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

| 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 \(S = X1 + \cdots + XN\)
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

Source & license

This open-source skill is cataloged on AgentStack and links to its original source — we do not rehost the code.

• Author: daemon-blockint-tech
• Source: daemon-blockint-tech/Agentic-Enteprises-Skill
• License: MIT

Install and usage instructions live in the source repository linked above.