Actuarial science Study Guide

📖 Core Concepts Actuarial Science – application of mathematics & statistics to quantify risk for insurance, pensions, finance, health, etc. Actuary – professional who evaluates & manages risk; must pass rigorous probability‑focused exams. Deterministic vs. Stochastic Models – deterministic use fixed values; stochastic incorporate probability distributions to capture uncertainty (standard since the 1980s). Life Table – tabulated probabilities of death/survival at each age; foundation for pricing life‑insurance, annuities, and pensions. Present Value (PV) – current worth of a future cash flow: $$PV = \frac{C}{(1+i)^n}$$ where C = future cash flow, i = discount rate, n = periods. Commutation Functions – pre‑calculated tables of summed discounted survival/death probabilities; speed up premium, reserve, and annuity calculations. Actuarial Control Cycle – iterative steps: problem definition → data collection → model building → result evaluation → monitoring. Arbitrage‑Free Principle – assets and liabilities with identical cash‑flow patterns must have identical prices; no risk‑free profit opportunities. 📌 Must Remember Key Exams – focus on probability & predictive analysis; passing them is required for professional credentialing. Life‑Insurance Core Tasks – mortality analysis → life tables → apply compound interest for pricing. Health‑Insurance Variables – disability, morbidity, mortality, fertility rates drive premium setting. Pension Influencers – bond rates, funded status, demographics, labor negotiations, tax code, macro‑economics. Property & Casualty Split – personal lines (auto, homeowners, etc.) vs. commercial lines (liability, workers’ comp, D&O). Stochastic Modelling Goal – estimate distribution of future losses/liabilities, not just a single “best‑guess”. Arbitrage‑Free Rule – identical cash‑flows ⇒ identical price; violates if different discount rates are arbitrarily chosen. 🔄 Key Processes Build a Life Table Gather mortality data → compute $qx$ (probability of death at age x) → derive $lx$ (survivors) → calculate $dx$, $px$, and $ex$ (expected remaining years). Price a Life‑Annuity (using commutation functions) Locate $Dx$ (present value of $1$ payable at death) → compute $a{\overline{n}|}$ = $Dx / D{x+n}$ → multiply by benefit amount. Run a Stochastic Simulation (e.g., Monte Carlo) Define probability distributions for key risks → generate many random scenarios → aggregate outcomes → derive percentiles for capital requirement. Actuarial Control Cycle Define problem → Collect data → Model (deterministic or stochastic) → Evaluate (PV, solvency, profit) → Monitor & adjust as experience emerges. 🔍 Key Comparisons Deterministic vs. Stochastic Deterministic: single point estimate, no randomness. Stochastic: distribution of possible outcomes, captures risk. Personal Lines vs. Commercial Lines (P&C) Personal: individuals, standard policies (auto, homeowners). Commercial: businesses, complex coverage (product liability, D&O). Life‑Insurance vs. Health‑Insurance Actuarial Work Life: mortality & longevity focus, long‑term cash flows. Health: morbidity, disability, utilization rates, shorter‑term expense patterns. ⚠️ Common Misunderstandings “Higher discount rate always lowers liabilities.” – True for PV, but using an unrealistic rate can violate the arbitrage‑free principle and misprice assets. “Deterministic models are obsolete.” – Still useful for quick approximations; stochastic adds depth but requires more data. “Commutation functions are only historical.” – They remain core for quick manual calculations and as checks on software outputs. 🧠 Mental Models / Intuition “Cash‑flow equivalence” – Treat any series of payments (assets or liabilities) as a single “stream”; if two streams are identical, they must cost the same. “Risk as a distribution, not a number.” – Visualize outcomes as a bell curve; the width (variance) matters as much as the mean. “Control Cycle as a feedback loop.” – Think of it like a thermostat: you set a target, measure, adjust, and repeat. 🚩 Exceptions & Edge Cases Discount Rate Choice – For pension liabilities, some regulators require a lower “risk‑free” rate (e.g., Treasury yield) rather than an assumption‑dependent rate. Catastrophe Modeling – Standard mortality tables do not apply; specialized severity/frequency models are needed. Health‑Insurance Pricing – Rapid changes in medical technology can render historical morbidity data less predictive. 📍 When to Use Which Deterministic vs. Stochastic – Use deterministic for quick pricing or regulatory reporting; switch to stochastic when assessing capital, solvency, or when risk distribution matters. Commutation Functions vs. Full Software – Use commutation tables for manual checks, small‑scale problems, or exam settings; rely on software for large portfolios or complex guarantees. Life Table vs. Experience Rating – Use standard life tables for new or low‑volume policies; apply experience rating when sufficient claim history exists. 👀 Patterns to Recognize “Same cash‑flow, different discount” → red flag for arbitrage violation. Increasing “mortality improvement” factors → signals need to adjust life‑insurance premiums downward over time. Spike in claim frequency coinciding with external events (e.g., hurricanes) → indicates catastrophe risk exposure. 🗂️ Exam Traps Choosing a higher discount rate to “make numbers look better.” – Exam questions often test understanding that the rate must be market‑consistent; an arbitrary high rate is wrong. Confusing $qx$ (death probability) with $px$ (survival probability). – Remember $px = 1 - qx$. Assuming “personal lines = low risk.” – Many exams include personal‑line catastrophes (e.g., hurricane damage) to test awareness of specialty risks. Mixing up commutation symbols – $Dx$ = discounted survival; $Cx$ = discounted death. Swapping them leads to incorrect annuity values. --- This guide pulls directly from the provided outline and is optimized for rapid review before an actuarial exam.