A general insurance company that writes Fine Art insurance has decided to use its historical loss data to replace the current model (mostly based on underwriters’

experience and judgment) with a generalised linear model calibrated using historical loss data, and you are in charge of this investigation.

As a result of this investigation, you have produced a frequency/severity model where the severity distribution comes from an MBBEFD exposure curve. As for the frequency, you have modelled it as a Poisson distribution with Poisson rate depending on the following rating factors:

• Territory X1 (5 levels: US and Canada, Latin America, Europe, Asia, Middle East and Africa)

• Security level X2 (3 levels: low, medium, high)

and proportional to the exposure X3 (sum insured).

i. Assuming the factors are non-interacting, write down an expression for the Poisson rate.

ii. Calculate the number of degrees of freedom of the model in (i).
According to the underwriters, the frequency of claims is not actually proportional to the exposure but sub-proportional. You then decide to explore a modified model in which the Poisson rate is proportional to X3 α , with α to be determined.
iii. Modify the expression for the Poisson rate to accommodate this modified model.
iv. Explain how you would test whether this change is actually an improvement using

(a) the Akaike Information Criterion (AIC), specifying what the relationship between the log-likelihood of the two models needs to be;

(b) a hold-out sample (i.e. a test data set that has not been used for calibration of the model).