Casualty Actuarial Society Credentials (ACAS/FCAS) Exam 8 — Advanced Ratemaking Syllabus

For a deductible $d$ with loss elimination ratio (LER), write the LER and the expected payment per loss.

$$\text{LER}(d)=\frac{E[X\wedge d]}{E[X]},\qquad E[\text{payment per loss}] = E[X]-E[X\wedge d]=E[X](1-\text{LER}(d)).$$ LER is the fraction of ground-up losses eliminated by the deductible.

Compare a straight (per-claim) deductible vs. a franchise deductible in terms of payment.

Straight deductible $d$ pays $\max(X-d,0)$ — the insurer pays loss minus $d$. Franchise deductible pays $0$ if $X\le d$ but pays the full $X$ if $X>d$. The franchise costs the insurer more above the threshold because no amount is subtracted once breached.

What is leveraged trend in excess layers, and why does trend leverage increase with the attachment point?

Leveraged trend means a uniform ground-up severity trend produces a higher effective trend in an excess layer. Because the fixed attachment $a$ does not inflate, applying trend $t$ pushes more dollars above $a$; the layer cost $E[(X-a)_+]$ grows faster than $E[X]$, and leverage rises with higher $a$ (and shrinks with a cap on the layer).

Why is loss development typically higher (slower-reporting) for excess layers than for primary layers?

Large losses take longer to be reported and to mature to their ultimate value, and case reserves on severe claims develop upward over time. Since excess layers are only penetrated by large claims, they exhibit longer development tails and higher development factors than primary layers.

Define the catastrophe modeling components: hazard, exposure (inventory), vulnerability, and financial module.

Hazard: physical event characteristics and frequency (e.g., windspeed, ground acceleration). Exposure/Inventory: insured properties, locations, characteristics. Vulnerability: damage functions mapping hazard intensity to damage ratio. Financial module: applies policy terms (deductibles, limits, reinsurance) to produce insured loss.

Define the Exceedance Probability (EP) curve, and distinguish OEP from AEP.

The EP curve gives the probability that loss exceeds a given level in a year. OEP (Occurrence EP) is based on the largest single event loss per year; AEP (Aggregate EP) is based on total annual losses summed across events. For a given return period, AEP $\geq$ OEP.

Define return period in catastrophe modeling and relate it to exceedance probability.

The return period is the average time between events of a given size: $$\text{Return Period}=\frac{1}{\text{annual exceedance probability}}.$$ A 100-year loss has a $1\%=0.01$ annual probability of being exceeded; it is not guaranteed to occur once per 100 years.

Distinguish the AAL (Average Annual Loss / pure premium from a cat model) from the PML/TVaR, and how each is used in pricing.

AAL is the mean modeled annual cat loss — used as the cat loss cost (pure premium) in the rate. PML / TVaR (e.g., the average loss beyond a return period) measures tail severity and is used for capital allocation and the cat risk load, not the expected loss component.

What are the two main sources of uncertainty captured in cat model output, and what does the term 'secondary uncertainty' mean?

Primary uncertainty: which events occur and how often (event frequency/occurrence). Secondary uncertainty: the variability of loss given an event occurs (the distribution of damage around the mean for a known event), captured by a damage distribution rather than a point estimate.

For specialty/professional liability lines written on a claims-made vs. occurrence basis, what is the key coverage trigger difference, and what is tail/nose coverage?

Occurrence policies cover claims for events that occur during the policy period regardless of when reported. Claims-made policies cover claims reported during the policy period (subject to a retroactive date). Tail (ERP) coverage extends reporting after a claims-made policy ends; nose coverage (prior-acts) covers occurrences before the current policy's inception.

Why does a claims-made program's pure premium increase across the first several 'maturities' before reaching a steady (mature) level?

A first-year claims-made policy only covers recently-occurred (short-reporting-lag) claims, so it is cheap. Each successive maturity adds another report-year of coverage from longer reporting lags, increasing expected losses until the program reaches the mature level equivalent to occurrence coverage.

State the general formula for the risk load (margin) using the standard deviation principle and the variance principle.

Standard deviation principle: $RL=k\,\sigma$ (load proportional to the standard deviation of losses). Variance principle: $RL=\lambda\,\sigma^{2}$ (load proportional to variance). Here $\sigma$ is the standard deviation of aggregate loss and $k,\lambda$ are selected risk parameters.

In Kreps' / marginal surplus framework, how is the risk load tied to required capital and a target return $y$?

Capital is held so that surplus covers losses to a chosen safety level (e.g., $C=z\sigma$). The risk load equals the cost of that capital: $$RL=\frac{y}{1+y}\,C,$$ where $y$ is the target return on the marginal surplus committed to the risk, charging the reluctance to hold capital.

Contrast a renewal (Kreps) additive risk load vs. a marginal (Mango) approach when adding a risk to a portfolio.

A marginal approach charges each risk the incremental capital/variance it adds, which depends on covariance with the existing portfolio and is order-dependent (sums may not equal the total). A renewal/additive method seeks a risk load that is allocable and renewal-additive so independent risks' loads sum to the portfolio load, addressing the order-dependence and renewal-additivity problems.

Define the Mango capital consumption / Shapley-value style allocation goals for risk load.

Capital consumption views each risk as consuming shared capital (rental cost for capacity occupied plus a penalty for capital actually consumed in adverse outcomes). Shapley allocation averages a risk's marginal contribution over all orderings of entry, producing an allocation that is fair (order-independent) and adds up to the total.

For pricing proportional (pro-rata) reinsurance, how do the premium and losses flow under a quota share treaty with cession rate $c$?

Under a quota share, the reinsurer assumes a fixed fraction $c$ of every risk: it receives $c$ of the premium and pays $c$ of every loss. The ceding company keeps $(1-c)$ of premiums and losses. The loss ratio is unchanged by the cession before commissions.

What is a ceding commission in proportional reinsurance and how does a sliding scale (or profit) commission work?

The ceding commission reimburses the cedent for acquisition/expenses on ceded premium. A sliding-scale commission varies inversely with the ceded loss ratio (lower loss ratio → higher commission), within min/max bounds, sharing favorable/unfavorable experience between cedent and reinsurer.

Contrast quota share vs. surplus share proportional reinsurance.

Quota share cedes the same fixed percentage of every risk. Surplus share cedes only the portion of each risk's limit above the cedent's retention (a 'line'), so the cession percentage varies by policy size; small policies within the retention are not ceded. Surplus share gives the cedent more flexibility to retain homogeneous small risks.

For excess-of-loss (XOL) reinsurance written as '$l$ xs $a$', define the reinsurer's payment per loss and the expected layer cost.

'$l$ xs $a$' = limit $l$ excess of attachment $a$. Per loss the reinsurer pays $\min(\max(X-a,0),\,l)$, expected cost $$E[X\wedge(a+l)]-E[X\wedge a].$$ Multiply by expected claim count to get expected aggregate layer losses.

How is the expected number of claims penetrating an XOL layer at attachment $a$ obtained from ground-up frequency $N$ and severity survival $S(a)$?

$$E[N_{>a}] = E[N]\cdot S(a)=E[N]\,[1-F(a)],$$ assuming frequency and severity are independent and severity is unaffected by reaching the layer. This thinned frequency drives the excess layer's expected loss and credibility.

Define the burning cost (experience rating) method for pricing an XOL layer and one major shortfall.

Burning cost = trended, developed historical losses in the layer divided by the trended subject premium (then loaded for expenses/profit): $$\text{Rate}=\frac{\text{trended developed layer losses}}{\text{trended subject premium}}.$$ Shortfall: high-layers have sparse/volatile data, so the experience may not be credible — it ignores exposure (the actual risk distribution) and tail potential beyond observed losses.

Contrast experience rating vs. exposure rating for XOL reinsurance, and when is each weighted more heavily?

Experience rating projects the cedent's actual historical layer losses forward. Exposure rating applies an industry severity curve (e.g., an exposure/first-loss-scale curve) to the cedent's premium and limits profile to estimate the layer share. Experience rating is weighted more for lower, well-populated layers; exposure rating for high, sparse layers with little credible loss history.

What is an exposure curve (first-loss scale) in property reinsurance, and what does its value $G(d)$ represent?

An exposure curve $G(d)$ gives the proportion of the full-value premium (expected loss) corresponding to a deductible/retention $d$ as a fraction of insured value: $$G(d)=\frac{E[X\wedge d]}{E[X]}.$$ It lets the reinsurer allocate premium to a layer using only policy limit profiles, without individual loss data.

How does an aggregate deductible (annual aggregate deductible, AAD) on an XOL treaty affect expected ceded losses relative to the per-occurrence layer?

An AAD requires the cedent to retain the first dollars of aggregate layer losses up to the AAD before the reinsurer pays. This lowers expected ceded losses by the expected losses falling under the aggregate retention, $E[\min(\text{Agg layer losses},\,\text{AAD})]$, and requires modeling the aggregate distribution, not just per-occurrence severity.

What is a reinstatement premium in XOL reinsurance and how does it affect the rate calculation?

A reinstatement restores exhausted layer limit after a loss, requiring the cedent to pay an additional (often pro-rata to amount and/or time) premium. With $k$ paid reinstatements the total available limit is finite; the expected reinstatement premium income partially offsets expected losses, lowering the upfront rate, and the aggregate limit caps total recoveries at $(1+k)\times$ the per-occurrence limit.

Define swing-rated (retrospectively rated) reinsurance and the role of the provisional, minimum, and maximum rates.

A swing-rated treaty's final premium varies with actual layer losses: $$\text{Rate}=\min\!\big(\max(\text{Loss Cost}\times\text{LCF},\,\text{Min}),\,\text{Max}\big),$$ where the loss conversion factor (LCF) loads losses for expenses. A provisional rate is paid upfront, then adjusted within the min/max collar once losses emerge, sharing risk between parties.

In ratemaking, state the fundamental insurance rate equation relating premium to losses, expenses, and profit (the pure premium and loss-ratio methods).

Pure premium method: $$\text{Rate}=\frac{\text{Pure Premium}+\text{Fixed Expense}}{1-V-Q},$$ where $V$ is variable expense ratio and $Q$ the profit/contingency provision. Loss-ratio method develops an indicated change: $$\text{Indicated Change}=\frac{\text{Loss \& LAE Ratio}+\text{Fixed Expense Ratio}}{1-V-Q}.$$

Define the underwriting profit provision target via a target return on the IRR/ROE framework, conceptually.

The profit provision $Q$ is set so the policy's expected cash flows — premiums (net of expenses), loss payments, and investment income on held reserves/surplus — produce the target return on the capital (surplus) supporting the policy. Lines with longer payout tails earn more investment income and can support a lower (even negative) underwriting profit provision.

What is the difference between systematic (non-diversifiable) and process (diversifiable) risk in capital/risk loading, and which justifies a risk load that does not vanish with volume?

Process (diversifiable) risk is independent random fluctuation that shrinks relatively as volume grows ($\sigma/\text{mean}\to 0$). Systematic/parameter risk is common across risks (e.g., trend, cat, parameter uncertainty) and does not diversify away. The non-diversifiable component justifies a persistent risk load even for large portfolios.

Explain why a higher excess layer generally requires a larger risk load per expected dollar of loss than a primary layer.

Excess layers have low frequency but high severity, producing a high coefficient of variation (volatility relative to mean) and heavy tails. Since risk load scales with volatility (e.g., $k\sigma$ or variance), and parameter/development/trend leverage uncertainty is greatest in the tail, the load per expected dollar increases as the attachment rises.