CAS Data Analysis and Interpretation 2 — Questions and Answers
Question 1: When fitting a generalized linear model (GLM) to insurance loss data, which link function is most commonly used for modeling claim frequency?
- Identity link
- Log link (Correct answer)
- Logit link
- Probit link
Correct answer: Log link
The log link is standard for Poisson-distributed claim frequency models because it ensures predicted frequencies remain positive.
Question 2: A claims analyst observes that residuals from a regression model exhibit a funnel shape when plotted against fitted values. This pattern most likely indicates:
- Autocorrelation in the residuals
- Heteroscedasticity in the error terms (Correct answer)
- Multicollinearity among predictors
- Omitted variable bias
Correct answer: Heteroscedasticity in the error terms
A funnel-shaped residual plot is the classic diagnostic for heteroscedasticity, where error variance is not constant across fitted values.
Question 3: In credibility theory, the Bühlmann credibility factor Z is defined as n/(n+k). As the number of observations n increases toward infinity, Z approaches:
- 0
- k
- 1 (Correct answer)
- k/(k+1)
Correct answer: 1
As n → ∞, the credibility factor Z = n/(n+k) approaches 1, meaning full weight is given to observed experience.
Question 4: Which of the following best describes the use of a Q-Q plot in actuarial data analysis?
- Comparing two time series for cointegration
- Assessing whether data follow a specified theoretical distribution (Correct answer)
- Measuring correlation between two continuous variables
- Testing for structural breaks in panel data
Correct answer: Assessing whether data follow a specified theoretical distribution
A Q-Q plot compares empirical quantiles of data against theoretical quantiles of a reference distribution to assess distributional fit.
Question 5: An insurer uses principal component analysis (PCA) on 20 rating variables. The first three principal components explain 85% of total variance. The primary benefit of using these three components instead of all 20 is:
- Eliminating all multicollinearity by design
- Reducing dimensionality while retaining most variance (Correct answer)
- Guaranteeing that each component is normally distributed
- Removing the need for cross-validation
Correct answer: Reducing dimensionality while retaining most variance
PCA's main benefit is dimensionality reduction — capturing most variance with far fewer uncorrelated components, which simplifies modeling.
Question 6: A dataset of 500 claims has a sample mean of $12,000 and sample standard deviation of $8,000. The 95% confidence interval for the population mean is constructed using the t-distribution rather than the normal because:
- The claim amounts are right-skewed
- The population standard deviation is unknown (Correct answer)
- The sample size is below 1,000
- The data contains outliers
Correct answer: The population standard deviation is unknown
The t-distribution is used when the population standard deviation is unknown and must be estimated from sample data.
Question 7: In a loss development triangle, the volume-weighted average link ratio for a development period is calculated by:
- Averaging all individual link ratios for that period with equal weight
- Dividing total cumulative losses at the later age by total cumulative losses at the earlier age (Correct answer)
- Multiplying the simple average link ratio by an exposure adjustment factor
- Using the median link ratio across all accident years
Correct answer: Dividing total cumulative losses at the later age by total cumulative losses at the earlier age
The volume-weighted (chain-ladder) link ratio divides the sum of later-age cumulative losses by the sum of earlier-age cumulative losses, giving more weight to larger values.
CAS Data Analysis and Interpretation 2
When fitting a generalized linear model (GLM) to insurance loss data, which link function is most commonly used for modeling claim frequency?