content_image/309782.jpg content_image/309783.jpg
because these are evaluated pointwise. The timing for Monte Carlo would be unchanged. The timing for Panjer would be roughly the proportion s$_{0}$/ $40 m of the total timing, because it is evaluated sequentially, from small to large values of s . 

## 6.3 Gamma thickening of the event losses 

We continue to consider one-year losses, but now treat the losses from each event as random, not fixed. For the simplest possible generalisation we use a Gamma distribution with a specified expectation x$_{i}$ and a common specified coefficient of variation, θ := σ$_{i}$/x$_{i}$ . The previous case of a fixed loss x$_{i}$ is represented by lim θ → 0, which we write, informally, as θ = 0. Solving 

gives the two Gamma distribution parameters as 

<img src='content_image/309782.jpg'>

<img src='content_image/309783.jpg'>

Figure 4 shows the effect of varying θ on a Gamma distribution with expec- tation $1 m. 

The only practical difficulty with allowing random losses for each event occurs for the Panjer method; we describe our approach in the Appendix. 

Figure 5 shows the exceedance probability curve with θ = 0 . 5: note that the horizontal scale now covers a much wider range of loss values than Figure 3. The timings are given in Table V: these are very similar to the 

