Institute and Faculty of Actuaries (IFoA) Exams General Insurance Specialism (SP7, SP8, SA3) Syllabus

What is bootstrapping in the context of stochastic reserving and what does it produce?

A simulation technique that resamples (with replacement) the residuals of a fitted reserving model to generate many pseudo-datasets, refitting each to build an empirical predictive distribution of the reserves. It produces estimates of reserve variability and percentiles (e.g. for risk margins / capital) without strong distributional assumptions.

Distinguish the standard error of prediction from the standard error of estimation in reserving.

Estimation error (parameter error) reflects uncertainty in the fitted parameters. Prediction error additionally includes process error (random variation of future claims about their mean). Mean square error of prediction $=$ process variance $+$ estimation variance, so prediction error $\geq$ estimation error.

What is a "risk margin" in reserving and why is it held?

An amount added to the best-estimate (mean) reserve to allow for uncertainty, so the total provision is more likely to be sufficient. Under Solvency II it is calculated using a cost-of-capital approach reflecting the cost of holding capital to support the run-off of the liabilities until settlement.

List several factors an actuary should consider that may distort a run-off triangle.

Changes in: claims-handling/reserving philosophy, mix of business, policy terms (limits, excesses), rate of settlement (speed-up/slow-down), inflation, large/catastrophe losses, court/legislative changes, and reinsurance arrangements. Each can break the assumption that past development predicts future development.

What is the difference between gross, net, and ceded claims reserves?

Gross: reserves before any reinsurance recoveries. Ceded: the reinsurers' share of those reserves. Net: gross minus ceded, i.e. the amount retained by the insurer. Reserving and capital must consider all three, including the credit (default) risk on reinsurance recoveries.

Define IBNR and IBNER, and state how they differ.

IBNR (incurred but not reported): claims for events that have occurred but not yet been reported to the insurer. IBNER (incurred but not enough reported): expected future development (up or down) on claims already reported and case-estimated. Pure IBNR plus IBNER make up the total reserve beyond case estimates.

What are the main components of an insurer's technical reserves (provisions)?

Outstanding (reported) claims reserve (case estimates), IBNR/IBNER, unearned premium reserve (UPR) for unexpired risk, additional unexpired risk reserve (AURR/URR) if premiums are inadequate, claims handling expense reserve, and any reinsurance recoveries shown separately.

Define the loss ratio, expense ratio, and combined ratio.

Loss ratio $=\dfrac{\text{incurred claims}}{\text{earned premium}}$; expense ratio $=\dfrac{\text{expenses (+ commission)}}{\text{written or earned premium}}$; combined ratio $=$ loss ratio $+$ expense ratio. A combined ratio above $100\%$ indicates an underwriting loss before investment income.

What is the fundamental risk premium relationship in general insurance pricing?

$$\text{Risk premium} = \text{frequency} \times \text{severity}$$ i.e. the expected claims cost per unit of exposure equals the expected number of claims times the expected cost per claim. The office premium then adds loadings for expenses, profit, contingencies and cost of capital, less investment income.

Describe the burning cost approach to pricing.

Burning cost $=\dfrac{\text{past claims}}{\text{past exposure}}$, often expressed per unit of exposure or as a rate on premium/sum insured. Past claims are adjusted (developed to ultimate, inflated, and re-based for current terms) and the resulting rate is loaded for expenses and profit. It is simple but data-hungry and ignores risk heterogeneity.

Explain the frequency-severity (risk-cost) modelling approach to pricing and its advantage over burning cost.

Claim frequency and claim severity are modelled separately, typically using generalised linear models (e.g. Poisson/negative binomial for frequency, gamma/lognormal for severity), then combined. The advantage is that the effect of rating factors on each component is captured explicitly, giving more accurate and granular risk-based prices.

What is a generalised linear model (GLM) and why is it the standard pricing tool?

A GLM relates a response (frequency or severity) to rating factors via a linear predictor and a link function: $g(E[Y])=\beta_0+\sum\beta_j x_j$, with the response from the exponential family. A log link gives a multiplicative rating structure. GLMs handle non-normal data, multiple correlated factors, and exposure offsets, producing statistically robust relativities.

Why is a log link function commonly used in frequency and severity GLMs?

The log link gives a multiplicative model: $E[Y]=\exp(\beta_0+\sum\beta_j x_j)=e^{\beta_0}\prod e^{\beta_j x_j}$. This matches how insurers build rating tables (base rate $\times$ relativities), keeps predicted values positive, and makes factor effects easy to interpret as percentage loadings/discounts.

Distinguish a rating factor from a risk factor.

A risk factor is any characteristic that genuinely affects the expected cost of claims. A rating factor is a measurable proxy actually used in the rating structure (must be objective, verifiable, practical and legally permissible). Some true risk factors cannot be used as rating factors (e.g. legally prohibited), so proxies are chosen.

What adjustments must be made to historical claims data before it can be used to set rates?

Develop claims to ultimate (IBNR/IBNER), adjust past payments for claims inflation to the future rating period, restate for changes in cover/terms and large/catastrophe loss loadings, allow for changes in mix and exposure measurement, and ensure consistency between the claims and exposure datasets (earned vs written, on-level premiums).

What is the purpose of capping or removing large losses when fitting a severity model, and how are they reintroduced?

Large/cat losses are volatile and would distort the modelling of attritional claims, so they are capped at a threshold or removed for fitting the GLM. They are then reintroduced via a separate large-loss loading (e.g. expected large-loss cost spread across the portfolio) so the total premium still reflects their expected cost.

Define exposure and credibility as used in pricing.

Exposure is the measure of the amount of risk (e.g. vehicle-years, sum insured, payroll) over which premium is charged. Credibility is the weight $Z$ given to the entity's own experience versus a wider base: $\text{rate}=Z\times(\text{own experience})+(1-Z)\times(\text{base})$, with $Z$ increasing with volume/stability of data.

How is non-proportional (excess-of-loss) reinsurance priced using the exposure rating method?

Exposure rating uses exposure (risk profile / sum insured bands) together with an industry exposure curve (e.g. an ILF or first-loss scale) to estimate the proportion of expected losses falling in the reinsured layer, without relying on the cedant's own large-loss history. It is useful when the cedant has little credible loss experience in the layer.

How does the experience rating method price an excess-of-loss layer?

It uses the cedant's own historical large losses, developed to ultimate and indexed for inflation, to estimate the expected cost of losses falling in the layer (the burning cost to the layer). It is credible where there is sufficient loss history in/around the layer; lower layers usually have enough data, higher layers do not.

What is an increased limit factor (ILF), and how is it used in reinsurance/liability pricing?

An ILF is the ratio of expected losses limited at a higher limit to those limited at a base limit: $\text{ILF}(L)=\dfrac{E[\min(X,L)]}{E[\min(X,B)]}$. It scales a known base-limit premium up to higher limits and lets actuaries derive the expected cost of a layer as the difference between ILFs at the layer's upper and lower bounds.

What is an exposure curve (e.g. an MBBEFD or first-loss curve) and what is it used for?

An exposure curve $G(d)$ gives the expected proportion of total loss retained by a deductible $d$ (as a fraction of sum insured), so $1-G(d)$ is the proportion ceded above $d$. It allows allocation of premium between primary and excess layers in proportional/per-risk reinsurance and in deductible/limit pricing when individual loss data is scarce.

How does the time value of money / investment income enter general insurance pricing?

Premiums are received before claims are paid, especially for long-tailed lines, so the insurer earns investment income on the float. The technical premium can be discounted: future expected claims and expenses are present-valued at an appropriate (risk-free) rate, reducing the premium needed relative to an undiscounted basis.

What loadings convert a risk premium into an office (gross) premium?

Office premium $=$ risk premium $+$ expenses (acquisition, administration, claims handling) $+$ commission $+$ profit/contingency loading $+$ cost of capital $-$ investment income credit, plus any reinsurance cost and allowance for the cost of options/guarantees and adverse selection. Taxes (e.g. IPT) are then added on top.

Why and how is reinstatement premium reflected when pricing an excess-of-loss treaty?

If a claim erodes the layer, a reinstatement premium is payable to restore cover, so the reinsurer effectively receives extra premium when losses occur. Pricing must allow for the expected reinstatement income (and the limited number of reinstatements), reducing the required up-front premium and capping the reinsurer's total exposure to (1 + number of reinstatements) $\times$ limit.

What is the purpose of capital modelling for a general insurer?

To quantify the capital needed to remain solvent at a chosen confidence level over a time horizon given all material risks, to support business decisions (pricing, reinsurance, capital allocation, risk appetite), and to satisfy regulatory and rating-agency requirements. Internal models aggregate risks allowing for dependencies and produce a full distribution of outcomes.

Define Value at Risk (VaR) and Tail Value at Risk (TVaR / Expected Shortfall).

$\text{VaR}_\alpha(X)$ is the $\alpha$-quantile of the loss distribution: the loss not exceeded with probability $\alpha$. $\text{TVaR}_\alpha(X)=E[X\mid X>\text{VaR}_\alpha]$ is the average loss in the tail beyond the VaR. TVaR is coherent (sub-additive) and captures tail severity, whereas VaR does not.

What are the main risk categories typically modelled in a general insurer's internal capital model?

Insurance/underwriting risk (premium/reserve/catastrophe risk), market risk (asset values, interest rates, FX), credit/counterparty risk (including reinsurer default), operational risk, and liquidity risk. These are modelled individually then aggregated allowing for dependencies (e.g. via correlation matrices or copulas).

What is a copula and why is it used in capital modelling?

A copula is a function that joins marginal distributions into a multivariate joint distribution, specifying the dependence structure separately from the marginals. It is used to aggregate risks with realistic, often tail-dependent, relationships (e.g. a Gumbel copula for joint extreme losses), avoiding the limitation of linear correlation under normality.

Under Solvency II, what are the two capital requirement levels and what do they represent?

The SCR (Solvency Capital Requirement) is the capital to withstand a 1-in-200 year (VaR at $99.5\%$ over one year) adverse outcome; breaching it triggers supervisory action. The MCR (Minimum Capital Requirement) is a lower floor (calibrated to roughly $85\%$ VaR) below which authorisation may be withdrawn; it lies between $25\%$ and $45\%$ of the SCR.

Describe the three-pillar structure of Solvency II.

Pillar 1: quantitative requirements (technical provisions, SCR, MCR, own funds). Pillar 2: qualitative requirements and supervisory review (governance, risk management, ORSA). Pillar 3: disclosure and reporting (SFCR to public, RSR to supervisor, QRTs). Together they ensure adequate capital, sound governance, and transparency.

Under Solvency II, how are technical provisions calculated for general insurers?

Technical provisions $=$ best estimate liability $+$ risk margin. The best estimate is the probability-weighted average of future cash flows (claims and expenses), discounted at the prescribed risk-free term structure. The risk margin uses a cost-of-capital approach (CoC rate $\times$ present value of future SCRs for the run-off of the business).

What is the ORSA under Solvency II Pillar 2?

The Own Risk and Solvency Assessment: the insurer's own forward-looking assessment of all material risks and the overall capital needed to meet its business strategy and risk appetite over its planning horizon. It is owned by the board, is not just the SCR, and must be embedded in business decision-making and regularly performed/reported to the supervisor.

What is the Solvency II standard formula SCR and how does it differ from an internal model?

The standard formula computes the SCR from prescribed risk modules (e.g. non-life underwriting, market, counterparty, operational) using set stresses and a correlation matrix to aggregate. An internal (or partial internal) model is the insurer's own model, calibrated to its specific risk profile and subject to regulatory approval, usually giving a more risk-sensitive (and often lower) SCR.

What are own funds under Solvency II, and how are they tiered?

Own funds are the capital resources available to meet the SCR/MCR, broadly assets minus liabilities (on the Solvency II valuation basis) plus eligible subordinated liabilities. They are tiered by quality/loss-absorbency: Tier 1 (highest, e.g. ordinary share capital, reconciliation reserve), Tier 2, and Tier 3, with limits on how much of each tier can count toward the SCR and MCR.

What is the difference between an underwriting (accounting) year and an accident year basis for organising claims data?

Accident (occurrence) year groups claims by the date the loss event occurred. Underwriting (policy) year groups them by the policy inception date, so a single underwriting year's claims can occur over up to 24 months. Underwriting year aligns claims with the premium that funds them but develops more slowly than accident year.

Distinguish written, earned, and unearned premium.

Written premium: total premium on policies incepted in the period. Earned premium: the portion relating to expired risk exposure in the period. Unearned premium (UPR): the portion relating to the unexpired period of cover, carried as a liability. For an annual policy written evenly, roughly half is earned in the calendar year of writing.

What is management information (MI) in a general insurance context, and give examples of key metrics.

MI is the set of regular, summarised data reports used by management to monitor and steer the business. Examples: loss ratios and combined ratios by line, premium volumes and rate change indices, renewal retention and new-business conversion, reserve movements/run-off, expense ratios, large-loss and catastrophe experience, and solvency coverage ratio.

Why might the technical (statutory) reserves differ from a Solvency II best-estimate liability?

Statutory/GAAP reserves are often undiscounted, may include implicit margins for prudence, and use case-based or actuarial best estimates without an explicit risk margin. The Solvency II best estimate is discounted, explicitly mean (no prudence), includes all future cash flows and events-not-in-data, and is shown with a separate cost-of-capital risk margin.

What is reinsurance bad debt (counterparty default) provision and why is it needed?

It is an allowance for the expected non-recovery of amounts due from reinsurers because of default or dispute. Even though reinsurance reduces net liabilities, the recoverable is only as good as the reinsurer's creditworthiness, so a provision (and capital under counterparty default risk) reflects the probability of default and loss given default.

How does claims inflation affect long-tailed reserving and what is "superimposed inflation"?

Long-tailed claims settle years after the event, so their ultimate cost is highly sensitive to future inflation; mis-estimating it materially mis-states reserves. Superimposed inflation is the excess of actual claims inflation over general/economic inflation (e.g. from court awards, legal trends, propensity to claim), which must be added explicitly when projecting claims.

What is the all-or-nothing impact of the "diversification benefit" in capital aggregation?

Diversification benefit is the reduction in total capital from combining risks that are not perfectly correlated, since extreme outcomes are unlikely to occur simultaneously: $\text{aggregate SCR} \leq \sum \text{standalone SCRs}$. It is the difference between the sum of standalone capital requirements and the aggregated requirement, and it is allocated back to lines/units for performance measurement.

Describe how a catastrophe model is used in pricing and capital modelling for property insurance.

A cat model combines a hazard module (event set with frequencies/severities), an exposure/vulnerability module (damage functions by location and construction), and a financial module (applying policy terms and reinsurance) to produce an exceedance probability (EP) curve. Outputs (AAL, OEP/AEP at return periods) feed cat loadings in pricing and catastrophe risk capital.

What is the average annual loss (AAL) and the occurrence/aggregate exceedance probability (OEP/AEP) curve from a cat model?

AAL is the expected loss per year from the modelled peril (the cat pure premium). The OEP curve gives the probability that the largest single event loss in a year exceeds a level; the AEP curve gives the probability that the aggregate annual loss exceeds a level. Return periods (e.g. 1-in-200) are read off these curves for pricing and capital.