Actuarial Notes

Data Types

Study guides: Exam MAS-I, Exam DISC-DA

Data types are the classes a variable falls into according to the values it can take — numeric (continuous or discrete) or categorical (nominal, ordinal or binary). A variable's type decides which summaries and EDA plots describe it, how it enters a model as a predictor, and which distribution can serve it as a response.

\begin{aligned} \text{Numeric} &: \text{continuous, discrete (count)} \\ \text{Categorical} &: \text{nominal, ordinal, binary} \end{aligned}

  • Continuous — any value in an interval: claim severity, amount of insurance, vehicle value. Plot with a Histogram for shape, a Box Plot for spread and outliers, and a QQ Plot for the tails. Insurance amounts are usually right-skewed, so a log scale is often the readable one
  • Discrete (count) — non-negative integers: claim counts, vehicles per policy. A histogram of a variable that is 90\% zeros is one tall bar; a bar chart of how often 0, 1, 2, \ldots occur is the right univariate plot
  • Nominal — unordered labels: territory, vehicle make, distribution channel. Ordinal — ordered labels whose spacing means nothing: credit tier, construction class, limit band. Binary — two levels: claim or no claim, renewed or lapsed. Plot each with a bar chart of counts or exposure by level; that is how thin levels are found before they destabilize a coefficient
  • The MAS-I outline counts differently. Its task C5 asks candidates to distinguish categorical, ordinal, and continuous predictors — three kinds, so there categorical means unordered (nominal) levels. This page uses the wider sense, in which nominal and ordinal are both levels of measurement of a qualitative (categorical) variable
  • Two variables at once: numeric against numeric is a Scatter Plot; categorical against numeric is a row of side-by-side box plots, one per level; categorical against categorical is a two-way table of counts or exposure
  • Storage type is not statistical type. A territory stored as the integers 1–12 is nominal, and a ZIP code is a label. Software reads what the file holds, so categorical variables must be declared — a factor in R, a category in pandas (Statistical Software) — or the model fits them as a single slope
  • Type also points to the model: counts to Poisson Regression, positive skewed amounts to a Gamma response, a binary outcome to Logistic Regression, and a spike at zero plus a skewed tail (pure premium) to the Tweedie Distribution. A nominal predictor enters as k - 1 indicator columns (Categorical Predictor)

Classifying a Personal Auto Data Set A modelling file has one row per policy-year with these columns: policy_id; veh_age (whole years, 0–25); territory (stored as the integers 1–8); credit_tier (A–E); n_claims; incurred (total incurred loss, \$0 on most rows); renewed (Y/N). Classify each column and name the plot you would start with.

Answer

  • policy_id — an identifier, not a variable to model. Use it to join tables and to check for duplicate rows
  • veh_age — discrete numeric, treated as continuous. A bar chart of its 26 values, then box plots of severity by age band
  • territory — nominal, despite being stored as integers. A bar chart of exposure by territory, then side-by-side box plots of severity
  • credit_tier — ordinal. A bar chart in tier order, and frequency by tier to see whether the effect is monotone
  • n_claims — count. A bar chart of 0, 1, 2, \ldots, and a comparison of its mean with its variance to check for overdispersion
  • incurred — continuous with a mass at zero. Report the share of zeros, then plot a histogram of \ln(\text{incurred}) for the positive rows
  • renewed — binary. The renewal rate overall and by level of each predictor

The trap is territory: left as an integer it is read as a measurement, and a model would assume territory 8 differs from territory 1 by seven equal steps.

A Territory Code Fitted as a Number A Poisson frequency GLM with a log link was fitted with territory (codes 1–8) left numeric. Its coefficient is \hat\beta = 0.05. What relativity does the model imply between territory 8 and territory 1, and what is wrong with it?

Answer

\begin{align*}
\text{Relativity}_{8:1} &= e^{0.05 \times (8 - 1)} \\
&= e^{0.35} \\
&= 1.419
\end{align*}

The model forces every territory to be e^{0.05} = 1.051 times the one numbered below it — a smooth, monotone trend across labels whose order was set by whoever assigned the codes. Territory 5 cannot be the worst unless 6 to 8 are worse still. Declared categorical, territory costs 7 coefficients instead of 1, but each is estimated from its own territory's data.

How it appears on the exam

  • DISC-DA covers data classifications in its first topic, planning an insurer data modeling project. Application questions describe the columns of an insurance data set and ask how a field should be classified, or what goes wrong when a code stored as a number is treated as a measurement. The same course later classifies data by structure (structured or unstructured) and by source (insurer operational data or external data).
  • MAS-I asks candidates to distinguish categorical, ordinal and continuous predictors in a GLM, so a question may give a variable and ask how it should enter the model.

Exam traps

  • A number is not always numeric: Territory codes, class codes and ZIP codes are labels. Averaging them or fitting them as a single slope treats an arbitrary numbering as a measurement.
  • Ordinal is not interval: An ordinal variable's levels are ordered, but the steps between them need not be equal, so coding credit tiers as 1 to 5 and fitting one slope assumes equal steps.
  • Count versus continuous: A claim count is a non-negative integer and calls for a bar chart and a count model, not a histogram of a continuous variable.