2.2.1. Climate module

The raw forecasts of climate variables (e.g., temperature, precipitation, and sea-level pressure) are derived from outputs of the Coupled Model Intercomparison Project Phase 6 (CMIP6). The findings are based on the CMIP6 models, which play a crucial role in informing the IPCC Sixth Assessment Report (Lee et al. Reference Lee, Calvin, Dasgupta, Krinmer, Mukherji, Thorne, Trisos, Romero, Aldunce, Barret, Blanco, Cheung, Connors, Denton, Diongue-Niang, Dodman, Garschagen, Geden, Hayward, Jones, Jotzo, Krug, Lasco, Lee, Masson-Delmotte, Meinshausen, Mintenbeck, Mokssit, Otto, Pathak, Pirani, Poloczanska, Pörtner, Revi, Roberts, Roy, Ruane, Skea, Shukla, Slade, Slangen, Sokona, Sörensson, Tignor, van Vuuren, Wei, Winkler, Zhai and Zommers2023). CMIP6 comprises a set of global climate model experiments that simulate historical, present, and future climate conditions under IPCC’s SSP scenarios (Eyring et al. Reference Eyring, Bony, Meehl, Senior, Stevens, Stouffer and Taylor2016). CMIP6 model outputs are typically provided as gridded datasets, representing climate variables across latitude–longitude grids over time, with spatial resolutions ranging from $1^{\circ}$ to $2.5^{\circ}$ . These outputs are aggregated by averaging across grid cells within defined regions, with the selection of regions for each hazard type discussed in detail in Section 3.1.

One limitation of the raw outputs from CMIP6 models is that they are deterministic in nature. As highlighted in Section 1.2, it is essential for actuarial applications, especially capital modelling, to incorporate the stochastic variability (i.e., aleatoric uncertainty) of climate forecasts. Additionally, model outputs can exhibit biases relative to observations (often due to resolution discrepancies) (Haug et al. Reference Haug, Dimakos, Vårdal, Aldrin and Meze-Hausken2011; Maraun Reference Maraun2013). Furthermore, uncertainties can also arise from limitations in the climate models used (i.e., model uncertainty) (Liu and Raftery Reference Liu and Raftery2021).

To address these biases and both aleatoric and model uncertainty, we adopt the following procedure for simulating future climate variables, building on the approach of Liu and Raftery (Reference Liu and Raftery2021):

1. 1. Model uncertainty: We use an ensemble of CMIP6 models to capture differences among future climate forecasts (Liu and Raftery Reference Liu and Raftery2021). This ensemble approach acknowledges that distinct models can yield varying projections.

2. 2. Bias correction: For each ensemble member, we correct bias by comparing model backcasts to historical observations via the quantile mapping method, a simple but effective technique frequently used in the literature (Haug et al. Reference Haug, Dimakos, Vårdal, Aldrin and Meze-Hausken2011; Maraun Reference Maraun2013; Sanabria et al. Reference Sanabria, Qin, Li and Cechet2022). Specifically, using a quantile mapping approach with a linear transformation function (Piani et al. Reference Piani, Weedon, Best, Gomes, Viterbo, Hagemann and Haerter2010; Qian and Chang Reference Qian and Chang2021), we estimate

   (2.1) \begin{equation} \hat{\theta}_{q} \;=\; \hat{\beta}_0^{(m)} \;+\; \hat{\beta}_1^{(m)} \,\hat{\theta}_{q}^{(m)},\end{equation}
   where $\hat{\theta}_{q}$ and $\hat{\theta}_{q}^{(m)}$ are the $q\text{th}$ quantiles of the historical observations and model m backcasts, respectively, over the same reference period.

3. 3. Aleatoric uncertainty: To incorporate inherent randomness, we collect residuals $\displaystyle z^{(m)}_t = \theta_t - \hat{\theta}_{t}^{(m),*}$ by comparing the bias-corrected model backcasts $\hat{\theta}_{t}^{(m),*}$ with actual historical data $\theta_t$ . We then calibrate a Normal distribution on the residuals (i.e., $z^{(m)}_t \sim \text{N}(0,\sigma^2_{(m)})$ ). We also acknowledge that, although this assumption is supported by normality tests (e.g., Shapiro–Wilk Yazici and Yolacan Reference Yazici and Yolacan2007) for most ensemble members in our calibration data (see Online Appendix C.2), alternative error distributions may better fit other regions. We therefore recommend validating the residual distributional assumptions when applying the method to new data, as indicated in Step 3 of the proposed flow diagram.

4. 4. Future projections and simulations: For each simulation path in the future projection period, we randomly select a CMIP6 model m to generate a deterministic forecast $\hat{\theta}_{t}^{(m)}$ . We apply the bias correction as $\hat{\theta}_{t}^{(m),*} \;=\; \hat{\beta}_0^{(m)} + \hat{\beta}_1^{(m)} \,\hat{\theta}_{t}^{(m)}$ and then draw one trajectory of residuals $\tilde{z}^{(m)}_t$ to account for aleatoric uncertainty. The final simulated climate variable is thus:

   (2.2) \begin{equation} \tilde{\theta}_{t} = \hat{\theta}_{t}^{(m),*} \;+\; \tilde{z}^{(m)}_t.\end{equation}

A schematic illustration of the process described above is shown in Figure 3.

Figure 3.

An illustrative diagram of climate variable simulations.

2.2.2. Hazards module

Based on the projected climate variables from the previous module, this section forecasts the frequency and severity of natural hazards. This constitutes a critical component of the DFA model, as the resulting hazard forecasts will be employed to model the general insurance assets and liabilities in subsequent sections. Numerous approaches exist for hazard modelling in the literature; however, as discussed in Section 1.2, balancing model interpretability and comprehensiveness is essential.

At one extreme, traditional Collective Risk Models (CRMs) (Klugman et al. Reference Klugman, Panjer and Willmot2012) offer a simplistic, intuitive means of modelling aggregate insurance losses, and it is also often used in traditional DFA applications (Kaufmann et al. Reference Kaufmann, Gadmer and Klett2001). Yet, their static assumption regarding insurance loss distributions neglects the dynamics introduced by climate change. At the other extreme, CAT models are sophisticated models that are usually capable of capturing the complex environmental process affected by climate change to generate hazard events based on advanced physical and mathematical models (Mitchell-Wallace et al. Reference Mitchell-Wallace, Jones, Hillier and Foote2017). However, these proprietary models usually have complex structures with modelling details usually not accessible by general insurers, making them less comprehensible for insurers (Weinkle and Pielke Reference Weinkle2017), leading to challenges in interpretability.

In light of the above considerations, we have opted for the weather-dependent CRMs (Haug et al. Reference Haug, Dimakos, Vårdal, Aldrin and Meze-Hausken2011) for modelling insurance losses. These models combine the high interpretability of traditional CRMs with the capability to incorporate climate effects by integrating meteorological variables in the modelling of insurance loss frequency and severity. In essence, the aggregate catastrophe loss ( $\tilde{X}_t$ ) is modelled as:

(2.3) \begin{equation} \tilde{X}_t = \sum_{i=1}^{I} \sum_{m=1}^{M_t^{(i)}} \tilde{X}_t^{(i), m},\end{equation}

where $M_t^{(i)}$ is the number of event of hazard type i in year t and $\tilde{X}_t^{(i), m}$ is the insurance loss associated with the mth event of hazard type i in year t. We further assume

(2.4) \begin{equation} M_t^{(i)} \sim F\big(\kappa_1 \big(\boldsymbol{\Theta}^{(i)}_t\big), \ldots, \kappa_p\big(\boldsymbol{\Theta}^{(i)}_t\big)\big), \quad X_t^{(i), m} \sim G\big(\vartheta_1\big(\boldsymbol{\Theta}^{(i)}_t\big), \ldots, \vartheta_q\big(\boldsymbol{\Theta}^{(i)}_t\big)\big),\end{equation}

where F denotes the frequency distribution with parameters $\kappa_1, \ldots, \kappa_p$ , and G denotes the severity distribution with parameters $\vartheta_1, \ldots, \vartheta_q$ . In both cases, the parameters are functions of the weather covariate vector $\boldsymbol{\Theta}_t^{(i)}$ associated with hazard type i. The variable $X_t^{(i),m}$ represents the normalised catastrophe loss, adjusted for both inflation and wealth exposure.Footnote ³ Typical choices for F include the Poisson or Negative Binomial distributions, while G is often specified as a heavy-tailed distribution (e.g., Log-Normal, Pareto, or Weibull) to capture catastrophic loss behaviour (Kaufmann et al. Reference Kaufmann, Gadmer and Klett2001).

We further specify

(2.5) \begin{equation} f\big(\kappa_{1,t}^{(i)}\big) = \boldsymbol{\beta}_{1,(i)}^{\prime}\boldsymbol{\Theta}_t^{(i)},\ldots, f\big(\kappa_{p,t}^{(i)}\big) = \boldsymbol{\beta}_{p,(i)}^{\prime}\boldsymbol{\Theta}_t^{(i)}; \;\; g\big(\vartheta^{(i)}_{1, t}\big) = \boldsymbol{\alpha}_{1,(i)}^{\prime}\boldsymbol{\Theta}_t^{(i)},\ldots,g\big(\vartheta^{(i)}_{q, t}\big) = \boldsymbol{\alpha}_{q,(i)}^{\prime}\boldsymbol{\Theta}_t^{(i)};\end{equation}

where the set of coefficients $\{\boldsymbol{\beta}_{1,(i)},\ldots,\boldsymbol{\beta}_{p,(i)}\}$ and $\{\boldsymbol{\alpha}_{1,(i)},\ldots,\boldsymbol{\alpha}_{q,(i)}\}$ are estimated via regression, and f and g denote the link functions.

We select the weather covariates based on the physical mechanisms driving each hazard type i and validate them statistically. A detailed illustrative example of the selection of weather covariates is provided in Section 3.1.

The hazard model presented in this section is designed to capture industry-level trends consistent with the scope of this paper. For more granular decision-making at the level of individual insurers (e.g., portfolio management), a higher-resolution modelling framework may be required. A discussion of how the current hazard model can be extended to regional projections and integrated with CAT model outputs is provided in Online Appendix G. Nonetheless, incorporating such extensions should be carefully balanced against the trade-off between precision and interpretability depending on the intended business applications, as discussed in Section 1.2.

2.3.1. Inflation rates

General inflation rates can influence both the liabilities and assets of general insurers by affecting claims inflation and nominal interest rates (Kaufmann et al. Reference Kaufmann, Gadmer and Klett2001). Baseline inflation rates are modelled following the common approaches in literature by using the mean-reverting AR(1) process (see, e.g., Chen et al. Reference Chen, Koo, Wang, O’Hare, Langrené, Toscas and Zhu2021; Bégin Reference Bégin2022):

(2.6) \begin{equation} i_t=\mu_i+a_i(i_{t-1}-\mu_i)+\sigma_i \epsilon_{i,t},\end{equation}

where $i_t$ denotes the inflation rate at time t, $\mu_i$ is the long-run mean inflation, $a_i$ is the autoregressive parameter, $\sigma_i$ is the volatility, and $\epsilon_{i,t}$ represents a standard error term.

Studies have shown that historical fluctuations in weather conditions – such as temperature shocks and increased temperature variability – can exert inflationary pressures on food, energy, and service prices (Faccia et al. Reference Faccia, Parker and Stracca2021; Mukherjee and Ouattara Reference Mukherjee and Ouattara2021; Ciccarelli et al. Reference Ciccarelli, Kuik and Hernández2023). This inflationary effect ultimately contributes to general inflation. Since climate change is expected to exacerbate weather fluctuations, it is crucial to account for its impact in modelling inflation rates (Kotz et al. Reference Kotz, Kuik, Lis and Nickel2024). To incorporate the influence of climate on inflation, we apply a climate overlay to the baseline inflation rates, following the methodology proposed by Kotz et al. (Reference Kotz, Kuik, Lis and Nickel2024). To the best of our knowledge, the study by Kotz et al. (Reference Kotz, Kuik, Lis and Nickel2024) is the first to quantitatively assess and project the effects of future climate change on both food and general inflation. Specifically, the climate-adjusted inflation rate is given by:

(2.7) \begin{equation} i_t^{\text{Clim}} = i_t + i_t^{\text{Clim-Impact}},\end{equation}

where $i_t^{\text{Clim-Impact}}$ captures the additional inflationary effects from climate change. Following the approach in Kotz et al. (Reference Kotz, Kuik, Lis and Nickel2024), the monthly climate impact on inflation is modelled as:

(2.8) \begin{equation} i_{m}^{\text{Clim-Impact}} = \sum_{L=0}^{11} \big(\alpha_{1+L} \Delta \bar{T}_{m-L}^{\text{NS}}+\beta_{1+L}\bar{T}_{m-L}^{\text{NS}} \cdot \Delta \bar{T}_{m-L}^{\text{NS}}\big),\end{equation}

where $\bar{T}_m^{\text{NS}}$ denotes the monthly average near-surface temperature over the selected country and $\Delta \bar{T}_m^{\text{NS}}$ represents the deviation of future monthly averages from the 1990–2021 baseline. This formulation assumes a one-year lag effect. The term $\beta_{1+L}\bar{T}_{m-L}^{\text{NS}} \cdot \Delta \bar{T}_{m-L}^{\text{NS}}$ is introduced to capture the interaction effect, whereby higher temperatures during hotter months lead to larger inflationary impacts (Faccia et al. Reference Faccia, Parker and Stracca2021; Kotz et al. Reference Kotz, Kuik, Lis and Nickel2024). For future projections, the monthly average near-surface temperature will be sourced from the outputs of the climate module described in Section 2.2.1. The annual climate impact on inflation rates for year t is then obtained by summing the monthly impacts (Kotz et al. Reference Kotz, Kuik, Lis and Nickel2024): $i_t^{\text{Clim-Impact}} = \sum_{m \in t} i_{m}^{\text{Clim-Impact}}$ .

2.3.2. Risk-free interest rates

Drawing inspiration from Laubach and Williams (Reference Laubach and Williams2003) to Holston et al. (Reference Holston, Laubach and Williams2017), we model the real risk-free short-term interest rate as:

(2.9) \begin{equation} r_t = \beta_0 + \beta_1 g_t + z_t,\end{equation}

which is closely related to the Ramsey’s equation (Ramsey Reference Ramsey1928), given by $r^* = \rho+\gamma g$ , where g denotes the growth rate of potential output, $\rho$ represents the rate of time preference, and $r^*$ denotes the natural rate of interest. A positive relationship between $r^*$ and g is expected, as higher potential growth enhances future income prospects, reducing households’ incentives to save today and thereby placing upward pressure on natural rate of interest (Mongelli et al. Reference Mongelli, Pointner and Van den End2024).

In our model, $g_t$ denotes the growth rate of potential real GDP. For calibration, these growth rates will be obtained from the World Bank Potential Growth Database (Kilic Celik et al. Reference Kilic Celik, Kose, Ohnsorge and Ruch2023). For future projections, $g_t$ will be derived from the GDP forecasts under each SSP scenario provided in the SSP database (Riahi et al. Reference Riahi, Van Vuuren, Kriegler, Edmonds, O’Neill, Fujimori, Bauer, Calvin, Dellink, Fricko, Lutz, Popp, Crespo Cuaresma, Samir, Leimbach, Jiang, Kram, Rao, Emmerling, Ebi and Tavoni2017). The residual term $z_t$ is assumed to follow an AR(1) process:

(2.10) \begin{equation} z_t = \mu_r + \phi_r (z_{t-1} - \mu_r) + \epsilon_r(t),\end{equation}

which captures residual factors not explained by the growth rate. The nominal risk-free rate is then derived by incorporating inflationary effects using Fisher’s equation: $\tilde{r}_t = r_t + i_t^{\text{Clim}}$ .

In summary, the key inputs for this model are the real GDP growth rate $g_t$ and the climate-adjusted inflation rate $i_t^{\text{Clim}}$ (as output from the inflation model; see Section 2.3.1). These inputs yield the nominal risk-free rate, $\tilde{r}_t$ , as the final output.

2.3.3. Equity module

We begin by considering the benchmark equity return model proposed by Ahlgrim et al. (Reference Ahlgrim, D’Arcy and Gorvett2005), which is given by:

(2.11) \begin{equation} r_t^{(S)} = \tilde{r}_t+x_t,\end{equation}

where $\tilde{r}_t$ is the nominal risk-free rates and $x_t$ is the excess equity return.

Under traditional DFA or ESG frameworks (see, e.g., Wilkie Reference Wilkie1995; Ahlgrim et al. Reference Ahlgrim, D’Arcy and Gorvett2005; Chen et al. Reference Chen, Koo, Wang, O’Hare, Langrené, Toscas and Zhu2021; Bégin Reference Bégin2022), excess equity returns are often modelled as independent stochastic processes, which is a self-contained approach that can enhance reliability in light of the considerable uncertainty surrounding exogenous variables over long-term horizons (Wilkie Reference Wilkie1995). However, relying solely on historical data limits the capacity to capture the forward-looking climate impacts and the evolving socio-economic conditions under different scenarios.

By contrast, factor models leverage a wide range of climate proxies – often at a granular level – to assess their influence on equity returns (see, e.g., Bansal et al. Reference Bansal, Kiku and Ochoa2019; Hong et al. Reference Hong, Li and Xu2019; Görgen et al. Reference Görgen, Jacob, Nerlinger, Riordan, Rohleder and Wilkens2020; Venturini 2022). While these models are effective for empirical, in-sample analyses of individual or portfolio assets, their extensive data requirements and focus on asset-specific rather than market-level returns pose challenges for long-term projections, whereas ESG or DFA typically focuses on market-level returns.

To strike a balance between these two approaches, we propose a partial self-contained framework that incorporates forward-looking climate considerations without relying on overly granular external data. Inspired by the climate-economic literature (see, e.g., Karydas and Xepapadeas Reference Karydas and Xepapadeas2022; Barnett Reference Barnett2023), our method channels climate’s influence on equity returns through climate-damaged consumption. A flow diagram illustrating the simulation of excess equity returns is presented in Figure 4.

Figure 4.

Illustrative diagram showing the simulation flow for equity excess returns.

In the first step, we obtain the simulated nominal market insurance catastrophe loss $\tilde{X_t}$ from the hazard module. Next, these insurance losses are scaled to represent uninsured economic damage, yielding $\eta \tilde{X}_t$ , where $\eta$ is the multiplier. The production available for consumption after climate damage is then computed as:

(2.12) \begin{equation} C_t = Y_t - \eta \tilde{X}_t,\end{equation}

where $Y_t$ is the real GDP projection under each SSP scenario.

To transmit the effects of economic growth and climate damage to equity returns, excess total equity returns are modelled as a function of corporate earnings (based on operating profits) growth, $\Delta \text{OP}_t$ , which is itself modelled as a function of consumption growth, $\Delta C_t$ . Specifically, we have

(2.13) \begin{equation} \Delta \text{OP}_t = \alpha_0 + \alpha_1 \Delta C_t + \epsilon_t^O, \;\;\; x_t = \beta_0 + \beta_1 \Delta \text{OP}_t + \epsilon_t^{\text{x}}\end{equation}

where the parameters are estimated through regression using historical data. The random shocks to excess equity returns and operating profit growth are represented by $\epsilon_t^{\text{x}} \sim N(0, \sigma^2_{\text{x}})$ and $\epsilon_t^O \sim N(0, \sigma^2_O)$ , with their volatilities calibrated from historical observations.

For the brown sector, we further apply a transition stress overlay factor (Grippa and Mann Reference Grippa and Mann2020) on its operating profit growth:

(2.14) \begin{equation} \Delta \text{OP}_t^B = \Delta \text{OP}_t + \beta \Delta Y_t^B,\end{equation}

where $\Delta Y_t^B$ represents the change in brown energy production and $\beta$ is the sensitivity of brown firms’ corporate profits to these changes. The adjusted operating profit growth for the brown portfolio is then used to simulate its excess equity returns based on (2.13).

Based on the outputs from the interest rate (Section 2.3.2) and equity modules, investment returns are calculated as: $r_t^I = w_{f} \tilde{r}_t + (1 - w_f)r_t^{(S)}$ , where $w_{f}$ is the proportion of the portfolio allocated to risk-free assets.

In summary, the key inputs to this model are the nominal risk-free rate $\tilde{r}_t$ , aggregate catastrophe losses $\tilde{X}_t$ , real GDP projections, and brown energy production for each SSP scenario ( $Y_t$ and $Y_t^B$ ), with $\tilde{r}_t$ and $\tilde{X}_t$ obtained from the respective interest rate and hazards modules. The model outputs are the equity returns for the general portfolio, $r_t^{(S,G)}$ , and for the brown portfolio, $r_t^{(S,B)}$ . By preserving the simplicity of traditional methods while integrating forward-looking climate damage projections, this partial self-contained approach captures key trends in evolving climate and socio-economic conditions without using high-dimensional external factors.

2.4.1. Insurance costs

Drawing on the hazard module outputs and assuming an aggregate excess-of-loss reinsurance contract, the net catastrophe loss allocated to insurer j is determined by:

(2.15) \begin{equation} \tilde{X}_{t,(\,j)}^{\text{net}} = w_j \tilde{X}_t - \text{min}((w_j \tilde{X}_{t} - d_{(\,j)})_+, L_{(\,j)}),\end{equation}

where $\tilde{X}_t$ represents the gross CAT losses adjusted for CPI and GDP growth, $w_j$ denotes the market share of insurer j, and $d_{(j)}$ and $L_{(\,j)}$ are the inflation and GDP-adjusted reinsurance excess and limit levels for insurer j. The second term in (2.15) represents the recoverables from reinsurers based on the excess-of-loss contract.

In addition to catastrophe losses, another key component of DFA liability modelling is non-catastrophe losses (Kaufmann et al. Reference Kaufmann, Gadmer and Klett2001). We model non-catastrophe losses per exposure unit with a Tweedie distribution (Jø rgensen and Paes De Souza Reference Jørgensen and Paes De Souza1994), which is a commonly used distribution assumption for modelling non-catastrophe loss. Specifically,

(2.16) \begin{equation} X_t^{\text{NC}} \sim \text{Tweedie}(\mu^{\text{NC}}, \phi),\end{equation}

where $\mu^{\text{NC}}$ is the location parameter and $\phi$ is the dispersion parameter, both calibrated using historical data. Using a Tweedie distribution implies that claim frequency follows a Poisson distribution, while claim severity follows a Gamma distribution, reflecting the typically high-frequency, low-severity nature of non-catastrophe losses. Some studies have investigated the influence of weather on non-catastrophe claims (e.g., McGuire and Actuaries Reference McGuire and Actuaries2008; Haug et al. Reference Haug, Dimakos, Vårdal, Aldrin and Meze-Hausken2011; Scheel et al. Reference Scheel, Ferkingstad, Frigessi, Haug, Hinnerichsen and Meze-Hausken2013; Reig Torra et al. Reference Reig Torra, Guillen, Pérez-Marn, Rey Gámez and Aguer2023), but these typically require high-resolution daily or monthly municipal-level data, exceeding the usual granularity of DFA. Moreover, the long-term effect of weather on non-catastrophe claims is uncertain. For instance, McGuire and Actuaries (Reference McGuire and Actuaries2008) found that same-day precipitation increases motor claims frequency, whereas lagged precipitation decreases it possibly due to cleaner road conditions (Eisenberg Reference Eisenberg2004; McGuire and Actuaries Reference McGuire and Actuaries2008). Consequently, we do not explicitly incorporate climate impacts in our modelling of non-catastrophe losses, and the integration of climate covariates is left for future research as more data become available.

For projections, the aggregate non-catastrophe loss is computed as

(2.17) \begin{equation} \tilde{X}_{t, (\,j)}^{\text{NC}} = X_t^{\text{NC}} \cdot \omega_t \cdot w_j \cdot \frac{\text{CPI}_t}{\text{CPI}_{s}},\end{equation}

where $\omega_t$ is the total number of risks, s denotes the reference year, and $\text{CPI}_t$ is the climate-adjusted CPI from the inflation module (see Section 2.3.1). The total number of risks is modelled as a linear function of population:

(2.18) \begin{equation} \hat{\omega}_t = \hat{\omega}_0 + \hat{\omega}_1 \text{Pop}_t,\end{equation}

where $\text{Pop}_t$ is the projected population for each climate scenario, and the parameters $\hat{\omega}_0$ and $\hat{\omega}_1$ are calibrated on historical data via linear regression. Under these assumptions, climate change does not directly affect non-catastrophe losses; however, it still influences them indirectly through population growth and inflation.

2.4.2. Insurance premiums

Based on the distribution assumptions of catastrophe and non-catastrophe losses, the insurance premium is then calculated using the standard deviations loading principle (Paudel et al. Reference Paudel, Botzen and Aerts2013; Paudel et al. Reference Paudel, Botzen, Aerts and Dijkstra2015; Tesselaar et al. Reference Tesselaar, Botzen and Aerts2020):

(2.19) \begin{equation} \pi_{t,(\,j)} = \underbrace{\text{E}\big(\tilde{X}_{t, (\,j)}\big) + \rho \sqrt{\text{Var}\big(\tilde{X}_{t, (\,j)}\big)}}_{\text{CAT premium}} + \underbrace{\text{E}\big(\tilde{X}_{t, (\,j)}^{\text{NC}}\big) + \rho \sqrt{\text{Var}\big(\tilde{X}_{t, (\,j)}^{\text{NC}}\big)}}_{\text{Non-CAT premium}},\end{equation}

where $\rho$ is the risk aversion parameter that reflects the level of insurer’s risk aversion towards the extreme nature of the risk (Paudel et al. Reference Paudel, Botzen and Aerts2013). The risk aversion parameter could be selected empirically. We adopt the assumed risk aversion parameter of 0.55 in Kunreuther et al. (Reference Kunreuther, Michel-Kerjan and Ranger2011) and Paudel et al. (Reference Paudel, Botzen and Aerts2013), which is based on an empirical survey analysis conducted by Kunreuther and Michel-Kerjan (Reference Kunreuther and Michel-Kerjan2011). The implication of this assumption on the projected premiums growth will also be examined in Section 3.3.3.

2.4.3. Reinsurance premiums

Based on the reinsurance structure specified in Section 2.4.1, the reinsurance premium is derived as:

(2.20) \begin{equation} \pi_{t,(\,j)}^{RI} = \text{E}\big[\! \min\big(\big(\tilde{X}_{t, (\,j)} - d_{(\,j)}\big)_+, L_{(\,j)}\big)\big]+ \rho \sqrt{\text{Var}\big(\!\min\big(\big(\tilde{X}_{t, (\,j)} - d_{(\,j)}\big)_+, L_{(\,j)}\big)\big)}.\end{equation}

Similarly, the second term in (2.20) represents the surcharge on the premium above the expected value of the loss, which is dependent on the variability of the reinsurance losses.

Reinsurers typically cover the extreme tail of insurers’ risk portfolios through excess-of-loss coverage, making them particularly vulnerable to large natural catastrophes. Such events can strain reinsurance capital and trigger hard markets with higher premiums, which is a trend expected to intensify as climate change increases catastrophe losses (Tesselaar et al. Reference Tesselaar, Botzen and Aerts2020). Capital constraint theory offers a common explanation, suggesting firms prefer to accumulate surplus through higher premiums rather than raise costly external capital (Winter Reference Winter1988; Winter Reference Winter1994; Dicks and Garven Reference Dicks and Garven2022).

To capture this relationship, we model reinsurance premiums as a function of reinsurance capital using a negative exponential form, inspired by Taylor (Reference Taylor2008):

(2.21) \begin{equation} \pi_{t,(\,j)}^{RI,*} = \max\Big(\pi_{t,(\,j)}^{RI}, \pi_{t,(\,j)}^{RI} e^{-k_1 \cdot(S_{t-1}-S_0)}\Big),\end{equation}

where $S_{t-1}$ denotes the solvency ratio at the end of period $t-1$ , $S_0$ represents the reference (or steady-state) solvency ratio, and $k_1$ is the premium-to-solvency sensitivity parameter. The solvency ratio is defined as the ratio of reinsurance capital to premium (i.e., $S_{t-1} = K_{t-1}^{\text{Re}}/\pi_{t-1}^{RI,*}$ ), where the reinsurance capital is derived as:

(2.22) \begin{equation} K_t^{\text{Re}} = \big(1 + \tilde{r}_t^{(I)}\big)\left(K_{t-1}^{\text{Re}} + \sum_{j=1}^J \pi_{t,(\,j)}^{RI,*}\right) - \sum_{j=1}^J \min\left[(\tilde{X}_{t, (\,j)} - d_{(\,j)})_+, L_{(\,j)}\right].\end{equation}

Here, we assume that reinsurers earn the same investment returns as direct insurers. The specification in (2.21) ensures that premiums increase as the solvency ratio decreases. Furthermore, due to the concave nature of the function, a decline in capital levels results in a more pronounced increase in premiums compared to the decrease in premiums when capital levels rise. This asymmetry aligns with the assumptions in Winter (Reference Winter1994), where insurers are assumed to be averse to the risk of bankruptcy.

2.6.1. Hazards model

Under the stationary specification, event frequency and severity follow time-invariant distributions (Kaufmann et al. Reference Kaufmann, Gadmer and Klett2001):

(2.27) \begin{equation} M_t^{(i)} \sim F(\kappa_1, \ldots, \kappa_p), \quad X_t^{(i), m} \sim G(\vartheta_1, \ldots, \vartheta_q),\end{equation}

where the parameters $\kappa_1, \ldots, \kappa_p$ and $\vartheta_1, \ldots, \vartheta_q$ are constant over time. This corresponds to the climate-dependent hazard model in Section 2.2.2 with all coefficients on climate variables set to zero.

2.6.2. Inflation rate

Inflation is modelled as a mean-reverting AR(1) process consistent with the baseline specification in (2.6). This is a special case of the climate-dependent inflation model obtained by setting the climate overlay component (i.e., $i_{m}^{\text{Clim-Impact}}$ ) in (2.7) to zero.

2.6.3. Interest rates

Interest rates are modelled using a mean-reverting AR(1) process, which can be viewed as a special case of the generalised one-factor CIR model in Kaufmann et al. (Reference Kaufmann, Gadmer and Klett2001) with $g = 0$ :

(2.28) \begin{equation} r_t = m_r + \varphi_r (r_{t-1} - m_r) + (r_{t-1})^g \epsilon_r(t), \quad \epsilon_r(t) \sim N(0, \sigma_r^2).\end{equation}

This corresponds to the climate-dependent specification in Section 2.3.2 when the coefficient on scenario-dependent economic growth is set to zero.

2.6.4. Equity model

Excess equity returns are modelled as a stationary Normal distribution:

(2.29) \begin{equation} x_t \sim N\big(\mu_{{x}}, \sigma_{{x}}^2\big).\end{equation}

This is a special case of the climate-dependent equity model in Section 2.3.3, obtained by muting sensitivity to scenario-dependent production growth and energy variables. Implicitly, this assumes that future economic conditions follow historical patterns.

2.6.5. Liabilities, premiums, and surplus

Adopting a similar approach to Kaufmann et al. (Reference Kaufmann, Gadmer and Klett2001), the exposure growth under the stationary setting is modelled as:

(2.30) \begin{equation} \delta_t = \mu_{\omega} + \phi_{\omega} (\delta_{t-1} - \mu_{\omega}) + \epsilon_t^{\omega},\end{equation}

where $\delta_t = \Delta \omega_t / \omega_{t-1}$ denotes the growth rate of exposure units, with its long-run mean denoted as $\mu_{\omega}$ .

Gross premiums and surplus are computed in the same manner as in the climate-dependent DFA, except that hazard distributions are assumed stationary and independent of climate variables. While the calculation of base reinsurance premiums is similar in both frameworks, the climate-dependent DFA incorporates the impact of climate change on reinsurance capital when deriving the solvency uplift applied to reinsurance premiums, whereas the stationary DFA does not. As a result, the difference in final reinsurance premiums between the two frameworks reflects not only changes in the underlying base premiums due to evolving climate risk, but also the additional solvency uplift arising from climate-related impacts on reinsurance capital.

3.3.1. Climate and hazards

The simulated climate variables derived from the CMIP6 model outputs (Copernicus Climate Change Service 2024a ) are presented in Appendix A. Most variables show an upward trend, especially under high-emission scenarios, indicating a general increase in climate risk. The projections also exhibit notable inter-annual variability and a high degree of uncertainty, both of which are important drivers for the results shown in the downstream modules.

Figure 5.

Simulation results of normalised hazard losses. Solid lines (and dotted lines under the stationary assumption) represent the average simulation paths under different climate scenarios, while dashed lines denote the $5^{\text{th}}$ and $95^{\text{th}}$ percentiles. Results are derived from simulated climate variables and calibrated hazard models. The simulations reveal an increasing trend in both the mean and volatility of hazard losses for most hazard types under high-emission scenarios.

Based on the simulated climate variables, catastrophe losses for each hazard type are generated using the calibrated relationships between climate variables and hazard frequency and severity (see Figure 5). Overall, the projected normalised losses exhibit an increasing trend for most hazards, particularly under high-emission scenarios, with notable increases in both the mean and the upper tails of the loss distributions. In addition, the projected losses exhibit substantial inter-annual variability. For example, a pronounced peak in simulated flood losses is observed around 2047, particularly at the $95^{\text{th}}$ percentile. This peak is primarily driven by the corresponding increase in extreme precipitation (see Online Appendix D.1), which may be explained by the projected rise in the frequency of extreme ENSO events under high-emission scenarios reported in the literature (see, e.g., Cai et al. Reference Cai, Wang, Santoso, McPhaden, Wu, Jin, Timmermann, Collins, Vecchi, Lengaigne, England, Dommenget, Takahashi and Guilyardi2015; Heede and Fedorov Reference Heede and Fedorov2023). Moreover, the results reveal considerable uncertainty, driven by both the variability in simulation of climate variables as shown in Appendix A and the heavy-tailed nature of catastrophe losses. The rising mean and volatility are expected to place upward pressure on both insurance and reinsurance premiums, as well as shocks to capital over time.

3.3.2. Investment returns

The simulation results for the cumulative returns on the total investment portfolios, presented on a logarithmic scale, are shown in Figure 6(a).Footnote ⁵ The uncertainty bounds in the figure account for historical fluctuations in interest rates and equity returns, as well as the variability in climate-related damage to production available for consumption, as illustrated in Figure 7(a) and derived via (2.12).

Figure 6.

Simulated (log) compounded investment returns (a) and projected compounded real GDP growth in Australia (b). Panel (a) shows the simulated compounded returns on the total investment portfolios generated from the economic growth assumptions and the simulated hazard losses underlying each scenario. Panel (b) presents compounded real GDP growth projections derived from the SSP database (Riahi et al. Reference Riahi, Van Vuuren, Kriegler, Edmonds, O’Neill, Fujimori, Bauer, Calvin, Dellink, Fricko, Lutz, Popp, Crespo Cuaresma, Samir, Leimbach, Jiang, Kram, Rao, Emmerling, Ebi and Tavoni2017).

Figure 7.

Ratios of uninsured (a) and total (b) catastrophe losses to GDP. The results are obtained by dividing the scaled outputs from the hazard module (see Section 2.3.3) by the corresponding GDP projections under each climate scenario.

The observed trends in cumulative investment returns are primarily driven by the economic growth assumptions underlying each SSP scenario (Figure 6(b)). Under SSP 8.5, the highest cumulative investment return is projected, driven by its robust economic growth assumption (O’Neill et al. Reference O’Neill, Kriegler, Ebi, Kemp-Benedict, Riahi, Rothman, Van Ruijven, Van Vuuren, Birkmann, Kok, Levy and Solecki2017), followed by SSP 2.6. Conversely, SSP 7.0 exhibits the weakest cumulative investment return, influenced by both its slow economic growth assumptions and high catastrophe damages, in line with its narrative (O’Neill et al. Reference O’Neill, Kriegler, Ebi, Kemp-Benedict, Riahi, Rothman, Van Ruijven, Van Vuuren, Birkmann, Kok, Levy and Solecki2017). The relatively high investment returns under SSP 8.5 and SSP 2.6 are likely to accelerate capital accumulation. By contrast, the lower returns, coupled with high catastrophe losses in SSP 7.0 (see Figures 7(b) and 5), are expected to slow the pace of capital accumulation under this scenario. A sensitivity analysis using alternative economic damage assumptions has also been conducted (see Online Appendix I), indicating broadly consistent patterns with those reported in this section.

A breakdown analysis of investment returns for the general and brown portfolios is provided in Online Appendix E.2. Overall, the transition stress remains relatively mild, as the SSP scenarios considered do not assume abrupt or disorderly transitions, as noted in Limitation 2.1.

3.3.3. Premiums and underwriting losses

The normalised gross premiums associated with catastrophe (CAT) coverage are presented in Figure 8(a). An overall increasing trend is found, driven by the rising risk of most hazards as shown in Figure 5. Gross CAT premiums reach their highest levels under SSP 8.5, particularly in the later projection period, followed by SSP 7.0, SSP 4.5, and SSP 2.6, in line with the physical risk narratives underlying those scenarios as described in Section 2.1. The increase in premiums may also create affordability issues and, as noted in Limitation 2.7, affect insurance demand. This effect is noted as a limitation of the current paper but is encouraged for future research. In addition to this general upward trend, significant inter-annual variability is observed, reflecting the internal climate variability of the underlying climate variables as shown in Appendix A.

Figure 8.

Gross premiums (normalised) associated with catastrophe cover, shown as dollar values (a) and as a proportion of total gross premiums (b). Panel (a) presents the normalised values of calculated gross catastrophe (CAT) premiums based on (2.19). Panel (b) shows the projected CAT premiums as a proportion of total premiums, compared with historically observed CAT losses as a share of total insurance losses, based on data from the APRA database and the ICA dataset. Both panels indicate an increasing trend in CAT premiums – both in dollar terms and as a share of total premiums – particularly under high-emission scenarios.

Moreover, the proportion of CAT premiums relative to total general insurance premiums is expected to increase relative to the historical levels, with a more pronounced increase under high-emission scenarios. This is illustrated in Figure 8(b), which compares projected CAT premium proportions with historically observed CAT loss proportions (used here as a proxy for historical CAT premium proportions) derived from the General Insurance Performance Database (APRA 2024b ) and the ICA dataset. These findings suggest that catastrophe losses will have an increasingly significant impact on the underwriting performance of general insurers under the influence of climate change. Consequently, managing CAT exposures will become a more critical component of portfolio management for general insurers.

A similar pattern to the gross insurance premiums emerges in the reinsurance premiums (Figure 9(a)), which show a general upward trend. The highest projected premium occurs under SSP 8.5, followed by SSP 7.0, while SSP 2.6 is expected to have the lowest premium.

Figure 9.

Total reinsurance premiums (normalised) (a) and relative difference between total and base reinsurance premiums (b). Panel (a) shows the normalised values of calculated total reinsurance premiums (including mark-up), averaged across simulation paths, based on (2.21). The results exhibit trends similar to those observed for gross CAT premiums. Panel (b) presents the relative difference between total reinsurance premiums and base reinsurance premiums, with the latter calculated using (2.20). The largest premium uplift is projected under the SSP 7.0 scenario.

To illustrate the role of reinsurance capital constraints, Figure 9(b) presents the average relative difference between the solvency-sensitive reinsurance premiums (in (2.20)) and the base premiums (in (2.21)). The largest uplift appears under SSP 7.0, likely due to low investment returns coupled with high insurance losses (Sections 3.3.1 and 3.3.2), resulting in slower capital accumulation and prolonged tight capacity. In contrast, SSP 8.5 exhibits the smallest uplift despite experiencing the highest catastrophe (CAT) losses, possibly because its stronger investment returns (see Section 3.3.2) accelerate capital accumulation and shorten tight-capacity periods. A similar pattern is also observed in the primary general insurance market (see Section 3.4).

The results also suggest that the percentage growth in reinsurance premiums exceeds that of primary insurance, reflecting both rising physical risks and adjustments to tightening reinsurance capital. The modelled uplift relative to the base level is considered as conservative due to the simplified assumptions outlined in Limitation 2.8. Nevertheless, it is assumed that primary insurers will continue to purchase reinsurance despite higher prices; however, rising premiums may dampen demand, although this effect could be partly offset by the strengthening risk-transfer demand as climate risk intensifies (Lloyd’s 2025). The dynamics of these effects are left for future research.

Using the calculated premiums and simulated catastrophe losses from the hazard module (Section 3.3.1), we derive the simulated underwriting losses, with normalised results presented in Figure 10. On average, the simulated underwriting loss remains below zero (indicating a positive underwriting profit) and is relatively consistent across scenarios, which could be explained by the premium loadings. At higher quantiles, however, underwriting losses remain similar across scenarios initially but rise substantially in later periods under high-emission pathways (SSP 8.5 and SSP 7.0) due to escalating climate risks. This upward shift in the tail of underwriting losses can be attributed to the increasing volatility and extreme percentile of hazard losses observed under high-emission scenarios (see Section 3.3.1). Consequently, while downside liability impacts on financial performance appear moderate at first, they are expected to intensify over longer projection horizons in the high-emission scenarios.

Figure 10.

Simulations of underwriting losses (normalised). Underwriting losses are calculated by subtracting the net premium received (i.e., gross premium received minus reinsurance premiums paid) from the net simulated insurance losses (i.e., gross insurance losses net of reinsurance recoverables). Solid lines (and dotted lines under the stationary assumption) represent the average simulation paths across climate scenarios, while dashed lines indicate the $5^{\text{th}}$ and $95^{\text{th}}$ percentiles. Across climate scenarios, the results are similar at the mean level but diverge at higher quantiles, with greater losses projected under high-emission scenarios.