CAS Exam 7: Estimation of Liabilities study guide

Exam 7 tests liability estimation beyond basic chain-ladder mechanics: selecting among expected-loss-ratio, chain-ladder, Bornhuetter-Ferguson (BF), Cape Cod and stochastic methods, diagnosing development patterns, and quantifying reserve uncertainty. Answers are judged on whether the method's assumptions match the data and whether uncertainty is communicated clearly.

Method mechanics

Chain ladder applies cumulative development factors to reported amounts; it is most credible for mature, stable accident years.
Expected loss ratio (ELR) sets ultimate as premium multiplied by ELR, independent of reported losses.
BF adds reported-to-date to ELR × premium × (1 − percent reported); it shrinks immature estimates toward the prior.
Cape Cod derives an implied ELR from reported losses divided by percent reported and premium, then applies BF; credibility weighting controls how much the year's own data affect the ELR.
Mack decomposes prediction error into process variance and parameter variance; bootstrap resampling produces a full predictive distribution.

Worked example

Accident year 2024 has premium 1,000, selected ELR 65%, reported loss 200 at 24 months, and selected percent reported 20%. Expected ultimate is 650. Chain ladder gives 200 / 0.20 = 1,000, a volatile result because only one fifth of the year is reported. BF IBNR is 650 × 0.80 = 520, and BF ultimate is 200 + 520 = 720. The BF result exceeds the prior 650 because reported losses already exceed the 130 expected at 20% reported, but it is far below chain ladder. If reported were 100 instead, chain ladder would be 500 while BF would be 620, showing BF's stabilising effect.

Traps

Mixing paid and incurred development factors without adjusting for case-reserve adequacy changes.
Double counting the tail factor when the selected CDFs already include tail development.
Using an ELR from a different mix, territory or on-level premium.
Treating Mack's standard error as a full predictive distribution; it is an approximation under independence assumptions.
Ignoring negative incremental losses or changing case-reserve practices when selecting factors.

Valuation

Exam 7 valuation questions test how actuaries convert expected cash flows and capital requirements into a fair value or appraisal value for insurance liabilities and blocks of business. The focus is discounted cash flow (DCF), cost-of-capital risk margins, and the distinction between a reserve estimate and economic value.

Building blocks

Start with expected cash flows by period, including losses, expenses, taxes and reinsurance recoveries, on a consistent basis.
Discount at a rate consistent with the risk already reflected in the cash flows; a risk-free rate plus an explicit risk margin avoids double counting.
The cost-of-capital method sets the risk margin as the discounted cost of holding required capital until the liability runs off.
Fair value equals the present value of expected cash flows plus the risk margin. Actuarial appraisal value also reflects the value of new business and adjusts for frictional costs and taxes.
IFRS 17 separates the fulfilment cash flows (expected present value plus risk adjustment) from the contractual service margin, which represents unearned profit.

Worked example

A block requires capital of 100, 60 and 20 at the end of years 1, 2 and 3. The cost of capital is 6% and the discount rate is 4%. The risk margin is 0.06 × (100/1.04 + 60/1.04² + 20/1.04³) = 0.06 × (96.15 + 55.47 + 17.78) = 0.06 × 169.40 = 10.16. If the present value of expected net outflows is 500, the fair value liability is 510.16. The capital amounts must follow the actual release pattern; holding 100 in every year would overstate the margin.

Traps

Discounting at a risk-adjusted rate and adding a risk margin for the same risk.
Applying the cost of capital to total invested assets rather than to required capital.
Using undiscounted reserves as fair value or ignoring the timing of reinsurance recoveries.
Forgetting taxes in an appraisal value, or mixing going-concern and run-off assumptions.
Treating the IFRS 17 contractual service margin as a risk adjustment.

Enterprise Risk Management

Exam 7 ERM tests whether candidates can connect risk appetite, capital, reinsurance and risk aggregation into a coherent framework rather than listing risk categories. Questions often ask for economic capital, diversification benefit, stress-test design or the limitations of a risk measure.

Framework and measurement

A risk appetite statement sets qualitative tolerances and quantitative limits for earnings volatility, capital adequacy, liquidity and concentration.
Economic capital is the capital needed to absorb losses at a chosen confidence level over a horizon; it is not the same as regulatory or accounting capital.
VaR is a quantile and is not subadditive; TVaR (expected shortfall) captures tail severity and is coherent.
Aggregation depends on dependence. A normal copula with low correlation can understate catastrophe and operational tail dependence.
Reinsurance reduces net risk but introduces counterparty credit risk, basis risk and reinstatement cost.

Worked example

Two lines have standalone economic capital of 100 and 80. With a linear correlation of 0.5, combined capital is √(100² + 80² + 2 × 0.5 × 100 × 80) = √24,400 = 156.2. The diversification benefit is 180 − 156.2 = 23.8, or 13.2% of standalone capital. If correlation rises to 0.9, combined capital is √(10,000 + 6,400 + 14,400) = 175.5 and the benefit falls to 4.5. A stress scenario with both lines in tail simultaneously produces a similar result without assuming a distribution.

Traps

Presenting VaR as subadditive or ignoring tail dependence in catastrophe aggregation.
Counting a reinsurance recoverable at full value without a counterparty credit adjustment.
Confusing risk appetite (what risk the firm wants) with risk tolerance (how much deviation is allowed).
Double counting reinsurance: reducing cash flows and also reducing capital for the same recoverable.
Using a single correlation matrix for all stresses, including extreme scenarios.