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

Exam 8 — Advanced Ratemaking syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Exam 8 — Advanced Ratemaking in Casualty Actuarial Society Credentials (ACAS/FCAS), not a summary of it.

1. 1

   Classification Plan Design and GLMs in Pricing

   3 topics

   • Multivariate Classification Ratemaking
     • GLMs for rating plan construction
     • Interaction effects and offset terms
     • Model validation and lift charts in pricing
   • Territory and Geographic Ratemaking
     • Spatial smoothing of territorial relativities
     • Geodemographic and external data
   • Credibility in Advanced Ratemaking
     • Credibility-blended class relativities
     • Generalized linear mixed models
2. 2

   Excess, Deductible, and Increased Limits Pricing

   3 topics

   • Increased Limits Factors
     • Size-of-loss distributions and limited expected value
     • Risk load and parameter uncertainty in ILFs
   • Deductible and Layer Pricing
     • Loss elimination ratios for deductible rating
     • Pricing excess layers
   • Trend and Development in Excess Pricing
     • Leveraged trend on excess layers
     • Development of large and excess losses
3. 3

   Catastrophe and Specialty Ratemaking

   3 topics

   • Catastrophe Modeling
     • Hazard, vulnerability, and financial modules
     • Exceedance probability curves and AAL
     • Loading catastrophe risk into rates
   • Specialty Lines Considerations
     • Pricing low-frequency, high-severity exposures
     • Credibility and external data for thin lines
   • Loading for Risk and Capital
     • Risk loads for catastrophe exposure
     • Capital allocation for peak perils
4. 4

   Reinsurance Pricing

   3 topics

   • Pricing Proportional Reinsurance
     • Quota share and surplus share treaties
     • Ceding commissions and sliding scales
   • Pricing Excess-of-Loss Reinsurance
     • Experience and exposure rating
     • Property per-risk and per-occurrence covers
     • Aggregate and stop-loss treaty pricing
   • Risk Load and Profitability
     • Risk margins in reinsurance pricing
     • Reinstatement premiums

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

22 of 50 cards from the Exam 8 — Advanced Ratemaking deck — real questions with worked answers.

1. In a multivariate classification GLM, what is the role of the link function $g(\cdot)$, and what link is standard for modeling claim frequency and severity?

   The link function connects the linear predictor to the mean response: $g(\mu_i) = \sum_j x_{ij}\beta_j$. For frequency and severity (multiplicative rating), the log link $g(\mu)=\ln(\mu)$ is standard, so $\mu_i = \exp(\sum_j x_{ij}\beta_j)$ produces a multiplicative rating structure.

2. For a GLM, what are the variance functions $V(\mu)$ for the Poisson and Gamma distributions, and which models frequency vs. severity?

   Poisson: $V(\mu)=\mu$ (used for claim frequency/counts). Gamma: $V(\mu)=\mu^{2}$ (used for claim severity). The Tweedie with $1<p<2$, $V(\mu)=\mu^{p}$, models pure premium directly.

3. Define the dispersion parameter $\phi$ in the exponential family variance form, and why does it not affect GLM coefficient estimates?

   The variance is $\mathrm{Var}(Y_i)=\frac{\phi\,V(\mu_i)}{w_i}$, where $w_i$ is the prior weight. Coefficient estimates depend only on the mean-variance relationship $V(\mu)$, so $\phi$ scales standard errors but not the fitted $\hat\beta$ values.

4. What is the difference between Type III deviance tests and the F-test when comparing nested GLMs, and when must the F-test be used?

   The likelihood-ratio / chi-square deviance test assumes $\phi$ is known. When $\phi$ must be estimated (e.g., Gamma, Tweedie), use the F-test: $F=\frac{(D_{small}-D_{big})/(\Delta\text{df})}{\hat\phi}$, which accounts for estimating the scale parameter.

5. In a GLM, what does an offset term represent and give a ratemaking example.

   An offset is a predictor with a fixed coefficient of 1, entering as $\ln(\text{exposure})$ in a Poisson frequency model so the model predicts counts proportional to exposure. It can also incorporate known relativities (e.g., a fixed ILF or prior territory factor) that are not re-estimated.

6. What is the purpose of including interaction terms in a classification GLM, and how is an interaction detected diagnostically?

   Interactions capture when the effect of one variable depends on the level of another (non-multiplicative behavior between two factors). They are detected via consistent patterns in residuals across a two-way tabulation of the variables, or by a significant improvement in deviance when the interaction term is added.

7. Define the Gini index / Lorenz curve as used to assess a ratemaking model's lift, and what value indicates better segmentation.

   Ordering risks by predicted loss cost, the Lorenz curve plots cumulative percent of exposures against cumulative percent of losses. The Gini index is twice the area between the Lorenz curve and the line of equality. A higher Gini indicates greater ability to differentiate (better lift/segmentation).

8. What is a double-lift chart and what does it show that a single-lift chart cannot?

   A double-lift chart sorts records by the ratio of two competing models' predictions (Model A / Model B) into buckets, then compares each model's average prediction to actual within each bucket. It directly shows which of two models more accurately predicts where they most disagree — a single-lift chart only validates one model in isolation.

9. In territory ratemaking, why are spatial smoothing methods needed, and name a distance-based and an adjacency-based approach.

   Geographic units have sparse/volatile data, so smoothing borrows information from nearby locations. Distance-based: weight observations by proximity (e.g., a kernel/distance decay). Adjacency-based: neighbors sharing a border influence each other (e.g., a spatial smoothing / Markov random field that pulls a unit toward its bordering units' estimates).

10. Distinguish spatially smoothed signals using a distance-based vs. adjacency-based method by Fu/Wu (or similar): which preserves discontinuities at borders?

    Adjacency-based smoothing respects natural boundaries and can preserve discontinuities between non-adjacent regions, since only bordering units influence each other. Distance-based smoothing blends across borders purely on geographic distance and tends to wash out sharp boundary effects.

11. In territory ratemaking, why must territory be modeled simultaneously with (or net of) other classification variables rather than univariately?

    Geographic loss differences can be partly explained by the distribution of other rating variables across regions (e.g., vehicle type, age). Modeling territory net of other variables (e.g., as a GLM residual signal) isolates the pure geographic effect and avoids double-counting overlapping signals.

12. State the classical (Bühlmann limited fluctuation) full-credibility standard for frequency, with $k$ and $P$ defined.

    For frequency claims to be within $\pm k$ of the mean with probability $P$: $$n_F=\left(\frac{z_{(1+P)/2}}{k}\right)^{2}=\lambda_0,$$ where $z_{(1+P)/2}$ is the standard normal percentile. Commonly $P=0.90,\,k=0.05$ gives $n_F\approx 1082$ claims.

13. Write the classical partial credibility (square-root rule) formula and state its key assumption.

    $$Z=\min\left(\sqrt{\frac{n}{n_F}},\,1\right),$$ where $n$ is observed claims and $n_F$ the full-credibility standard. Key assumption: the standard deviation of the estimator is proportional to $\frac{1}{\sqrt{n}}$ (Poisson-like) and the process is approximately normal.

14. Give the Bühlmann (greatest accuracy / least-squares) credibility factor and define EPV and VHM.

    $$Z=\frac{n}{n+k},\qquad k=\frac{\text{EPV}}{\text{VHM}}=\frac{E[\mathrm{Var}(X\mid\Theta)]}{\mathrm{Var}(E[X\mid\Theta])}.$$ EPV is the Expected Process Variance (within-risk noise); VHM is the Variance of the Hypothetical Means (between-risk signal). $n$ is the number of exposures/years.

15. How does Bühlmann-Straub credibility differ from basic Bühlmann credibility?

    Bühlmann-Straub allows the number of exposures $m_i$ to vary by period/risk. The credibility is $Z=\frac{m}{m+k}$ with $m=\sum_i m_i$, and the estimate uses exposure-weighted averages, accommodating unequal exposure volumes across observations.

16. Define an Increased Limits Factor (ILF) for limit $L$ relative to a basic limit $B$, in terms of the limited expected value.

    $$\text{ILF}(L)=\frac{E[X\wedge L]}{E[X\wedge B]},$$ the ratio of the limited expected value (limited average severity) at limit $L$ to that at the basic limit $B$. It assumes frequency and severity are independent of the limit and excludes loadings unless stated.

17. State the consistency (decreasing marginal ILF) test for increased limits factors and what a violation implies.

    ILFs must be increasing in $L$ but with decreasing marginal increases: the incremental cost per additional layer must not rise. Mathematically the marginal rate $\frac{d\,\text{ILF}}{dL}$ should be non-increasing. A violation implies an arbitrage opportunity — a buyer could combine policies to obtain higher coverage more cheaply, signaling a data or fitting error.

18. In ILF calculation, how do allocated loss adjustment expenses (ALAE) and a risk load typically enter the formula?

    With ALAE and risk load: $$\text{ILF}(L)=\frac{E[X\wedge L]+\text{ALAE}+RL(L)}{E[X\wedge B]+\text{ALAE}+RL(B)}.$$ ALAE is often added (pro-rata or in full) to losses, and a risk/parameter load $RL$ that grows with limit reflects greater volatility in higher layers.

19. What is the limited expected value $E[X\wedge d]$ for a severity distribution with density $f(x)$, expressed as an integral?

    $$E[X\wedge d]=\int_0^{d}x\,f(x)\,dx + d\,[1-F(d)]=\int_0^{d}[1-F(x)]\,dx,$$ the average loss after capping each loss at $d$. The right form (survival-function integral) is often most convenient.

20. Define the layer of loss from attachment $a$ to limit $a+l$ and express its expected cost.

    The layer $(a,\,a+l]$ pays $\min(\max(X-a,0),\,l)$. Its expected cost is $$E[X\wedge(a+l)]-E[X\wedge a],$$ the difference of limited expected values at the top and bottom of the layer.

21. 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.

22. 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.

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