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name: advanced-long-term-actuarial-mathematics description: | Guides advanced long-term actuarial mathematics (SOA ALTAM)—survival models, life insurance and annuity APVs, premiums and reserves (equivalence principle, Thiele), multiple decrement and Markov states, yield-curve discounting, mortality improvement, longevity risk, profit testing, and mortality graduation. Tool-agnostic, concept-first. Use when the user mentions advanced long-term actuarial mathematics, ALTAM, survival model, life insurance reserve, annuity valuation, equivalence principle, Thiele equation, multiple decrement, force of mortality, longevity risk, mortality improvement, actuarial present value, or net premium reserve—not ASTAM/P&C (advanced-short-term-actuarial-mathematics), workpapers only (actuarial-analyst), appointed actuary (appointed-chief-actuary), assumption governance (assumption-setting), ALM detail (asset-liability-management), or exam-only deliverables.

Advanced Long-Term Actuarial Mathematics

When to Use

• Build and interpret survival models: (l_x), (q_x), (\mu_x), select vs ultimate mortality
• Value life insurance and annuity benefits (term, whole life, endowment, temporary/whole life annuities)
• Derive premiums and reserves via equivalence principle, Thiele, prospective vs retrospective views
• Model multiple decrements and Markov/multi-state paths (disability, pension, active/retired)
• Apply yield curves and discounting for long-dated liability cash flows
• Frame mortality improvement and longevity risk at a technical level
• Outline profit testing and emerging cost for life products (conceptual)
• Estimate and graduate mortality tables; document data and smoothing choices
• Connect math to pricing, reserving, and pension liability measurement; hand execution to actuarial-analyst

When NOT to Use

• P&C frequency-severity, aggregate loss, credibility, or short-tail reserving math → advanced-short-term-actuarial-mathematics
• Triangle workbooks, exhibits, 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
• ALM duration matching, LDI hedge design, and investment policy depth → asset-liability-management
• Life/health product mechanics, underwriting, or claims without contingency math focus → life-health-insurance
• Pension ERISA, PBGC, fiduciary, or plan design narrative without contingency formulas → pension-retirement-funds
• 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 life-contingency framing → data-scientist, quantitative-researcher
• Credential pathway and exam strategy only → associate-actuary

Related skills

Need	Skill
Workpapers, model I/O, exhibits, 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
Life/health products, benefits, and distribution context	life-health-insurance
DB/DC pensions, funding policy, liability overview	pension-retirement-funds
ALM, duration, LDI, and asset–liability strategy	asset-liability-management
Short-term loss models (ASTAM)	advanced-short-term-actuarial-mathematics
Statistical/ML modeling beyond standard actuarial methods	quantitative-researcher
General ML and predictive pipelines	data-scientist

Core Workflows

1. Problem framing (ALTAM-aligned)

Before deriving formulas:

1. Horizon — Long-duration; valuation vs pricing vs funding purpose
2. State space — Single life, joint life, or multi-state (active/disabled/retired/dead)
3. Benefit/premium basis — Net vs gross; continuous vs discrete; payment timing
4. Mortality basis — Ultimate vs select; table vintage; improvement scale (if any)
5. Interest/discount — Flat rate vs term structure; real vs nominal
6. Deliverable — Formula spec, APV tables, reserve recursion, sensitivity—not filing sign-off
7. Peer execution — Route spreadsheet builds and filing exhibits to actuarial-analyst

See references/altam_scope_and_principles.md.

2. Survival models and mortality

1. Define life table functions: (l_x), (d_x), (q_x), (p_x), (\mu_x)
2. Distinguish ultimate vs select mortality and duration since selection
3. State independence assumptions for joint lives unless modeling dependence
4. Document radix, age basis, and table source (regulatory, industry, experience)
5. Link force of mortality to discrete (q_x) under stated assumptions

See references/survival_models_and_mortality.md.

3. Life insurance and annuities

1. Map benefit types: level term, whole life, endowment, deferred benefits
2. Write actuarial present value (APV) with correct survival and payment timing
3. Cover annuities: temporary, whole life, due vs immediate, varying payments
4. Address joint-life and last-survivor structures at formula level
5. Flag riders and options for product context → life-health-insurance

See references/life_insurance_and_annuities.md.

4. Premiums and reserves

1. Apply equivalence principle for net/gross premiums; define expense and profit loadings
2. Build prospective and retrospective reserves; reconcile where standard
3. Use Thiele's equation for continuous reserves; discrete analogs where needed
4. Distinguish net level premium vs gross premium reserves
5. Summarize deficiency and emerging cost implications (conceptual)

See references/premiums_and_reserves.md.

5. Multiple decrement and state models

1. Set up multiple decrement table: dependent vs independent decrements
2. Compute probabilities of single and combined decrements; associated single decrement rates
3. Extend to Markov multi-state models (transition intensities, Kolmogorov forward equations)
4. Apply to disability, pension, and long-term care state spaces (math only)
5. Hand pension plan law and governance narrative to pension-retirement-funds

See references/multiple_decrement_and_state_models.md.

6. Longevity, improvement, estimation, and discounting

1. Frame mortality improvement scales and longevity risk (framework-level)
2. Describe estimation from experience; graduation and smoothing methods
3. Apply yield curves to long cash flows; note ALM interaction at high level → asset-liability-management
4. Outline profit testing cash flows and sensitivity to mortality and interest
5. Package assumptions for actuary review; governance → assumption-setting

See references/longevity_improvement_and_estimation.md.

Deliverable standards

Deliverable	Minimum content
Model specification	States, benefits, premium type, mortality/interest basis, payment timing
Formula sheet	APV, premium, reserve recursions with defined notation
Numerical illustration	Small worked example or table with stated assumptions
Sensitivity	Mortality, interest, improvement, or lapse drivers (as relevant)
Limitations	Table vintage, extrapolation, select period, data volume

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

Assignment type matrix

Trigger phrase	Primary workflow	Lead reference
survival model / force of mortality	Life table and (\mu_x)	survival_models_and_mortality.md
life insurance APV / endowment	Benefit APVs	life_insurance_and_annuities.md
annuity valuation	Annuity due/immediate	life_insurance_and_annuities.md
equivalence principle / net premium	Premium derivation	premiums_and_reserves.md
Thiele equation / reserve	Reserve recursion	premiums_and_reserves.md
multiple decrement	Decrement table	multiple_decrement_and_state_models.md
Markov / disability / pension states	State models	multiple_decrement_and_state_models.md
mortality improvement / longevity risk	Improvement frameworks	longevity_improvement_and_estimation.md
graduation / mortality table	Estimation and smooth	longevity_improvement_and_estimation.md
yield curve / discount long cash flows	Discounting + ALM cross-ref	longevity_improvement_and_estimation.md

When to load references

• Scope, ALTAM alignment, principles → references/altam_scope_and_principles.md
• Survival models and mortality → references/survival_models_and_mortality.md
• Life insurance and annuities → references/life_insurance_and_annuities.md
• Premiums and reserves → references/premiums_and_reserves.md
• Multiple decrement and state models → references/multiple_decrement_and_state_models.md
• Longevity, improvement, estimation, discounting → references/longevity_improvement_and_estimation.md