Casualty Actuarial Society Credentials (ACAS/FCAS) Exam 7 — Estimation of Policy Liabilities, Insurance Company Valuation, and Enterprise Risk Management Syllabus

Exam 7 — Estimation of Policy Liabilities, Insurance Company Valuation, and Enterprise Risk Management syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Exam 7 — Estimation of Policy Liabilities, Insurance Company Valuation, and Enterprise Risk Management in Casualty Actuarial Society Credentials (ACAS/FCAS), not a summary of it.

1. 1

   Advanced Reserving Techniques

   3 topics

   • Stochastic Reserving Models
     • Mack model and variability of chain ladder estimates
     • Bootstrap methods for reserve distributions
     • Over-dispersed Poisson and GLM reserving
   • Bayesian and Credibility Reserving
     • Clark's growth curve / LDF and Cape Cod methods
     • Bayesian reserve estimation approaches
   • Reserve Ranges and Uncertainty
     • Process vs. parameter vs. model risk
     • Quantifying and communicating reserve variability
2. 2

   Reinsurance Reserving and Pricing

   3 topics

   • Reinsurance Loss Reserving
     • Challenges of long-tail reinsurance triangles
     • Estimating ceded and assumed reserves
   • Excess and Layered Losses
     • Development of excess layers
     • Allocating losses across reinsurance layers
   • Reinsurance Program Considerations
     • Treaty structures and their reserving impact
     • Commutations and finality of reinsurance liabilities
3. 3

   Insurance Company Valuation

   3 topics

   • Valuation Frameworks
     • Embedded value and appraisal value
     • Economic value of in-force business
   • Cost of Capital and Financial Pricing
     • Cost of capital allocation
     • Discounting liabilities and risk margins
   • Earnings and Performance Measurement
     • Sources of earnings analysis
     • Return on equity and value creation metrics
4. 4

   Enterprise Risk Management

   3 topics

   • ERM Frameworks and Risk Identification
     • Risk taxonomy and risk appetite
     • ORSA (Own Risk and Solvency Assessment)
   • Risk Measurement
     • Value at Risk and Tail Value at Risk
     • Economic capital modeling
   • Solvency Frameworks and Regulation
     • Solvency II and internal models
     • Rating agency capital models

Exam 7 — Estimation of Policy Liabilities, Insurance Company Valuation, and Enterprise Risk Management flashcards for Casualty Actuarial Society Credentials (ACAS/FCAS)

20 of 61 cards from the Exam 7 — Estimation of Policy Liabilities, Insurance Company Valuation, and Enterprise Risk Management deck — real questions with worked answers.

1. In the Mack (1993) distribution-free chain-ladder model, what are the three key assumptions?

   (1) Expected cumulative losses scale with the prior period: $E[C_{i,k+1}\mid C_{i,1},\dots,C_{i,k}] = f_k\, C_{i,k}$. (2) Variance proportional to the prior: $\mathrm{Var}[C_{i,k+1}\mid \cdots] = \sigma_k^{2}\, C_{i,k}$. (3) Accident years $i$ are independent.

2. State the Mack estimator of the age-to-age development factor $f_k$ and its variance parameter $\sigma_k^{2}$.

   $\hat{f}_k = \dfrac{\sum_{i=1}^{n-k} C_{i,k+1}}{\sum_{i=1}^{n-k} C_{i,k}}$ (volume-weighted). And $\hat{\sigma}_k^{2} = \dfrac{1}{n-k-1}\sum_{i=1}^{n-k} C_{i,k}\left(\dfrac{C_{i,k+1}}{C_{i,k}} - \hat{f}_k\right)^{2}$.

3. How does Mack estimate $\sigma_{n-1}^{2}$ for the last development period when the usual formula is undefined?

   Use the extrapolation $\hat{\sigma}_{n-1}^{2} = \min\!\left(\dfrac{\hat{\sigma}_{n-2}^{4}}{\hat{\sigma}_{n-3}^{2}},\ \hat{\sigma}_{n-3}^{2},\ \hat{\sigma}_{n-2}^{2}\right)$, assuming the $\sigma_k^{2}$ decrease geometrically.

4. What is the conceptual difference between process variance, parameter variance, and total prediction (mean square) error in reserve estimation?

   Process variance is the inherent randomness of future outcomes around the true mean; parameter variance (estimation error) reflects uncertainty in the estimated parameters; total prediction error (MSEP) combines both: $\text{MSEP} = \text{process variance} + \text{estimation variance}$.

5. In the over-dispersed Poisson (ODP) model used for the bootstrap, what is the variance structure and how does it relate to incremental claims?

   Incremental claims $q_{ij}$ satisfy $E[q_{ij}]=m_{ij}$ and $\mathrm{Var}[q_{ij}]=\phi\, m_{ij}$, where $\phi$ is the dispersion (scale) parameter. It reproduces chain-ladder reserve point estimates while allowing a full predictive distribution.

6. Describe the two-step structure of the ODP bootstrap for reserve variability (Shapland/England-Verrall).

   Step 1 — resample the (adjusted) Pearson residuals to create pseudo-triangles and refit, generating the distribution of parameter/estimation error. Step 2 — for each bootstrap point estimate, simulate the process error by drawing future incrementals from a process distribution (e.g., gamma/ODP). Combining gives the full predictive distribution of unpaid claims.

7. Define the unscaled Pearson residual and the degrees-of-freedom adjustment factor used in the ODP bootstrap.

   Pearson residual: $r_{ij} = \dfrac{q_{ij} - \hat{m}_{ij}}{\sqrt{\hat{m}_{ij}}}$. The DoF adjustment is $f^{DoF} = \sqrt{\dfrac{N}{N - p}}$, where $N$ is the number of data points and $p$ the number of parameters, applied to scale residuals upward.

8. What does a Bayesian credibility approach add to reserving compared with a purely frequentist chain ladder?

   It blends prior beliefs (e.g., an a priori loss ratio or industry benchmark) with observed data through a posterior distribution, yielding a credibility-weighted estimate and a full predictive distribution of reserves rather than a single point. It naturally stabilizes immature/volatile years.

9. Express the Bühlmann credibility estimate and identify the credibility factor $Z$.

   Estimate: $\hat{\mu} = Z\,\bar{X} + (1-Z)\,\mu$, with $Z = \dfrac{n}{n + k}$ and $k = \dfrac{\text{EPV}}{\text{VHM}} = \dfrac{E[\mathrm{Var}(X\mid\Theta)]}{\mathrm{Var}(E[X\mid\Theta])}$, where $n$ is the number of observations.

10. How is the Bornhuetter-Ferguson (BF) method a credibility blend, and what weights does it apply?

    $\text{Ultimate} = \text{Actual reported} + (1 - \%\text{reported})\times \text{a priori ultimate}$. It gives weight $Z = \%\text{reported}$ to the chain-ladder (development) estimate and $(1-Z)$ to the a priori expected losses, so immature years lean on the prior.

11. In Clark's (2003) growth-curve reserving model (LDF and Cape Cod methods), what two growth curves are commonly used and what parameters describe them?

    The Loglogistic $G(x)=\dfrac{x^{\omega}}{x^{\omega}+\theta^{\omega}}$ and the Weibull $G(x)=1-\exp\!\left(-(x/\theta)^{\omega}\right)$, where $\theta$ is the scale and $\omega$ the shape parameter. The model uses maximum likelihood under an over-dispersed Poisson assumption.

12. In Clark's model, how is the variance/scale parameter $\sigma^{2}$ estimated and what is its interpretation?

    $\sigma^{2}=\dfrac{1}{n-p}\sum \dfrac{(c_i - \mu_i)^{2}}{\mu_i}$, the over-dispersion factor relating variance to mean ($\mathrm{Var}=\sigma^{2}\mu$). It scales the Poisson variance to match the observed scatter of incremental losses.

13. What are the two components of reserve estimation error in Clark's model, and how does parameter variance behave relative to process variance over time?

    Process variance ($\sigma^{2}\times\text{reserve}$) and parameter variance (from the covariance matrix of the MLE parameters). For mature/older accident years parameter variance dominates; reserve estimates with longer tails carry more parameter uncertainty.

14. Distinguish a reserve 'range' from a 'range of reasonable estimates' from a 'distribution of possible outcomes'.

    A range of reasonable estimates spans defensible point estimates a reserving actuary could select (a narrow band of central estimates). A distribution of possible outcomes captures the full variability of actual results (much wider). A range of reasonable estimates is NOT the same as a confidence interval of outcomes.

15. According to the CAS (Working Party on Quantifying Variability), what are the main sources of reserve uncertainty?

    Process risk (random future claim variation), parameter risk (error in estimated parameters), model risk (wrong model/specification), and systemic/internal & external systematic risks. Total uncertainty must reflect all of these, not just process risk.

16. Why can a method that reproduces the chain-ladder point estimate still produce materially different reserve ranges?

    The point estimate depends only on the mean structure (development factors), whereas the range depends on the assumed variance/correlation structure, tail behavior, and treatment of parameter and model risk—so different stochastic models (Mack, ODP bootstrap, Clark) can agree on the mean but differ widely on the spread.

17. What is the key driver that makes excess/reinsurance loss reserving harder than primary reserving (per Siewert / Patrik)?

    Excess layers have much longer reporting and payment tails, lower claim frequency with high severity (sparse, volatile data), greater leverage from trend and development, and report lags—so development factors are larger, less stable, and require exposure-based or industry benchmarks.

18. In reinsurance reserving, define the difference between 'reporting pattern' and 'settlement/payment pattern' and why the report lag matters.

    The reporting pattern describes how claims are reported to the reinsurer over time (often delayed because primary insurer must breach the retention first); the payment pattern describes cash settlement. Long report lags mean a large share of ultimate losses are pure IBNR for many years, increasing reliance on a priori methods like BF.

19. What is the 'leveraged effect' of trend (inflation) on excess layers?

    Loss trend inflates excess-of-retention losses at a higher rate than ground-up losses, because a uniform percentage increase in all claim sizes pushes more claims over a fixed retention and increases the excess portion disproportionately. The excess trend rate exceeds the ground-up trend rate.

20. Give the formula for the expected loss in an excess layer with attachment $a$ and limit $l$ (layer $a$ xs $a$ to $a+l$) using the limited expected value function.

    Layer losses $= E[(X\wedge (a+l))] - E[(X\wedge a)]$, where $E[X\wedge d]=\int_0^{d} S(x)\,dx = \int_0^{d}\big(1-F(x)\big)\,dx$ is the limited expected value at limit $d$.

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