can enrich with survivor minimums, GMP guarantees and more involved cash waterfalls.

Existing actuarial valuation theory gives the financial identities motivating these new formal obligations, but the exact original Lean declarations have no accepted proofs yet. The proposed capstone is nontrivial because arbitrary benefit dates, positivity hypotheses and boundary cases cannot be excluded or replaced with a degenerate zero-cashflow special case. The algebraic components are reusable in future UK scheme valuation and administration missions. For source interpretation, a valuation basis means the explicit package assumptions and a benefit constraint means a scheme-specific rule, not an implicit mathematical axiom.

The pinned Mathlib revision is 0df444a360eaa60ab8c11dca51a86af692955474. All source declarations use targeted imports, with no unrelated namespace opening. An authoring compilation with open sorry targets checks source elaboration but does not prove a mathematical statement. The full actuarial benefit schedule may require further scheme-specific data beyond this theorem graph.

Selected references

• HM Revenue & Customs, Pensions Tax Manual PTM063240, defined-benefit PCLS applicable amount, updated (2026). https://www.gov.uk/hmrc-internal-manuals/pensions-tax-manual/ptm063240
• HM Revenue & Customs, Pensions Tax Manual PTM063230, available allowances, updated (2026). https://www.gov.uk/hmrc-internal-manuals/pensions-tax-manual/ptm063230
• Dickson, Hardy and Waters, Actuarial Mathematics for Life Contingent Risks (2009). https://doi.org/10.1017/CBO9780511800146

Motivation

Defined benefit pension schemes may contain multiple legally different benefit tranches. Their pension increases, retirement ages, survivor benefits, guarantees and death-in-deferment provisions need not be identical. Applying one arithmetic average factor to a total pension would ignore differences in complete present value. The controlled UK DB Retirement Factors Mathematical Framework (2026, Chapter 18, equations 58–60) specifies a separately derived factor for each tranche, and treats a shared factor only when scheme provisions actually permit it. Dickson, Hardy and Waters (2009, Pension mathematics, Chapter 9, https://doi.org/10.1017/CBO9780511800146) explain pension cashflows and service-table valuation. This mission defines formal monetary equivalence for every tranche separately, retaining each actuarial package.

Setting

Take a finite number n of pension tranches. For each index i, P(i) denotes accrued annual pension measured in pounds per year, A(i) the complete early-date immediate retirement package value in annuity years, and D(i) the alternative normal-retirement comparator in the same units. Each package can itself be the sum of discounted expected payments at tranche-specific dates. The individual factor f(i)=D(i)/A(i) is applied only to the original pension in tranche i, giving R(i)=P(i)f(i). Its immediate monetary value is R(i)A(i). Total retirement pension and total actuarial liability must be formed by summing their respective tranche contributions. The model never substitutes a pension-only average factor.

Formalization targets

The following declarations are original derived results, not direct quotations of numbered textbook or statutory theorems. Exact hypotheses and corner cases are fixed by the formal Lean statements.

Fundamental boundary

The named result ActuarialValuation.trancheAnnuityLedgerValue_zero captures the relevant actuarial relationship. Original derived theorem for UK DB pension valuation. A tranche with no ledger payment dates has zero immediate package value. In the published mission the exact Lean binders and all positivity conditions determine the actuarial model. This proposition is a new derived finite or real-algebraic statement and is not asserted to be a numbered theorem in a pension statute.

$$A_i(0)=0$$

Intermediate valuation relation

The named result ActuarialValuation.trancheSchemePension_zero captures the relevant actuarial relationship. Original derived theorem for UK DB pension valuation. No tranches means no payable scheme pension. In the published mission the exact Lean binders and all positivity conditions determine the actuarial model. This proposition is a new derived finite or real-algebraic statement and is not asserted to be a numbered theorem in a pension statute.

$$P_{\rm total}(0)=0$$

Main actuarial capstone

The named result ActuarialValuation.separateTrancheFactors_fundamental captures the relevant actuarial relationship. Original derived theorem for UK DB pension valuation. The capstone asserts per-tranche and aggregate monetary equivalence simultaneously, without computing or applying a common averaged factor. In the published mission the exact Lean binders and all positivity conditions determine the actuarial model. This proposition is a new derived finite or real-algebraic statement and is not asserted to be a numbered theorem in a pension statute.

$$(\forall i<n:\ P_if_iA_i=P_iD_i)\land V_{\rm adjusted}=V_{\rm comparator}$$

Additional supporting theorems establish cancellations, finite-sum identities, positivity and uniqueness. In all cases the generated formal theorem type rather than an informal diagram is authoritative.

Significance

Tranche-specific monetary valuation is crucial for pension administration, where two pensions with identical nominal amounts may carry different spouse percentages, indexation and normal retirement dates. The actuarial cost of each unit of pension can consequently vary, and the conditions for equivalence differ. Dickson, Hardy and Waters, pension mathematics (2009, https://doi.org/10.1017/CBO9780511800146), provides the standard actuarial valuation background, while the UK retirement-factor framework 5.3.0 demonstrates the particular multi-tranche application. The resulting identity is useful in migration audits, administering benefit rectifications, reconciling aggregate valuation, and checking that rounding or source ordering has not silently changed benefit values. Separate original benefit rights remain authoritative.

Difficulty

The denominator A(i) must not vanish when cancelling the value ratio. In particular, the real-number convention that division by zero has a defined total Lean value does not make its ordinary actuarial quotient meaningful. Even when total monetary values happen to match, separate per-tranche equality cannot be inferred without additional assumptions. The formal task must retain all individual tranche indices and not count a benefit twice. Zero pension tranches, empty pension schedules and appending another tranche are meaningful cases, rather than mathematical errors. Every unit package must be valued on its own scheme-rule basis, including its relevant discount and contingent cashflows.

Formalization scope

The declarations use exact real arithmetic and finite sums. In particular, the discount, survival, pension increases, guarantee and survivor entitlement details may be assembled into each package ledger as inputs. Their stochastic derivation is an additional task and not silently asserted by a generic real-valued annuity factor. The central identity is P(i)f(i)A(i)=P(i)D(i), and the whole scheme result sums these monetary equivalence equations over finite tranches. The derived calculation remains exact before administration rounding, with no common average factor theorem and no claim that all tranches have the same normal retirement age. It is a reusable base for more detailed pension option missions.

Existing actuarial valuation theory gives the financial identities motivating these new formal obligations, but the exact original Lean declarations have no accepted proofs yet. The proposed capstone is nontrivial because arbitrary benefit dates, positivity hypotheses and boundary cases cannot be excluded or replaced with a degenerate zero-cashflow special case. The algebraic components are reusable in future UK scheme valuation and administration missions. For source interpretation, a valuation basis means the explicit package assumptions and a benefit constraint means a scheme-specific rule, not an implicit mathematical axiom.

The pinned Mathlib revision is 0df444a360eaa60ab8c11dca51a86af692955474. All source declarations use targeted imports, with no unrelated namespace opening. An authoring compilation with open sorry targets checks source elaboration but does not prove a mathematical statement. The full actuarial benefit schedule may require further scheme-specific data beyond this theorem graph.

Selected references

• Dickson, Hardy and Waters, Actuarial Mathematics for Life Contingent Risks, Cambridge University Press (2009). https://doi.org/10.1017/CBO9780511800146
• UK DB Retirement Factors Project, Mathematical Framework v5.3.0, controlled pension valuation manuscript (2026). Background: https://doi.org/10.1017/CBO9780511800146

Motivation

Retiring earlier changes the number of pension payments and may introduce payments before normal retirement age. Retiring later removes some payments but may entitle the member to an uplift and missed increases. A defensible actuarial equivalence calculation compares complete benefit packages, including contingent benefits, at the same valuation date. The UK DB Retirement Factors Mathematical Framework 5.3.0 (2026, Chapters 8 to 11) explicitly treats early and late benefit routes, guaranteed periods, pension increases and pre-commencement death. Dickson, Hardy and Waters (2009, Chapter 9, https://doi.org/10.1017/CBO9780511800146) provide independent actuarial pension valuation background. This mission first formalises a carefully delimited factor-independent death-benefit model and retains explicit differences between pure and total late-retirement factors.

Setting

For early retirement, A_E is the immediate retirement package, D_E the normal retirement comparator, and F_E=D_E/A_E the unrounded reduction factor. The comparator may contain a death-before-retirement component plus survival-to-normal-age probability p times an appropriate discount v and normal pension annuity A_N. For late retirement N is the value of the benefit package at normal retirement, D the value of death benefits payable while commencement is postponed, and A_L the immediate package at the late date. The positive multiplier m represents pension increases that would otherwise have accrued while in payment. The pure LRF F_L solves N=D+pvm A_L F_L. The total pension multiplier M_L=mF_L includes both late uplift and missed increases.

Formalization targets

The following declarations are original derived results, not direct quotations of numbered textbook or statutory theorems. Exact hypotheses and corner cases are fixed by the formal Lean statements.

Fundamental boundary

The named result ActuarialValuation.pensionEarlyComparator_nonneg captures the relevant actuarial relationship. Original derived theorem for UK DB pension valuation. Nonnegative disjoint death and surviving deferred-annuity route values yield a nonnegative comparator. In the published mission the exact Lean binders and all positivity conditions determine the actuarial model. This proposition is a new derived finite or real-algebraic statement and is not asserted to be a numbered theorem in a pension statute.

$$D_E\ge0$$

Intermediate valuation relation

The named result ActuarialValuation.pensionLatePureFactor_nonneg captures the relevant actuarial relationship. Original derived theorem for UK DB pension valuation. The net value available for deferred pension is nonnegative once factor-independent death costs are deducted. In the published mission the exact Lean binders and all positivity conditions determine the actuarial model. This proposition is a new derived finite or real-algebraic statement and is not asserted to be a numbered theorem in a pension statute.

$$N\ge D,\ pv m A>0\Longrightarrow F_L\ge0$$

Main actuarial capstone

The named result ActuarialValuation.pensionRetirementEquivalence_fundamental captures the relevant actuarial relationship. Original derived theorem for UK DB pension valuation. The capstone validates both early and late monetary equivalence and proves that the pure factor and total missed-increase multiplier represent the same late pension liability. In the published mission the exact Lean binders and all positivity conditions determine the actuarial model. This proposition is a new derived finite or real-algebraic statement and is not asserted to be a numbered theorem in a pension statute.

$$F_E A_E=D_E,\quad D+pv\,m A_LF_L=N,\quad D+pv A_L M_L=N$$

Additional supporting theorems establish cancellations, finite-sum identities, positivity and uniqueness. In all cases the generated formal theorem type rather than an informal diagram is authoritative.

Significance

The distinction prevents a common implementation error: treating the total missed-increase pension multiplier as the pure actuarial LRF and then applying increases a second time. The financial basis and cashflow boundaries must be stated before taking an algebraic ratio. The UK retirement factor methodology v5.3.0 details the pension ledger and explicitly separates pure and total late multipliers. Pension mathematics and contingent annuities in Dickson, Hardy and Waters (2009, https://doi.org/10.1017/CBO9780511800146) explain the broader framework. The resulting small formal theorem can be combined with exact-dated benefit ledgers and numerical models to test early and late valuation reconciliation before rounding. That reuse is important when individual tranches have different dates and benefit conditions.

Difficulty

Both ratio definitions need nonzero annuity packages or survival-weighted denominators. If normal survival probability is zero, a late pension cannot compensate financially in the ordinary quotient model, and Lean division must not hide that failure. The death benefit during late postponement is assumed independent of the eventual pure factor, an explicit limitation. Factor-dependent death benefits require solving a different potentially nonlinear root equation; simply dividing a fixed numerator would be wrong. Increases accrued while waiting are distinct from in-payment increases already present in A_L. All route components are valued at a common date, and different death and survival outcomes are treated as disjoint events.

Formalization scope

The theorem graph establishes early comparator positivity, factor existence and uniqueness, and exact equivalence when an annuity denominator is nonzero. It separately proves late pension value and uniqueness under the corresponding nonzero survival-discount-increase-annuity product. The capstone reconciles the early option and the two equivalent late expressions: using the pure late factor and using the total multiplier mF_L. The underlying cashflow package values are abstract real inputs, which enables later missions to supply actual discounted mortality-weighted schedules. The current restricted identity does not claim to model every dependant, guarantee, calendar increase or factor-linked death benefit. Scheme-specific cashflow rights remain controlling.

Existing actuarial valuation theory gives the financial identities motivating these new formal obligations, but the exact original Lean declarations have no accepted proofs yet. The proposed capstone is nontrivial because arbitrary benefit dates, positivity hypotheses and boundary cases cannot be excluded or replaced with a degenerate zero-cashflow special case. The algebraic components are reusable in future UK scheme valuation and administration missions. For source interpretation, a valuation basis means the explicit package assumptions and a benefit constraint means a scheme-specific rule, not an implicit mathematical axiom.

The pinned Mathlib revision is 0df444a360eaa60ab8c11dca51a86af692955474. All source declarations use targeted imports, with no unrelated namespace opening. An authoring compilation with open sorry targets checks source elaboration but does not prove a mathematical statement. The full actuarial benefit schedule may require further scheme-specific data beyond this theorem graph.

Selected references

• Dickson, Hardy and Waters, Actuarial Mathematics for Life Contingent Risks, Cambridge University Press (2009). https://doi.org/10.1017/CBO9780511800146
• UK DB Retirement Factors Project, Mathematical Framework v5.3.0, controlled retirement factors manuscript (2026). Background: https://doi.org/10.1017/CBO9780511800146

Motivation

The Bühlmann–Straub credibility method allocates weight to individual observed claims experience according to total exposure and variability. It balances the expected value of process variance EPV with the variance of hypothetical means VHM. More exposure raises the statistical information carried by the policyholder's observed rate and should weakly improve the resulting optimal linear prediction error. The established credibility formula and its interpretation are developed in Loss Data Analytics (https://openacttexts.github.io/LDAVer2/ChapCredibility.html). The new mission formalises how these exact deterministic weight and risk formulae change with exposure.

Setting

Let P be a nonnegative real representing total risk exposure. EPV is strictly positive, while VHM is nonnegative and may equal zero for a homogeneous risk class. The credibility factor Z(P) is P times VHM divided by the sum of P times VHM and EPV. A credibility premium combines the policyholder's experience rate with the collective mean using weights Z(P) and 1-Z(P). A normalised optimal quadratic prediction error is EPV times VHM over the same positive denominator. A scaled quadratic expression captures the trade-off between random process noise and between-risk heterogeneity.

Formalization targets

The formal targets progress from structural boundary identities to financial sensitivity and the full capstone. Supporting theorems remain genuine Open declarations until accepted Lean proofs are submitted; no previously accepted proof is silently assumed.

Boundary identity

The exact theorem name is ActuarialValuation.exposureCredibilityWeight_zero. Without any exposure, the experience credibility weight must be zero, provided the process variance is strictly positive.

$$\mathrm{EPV}>0\Longrightarrow Z(0)=0$$

Structural intermediate target

The exact theorem name is ActuarialValuation.exposureCredibilityPremium_between. As the credibility weight lies in the unit interval, a lower observed claim rate produces a premium bounded by the observed and collective rates.

$$\bar x\le\mu\Longrightarrow \bar x\le\widehat\mu(P)\le\mu$$

Main capstone

The exact theorem name is ActuarialValuation.exposureCredibilityAccuracy_fundamental. The capstone reconciles exposure-driven increases in credibility with the monotone decline of minimum expected squared error, including the zero heterogeneity boundary.

$$Z(P_1)\le Z(P_2),\quad G(P_1,P_2)\ge0,\quad R_{\min}(P_2)\le R_{\min}(P_1)$$

The graph also contains separately stated supporting inequalities and exact definitions, with the intended original derived mathematical identity written out in every milestone.

Significance

Experience-rating systems can be checked for sensible monotonicity when an additional policy year or risk exposure enters the rating base. Larger exposure should not decrease the weight assigned to otherwise unchanged policy experience and should not increase the theoretically optimal error. The classical structure of Bühlmann credibility and its use of EPV and VHM are described by Loss Data Analytics (https://openacttexts.github.io/LDAVer2/ChapCredibility.html). These precise algebraic statements are reusable for downstream weighted risk models, rating systems and numerical sensitivity checks.

Difficulty

Credibility is a rational function whose denominator could vanish under an invalid zero-process-variance model. The hypotheses impose strictly positive EPV and retain zero VHM as a legitimate boundary. Monotonicity is therefore weak, not strict, since a homogeneous portfolio provides no justification for relying more on experience. Comparisons between two exposures must use the same structural parameters, otherwise a change of class could reverse the effect. The theorem relating quadratic risk and the credibility weight is a genuine completing-square identity, while the mission does not assert that an arbitrary Bayesian posterior has linear form.

Formalization scope

Lean uses exact real arithmetic for exposure, EPV, VHM, premium and risk formulae. The finite statistical model is reduced to its deterministic structural-parameter consequences; estimation of the parameters and robustness of individual claims are outside the scope. The theorem graph begins with positivity and zero-exposure boundaries, continues through weight and premium identities, and ends with a conjunction linking increasing credibility to decreasing minimal prediction error. The mathematical framework is known, but the newly stated Lean specialisations have to be proved and exclude vacuous denominator cases.

The pinned Mathlib revision and the precise Lean type of each definition or proposition are authoritative. The imported source must use targeted Mathlib modules; one large aggregate compiler check never substitutes for testing individual definition imports and obsolete namespace openings. The source-specific intermediate statements separate units and hypotheses. The finite index always counts claims, scenarios, or observations according to the declaration, not an implicit probability measure. In an insurance pricing setting the difference between a gross loss, a ceded payment, a net premium and a credibility risk estimate is financially important. A positive coefficient does not by itself imply the coefficients sum to one. Likewise a deterministic algebraic identity does not grant stochastic independence or a forecast model. These restrictions preserve the distinction between a purely algebraic result and a probabilistic interpretation. The formal theorem declarations have to maintain exactly the same quantifiers, arithmetic operations and endpoint conventions as the descriptions, including meaningful boundary cases. Future extensions can introduce richer distributions and dependence without altering the precise finite identities established here.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015, printed page 489. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Frees and colleagues, Loss Data Analytics: Experience Rating Using Credibility Theory, 2024. https://openacttexts.github.io/LDAVer2/ChapCredibility.html

Motivation

The stop-loss transform is the expected value of the positive excess of a claim above an attachment, an important building block of actuarial risk ordering. Increasing retention should never increase an otherwise unchanged pure reinsurance premium. A second useful criterion is that the decrease in price should not exceed the increase in retention, once scenario probabilities are normalised. Insurance mathematics describes stop-loss premiums as decreasing functions of the deductible (https://www.researchgate.net/publication/4958878_The_Concept_of_Comonotonicity_in_Actuarial_Science_And_Finance_Theory). This mission additionally identifies their convexity in retention and develops finite verification targets for the exact bound.

Setting

A real loss X(s), a real attachment d and a real scenario weight w(s) determine a positive excess payment max(X(s)-d,0). The pure stop-loss premium at d is the weighted finite sum across s in the range from zero through bound B. Scenario weights are nonnegative for premium-order and convexity theorems. Their finite total W may be arbitrary, while W=1 expresses a genuine probability distribution and yields a sharp unit Lipschitz constant. Two attachments a and b can be combined by d(theta)=(1-theta)a+theta b, with interpolation weight theta between zero and one.

Formalization targets

The formal targets progress from structural boundary identities to financial sensitivity and the full capstone. Supporting theorems remain genuine Open declarations until accepted Lean proofs are submitted; no previously accepted proof is silently assumed.

Boundary identity

The exact theorem name is ActuarialValuation.retentionExcessPositive_nonneg. An excess loss payment cannot be negative even for signed real inputs, since the positive part is the maximum against zero.

$$(x-d)_+\ge0$$

Structural intermediate target

The exact theorem name is ActuarialValuation.retentionStopLossPremium_zero_weight. With no scenario mass, a finite net premium is zero regardless of loss amounts and attachment.

$$w_s\equiv0\Longrightarrow\Pi(d)=0$$

Main capstone

The exact theorem name is ActuarialValuation.retentionStopLossSensitivity_fundamental. The capstone states the three structural constraints on an actuarially valid premium curve, including convex interpolation, when finite scenario weights are normalized.

$$0\le\Pi(a)-\Pi(b)\le b-a,\qquad\Pi(d_\theta)\le(1-\theta)\Pi(a)+\theta\Pi(b)$$

The graph also contains separately stated supporting inequalities and exact definitions, with the intended original derived mathematical identity written out in every milestone.

Significance

These conditions provide powerful reasonableness checks for treaty pricing functions. An actuary can diagnose a premium that rises with retention, or an implausibly large fall inconsistent with maximum ceded-payment change, before relying on a stochastic model. The classical stop-loss transform is extensively studied in insurance finance (https://www.researchgate.net/publication/4958878_The_Concept_of_Comonotonicity_in_Actuarial_Science_And_Finance_Theory). The reusable finite inequalities connect claim severity, scenario weights and contract terms, and can support downstream optimisation, layered reinsurance sensitivity and refinement of discrete tail distributions.

Difficulty

The positive part is a maximum of two affine functions, which is convex but not differentiable when the claim equals the retention. The finite theorem must not assume every claim is above a or b: a distribution can straddle both attachment points. The premium derivative is not required, since pointwise algebraic inequalities suffice to justify convex interpolation and monotone order. The mass-dependent Lipschitz bound becomes one only when total mass is one, so probability normalisation must be stated. Claim severity need not be nonnegative for the algebraic result, while weights must be nonnegative for order preservation.

Formalization scope

The Lean representation uses real numbers for claim severities, deductibles, masses and convex interpolation. Finite scenarios are represented by natural indices and Finset.range. Positive excess is a maximum of two real numbers. All inequalities are original derived finite specialisations of well-known stop-loss facts, with explicit mathematical types and numerical boundary conventions. The formalisation intentionally excludes continuous distributions, derivatives, infinite-dimensional weak convergence and loaded commercial premium assumptions. The main goal is nontrivial because it demands the general weighted convexity and Lipschitz properties for an arbitrary finite loss table.

The pinned Mathlib revision and the precise Lean type of each definition or proposition are authoritative. The imported source must use targeted Mathlib modules; one large aggregate compiler check never substitutes for testing individual definition imports and obsolete namespace openings. The source-specific intermediate statements separate units and hypotheses. The finite index always counts claims, scenarios, or observations according to the declaration, not an implicit probability measure. In an insurance pricing setting the difference between a gross loss, a ceded payment, a net premium and a credibility risk estimate is financially important. A positive coefficient does not by itself imply the coefficients sum to one. Likewise a deterministic algebraic identity does not grant stochastic independence or a forecast model. These restrictions preserve the distinction between a purely algebraic result and a probabilistic interpretation. The formal theorem declarations have to maintain exactly the same quantifiers, arithmetic operations and endpoint conventions as the descriptions, including meaningful boundary cases. Future extensions can introduce richer distributions and dependence without altering the precise finite identities established here.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015, printed page 409. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Denuit and colleagues, The Concept of Comonotonicity in Actuarial Science and Finance: Theory, 2002. https://www.researchgate.net/publication/4958878_The_Concept_of_Comonotonicity_in_Actuarial_Science_And_Finance_Theory

Motivation

Sampling an entire finite-state execution is a different computational task from recognizing a supplied word. A membership circuit must answer correctly on its given input. A sampler instead chooses the input together with its associated output, while preserving their joint distribution. This distinction is relevant when finite-state models describe allowable executions and when parallel generation should avoid a sequential pass through every transition. The source paper establishes constant-depth sampling results for regular languages and finite Markov chains in Appendix E (Zhang et al., 2026).

The mission concerns that appendix's finite-state sampling results. It formalizes all six numbered results E.1–E.6, together with the seed bounds stated in the surrounding text. Each milestone points to a published theorem with a separately checked proof. The objects to sample are complete words, word–acceptance pairs, and complete state trajectories. Sampling only the final state would lose the joint temporal information required by the targets.

Finite-state models and fair-bit circuits

A binary deterministic finite automaton has finitely many states, a designated start state, a transition for each state and bit, and a set of accepting states. Starting from its initial state, a binary word of length $n$ determines a run. Let $L$ be the set of words whose final state is accepting. Put $L_n=L\cap\{0,1\}^n$. An arbitrary such automaton represents an arbitrary binary regular language, with no assumption that membership belongs to $\mathrm{AC}^0$.

A finite Markov chain consists of a nonempty finite state set $Q$, an initial probability vector $\mu$, and a row-stochastic transition matrix $P$. Entries are nonnegative real numbers, not restricted to rational numbers. For $\gamma=(x_0,\ldots,x_n)$, its probability is $p_M(\gamma)=\mu(x_0)\prod_{i=1}^{n}P(x_{i-1},x_i)$. The distribution of all $n+1$ states is denoted $\mathrm{Path}_n(M)$. For $F\subseteq Q$, conditioning is allowed when the probability that $X_n\in F$ is positive.

A randomized $\mathrm{AC}^0$ sampler is a deterministic Boolean circuit supplied with independent, unbiased random bits. AND and OR gates have unbounded fan-in; NOT gates and Boolean constants are permitted. Its size is polynomial and its depth has a constant bound independent of trajectory length. Approximate families are nonuniform in both $n$ and their real error tolerance $\epsilon$: existence does not require efficiently computing circuit descriptions or numerical tables from a supplied transition matrix.

Formalization targets

The main goal is the conjunction of all results E.1–E.6, including the linear seed guarantee for E.5. This organizational theorem is not an additional numbered result attributed to the paper. All six milestones must be proved for the goal to hold. The exact Markov sampling criterion in Theorem E.5 is one component. A nonnegative real number is dyadic if it equals $a/2^k$ for natural numbers $a,k$, including zero. A chain is path-dyadic when every finite trajectory, including a trajectory of length zero, has a dyadic probability. The target is

$$\mathrm{Path}_n(M)\text{ has an exact randomized }\mathrm{AC}^0\text{ sampler family} \quad\Longleftrightarrow\quad \forall n\geq0\;\forall\gamma\in Q^{n+1},\ p_M(\gamma)\text{ is dyadic}.$$

On the left, the family has polynomial size in $n+1$ and depth bounded independently of $n$. A supporting target records the linear fair-bit budget $O_M(n+1)$ when path-dyadicity holds. Lemma E.4 supplies another target: every path-dyadic chain is the coordinatewise projection of a finite chain whose initial and transition probabilities are all dyadic. Path-dyadicity must not be replaced by the stronger requirement that the original transition matrix itself be dyadic (Appendix E.3).

Theorem E.1 concerns exact sampling of a uniformly random word together with its acceptance indicator:

$$(X,\mathbf 1[X\in L]),\qquad X\sim\operatorname{Unif}(\{0,1\}^n).$$

The sampler uses exactly $n$ fair bits, polynomial size, and depth depending only on the fixed automaton. This is a sampling assertion and does not assert that membership can be computed in constant depth on an arbitrary externally supplied word.

Theorem E.2 concerns the uniform law on nonempty $L_n$. For every $0<\epsilon\leq1/2$, the target is

$$d_{\mathrm{TV}}(Q_{n,\epsilon},\operatorname{Unif}(L_n))\leq\epsilon, \qquad r_{n,\epsilon}\leq n+C_L\log_2(1/\epsilon).$$

Every possible output is accepted. Size is polynomial in $n+1$ and $1/\epsilon$, and depth depends only on the fixed language or its automaton. In particular, the leading coefficient of the $n$ term in the random-bit bound is one (Appendix E.2).

Theorem E.6 is the corresponding approximate target for trajectories conditioned on $X_n\in F$. It requires total variation error at most $\epsilon$, only positive-probability outputs ending in $F$, $O_M(n+\log(1/\epsilon))$ fair bits, polynomial size in $n+1$ and $1/\epsilon$, and depth depending only on the chain. There is no lower bound on the conditioning probability beyond strict positivity (Appendix E.4).

Lemma E.3 supplies the exact representation used in the Markov construction. A transition-dyadic chain is represented by one binary DFA and a fixed state projection. On a uniform input of length $v+ns$, with positive constants $v,s$, the states observed after $v,v+s,\ldots,v+ns$ bits project to exactly the complete trajectory law of the chain. The same DFA and projection work for every length, including zero.

Significance of the targets

These results separate sampling an input together with its finite-state behavior from computing that behavior on a prescribed input. They also identify a precise obstruction to exact sampling with a finite fair-bit seed: each output probability must be dyadic. Approximate sampling allows arbitrary real transition probabilities while retaining the valid trajectory support. The complete trajectory requirement preserves correlations among all time coordinates.

The formalization makes the circuit model, seed accounting, support guarantees and asymptotic quantifiers explicit and reusable. The theorem statements and their separately submitted proofs have passed the platform verifier. The goal proof imports all six milestones; its E.5 milestone in turn imports both the equivalence and the linear seed theorem. Thus the platform dependency graph covers the entire appendix. Platform verification establishes the formal statements; semantic correspondence is documented in the statement descriptions and source references.

Difficulty

A literal step-by-step implementation of a run makes circuit depth grow with $n$, so it does not meet the constant-depth target. Recognition results alone do not furnish the required joint sampler. For approximate word sampling, an error estimate alone is insufficient: the output must remain in the accepted support, and the coefficient-one seed bound must also hold. For exact Markov sampling, unreachable states and non-dyadic individual transitions prevent identifying the desired criterion with entrywise dyadicity. Conditioning on a rare terminal event must not introduce an unstated dependence on its probability.

Formalization scope

States are indexed by finite types $\mathrm{Fin}(q)$. Binary words have type $\mathrm{Fin}(n)\to\mathrm{Bool}$; trajectories have $n+1$ coordinates. Markov vectors and matrices are real-valued and carry normalization and nonnegativity proofs. Sampling probabilities are defined by counting all seeds that give a particular encoded outcome and dividing by $2^r$. Total variation is half the finite sum of absolute probability differences.

The circuit representation is an explicit topologically ordered gate list with checked wire references. Size counts input bits, gates, gate-input wires and output wires; this yields the usual polynomial-size class. State trajectories use a fixed one-hot encoding at each time coordinate. The invalid-output predicate is checked over every seed, so neither arbitrary decoders nor unrepresented outcomes can satisfy an exact or support-preserving claim. The encodings used by the targets are injective. The circuit family is existentially quantified after the fixed finite-state model; the depth and polynomial constants precede $n$ and $\epsilon$.

Zero-length words and zero-transition Markov paths are included. The word target excludes empty accepted slices. The conditioned target excludes zero-probability conditioning events. Definitions of division by zero still exist as total Lean functions, but the hypotheses exclude those cases from approximate claims. Extensions of these results, reusable finite-state probability lemmas, and further circuit constructions are welcome. No sampler-existence axiom, vacuous model or final-state-only replacement is part of the interface.

Selected references

• Jiarui Zhang, Chengwei Liang, Haozhe Jiang, Binhang Yuan and Jingzhao Zhang. Revision Provably Reduces Sequential Computation in Diffusion Language Models. Zenodo preprint, 2026. DOI 10.5281/zenodo.23105365, Appendix E, Theorems E.1, E.2, E.5, E.6 and Lemmas E.3, E.4, printed pages 33–36.

Motivation

Per-occurrence excess and aggregate excess have different financial triggers. One aggregate deductible applies once to the sum of claims, while a per-risk retention is charged against each loss. Combining many small claims may exhaust an aggregate deductible without producing a payment under any separate per-claim policy. HMRC explains how non-proportional reinsurance differs across individual risks and catastrophe or aggregate protections (https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8060). The valuation question is how the ceded amounts and their pure expected costs are ordered when otherwise identical annual experience and deductibles are used.

Setting

The individual claims x(i), their finite number n, and the deductible d are natural-number monetary quantities. Gross aggregate loss S(n) is the sum of the first n claims. Aggregate recovery is positive excess of S(n) above d; separate per-claim recovery is the sum of each claim's positive excess over d. Finite scenario weights w(s) are real. A scenario maps to an entire loss sequence, and its total claim portfolio is evaluated consistently under both treaty designs. Weighted claims are aggregated for indices from zero to a finite bound B, allowing the scenario count itself to be zero or the probability weights to lack normalisation.

Formalization targets

The formal targets progress from structural boundary identities to financial sensitivity and the full capstone. Supporting theorems remain genuine Open declarations until accepted Lean proofs are submitted; no previously accepted proof is silently assumed.

Boundary identity

The exact theorem name is ActuarialValuation.aggregateLossTotal_zero. An empty collection of claims has zero gross claims and therefore defines the base case for deductible comparisons.

$$S_0=0$$

Structural intermediate target

The exact theorem name is ActuarialValuation.aggregateDeductibleCeded_zero_attachment. With attachment zero, the aggregate excess treaty covers every unit of gross annual claims.

$$C_{\mathrm{agg}}(0)=S_n$$

Main capstone

The exact theorem name is ActuarialValuation.aggregateVsPerClaim_fundamental. The capstone joins the nontrivial pointwise reinsurance coverage inequality to the same ordering of expected pure premiums, without conflating a whole-year treaty with a per-claim deductible.

$$C_{\mathrm{individual}}\le C_{\mathrm{agg}},\quad0\le\Pi_{\mathrm{agg}},\quad\Pi_{\mathrm{individual}}\le\Pi_{\mathrm{agg}}$$

The graph also contains separately stated supporting inequalities and exact definitions, with the intended original derived mathematical identity written out in every milestone.

Significance

The comparison can diagnose material pricing errors, since an aggregate deductible and a per-risk deductible with the same quoted monetary value should not normally have the same technical premium. HMRC's treatment of non-proportional reinsurance (https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8060) makes this distinction a contractual reality. The formal theorem is useful for ceded reserve reconciliation, claims-data audit, treaty design, and future models including several occurrence covers within a separate annual limit. In each case exact mathematical relationships are more valuable than approximate simulation agreement.

Difficulty

The algebra of positive-part natural subtraction differs from ordinary subtraction over integers or real values. The aggregate deductible is taken once, whereas the per-event premium subtracts the same amount separately from each positive claim. The theorem is a weak inequality and must remain valid even when every event is below the separate retention but the aggregate exceeds the annual attachment. The model explicitly includes zero claims, zero deductible, empty portfolios and nonnegative scenario weights. Signed weights would make the expectation order invalid and are not silently permitted in that result.

Formalization scope

Lean treats all claim amounts, event counts, attachment values and scenario bounds as natural numbers. Real numbers occur only in the finite expected ceded amounts. Each individual-claim sum uses Finset.range and the pure premium is a further finite sum over scenarios. This is an exact deterministic relationship between two reinsurance structures rather than an assumption of independence or identical event distributions. No catastrophe frequency model, claims adjustment expenses or risk load is included. The established contractual idea motivates original derived finite relations, and the formal statements remain open until proved.

The pinned Mathlib revision and the precise Lean type of each definition or proposition are authoritative. The imported source must use targeted Mathlib modules; one large aggregate compiler check never substitutes for testing individual definition imports and obsolete namespace openings. The source-specific intermediate statements separate units and hypotheses. The finite index always counts claims, scenarios, or observations according to the declaration, not an implicit probability measure. In an insurance pricing setting the difference between a gross loss, a ceded payment, a net premium and a credibility risk estimate is financially important. A positive coefficient does not by itself imply the coefficients sum to one. Likewise a deterministic algebraic identity does not grant stochastic independence or a forecast model. These restrictions preserve the distinction between a purely algebraic result and a probabilistic interpretation. The formal theorem declarations have to maintain exactly the same quantifiers, arithmetic operations and endpoint conventions as the descriptions, including meaningful boundary cases. Future extensions can introduce richer distributions and dependence without altering the precise finite identities established here.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015, printed page 389. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• HM Revenue & Customs, General Insurance Manual: Non-proportional reinsurance, GOV.UK, 2023. https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8060

Motivation

Hattendorff's theorem allocates the variance of an insurer's life-contingent loss to policy years. In a fully discrete life insurance model, each policy year's contribution depends on the discounted net amount at risk, the probability of dying in that year, and the conditional probability of surviving the year. Shiu and Xiong (1) derive the classical loss-at-issue formula and the corresponding discrete Hattendorff variance through reserve identities and a summation-by-parts argument. Gerber [2, §6.7] discusses the actuarial allocation of total loss to individual policy years.

The related Missions XVI and XIX address two complementary parts of that argument. XVI specifies death/survival masses for a finite lifetime distribution and proposes proofs of centred annual mortality innovations, cross-year orthogonality and variance. XIX describes a pathwise, year-by-year accounting identity expressing insurer loss, including the terminal reserve, as the opening reserve plus discounted mortality innovations. The present mission is the mathematical bridge combining these structures.

It targets the full finite-horizon reserve-loss expectation and variance theorem under consistent conditional mortality probabilities and the annual prospective reserve recurrence. Unlike a purely algebraic zero-covariance lemma, the mortality model supplies the orthogonality. Unlike a standalone cashflow telescoping lemma, the result calculates the variance of the insurer's actual policy loss.

Setting

Fix a finite scenario space $\Omega$, with nonnegative probabilities $w_\omega$ summing to one, and a curtate future lifetime $K:\Omega\to\mathbb N$. An insured with $K(\omega)=t$ dies during policy year $t$.

Define the in-force and death-year probability masses

$$S_t=\sum_{\omega\in\Omega}w_\omega\mathbf1_{\{K(\omega)\ge t\}}, \qquad D_t=\sum_{\omega\in\Omega}w_\omega\mathbf1_{\{K(\omega)=t\}}.$$

The disjoint survival/death partition gives $S_t=D_t+S_{t+1}$. For each year covered by the policy valuation horizon, assume positive in-force mass, $S_t>0$, so that the conditional death and survival rates are well-defined:

$$q_t=\frac{D_t}{S_t}, \qquad p_t=\frac{S_{t+1}}{S_t}, \qquad q_t+p_t=1.$$

These rates are derived from one joint lifetime distribution. The annual death indicators are therefore linked across years and are not independent. The zero-covariance result must be derived from the nested survival-event structure rather than assumed.

Let $n$ denote the finite number of years covered, $v$ a deterministic annual discount factor, $\pi_t$ the beginning-of-year premium, $b_{t+1}$ the death benefit payable at time $t+1$, and $V_t$ the prospective reserve at time $t$ for a policy still in force.

Assume that, for each year $0\le t<n$, the prospective reserve satisfies

$$\boxed{ V_t+\pi_t=v\left(p_tV_{t+1}+q_tb_{t+1}\right). }$$

The realised insurer loss, valued at issue, is

$$\begin{aligned} L_n(\omega) ={}&\sum_{0\le t<n}\left[ v^{t+1}b_{t+1}\mathbf1_{\{K(\omega)=t\}} -v^t\pi_t\mathbf1_{\{K(\omega)\ge t\}} \right]\\ &+v^nV_n\mathbf1_{\{K(\omega)\ge n\}}. \end{aligned}$$

The terminal reserve is explicitly included if the insured survives beyond the chosen horizon. Thus $L_0=V_0$, a life may remain insured after year $n$, and neither $V_0$ nor $V_n$ is assumed zero. The original Shiu–Xiong complete-life issue-loss formula is recovered only under appropriate terminal/residual liability conditions; the present finite-horizon formula is the reserve-inclusive extension needed for a finite theorem.

Formalization targets

Mortality innovation and pointwise reserve loss

The year-$t$ mortality innovation is

$$I_t(\omega)= \mathbf1_{\{K(\omega)=t\}} -q_t\mathbf1_{\{K(\omega)\ge t\}}.$$

The discounted net amount at risk is

$$\rho_t=v^{t+1}(b_{t+1}-V_{t+1}).$$

By the annual reserve recurrence and the in-force event partition, the change in the insurer's loss across a policy year equals $\rho_tI_t$. Summing over years gives the exact pathwise representation

$$\boxed{ L_n(\omega)=V_0+\sum_{0\le t<n}\rho_tI_t(\omega). }$$

Mission XIX describes this pointwise reserve-loss identity. The present formalisation must bridge XIX's realised-death-year indicator and reserve-derived innovation to XVI's scenario-based mortality innovation. They represent the same object when the death probabilities are calculated from the same $w,K$ model.

From the finite scenario lifetime law and $S_t>0$, the annual innovations satisfy

$$\mathbb E_w[I_t]=0,\qquad \mathbb E_w[I_iI_j]=0\quad(i\ne j),\qquad \mathbb E_w[I_t^2]=D_tp_t.$$

The final factor $p_t$ is the conditional one-year survival probability and is essential to the fully discrete variance formula. It must not be replaced by an assumed independence condition or omitted by importing a continuous-time formula.

Full expectation and variance capstone

Taking expectations of the pathwise identity, using $\sum_\omega{} w_\omega=1$, yields

$$\boxed{\mathbb E_w[L_n]=V_0.}$$

The variance of an insurer loss is unaffected by adding a deterministic opening reserve when the probability weights have unit total mass. Consequently,

$$\begin{aligned} \operatorname{Var}_w(L_n) &=\operatorname{Var}_w\left(\sum_{0\le t<n}\rho_tI_t\right)\\ &=\sum_{0\le t<n}\rho_t^2\mathbb E_w[I_t^2]\\ &=\boxed{\sum_{0\le t<n}v^{2(t+1)} (b_{t+1}-V_{t+1})^2D_tp_t}. \end{aligned}$$

The capstone combines these two results, with no unproved independence assumption. Its mathematical hypotheses explicitly include nonnegative, normalised scenario weights, positive in-force probability for every year in the range, and the one-year reserve recurrence with conditional rates computed from the given lifetime distribution.

The theorem remains valid at $n=0$: the insurer's finite-horizon loss is deterministically $V_0$, and the variance sum is empty and equals zero.

The exact final theorem is ActuarialValuation.finiteHattendorffReserveLoss_fundamental. This result is nontrivial at positive horizons. The formal model does not assert independence, fix the opening reserve to zero, or substitute a deterministic insurer loss.

Significance

The annual decomposition separates valuation assumptions from stochastic fluctuations in mortality. Shiu and Xiong's classical fully discrete treatment (https://doi.org/10.1007/s13385-020-00256-9) emphasises that a survival factor remains in each annual variance contribution. Preserving this factor makes the result useful for actuarial risk allocation and for reconciling the valuation reserve with actual finite-period insurer loss.

The finite-scenario definitions are reusable in downstream insurance valuation missions, for example multiple-decrement reserves and explicit conditional mortality shock models. The result also supplies finite boundary tests for nonzero opening reserves, death exactly at the horizon, and survival beyond the final premium year.

Difficulty

Annual death events arise from the same curtate lifetime, so their indicator variables have nested dependence. Their centred innovations have vanishing cross-year covariance only under the correct conditional in-force rates and probability-weighting conventions. Incorrect denominators or an unproved independence assumption would change the result.

The pathwise identity also accounts for different timing of premiums and benefits, and includes a terminal reserve for survivors. At the last covered year a policy death and survival are disjoint, so the last benefit and terminal reserve must not both be counted. The exact Lean imports must retain the definitions and parameter order of the previously published mortality and loss modules.

Formalization scope

The Lean 4 model uses a finite scenario type with Fintype, real-valued weights, a natural-number curtate lifetime, and the published definition modules from related actuarial missions. Four new definitions and 13 Open theorem obligations form the formal graph, including 12 supporting targets before the conjunction capstone. The accepted proof status of another theorem is never inferred merely from its publication.

The result is finite horizon, finite probability space and fully discrete. The source's complete-life identity would additionally require justified limiting arguments, the disappearance (or valuation) of terminal reserves, integrability and interchange of expectation, variance and limit. None of those conditions is silently assumed here. There is no continuous-time martingale or Lévy-process result in this mission.

The classical full-life result is known mathematically, while the specific finite-horizon Lean declarations remain open proof targets. A compiler-accepted declaration with a sorry body does not establish a proof.

Selected references

• [1] Shiu, Elias S. W. and Xiong, Xiaoyi (2021), “An elementary derivation of Hattendorff's theorem”, European Actuarial Journal, 11, pp. 319–323, equations (1)–(5). Source for the discrete variance survival factor and the reserve-based derivation.
• [2] Gerber, Hans U. (1997), Life Insurance Mathematics, third edition, Springer, Chapter 6, §6.7, Allocation of the Overall Loss to Policy Years.
• [3] Gerber, Hans U., Leung, Bartholomew P. K. and Shiu, Elias S. W. (2003), “Indicator Function and Hattendorff Theorem”, North American Actuarial Journal, 7(1), pp. 38–47. Relates mortality indicator functions, actuarial loss variance and orthogonal yearly increments.

Motivation

A small insurance fund or participant with limited resources can gain or lose capital repeatedly and be forced to stop at one of two barriers. This is the familiar gambler's ruin problem, useful as a discrete illustration of capital adequacy, finite-state absorption and the distinction between ruin and eventual success. Promislow's treatment derives fair and biased cases using finite capital and a bounded opponent, and Grinstead and Snell present related random-walk absorption probabilities (https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin). The practical distinction is whether a probability refers to absorption at some eventual time or only to absorption within a fixed number of additional bets.

Setting

A unit-stake capital process occupies a natural-number capital state i between zero and a positive upper barrier N. With probability p the next bet raises capital by one unit; with complementary probability one minus p it lowers capital by one unit. Capital zero and capital N are absorbing. The fair case has p equal to one-half and a linear harmonic success function. In the biased case, a down-to-up odds ratio r is defined by (1−p)/p. An additional finite-horizon model records separately the chance of hitting the upper barrier and the chance of hitting zero before a given step count, leaving unresolved paths in the interior.

The fair finite-capital eventual upper-absorption harmonic candidate is named actuarial_gamblerFairSuccess in the Lean development. A fair unit-stake random walk on capital states zero through positive target N has eventual upper-barrier success probability i/N when starting from integer capital i in the bounded interval. This source defines the harmonic candidate, not the underlying stochastic stopping-time construction.

The downward-to-upward transition probability ratio is named actuarial_gamblerOddsRatio in the Lean development. The fixed probability of an up step is p; the probability of a down step is 1−p. Their ratio (1−p)/p controls the biased harmonic solution and requires p>0 to be interpreted as transition odds.

The biased finite-capital upper-absorption harmonic candidate is named actuarial_gamblerBiasedSuccess in the Lean development. For a biased unit-stake walk with r=(1−p)/p different from one and a positive upper barrier N, the formula gives the eventual probability of hitting N before zero: (1−r^i)/(1−r^N). The denominator must be nonzero. The fair p=1/2 case requires the separate linear formula.

The probability of hitting upper capital target within finitely many steps is named actuarial_gamblerFiniteSuccess in the Lean development. The bounded random walk makes unit upward jumps with probability p and unit downward jumps with probability 1−p. It is absorbed at capital zero or at or above target N. This finite-horizon recursion records only the probability of reaching the success barrier by time n; ongoing paths at the horizon remain unresolved.

The probability of reaching zero capital within finitely many steps is named actuarial_gamblerFiniteRuin in the Lean development. The second finite-horizon recursion tracks absorption at zero capital, the ruin boundary. Successful absorption at the upper target contributes zero ruin probability. At an interior state it mixes successor ruin probabilities, and paths still active at horizon n contribute to neither finite success nor finite ruin.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.gamblerFairSuccess_zero states the following relation. The fair candidate's numerator is zero at the lower absorbing boundary, so the proposed upper-barrier success probability is zero even if the target denominator is algebraically zero.

$$H_{\rm fair}(0)=0$$

Stronger structural properties

The declaration ActuarialValuation.gamblerFiniteRuin_nonneg states the following relation. Ruin starts at the boundary indicators zero or one and evolves through nonnegative weighted averages at every interior capital state. Therefore finite-horizon absorption-at-zero probability cannot be negative under valid win probabilities.

$$0\le p\le1\Rightarrow R_n(i)\ge0$$

Capstone valuation statement

The declaration ActuarialValuation.gamblerBoundedRuin_fundamental states the following relation. The capstone formalises the correct first-step harmonic equations for the fair and biased finite-capital random walk and nonnegative bounded finite-horizon absorption-event probabilities. Unlike an infinite-horizon gambler's ruin theorem, it does not assert that every path has already been absorbed by a finite horizon or identify finite success directly with the harmonic candidate.

$$H_{1/2}(i)=\tfrac12(H_{1/2}(i+1)+H_{1/2}(i-1)),\quad H_p(i)=pH_p(i+1)+(1-p)H_p(i-1)$$

Significance

The exact harmonic functions identify candidate upper-absorption probabilities for fair and biased walks, while the finite-horizon recursions quantify how much absorption can have occurred after a specified number of steps. The latter distinction matters for capital planning because some solvent paths may still be active when an assessment period ends. Standard treatments show the importance of the bias ratio and the two absorbing barriers (https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin). The mathematical identities themselves are known; a Lean representation offers checkable conventions around capital zero, positive target, finite horizon and the singular fair limit. Its state-recursion definitions can be reused in finite Markov reward and insurance capital models.

Difficulty

The fair success expression cannot simply be obtained by substituting p=one-half into the biased quotient: both its numerator and denominator become zero. The separate linear expression needs its own fair-case convention. A second easy mistake is to regard finite-horizon success and ruin as exhaustive complementary events. They are disjoint, but an interior path may reach neither barrier during the available number of steps. At extreme probabilities zero or one, the odds ratio may be undefined or degenerate, and an eventual absorption assertion requires hypotheses not present in a finite first-step equality. These boundary distinctions are central to the stated target.

Formalization scope

Lean uses natural numbers for capital, positive upper barrier and finite horizon, with real-valued probabilities and harmonic candidate functions. Both boundary states are absorbing; ruin is capital equal to zero, unlike the strictly negative surplus convention in another insurance reserve model. The biased quotient is used only when its normalising denominator is nonzero, and the fair case is represented separately. The source does not assert almost-sure absorption or equality of finite-horizon success with an eventual harmonic function, and it does not equate unresolved paths with failure. Mathlib finite recursion and rational function arithmetic support the scope. Reusable contributions include independent success/ruin state operators, boundary-state lemmas and nonnegative weighted continuation bounds.

The five local definitions form the available mathematical vocabulary: actuarial_gamblerFairSuccess, actuarial_gamblerOddsRatio, actuarial_gamblerBiasedSuccess, actuarial_gamblerFiniteSuccess, actuarial_gamblerFiniteRuin. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society, 1997. https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin

Motivation

Non-proportional reinsurance allows an insurer to retain ordinary claims while transferring part of the cost of larger claims to a reinsurer. Such contracts specify both an attachment point and a maximum amount the reinsurer will pay. Confusing the retention with the limit changes ceded payments, claims reporting and the net expected treaty cost. HMRC describes common non-proportional arrangements and their layers in its general-insurance technical guidance (https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8080). Pricing systems must distinguish an unlimited excess claim from the finite reimbursement available under one layer, including claims exactly at the attachment or at the upper layer boundary.

Setting

A gross claim x, attachment d and layer width L are nonnegative natural-number amounts in the same currency unit. An unlimited excess contract pays only the positive part of x minus d. A capped ceded layer pays the smaller of that gross excess and L. Amounts beyond the top of the layer, d plus L, form a separate higher excess. The expected treaty cost is evaluated under a bounded severity distribution with nonnegative mass w(x) at each integer claim amount from zero through the finite bound B. An unlimited stop-loss expected cost uses the same distribution without a layer cap.

The single-claim positive excess over an attachment is named actuarial_layerExcessLoss in the Lean development. The claim amount remaining above the insurer's retained attachment is natural-number positive-part subtraction. Below attachment it is zero, matching the gross excess payment of an unlimited stop-loss treaty.

The capped single-layer reinsurance payment is named actuarial_layerCededPayment in the Lean development. For a claim amount x, a reinsurance layer of width or limit L attaching at d reimburses the lesser of excess x−d positive part and the layer limit. It pays zero below attachment and cannot pay more than L.

The remaining gross excess above the top of the layer is named actuarial_layerHigherExcess in the Lean development. Any part of the claim above the layer's upper attachment d+L is left for higher reinsurance layers or the ultimate insurer. It is the positive excess loss above the sum of the lower attachment and the stated layer width.

The finite aggregate expected ceded loss cost of a layer is named actuarial_layerExpectedCeded in the Lean development. The finite bounded claim-severity mass model computes the net expected excess-of-loss reinsurance premium by weighting each loss payment of the capped layer by its claim probability. Expected ceded cost excludes risk loading and expenses.

The finite unlimited stop-loss expected payment is named actuarial_layerExpectedExcess in the Lean development. The net stop-loss premium above an attachment d is the finite weighted expected excess claim amount without an upper layer cap. The difference of this value at d and at d+L will equal the capped layer's expected ceded cost.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.layerExcessLoss_below states the following relation. When the claim amount does not exceed the attachment, no monetary loss lies above the retention. Saturating natural-number subtraction gives zero excess exactly.

$$x\le d\Rightarrow E_d(x)=0$$

Stronger structural properties

The declaration ActuarialValuation.layerExpectedCeded_stoploss_difference states the following relation. The expected capped ceded payment equals the expected unlimited excess above attachment d, less the excess above the upper layer attachment d+L. The identity follows termwise from the exact pointwise loss partition and finite linearity, with no distribution-normalisation assumption.

$$\mathrm{EL}_{d,L}=\mathrm{SL}(d)-\mathrm{SL}(d+L)$$

Capstone valuation statement

The declaration ActuarialValuation.layerCededValuation_fundamental states the following relation. The capstone unifies a bounded single-claim excess-of-loss treaty's payment decomposition with the exact net layer premium as difference of two unlimited stop-loss premiums, and establishes its nonnegative bound against gross excess. It does not assert an expense-loaded reinsurance price or infinite-support integration.

$$E_d(x)=C_{d,L}(x)+H_{d,L}(x),\quad \mathrm{EL}_{d,L}=\mathrm{SL}(d)-\mathrm{SL}(d+L)$$

Significance

The difference between net unlimited excess payments at two attachments gives an immediately auditable valuation of a finite reinsurance layer. A treaty administrator can reconcile the covered amount, the higher excess and the original unlimited amount without making assumptions about claim frequencies or total policyholder losses. This separation is useful in exposure rating and in modelling multiple reinsurance towers, including arrangements described in the HMRC guidance at https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8080. The identity is already a standard insurance-payment relation; formalising its finite version would produce reusable payment functions and expected-cost inequalities rather than announcing a new result about treaty markets. It would also distinguish claim indemnities from premiums loaded for operating costs and capital.

Difficulty

An intuitive subtraction of the reinsurance limit from the gross excess can become negative and cannot describe the remaining claim burden below the top of a layer. Likewise, a formula that gives every loss above attachment an unlimited reimbursement silently overstates the provider's maximum liability. Natural-number subtraction in Lean saturates at zero, so any stated decomposition must respect positive-part semantics at both the lower and upper treaty thresholds. There are also two independent directions of monotonicity: increased layer width raises coverage, while a larger retention generally reduces the covered part. The capstone fixes only the former and the precise difference of stop-loss expected amounts.

Formalization scope

The source uses natural numbers for money, including the deductible and treaty limit, and real numbers for bounded claim coefficients and expected ceded payment. There is no independent hidden frequency distribution, and the finite claim grid can include zero severity and zero layer width. Nonnegative weights are explicitly required for order and positivity conclusions, while pointwise payment identities hold without probability normalisation. The formalisation does not collapse a capped layer into a trivial unlimited excess contract, nor assume an expense-loaded market premium is equal to an actuarial expectation. Reusable contributions include positive-part payments, min-limited cover, finite layer aggregation and higher-attachment functions. Later models may combine several layers or add claims-inflation adjustments.

The five local definitions form the available mathematical vocabulary: actuarial_layerExcessLoss, actuarial_layerCededPayment, actuarial_layerHigherExcess, actuarial_layerExpectedCeded, actuarial_layerExpectedExcess. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• HM Revenue & Customs, General Insurance Manual: Non-proportional reinsurance, UK government technical guidance, accessed 2026. https://www.gov.uk/hmrc-internal-manuals/general-insurance-manual/gim8080

Motivation

An insurer can expect premium income to exceed claims and nevertheless become insolvent because the timing and size of claim payments are uncertain. Ruin theory asks about the chance of crossing an insolvency boundary before a specified date or eventually. The classic actuarial approach uses the distribution of yearly net cashflows and, when it exists, an exponential adjustment condition to bound risk as starting capital increases. Finite-state random walks provide a related illustration of first-passage probability and absorption (https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin). The mission isolates the finite-horizon inequality that can be stated without imposing the measure-theoretic hypotheses of an infinite stopping-time theorem.

Setting

An annual aggregate claim X is a nonnegative integer taking values zero through a fixed maximum B, with annual probabilities w(k). The insurer starts at signed integer surplus u, receives nonnegative integer premium c each year and pays an independent claim from the same finite distribution. The next signed surplus is current surplus plus c minus that claim. Strict ruin occurs as soon as surplus becomes negative, including at time zero if the initial reserve is negative. A finite-horizon ruin probability assigns one to an already-ruined state, zero to a solvent state at zero horizon, and a probability-weighted continuation at other states.

The one-period integer insurance surplus update is named actuarial_ruinNextSurplus in the Lean development. The insurer begins a period with an integer surplus, receives a fixed nonnegative integer premium, then pays the period's aggregate integer claim. The next surplus can be negative, so it is represented as an integer rather than a natural number with saturating subtraction.

The total annual claim probability on a bounded integer grid is named actuarial_ruinClaimMass in the Lean development. The finite annual aggregate-claim model has possible integer loss amounts 0 through B. Its total probability is the sum of those coefficients, required to equal one in the genuine probability interpretation of the recursion.

The exponential moment of one period's net insurance loss is named actuarial_ruinAdjustmentMoment in the Lean development. The adjustment coefficient R compares the annual aggregate claim X with the premium c via the exponential moment E[exp(R(X−c))]. When this quantity is no greater than one and R≥0, the exponential capital bound becomes a supermartingale-style one-step estimate.

The lundberg exponential benchmark for an integer initial surplus is named actuarial_ruinExponentialBound in the Lean development. The benchmark at current surplus u is exp(−Ru). For R≥0 this is at least one if u is negative, allowing it to bound an already-ruined state whose ruin indicator equals one; for a nonnegative initial reserve it falls exponentially with additional starting capital.

The first-passage ruin probability over a finite number of annual periods is named actuarial_ruinProbabilityFinite in the Lean development. The probability of ruin by horizon n records whether surplus was ever strictly negative. Already-ruined integer states have probability one at any horizon. Otherwise the next-year claim distribution mixes the ruin probabilities of all possible updated signed surpluses, recursively for n further years. The horizon is finite and annual claims are modelled as conditionally independent copies of w.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.ruinNextSurplus_zero_claim states the following relation. If the insurance portfolio generates no claim payment during a period, only the fixed premium c affects the insurer's capital. The signed next-period surplus is therefore the old surplus plus the earned premium.

$$U'=U+c$$

Stronger structural properties

The declaration ActuarialValuation.ruinProbabilityFinite_le_one states the following relation. With an annual claim mass function summing to one, every solvent state averages predecessor-horizon ruin probabilities with nonnegative weights of total one. Since the base and already-ruined values lie at most one, induction establishes that the finite-horizon ruin probability never exceeds one.

$$w\ge0,\ \sum_kw_k=1\Rightarrow\psi_n(u)\le1$$

Capstone valuation statement

The declaration ActuarialValuation.ruinFiniteHorizon_fundamental states the following relation. The capstone formalises a bounded annual integer-claim surplus recursion with an absorbing strict-negative ruin condition. Proper nonnegative annual claim probabilities keep finite-horizon ruin probability within the unit interval. A nonnegative exponential adjustment parameter satisfying E[exp(R(X−c))]≤1 then bounds it by exp(−Ru) for every finite horizon and signed starting capital.

$$0\le\psi_n(u)\le1,\quad M(R)\le1\Rightarrow\psi_n(u)\le e^{-Ru}$$

Significance

A valid exponential upper bound gives a transparent relationship between initial capital and probability of failure over every finite number of years. It can be used to compare capital buffers or to test an insurer's response to a changed premium and annual claims distribution. The bounded recursive setting is particularly relevant when annual severity is itself computed by exact portfolio convolution. Classical first-passage models distinguish eventual absorption from a finite-horizon crossing event (https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin). The formal target is the known exponential comparison property, not a new adjustment-root existence result. It yields reusable recursion and exponential-moment definitions for subsequent finite insurance reserve analysis.

Difficulty

A tempting representation of surplus as a natural number is incorrect for insolvency, because saturating subtraction hides the negative state that defines ruin. A second mistake is to present the exponential capital function as an unconditional bound. It can fail when the claim-loss exponential moment exceeds one, even in a deterministic one-year example. The argument also needs the negative-surplus boundary, at which ruin probability is already one. These complications are visible at finite horizons and cannot be repaired merely by assuming that premium income has a positive expected margin.

Formalization scope

The Lean source uses signed integers for surplus and natural numbers for annual claim amounts, premium, severity bound and horizon. Annual probabilities and adjustment exponent are real. The event is strictly negative first-passage ruin, distinct from reaching exactly zero in a gambler's ruin model. A nonnegative adjustment exponent and an explicit upper bound of one on its net-claim exponential moment are mandatory. The source gives the finite recursive quantity algebraically, not an independently constructed probability space, an infinite-horizon supremum or a theorem that a positive adjustment root exists. Reusable developments include finite weighted expectation operators, signed reserve transitions and nonnegative probability bounds; meaningful contributions may extend these to random premiums or general annual claim distributions.

The five local definitions form the available mathematical vocabulary: actuarial_ruinNextSurplus, actuarial_ruinClaimMass, actuarial_ruinAdjustmentMoment, actuarial_ruinExponentialBound, actuarial_ruinProbabilityFinite. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society, 1997. https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/12%3A_Random_Walks/12.02%3A_Gambler%27s_Ruin

Motivation

Insurers model annual claim frequency to price collections of policies and assess how often unusually large numbers of claims may arrive together. A Poisson count model has equal mean and variance, but heterogeneity between policyholders can produce materially more dispersion. The negative-binomial family is an established alternative, including as a marginal count law obtained by mixing the Poisson intensity over a Gamma distribution. The mathematical relationship, with explicit Gamma shape and rate parameters, is set out in the Stanford discussion of mixture models (https://web.stanford.edu/class/bios221/MixtureLecture.html). A reliable formal description must specify which negative-binomial probability parameter is being used, since actuarial and statistical literature adopt incompatible success and failure conventions.

Setting

A negative-binomial count variable N records a nonnegative integer number of claims. The integer shape parameter r is strictly positive, and a real count probability parameter p lies between zero and one, with the upper endpoint excluded when calculating moments. In this convention p is associated with powers of the claim count n, while 1 minus p is associated with powers of the shape r. The Gamma-mixed Poisson model starts from a Gamma-distributed intensity with positive rate beta and integer shape r; at one future exposure unit its negative-binomial count parameter equals the reciprocal of beta plus one.

The negative-binomial claim count coefficient is named actuarial_negBinCountMass in the Lean development. Defines a negative-binomial frequency coefficient for n claims and positive integer shape r, with p the count probability parameter and 1−p the Bernoulli success probability. The finite choose expression is mathematically the conventional formula only for shape r≥1.

The expected claim count under nb size and count parameter is named actuarial_negBinCountMean in the Lean development. For 0≤p<1 and positive integer shape, the negative-binomial mean count equals r p divided by one minus p. This p denotes count rather than Bernoulli success probability, so the formula differs from sources that reverse the parametrisation.

The negative-binomial count variance is named actuarial_negBinCountVariance in the Lean development. For valid negative-binomial parameters, the variance exceeds the mean because uncertainty in the Poisson event frequency introduces additional dispersion. The exact variance is r p divided by (1−p) squared.

The unit-exposure gamma-poisson predictive count parameter is named actuarial_negBinGammaProbability in the Lean development. A Gamma mixing distribution with positive rate beta and positive integer shape yields a negative-binomial mixed-Poisson count law at one exposure unit with the count parameter p=1/(beta+1). This is the reciprocal rate-plus-one, not beta/(beta+1), which is the complementary success parameter.

The integer-shape gamma-poisson predictive count coefficient is named actuarial_negBinGammaPredictive in the Lean development. The model uses a shape/rate Gamma prior for a Poisson rate at unit exposure. The predictive count mass is the integer-shape negative-binomial coefficient evaluated at the rate-derived count parameter. The declaration encodes the known mixture result algebraically without asserting an integral identity.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.negBinGammaProbability_pos states the following relation. Prior Gamma rate b positive implies denominator b+1 strictly positive, so its reciprocal is a valid strictly positive negative-binomial count parameter.

$$\beta>0\Rightarrow p>0$$

Stronger structural properties

The declaration ActuarialValuation.negBinGammaMean_unit_exposure states the following relation. The frequency mean for the negative-binomial predictive law derived from a Gamma shape/rate prior at one future exposure is shape divided by rate. This agrees with the prior Gamma mean Poisson intensity, not shape times rate.

$$\mu=r/\beta$$

Capstone valuation statement

The declaration ActuarialValuation.negBinClaimFrequency_fundamental states the following relation. The capstone combines the exact integer-shape negative-binomial count-mass recursion and the unit-exposure Gamma-Poisson mixture parameterisation. It verifies both predictive moments and coefficient nonnegativity, without claiming an infinite PMF normalisation proof or a Gamma mixture integral in Lean.

$$(n+1)g(n+1)=(n+r)pg(n),\quad\mu=r/\beta,\quad\sigma^2=\mu+r/\beta^2$$

Significance

The ability to compute neighbouring frequency coefficients from one another supports actuarial claim-table construction without repeatedly evaluating large factorial ratios. The difference between count variance and mean provides a direct overdispersion diagnostic when comparing heterogeneous risks with a Poisson benchmark. Under a Gamma prior, the mixed frequency mean and variance recover the Gamma shape/rate moments, linking pricing assumptions to uncertainty in latent risk intensity. Panjer's work on recursive compound distribution evaluation shows why stable frequency recurrences matter for downstream aggregate insurance claims (https://doi.org/10.1017/S0515036100006796). The results are known mathematics; an exact formalisation would make the parameter conventions and real-valued boundary assumptions machine checkable and provide coefficient lemmas reusable by collective-risk computations.

Difficulty

The main hazard is a parameterisation mismatch. Some publications place p on the Bernoulli success count and others put p on the recorded total of failures or claims; the mean and Gamma mapping change accordingly. The integer-binomial coefficient must also be interpreted with positive shape: allowing r=0 while retaining the expression choose(n+r−1,n) introduces natural subtraction at a degenerate boundary. Moment formulas include one-minus-p in a denominator and therefore cannot be stated as ordinary probability facts at p=1. Finally, the algebraic coefficient sequence alone does not supply a probability-space proof that Gamma integration produces the count law.

Formalization scope

The Lean representation uses natural numbers for integer shape and count, and real numbers for the probability parameter, Gamma prior rate and moment expressions. The combinatorial coefficient is Nat.choose, and all arithmetic is within Mathlib. The zero-shape index is excluded from the conventional distribution claim, although some sign statements remain algebraically valid there. This mission does not assert an infinite normalisation sum, an unrestricted non-integer shape model or a vacuous Gamma-mixture identity proved by defining the mixture as the desired result. Reusable contributions include count-ratio recurrences, nonnegative binomial coefficient calculations and parameter conversion lemmas. Future work could connect those components to a genuine Gamma integral or a separately formalised Panjer aggregate recursion.

The five local definitions form the available mathematical vocabulary: actuarial_negBinCountMass, actuarial_negBinCountMean, actuarial_negBinCountVariance, actuarial_negBinGammaProbability, actuarial_negBinGammaPredictive. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Harry H. Panjer, Recursive evaluation of a family of compound distributions, ASTIN Bulletin, 1981. https://doi.org/10.1017/S0515036100006796

Motivation

A pair of insurance portfolios can have the same expected aggregate claim while requiring different ceded premiums or capital to withstand severe outcomes. Expected loss alone cannot identify the difference: the distribution of loss across small and large events also matters. The stop-loss order compares expected loss above every retention level and provides a concrete way to rank riskiness for excess-of-loss contracts. Müller studied wider relationships among such risk orders and their stop-loss transforms (https://doi.org/10.1016/0167-6687(96)90002-5). The present object is a local, explicit change in an integer loss distribution that preserves its first moment but shifts probability from a middle amount towards two neighbouring outcomes.

Setting

A bounded aggregate-loss distribution has nonnegative coefficients w(s) for integer losses s between zero and a finite maximum B, with total coefficient mass usually equal to one. Its gross expected claim is the weighted sum of loss amounts and its stop-loss premium at integer deductible d is the expected positive excess above d. A proposed new coefficient array moves an amount delta out of one interior integer claim centre c, placing equal halves at the adjacent claim sizes c minus one and c plus one. Strictly positive centre and a top neighbour no higher than B keep the change within the declared support.

The bounded discrete expected excess above an integer attachment is named actuarial_orderStopLossPremium in the Lean development. The actuarial stop-loss transform multiplies each aggregate loss's excess over the attachment point by its real coefficient and sums over the finite covered loss grid. A nonnegative probability mass law gives its net excess-of-loss reinsurance premium.

The bounded aggregate loss expectation is named actuarial_orderAggregateMean in the Lean development. The finite aggregate claims mean uses the claim amount at each integer index as its payment and multiplies it by its probability mass. Comparing mean claims is necessary when interpreting stop-loss order as mean-preserving convex order.

The aggregate probability within a finite loss bound is named actuarial_orderAggregateMass in the Lean development. The sum of all coefficients on the finite aggregate-loss support is the bounded probability mass, whose value is one for a normalised distribution. An explicit mass operator makes the mean-preserving spread's total-mass conservation separately checkable.

The mean-preserving symmetric transfer from one claim amount to adjacent values is named actuarial_orderNeighbourSpread in the Lean development. A fixed amount δ of probability mass is moved away from an interior integer claim level centre: half is allocated one unit below, and half one unit above. Under 1≤centre and centre+1≤B these three loss levels lie within the support and the transfer preserves the aggregate mean.

The finite-grid stop-loss ordering of aggregate risks is named actuarial_orderStopLossDominates in the Lean development. The first aggregate loss distribution is no riskier in stop-loss order than the second when its expected excess payment never exceeds the other's for any integer attachment d. Equal means and unit total masses can be imposed separately to obtain a mean-preserving spread interpretation.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.orderStopLossPremium_zero states the following relation. Since every nonnegative integer loss exceeds or equals attachment zero, the stop-loss payment equals the gross loss in each scenario. Thus the expected excess premium at attachment zero coincides exactly with the original aggregate expected claim.

$$\Pi_w(0)=\mu_w$$

Stronger structural properties

The declaration ActuarialValuation.orderNeighbourSpread_stoploss states the following relation. The payment function (s−d) positive-part is convex in the claim amount s. Transferring nonnegative mass symmetrically from centre to its adjacent lower and higher losses can only increase its expected value, with a difference equal to δ/2 times the nonnegative discrete second difference at c.

$$\Pi_{w'}(d)-\Pi_w(d)=\frac{\delta}{2}[(c-1-d)_++(c+1-d)_+-2(c-d)_+]\ge0$$

Capstone valuation statement

The declaration ActuarialValuation.orderNeighbourSpread_fundamental states the following relation. The capstone shows that a feasible symmetric probability transfer to adjacent losses produces a valid nonnegative bounded coefficient distribution, preserves the aggregate total probability and mean, yet weakly raises every stop-loss expected excess premium. This is an explicit constructive mean-preserving spread and a finite-lattice convex-order result.

$$w'\ge0,\ M_{w'}=M_w,\ \mu_{w'}=\mu_w,\ w\preceq_{\rm sl}w'$$

Significance

A mean-preserving spread separates the effect of greater claim variability from a change in the underlying expected loss. By conserving total mass and gross mean, the comparison can be attributed to the redistribution of outcomes rather than an increase in total claim cost. Its downstream consequence is a weak rise in the expected ceded payout for every positive-part reinsurance retention. This connects aggregate frequency/severity valuation to ordered risks and tail measures, consistent with wider stop-loss ordering results studied in https://doi.org/10.1016/0167-6687(96)90002-5. Formalising the finite construction provides a reusable bounded-grid example of stochastic-order consequences without requiring an axiomatic treatment of all convex functions.

Difficulty

The obvious claim that any symmetric redistribution increases expected excess is false if the transfer is allowed to cross the boundary of the support or remove more probability than the central cell contains. At centre zero the intended lower neighbouring loss is negative and no longer exists in a natural-number grid. If delta exceeds the centre's initial coefficient, the resulting array is not a probability distribution. Equality of expected claim amounts by itself also does not imply stop-loss order. The formal statement must account for the actual positive-part payment at each possible deductible, including deductions outside the claim support.

Formalization scope

Lean represents loss indices, attachment points and the spread centre as naturals, and probability coefficients and expected claims as reals. The claim grid is finite, and the full probability-normalisation equation is separate from local mass conservation. The capstone explicitly requires nonnegative starting coefficients and a feasible interior transfer. It does not substitute a trivial comparison between identical distributions, nor claim general continuous convex order or a complete converse representation of risk ordering. Mathlib finite sums and arithmetic over nonnegative parts are enough for the declared scope. Reusable infrastructure includes finite aggregate means, bounded mass totals and stop-loss order on arrays, suitable for later reinsurance layers and capital-risk measures.

The five local definitions form the available mathematical vocabulary: actuarial_orderStopLossPremium, actuarial_orderAggregateMean, actuarial_orderAggregateMass, actuarial_orderNeighbourSpread, actuarial_orderStopLossDominates. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• Alfred Müller, Orderings of risks: A comparative study via stop-loss transforms, Insurance: Mathematics and Economics, 1996. https://doi.org/10.1016/0167-6687(96)90002-5

Motivation

Insurers routinely estimate next-period frequency from policyholders who have accumulated unequal historical exposure. A raw observed rate can swing markedly when the exposure base is small, while a collective rate may ignore evidence of persistent individual risk differences. The Gamma–Poisson model gives a precisely defined compromise: a posterior expected claim intensity that responds to observed counts and earned exposure. The literature on credibility analysis documents the distinction between process variation and variation across risks, including the role of Bayesian conjugacy (https://openacttexts.github.io/LDAVer2/ChapCredibility.html). The relevant question is how the posterior parameters and credibility premium behave under sequential reporting, zero-data boundaries and different exposure amounts.

Setting

The latent claim frequency is measured in expected events per insured exposure unit. A Gamma prior has shape alpha greater than zero and rate beta greater than zero, so beta is the reciprocal of the Gamma scale. Conditional on this frequency, the observed number of claims C is Poisson with mean equal to exposure E times the intensity. Distinct observed blocks contribute counts C and D and exposures E and F. The formal declarations use natural numbers for counts and real numbers for rates, exposures and expected frequencies. The posterior shape/rate update and credibility-weight equations are the objects of the stated algebraic model.

The gamma posterior shape after observing claim count is named actuarial_gammaPoissonPosteriorShape in the Lean development. The Gamma prior uses the shape-and-rate parameterisation and starts with prior shape α. Observing a nonnegative integer aggregate count C increases the posterior shape to α+C. This is an algebraic conjugacy update rather than a complete formalisation of the Gamma probability density.

The gamma posterior rate after exposure volume is named actuarial_gammaPoissonPosteriorRate in the Lean development. The Gamma distribution's second parameter is a rate, not a scale. The posterior rate is prior rate β plus observed insured exposure E; this explicit convention prevents the common reciprocal-scale error in Poisson-Gamma credibility.

The posterior expected claim frequency per exposure unit is named actuarial_gammaPoissonPosteriorMean in the Lean development. For a Poisson count model and shape/rate Gamma prior, the posterior expected frequency per unit exposure is updated shape divided by updated rate. Positive prior rate and nonnegative exposure ensure an ordinary positive denominator.

The exposure-dependent exact bayesian credibility coefficient is named actuarial_gammaPoissonExperienceWeight in the Lean development. The exact credibility weight in the Poisson-Gamma conjugate model is E divided by β+E. When β>0 and E≥0 this lies between zero and one, approaching greater reliance on claim experience as accumulated exposure increases.

The weighted experience and collective frequency is named actuarial_gammaPoissonCredibilityBlend in the Lean development. The credibility premium combines observed aggregate claims per unit exposure C/E and the Gamma prior population frequency α/β, with respective weights E/(β+E) and β/(β+E). The usual representation requires E>0 and β>0; at zero exposure the claim rate C/E is not a valid observed experience statistic.

Formalization targets

The formal targets are ordered from elementary boundary and consistency claims to their combined capstone. The exact Lean identifiers below are the authoritative local targets; none of the listed theorem bodies is already accepted as a machine-checked proof. The mathematical results themselves are established actuarial or probability identities.

Baseline identities

The declaration ActuarialValuation.gammaPoissonPosteriorShape_zero states the following relation. When no claims are observed, the sufficient statistic added to the shape is zero. The updated Gamma posterior shape therefore equals its prior shape, even before imposing positivity required by the probabilistic model.

$$\alpha+0=\alpha$$

Stronger structural properties

The declaration ActuarialValuation.gammaPoissonCredibilityBlend_exact states the following relation. With positive prior rate and positive observed exposure, the observed claim frequency and collective mean are both well-defined. Substituting weight E/(β+E) and simplifying proves the exact Bayesian credibility decomposition of posterior mean (α+C)/(β+E).

$$Z\,C/E+(1-Z)\alpha/\beta=(\alpha+C)/(\beta+E)$$

Capstone valuation statement

The declaration ActuarialValuation.gammaPoissonCredibility_fundamental states the following relation. The capstone combines exact Gamma-Poisson experience credibility at positive exposure with associative sequential count/exposure updates and admissibility of the Bayesian experience weight. It formalises the conjugate-model parameter and premium algebra without claiming a full measure-theoretic Gamma likelihood integral.

$$\hat\lambda=ZC/E+(1-Z)\alpha/\beta,\ \alpha'=\alpha+C,\ \beta'=\beta+E,\ 0\le Z\le1$$

Significance

Exact agreement between the posterior mean and the collective/individual premium blend allows underwriting calculations to be reconciled across reporting intervals. Exposure aggregation must not alter the end-of-period estimate when the underlying sufficient statistics are unchanged. The additional Gamma predictive parameter variance explains why a future claim count can exhibit dispersion beyond that of a Poisson distribution with deterministic intensity. These are established properties of conjugate credibility models, rather than new claims about the existence of a posterior distribution; the mathematical background is described in https://openacttexts.github.io/LDAVer2/ChapCredibility.html. The formal output would certify the numeric parameter identities and boundary assumptions in Lean, allowing subsequent developments to reuse the same frequency and credibility conventions.

Difficulty

The common failure is to confuse a Gamma rate with a Gamma scale: this reverses the update and the ratio used as the experience credibility weight. Another false shortcut is to take the product of a zero exposure weight and an undefined observed count divided by zero as a meaningful zero experience contribution. Those algebraic terms are distinguishable in Lean's totalised division and in actuarial modelling. The exact mixture identity needs positive observed exposure, whereas the no-data statement requires both zero exposure and zero observed claims. Strict positivity is also essential when the total rate appears in a denominator.

Formalization scope

The Lean development uses Mathlib real arithmetic and natural claim counts. The source represents the posterior parameter update directly and does not formalise an underlying Gamma density, conditional-expectation kernel or integration proof of conjugacy. In particular, no vacuous statistical assumption is allowed to stand in for a parameter identity, and the nonzero-exposure requirement is retained where observed frequency occurs. Reusable contributions include real-valued exposure-ratio identities, independent positivity facts, sequential sufficient-statistic updates and sharply stated data boundaries. Further work could construct the probability-space interpretation or the negative-binomial predictive count law without changing the current capstone.

The five local definitions form the available mathematical vocabulary: actuarial_gammaPoissonPosteriorShape, actuarial_gammaPoissonPosteriorRate, actuarial_gammaPoissonPosteriorMean, actuarial_gammaPoissonExperienceWeight, actuarial_gammaPoissonCredibilityBlend. Their support conditions, nonzero denominators, and variable types are fixed by the Lean declarations rather than inferred from examples. The nontrivial mathematical content must not be replaced by a vacuous premise or by a special case that bypasses the model boundaries. The existing paper mathematics is known, while these particular theorem declarations remain formalisation targets with no accepted proofs. Contributions that strengthen reusable finite identities or expose missing boundary hypotheses are appropriate, provided they preserve the exact authoritative theorem types.

Selected references

• S. David Promislow, Fundamentals of Actuarial Mathematics, third edition, Wiley, 2015. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/

• OpenActTexts contributors, Loss Data Analytics: Experience Rating Using Credibility Theory, online actuarial textbook, accessed 2026. https://openacttexts.github.io/LDAVer2/ChapCredibility.html