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

Risk Classification 67
The computation is easier if we change the likelihood to the log-likelihood which is then
given by
L = ln =
n∑
i=1
(
− ln k i !+k i ln i − i
)
(2.2)
The maximum likelihood estimators ̂ 0 and the ̂ j s are the solutions of the following
likelihood equations that are obtained by making the first derivatives of the log-likelihood
with respect to the regression coefficients equal to zero:
0
L = 0 ⇔
n∑
k i − i = 0 (2.3)
i=1
j
L = 0 ⇔
n∑
x ij k i − i = 0 j = 1p (2.4)
i=1
2.3.6 Interpretation of the Likelihood Equations
Equation (2.3) has an obvious interpretation: the fitted total number of claims ∑ n
i=1̂ i is
equal to the observed total number of claims ∑ n
i=1 k i. Therefore, provided that an intercept
0 is included in the score, the total claim number predicted by the regression model equals
its observed counterpart. Note that this equality holds for the observation period and not
necessarily for the future, when the ratemaking will be implemented in practice. In other
words, we cannot be sure that ∑ n
i=1̂ i claims will be filed in the future, just that the actual
number of claims should be close to ∑ n
i=1̂ i if the yearly number of claims filed to the
company remains stable over time.
The interpretation of the second likelihood Equation (2.4) is as follows: In Example 2.2
with Portfolio A, Equation (2.4) for j = 4 gives
∑
females
k i = ∑
females
Therefore, the model fits exactly the total number of claims filed by female policyholders.
There is no cross-subsidies between men and women. The conclusion is similar for the other
values of j. For j = 1 2 3 for instance, the Equations (2.4) thus ensure that the sum of all
the claims reported for each age category is exactly reproduced by the model.
̂i
2.3.7 Solving the Likelihood Equations with the Newton–Raphson
Algorithm
The likelihood equations do not admit explicit solutions and must therefore be solved
numerically. Let U be the gradient vector of the log-likelihood L = ln defined