Application of uncertain hurricane climate change projections to catastrophe risk models

Abstract

There is great interest in trying to understand how to take climate model projections of possible changes in hurricane behaviour due to climate change and apply them to hurricane risk models. In Knutson et al. (Bull Am Meteorol Soc 101:E303–E322, 2020), projections from many climate models were combined to form distributions of possible changes in hurricane frequency and intensity. It has been shown that propagating the uncertainty represented by these distributions is necessary to estimate the impact on risk correctly. Building on these results, we now consider how distributions of changes in hurricane frequency and intensity can be applied to hurricane risk models that are formulated in the standard event loss table and year loss table formats. Because of the uncertainty, this requires the use of novel simulation and weighting techniques that extend standard methods for adjusting risk models. We demonstrate that these novel techniques work in a simple hurricane risk model. We also present new analytical solutions that show how means and variances of risk change due to the application of uncertain adjustments. Finally, we use emulators to explore how the output from just a single evaluation of a hurricane risk model can be used to derive sensitivity estimates that would otherwise require a large number of evaluations of the model. The methods we present could readily be applied to full complexity hurricane risk models and will hopefully contribute to efforts to quantify the possible effects of climate change on present and future hurricane risk.

Appendices

Appendix A

In this appendix we define the various quantities used in the derivations in appendices B, C and D

$N_{events} =$ number of events in the event set.

$i = 1 \ldots n =$ index for the events.

$l_{i} =$ loss for event $i$.

${\upmu }_{{l_{i} }} ,{\upsigma }_{{l_{i} }}^{2} =$ mean and variance of $l_{i}$.

$n_{i} =$ number of occurrences of event $i$ in a year.

${\upmu }_{{n_{i} }} ,{\upsigma }_{{n_{i} }}^{2} =$ mean and variance of $n_{i}$.

$\lambda_{i} =$ mean and variance of $n_{i}$ when Poisson distributed.

$r_{j} \ge 0$ = the scaling applied to ${\upmu }_{{n_{i} }}$ in the $j$′th subset of the event set.

$s_{j} \ge 0 =$ the scaling applied to $\sigma_{{n_{i} }}^{2}$ in the $j$′th subset of the event set. For the special case of a Poisson distribution, $s_{j} = r_{j}$.

${\upmu }_{{r_{j} }} , {\upsigma }_{{r_{j} }}^{2} =$ mean and variance of $r_{j}$.

${\upmu }_{{s_{j} }} , {\upsigma }_{{s_{j} }}^{2} =$ mean and variance of $s_{j}$.

$\Sigma$ is the sum over all events in the event set.

$\Sigma_{j}$ is the sum over events in the $j$′th subset of the event set.

$\Sigma_{ss}$ is the sum over subsets.

$N = \Sigma n_{i} = \Sigma_{ss} \Sigma_{j} n_{i} =$ number of events in one year.

$L = \Sigma n_{i} l_{i} = \Sigma_{ss} \Sigma_{j} n_{i} l_{i} =$ loss in one year.

$E, V =$ expectation and variance over all random variables.

$E_{rs} ,V_{rs} =$ expectation and variance over $r$ and $s$, holding the $n_{i}$ and $l_{i}$ fixed.

$E_{n} ,V_{n} =$ expectation and variance over $n_{i}$, holding $r_{j}$ and $s_{j}$ fixed.

$E_{nl} ,V_{nl} =$ expectation and variance over $n_{i}$ and $l_{i}$, holding $r_{j}$ and $s_{j}$ fixed.

$AAE_{base} ,AAE_{adj} =$ expectation of number of events in a year for the base and adjusted models.

$AAE_{j} =$ expectation of number of events in a year in the $j$′th subset of the event set in the base model.

$AAL_{base} ,AAL_{adj} =$ expectation of loss in a year (a.k.a. average annual loss) for the base and adjusted models.

$AAL_{j} =$ expectation of loss in a year (a.k.a. average annual loss) in the $j$′th subset of the event set in the base model.

$VAE_{base} ,VAE_{adj} =$ variance of the number of events in a year for the base and adjusted models.

$VAE_{j} =$ variance of the number of events in a year in the $j$′th subset of the event set in the base model.

$VAL_{base} ,VAL_{adj} =$ variance of the loss in a year for the base and adjusted models.

$VAL_{j} =$ variance of the loss in a year in the $j$′th subset of the event set in the base model.

Appendix B

We now derive expressions for the AAE, AAL, VAE and VAL, based on the properties of the events in an ELT. The derivations use the standard properties of the expectation and variance operators, including expectation and variance of products. These expressions are used, with the secondary uncertainty set to zero, to calculate the first column in each panel in Fig. 3.

$$AAE_{base} = E\left( N \right) = E\left( {\Sigma n_{i} } \right) = \Sigma E(n_{i} ) = \Sigma {\upmu }_{{n_{i} }}$$

(A1)

$$VAE_{base} = V\left( N \right) = V\left( {\Sigma n_{i} } \right) = \Sigma V(n_{i} ) = \Sigma {\upsigma }_{{n_{i} }}^{2}$$

(A2)

$$AAL_{base} = E\left( L \right) = E\left( {\Sigma n_{i} l_{i} } \right) = \Sigma E(n_{i} l_{i} ) = \Sigma E(n_{i} )E(l_{i} ) = \Sigma {\upmu }_{{n_{i} }} {\upmu }_{{l_{i} }}$$

(A3)

$$\begin{gathered} VAL_{base} = V\left( L \right) = V\left( {\Sigma n_{i} l_{i} } \right) = \Sigma V(n_{i} l_{i} ) = \Sigma V(n_{i} )V\left( {l_{i} } \right) + \Sigma V(n_{i} )E(l_{i} )^{2} + \Sigma E(n_{i} )^{2} V\left( {l_{i} } \right) \hfill \\ = \Sigma \sigma_{{n_{i} }}^{2} \sigma_{{l_{i} }}^{2} + \Sigma \sigma_{{n_{i} }}^{2} \mu_{{l_{i} }}^{2} + \Sigma \mu_{{n_{i} }}^{2} \sigma_{{l_{i} }}^{2} \hfill \\ \end{gathered}$$

(A4)

Appendix C

We now derive expressions for the AAE, VAE, AAL and VAL for the more general case that includes adjustments to frequency distributions of subsets of the event set. The derivations use the laws of total expectation and variance. These expressions are used, with the secondary uncertainty set to zero, to calculate the third column in each panel in Fig. 3.

$$\begin{gathered} AAE_{adj} = E\left( N \right) = E\left( {\Sigma_{ss} \Sigma_{j} n_{i} } \right) = \Sigma_{ss} \Sigma_{j} E(n_{i} ) = \Sigma_{ss} \Sigma_{j} E_{rs} \left( {E_{nl} \left( {n_{i} |r_{j} ,s_{j} } \right)} \right) = \Sigma_{ss} \Sigma_{j} E_{rs} \left( {r_{j} {\upmu }_{{n_{i} }} } \right) \hfill \\ = \Sigma_{ss} \Sigma_{j} {\upmu }_{{r_{j} }} {\upmu }_{{n_{i} }} = \Sigma_{ss} {\upmu }_{{r_{j} }} \Sigma_{j} {\upmu }_{{n_{i} }} \hfill \\ \end{gathered}$$

(A5)

$$\begin{gathered} VAE_{adj} = V\left( N \right) = V\left( {\Sigma_{ss} \Sigma_{j} n_{i} } \right) = E_{rs} V_{nl} \Sigma_{ss} \Sigma_{j} (n_{i} |r_{j} ,s_{j} ) + V_{rs} E_{nl} \Sigma_{ss} \Sigma_{j} (n_{i} |r_{j} ,s_{j} ) \hfill \\ = E_{rs} \Sigma_{ss} \Sigma_{j} V_{nl} (n_{i} |r_{j} ,s_{j} ) + V_{rs} \Sigma_{ss} \Sigma_{j} E_{nl} (n_{i} |r_{j} ,s_{j} ) = E_{rs} \Sigma_{ss} \Sigma_{j} s_{j} {\upsigma }_{{n_{i} }}^{2} + V_{rs} \Sigma_{ss} \Sigma_{j} r_{j} {\upmu }_{{n_{i} }} \hfill \\ = E_{rs} \left( {\Sigma_{ss} s_{j} \Sigma_{j} {\upsigma }_{{n_{i} }}^{2} } \right) + V_{rs} \left( {\Sigma_{ss} r_{j} \Sigma_{j} {\upmu }_{{n_{i} }} } \right) \hfill \\ = \Sigma_{ss} {\upmu }_{{s_{j} }} \Sigma_{j} {\upsigma }_{{n_{i} }}^{2} + \Sigma_{ss} \Sigma_{ss} cov(r_{j} ,r_{k} )\left( {\Sigma_{j} {\upmu }_{{n_{i} }} } \right)\left( {\Sigma_{k} {\upmu }_{{n_{i} }} } \right) \hfill \\ \end{gathered}$$

(A6)

$$\begin{gathered} AAL_{adj} = E\left( L \right) = E\left( {\Sigma_{ss} \Sigma_{j} n_{i} l_{i} } \right) = \Sigma_{ss} \Sigma_{j} E(n_{i} l_{i} ) = \Sigma_{ss} \Sigma_{j} E_{rs} \left( {E_{nl} \left( {n_{i} l_{i} |r_{j} ,s_{j} } \right)} \right) \hfill \\ = \Sigma_{ss} \Sigma_{j} E_{rs} \left( {r_{j} {\upmu }_{{n_{i} }} l_{i} } \right) = \Sigma_{ss} \Sigma_{j} {\upmu }_{{r_{j} }} {\upmu }_{{n_{i} }} {\upmu }_{{l_{i} }} = \Sigma_{ss} {\upmu }_{{r_{j} }} \Sigma_{j} {\upmu }_{{n_{i} }} {\upmu }_{{l_{i} }} \hfill \\ \end{gathered}$$

(A7)

$$\begin{gathered} VAL_{adj} = V\left( N \right) = V\left( {\Sigma_{ss} \Sigma_{j} n_{i} l_{i} } \right) = E_{rs} V_{nl} \Sigma_{ss} \Sigma_{j} (n_{i} l_{i} |r_{j} ,s_{j} ) + V_{rs} E_{nl} \Sigma_{ss} \Sigma_{j} (n_{i} l_{i} |r_{j} ,s_{j} ) \hfill \\ = E_{rs} \Sigma_{ss} \Sigma_{j} V_{nl} (n_{i} l_{i} |r_{j} ,s_{j} ) + V_{rs} \Sigma_{ss} \Sigma_{j} E_{nl} (n_{i} l_{i} |r_{j} ,s_{j} ) \hfill \\ = E_{rs} \Sigma_{ss} \Sigma_{j} s_{j} {\upsigma }_{{n_{i} }}^{2} {\upsigma }_{{l_{i} }}^{2} + E_{rs} \Sigma_{ss} \Sigma_{j} r_{j}^{2} {\upmu }_{{n_{i} }}^{2} {\upsigma }_{{l_{i} }}^{2} + E_{rs} \Sigma_{ss} \Sigma_{j} s_{j} {\upsigma }_{{n_{i} }}^{2} {\upmu }_{{l_{i} }}^{2} + V_{rs} \Sigma_{ss} \Sigma_{j} r_{j} {\upmu }_{{n_{i} }} {\upmu }_{{l_{i} }} \hfill \\ = \Sigma_{ss} {\upmu }_{{s_{j} }} \Sigma_{j} {\upsigma }_{{n_{i} }}^{2} {\upsigma }_{{l_{i} }}^{2} + \Sigma_{ss} E_{rs} \left( {r_{j}^{2} } \right)\Sigma_{j} {\upmu }_{{n_{i} }}^{2} {\upsigma }_{{l_{i} }}^{2} + \Sigma_{ss} {\upmu }_{{s_{j} }} \Sigma_{j} {\upsigma }_{{n_{i} }}^{2} {\upmu }_{{l_{i} }}^{2} + \Sigma_{ss} {\upsigma }_{{r_{j} }}^{2} (\Sigma_{j} {\upmu }_{{n_{i} }} {\upmu }_{{l_{i} }} )^{2} \hfill \\ = \Sigma_{ss} {\upmu }_{{s_{j} }} \Sigma_{j} {\upsigma }_{{n_{i} }}^{2} {\upsigma }_{{l_{i} }}^{2} + \Sigma_{ss} ({\upsigma }_{{r_{j} }}^{2} + {\upmu }_{{r_{j} }}^{2} )\Sigma_{j} {\upmu }_{{n_{i} }}^{2} {\upsigma }_{{l_{i} }}^{2} \hfill \\ + \Sigma_{ss} {\upmu }_{{s_{j} }} \Sigma_{j} {\upsigma }_{{n_{i} }}^{2} {\upmu }_{{l_{i} }}^{2} + \Sigma_{ss} \Sigma_{ss} cov(r_{j} ,r_{k} )\left( {\Sigma_{j} {\upmu }_{{n_{i} }} {\upmu }_{{l_{i} }} } \right)\left( {\Sigma_{k} {\upmu }_{{n_{i} }} {\upmu }_{{l_{i} }} } \right) \hfill \\ \end{gathered}$$

(A8)

Appendix D

By combining the expressions in Appendix B and Appendix C, with no secondary uncertainty, we can derive expressions for the changes in the four metrics due to the application of uncertain frequency changes. These expressions allow us to calculate the diagnostics for an adjusted YLT, in terms of the diagnostics for the base YLT and the mean and variance of the adjustments, without having to resimulate or reweight the YLT.

$$AAE_{adj} = \Sigma_{ss} {\upmu }_{{r_{j} }} AAE_{j}$$

(A9)

$$VAE_{adj} = \Sigma_{ss} {\upmu }_{{r_{j} }} VAE_{j} + \Sigma_{ss} \Sigma_{ss} cov\left( {r_{j} ,r_{k} } \right)\left( {AAE_{j} } \right)\left( {AAE_{k} } \right)$$

(A10)

$$AAL_{adj} = \Sigma_{ss} {\upmu }_{{r_{j} }} AAL_{j}$$

(A11)

$$VAL_{adj} = \Sigma_{ss} {\upmu }_{{r_{j} }} VAL_{j} + \Sigma_{ss} \Sigma_{ss} cov\left( {r_{j} ,r_{k} } \right)\left( {AAL_{j} } \right)\left( {AAL_{k} } \right)$$

(A12)