2.1 Types of DRF instruments

Following the World Bank classification (Mahul et al., Reference Mahul, Signer and Krishna2018) and a substantial body of DRF literature (Kashiwagi, Reference Kashiwagi2011; Zelinschi et al., Reference Zelinschi, Domide and Vîrban2013; WorldBank, 2018), we group DRF instruments into three broad categories.

• • Budgetary Measures: These measures refer to the allocation of funds by the government for disaster-related expenditures and are typically risk-retention strategies. These funds can be regular injections (e.g., a proportion of GDP) into reserve funds or reallocations from other programs. For ex-ante reserve funds, a sum of money is set aside for disaster relief, allowing the government to react swiftly when disaster strikes. On the other hand, ex-post borrowing and reallocating funds require no preparatory planning but carry high financial and opportunity costs since interest rates will be extremely high after a disaster, and the government must sacrifice otherwise potentially lucrative and valuable development and improvement projects.

• • Market-based Instruments: These instruments are usually risk-transfer mechanisms, including insurance and insurance-linked securities such as catastrophic bonds and swaps. Adopting such instruments is imperative for a risk-averse decision-maker since they help remove uncertainty. These instruments transfer risk away from the government to third parties, such as the (re)insurance industry or the catastrophe bond capital market.

• • Contingent Credit: These instruments are specific to DRF and combine elements of risk retention and risk transfer. An external third party, typically a well-established supranational organizationFootnote 2, agrees to lend a pre-agreed amount of funds to the participating countries in the event of a disaster. The loan typically carries a much lower interest rate compared to ex-post financing. However, countries must determine the maximum drawdown amount and pay a small upfront fee, both at the outset and at recurring intervals during the agreement, based on the maximum available drawdown before a disaster occurs.

Lastly, governments can turn to ex-post borrowing (from the public or other nations) and humanitarian aid (relying on donor countries for assistance) as a last resort to finance disaster losses. However, these ex-post financing options are highly uncertain and costly. The Disaster Fund focuses on the most representative and widely used DRF instruments in each category – reserve fund, insurance, CC, and ex-post borrowing.

2.2 Real-world considerations

In this section, we identify several practical constraints and considerations that governments face in managing disaster risk, which will subsequently be incorporated into the construction of the Disaster Fund model.

Funds injections into the Disaster Fund have competing uses and opportunity costs, such as investing in development and improvement projects. Hence, it cannot be inexhaustible, making storing huge amounts of money to self-insure against disaster risk practically infeasible. Political opposition and protests are also likely to occur if a disproportionately large amount of cash is locked away for some future probabilistic event instead of improving citizens’ current welfare. For example, Baratz & Moskowitz (Reference Baratz and Moskowitz1978) note that California’s Proposition 13, which limited tax rates on citizens, was partly motivated by public discontent over a $\$$ 5 billion surplus held by the government while taxpayers felt overburdened by excessive taxation. Consequently, the limited reserves that the government can accumulate are often insufficient to cover significant disaster losses, requiring the government to explore other funding sources.

A government typically does not protect itself from the entire spectrum of disaster risk, particularly when the risk involves exceptionally large losses. For example, within the past 10 years (2010–2020), there have been 8,235 emergency declarations and 14,539 major disaster declarations by U.S. states, which are instances where the individual states face disaster damages beyond their planned capability and resources and, thus, request federal assistance. (FEMA, 2023). Similarly, the National Flood Insurance Program (NFIP) in the U.S. relies on federal borrowings to cover extreme flood losses, as its purchase of public reinsurance and issuance of catastrophe bonds only provide coverage up to a certain threshold (Horn, Reference Horn2024). There are several possible explanations for a government to adopt a limited risk horizon. For enormous losses, governments may consider such disasters too unlikely to occur, or they may believe that securing protection against such remote events is not justified given the associated effort and opportunity costs. Budget constraints (insurance premiums covering the whole disaster loss are impractically high) and a myopic mindset (it is highly unlikely for a disaster to occur within the next few years) are common reasons to forsake planning efforts for such massive losses. The scarcity of market participants willing to absorb high-severity, low-likelihood disaster risk further exacerbates the issue. Additionally, moral hazard may play a role, as governments might expect humanitarian aid from other countries or organizations in the event of catastrophic disasters.

A critical consideration often overlooked in many DRF papers is the government’s risk aversion. Stewart et al. Reference Stewart, Ellingwood and Mueller(2011) suggest that for non-catastrophic events, governments typically adopt a net present value (NPV) approach to evaluate projects, implying risk neutrality. However, this NPV method fails to account for many of the risk-averse actions governments take regarding catastrophic lossesFootnote 3. Kaufman (Reference Kaufman2014) discusses why many governments tend to overlook the risk-reduction benefits of their actions by failing to account for uncertainties, and strongly recommends that policymakers adjust their decision-making processes to include risk-aversion analysis. Harris (Reference Harris2014) further highlights that governments often exhibit risk-neutrality in response to climate events due to political factors. By framing climate change as “unpredictable, unavoidable, or simply natural,” governments may justify their inaction. However, this mindset can be detrimental, as “by the time climate change impacts are bad enough for policymakers to react effectively, it will probably be too late” (Harris, Reference Harris2014). These examples underscore the importance of adopting a utility-based framework in disaster risk management that accounts for the government’s risk aversion.

Governments typically define acceptable risk levels to ensure they can address essential disaster response requirements while maintaining balanced risk exposure. To achieve this, they establish targets based on preferred risk measures, selected to align with specific objectives such as solvency or capital adequacy requirements (Bernard & Tian, Reference Bernard and Tian2009; Chen et al., Reference Chen, Li and Li2010; Melnikov & Smirnov, Reference Melnikov and Smirnov2012). Among these, VaR and TVaRFootnote 4 are the most commonly employed risk measures.

Under the VaR constraint, the government necessitates that the terminal value of the Disaster Fund remains above a pre-determined threshold after a disaster loss at or below the $p$ -th percentile if its distribution occurs. For example, insurers under Solvency II must meet a 99.5% VaR requirement (BIS, 2019), while banks under Basel III adhere to a 99.9% VaR standard (EU Commission, 2015). The VaR measure is widely adopted due to its simplicity, interpretability, and ease of communication, making it a popular choice among regulators and financial institutions for quantifying risk.

Conversely, a TVaR constraint limits exposure to extreme disaster losses by ensuring that the conditional expected terminal value of the Disaster Fund in the tail of the distribution remains above a specified threshold. The TVaR measure is highly relevant for DRF, given that disaster losses are often characterized by high skewness and fat tails. Unlike VaR, which only considers losses up to a certain quantile, TVaR accounts for the average severity of losses beyond this threshold, addressing the limitations of VaR in ignoring extreme tail risks. In the financial sector, the significance of TVaR has been increasingly recognized. Basel IV, implemented starting in 2023, mandates the use of TVaR at the 97.5% confidence level as a primary risk measure, underscoring its effectiveness in providing a comprehensive assessment of extreme risk scenarios (pwc, 2016; FTSE, 2022).

Under many optimal insurance problems, the excess-of-loss indemnity structure often emerges as the optimal structure (see, e.g., Arrow, Reference Arrow1974; Denuit & Vermandele, Reference Denuit and Vermandele1998; Kaluszka & Okolewski, Reference Kaluszka and Okolewski2008). The superiority of the excess-of-loss structure arises from its flexibility to allow the policyholder to self-insure and avoid paying insurance loading for low-severity risk while removing high-severity risk, which is more volatile and uncertain, especially when the policyholder is highly risk-averse relative to the insurer.

However, practical considerations may render the proportional insurance structure preferable in certain contexts. For example, Huberman et al. Reference Huberman, Mayers and Smith(1983) argue that deductible contracts are suboptimal when there exist economies of scale in cost management. Lampaert & Walhin (Reference Lampaert and Walhin2006) highlight that proportional insurance can reduce moral hazard and is simpler to price. Additionally, Raviv (Reference Raviv1992) shows that co-insurance arrangements are optimal when insurers exhibit risk aversion or when insurance costs are non-linear. Similarly, our study’s inclusion of numerous real-world considerations in designing the DRF fund may suggest that the excess-of-loss structure is not necessarily optimal. These considerations introduce complexities that can alter the balance of costs and benefits associated with each insurance structure, potentially favoring the proportional structure under specific circumstances.

Variability-based premium principles have been extensively explored in the actuarial literature (see, e.g., Kaluszka, Reference Kaluszka2001; Chi, Reference Chi2012; Liang and Yuen, Reference Liang and Yuen2016b). Notably, Zeng & Luo (Reference Zeng and Luo2013) demonstrate that under the standard deviation or variance premium principle, the proportional insurance structure emerges as optimal. Furthermore, Landsman & Sherris (Reference Landsman and Sherris2001) critique that the expected value principle does not preserve a consistent risk ordering, as it neglects the variability of risks by assuming that two risks are indifferent as long as their expected payouts are equivalent – an oversight that is particularly significant in catastrophic loss scenarios. Lane & Mahul (Reference Lane and Mahul2008) also provide empirical evidence showing that catastrophe bonds are priced with up to a 44.9% premium loading on the standard deviation of losses, in addition to accounting for expected loss. In this paper, we adopt a hybrid approach that combines both the expected loss and standard deviation premium principles, ensuring a more realistic pricing mechanism. Premium loadings on expected loss help cover claims payouts, administrative charges, and profit margins of the insurer, while loadings on the standard deviation of losses compensate for the high uncertainty and risk borne by the insurer in relation to non-catastrophic risks.

2.3 Disaster Fund

Consider a government seeking to establish a Disaster Fund for effective disaster risk management. The design of the Disaster Fund must incorporate the various instruments discussed in Section 2.1, while also adhering to the constraints and considerations outlined in Section 2.2.

Firstly, since the reserve fund is the most cost-effective form of capital, the government will prioritize its use. Therefore, for low-severity losses up to a certain endogenous threshold $L$ , the government will opt to use the reserve funds over other instruments. This approach is akin to self-insuring against the first portion of disaster risk from $0$ to $L$ . To integrate Observation1 into the Disaster Fund model, we impose a cap on the availability of the reserve fund, denoted as $k_L$ , which is exogenously determined. Once the reserve fund is depleted, the government must then rely on alternative methods to finance the remaining lossesFootnote 5.

From Observation2, there exists an upper threshold loss amount $U$ such that losses above $U$ beyond which the government does not plan in advance to cover disaster losses, as these losses exceed its defined risk horizon. For losses surpassing $U$ , the government will resort to ex-post financing methods, such as borrowing or reallocating budget from other sources, since ex-post financing generally comes with extremely high interest rates and significant opportunity costs, making it a last resort option after all ex-ante alternatives have been exhausted. Furthermore, we introduce an exogenous upper bound $k_U$ for $U$ where the government will cease arranging additional DRF instruments. This limit could be driven by external factors, such as the unavailability of insurance markets or CC facilities willing to cover extremely severe losses.

Lastly, for the remaining losses of moderate loss severity between $L$ and $U$ , the government will adopt a combination of risk retention and risk transfer strategies – specifically, insurance and CC – to cover disaster losses. This approach aligns with the World Bank’s recommended tiers of risk management: (i) self-retention to finance small but recurrent disasters, (ii) CC mechanisms for less frequent but more severe events, and (iii) disaster risk transfer, such as insurance, to cover major natural disasters (GFDRR, 2015). Our primary focus will be on this layer of risk management. The left diagram of Fig. 1 summarizes the key elements of the model discussed thus far.

Figure 1 The Disaster Fund model. We consider two strategies, which consist of three distinct layers for losses with various severities, along with the parameters to optimize.

Let

• • $R_0 \in \mathbb{R}$ denotes the initial wealth allocated to the fund by the government,

• • $F_{CC},r_{CC},r_e \geq 0$ denote the per-unit upfront administrative charge of CCFootnote 6, rate of interest for CC, and the rate of interest for ex-post borrowing, respectively,

• • $X$ denotes a continuousFootnote 7, non-negative random variable modeling the disaster loss with a cumulative distribution function $F_X(x)$ and a density function $f_X(x)$ ,

• • $Y_I(X),Y_{CC}(X),Y_e(X) \geq 0$ denote the cashflows from insurance claims, CC, and ex-post financing, respectively, and

• • $C_I, C_{CC} \geq 0$ denote the insurance premium and upfront cost for arranging CC, respectively.

To establish the Disaster Fund, the government determines the allocation of each DRF instrument to adopt across various loss severities, governed by endogenous variables $L,M,U$ for the CC-I layer structure and $L,\alpha ,U$ for the proportional structure. At the fund’s inception, the government pays the insurance premium $C_I$ and the upfront cost of the CC $C_{CC}$ . Upon the occurrence of a disaster event, the government raises the realized disaster loss amount $X$ in full according to the predetermined Disaster Fund strategy. The realized loss will first be raised through the reserve fund $L$ , followed by a mix of insurance payout $Y_I(X)$ and CC $Y_{CC}(X)$ if the reserve fund is insufficient to cover the entire disaster loss, and finally, the government resorts to ex-post financing $Y_{e}(X)$ as a last resort. Subsequently, if CC facilities or ex-post financing are utilized, the government is obligated to repay the loans with interest charged at $r_{CC}$ and $r_e$ , respectively. Thus, the terminal Disaster Fund value after disaster occurrence and loan repayments is

(1) \begin{align} R_1(X) &= R_0 - C_I - C_{CC} \kern7pt\qquad\qquad\qquad\qquad\qquad\qquad \text{(fund initiation)} \\ &\quad -X+Y_I(X)+Y_{CC}(X)+Y_e(X) \qquad \qquad \text{ (disaster occurrence) } \nonumber \\ & \quad - (1+r_{CC})Y_{CC}(X) - (1+r_e)Y_e(X). \qquad \qquad \text{ (loan repayment) }\nonumber \end{align}

(2) \begin{align} &= R_0 - C_I - C_{CC} - X + Y_I(X) \\ & \quad - r_{CC} Y_{CC}(X) - r_{e} Y_{e}(X).\nonumber \end{align}

Equation (1) groups cashflows occurring at different points in time. In contrast, disregarding the terms $C_{\text{CC}}$ , $Y_{\text{CC}}(X)$ , and $Y_{\text{e}}(X)$ , Equation (2) coincides with the utility maximization problem of the renowned Arrow (Reference Arrow1974) model. However, disaster losses are often extraordinarily large, making it impractical for governments to self-retain risk, even when this may be the utility-maximizing solution. Consequently, governments must resort to borrowing to manage retained risk, incurring additional interest payments compared to the zero interest cost associated with using initial wealth. For each DRF instrument, a marginal unit of disaster loss is financed by

1. (i) one dollar of initial wealth,

2. (ii) one dollar of insurance payoff,

3. (iii) $(1+r_{CC})$ dollars from a CC loan, or

4. (iv) $(1+r_{e})$ dollars from an ex-post loan.

In an unconstrained scenario, option (i) is the most cost-effective, followed by (iii), with (iv) being the least favorable. Consequently, the government will naturally prioritize funding disaster losses using its own wealth, then CC, and finally ex-post borrowing. To reflect real-world constraints, the limited reserve constraint caps the wealth allocated as a reserve at $k_L$ , and the limited risk horizon constraint imposes an upper limit $k_U$ on the combined use of CC and insurance. Any self-retained risk exceeding $k_L$ must be financed through CC, and losses beyond $k_U$ necessitate ex-post financing.

In summary, our model is a one-period static framework based on Arrow (Reference Arrow1974)’s setup, with modifications tailored to the context of disaster losses. Specifically, self-retention becomes costly beyond a certain threshold as borrowing is required. The rising cost of capital discourages self-retention and promotes the purchase of insurance.

We assume that initial wealth is not a limiting constraint, meaning the initial fund is sufficiently large to cover at least the reserve fund, the insurance premium, and the CC upfront fee, such that $R_0 \gt L + C_I + C_{CC}$ . In practice, loan repayments often occur over future periods and may extend across multiple years. However, for analytical convenience, we simplify the model into a single-period framework. Heuristically, the interest rates $r_{CC}$ and $r_e$ can be interpreted as adjusted rates, reflecting the net impact of the discounted time value of money and the potential investment returns of the fund over the repayment period.

2.4 Insurance structures and premium principles

Based on Observation5, we consider two distinct structures for combining insurance and CC to address mid-severity losses between $L$ and $U$ . Under the excess-of-loss (Layer CC-I) structure, CC is arranged to cover losses within the interval $[L,M]$ , while insurance is used to cover the remaining losses within the range $[M,U]$ . In contrast, the proportional (prop) structure entails a fixed proportional allocation parameter $\alpha \in [0,1]$ , where insurance covers $\$\alpha$ of each marginal unit of disaster loss and CC covers the remaining $\$(1-\alpha )$ . As a result, insurance provides protection for up to $\alpha (U-L)$ of the total disaster loss, while CC is responsible for $(1-\alpha )(U-L)$ .

Furthermore, in line with Observation6, we assume that insurance premiums are loaded by two factors: a multiplier $\rho _1 \geq 0$ on the expected payout and a multiplier $\rho _2 \geq 0$ on the standard deviation of the payout. Overall, the terms in Equation (1) are governed by

(3) \begin{align} C_I &= (1+\rho _1)\mathrm{E}(Y_I(X))+\rho _2 \sqrt {\mathrm{Var}(Y_I(X) )},\\[-6pt]\nonumber \end{align}

(4) \begin{align} C_{CC} & = \begin{cases} (1-\alpha ) (U - L) F_{CC} &\mbox{for proportional strategy}\\ (M - L) F_{CC} &\mbox{for layer CC-I strategy} \end{cases} ,\\[-6pt]\nonumber \end{align}

(5) \begin{align} Y_I(X) & = \begin{cases} \alpha [(X - L)^+ \wedge (U-L)] &\mbox{for proportional strategy}\\ (X - M)^+ \wedge (U-M) &\mbox{for layer CC-I strategy} \end{cases} ,\\[-6pt]\nonumber \end{align}

(6) \begin{align} Y_{CC}(X) & = \begin{cases} (1-\alpha ) [(X - L)^+ \wedge (U-L)] &\mbox{for proportional strategy}\\ (X - L)^+ \wedge (M-L) &\mbox{for layer CC-I strategy} \end{cases} ,\\[-6pt]\nonumber \end{align}

(7) \begin{align} Y_e(X) & = (X - U)^+, \end{align}

where $(x)^+\,:\,x \mapsto \max (0,x)$ , $x \wedge y\,:\,(x,y) \mapsto \min (x,y)$ , and the decision variables satisfy

(8) \begin{eqnarray} \begin{cases} L \leq U, \quad 0 \leq \alpha \leq 1, &\text{for proportional structure} \\ L \leq M \leq U, &\text{for CC-I structure} \end{cases} . \end{eqnarray}

2.5 VaR and TVaR constraints

Referring to Observation4, we consider a government that seeks to limit the risk associated with the terminal value of the Disaster Fund by using the VaR and TVaR measures. Let $p \in (0,1)$ denote the confidence level. We define

(9) \begin{align} \mathrm{VaR}_p(R_1) &\;:=\;F^{-1}_{R_1}(1-p),\quad \text{and}\\[-6pt]\nonumber\end{align}

(10) \begin{align} \mathrm{TVaR}_p(R_1) &\;:=\;\mathrm{E} [R_1 | R_1 \leq \mathrm{VaR}_p(R_1) ], \end{align}

where $F_{R_1}(\cdot )$ is the cumulative distribution function of $R_1$ Footnote 8. Therefore, under the VaR and TVaR constraints, the government requires the following conditions to hold

(11) \begin{align} \mathrm{VaR}_p(R_1) & \geq k_{\mathrm{VaR}}, & &\text{ for VaR constraint, and, } \\[-6pt]\nonumber\end{align}

(12) \begin{align} \mathrm{TVaR}_p(R_1)& \geq k_{\mathrm{TVaR}}, & & \text{ for TVaR constraint, } \end{align}

for some $k_{\mathrm{VaR}} \in \mathbb{R}$ and $k_{\mathrm{TVaR}} \in \mathbb{R}$ exogenously determined by the government.

2.6 Expected utility maximization problem

To simplify notations, let $\Theta$ denote the feasible parameter space for the decision variables under the various insurance structures. Specifically, we have $\theta = (L,\alpha ,U) \in \Theta$ for proportional structure and $\theta = (L,M,U) \in \Theta$ for layer CC-I structure. To address the risk aversion behavior of governments as outlined in Observation3, we assume the existence of a convex utility function $\mathbb{U}(w;\,\gamma )$ that reflects the government’s preferences, where $\gamma \geq 0$ is the risk aversion parameter, and $\mathbb{U}'(w) \gt 0$ , $\mathbb{U}''(w) \lt 0$ .

To determine the optimal set of parameter values $\theta$ , we aim to maximize the expected utility of the terminal Disaster Fund value, as expressed in Equation (1), subject to the constraints discussed in Sections 2.3 and 2.5. Thus, the maximization problem is

(13) \begin{align} \max _{\theta \in \Theta } \mathrm{E} \Big (\mathbb{U}(R_1 (X;\,\theta);\,\gamma )\Big ) \qquad \text{ s.t. } \qquad \begin{array}{cc} L \lt k_L, & \mathrm{VaR}_p(R_1) \geq k_{\mathrm{VaR}}, \\[3pt] U \lt k_U & \mathrm{TVaR}_p(R_1) \geq k_{\mathrm{TVaR}} \end{array} \end{align}

Intuitively, under utility maximization, the government aims to maximize the terminal value of the Disaster Fund after a disaster event, while simultaneously minimizing the uncertainty associated with that value. The trade-off between these two competing objectives is governed by the risk aversion parameter $\gamma$ . A higher value of $\gamma$ indicates a stronger preference for minimizing uncertainty, thus prioritizing stability over potential returns.

3.1 Graphical interpretation of Disaster Fund

Figure 2 Graphs of terminal Disaster Fund value $R_1(x)$ as a function of realized disaster losses $x$ under the (a) proportional and (b) layer CC-I insurance structures. In each graph, the blue curve illustrates the Disaster Fund value with lower insurance coverage and higher contingent credit (CC) arrangements relative to the green curve, which depicts an alternative Disaster Fund with higher insurance and lower CC.

Fig. 2 illustrates the terminal value of the Disaster Fund across various disaster loss severities. The vertical intercept represents the terminal Disaster Fund value when there are no disaster losses. A larger initial reserve $R_0$ shifts the entire graph up vertically. A higher level of insurance coverage results in a greater initial premium payment, thereby lowering the vertical intercept. For both insurance structures, the loss magnitude is divided into three distinct regions $[0,L)$ , $[L,U)$ and $[U,\infty )$ , separated by kinks at the attachment point $L$ and exhaustion point $U$ of the mid-severity insurance layer. For small losses within the range $0$ to $L$ , the reserve fund fully covers each incremental dollar of disaster loss, producing a linear decline with a gradient of $-1$ . For losses exceeding $U$ , the steep gradient reflects the high cost of ex-post financing, characterized by a gradient of $-(1+r_e)$ .

The shape of the graph for intermediate losses between $L$ and $U$ varies depending on the specific insurance structure employed. Under the proportional structure illustrated in Fig. 2(a), each marginal dollar of loss is split such that a fraction $\$$ $\alpha$ is funded by the insurer, while the remaining $\$$ $(1-\alpha )$ is financed through borrowing from the CC facility, resulting in a gradient of $(1-\alpha )(1+r_{CC})$ for the mid-severity layer, which is generally less steep compared to the gradient for losses between $0$ and $L$ . On the other hand, there are two distinct sub-layers in the CC-I structure in Fig. 2(b). For losses between $L$ and $M$ , the gradient $1+r_{CC}$ is slightly steeper than $-1$ due to the interest charged by CC. In contrast, since the insurance payout covers the entire disaster losses for losses between $M$ and $U$ , the terminal Disaster Fund value is unaffected, resulting in a horizontal segment with a zero gradient. The point $M$ marks the boundary between the two sub-layers, delineating the transition from CC financing to full insurance coverage within the intermediate loss range.

Lastly, we examine how the imposition of various constraints impacts the terminal Disaster Fund value. The constraint $L\lt k_L$ (respectively, $U\lt k_U$ ) restricts the attachment point $L$ (respectively, the exhaustion point $U$ ) on the horizontal axis to be positioned leftward of the threshold $k_L$ (respectively, $K_U$ ). In contrast, the VaR constraint asserts that the graph’s height at the $p$ -th quantile of the loss distribution must remain above the threshold $k_{\mathrm{VaR}}$ . Similarly, the TVaR constraint can be heuristically interpreted as necessitating that the weighted average height of the graph for losses exceeding the $p$ -th quantile is greater than the threshold $k_{\mathrm{TVaR}}$ .

Propositions1 and 2 highlight several interesting analytical characteristics of our Disaster Fund model concerning the VaR and TVaR constraints. Let $p_U = F_X(U)$ .

Proposition1 simplifies the consideration of the VaR constraint for high percentile values $p\gt p_U$ by allowing it to be re-expressed at a lower percentile. When combined with the constraint $U \leq k_U$ , it suffices to focus on $p = p_{k_U}=F_X(k_U)$ , as any VaR constraint with a larger $p$ can be equivalently reformulated as a constraint on $\mathrm{VaR}_{p_{k_U}}(R_1)$ . Intuitively, for losses beyond $U$ , all strategies converge to relying exclusively on ex-post financing. Consequently, imposing a threshold on the terminal Disaster Fund value for one segment of losses beyond $U$ is not unique; the constraint can effectively be “shifted” anywhere within the range $[U,\infty )$ . Graphically, $\mathrm{VaR}_{p_1}(R_1) - \mathrm{VaR}_{p_2}(R_1)$ is the vertical distance in the ex-post financing layer, $1+r_e$ corresponds to the gradient and $F^{-1}_X(p_1) - F^{-1}_X(p_2)$ is the horizontal distance of the ex-post financing layer. This relationship highlights the linear dependence of the ex-post financing segment, regardless of the type of insurance structure adopted for lower severity losses.

Proposition2 establishes that, for a given loss distribution, the VaR and TVaR constraints are equivalent. This equivalence is particularly advantageous since the TVaR threshold is typically more difficult to determine and quantify in practice. Consequently, the government can limit its focus to the VaR measure without losing analytical generality. In conjunction with Proposition1, the TVaR and VaR constraints can jointly be simplified to the form $ \mathrm{VaR}_{p_{k_U}}(R_1) \geq \bar {k}_{\mathrm{VaR}}$ , where $ p_{k_U} = F_X(k_U)$ and $\bar {k}_{\mathrm{VaR}} = \max \{k_{\mathrm{VaR}}, k^*_{\mathrm{VaR}} \}$ . This simplification significantly reduces the complexity of the maximization problem by eliminating the need to separately consider VaR or TVaR constraints for higher percentiles.

For commonly used parametric loss distributions $X$ , the analytical expression for the TVaR $\mathrm{TVaR}_p(X)$ can often be derived explicitly, facilitating the calculation of $k^*_{\mathrm{VaR}}$ in Proposition2. For instance, when the loss follows an exponential distribution $X \sim \mathrm{Exp}(\lambda )$ , $\mathrm{TVaR}_p(X)= \frac {1}{\lambda } (1-\ln (1-p))$ , while for a lognormal distribution $X \sim \mathrm{LN}(\mu ,\sigma ^2)$ , we have $\mathrm{TVaR}_p(X)= \frac {1}{1-p}e^{\mu +0.5\sigma ^2} \Phi \left ( \sigma - \Phi^{-1} (p) \right )$ , where $\Phi (\cdot )$ is the CDF of the standard normal distribution $\mathrm{N}(0,1)$ .

3.2 Premium principles

Building upon the extensive literature on optimal insurance structures, the optimal design for the Disaster Fund’s insurance coverage depends on the pricing methodology of the insurance contract. Consider a simplified scenario in which the VaR and TVaR constraints are not binding, and the upfront cost of CC is negligibleFootnote 9, then Propositions3 and 4 outline the implications of various insurance structures on the Disaster Fund.

Proposition3 indicates that when the insurance premium principle does not account for the volatility of the payout, it is always preferable to adopt the layered CC-I structure for the Disaster Fund. This is because higher-severity losses lead to a greater reduction in utility for the risk-averse government compared to lower-severity losses. In contrast, Proposition4 suggests that when the premium includes a loading based on the standard deviation of the payout, the proportional structure may be more advantageous than the CC-I structure.

In practice, due to the substantial uncertainty in characterizing disaster losses and the potentially catastrophic magnitudes of the associated payouts, insurers typically impose significant loadings on the variability of disaster risk payouts. Consequently, when $\rho _2$ is significant, we may observe the proportional structure outperforming the CC-I structure.

3.3 Comparative statics

In this section, we perform a comparative static analysis to assess the impact of key external factors on the optimal solution, assuming that all constraints are non-binding, in order to focus on the direction of influence. To facilitate this analysis, we first introduce Lemma1.

Lemma1 provides insights into how changes in $\zeta$ affect the optimal solution $\varphi ^*$ . It is sufficient to examine the change in $ \frac {\partial \mathrm{E}\mathbb{U}}{\partial \varphi }$ as $\zeta$ varies to deduce the impact of $\zeta$ on $\varphi ^*$ . Table 1 presents the comparative statics for the optimal parameters under both the proportional and CC-I structuresFootnote 10.

Table 1. Comparative statics for the optimal parameters $\varphi \in \theta ^*$ under the proportional and CC-I structures in response to changes in exogenous variable $\xi$ . As $\xi$ increases, the optimal parameter $\varphi$ will either increase $\uparrow$ , decrease $\downarrow$ , or the direction of influence is indeterminate $\uparrow \downarrow$ . Since without imposing constraints on $L$ , $L^*_{\text{CC-I}} = M^*_{\text{CC-I}}$ under the layer CC-I structure, the column for $L^*_{\text{CC-I}}$ is excluded

As insurance premium becomes more expensive (i.e., higher $\rho _1, \rho _2$ ), the government tends to adopt less insurance, reflected in higher values of $L^*_{\text{prop}}$ and $M^*_{\text{CC-I}}$ and lower values of $\alpha ^*_{\text{prop}}$ , $U^*_{\text{prop}}$ and $U^*_{\text{CC-I}}$ . Moreover, when the upfront fee $F_{CC}$ for arranging CC is high, the government will generally opt for less CC, leading to a higher $L^*_{prop}$ and lower $U^*_{\text{prop}}$ and $U^*_{\text{CC-I}}$ . However, the effect of $F_{CC}$ on $\alpha ^*_{\text{prop}}$ and $M^*_{\text{CC-I}}$ is inconclusive as higher $F_{CC}$ can (i) reduce preference for CC due to higher cost, leading to a higher $\alpha ^*_{\text{prop}}$ and lower $M^*_{\text{CC-I}}$ , and (ii) deplete terminal wealth, prompting the government to reduce insurance purchase to lower premium payments and offset the wealth reduction caused by the higher upfront cost causing a lower $\alpha ^*_{\text{prop}}$ and higher $M^*_{\text{CC-I}}$ .

Similarly, the net impact of an increase in $r_{CC}$ on the optimal $L^*_{\text{prop}}, U^*_{\text{prop}}$ and $M^*_{\text{CC-I}}$ is indeterminate since a higher interest rate on CC has two opposing effects: (i) deter the use of CC and propel the government to adopt insurance, which lowers $L^*_{\text{prop}}$ and $M^*_{\text{CC-I}}$ and increases $U^*_{\text{prop}}$ , and (ii) deplete terminal wealth due to higher loan repayment, causing the government to lower insurance purchase to in an effort to stabilize the terminal fund value, which in turn increases $L^*_{\text{prop}}$ and $M^*_{\text{CC-I}}$ while decreasing $U^*_{\text{prop}}$ .

A higher ex-post borrowing interest rate $r_e$ has an unambiguous effect of increasing insurance uptake to avoid incurring substantial costs to secure additional funding for extremely large losses. Similarly, a higher risk aversion $\gamma$ motivates the government to rely more on insurance over CC and ex-post borrowing, driven by a preference to minimize volatility in the terminal Disaster Fund value.

3.4 The VaR constraint

While the constraints $L \leq k_L$ and $U \leq k_U$ directly affect the optimal decision variables, the influence of the VaR constraint on these variables is more complex. To facilitate discussion, let $\varphi \in \{L,M,U \}$ denote a decision variable of interest and let $\varphi ^*$ and $\varphi ^\#$ denote the optimal parameters for our optimization problem in Equation (13) obtained without and with VaR constraint ( $\mathrm{VaR}_p(R_1) \geq k_{\mathrm{VaR}}$ ), respectively.

Proposition5 outlines the ramifications of enforcing a binding VaR constraint. Intuitively, if the VaR constraint is active and $p_{\mathrm{VaR}}$ -th quantile loss $F^{-1}_X(p_{\mathrm{VaR}})$ is high (resp. low), the optimal strategy involves increasing (resp. decreasing) insurance coverage by reducing (resp. raising) $L$ and raising (resp. lowering) $U$ .

3.5 The case of CARA utility

The Constant Absolute Risk Aversion (CARA) utility function, widely favored in the literature for its broad adoption and mathematical tractability, is expressed as

\begin{equation*} \mathbb{U}(w;\,\gamma ) = -\mathrm{e}^{-\gamma w}, \text{ for } \gamma \gt 0. \end{equation*}

Due to the multiplicative property of exponential utility $\mathbb{U}(w_1 + w_2) = \mathbb{U}(w_1)\mathbb{U}(w_2)$ , the optimization problem becomes independent of the initial fund injection $R_0$ when there are no VaR or TVaR constraints. In other words, the magnitude of $R_0$ does not affect the values of the optimal decision variables $\theta ^*$ , and it can be omitted when VaR or TVaR constraints are absent.

Alternatively, if a government’s preference is governed by a CARA utility function, it is possible to endogenize the initial fund injection $R_0$ and determine the minimum value of $R_0$ necessary to satisfy the government’s risk requirement, as quantified by a VaR constraint with $p_{\mathrm{VaR}}$ above the exhaustion point $U$ . The government can first solve the expected utility maximization problem without the VaR constraint to obtain the optimal decision variables $\theta ^*$ . Subsequently, the minimum $R_0$ required to satisfy the VaR requirement is

(14) \begin{align} R_0 & = \inf \Big \{ r_0 \gt 0 \ | \ \mathrm{VaR}_{p_{\mathrm{VaR}}}(R_1(X);\,\theta ^*, r_0) \geq k_{\mathrm{VaR}} \Big \} \nonumber \\ & = \inf \left \{ r_0 \gt 0 \ \Bigg | \ \begin{array}{l} r_0 - C_I - C_{CC} - F^{-1}_{X}(p_{\mathrm{VaR}}) +Y_I\left (F^{-1}_{X}(p_{\mathrm{VaR}})\right ) \\ -r_{CC} Y_{CC}\left (F^{-1}_{X}(p_{\mathrm{VaR}}) \right ) -r_e Y_e\left (F^{-1}_{X}(p_{\mathrm{VaR}})\right ) \end{array} \geq k_{\mathrm{VaR}} \right \} \nonumber \\ & = k_{\mathrm{VaR}} + C_I + C_{CC} + F^{-1}_{X}(p_{\mathrm{VaR}}) - Y_I\left (F^{-1}_{X}(p_{\mathrm{VaR}})\right ) \nonumber \\ & \quad + r_{CC} Y_{CC}\left (F^{-1}_{X}(p_{\mathrm{VaR}})\right ) + r_e Y_e\left (F^{-1}_{X}(p_{\mathrm{VaR}})\right ). \end{align}

Intuitively, $R_0$ represents the minimum initial fund injection required to satisfy the VaR constraint $\mathrm{VaR}_{p_{\mathrm{VaR}}}(R_1) \geq k_{\mathrm{VaR}}$ . Graphically, as illustrated in Fig. 2, adjusting $R_0$ shifts the entire graph vertically upwards or downwards, while leaving the structure of the different layers (such as the reserve fund, insurance, CC, and ex-post financing) unchanged. In this context, $R_0$ functions as a vertical adjustment that alters the terminal fund value but does not impact the optimization procedure. The government can first determine the optimal Disaster Fund structure independently of $R_0$ and calculate the necessary $R_0$ to inject to meet the required VaR level $k_{\mathrm{VaR}}$ .

4.1 Data

Using the loss data from OpenFEMAFootnote 12, we employ the NFIP as a case study to explore the construction of the Disaster Fund for flood risk management under various practical constraints and considerations. To ensure relevance and avoid potential seasonality, we aggregate all individual claims and policies at the annual level and restrict the data to the period from 2002 to 2020Footnote 13. While most flood insurance policies are single-year contracts, a subset of policies spans multiple years; for these, we distribute the total premium amount evenly across all applicable years. Furthermore, to ensure the claims data possess similar exposures across years, we calculate the annual premium growth rate over the last 12 years (2009 to 2020), which is approximately 4%, and inflate the annual claims data by a factor of $1.04^{(2020 - Year)}$ to bring all losses to 2020 exposure levels. By scaling all claims data to 2020 levels, we account for both the growing number of policyholders and the effects of inflation. Additionally, we conduct several time-series analyses (at $5\%$ significance level) to ensure that the loss data (i) is homoskedastic (Breusch-Pagan test), (ii) has no autocorrelation (Breusch-Godfrey and Ljung-box test), (iii) is stationary (ADF and KPSS test), and (iv) is absent of time trend (regression slope t-test). Moreover, we rescale the loss data by dividing by $\$$ 10 billion for brevity. Next, we fit the loss data to several candidate distributions – including Weibull, gamma, and lognormal – using the maximum likelihood estimation method. Among all distributions, the lognormal distribution emerges as the best fit, as it achieves the lowest Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling test statistics, as well as the lowest AIC and BIC values. The fitted loss distribution is $X \sim LN(\mu = -1.466,\sigma ^2 = 1.096^2)$ .

We set $k_U$ at the 99.8th percentile of the loss distribution, corresponding to the highest recurrence interval of a 1-in-500-year event, as documented by FEMA (e.g., FEMA, 2021). Additionally, we set $k_L$ to align with the common rule of thumb in literature and practice, where 1-in-10-year period losses are retained. Both $p_{\mathrm{VaR}}$ and $p_{\mathrm{TVaR}}$ are set at 99.8% to be consistent with $U$ , following Proposition1. Without the loss of generality, we focus on the VaR constraint and set $k_{\mathrm{VaR}} = 26$ for illustrative purposes to effectively demonstrate the functionality of the Disaster Fund modelFootnote 14.

For the simulation exercise, we adopt the CARA utility with an initial fund injection of $R_0 = 29.181$ , which corresponds to the 99.9995th percentile of the loss distribution. Under expected utility theory, it is widely known that wealth and risk aversion are indistinguishable (Yaari, Reference Yaari1987). We will put less emphasis on $R_0$ and focus on highlighting the characteristics of the DRF model. Moreover, as established in Section 3.5, when a CARA utility function is adopted and the VaR constraint is non-binding, $R_0$ does not influence the optimal decision parameters. Next, we set the necessary input parameters, referring to Clarke et al. Reference Clarke, Mahul, Poulter and Teh(2017) for guidance. In practice, governments tailor these parameters to reflect their specific requirements, economic conditions, and risk profiles. A summary of all parameter values can be found in Table 2.

Table 2. Parameters for the Disaster Fund model for empirical simulations

4.2 Empirical findings

Figure 3 (a) and (b) show the optimal parameters $\theta ^*$ ; (c) shows the differences in expected utility between the insurance structures; (d) and (e) show the amount of insurance and contingent credit; and (f) shows the $VaR_{99.5\%}$ value for both insurance structures. All plots depict risk aversion parameter $\gamma \in [0,10]$ and premium loading $\rho _2 \in [0,1]$ . Optimization adopts CARA utility and parameters from Table 2. To make $\alpha ^*$ span the entire vertical axis for illustrative purposes, we multiply $k_U=5.4$ (upper limit of the plot) to $\alpha ^* \in [0,1]$ so that its new range becomes $[0,5.4]$ .

4.2.3 Influence of higher-order moments

Disaster losses are often characterized by significant right skewness and fat tails, reflecting the frequent occurrence of extreme events. While the impact of the mean and variance of the loss distribution on optimal insurance decisions is well-established and intuitive, the influence of higher-order moments, such as skewness and kurtosis, is less direct and has not been as extensively studied. In this section, we investigate how the higher moments of the disaster loss distribution affect the optimal construction of the Disaster Fund.

Under fat-tailed losses, the government tends to purchase more insurance. When comparing Figs. 4(a)–(d) with Figs. 3(a)–(b), we observe that higher-order moments cause the region where insurance and CC change rapidly (referred to as the middle region in Section 4.2.1) to shift toward higher values of $\gamma$ and lower values of $\rho _2$ . This trend is also apparent in Figs. 4(e)–(f), where the amount of insurance is lowest under $X_3$ , followed by $X_2$ , and then $X_1$ for any given ( $\gamma , \rho _2$ )-pair.

Moreover, the region where insurance and CC change rapidly is narrowest under the lognormal distribution, indicating that, under highly skewed and fat-tailed losses, the government is more likely to lien toward either one extreme – (I) arrange for insurance without any CC or (2) purchase the minimum amount of insurance necessary to meet the VaR constraint – and less probable to adopt a mix of insurance and CC above the VaR requirement.

Table 3. Three illustrative loss distributions, along with their first four moments. $X_1$ corresponds to the best-fit loss distribution in Section 4.1. $X_2$ and $X_3$ are two alternative distributions fitted with the same first two moments but possess smaller higher-order moments

Figure 4 Similar plot to Fig. 3 under distributions $X_1, X_2, X_3$ (refer to Table 3).

4.3 Optimal parameters under the Disaster Fund model

Figure 5 Optimal decision variables across risk aversion parameter $\gamma$ under the (a) proportional and (b) layer CC-I structure, with $\rho _2 = 0.2$ and all remaining parameter values from Table 2. The top red line denotes $U^*$ , and the bottom green line denotes $L^*$ . The middle gray line represents $\alpha ^*$ for the (a) proportional structure and $M^*$ for the (b) CC-I structure. The red area denotes the proportion (for proportional structure) or amount (for CC-I structure) of insurance, while the green area represents the proportion/amount of contingent credit. The dashed line highlights $\gamma = 0.2$ , which is used in the discussion.

Finally, we apply our Disaster Fund model to the NFIP to analyze its optimal combination of disaster financing tools. For illustration, in addition to the parameters in Table 2, we assume that insurance is available to NFIP at $\rho _2 = 0.2$ , and that NFIP’s preferences are represented by a CARA utility function with a risk aversion parameter of 0.2.

From Fig. 5(a), under the proportional strategy, the government should arrange insurance and CC to cover losses between $\$$ 9 billion and $\$$ 47 billion, with insurance covering 65% of the losses. The expected terminal value of the Disaster Fund in this case is $\$$ 287 billion, with a standard deviation of $\$$ 2.01 billion.

Similarly, Fig. 5(b) shows that under the CC-I strategy, the optimal configuration is to arrange CC for losses from $\$$ 9 billion ( $L=0.9$ ) to $\$$ 22 billion ( $M=2.2$ ) and purchase insurance for losses between $\$$ 22 billion and $\$$ 46 billion ( $U^*=4.6$ ). This corresponds to $\$$ 13 billion of CC and $\$$ 24 billion of insurance coverage. The expected terminal value of the Disaster Fund remains $\$$ 287 billion, while the standard deviation increases to $\$$ 2.82 billion.

For the VaR constraint to become binding, the threshold $k_{\mathrm{VaR}}$ must be set above $\$$ 244.5 billion.

4.4 An alternative budget constraint

As robustness check, we consider the budget constraint $L + C_{\text{I}} + C_{\text{CC}}$ as an alternative specification to our limited reserve constraint $L \leq k_L$ . This budget constraint caps the total initial monetary outlay prior to loan repayments at $k_L$ , thereby imposing an upper limit on the total insurance premium and the upfront cost of arranging CC, with all remaining funds allocated to the costless reserve fund.

Figure 6 Base case results replicating Fig. 3 with the alternative constraint $L+C_I+C_{CC} \leq k_L$ .

Fig. 6 replicates Fig. 3 after replacing the limited reserve constraint $L \leq k_L$ with the budget constraint $L + C_I + C_{CC} \leq k_L$ . Our empirical findings remain largely consistent under the budget constraint framework, as evidenced by the close resemblance between Figs. 3 and 6. The key distinction introduced by the budget constraint specification is the minor reallocation of funds, characterized as a small transfer from CC adoption to the costless reserve fund. However, the magnitude of these changes is relatively insignificant.

For the layer CC-I strategy, in the absence of the VaR constraint, the optimal $M^*$ and $U^*$ are identical to those obtained under the limited reserve constraint, indicating that the amount of insurance and the severity of insured losses remain unchanged. For low levels of risk aversion $\gamma$ and the standard deviation loading factor $\rho _2$ , the optimal $L^*$ also remains across both constraints. However, for higher values of $\gamma$ and $\rho _2$ , the budget constraint slightly increases the optimal $L^*$ , as it induces a substitution effect from CC toward the reserve fund. Given that the reserve fund does not incur an upfront cost, the government prioritizes its full utilization before resorting to CC. Nonetheless, since the upfront cost of CC $F_{text{CC}}$ is relatively small, the overall impact is minimal.

On the other hand, when the VaR constraint is binding, $L^*$ becomes marginally higher compared to the non-binding VaR constraint case, particularly for low $\gamma$ and $\rho _2$ values. Since the amount of insurance is fixed at a minimum level to satisfy the VaR requirement, the new budget constraint allows slightly more CC to be substituted with reserve fund. Moreover, as the total budget $k_L$ increases under the new constraint, the allocation to the reserve fund $L^*$ rises, especially when $\gamma$ is low and $\rho _2$ is high. In contrast, the value of $M^*$ exhibits only a marginal increase and exhaustion point of $U^*$ remains relatively constant.Footnote 15 Intuitively, the increase in $k_L$ induces only a minor reallocation from CC to the costless reserve fund, with negligible impact on the overall balance between risk transfer (insurance) and risk retention (CC and reserve fund) mechanisms.

A similar pattern emerges under the proportional strategy. Under the new budget constraint, both the optimal $L^*$ and $\alpha ^*$ increase, while optimal $U^*$ remains relatively stable. Consistent with the observations under the layer CC-I strategy, the model favors greater reliance on the reserve fund at the expense of CC, with insurance coverage remaining essentially unchanged relative to the limited reserve constraint. Additionally, there is a modest shift toward insuring more severe losses, with both $L^*$ and $\alpha ^*$ rising to preserve the same total insurance coverage. The presence of the VaR constraint leads to a more pronounced increase in $L^*$ and $\alpha ^*$ , while $U^*$ decreases slightly to preserve the total amount of insurance coverage. As the constraint is relaxed through higher values of $k_L$ , $L^*$ increases, $U^*$ stays constant, and $\alpha ^*$ increases slightly, resulting in a negligible influence on the amount of insurance uptake and a slight decline in CC arrangement.

Overall, the differences in the optimal DRF structure between the limited reserve constraint and the budget constraint are negligible.