A new simulation algorithm for more precise estimates of change in catastrophe risk models, with application to hurricanes and climate change

Abstract

Catastrophe models are widely used for assessing extreme weather risks. One way catastrophe models are applied is to run an analysis, and then run it again with adjusted hazard. This allows, for example, estimation of the impact of seasonal forecasts or climate change projections. One way to implement this approach involves resimulating a year loss table (YLT) from a list of events with adjusted frequencies. The new YLT is usually simulated independently from the original YLT, but this leads to differences between the two YLTs, even if the frequencies have not been adjusted, due to simulation noise. The simulation noise reduces the precision of all estimates of change. We present a new algorithm that attempts to reduce this problem. We create the new YLT incrementally by copying the original YLT and then adding or removing just enough events to capture the change. This algorithm does not involve any approximations, and is no slower. We test the algorithm using a number of simple catastrophe models for U.S. hurricane loss, to which we apply adjustments for climate change. We find that the incremental simulation method is much more precise than independent simulation in all our tests. For example, the estimates of the impact of climate change so far are three times more precise. This equates to the increase in precision that would occur from using nine times as many years of simulation, but for no extra cost.

Appendices

Appendix A

In the incremental simulation algorithm, for events with frequency adjustment $d\left({{i}},{{j}}\right)$ in the range $\left(\mathrm{0,1}\right),$ we copy the events from the original simulation to the adjusted simulation with probability ${{p}}=d\left({{i}},{{j}}\right).$ As a result, the number of occurrences of an event in the incremental simulation is given by a binomial distribution with number of trials given by the number of occurrences of the event in the original simulation, which is a sample from the original Poisson distribution ${{P}}{{o}}({{r}})$, and with probability of success ${{p}}$. The unconditional probability of ${{k}}$ events occurring in the incremental simulation can therefore be written in terms of the probabilities of ${{n}}$ events from the original Poisson distribution using the law of total probability as:

$$p(k)={\sum}_{n=k}^{\infty}{p}(k|n)p(n)={\sum}_{n=k}^{\infty}(nk)p^k(1-p)^{nk}\frac{e^{-r}r^n}{n!}=\frac{e^{-pr}(pr)^k}{k!}$$

We see that the frequency distribution of events in the incremental simulation is a Poisson distribution with mean ${{pr}}=d\left({{i}},{{j}}\right){{r}}$, which is the same as the frequency distribution of events in the independent simulation. This justifies the methodology of deleting events with probability ${{p}}.$

Appendix B

For AAL as a metric, we can gain more insight into $Cov(B,A)$ as follows. Writing ${{{n}}}_{{{i}}{{j}}},{{{n}}}_{{{i}}{{j}}}{^{\prime}}$ as the number of events simulated in the original and incremental simulations respectively, the estimates $A,B$ are given by ${{A}}=\frac{1}{{{{N}}}_{{{y}}}}{\sum }_{{{i}}=1}^{{{{N}}}_{{{y}}}}{\sum }_{{{j}}=1}^{{{{N}}}_{{{e}}}}{{{n}}}_{{{i}}{{j}}}{{{l}}}_{{{j}}}$ and ${{B}}=\frac{1}{{{{N}}}_{{{y}}}}{\sum }_{{{i}}=1}^{{{{N}}}_{{{y}}}}{\sum }_{{{j}}=1}^{{{{N}}}_{{{e}}}}{{{n}}}_{{{i}}{{j}}}{\prime}{{{l}}}_{{{j}}}$, where ${{{N}}}_{{{y}}}$ is the number of years of simulation, ${{{N}}}_{{{e}}}$ is the number of events in the event set and ${{{l}}}_{{{j}}}$ is the loss for event ${{j}}$. We assume that ${{{n}}}_{{{i}}{{j}}}$ is non-zero for at least one pair $\left({{i}},{{j}}\right).$ We then have

$$Cov(B,A)=Cov\left(\frac{1}{{{{N}}}_{{{y}}}}{\sum }_{{{i}}=1}^{{{{N}}}_{{{y}}}}{\sum }_{{{j}}=1}^{{{{N}}}_{{{e}}}}{{{n}}}_{{{i}}{{j}}}{{{l}}}_{{{j}}},\frac{1}{{{{N}}}_{{{y}}}}{\sum }_{{{i}}=1}^{{{{N}}}_{{{y}}}}{\sum }_{{{j}}=1}^{{{{N}}}_{{{e}}}}{{{n}}}_{{{i}}{{j}}}{\prime}{{{l}}}_{{{j}}}\right)={\left(\frac{1}{{{{N}}}_{{{y}}}}\right)}^{2}\left({\sum }_{{{i}}=1}^{{{{N}}}_{{{y}}}}{\sum }_{{{j}}=1}^{{{{N}}}_{{{e}}}}{{{{{l}}}_{{{j}}}}^{2}Cov({{n}}}_{{{i}}{{j}}},{{{n}}}_{{{i}}{{j}}}{\prime})\right)$$

We will write ${{{n}}}_{{{i}}{{j}}}^{{\prime}}={{{{\epsilon}}}_{{{i}}{{j}}}{{n}}}_{{{i}}{{j}}}+{{\alpha }}_{{{i}}{{j}}}$ where ${{{\epsilon}}}_{{{i}}{{j}}}$ $\in$ (0,1) and represents the events being copied or not (for frequency decreases), and ${\alpha }_{{{i}}{{j}}}$ represents additional events being added (for frequency increases). This gives

$${Cov({{n}}}_{{{i}}{{j}}},{{{n}}}_{{{i}}{{j}}}{\prime})={Cov({{n}}}_{{{i}}{{j}}},{{{{\epsilon}}}_{{{i}}{{j}}}{{n}}}_{{{i}}{{j}}}+{\alpha }_{{{i}}{{j}}})={Cov({{n}}}_{{{i}}{{j}}},{{{{\epsilon}}}_{{{i}}{{j}}}{{n}}}_{{{i}}{{j}}})+{Cov({{n}}}_{{{i}}{{j}}},{\alpha}_{{{i}}{{j}}})$$

When additional events are simulated, the simulation is performed independently of the events in the original event set, and so ${Cov({{n}}}_{{{i}}{{j}}},{\alpha}_{{{i}}{{j}}})=0.$ As a result, ${Cov({{n}}}_{{{i}}{{j}}},{{{n}}}_{{{i}}{{j}}}^{\prime})={Cov({{n}}}_{{{i}}{{j}}},{{{{\epsilon}}}_{{{i}}{{j}}}{{n}}}_{{{i}}{{j}}}).$Using the law of total covariance, and conditioning on ${{{\epsilon}}}_{{{i}}{{j}}}$, we find that ${Cov({{n}}}_{{{i}}{{j}}},{{{{\epsilon}}}_{{{i}}{{j}}}{{n}}}_{{{i}}{{j}}})={{{E}}\left({{{\epsilon}}}_{{{i}}{{j}}}\right){{V}}({{n}}}_{{{i}}{{j}}})={{p}}{{{V}}({{n}}}_{{{i}}{{j}}}).$ Since ${{p}}{{{V}}({{n}}}_{{{i}}{{j}}})$ is greater than or equal to zero, we conclude that ${Cov({{n}}}_{{{i}}{{j}}},{{{{\epsilon}}}_{{{i}}{{j}}}{{n}}}_{{{i}}{{j}}})\ge 0,$ and so ${Cov({{n}}}_{{{i}}{{j}}},{{{n}}}_{{{i}}{{j}}}\prime)\ge 0,$ implying that $Cov(B,A)\ge0.$ The special case $Cov(B,A)=0$ only occurs when ${{{\epsilon}}}_{{{i}}{{j}}}=0$ for all $\left({{i}},{{j}}\right)$, which is the case where none of the events in the baseline event set are copied. For a large enough number of years of simulation, this is only likely to happen if the frequency changes are zero or very close to zero i.e., that climate change is projected to lead to almost no events in the future.