MATH MODELING TO SUPPORT REGIONAL NATURAL DISASTER RISK MANAGEMENT

Transcription

1 10NCEE Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21-25, 2014 Anchorage, Alaska MATH MODELING TO SUPPORT REGIONAL NATURAL DISASTER RISK MANAGEMENT J. Peng 1, Y. Kesete 2, Y. Gao 2, X. Shan 3, R. A. Davidson 4, L. K. Nozick 5, and J. Kruse 6 ABSTRACT We apply a new mathematical modeling framework to examine how the magnitude and nature of the natural disaster risk being managed affects insurer and homeowner risk management decisions and outcomes. The framework includes three interacting models representing the insurer s pricing and risk transfer decisions, each homeowner s insurance and retrofit decisions, and regional hurricane loss. By comparing runs that consider only wind-related damage to those that consider only storm surge flood-related damage, the analysis demonstrates how differences in size and geographic extent of the insurance market, loss distributions of the individual homes and entire region, and available retrofit alternatives affect the optimal insurer and homeowner choices and outcomes. The framework could be adapted for earthquake or multihazard application. 1 Graduate student, Dept. of Civil and Environmental Engineering, University of Delaware, Newark, DE Graduate student, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY Post-doc, Dept. of Civil and Environmental Engineering, University of Delaware, Newark, DE Professor, Dept. of Civil and Environmental Engineering, University of Delaware, Newark, DE Professor, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY Professor, Department of Economics, East Carolina University, Greenville, NC Peng J, Kesete Y, Gao Y, Shan X, Davidson RA, Nozick LK, and Kruse J. Math modeling to support regional natural disaster risk management. Proceedings of the 10 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

2 Math Modeling to Support Regional Natural Disaster Risk Management J. Peng 1 3, Y. Kesete2, Y. Gao 2, X. Shan 3, R. A. Davidson 4, L. K. Nozick 5, and J. Kruse 6 ABSTRACT We apply a new mathematical modeling framework to examine how the magnitude and nature of the natural disaster risk being managed affects insurer and homeowner risk management decisions and outcomes. The framework includes three interacting models representing the insurer s pricing and risk transfer decisions, each homeowner s insurance and retrofit decisions, and regional hurricane loss. By comparing runs that consider only wind-related damage to those that consider only storm surge flood-related damage, the analysis demonstrates how differences in size and geographic extent of the insurance market, loss distributions of the individual homes and entire region, and available retrofit alternatives affect the optimal insurer and homeowner choices and outcomes. The framework could be adapted for earthquake or multihazard application. Introduction This paper describes an application of a new mathematical modeling framework to examine how insurance and retrofit can be used to manage risk. The framework includes three interacting models: (1) a stochastic programming optimization representing the insurer s pricing and risk transfer decisions, (2) a utility-based model of each homeowner s insurance and retrofit decisions, and (3) a regional catastrophe loss model to represent the risk and the effect of alternative retrofit strategies. Reinsurer and government roles are represented as inputs to the insurer-insured interactions. The framework was developed for hurricane risk, but could be adapted for earthquake or even multihazard applications. In [1], the framework is applied considering combined effects of hurricane wind and storm surge flooding. In this paper, we use it to explore how the peril (or magnitude and nature of the risk being managed) affects insurer and homeowner decisions and outcomes. Specifically, we compare runs that consider only wind-related damage and only storm surge flood-related damage, first considering just insurance, then retrofit as well. In the modeling, the perils differ in terms of the size and geographic extent of the insurance market, loss distributions of the individual homes and region as a whole, and costs and benefits of available retrofit alternatives. 1 Graduate student, Dept. of Civil and Environmental Engineering, University of Delaware, Newark, DE Graduate student, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY Post-doc, Dept. of Civil and Environmental Engineering, University of Delaware, Newark, DE Professor, Dept. of Civil and Environmental Engineering, University of Delaware, Newark, DE Professor, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY Professor, Department of Economics, East Carolina University, Greenville, NC Peng J, Kesete Y, Gao Y, Shan X, Davidson RA, Nozick LK, and Kruse J. Math modeling to support regional natural disaster risk management. Proceedings of the 10 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

3 Modeling Approach Scope and Main Assumptions The modeling framework, explained more fully in [1], is designed to improve understanding of how stakeholders decisions interact when the interrelated risk management strategies of insurance and retrofit are considered simultaneously under different policy configurations. Building inventory. The framework addresses single-family residential buildings only. The building inventory is divided into groups, each defined by its geographic area unit or location i (e.g., census tract), building category m, resistance level c, and risk region. Building categories m are defined based on architectural features and assumed to perform similarly and have similar value (e.g., one-story home with a garage and hip roof). Each building is defined as a collection of components represented explicitly in the loss modeling (e.g., openings). Each component in turn is made of many component units (e.g., a single window). For each component, a few possible physical configurations are defined, each with an associated mean component resistance, treated as a random variable. The building resistance c of each building is then the vector of mean resistances of its components, and a retrofitting alternative cc is defined as changing a building from building resistance c to a better building resistance c. Risk regions are larger geographic regions made up of many area units i, defined to allow insurer premiums and homeowner risk attitudes to vary geographically, but at greater aggregation than area units. Stakeholders and time. Since homeowners differ based on their homes type (defined by ) and therefore risk and possibly risk attitude, the models capture their heterogeneous behavior. We assume one primary insurer and one single layer of catastrophe risk excess of loss reinsurance. The durations of the time steps t vary (a few days to a few weeks). They are defined to be short enough to reasonably assume no two hurricanes occur in same time period and so the probability a hurricane occurs in one time period is equal across time periods. Time periods are shorter, therefore, in the months when hurricanes are more likely. Hazard. The hurricane hazard is represented by an efficient set of probabilistic hurricane scenarios ( ), defined as tracks with along-track parameters that determine the intensity. Each hurricane scenario has an associated adjusted annual occurrence probability P h such that when probabilistically combined, the set of hurricane scenarios represents the regional hazard [2]. For each hurricane, wind speeds and surge depths are estimated throughout the study area. Since a series of hurricanes in quick succession can create very different outcomes for an insurer than the same hurricanes evenly spread over a long time, we define a long-term (say, 30- year) timeline of hurricanes as a scenario ( ). Each scenario is a vector, where is number of time periods in a scenario. For each time period, either one of the possible hurricanes h occurs, or no hurricane occurs ( is the no hurricane case). Each scenario has an occurrence probability, and. The complete set of scenarios is defined so it has the same key characteristics as the full set of ( ) scenarios theoretically possible [1]. Overall Framework The framework includes three models and represents four main players (Fig. 1). The loss model

4 is a simulation combining hazard, inventory, and damage modules to compute a probability distribution of losses for each group of buildings in the study area (defined by ) in each possible hurricane h. The primary insurer and homeowners play a Stackelberg game in which the insurer determines what premiums to charge for policies at a specified deductible, and what reinsurance to purchase. Each homeowner (defined by ) responds by choosing from a menu of insurance and/or retrofit options. Specifically, the primary insurer model is a two-stage stochastic optimization in which the objective is to maximize profit while avoiding insolvency and maintaining sufficient yearly profitability and capacity. The homeowners decision-making is modeled as a utility maximization. The reinsurer offers reinsurance at a specified price, and the government may set constraints on the insurer and/or homeowners. Figure 1. Structure of interacting models. In the event of a hurricane h, the loss to insured buildings is divided among the stakeholders. The homeowners pay the loss up to the deductible; the reinsurer pays % of any loss above the attachment point A and up to % of (M-A), where and M are the co-participation percentage and maximum limit of the reinsurance treaty, respectively; and the primary insurer pays the remaining loss. The loss to uninsured buildings falls on their owners. The model outputs describe the recommended primary insurer and homeowner actions, and a probabilistic characterization of the resulting outcomes for the primary insurer, homeowners, and reinsurer, as well as the proportion of the total hurricane losses borne by each of the stakeholders. Primary Insurer Formulation The primary insurer s objective is to maximize total profit averaged over all scenarios S (Eq. 1), subject to Constraints (2) to (16). The decision variables define the premium pricing (i.e., profit loading factor ) and reinsurance strategy (i.e., attachment A and maximum M). The analysis is conducted on an annual basis, with shorter time steps t to allow for multiple hurricanes in a year. ( ) ( ) (5) (1) (2) (3) (4) (6)

5 ( ) (8) ( )[ ] (9) [ ( )] ( ) [ ] (10) [ ( )] ( ) ( ) (7) (11) (12) ( ) (13) ( ) Eq. 2 defines, the total loss to insured buildings in hurricane h, where is loss to a building of type in hurricane h (from the loss model); is the number of buildings of type ; and are binary decision variables output from the homeowner model that equal one if homeowners of type buy insurance and zero otherwise. Eqs. 3 and 4 define, the deductible an insured building of type actually pays in hurricane h, and the total deductible actually paid in hurricane h, where is the dollar value of the deductible. The annual homeowner premium for an insured building of type is defined in Eq. 5 as the expected value of the loss to insured buildings of type less the deductible, multiplied by one plus plus. The loading factors and represent the primary insurer s administrative cost and profit margin, respectively. Eq. 6 defines the total annual premium homeowners pay. Eqs. 7 and 8 define and as the loss that is above A and below M for hurricane and for scenario s in year y, respectively. The is an indicator variable that is one if hurricane h happens in scenario s at time t and zero otherwise. The set ( ) defines the time periods t in year y. In each year y, the primary insurer pays the reinsurer a base premium b, and if hurricane h happens, it also pays a reinstatement premium. The base premium is computed as the reinsurer s expected loss multiplied by one plus a loading factor, plus the standard deviation of the net reinsurer loss multiplied by a constant g representing the reinsurer s risk aversion (Eq. 9) [3]. The reinstatement premium is a pro rata amount of the expected reinsurer loss (Eq. 10). Eq. 11 defines the insurer s net profit,, in scenario and year as, in turn, the total homeowners premiums collected, minus the administrative cost portion of the premiums collected, minus the total actual loss, plus the actual deductibles homeowners pay, plus the actual loss recovered from the reinsurer, minus the reinsurance premium. We assume the insurer s initial surplus equals k times the expected annual premium, where k is a user-specified constant, and the insurer does not retain a surplus greater than this in any year y (Eqs. 12, 13). If the accumulated surplus in year y equals zero or less, the insurer becomes insolvent, and the profit and surplus are set to zero for the remaining years ( ) of the scenario s. (14) (15) (16)

6 Eq. 14 ensures that the probability of insolvency may not be larger than a user-specified constant, where is a binary indicator variable that is one if the insurer becomes insolvent at any time in scenario s and zero otherwise. Eq. 15 ensures that the capacity ratio, defined as the net written premiums divided by the policyholder surplus, does not exceed a user-specified constant for any scenario s or year y. Eq. 16 ensures the average annual return on equity for the years Z that the insurer is solvent is at least a user-specified constant. Homeowner Formulation Homeowners choose some combination of insurance and retrofit, or neither. We let the binary index n indicate the insurance purchasing choice one if a homeowner purchases insurance, or zero if not. If he chooses to retrofit, he can choose which retrofit alternative to do, each of which represents a physical modification of the building that requires a cost to implement and reduces the probability of damage. The case of corresponds to no retrofit. We assume the insurance policy applies to the building after any retrofit. The model is defined by objective function (17) and constraints (18) to (24), with binary decision variables equal to one if a homeowner of type makes the insurance choice n and implements a retrofit that changes building resistance from c to resistance, and zero otherwise. The analysis is conducted on an individual building and annual basis, and is run separately for each group. [ { ( )}] [ { ( )}] (17) (18) (19) (20) (21) (22) (23) (24) We assume the decision is made by maximizing utility function ( ), where is the risk aversion coefficient assigned to homeowners in risk region. The homeowner s objective function (17) is to maximize expected utility over all possible hurricane scenarios h. If the homeowner buys insurance (first term), he pays the premium, ; loss up to the deductible, ; and cost to retrofit a building of category from to,. If the homeowner does not buy insurance, he pays the cost to retrofit and loss due to building damage,. Note that when, no hurricane occurs, and since expected losses depend on the retrofit decision, the premium, deductible, and loss are computed for. We assume each homeowner has a maximum budget for homeowner insurance equal to a specified percentage of his total home value, and an insurer will not offer insurance for a premium less than a minimum threshold, (Eqs. 18 and 19). We define as the set of resistances c to which a building with resistance c can be retrofitted (e.g., only gable roof buildings can strengthen the gable end). Eq. 20 ensures that only permitted retrofits are selected.

7 Each homeowner chooses exactly one combination of insurance-retrofit strategy - (Eq. 21). Eq. 23 defines, the building inventory after any retrofits take place, where defines the set of resistances c which can be retrofitted to resistance c. When the solution procedure iterates back to the primary insurer model, the values of replace the in the insurer model to reflect the updated building inventory after retrofit. Eq. 24 defines, a binary variable that equals one if buildings of type are insured, no matter whether they were retrofitted or not since that is what the primary insurer must consider. Scope and Inputs Case Study Application The case study focuses on single-family wood-frame homes in Eastern North Carolina from the coast to Raleigh. The 2010 census tracts are the basic area unit i, but those that touch the coast were divided into three zones resulting in 732 area units (143 within <1 mile of coast, miles from coast, and 454 >2 miles from coast). We define two risk regions based on location 0-2 miles from the coast (higher risk) or not (lower risk); and eight building categories m. With six building components (roof cover, roof sheathing, roof-to-wall connections, openings, walls, and flood susceptibility), and 2-4 configurations for each (e.g., regular or high wind shingles), there are 192 possible building resistance levels c, and up to 143 possible retrofits. Building values were estimated using R.S. Means and retrofit costs were estimated using [4, 5] and expert opinion. We divide the retrofit cost by thirty so the costs and benefits of the retrofits and insurance are normalized to a constant basis. The total initial (pre-retrofit) building inventory was estimated based on census data and year built relative to major building code and construction practice changes. The final building inventory included 649,012 (70%) buildings in low risk region and 282,890 (30%) in the high risk region. The component-based building loss model is a modified version of the Florida Public Hurricane Loss Model [6], extended to include flood damage based on results from the component-based flood damage simulation model in [7, 8]. The model includes losses due to damage to structural, non-structural, interior, electrical, mechanical, and plumbing components. Damage to home contents, relocation expenses, disruption to occupants lives, or other indirect costs are not included. Using the set of probabilistic hurricane scenarios developed in [2], we developed thirty-year scenarios s [1]. With 20 time steps per year, time steps per scenario s. The risk aversion parameters ( ) and ( ) for high and low risk areas, respectively, were estimated in [9] using NFIP data. In addition,,,,,,,,, =0.05,,, and. The models were solved with the genetic algorithm solution procedure in [1]. Runs We conducted four runs to examine the effect of the risk being managed with and without retrofit allowed; and considering only wind damage and only flood damage. In Runs 1 and 2, homeowners only choices are to buy insurance at the premium offered, or not. In Runs 3 and 4, a homeowner can choose to buy insurance and/or undertake one of the many retrofits possible for his home. In Runs 1 and 3, we consider only wind-related risk (i.e., we assume the flood depth is zero for all homes in every hurricane h, so homeowners and the insurer are concerned only with managing wind risk). In Runs 2 and 4, we similarly consider only flood-related risk.

8 The wind and flood runs represent applications of the modeling framework to situations with different regional risk, keeping most of the rest of the analysis constant (e.g., other inputs, data quality). In terms of the modeling, there are five primary related differences between the wind and flood analyses the size and geographic extent of the insurance market, loss distributions of the individual homes and region as a whole, and costs and benefits of available retrofit alternatives. First, in the flood runs, there are approximately 212,000 homes that could potentially buy insurance (i.e., that experience flood-related loss in at least one hurricane scenario), of which 85% are in the high risk region. In the wind runs, there are approximately 932,000 homes that could potentially buy insurance. The geographic distributions of the markets differ as well with the floodprone homes all being within a few miles of the coast, which can be important because it affects the correlation of losses and therefore an insurer s ability to geographically diversify its portfolio. Related to these points, the loss distributions of the individual homes differ depending on which peril is considered (Fig. 2a), which directly affects the choices each homeowner makes. In this case study, the average home has a much higher probability of a large loss in the flood runs than in the wind runs (Fig. 2a). The total loss distributions for all the homes prone to the peril (wind or flood) differ as well (Fig. 2b), which affects the insurer s decisions. The probability of a large regional loss is more likely in the wind case than the flood case because there are more homes exposed. Finally, in Runs 3 and 4 (with retrofit), the effect of retrofit differs by peril since each has a different set of possible retrofits with different costs, benefits, and set of buildings to which they can be applied. Figure 2. Annual probability of exceedence vs. loss for (a) an average building, and (b) the entire study region. Results with Retrofit Not Allowed (Runs 1 and 2) We first compare Runs 1 and 2, considering only insurance, not retrofit. Table 1 summarizes the insurer s and homeowners decisions. It shows that in the high risk area, the insurer chooses a much higher profit loading factor in the wind case than the flood case ( vs. ), and relies more heavily on reinsurance in the flood case, using it on average 4.5 times vs. 0.8 times in thirty years. Interestingly, even with a profit loading factor six times smaller, a smaller percentage of homeowners insure in the flood case (Run 2) than the wind case (Run 1). Looking at the disaggregated results shows this is largely due to differences in the loss distributions of the individual homes in each case. For the flood case, many cannot buy insurance because their

9 annual premium would be either below the minimum $100 threshold (51%) or above their budget even with such a small profit loading factor (7%). In the wind case, by contrast, those percentages are 10% and 0%, respectively. For most of the remaining homeowners in Run 2, the utility of doing nothing is better than the utility of buying insurance. Insurance is most appealing when a home has a relatively low expected annual loss (which results in a low premium, see Eq. 5) and relatively high variability, because the relative benefit insurance provides in cutting off the tail of the loss distribution is greater [1]. Given that understanding, Fig. 3 suggests why insurance is more appealing in the wind case (Run 1) than the flood case (Run 2). For Figs. 3a and 3c, each home making the corresponding decision do nothing or insure was plotted as a point on the graph based on the coefficient of variation (COV) and annual mean of its loss distribution. Similarly for Figs. 3b and 3d. Fig. 3 shows that in the wind case, there are relatively more homes with the lower expected annual loss and higher COV of loss that makes insurance appealing, and that those are the homes that choose to purchase insurance. Run Profit loading factor, l High risk Low risk Table 1. Reinsurance usage* Insure Summary of results for each run. Penetration rate Retrofit Nothing Insure only Percentage of homeowners who make each choice High risk Low risk Retrofit only Both Nothing Insure only Retrofit only 1. No retrofit-wind % % 9% % 8% No retrofit-flood % % 7% % 0% Retrofit-Wind % 34.2% 33% 0% 54% 12% 77% 3% 17% 3% 4. Retrofit-Flood % 9.7% 71% 2% 25% 1% 97% 0% 3% 0% * Expected number of times reinsurance is triggered (i.e., loss exceeds attachment point A) in 30 years Both Figure 3. Coefficient of variation (COV) of loss vs. mean loss per home, by decision made in Run 1 (a,c) and Run 2 (b,d), without retrofit. Table 2 summarizes the outcomes for the insurer and homeowners on an average annual

10 basis. On average, the insurer makes a positive annual profit for both the wind and flood cases (Runs 1 and 2). The insurer makes 10% more in Run 1 than Run 2 on a per eligible home basis though, because of the higher profit loading factor in Run 1 and reduced need to pay for reinsurance. Since the market is more than four times larger in the wind only case, the total profit is five times more for wind only than flood only, $50.8 vs. $10.6 million. Table 2. Outcomes by stakeholder, for each run, on average annual basis. Primary insurer net profit Average per home expenditures* Run Per peril- High risk Low risk Total prone home Insured Uninsured Insured Uninsured ($M) ($) ($) ($) ($) ($) 1. No retrofit-wind No retrofit-flood N/A** Retrofit-Wind Retrofit-Flood N/A 26 * Expenditures include any premium, deductible, retrofit cost, or loss due to damage ** N/A means not applicable because there are no insured homes in the low risk region On average, the homeowners pay more in the flood case (e.g., $6331 vs. $3426 for insured homeowners in the high risk region), because the loss distributions for each home are higher in the flood case, so for insured homeowners, that translates to higher premiums, and for the uninsured, higher losses. It is important to note, however, that in both Runs 1 and 2, most of the losses (more than 80% of the average annual losses) are paid by uninsured homeowners, so from a societal perspective, the insurance market does not fully manage the risk. Results with Retrofit Allowed (Runs 3 and 4) Runs 3 and 4 are the same as Runs 1 and 2, respectively, but we allow homeowners to retrofit their homes in addition to or instead of insuring them. In both cases, the total losses are actually reduced (26% for wind, 17% for flood), unlike in Runs 1 and 2, where insurance only spreads the overall losses among stakeholders. Interestingly however, retrofit appears to play a somewhat different role in the wind and flood cases. In the wind case (Run 3) most homeowners (54%) Figure 4. Percentage of homeowners who make each decision in Run 1 and Run 2 for (a) wind only and (b) flood only runs.

11 who were insured in Run 1 add retrofit but keep their insurance (i.e., switch from insure to both), whereas only 12% drop their insurance and retrofit instead (Fig. 4a). By contrast, in the flood case, only 11% of those who were insured in Run 2 add retrofit (i.e., switch to both), while 26% drop the insurance and retrofit instead (Fig. 4b). In other words, retrofit is acting as a complement to a greater extent in the wind case, and as a substitute for insurance to a greater extent in the flood case. This is again because insurance tends to be preferred when the mean loss is lower and the variability is higher, which does not occur much in the flood case; whereas retrofit is preferred for a higher mean loss, which occurs more often in the flood case. Conclusions In this paper, we demonstrate application of a new natural disaster risk modeling framework to examine how the magnitude and nature of risk being managed affects homeowner and insurer decisions and resulting outcomes. Comparing a case in which the only hazard is hurricane storm surge flooding, with 212,000 homes exposed mostly within two miles of the coastline to a case in which the only hazard is hurricane wind damage, with 932,000 exposed homes over half the state of North Carolina, the analyses showed some of the differences caused by the differences in size and geographic extent of the insurance market, loss distributions of the individual homes and the region as a whole, and available retrofit alternatives. Acknowledgments This publication was prepared by the Univ. of Delaware, Cornell Univ., and East Carolina Univ. using Federal funds under award 60NANB10D016 from the National Institute of Standards and Technology, U.S. Department of Commerce. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of the National Institute of Standards and Technology or the U.S. Department of Commerce. References 1. Peng J. Modeling Natural Disaster Risk Management: Integrating the Roles of Insurance and Retrofit, and Multiple Stakeholder Perspectives. Dissertation, University of Delaware, Newark, DE, Apivatanagul P, Davidson R, Blanton B, Nozick L. Long-term regional hurricane hazard analysis for wind and storm surge. Coastal Engineering 2011; 58(6): Kunreuther H, Michel-Kerjan EO. At War with the Weather: Managing Large-Scale Risks in a New Era of Catastrophes. The MIT Press: New York, NY, National Estimator. National Renovation and Insurance Repair Estimator software. Version Craftsman Book Company, Available from 5. Institute for Business and Home Safety (IBHS). Shutter Selection Guide. IBHS: Tampa, FL, Florida Public Hurricane Loss Model (FPHLM). Engineering Team Final Report, Vol.I, II, and III. Florida International University, Available from (Nov.16, 2010). 7. Taggart M, van de Lindt J. Performance-based design of residential wood-frame buildings for flood based on manageable loss. Journal of Performance of Constructed Facilities 2009; 23(2): van de Lindt J, Taggart M. Fragility analysis methodology for performance-based analysis of wood-frame buildings for flood. Natural Hazards Review 2009; 10(3): Gao Y. Modeling the Natural Catastrophe Loss Insurance Market. Dissertation, Cornell U., Ithaca, NY, in prog.