Tweedie double GLM loss triangles with dependence within and across business lines

Abstract

We propose a stochastic model allowing property and casualty insurers with multiple business lines to measure their liabilities for incurred claims risk and calculate associated capital requirements. Our model includes many desirable features which enable reproducing empirical properties of loss ratio dynamics. For instance, our model integrates a double generalized linear model relying on accident semester and development lag effects to represent both the mean and dispersion of loss ratio distributions, an autocorrelation structure between loss ratios of the various development lags, and a copula-based aggregation of risks model driving the dependence across the various business lines. Our work is the first in the literature to combine all such advantageous features within a loss triangle model. The model allows for a joint simulation of loss triangles and the quantification of the overall portfolio risk through risk measures. Consequently, a diversification benefit associated with the economic capital requirements can be measured, in accordance with IFRS 17 standards which allow for the recognition of such benefit. The allocation of capital across business lines based on the Euler allocation principle is then illustrated. The implementation of our model is performed by estimating its parameters based on a car insurance data obtained from the General Insurance Statistical Agency (GISA), and by conducting numerical simulations whose results are then presented.

Appendices

Appendix 1: Parameter estimates for marginal business lines

Estimated parameters for the accident semester and development lag effects which are denoted AS and DL, respectively. Furthermore, the lines of business are Personal Auto (PA) and Commercial Auto (CA) for the three regions of Ontario (ON), Alberta (AB) and Atlantic Canada (ATL) (Tables 5, 6, 7, 8, 9).

Table 5 Mean model—accident semester effects

Table 6 Mean model—development lag effects

Table 7 Dispersion submodel—development lag effects

Table 8 Estimated correlation parameter $\rho _k$ for each business line k

Table 9 Tweedie distribution index parameters $p_k$ for each business line k

Appendix 2: Selection of the Tweedie index

Estimating the Tweedie index $p_k$ is not a trivial endeavour, and therefore a procedure inspired from Dunn and Smyth [13] is considered in the current work. A set of fixed values of $p_k$, namely $p_k=\{1.105,1.110,1.115,\ldots ,1.900\}$, is considered. For each of these values, DGLM parameters $\Theta ^{(\mu )}_k$ and $\Theta ^{(\phi )}_k$ are estimated through maximum likelihood while assuming a null development lag correlation i.e. $\rho _k=0$, the latter assumption considerably simplifying the estimation. The value of $p_k$ for which the loglikelihood is maximized is the value selected as the parameter estimate.

Recall that values of $p_k$ must lie in the (1, 2) interval. However, for stability considerations, values below 1.1 are not considered since values very close to 1 tend to make the distribution multimodal; this complicates the estimation procedure and creates convergence issues. Moreover, values close to 2 were also disregarded for numerical considerations; when $p_k$ is close to 2, the infinite sum approximation embedded in the Tweedie distribution includes a large number of terms that are materially different from zero, which makes computations more cumbersome.

Appendix 3: Goodness-of-fit for parametric copulas

To select appropriate bivariate copula families within the global dependence structure, a Crámer–Von Mises test applicable to copula families is used. Its test statistic is defined as

$${S_n=n\int_{[0,1]^2}\left(C_n({\mathbf{u}})-C_{\theta_n}({\mathbf{u}})\right)^2 \text{d}C_n({\mathbf{u}}),}$$

where $C_n$ is the empirical copula, $\bf{u}=(u_1,u_2) \,{\text{for}}\,\text{all}\, \bf{u} \in [0,1]^2$ $C_{\theta _n}$ is the parametric copula and $\theta _n$ is its rank-based estimated parameter. The null hypothesis of the test is $H_0: C \in C_{\theta }$, i.e., the copula C is a member of the parametric family $C_{\theta }$. p values from the Cramér–Von Mises test are obtained by parametric bootstrapping according to the methodology proposed in Genest and Rémillard [14].

Table 10 presents the test statistic $S_n$, the estimated dependence parameter $\theta _n$ and the p value for the various combinations of business lines subgroups which exhibit statistical evidence of dependence. The Cramér–Von Mises test is applied to six different copula families: Clayton, Gumbel, Frank, Plackett, Gaussian and student-t. The first three are Archimedean copulas, the Plackett copula only exists in a bivariate setting and the Gaussian and t-copula are elliptical copulas. The t-copula family is retained due to its higher associated test statistics $S_n$ for all aggregation steps, except for the aggregation of Alberta and Atlantic Canada where it is the second best. It is nevertheless relevant to note that except for the Clayton family, other copulas are never rejected by the test at a $90\%$ confidence level. Other copula families could therefore also have been adequate choices.

Table 10 Goodness-of-fit test results for alternative parametric families

Appendix 4: The Iman–Conover procedure

Figure 2 provides an illustration of the modeled dependence structure of the GISA dataset lines of business, which is based on a hierarchical aggregation of risks model. The Iman–Conover reordering algorithm used to simulate from the dependence structure goes as follows:

• 1. Simulate k independent samples of size $m>>N$⁸ composed of independent standard normal random variables:

  $$\begin{aligned} {\mathbf {U}}^{(k)} \sim N(0,1), \quad k=\{1,2,3,4,5,6\}. \end{aligned}$$

• 2. Simulate independent copula samples of size m from each bivariate copula $C_1,\ldots ,C_5$.

• 3. Reorder the samples of each bivariate vector by merging the observed marginal ranks with the joint ranks in the copula sample (Table 11). A brief example follows for the first node of the dependence structure.

  Then, the reordered data is a sample from the copula $\left( {\mathbf {U}}^{(1)},{\mathbf {U}}^{(2)}\right) \sim C_1$.

• 4. Repeat step 3 for the first level copulas $C_2$ and $C_3$.

• 5. Aggregate the reordered data following the dependence structure to obtain samples from ${\mathbf {U}}^{(1)}+{\mathbf {U}}^{(2)}$ and respectively for ${\mathbf {U}}^{(3)}+{\mathbf {U}}^{(4)}$ and ${\mathbf {U}}^{(5)}+{\mathbf {U}}^{(6)}$.

• 6. Repeat step 3 to obtain sample from $\left( {\mathbf {U}}^{(3)}+{\mathbf {U}}^{(4)},{\mathbf {U}}^{(5)}+{\mathbf {U}}^{(6)}\right) \sim C_4$.

• 7. Aggregate the reordered sample from $C_4$ to obtain a sample from ${{\sum}_{k=3}^{6}{\mathbf{U}}^{(k)}}$, and repeat step 3 for ${\left({\mathbf{U}}^{(1)}+{\mathbf{U}}^{(2)},{\sum}_{k=3}^{6}{\mathbf{U}}^{(k)}\right)\sim C_5}$.

• 8. To obtain a joint sample of $\left( {\mathbf {U}}^{(1)}, {\mathbf {U}}^{(2)}, {\mathbf {U}}^{(3)}, {\mathbf {U}}^{(4)}, {\mathbf {U}}^{(5)}, {\mathbf {U}}^{(6)}\right)$, perform the permutations applied to ${\mathbf {U}}^{(1)}+{\mathbf {U}}^{(2)}$ back to ${\mathbf {U}}^{(1)}$ and ${\mathbf {U}}^{(2)}$, the permutations applied to ${\mathbf {U}}^{(3)}+{\mathbf {U}}^{(4)}$ back to ${\mathbf {U}}^{(3)}$ and ${\mathbf {U}}^{(4)}$, and finally, the permutations applied to ${\mathbf {U}}^{(5)}+{\mathbf {U}}^{(6)}$ back to ${\mathbf {U}}^{(5)}$ and ${\mathbf {U}}^{(6)}$.

• 9. Get a subsample of size N from the reordered sample of size m.

Table 11 Iman–Conover reordering algorithm example for the first node of the dependence structure from Fig. 2

Appendix 5: The conditional distribution of simulated scaled observations

The assumption made in the current paper’s model based on (3) is that for a given accident semester i and business line k, the scaled observations vector $\tilde{{\mathbf {Y}}}^{(k)}_i$ are approximately multivariate normal with a null mean vector and covariance matrix $R_{k,J}$ as defined in (4).

A classic result on multivariate normal distributions is first recalled. Consider a multivariate normal random column vector $\mathbf{X }$ which is decomposed into two blocks (i.e. two stacked random vectors): $\mathbf{X } = [(\mathbf{X }^{(1)})^\top \,\, (\mathbf{X }^{(2)})^\top ]^\top$. Denote respectively the mean vector and covariance matrix of the entire vector $\mathbf{X }$ and of each of the two blocks $\mathbf{X }^{(1)}$ and $\mathbf{X }^{(2)}$ by

$$\begin{aligned} \pmb {\mu } = \left[ \begin{array}{c} \pmb {\mu }^{(1)} \\ \pmb {\mu }^{(2)} \end{array} \right] , \quad \Sigma = \left[ \begin{array}{cc} \Sigma ^{(1,1)} &{} \Sigma ^{(1,2)} \\ \Sigma ^{(2,1)} &{} \Sigma ^{(2,2)} \end{array} \right] . \end{aligned}$$

Then, the conditional distribution of $\mathbf{X }^{(2)}$ given $\mathbf{X }^{(1)}$ is multivariate normal with mean $\pmb {\mu }^{(2)} + \Sigma ^{(2,1)} \left[ \Sigma ^{(1,1)}\right] ^{-1} \left( \mathbf{X }^{(1)}-\pmb {\mu }^{(1)}\right)$ and variance $\Sigma ^{(2,2)} - \Sigma ^{(2,1)} \left[ \Sigma ^{(1,1)}\right] ^{-1} \Sigma ^{(1,2)}$.

We can decompose the scaled innovation vector $\tilde{{\mathbf {Y}}}^{(k)}_i$ into two blocks: the unobserved one $\mathbf{X }^{(1)} = \tilde{{\mathbf {Y}}}^{(k)}_{i,J+2-i:J} \equiv [Y^{(k)}_{i,J+2-i} , \ldots , Y^{(k)}_{i,J}]^\top$ and the observed one $\mathbf{X }^{(2)} = \tilde{{\mathbf {Y}}}^{(k)}_{i,1:J+1-i} \equiv [Y^{(k)}_{i,1} , \ldots , Y^{(k)}_{i,J+1-i}]^\top$. In other words,

$$\begin{aligned} \tilde{{\mathbf {Y}}}^{(k)}_i = \left[ \begin{array}{c} \tilde{{\mathbf {Y}}}^{(k)}_{i,1:J+1-i} \\ \tilde{{\mathbf {Y}}}^{(k)}_{i,J+2-i:J} \end{array} \right] . \end{aligned}$$

Since, the covariance matrix of $\tilde{{\mathbf {Y}}}^{(k)}_i$ can be decomposed as

$$\begin{aligned} R_{k,J} = \left[ \begin{array}{cc} R_{k,J+1-i} &{} R^{(1,2)}_{k,i} \\ (R^{(1,2)}_{k,i})^\top &{} R_{k,i-1} \end{array} \right] \end{aligned}$$

with

$$\begin{aligned} R^{(1,2)}_{k,i} \equiv \begin{bmatrix} \rho ^{J+1-i}_k &{} \rho ^{J+2-i}_k &{} \rho ^{J+3-i}_k &{} \ldots &{}\rho ^{J-1}_k \\ \vdots &{} \vdots &{} \vdots &{}\ddots &{}\vdots \\ \rho ^{2}_k &{} \rho ^{3}_k &{} \rho ^{4}_k&{}\ldots &{}\rho ^{i}_k \\ \rho _k &{} \rho ^{2}_k &{} \rho ^{3}_k&{}\ldots &{} \rho ^{i-1}_k \\ \end{bmatrix}. \end{aligned}$$

Setting $\pmb {\mu } = \pmb {0}$ in the previous result along with $\Sigma ^{(1,1)} = R_{k,J+1-i}$, $\Sigma ^{(2,2)} = R_{k,i-1}$ and $\Sigma ^{(1,2)} = R^{(1,2)}_{k,i}$ leads to approximate the conditional distribution of unobserved scaled observations $\tilde{{\mathbf {Y}}}^{(k)}_{i,J+2-i:J}$ given observed ones $\tilde{{\mathbf {Y}}}^{(k)}_{i,1:J+1-i}$ by a multivariate normal with mean vector and covariance matrix being respectively:

$$\begin{aligned} \breve{{\mathbf {M}}}^{(k)}_{i} \equiv {\mathbb {E}} \left[ \tilde{{\mathbf {Y}}}^{(k)}_{i,J+2-i:J} \vert \tilde{{\mathbf {Y}}}^{(k)}_{i,1:J+1-i} \right]= & {} (R^{(1,2)}_{k,i})^\top \left[ R_{k,J+1-i} \right] ^{-1} \tilde{{\mathbf {Y}}}^{(k)}_{i,1:J+1-i}, \end{aligned}$$

(8)

$$\begin{aligned} \breve{V}^{(k)}_{i} \equiv \text {Cov} \left[ \tilde{{\mathbf {Y}}}^{(k)}_{i,J+2-i:J} \vert \tilde{{\mathbf {Y}}}^{(k)}_{i,1:J+1-i} \right]= & {} R_{k,i-1} - (R^{(1,2)}_{k,i})^\top \left[ R_{k,J+1-i} \right] ^{-1} R^{(1,2)}_{k,i} \!\!. \end{aligned}$$

(9)

Appendix 6: Capital requirements based on alternative copulas

Tables 12 and 13 present values for the economic capital and the risk adjustment for non-financial risks under various choices of parametric copulas in the global dependence structure depicted by Fig. 3. The first row (t) provides the results using the t-copula structure exactly as obtained in Sect. 5.1.2. The second row (Gaussian) and third row (Independence) respectively present the results obtained when replacing the t-copulas with either Gaussian copulas or the independence copula.

Table 12 Economic capital and allocation to business lines based on the Euler allocation principle (millions CAD)

Table 13 Risk adjustments for non-financial risks and allocation to business lines based on the Euler allocation principle (millions CAD)

As expected, the risk adjustment is lower when using the independence copula rather than the elliptical copulas. Furthermore, the t-copula structure is slightly more conservative due to its heavy dependence in the tails. The difference between results stemming from the three choices of copula families is nevertheless small. This is mainly due to two reasons. First, the exposure is heavily skewed towards the Ontario personal auto branch in the GISA dataset of the current study, which limits the impact of the dependence model. Second, in our framework, the data contains industry aggregates rather than company-specific losses, which lead to small variability due to the law of large numbers.