By Yuri M. Ermoliev, Stephen M. Robinson, Elena A. Rovenskaya, and Tatiana Y. Ermolieva
Humans continually face catastrophes involving natural disasters, such as floods, droughts, hurricanes, and large-scale fires. In today’s highly interconnected world, losses from such incidents have increased greatly due to growing population densities, asset concentration in disaster-prone areas, and environmental change from anthropogenic impacts.
Catastrophic natural disasters are random events that are rare but very impactful. Traditionally, most catastrophic losses are paid ex-post (adaptively) by individuals (property owners), government agencies, insurers and reinsurers, charity institutions, and international organizations, rather than through explicit ex-ante (forecast-based) arrangement via long-term strategic decisions [7].
Moreover, there is typically little or no prior agreement as to who should bear what portions of the monetary cost. In anticipation of the need to cover potentially large losses in an ad-hoc way, responsible agencies retain certain budget resources for this purpose. However, such retention reduces the options for profitable investment; in the case of large funds, it can potentially stifle economic growth.
We propose that intensification of ex-ante measures—combined with a more intelligent method for setting aside resources to build adaptive capacities for ex-post compensations, contingent credits, catastrophic bonds, monitoring, and regulation—can significantly reduce the overall burden on national economies and strike a healthy balance between economic growth and security. Integrated long-term approaches to risk management and economic development, with an explicit emphasis on the possibility of rare high-consequence catastrophes, enable effective decisions in this context. This tactic requires one to account for the dependence between decisions and risk distributions.
Existing observations demonstrate the increasing magnitude and variability of risks, indicating that one cannot assume catastrophic risk distribution to be Gaussian; in fact, they are skewed and have fat tails. Their focus on tails makes quantile-based risk measures—e.g., value at risk (VaR) and conditional value at risk (CVaR)—more appropriate than variance-based measures applicable only to Gaussian distributions. We have developed and applied a new approach to stochastic optimization in a number of case studies. Our strategy allows us to include quantile-based performance functions in decision support models for integrated catastrophic risk management. These models are characterized by complex nested distributions shaped by the decisions of policymakers. Here we briefly outline this approach, its advantages, and problems to which one can effectively apply it.
Optimization under Chance Constraints
We consider maximization of a prescribed objective function—such as an insurer’s expected profit or a country’s social welfare—defined in a feasible set under chance constraints. These constraints can specify the desired or accepted probability of a system’s default, or the violation of certain security constraints (e.g., exceeding a prescribed emission level). The initial problem of maximizing an expected utility under chance constraints is equivalent to including the expected utility combined with a nonsmooth function penalizing constraint violation.
The solution to such an augmented problem is often called a robust solution, as it is “reasonably good” for most realizations of the random input. The equivalence between the two problems holds true for a rather general class of problems [3]. Specifically, the penalty term in the equivalent problem emerging from the problem’s transformation with chance constraints is essentially the expected shortfall, or CVaR risk measure.
The robust solutions derived by this approach combine ex-ante and ex-post decisions, where ex-ante measures are typically long-term investments in preventive actions (e.g., dams to inhibit flooding, earthquake-resistant buildings, or water and energy infrastructure). Ex-post practices are flexible short-term actions in response to random events (e.g., reconstruction of damaged infrastructure). Design of a robust mix of ex-ante and ex-post policies aims to invest in long-term precautionary procedures enabling optimal adaptive capacity. Application of robust solutions affords security for large quantities of resources, as we observed in our case studies.
Figure 1. Geographical distribution of robust premiums as percentage of the 100-year flood damages. Figure courtesy of [6].
Transformation of the maximization problem (with discontinuous “hit-or-miss” type chance constraints) into one with the expected shortfall as penalty in the objective function renders the resulting optimization problem nonsmooth. Standard gradient-based solution methods are thus inapplicable. Another fundamental complexity arises from catastrophic events’ dependence on agent decisions, eliminating conventional independent scenario simulations and optimization procedures. Brute-force approaches quickly become computationally infeasible, even for problems of realistic dimension. For example, straightforward joint evaluations of \(n=10\) location-specific decisions and \(m=10\) independent scenarios for each location with only one second per evaluation could require \(10^{10}\) seconds — more than 317 years.
A numerical method that solves this problem efficiently combines a Monte Carlo-based catastrophe generator that produces realizations of random inputs/variables (e.g., insolvency of an insurance system or damages to critical infrastructure) and a specific iterative stochastic optimization quasi-gradient procedure [1] with random stopping time moments. Such moments define catastrophe arrivals and induce long-term catastrophe-related social discounting [2].
Applications: The Value of an Integrated Catastrophic Risk Management Approach
Figure 2. Insurers’ balance between premiums and coverage (in millions of euros) for 10-, 100-, and 1000-year floods for robust and conventional—average annual loss (AAL)—premiums. Large positive numbers in AAL cases indicate the level of overpayment. Figure courtesy of [6].
Researchers have applied the aforementioned strategy in case studies of floods, earthquakes, windstorms, energy and information infrastructure networks, and homeland security [5]. For example, Ermolieva et al. considered the flood risks in a flood-prone area around Rotterdam in the Netherlands [6]. Due to the elevated risk, insurers set high premiums that many firms and households could not afford; this left a large number of assets uninsured. One derives high insurance premiums using the traditional actuarial average annual loss approach, which ignores heterogeneous exposures and sets the same premium for all regional contracts. Most importantly, it averages the expected losses over a period of years. Ermolieva et al. compared this technique to robust premiums computed using the proposed approach to integrated catastrophic risk management, which explicitly accounts for the fat-tailed distribution of flood losses over the years and the regional differences in exposure (see Figure 1). Numerical results demonstrated that the high premiums computed and applied in the region are unwarranted; insurers are essentially overpaid. Quantile-based stochastic optimization suggests lower premiums, which can ensure the insurers’ solvency under all flood scenarios relevant to national flood safety standards (see Figure 2). This optimizes the balance between the interests of insurers and the insured.
Many researchers have adopted and used the quantile-based approach. Our method is novel in that it integrates geographically-explicit modeling of dependent catastrophic risks with quantile-based stochastic optimization for robust ex-ante and ex-post disaster risk management. It complements the standard risk-pooling concepts, extreme value theory, and mean-variance approach, all of which are valid and useful for independent, frequent, low-consequence risks like car accidents. Due to the skewness of natural disasters’ loss distribution, application of variance-based risk measures, for instance, would result in an underestimation of high-magnitude risks, which can lead to disastrous societal consequences [4]. The approach we present is capable of handling non-Gaussian, decision-dependent risks that are interdependent in space and time; such features are applicable to a variety of applications, from floods and other natural disasters to terrorist attacks.
References
[1] Ermoliev, Y. (2001). Stochastic Quasigradient Methods: General Theory and Applications. In C.A. Floudas & P.M. Pardalos (Eds.), Encyclopedia of Optimization. Dordrecht, Netherlands: Kluwer Academic Publishers.
[2] Ermoliev, Y., Ermolieva, T., Fischer, G., & Makowski, M. (2010). Extreme events, discounting and stochastic optimization. Ann. Oper. Res., 177(1), 9-19.
[3] Ermoliev,Y., Ermolieva, T., MacDonald, G., & Norkin, V. (2000). Stochastic Optimization of Insurance Portfolios for Managing Exposure to Catastrophic Risks. Ann. Oper. Res., 99, 207-225.
[4] Ermoliev, Y., & Hordijk, L. (2006). Facets of robust decisions. In K. Marti, Y. Ermoliev, M. Makovskii, & G. Pflug. (Eds.), Coping with Uncertainty: Modeling and Policy Issues. Berlin, Germany: Springer Verlag.
[5] Ermoliev, Y., & von Winterfeldt, D. (2012). Systemic risk and security management. In Y. Ermoliev, M. Makowski, & K. Marti (Eds.), Managing Safety of Heterogeneous Systems: Lecture Notes in Economics and Mathematical Systems (pp. 19-49). Berlin, Germany: Springer Verlag.
[6] Ermolieva, T., Filatova, T., Ermoliev, Y., Obersteiner, M., de Bruijn, K., & Jeuken, A. (2017). Flood Catastrophe Model for Designing Optimal Flood Insurance Program: Estimating Location-Specific Premiums in the Netherlands. Risk Anal., 37(1), 82-98.
[7] Froot, K. (1997). The Limited Financing of Catastrophe Risk: An Overview. NBER Working Paper Series, National Bureau of Economic Research.
Yuri M. Ermoliev is an academician of the National Academy of Sciences of Ukraine, an Institute Scholar at the International Institute for Applied Systems Analysis (IIASA), and a senior research scholar in the Advanced Systems Analysis (ASA) Program at IIASA. Stephen M. Robinson is professor emeritus of industrial and systems engineering at the University of Wisconsin-Madison, a member of the National Academy of Engineering, and a senior research scholar in the ASA Program at IIASA. Elena A. Rovenskaya is director of the ASA Program at IIASA and a research scholar in the Department of Optimal Control at the Faculty of Computational Mathematics and Cybernetics at Lomonosov Moscow State University in Russia. Tatiana Y. Ermolieva is a research scholar at IIASA’s Ecosystems Services and Management Program.