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name: advanced-short-term-actuarial-mathematics description: | Guides advanced short-term actuarial mathematics aligned with SOA ASTAM and P&C/health-adjacent modeling—severity and frequency distributions, aggregate and compound loss models, Bühlmann and Bühlmann-Straub credibility, ratemaking and experience rating, short-term reserving at the math level, MLE and goodness-of-fit, and risk measures (VaR, TVaR). Tool-agnostic and concept-first. Use when the user mentions advanced short-term actuarial mathematics, ASTAM, severity model, frequency model, aggregate loss, compound distribution, Bühlmann credibility, experience rating, ratemaking, pure premium, negative binomial frequency, tail factor, TVaR, or short-term actuarial models—not life contingencies (life-health-insurance), Excel workpapers only (actuarial-analyst), appointed actuary sign-off (actuary, appointed-chief-actuary), assumption governance (assumption-setting), P&C legal/operations depth (property-casualty-insurance), or general ML (data-scientist, quantitative-researcher).

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

Need	Skill
Workpapers, triangles, exhibits, model I/O, analyst QA	actuarial-analyst
Sign-off, capital overview, governance memos	actuary
Appointed actuary / chief actuary regulatory framing	appointed-chief-actuary
ASA/FSO exam pathways and professional standards	associate-actuary
Assumption governance and enterprise change control	assumption-setting
P&C lines, underwriting, claims, and policy mechanics	property-casualty-insurance
Statistical/ML modeling beyond standard actuarial methods	quantitative-researcher
General ML and predictive pipelines	data-scientist
Charts, dashboards, and visual design	data-visualization

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

Deliverable	Minimum content
Model specification	Random variables, independence, censoring/truncation, segment definition
Parameter table	Estimates, method, uncertainty, stability notes
Diagnostics	QQ/PP, GOF tests, tail plot, A/E if applicable
Business bridge	Pure premium, credibility blend, indicated change or reserve factor (math only)
Limitations	Data volume, tail extrapolation, regime change, outlier treatment

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

Assignment type matrix

Trigger phrase	Primary workflow	Lead reference
severity model / tail behavior	Severity families and selection	severity_and_frequency_models.md
frequency model / negative binomial	Frequency and dispersion	severity_and_frequency_models.md
aggregate loss / compound distribution	Compound (S)	aggregate_loss_models.md
Bühlmann credibility	Credibility weights	credibility_and_experience_rating.md
experience rating / pure premium	Rating blend	credibility_and_experience_rating.md
ratemaking / trend / on-level	Indication math	ratemaking_and_trend.md
chain ladder / ELR (math)	Reserving formulas	ratemaking_and_trend.md
MLE / goodness-of-fit	Estimation and GOF	estimation_diagnostics_and_risk_measures.md
VaR / TVaR	Risk measures	estimation_diagnostics_and_risk_measures.md

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