content_image/299995.jpg
Now it is crucial that the f$_{i}$ have standard forms with simple known MGFs. 

In an unlimited optimisation, the Chernoff bound will never outperform the Moment bound (Philips and Nelson, 1995). In practice, however, con- straints on the optimisation of the Moment bound may result in the best available Chernoff bound being lower than the best available Moment bound. But there is another reason to include the Chernoff bound, from large de- viation theory; see, e.g., Whittle (2000, sec. 15.6 and ch. 18). Let t be an integer number of years, and define S$_{1}$ as the loss from one year, so that M$_{S}$$_{t}$ ( k ) = { M$_{S}$$_{1}$ ( k ) } $^{t}$. Then large deviation theory states that 

<img src='content_image/299995.jpg'>

and so as t becomes large the Chernoff upper bound becomes exact. Very informally, then, the convergence of the Chernoff bound and the Moment bound suggest, according to a squeezing argument, that both bounds are converging from above on the actual probability. 

## 4 TWO ‘EXACT’ APPROACHES 

There are several approaches to computing Pr( S$_{t}$ ≥ s ) to arbitrary accu- racy, although in practice this accuracy is limited by computing power (see Shevchenko, 2010, for a review). We mention two here. 

Monte Carlo simulation. One realisation of S$_{t}$ for a fixed time-interval can be generated by discrete event simulation, also known as the Gillepsie algorithm (see, e.g., Wilkinson, 2012, sec. 6.4). Many such simulations can 

