## Predicting annual losses

Results from the previous section provide the needed background for building a preseason model capable of predicting the annual expected loss. The model uses a hierarchical Bayesian specification. The final form of the model was based on comparison of the deviance information criterion (DIC) using several different models involving the three covariates. The DIC is a generalization of the Akaike information criterion (AIC) and Bayesian information criterion (BIC). It is useful in Bayesian model selection where the posterior distributions of the models are obtained by Markov chain Monte Carlo (MCMC) simulation. Like the AIC and BIC, it is an asymptotic approximation as the sample size becomes large. It is only valid when the posterior distribution is close to multivariate normal. We chose the model with the lowest value of DIC.

A schematic of the hierarchical model is shown in Figure 10.6. The predicted annual loss (TL) is the sum of the individual loss amounts (both large [LLl] and small [LLS] amounts) multiplied by the respective number of large (Nl) and small (NS) loss counts. Given the mean (^L) and standard deviation (sL) of the logarithm of large losses, the logarithm of large loss follows a truncated normal distribution. Small loss amounts are also specified by using a truncated normal distribution, although the mean is not a function of any of

Figure 10.6. Hierarchical graph illustrating our strategy for simulating annual insured losses based on preseason values of the NAO and Atlantic SST. The connection between nodes is either stochastic (thick arrow) or logical (thin arrow). Node 1L (1S) is the mean annual rate of large (small) losses, Nl (Ns) is the annual count of large (small) loss events, mL (mS) is the mean amount of large (small) loss on a log scale, aL (aS) is the standard deviation of large (small) loss amounts, LLL (LLS) is the logarithm of large (small) loss amount, and TL is the total loss. Figure 10.6. Hierarchical graph illustrating our strategy for simulating annual insured losses based on preseason values of the NAO and Atlantic SST. The connection between nodes is either stochastic (thick arrow) or logical (thin arrow). Node 1L (1S) is the mean annual rate of large (small) losses, Nl (Ns) is the annual count of large (small) loss events, mL (mS) is the mean amount of large (small) loss on a log scale, aL (aS) is the standard deviation of large (small) loss amounts, LLL (LLS) is the logarithm of large (small) loss amount, and TL is the total loss.

the covariates. Given a mean annual rate of large losses (1L), the annual number of large losses follows a Poisson distribution with the natural logarithm of the rate given as a linear function of the NAO. Similarly, given a mean annual rate of small losses (1S), the annual number of small losses follows a Poisson distribution with the natural logarithm of the rate given as a separate function of the NAO.

Samples of the annual losses are generated using WinBUGS (Windows version of Bayesian inference Using Gibbs Sampling) developed at the Medical Research Council in the United Kingdom (Gilks et al, 1996; Spiegelhalter et al., 1996). WinBUGS chooses an appropriate MCMC sampling algorithm based on the model structure. In this way, annual losses are samples conditional on the model coefficients and the observed values of the covariates. The cost associated with a Bayesian approach is the requirement to formally specify prior beliefs. Here we take the standard route and assume noninformative priors, which as the name implies provide little information about the parameters of interest. Markov chain Monte Carlo analysis, in particular Gibbs sampling, is used to sample the parameters given the data, since no closed form distribution exists for the truncated normal (or for the generalized Pareto distribution [GPD] used in the next section).

We check for mixing and convergence by examining successive sample values of the parameters. Samples from the posterior distributions of the parameters indicate relatively good mixing and quick settling as two different sets of initial conditions produce sample values that fluctuate around a fixed mean. Based on these diagnostics, we discard the first 10,000 samples and analyze the output from the next 10,000 samples. The utility of the Bayesian approach for modeling the mean number of coastal hurricanes is described in Elsner and Jagger (2004).

Figure 10.7 shows the predictive posterior distributions of losses for two different climate scenarios. The first scenario is characterized by preseason conditions featuring a combination of high NAO values and low SST values. To offer a strong contrast, we set the values to their maximum and minimum, respectively, over the 106-year period (1900-2005; NAO =+2.9 s.d. and SST = —0.61 °C). This situation is unfavorable for hurricane activity along the US coast (Elsner and Jagger, 2006). Simulation results show that the probability of no loss (47%) is close to the probability of at least some loss (53%). This result contrasts with those from the second scenario, which is characterized by conditions favorable for hurricane activity (NAO =—2.7 s.d. and SST =+0.55 °C). Here the probability of at least some loss is 94%.

Perhaps more useful is the predictive distribution of losses, given that at least some loss occurs. Here the distributions are shown for the logarithm

CL "cB

Loss No Loss 