Frequency-Severity Models

Study guide: Exam 8

Frequency-Severity Models build a pure premium from separate component models — one for claim frequency (claims per exposure) and one for claim severity (loss per claim) — and combine them, rather than modelling pure premium in a single step.

$\widehat{PP}_i = \hat{f}_i \times \hat{s}_i$

$R^{PP}_k = R^{f}_k \times R^{s}_k = e^{\hat\beta^{f}_k + \hat\beta^{s}_k}$

$\hat f_i$ and $\hat s_i$ are the predicted frequency and severity for risk $i$ . $R_k$ is the relativity for level $k$ of a variable, and $\hat\beta_k$ its fitted coefficient. The usual GLM setup is a Poisson (or overdispersed Poisson or negative binomial) frequency model with a log link and a $\ln(\text{exposure})$ offset, and a Gamma severity model with a log link, weighted by claim count. With log links on both, relativities multiply and coefficients add. The one-step alternative is a Tweedie pure premium model.
Why separate (Goldburd et al.). First, insight: you see whether an effect is frequency- or severity-driven, and a variable that raises frequency while lowering severity equally would look like no effect at all in pure premium. Second, stability: each component is less noisy than pure premium, so a frequency-only effect is not drowned by severity noise, which would cause underfitting. Third, no spurious fit: a pure premium model forced to fit a frequency effect also fits the severity noise, which is overfitting. Fourth, the Tweedie implicitly assumes frequency and severity move in the same direction.
Costs. Separate models need claim counts and amounts and double the modelling work, which matters when many segments or perils must each be modelled.
Other component splits. Model each coverage or peril separately (fire, wind and hail, all other), or each claim type (workers compensation injury types), or capped losses with a separate excess load. To combine by-peril models, score them on data with the expected mix of business, add the peril predictions, and fit one multiplicative model to the sum using the union of their predictors. A sum of multiplicative models is not itself multiplicative, so the refit's factors depend on the mix.
The combined predictions are then tested on a holdout like any other candidate: lift, and a double lift chart against the current plan (see Model Fit).

Combining Frequency and Severity Relativities A log-link frequency GLM has intercept $\ln 0.05$ and a youthful-driver coefficient of $0.40$ . The companion severity GLM has intercept $\ln 8{,}000$ and a youthful coefficient of $-0.10$ . Find the youthful pure premium relativity and the youthful pure premium.
Answer
 $\begin{align*} R^{f} &= e^{0.40} \\ &= 1.492 \\[4pt] R^{s} &= e^{-0.10} \\ &= 0.905 \\[4pt] R^{PP} &= e^{0.40 - 0.10} \\ &= e^{0.30} \\ &= 1.350 \\[6pt] \widehat{PP}_{\text{youthful}} &= (0.05 \times 8{,}000) \times 1.350 \\ &= \$539.94 \end{align*}$ 
Check: $0.074591 \times \$7{,}238.70 = \$539.94$ . Youthful drivers have $49\%$ more claims, but those claims are $10\%$ smaller. A Tweedie model would report only a $+35\%$ effect, and it would fit that effect as if frequency and severity rose together. The split also tells underwriting and claims where the problem lies.

Combining Two Peril Models A homeowners fire model gives a base loss cost of $\$100$ and a frame-construction relativity of $1.50$ . Fire has no territory effect. A separate wind model gives a base of $\$60$ and a coastal relativity of $2.00$ . Wind has no construction effect. Exposure mix: brick/inland $40\%$ , frame/inland $30\%$ , brick/coastal $20\%$ , frame/coastal $10\%$ . Build an all-peril multiplicative plan.
Answer Add the peril loss costs in each cell: brick/inland $160$ , frame/inland $210$ , brick/coastal $220$ , frame/coastal $270$ .

Reading the factors off the base cell gives frame $210/160 = 1.3125$ and coastal $220/160 = 1.375$ . The frame/coastal cell then comes out at
 $\begin{align*} 160 \times 1.3125 \times 1.375 &= 288.75 \\ \frac{288.75}{270} - 1 &= +6.9\% \end{align*}$ 
The whole error lands in one cell. Instead, fit an exposure-weighted log-link model to the four summed cells. It gives a base of $161.85$ , frame $1.282$ and coastal $1.337$ , and the fitted cells are

$161.85,\ 207.54,\ 216.31,\ 277.38$

The errors are now $+1.2\%$ , $-1.2\%$ , $-1.7\%$ and $+2.7\%$ , and the book total is exact at $\$198$ per exposure. The refit spreads the unavoidable non-multiplicative error across the cells in proportion to the mix, which is why the data used for it should reflect the expected mix of business.