Casualty Actuarial Society Credentials (ACAS/FCAS) Exam 5 — Basic Techniques for Ratemaking and Estimating Claim Liabilities Syllabus

State the chain-ladder (development) method estimate of ultimate losses.

$$\hat{U}_i = C_{i,\text{latest}} \times \text{CDF}_i$$ Ultimate for accident year $i$ equals the latest reported (or paid) losses multiplied by the cumulative development factor to ultimate. It assumes future development is proportional to losses reported to date.

What are the key assumptions and weaknesses of the chain-ladder development method?

Assumption: future development is proportional to reported-to-date losses and historical development patterns persist. Weaknesses: highly leveraged for immature years (small CDF errors hugely affect estimates), distorted by changes in case reserving, mix, or payment patterns, and unreliable for the most recent (greenest) years.

State the Bornhuetter-Ferguson (BF) method formula and its core idea.

$$\hat{U}_i = C_i + \widehat{ELR} \times P_i \times \left(1 - \frac{1}{\text{CDF}_i}\right)$$ Reported losses plus expected unreported losses, where expected losses $= \widehat{ELR}\times P_i$ and the unreported portion is $(1 - 1/\text{CDF})$. It credibility-blends actual reported losses with an a priori expectation, stabilizing immature years.

How does the Bornhuetter-Ferguson method relate to the development and expected-loss methods?

BF is a weighted average: $\hat{U} = Z \cdot (\text{development estimate}) + (1-Z)\cdot(\text{expected-loss estimate})$, with $Z = 1/\text{CDF}$ (percent reported). When $\text{CDF}=1$ (fully developed) BF equals the development method; for very immature years it approaches the a priori expected-loss method.

Describe the expected loss ratio (a priori) method for estimating ultimate losses.

$$\hat{U}_i = \widehat{ELR} \times P_i$$ Ultimate losses equal an a priori expected loss ratio multiplied by premium, ignoring actual reported losses entirely. It is stable but unresponsive to emerging experience — useful for the greenest years or new lines.

What is the Cape Cod (Stanard-Bühlmann) method and how does it differ from BF?

Cape Cod is like BF but the expected loss ratio (ELR) is estimated from the data itself rather than selected a priori: $$\widehat{ELR} = \frac{\sum_i C_i}{\sum_i P_i \cdot (1/\text{CDF}_i)}$$ i.e., total reported losses divided by total 'used-up' (on-level, developed) premium. This grounds the prior in the experience, reducing reliance on judgment.

What distinguishes Allocated Loss Adjustment Expenses (ALAE) from Unallocated LAE (ULAE)?

ALAE (now often DCC — Defense & Cost Containment) can be assigned to a specific claim (e.g., defense attorney fees, expert witnesses). ULAE (now A&O — Adjusting & Other) cannot be tied to a specific claim (e.g., claims-department salaries, overhead) and is typically estimated as a ratio to losses.

How is ULAE commonly estimated and provided for in ratemaking?

ULAE is typically estimated as a ratio of ULAE payments to paid losses (or to losses + ALAE) from calendar-year data, then applied to projected losses. The classic ratio approach assumes a portion of ULAE is incurred when a claim opens and the remainder as it pays.

How is ALAE/DCC typically incorporated into loss development and ratemaking?

ALAE is usually combined with losses (developed and trended together via loss+ALAE triangles) because it follows claim development patterns. Alternatively it can be developed in its own triangle or loaded as a ratio to losses. ULAE, lacking claim detail, is applied as a factor.

State the pure premium (loss cost) method for the indicated rate.

$$\text{Indicated Rate} = \frac{\text{Pure Premium} + \text{Fixed Expense per Exposure}}{1 - V - Q}$$ where pure premium = projected (trended, developed) losses+LAE per exposure, $V$ = variable expense ratio, and $Q$ = target profit (and contingencies) provision.

State the loss ratio method for the indicated rate change.

$$\text{Indicated Change} = \frac{\frac{L+E_L}{P} + F}{1 - V - Q} - 1$$ More commonly: $$\text{Indicated Rate Change Factor} = \frac{\text{Experience Loss \& LAE Ratio} + \text{Fixed Expense Ratio}}{1 - V - Q}$$ It indicates a change to current rates rather than an absolute rate.

Contrast the pure premium method and the loss ratio method for rate indication.

Pure premium method produces an indicated rate (absolute dollar amount per exposure) and requires well-defined exposures; uses loss costs per exposure. Loss ratio method produces an indicated change to existing rates and requires on-level premium; needs no exposure definition. They produce identical results given consistent data. Loss ratio is used when exposures are not well-defined; pure premium for new lines without existing rates.

In rate indications, distinguish fixed expenses from variable expenses.

Variable expenses vary directly with premium (e.g., commissions, premium taxes) and are expressed as a percentage of premium ($V$). Fixed expenses are roughly constant per exposure/policy regardless of premium (e.g., general expenses, some underwriting), expressed as dollars per exposure ($F$) or a ratio to premium.

What is the permissible loss ratio (PLR)?

$$\text{PLR} = 1 - V - Q$$ The portion of each premium dollar available to pay losses and LAE after providing for variable expenses ($V$) and target profit/contingencies ($Q$). Used in the loss ratio method as the denominator.

What is the all-variable expense assumption, and why is the fixed/variable split important?

The all-variable assumption treats all expenses as a percent of premium. Splitting fixed vs variable matters because treating fixed expenses as variable overstates expense loads for high-premium risks and understates them for low-premium risks, distorting equity. The fixed/variable split produces more accurate per-policy expense provisions.

What is a profit and contingencies provision, and what does the contingencies portion address?

The profit & contingencies provision ($Q$) is the target underwriting profit margin built into rates. The contingencies portion provides for the expectation that actual costs may systematically exceed expected costs (a load for adverse deviation), distinct from the intended profit, and it should be unbiased over time.

What is the goal of classification ratemaking, and what underwriting principle does it support?

Classification ratemaking groups risks with similar expected costs into classes charged similar rates, achieving rate equity (each insured pays in proportion to expected cost). It supports adverse selection avoidance: failing to differentiate lets low-risk insureds leave and high-risk insureds remain, eroding rate adequacy.

List criteria for selecting/evaluating classification rating variables.

Statistical (significant cost differences, homogeneity within class, credibility, reliability over time), operational (objective, inexpensive/verifiable, non-manipulable), social (privacy, causality, affordability, public acceptability), and legal (compliant with regulation/anti-discrimination law).

Describe the pure premium (loss cost) approach to determining class relativities.

Compute each class's pure premium (loss & LAE per exposure), then express each as a relativity to a base class: $$\text{Relativity}_i = \frac{\text{Pure Premium}_i}{\text{Pure Premium}_{\text{base}}}.$$ Requires accurate exposure data and is unaffected by historical rate differences.

Describe the loss ratio approach to determining class relativities.

Compute each class's loss ratio relative to the base class's loss ratio, then multiply by the base class's current relativity: $$\text{Indicated Relativity}_i = \frac{\text{Loss Ratio}_i}{\text{Loss Ratio}_{\text{base}}}\times \text{Current Relativity}_i.$$ It adjusts existing relativities and needs no exposure data, but premium must be on-level.

What problem does the minimum-bias / generalized linear model (GLM) approach solve in classification ratemaking?

When multiple rating variables are correlated, one-way (univariate) relativities double-count overlapping effects and produce biased relativities. Minimum-bias procedures and GLMs estimate multivariate relativities simultaneously, isolating each variable's independent effect on expected cost and removing distributional bias.

Distinguish multiplicative and additive rating structures for combining class relativities.

Multiplicative: $\text{Rate} = \text{Base} \times R_1 \times R_2 \times \cdots$ (relativities compound; common, avoids negative rates). Additive: $\text{Rate} = \text{Base} \times (1 + a_1 + a_2 + \cdots)$. Choice affects interaction handling; multiplicative assumes effects compound, additive assumes they sum independently.

What is the purpose of individual risk rating, and name its main forms.

Individual risk rating modifies manual (class) rates to reflect an individual risk's own experience or characteristics, improving equity for large/heterogeneous risks. Main forms: schedule rating, experience rating, retrospective rating, and composite rating.

Contrast prospective experience rating with retrospective rating.

Experience (prospective) rating adjusts the upcoming policy's premium based on the insured's past loss experience (a credibility-weighted modification of the manual premium). Retrospective rating adjusts the current policy's final premium after the period based on that policy's own actual losses, subject to a maximum and minimum premium.

State the basic experience rating modification formula and the role of credibility.

$$\text{Mod} = \frac{Z \cdot A + (1-Z)\cdot E}{E}$$ where $A$ = actual losses, $E$ = expected losses, and $Z$ = credibility. Mod $>1$ surcharges, $<1$ credits. Larger risks get higher $Z$ (more weight on their own experience).

State the retrospective rating premium formula.

$$\text{Retro Premium} = (\text{Basic Premium} + \text{Converted Losses} \times \text{LCF} + \text{Excess Loss Premium}) \times \text{Tax Multiplier}$$ subject to a maximum and minimum premium. Converted losses = actual losses × loss conversion factor (for ULAE); LCF loads losses; the result is capped between min and max.

What is the difference between a claim's case reserve, IBNR, and IBNER?

Case reserve: the adjuster's estimate for a specific known open claim. IBNR (Incurred But Not Reported): provision for claims that have occurred but not yet been reported. IBNER (Incurred But Not Enough Reported): development on known claims (changes in case reserves). 'Broad' IBNR often includes both pure IBNR and IBNER.

Define the components: paid losses, case-incurred losses, and ultimate losses.

Paid losses = amounts actually paid to date. Case-incurred (reported) losses = paid + case reserves. Ultimate losses = the final value when all claims are closed = case-incurred + IBNR (including IBNER). Total reserve (unpaid) = Ultimate − Paid.

When comparing paid-loss vs reported (incurred)-loss development methods, what are the trade-offs?

Paid development is unaffected by changes in case-reserve adequacy but develops slowly with higher leverage (long tail) and ignores known case information. Reported (incurred) development uses case reserves so it develops faster and is more responsive, but is distorted by any changes in case-reserving practices or adequacy.

Describe the frequency-severity (disposal/ultimate counts × severity) approach to reserving.

Project ultimate claim counts and ultimate average severity separately, then $$\hat{U} = \hat{N} \times \hat{S}.$$ Counts are developed via a count triangle (or reporting pattern) and severities via a severity triangle (trended). Useful when frequency and severity have distinct trends/patterns and gives insight obscured by aggregate methods.

How are diagnostics used to evaluate and reconcile competing ultimate estimates?

Compare estimates from multiple methods (paid CL, incurred CL, BF, Cape Cod, frequency-severity) and examine diagnostics: paid-to-incurred ratios, ratios of paid/reported to ultimate, average case reserves, closure (disposal) rates, and ultimate loss ratios by year. Reconcile by selecting/weighting methods based on data quality, maturity, and stability — relying on more stable methods (BF/Cape Cod) for immature years and development methods for mature years, and investigating outliers for changes in operations.