CAS Exam 7: Estimation of Policy Liabilities and ERM

What Exam 7 Covers

CAS Exam 7 is a fellowship-level exam that covers estimation of policy liabilities (advanced reserving) and enterprise risk management (ERM). The exam uses a written-answer format, requiring candidates to show their work, explain their reasoning, and justify their assumptions. This format tests not just technical competence but also communication skills. Exam 7 builds on the foundational reserving material from Exam 5 and adds stochastic methods, risk measures, and capital management concepts.

Stochastic Reserving Methods

While Exam 5 focuses on deterministic methods like chain ladder and Bornhuetter-Ferguson, Exam 7 introduces stochastic approaches that quantify the uncertainty around reserve estimates. Mack's model is the most important stochastic reserving method on the syllabus. It provides a formula for the standard error of the chain ladder reserve estimate without requiring distributional assumptions beyond the first two moments of the development factors.

In Mack's model, the variance of the development factors for accident year i at development period k is estimated as sigma_k^2 / C_{i,k}, where C_{i,k} is the cumulative loss at that point. The total reserve variability for a single accident year combines the process variance (inherent randomness in future development) and the estimation variance (uncertainty in the estimated development factors). The formula for the mean squared error of the reserve for accident year i involves summing these components across all remaining development periods.

Bootstrap methods provide another approach to generating reserve distributions. The procedure involves fitting a chain ladder model to the original triangle, computing standardized residuals, resampling those residuals with replacement to create new pseudo-triangles, applying the chain ladder to each pseudo-triangle to produce a reserve estimate, and repeating thousands of times to build a distribution of reserve outcomes. The resulting distribution can be used to compute percentiles, confidence intervals, and risk measures for the reserve.

Complications in Reserving

Real-world reserving involves complications that standard methods do not directly address. Changing settlement patterns can distort development factors if claims are being settled faster or slower than historical norms. The exam tests your ability to detect such changes by analyzing development factor patterns and to adjust your methods accordingly, perhaps by giving more weight to recent experience or by using a different development pattern for recent accident years.

Large losses present challenges because they can dominate development patterns and create volatility in reserve estimates. Common approaches include capping individual losses at a threshold, reserving large losses separately, or using limited development factors. The exam may present a triangle with an obvious large loss and ask you to demonstrate how your reserve estimate changes when you treat it separately.

Aggregate cover pricing requires estimating the distribution of aggregate losses, not just the expected value. This connects reserving to ERM concepts, since the aggregate loss distribution determines the likelihood of adverse outcomes that affect capital requirements.

Enterprise Risk Management

The ERM section of Exam 7 covers how insurers identify, measure, and manage risks at the enterprise level. Risk categories include insurance risk (underwriting and reserving), market risk (interest rates, equity prices), credit risk (reinsurer defaults, investment defaults), and operational risk (process failures, fraud, technology outages). ERM provides a framework for understanding how these risks interact and affect the company as a whole.

Economic capital models estimate the amount of capital an insurer needs to survive adverse scenarios at a specified confidence level. The process involves modeling the distribution of losses for each risk type, specifying the dependence structure between risk types (often using correlation matrices or copulas), aggregating the risks to produce a total loss distribution, and computing the capital requirement at the chosen confidence level (often the 99th or 99.5th percentile).

Risk Measures: VaR and TVaR

Value at Risk (VaR) at confidence level alpha is the loss amount such that losses exceed that amount only (1 - alpha) percent of the time. For example, VaR at 99% is the 99th percentile of the loss distribution. VaR is simple to communicate but has limitations: it provides no information about the severity of losses beyond the threshold, and it is not subadditive, meaning the VaR of a combined portfolio can exceed the sum of the individual VaRs.

Tail Value at Risk (TVaR), also called Conditional Tail Expectation (CTE), is the expected loss given that the loss exceeds the VaR threshold. TVaR addresses VaR's limitations by capturing information about the tail of the distribution. TVaR is always greater than or equal to VaR at the same confidence level, and it is a coherent risk measure (satisfying subadditivity, monotonicity, translation invariance, and positive homogeneity). The exam frequently tests candidates on comparing VaR and TVaR, computing them from given distributions, and explaining why TVaR is generally preferred for capital adequacy assessment.

Capital Allocation

Once the total economic capital requirement is determined, it must be allocated to business units or lines of business for performance measurement and pricing decisions. Common allocation methods include proportional allocation (allocating capital in proportion to each unit's standalone risk), the Euler method (allocating based on marginal contributions to total risk), and the co-TVaR method (allocating based on each unit's expected contribution to tail losses). Each method has different properties, and the exam may ask you to compute allocations under different methods and discuss their advantages.

The Euler method is particularly important because it produces allocations that sum exactly to the total capital requirement (full allocation) and reflect each unit's marginal contribution to risk. For a portfolio with risk measure rho, the Euler allocation to business unit i is: allocated_i = E[X_i | X_total = VaR_alpha(X_total)], where X_i is the loss from unit i and X_total is the total loss. In practice, this conditional expectation is estimated from simulation output.

Study Tips for Exam 7

Since Exam 7 uses a written-answer format, practice writing clear, organized responses. Start each answer with a brief statement of your approach, show your calculations step by step, and state your conclusion explicitly. Time management is critical because written responses take longer than multiple-choice answers. Practice with past exams under timed conditions. For the reserving section, focus on Mack's model formulas and bootstrap procedures, and practice working through numerical examples. For the ERM section, understand the definitions and properties of VaR and TVaR, know how to compute economic capital from a loss distribution, and be prepared to discuss capital allocation methods. Allocate approximately 400 to 500 study hours for this fellowship-level exam.