IFoA SP1 cheat sheet

Gross premium for a health product = expected claim cost + expected expense load + risk margin + profit margin − expected investment return on reserves; each element is a cash-flow projection, not a single-year figure.

Morbidity vs mortality: health business is driven by inception/continuance rates (incidence of sickness, recovery, lapse), so a multiple-decrement or multiple-state model is used rather than a single life table.

PMI (private medical insurance) pricing is dominated by claim inflation, age-band mix, medical trend, and provider/consultant fee schedules; community rating requires pooling across the whole insured group.

Critical illness pricing requires the incidence rate of each defined condition, the survival period (e.g. 14 or 28 days), and the claims ratio at diagnosis; dread-disease models are stand-alone or accelerated.

Income protection: benefit is usually a proportion of pre-disability earnings (commonly 50-75%) with a deferred/waiting period and an offset for state benefits; deferred periods reduce claim cost materially.

Long-term care: incidence rises steeply with age, so a multi-state model with states (healthy, disabled, dead) is standard; claim cost depends on the assumed cost of care and the average duration of dependency.

Health insurers face regulatory capital and reserving constraints (e.g. solvency capital for premium and reserve risk, catastrophe risk); reserves are typically held on a best-estimate plus risk-margin basis.

Experience monitoring: analyse claims per life-year by age/sex/duration, split into frequency and severity; a rising claims ratio can reflect either higher incidence or higher average claim cost.

Net premium reserve (prospective) = EPV(future benefits) − EPV(future net premiums); the retrospective reserve = accumulated past premiums − accumulated past benefits, and the two agree on the same basis.

Recursive reserve relation: (V_t + P)(1+i) = q·S + p·V_{t+1}, where V is the reserve, P the premium, S the sum assured, q the mortality rate; solve forward or backward for reserves.

Gross premium is found from EPV(gross premiums) = EPV(benefits) + EPV(expenses) + EPV(profit/risk margin), using the pricing basis (best-estimate or prudent assumptions).

Unit-linked business: the reserve is the unit fund value plus a non-unit reserve for expected future expense and mortality strains; the insurer bears expense and persistency risk, the policyholder bears investment risk.

With-profits: asset shares accumulate premiums net of charges and cost of cover with actual investment returns; smoothing and estate distribution determine payouts, and the estate must be managed for solvency.

Profit testing uses a risk discount rate to compute the NPV of projected cash flows; the internal rate of return is the discount rate giving zero NPV, and the break-even year is when cumulative cash flow turns positive.

Mortality improvement: use a projection model (e.g. cohort-based) rather than a static table; annuity pricing is very sensitive to the assumed rate of future improvement.

Options and guarantees (e.g. guaranteed annuity options, minimum guaranteed returns) must be valued with stochastic or option-pricing methods, not deterministic best-estimate projections alone.

Technical provisions = best-estimate liabilities + risk margin; best estimate includes outstanding claims (reported but not settled) plus IBNR (incurred but not reported) plus UPR for unexpired risk.

Chain-ladder: develop cumulative paid or incurred claims by development factor f_k = Σ C_{i,k+1} / Σ C_{i,k}; ultimate = latest cumulative × product of remaining factors.

Bornhuetter-Ferguson blends an a-priori expected loss ratio with reported claims: ultimate = reported + (expected ultimate × % unreported), which stabilises estimates for immature years.

Expected loss ratio method: ultimate = earned premium × selected loss ratio; used when little credible claims data exists (new classes, thin triangles).

Premium = expected loss cost (frequency × severity) + expense load + risk/profit margin; for large commercial risks, experience rating and exposure rating are combined.

Catastrophe risk is modelled with frequency-severity or event-based simulation; reinsurance (quota share, surplus, excess of loss, stop loss) changes the net retained distribution and must be reflected in capital and pricing.

Discounting of claim reserves is permitted only where cash flows are reliably predictable; otherwise undiscounted best estimates are used, and the risk margin is added separately.

Underwriting cycle: soft markets compress margins as capacity grows; combined ratio = losses/earned premium + expenses/earned premium, and a ratio above 100% means an underwriting loss.

Duration (Macaulay) D = Σ t·PV(CF_t) / Σ PV(CF_t); modified duration D* = D/(1+i) and approximates the % price change for a small change in yield.

Convexity corrects duration: ΔP/P ≈ −D*·Δy + ½·C·(Δy)²; positive convexity means gains from yield falls exceed losses from equal yield rises.

Immunisation requires matching present value, duration (and ideally convexity) of assets and liabilities; it is exact only for small, parallel yield shifts.

No-arbitrage pricing: a derivative's price equals the cost of a replicating portfolio; under the risk-neutral measure, expected discounted payoffs equal today's price.

Capital Asset Pricing Model: E[R_i] = r_f + β_i·(E[R_m] − r_f), where β = Cov(R_i,R_m)/Var(R_m); only systematic risk is priced.

Put-call parity: C − P = S − K·e^{−rT} for European options on a non-dividend-paying stock; a violation implies an arbitrage.

Term structure: spot rates, forward rates and par yields are linked by (1+s_n)^n = Π(1+f_k); the expectations, liquidity preference and market segmentation theories explain the shape.

Risk measures: VaR is a quantile of the loss distribution and ignores the tail beyond it; TVaR/expected shortfall is the average loss beyond VaR and is sub-additive.

DB funding: contribution = benefits paid + expenses − investment income + change in deficit, where deficit = past-service liability − assets, valued on the scheme's funding basis.

Accrual methods: final salary (benefit linked to final pay) vs career average (CARE, each year's accrual revalued); the choice changes cost distribution across ages and salary-inflation sensitivity.

Funding valuation discount rate: a higher assumed discount rate lowers the liability; the rate should reflect the scheme's expected asset return or a risk-free/gilt-based rate depending on the funding objective.

Scheme funding (e.g. UK-style) requires a technical provisions calculation, a recovery plan for any deficit, and a statement of funding principles; the employer covenant underpins the recovery plan.

Risk transfer: buy-in (bulk annuity held as an asset) vs buy-out (liabilities and assets transferred to an insurer); both convert investment and longevity risk into insurer credit risk.

Longevity risk is the dominant DB risk; it is managed via longevity swaps, reinsurance, or scheme design (e.g. caps on pensionable pay increases).

State pension schemes are usually pay-as-you-go: current contributions fund current benefits, with an unfunded liability dependent on demographic and economic assumptions.

Auto-enrolment / DC adequacy: replacement income depends on contribution rate, investment return, charges and annuity/decumulation terms; small charge differences compound materially over a working life.