Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

144 Actuarial Modelling of Claim Counts
Numerical Illustration
Since we worked with mixing distributions with unbounded support, we cannot use the
maximal variables described above. In practice, setting b equal to a high quantile (99.99 %,
say) of the mixing distribution is expected to give good results.
Let us consider the Negative Binomial model for the annual claim number in Portfolio A.
In this case, i ∼ ama a, with estimated parameter â = 1065. The estimated moments
of i are
̂ 1 = 1
̂ 2 = â + 1 = 1939
â
â + 1â + 2
̂ 3 = = 5580
̂ 4 =
â 2
â + 1â + 2â + 3
â 3
= 21299
Applying formula (3.7) with discrete approximation given by Table 3.5, we can approximate
i with the help of the 4-convex minimum with two support points: r + = 32883 with
probability 01521 and r − = 05897 with probability 08479, or with the 5-convex minimum
with three support points: t + = 45218 with probability 00474 and t − = 12341 with
probability 06366 and 0 with probability 03160. The accuracy of these approximations
decreases with T i and k • . Discrete approximations are useful for short claim history and
rather good driving record. Since the Gamma distribution (playing the role of the mixing
distribution in the Negative Binomial model) has a unimodal probability density function,
we can also use a mixture of uniform distributions to approximate i in Portfolio A. It turns
out that the approximation is satisfactory, except for high values of k • . Again, this is due to
the fact that the approximation uses s-convex minima that underestimate the riskiness of the
worse drivers.
To get accurate approximations we need to use a mixture (3.9) of the improved 4-convex
extrema in the unimodal case taking for b the 99.99th quantile of the Gamma distribution.
This gives the results displayed in Table 3.10 for a good driver, in Table 3.11 for an average
driver and in Table 3.12 for a bad driver. We can see that the approximations are now very
satisfactory for the vast majority of the combinations T i –k • .
3.3.9 Linear Credibility
Bayesian statistics offer an intellectually acceptable approach to credibility theory. Bayes
revision E i N i• of the heterogeneity component is theoretically very satisfying but is
often difficult to compute (except for conjugate distributions or discrete approximations).
Practical applications involve numerical methods to perform integration with respect to a
posteriori distributions, making more elementary approaches desirable (at least to get a first
easy-to-compute approximation of the result). Because we have observed N i1 N iTi , one
suggestion is to approximate E i N i1 N iTi by a linear function of the N it s. Basically,
the actuary still resorts to a quadratic loss function but the shape of the credibility predictor