Advanced Short Term Actuarial Mathematics

Advanced Short-Term Actuarial Mathematics

When to Use

• Select and justify severity families (parametric tails, mixtures) and frequency models (Poisson, negative binomial, mixtures)
• Build aggregate loss models: compound distributions, normal approximation limits, FFT/simulation concepts
• Apply credibility (Bühlmann, Bühlmann-Straub, limited fluctuation) and experience rating math
• Structure ratemaking: pure premium, loss ratio, trend, on-level, indicated change logic
• Explain short-term reserving at the mathematical level (chain ladder factors, expected loss ratio)
• Estimate parameters (MLE), run goodness-of-fit and diagnostics, interpret residuals and tail fit
• Compute risk measures (VaR, TVaR) and relate them to capital concepts at a technical level
• Connect modeling choices to pricing and reserving workflows; hand execution to actuarial-analyst

When NOT to Use

• Life insurance, annuities, long-term care, or life contingencies (mortality, reserves by policy) → life-health-insurance or longevity-focused skills
• Triangle workbooks, exhibit production, statutory tie-outs, or model run packs only → actuarial-analyst
• Appointed actuary opinions, regulatory sign-off, or enterprise capital policy → actuary, appointed-chief-actuary
• Enterprise assumption governance, assumption papers, and change control → assumption-setting
• P&C coverage wording, claims handling, underwriting authority, or DOI filing narrative → property-casualty-insurance
• Exam cram or past-exam solutions as the sole deliverable (support professional application; exam study is secondary)
• General data science, ML pipelines, or quant research without actuarial loss-model framing → data-scientist, quantitative-researcher
• Chart design and dashboard craft only → data-visualization
• Credential pathway and exam strategy only → associate-actuary

Related skills

Core Workflows

1. Problem framing (ASTAM-aligned)

Before fitting distributions:

1. Horizon — Short-term (annual or shorter); accident vs calendar year; prospective period for pricing 2. Random variables — Severity \(X\), frequency \(N\), aggregate \(S=\sum X_i\); clarify i.i.d. assumptions 3. Data grain — Claim-level vs policy-period; censoring/truncation (deductibles, limits) 4. Deliverable — Model spec, parameter estimates, diagnostics, business interpretation—not filing sign-off 5. Peer execution — Route spreadsheet builds and filing exhibits to actuarial-analyst

See `references/astam_scope_and_principles.md`.

2. Severity and frequency modeling

1. Explore severity empirical tail; candidate families (exponential, gamma, lognormal, Pareto, generalized Pareto for tail) 2. Explore frequency dispersion; test Poisson vs negative binomial vs mixtures 3. Document moments, tail indices, and parameter stability across segments 4. State dependence assumptions (usually independence for standard compound model; flag if copula needed → escalate) 5. Summarize model selection criteria (AIC/BIC, Anderson–Darling, QQ plots)—not a single automatic pick

See `references/severity_and_frequency_models.md`.

3. Aggregate and compound losses

1. Define compound model \(S = X_1 + \cdots + X_N\) 2. Apply normal approximation when conditions hold; state when it fails (heavy tail, low frequency) 3. Outline FFT and simulation approaches for discrete/continuous severity (conceptual steps) 4. Relate percentiles of \(S\) to risk measures and reinsurance layers (technical only)

See `references/aggregate_loss_models.md`.

4. Credibility and experience rating

1. Choose limited fluctuation, Bühlmann, or Bühlmann-Straub per homogeneity and data structure 2. Compute credibility weights \(Z\); define complement (manual, industry, prior) 3. Blend observed experience with complement for pure premium or loss ratio 4. Document heterogeneity across classes/years and structural parameters

See `references/credibility_and_experience_rating.md`.

5. Ratemaking and short-term reserving (math level)

1. Pure premium indication: frequency × severity with documented adjustments 2. Loss ratio and on-level premium; trend to prospective period 3. Indicated change vs constraints; distinguish technical indication from implemented rate 4. Reserving: chain-ladder factor algebra, expected loss ratio method—link full triangle work to actuarial-analyst

See `references/ratemaking_and_trend.md`.

6. Estimation, diagnostics, and risk measures

1. Fit via MLE (or method of moments where standard); report standard errors when available 2. Run goodness-of-fit and tail diagnostics; document limitations 3. Compute VaR and TVaR at stated confidence levels; interpret for capital layers (non-regulatory) 4. Package assumptions, alternatives, and sensitivity for actuary review

See `references/estimation_diagnostics_and_risk_measures.md`.

Deliverable standards

Label output as technical modeling support, not actuarial opinion, legal advice, or filed regulatory submission.

Assignment type matrix

When to load references

• Scope, ASTAM alignment, principles → references/astam_scope_and_principles.md
• Severity and frequency → references/severity_and_frequency_models.md
• Aggregate and compound losses → references/aggregate_loss_models.md
• Credibility and experience rating → references/credibility_and_experience_rating.md
• Ratemaking, trend, reserving math → references/ratemaking_and_trend.md
• Estimation, GOF, VaR/TVaR → references/estimation_diagnostics_and_risk_measures.md

Aggregate loss models

Table of contents

1. Compound model definition 2. Moments of aggregate S 3. Normal approximation 4. Exact and numerical methods 5. Percentiles and reinsurance 6. Dependence extensions

Compound model definition

Collective risk model:

\[ S = \begin{cases} 0 & N=0 \\ X_1 + \cdots + X_N & N \ge 1 \end{cases} \]

Standard assumptions:

• \(N\) independent of \(\{X_i\}\)
• \(X_i\) i.i.d. and independent of \(N\)

Individual risk model (optional extension): sum over policies with policy-specific \(N_j, X_{jk}\)—use when heterogeneity is policy-specific and not pooled.

Moments of aggregate S

When \(N\) is independent of \(X_i\):

\[ \mathrm{E}[S] = \mathrm{E}[N] \cdot \mathrm{E}[X] \]

\[ \mathrm{Var}(S) = \mathrm{E}[N]\mathrm{Var}(X) + \mathrm{Var}(N)(\mathrm{E}[X])^2 \]

These formulas underpin pure premium (\(\mathrm{E}[S]/\)exposure) and normal approximation variance.

For compound Poisson with Poisson mean \(\lambda\):

\[ \mathrm{Var}(S) = \lambda \mathrm{E}[X^2] = \lambda(\mathrm{Var}(X) + (\mathrm{E}[X])^2) \]

Normal approximation

Approximate:

\[ S \approx \mathcal{N}(\mathrm{E}[S], \mathrm{Var}(S)) \]

When reasonable:

• \(\mathrm{E}[N]\) moderately large
• Severity not extremely heavy-tailed relative to the question (e.g., central percentiles)

When weak:

• Low \(\mathrm{E}[N]\) (few claims)
• Heavy-tailed severity for high quantiles (VaR/TVaR)
• Highly skewed frequency

Report continuity correction only for discrete severity masses when relevant.

Exact and numerical methods

Simulation workflow (conceptual):

1. Fit frequency and severity separately (or joint if specified) 2. For \(b=1,\ldots,B\): draw \(N^{(b)}\), then \(X_1^{(b)},\ldots,X_{N^{(b)}}^{(b)}\); set \(S^{(b)}\) 3. Estimate percentiles and TVaR from \(\{S^{(b)}\}\)

State seed, B, and whether reinsurance transforms \(X\) before aggregation.

Percentiles and reinsurance

Reinsurance transforms severity (e.g., excess-of-loss \(Y=\min(\max(X-a,0), m)\)) before summing—aggregate model must apply transforms at claim level, not on \(S\) unless structure allows.

Link percentile of \(S\) to:

• Risk measures (see estimation_diagnostics_and_risk_measures.md)
• Occurrence limits and aggregate caps (may require separate treatment of frequency caps)

Dependence extensions

Default independence is a material assumption. If:

• Catastrophe drives simultaneous claims → consider common shock or frequency spike models
• Inflation correlates severities across claims → time-series methods may exceed ASTAM scope

Flag dependence needs to actuary; copula or hierarchical models route toward quantitative-researcher with actuarial oversight.

ASTAM scope and principles

Table of contents

1. Purpose and alignment 2. Short-term vs long-term 3. Core random structures 4. Modeling principles 5. Boundaries with peer skills 6. Ethics and reliance

Purpose and alignment

This reference supports advanced short-term actuarial mathematics in the spirit of SOA ASTAM (Advanced Short-Term Actuarial Models): stochastic models for non-life and short-duration health-adjacent coverages where claim counts and severities drive aggregate outcomes.

The skill is tool-agnostic and concept-first. Implementation in R, Python, Excel, or vendor tools belongs in actuarial-analyst unless the user explicitly wants formulas and interpretation only.

Short-term vs long-term

Route life/contingency work to life-health-insurance and qualified actuary review.

Core random structures

Standard collective risk setup:

• Severity \(X\) — payment per claim (possibly censored/truncated by deductible/limit)
• Frequency \(N\) — claim count in a period (often Poisson or negative binomial)
• Aggregate \(S = X_1 + \cdots + X_N\) with \(N\) independent of \(\{X_i\}\) unless modeling dependence explicitly

Policy random variables (per-policy frequency/severity) appear in some curricula; default to collective model unless data structure requires otherwise.

Modeling principles

1. Define the estimand — Pure premium, loss ratio, percentile of \(S\), IBNR factor—not “the model” in isolation 2. Document censoring/truncation — Deductibles and limits change the effective severity distribution 3. Segment homogeneity — Split classes before fitting; credibility handles partial pooling later 4. Tail honesty — Parametric tails extrapolate; justify with diagnostics and sensitivity 5. Prospective vs retrospective — Trend and on-level bridge historical experience to future periods 6. Reproducibility — State data cuts, exclusion rules, and random seeds for simulation

Boundaries with peer skills

Ethics and reliance

• Present results as technical modeling pending actuary review
• Do not substitute for appointed actuary statements or rate filing legal adequacy
• Cite data limitations, regime changes, and immature years in limitations
• Exam preparation may use this material but should not reduce deliverables to memorized exam templates without business context

Credibility and experience rating

Table of contents

1. Experience rating overview 2. Limited fluctuation credibility 3. Bühlmann model 4. Bühlmann-Straub 5. Credibility-weighted estimators 6. Diagnostics and governance

Experience rating overview

Experience rating updates a manual or prior indication with observed loss experience for a class, employer, or policy group.

Standard linear blend:

\[ \hat{\theta}_Z = Z \cdot \bar{X} + (1-Z) \cdot \mu \]

where:

• \(\bar{X}\) — observed experience statistic (pure premium, loss ratio, etc.)
• \(\mu\) — complement (manual rate, industry benchmark, prior approved)
• \(Z \in [0,1]\) — credibility weight

Distinguish gross observed from credibility-weighted final—underwriters implement constraints separately (actuarial-analyst).

Limited fluctuation credibility

Full credibility rules (conceptual): find minimum exposure \(n_0\) such that \(P(|\bar{X}-\mu| \le k\mu) \ge p\) under stated distributional assumptions.

Examples (Poisson frequency, normal severity approximations—verify formulas against company standard):

• Frequency full credibility thresholds on claim counts
• Severity full credibility on claim counts above a minimum

Partial credibility:

\[ Z = \min\left(1, \sqrt{\frac{n}{n_0}}\right) \]

(variants exist—document company formula).

Pros: Simple, transparent Cons: Ignores structural heterogeneity across years/classes

Bühlmann model

Structural model for contract \(i\):

\[ X_{ij} = \theta_i + \varepsilon_{ij} \]

\(\theta_i\) — random risk parameter; \(\varepsilon_{ij}\) — noise.

Homogeneous Bühlmann credibility factor:

\[ Z = \frac{k}{k + n} \]

where \(k\) is a structural parameter estimated from data (ratio of process variance to parameter variance) and \(n\) is exposure measure (e.g., claim counts or earned exposure).

Credibility premium for risk \(i\):

\[ \hat{\theta}_i = Z_i \bar{X}_i + (1-Z_i) \hat{\mu} \]

with \(\hat{\mu}\) the collective estimate.

Report estimated \(k\), within- and between-group variances, and sensitivity of \(Z\) to \(k\).

Bühlmann-Straub

Extends Bühlmann when exposure varies across observations (different payroll, car-years):

• Weights \(w_{ij}\) on observations
• Credibility formulas incorporate weighted means and effective exposures

Use when experience periods have unequal exposure bases—common in commercial lines.

Credibility-weighted estimators

Loss ratio form often pairs with on-level premium and trend (see ratemaking_and_trend.md).

Show before/after credibility table:

Diagnostics and governance

• Credibility too high with thin data → unstable rates; check \(Z\) caps
• Complement stale → biased toward outdated manual
• Outliers in experience → document removal rules before credibility
• Correlation across segments → Bühlmann may understate uncertainty

Assumption changes to complement or \(k\) belong in assumption-setting with actuary approval.

Execution of rating worksheets → actuarial-analyst.

Estimation, diagnostics, and risk measures

Table of contents

1. Estimation methods 2. Maximum likelihood 3. Goodness-of-fit 4. Model diagnostics 5. Risk measures 6. Capital implications (technical)

Estimation methods

Always state sample period, filters, and censoring handling.

Maximum likelihood

For i.i.d. sample \(x_1,\ldots,x_n\) with parameter \(\theta\):

\[ L(\theta) = \prod f(x_i;\theta), \quad \ell(\theta) = \log L(\theta) \]

Severity on censored data: use survival function contributions for observations above deductible or below limit.

Frequency with exposure \(e_i\):

\[ N_i \sim \text{Pois}(\lambda e_i) \Rightarrow \ell(\lambda) = \sum \left( n_i \log(\lambda e_i) - \lambda e_i - \log n_i! \right) \]

Report:

• Point estimates
• Standard errors (Fisher information or bootstrap)
• Correlation across parameters if multivariate

Goodness-of-fit

Discretization for continuous severity affects chi-square—document bin widths.

Overdispersion tests for frequency (variance/mean ratio; formal tests as available).

Do not accept GOF at \(\alpha=0.05\) alone when business tail fails visually.

Model diagnostics

• Residuals — Pearson/deviance for GLM-style fits; outlier flags
• A/E by segment — observed vs model-predicted counts or dollars
• Stability — refit by accident year; parameter drift
• Influence — large claims pull tail parameters
• Cross-validation — holdout years for predictive check (conceptual)

List limitations explicitly: sparse tail, COVID-era distortion, benefit changes (health).

Risk measures

For loss random variable \(S\):

Value at Risk at confidence \(1-\alpha\):

\[ \mathrm{VaR}_\alpha(S) = \inf\{ s : P(S \le s) \ge \alpha \} \]

Tail Value at Risk (TVaR / CTE):

\[ \mathrm{TVaR}_\alpha(S) = \mathrm{E}[S \mid S > \mathrm{VaR}_\alpha(S)] \]

Properties:

• TVaR is coherent subadditive in standard treatments; VaR is not
• Heavy tails: TVaR materially exceeds VaR
• Estimate from simulated \(S\) or parametric tail when model trusted

Report confidence level (e.g., 99.5%) and time horizon (annual aggregate vs per occurrence).

Capital implications (technical)

At technical level only (regulatory capital frameworks → actuary / appointed actuary):

• Higher TVaR relative to mean indicates need for risk load or reinsurance
• Compare TVaR to retention and reinsurance attachment
• Parameter risk not captured in single fitted distribution—sensitivity bands or Bayesian predictive distributions

Do not map VaR/TVaR to RBC or Solvency II charges without qualified regulatory context.

Link aggregate percentiles to aggregate_loss_models.md simulation output.

Ratemaking and trend

Table of contents

1. Pure premium ratemaking 2. Loss ratio and balance point 3. On-level and trend 4. Indicated rate change 5. Short-term reserving math 6. Link to analyst execution

Pure premium ratemaking

Pure premium per exposure unit:

\[ \hat{p} = \hat{\mathrm{E}}[S] / e = \hat{\mathrm{E}}[N] \cdot \hat{\mathrm{E}}[X] / e \]

or credibility-weighted:

\[ \hat{p}_Z = Z \cdot \hat{p}_{\text{obs}} + (1-Z) \cdot p_{\text{manual}} \]

Document:

• Exposure definition (car-year, $1,000 payroll, etc.)
• Loss basis (paid, incurred, reported)
• Reinsurance and large-loss treatment

Loss ratio and balance point

Loss ratio \(LR = L / P\) (losses over premium).

Balance point rate change (conceptual, no expenses):

\[ \text{Indicated factor} \approx \frac{LR_{\text{target}}}{LR_{\text{experience on-level trended}}} \]

Expense-loaded indications add fixed and variable expense ratios and profit provisions—state formula explicitly.

On-level and trend

On-level factor for year \(t\):

\[ OLF_t = \frac{\text{current rate level}}{\text{rate level in year }t} \]

Trend (compound):

\[ TF = (1 + \tau)^{(t_{\text{pros}} - t_{\text{base}})} \]

Document index choice (CPI, industry, internal) and from/to dates.

Sensitivity: ±1 point annual trend on indicated change when material.

Indicated rate change

Present:

1. Gross experience indication 2. Credibility-weighted indication 3. Constraints (caps, floors, competitive) — note but do not treat as actuarial necessity 4. Implemented rate — operational; may differ from indication

\[ \text{Indicated change} = \frac{\hat{p}_Z}{p_{\text{current}}} - 1 \]

or loss-ratio equivalent.

Short-term reserving math

At mathematical level (full triangles → actuarial-analyst):

Chain ladder (volume-weighted age-to-age factor):

\[ \hat{f}_j = \frac{\sum_i C_{i,j+1}}{\sum_i C_{i,j}} \]

Ultimate for origin \(i\):

\[ \hat{U}_i = C_{i,n} \cdot \hat{f}_j \cdots \hat{f}_{n-1} \]

Expected loss ratio (ELR) method:

\[ \hat{U}_i = \text{Earned premium}_i \times ELR_{\text{selected}} \]

Bornhuetter–Ferguson blends ELR prior with observed development—state weighting conceptually.

Tail factor — extension beyond last observed age; sensitivity critical for long-tail lines.

Relate IBNR to ultimate minus reported/paid at valuation date—do not conflate with severity model IBNR noise in pricing data.

Link to analyst execution

This skill supplies formulas and interpretation; actuarial-analyst supplies workpapers and tie-outs.

Severity and frequency models

Table of contents

1. Severity modeling 2. Frequency modeling 3. Mixtures and heterogeneity 4. Censoring and truncation 5. Model selection 6. Common pitfalls

Severity modeling

Goal: Model payment \(X\) per claim (or per loss event) with emphasis on body and tail behavior.

Moments drive aggregate approximations: \(\mathrm{E}[X]\), \(\mathrm{Var}(X)\), and skewness affect normal approximation quality.

Tail behavior:

• Compare empirical tail (mean excess plot, Hill estimator conceptually) to parametric tail
• Document threshold where tail model starts if using GPD-style reasoning
• Sensitivity: shift in high percentiles when tail parameter changes

Frequency modeling

Goal: Model claim count \(N\) in a fixed exposure period.

Poisson with exposure \(e\): often \(N \sim \mathrm{Pois}(\lambda e)\) or rate per unit exposure.

Negative binomial parametrizations differ by software—always state \(\mathrm{E}[N]\) and \(\mathrm{Var}(N)\) in output.

Mixtures and heterogeneity

Mixture models capture unobserved heterogeneity:

• Discrete mixture: e.g., “good” vs “bad” risk subpopulations
• Continuous mixture: gamma–Poisson yields negative binomial frequency

Document identifiability limits: mixtures can fit similarly with different interpretations.

Censoring and truncation

Misapplying uncensored fits to limited data biases severity downward and distorts frequency–severity splits.

Model selection

Use multiple lenses:

1. Graphical — Histogram, density, QQ plot (log scale for severity) 2. Goodness-of-fit — Chi-square (binned), Anderson–Darling, Kolmogorov–Smirnov (note sensitivity to tail) 3. Information criteria — AIC/BIC for nested models; not sufficient alone for tail choice 4. Business — Stability across accident years; interpretability for pricing

Prefer parsimony when tail differences are immaterial to the estimand (e.g., median severity vs 99th percentile of \(S\)).

Common pitfalls

• Fitting severity on incurred including IBNR noise as if claim-level payments
• Ignoring large loss caps or reinsurance recoveries in the severity sample
• Using Poisson when variance clearly exceeds mean
• Extrapolating Pareto tail without sufficient large-loss data
• Mixing accident year and calendar year counts in frequency