CAS Exam 7 cheat sheet

Expected loss ratio method: ultimate = earned premium × selected loss ratio; appropriate when premium is credible but loss experience is thin, as in new or recently re-underwritten lines.

Chain-ladder (development) method: ultimate = reported-to-date × product of age-to-ultimate development factors; assumes past development patterns persist and that case reserves are adequate.

Bornhuetter-Ferguson: ultimate = reported-to-date + expected unreported, where expected unreported = expected ultimate loss × (1 − 1/age-to-ultimate factor); blends observed emergence with an a priori expectation and stabilizes immature years.

Cape Cod (Stanard-Bühlmann) derives the a priori loss ratio from the data itself: prior loss ratio = Σ reported losses / Σ (premium × reported-to-ultimate percentage), then applies BF logic using that ratio.

Mack model gives a distribution-free estimate of chain-ladder standard error: process variance plus parameter (estimation) variance; it does not assume a distribution but does assume development factors are independent across accident years.

Berquist-Sherman adjustments restate historical data before projecting: adjust reported losses for changes in case-reserve adequacy (via disposal rates or paid-to-incurred ratios) and adjust earned premium for rate-level changes.

Frequency-severity method: ultimate = expected claim count × expected severity; select each component separately, using exposure-based frequency (counts per unit of exposure) and trended severity.

IBNR vs. case reserves: IBNR covers unreported claims and development on known claims; total reserves = case + IBNR; the indicated IBNR is ultimate minus paid minus case, not ultimate minus paid alone.

Test reasonableness of a selected ultimate against implied loss ratios, implied loss ratio by year, and implied development patterns; a selection that requires an implausible loss-ratio trend is suspect.

Discounted reserve = Σ (undiscounted cash flow at time t) / (1 + i)^t; discounting reduces carried reserves and creates equity, so the discount rate and payout pattern must be disclosed and justified.

Investment income on reserves is often allocated to lines or years in ratemaking; the indicated rate level depends on whether income is recognized and on the assumed payment lag.

Calendar-year vs. accident-year effects: reserve strengthening in a calendar year raises that year's incurred losses without changing ultimate for prior accident years; distinguish the two when analyzing trends.

Loss reserve adequacy tests compare carried reserves to a range of indications (e.g., chain-ladder, BF, frequency-severity) and to the actuary's best estimate; a carried reserve below the best estimate implies a deficiency.

Premium valuation: for a portfolio of unearned premium, economic value = unearned premium × (1 − expense ratio) minus the present value of expected losses and expenses; the result can be negative for underpriced business.

Surplus notes and other capital instruments are valued considering subordination, interest deferral, and regulatory approval requirements; treat them as debt-like when assessing financial leverage.

Fair value of reserves under a market-consistent framework uses a current estimate of expected cash flows plus a risk margin for non-hedgeable risk; the risk margin is not the same as a prudence margin in statutory reserves.

Reinsurance recoverables are valued net of collectibility risk; a reinsurer's credit standing and collateral affect the valuation of the recoverable, not just the ceded reserve amount.

Segment-level valuation requires allocated capital: return on capital for a line = (underwriting profit + investment income) / allocated capital, where allocated capital reflects the line's contribution to risk.

Risk categories in an insurer ERM framework: underwriting (reserve and pricing), credit, market, liquidity, operational, and strategic; each needs identification, measurement, monitoring, and reporting.

Economic capital is the amount needed to cover unexpected losses at a chosen confidence level over a horizon; it is not the same as regulatory or rating-agency capital, which use prescribed stresses.

Tail measures: VaR at confidence level p is the loss not exceeded with probability p; TVaR (expected shortfall) is the average loss beyond VaR and is subadditive, so it is coherent while VaR is not.

Risk aggregation: with correlation ρ between two risks, combined volatility = sqrt(σ1² + σ2² + 2ρσ1σ2); perfect correlation gives the sum, independence gives the square-root sum, and diversification benefit is the difference.

Copulas model dependence between risk factors; the choice of copula (Gaussian vs. t) changes tail dependence even when marginals are identical, so tail dependence must be tested, not assumed.

Scenario and stress testing complements models: reverse stress tests identify the scenario that exhausts capital, and scenario selection should target concentrations and tail events the model may understate.

Risk appetite framework links tolerance to limits: risk appetite is the aggregate statement, risk tolerances are the quantified boundaries, and risk limits are the operational triggers with escalation procedures.

ERM governance requires board-approved risk policies, a chief risk officer with independence from profit centers, and reporting that escalates breaches to the board or risk committee.

Catastrophe models produce an exceedance probability curve; key outputs are AAL (average annual loss), OEP/ AEP at return periods, and PML at a chosen return period; model uncertainty must be communicated with results.

Expected loss ratio method: ultimate = earned premium × selected loss ratio; appropriate when premium is credible but loss experience is thin, as in new or recently re-underwritten lines.

Chain-ladder (development) method: ultimate = reported-to-date × product of age-to-ultimate development factors; assumes past development patterns persist and that case reserves are adequate.

Bornhuetter-Ferguson: ultimate = reported-to-date + expected unreported, where expected unreported = expected ultimate loss × (1 − 1/age-to-ultimate factor); blends observed emergence with an a priori expectation and stabilizes immature years.

Cape Cod (Stanard-Bühlmann) derives the a priori loss ratio from the data itself: prior loss ratio = Σ reported losses / Σ (premium × reported-to-ultimate percentage), then applies BF logic using that ratio.

Mack model gives a distribution-free estimate of chain-ladder standard error: process variance plus parameter (estimation) variance; it does not assume a distribution but does assume development factors are independent across accident years.

Berquist-Sherman adjustments restate historical data before projecting: adjust reported losses for changes in case-reserve adequacy (via disposal rates or paid-to-incurred ratios) and adjust earned premium for rate-level changes.

Frequency-severity method: ultimate = expected claim count × expected severity; select each component separately, using exposure-based frequency (counts per unit of exposure) and trended severity.

IBNR vs. case reserves: IBNR covers unreported claims and development on known claims; total reserves = case + IBNR; the indicated IBNR is ultimate minus paid minus case, not ultimate minus paid alone.

Test reasonableness of a selected ultimate against implied loss ratios, implied loss ratio by year, and implied development patterns; a selection that requires an implausible loss-ratio trend is suspect.

Discounted reserve = Σ (undiscounted cash flow at time t) / (1 + i)^t; discounting reduces carried reserves and creates equity, so the discount rate and payout pattern must be disclosed and justified.

Investment income on reserves is often allocated to lines or years in ratemaking; the indicated rate level depends on whether income is recognized and on the assumed payment lag.

Calendar-year vs. accident-year effects: reserve strengthening in a calendar year raises that year's incurred losses without changing ultimate for prior accident years; distinguish the two when analyzing trends.

Loss reserve adequacy tests compare carried reserves to a range of indications (e.g., chain-ladder, BF, frequency-severity) and to the actuary's best estimate; a carried reserve below the best estimate implies a deficiency.

Premium valuation: for a portfolio of unearned premium, economic value = unearned premium × (1 − expense ratio) minus the present value of expected losses and expenses; the result can be negative for underpriced business.

Surplus notes and other capital instruments are valued considering subordination, interest deferral, and regulatory approval requirements; treat them as debt-like when assessing financial leverage.

Fair value of reserves under a market-consistent framework uses a current estimate of expected cash flows plus a risk margin for non-hedgeable risk; the risk margin is not the same as a prudence margin in statutory reserves.

Reinsurance recoverables are valued net of collectibility risk; a reinsurer's credit standing and collateral affect the valuation of the recoverable, not just the ceded reserve amount.

Segment-level valuation requires allocated capital: return on capital for a line = (underwriting profit + investment income) / allocated capital, where allocated capital reflects the line's contribution to risk.

Risk categories in an insurer ERM framework: underwriting (reserve and pricing), credit, market, liquidity, operational, and strategic; each needs identification, measurement, monitoring, and reporting.

Economic capital is the amount needed to cover unexpected losses at a chosen confidence level over a horizon; it is not the same as regulatory or rating-agency capital, which use prescribed stresses.

Tail measures: VaR at confidence level p is the loss not exceeded with probability p; TVaR (expected shortfall) is the average loss beyond VaR and is subadditive, so it is coherent while VaR is not.

Risk aggregation: with correlation ρ between two risks, combined volatility = sqrt(σ1² + σ2² + 2ρσ1σ2); perfect correlation gives the sum, independence gives the square-root sum, and diversification benefit is the difference.

Copulas model dependence between risk factors; the choice of copula (Gaussian vs. t) changes tail dependence even when marginals are identical, so tail dependence must be tested, not assumed.

Scenario and stress testing complements models: reverse stress tests identify the scenario that exhausts capital, and scenario selection should target concentrations and tail events the model may understate.

Risk appetite framework links tolerance to limits: risk appetite is the aggregate statement, risk tolerances are the quantified boundaries, and risk limits are the operational triggers with escalation procedures.

ERM governance requires board-approved risk policies, a chief risk officer with independence from profit centers, and reporting that escalates breaches to the board or risk committee.

Catastrophe models produce an exceedance probability curve; key outputs are AAL (average annual loss), OEP/ AEP at return periods, and PML at a chosen return period; model uncertainty must be communicated with results.