Catastrophe Reinsurance Pricing Science, Art or Both? By Joseph Qiu, Ming Li, Qin Wang and Bo Wang Insurers using catastrophe reinsurance, a critical financial management tool with complex pricing, can benefit from the use of in-depth analytics to help identify market opportunity and achieve pricing accuracy. Catastrophe reinsurance is an important tool for insurance companies to hedge extreme risk and manage capital, but its price varies much more than general insurance (Figure 1). Reasons for this market phenomenon include the great uncertainty associated with catastrophe risk, the capitalintensive nature of catastrophe business and the relatively low level of agreement on catastrophe risk pricing. A better understanding of catastrophe reinsurance pricing benefits the whole market because it will help (re)insurers manage underwriting and deploy capital more efficiently. This article clarifies both market- and academic-pricing techniques. Catastrophe reinsurance is priced mainly on exposure rather than experience. In part, the low-frequency nature of catastrophes makes it difficult to collect sufficient historical data for a credible analysis. The dynamics of catastrophe risks (e.g., weather change due to global warming and terrorism attacks launched by Al-Qaeda) further reduce the predictive power of already-limited historical data. So the market usually focuses on exposure analysis to determine the catastrophe reinsurance premium, while loss experience mainly serves as an adjustment factor. Pricing techniques for catastrophe reinsurance can be grouped into three categories: Market approaches based on catastrophe modeling output Market approaches based on non-modeling factors Academic approaches based on financial and actuarial theories Figure 2 is a summary of the elements under each category. Among these pricing methods, approaches based on catastrophe modeling output require the lowest level of subjective input but are inevitably associated with model risk (i.e., how accurately catastrophe models capture loss distributions). If the market is dominated by these modeling approaches, there will be less room for price negotiation because different parties will achieve similar modeling results using the same data set. But if price is determined purely by non-modeling factors, model risk will be reduced and Figure 1. Annual price change comparisons 80% 70% 60% 50% 40% 30% 20% 10% 0% 10% 20% 9 14 2003 1 11 2004 2 6 2005 76 2006 6 9 2007 Commercial line insurance price change Property catastrophe reinsurance price change Source: Towers Watson 16 2008 0 10 2009 Figure 2. Major catastrophe reinsurance pricing approaches Market approaches based on catastrophe modeling output Market approaches based on non-modeling factors Academic approaches based on financial and actuarial theories 1 12 2010 2 2011 Expected loss of a reinsurance layer, standard deviation of layer loss (or other statistics that measure uncertainty), loss on line, probabilities of layer activation and exhaustion, deterministic loss, risk concentration, data quality, peer comparison, proprietary model, etc. Territory, non-modeled exposures, cedant s loss experience, reinsurance relationship, market cycle, reinsurer s underwriting strategy and capital adequacy, etc. Option pricing theory, arbitrage pricing theory, utility optimization theory, etc. Emphasis 2012/2 11
Figure 3. Market pricing approaches comparison High Model risk Low Pricing based on catastrophe model Reduce model risk Ideal pricing method Low Reduce subjective input Subjectivity subjectivity will increase significantly. An ideal pricing method should minimize both model risk and subjectivity (Figure 3). Unfortunately, a mixed approach is currently being used to price catastrophe reinsurance, and it is the trade-off that the (re)insurance market has to manage. Pricing Approaches Pricing based on non-modeling factors High Catastrophe Modeling Output Before modern catastrophe models (e.g., RMS, AIR, EQE) were developed, (re)insurers often used exposure curves to evaluate catastrophe reinsurance. Exposure curves detail the relationship between the accumulation of insured values and the accumulation of estimated loss. Exposure information, coupled with exposure curves, can infer loss estimates as well as catastrophe reinsurance prices. Even so, because of the complexities that are naturally associated with catastrophes, it is not surprising that simple exposure curves are hardly sufficient to reflect the full spectrum of catastrophe risks. Modern catastrophe models were created to provide a more scientific view. These models take advantage of the A thorough understanding of pricing techniques helps reconcile market discrepancies and achieve efficiency. most recent science and engineering knowledge, employ vast computing power made possible by recent IT advances and are frequently configured using new catastrophe events. Catastrophe models can analyze risks at a location level and then build the location-level results up to a portfolio level. This differs from the exposure curve approach, which is based on aggregate exposures. Figure 4 shows the framework of modern catastrophe models that can calculate loss distribution for given exposures. Although loss distributions can be produced, it is how they are used to price catastrophe reinsurance that remains so interesting. Various approaches can lead to dramatically different prices, and a thorough understanding of pricing techniques helps reconcile market discrepancies and achieve efficiency. For instance, the use of excess-of-loss catastrophe reinsurance covers losses from layer retention up to a certain limit. Generally, the most important modeling output for pricing is the expected layer loss. If multiple perils can trigger the layer (hurricane, earthquake, tornado/hail, among other catastrophes), then each of these perils needs to be modeled, and total expected layer loss should be calculated. Still, no reinsurer would accept expected layer loss as the final price. Because of the significant uncertainty of catastrophe risks, catastrophe reinsurance requires large amounts of capital to be allocated to support the underwriting of this type of business. And uncertainty, which can be partially reflected by standard deviation of layer loss, also needs to be properly priced. The market has used the following pricing formula: Layer premium = Expected layer loss + Risk load factor Standard deviation In this equation, the expected layer loss and standard deviation can be calculated by catastrophe models, but the risk load factor is a subjective entry that depends on risk perception, reinsurers capital position and (re)insurance market conditions, among other considerations. Reinsurers may also load non-modeled loss factors, expense ratios and target profit ratios on top of the equation. A report from the Bermuda Monetary Authority finds that 81% of Bermuda (re)insurers add loads to modeling results. 12 towerswatson.com
Figure 4. Framework of modern catastrophe models Stochastic events: California earthquake, Florida hurricane, etc. Hazard parameters: Ground motion attenuation, wind speed, etc. Exposure information: Insured value, location, construction, year built, etc. Economic loss estimate: Building, contents, time element, etc. Insured loss distribution: Annual average loss, standard deviation, exceeding probability, etc. The discussion so far provides a framework for how catastrophe modeling can be used to price reinsurance. But in the real market, many other modeling outputs need to be considered to comprehensively evaluate pricing adequacy. First, expected layer loss as a percentage of layer limit, often called loss on line, can reflect a layer s riskiness: The higher the loss on line, the higher the reinsurance price. Second, probabilities that a layer will be triggered and exhausted reflect how frequently the cedant may receive recovery. These probabilities can be calculated by applying the layer in catastrophe models or using a simulation approach. Higher probabilities usually make it harder for the cedant to receive a low price because reinsurers are managing their own risks and are generally very cautious about low-attachment layers. Third, a deterministic approach is often used to supplement stochastic modeling. Instead of using tens of thousands of stochastic events to generate loss distribution, a deterministic approach is based on a single event and focuses on exposure vulnerability. It is an intuitive and easy-to-manage approach. Lloyd s of London issues realistic disaster scenarios to evaluate the risk management and capital adequacy of its syndicates. So whether a reinsurance layer will be hit by these deterministic events is a meaningful element of a price quoted by Lloyd s. Fourth, geographical concentration of a cedant s risks can be evaluated by a catastrophe model s accumulation analysis and will be carefully reviewed by reinsurers. The rationale behind risk concentration analysis is that a catastrophe event may well affect a large number of risks within a geographical area. A high risk concentration increases correlation, reduces diversification and can lead to an extra charge for the reinsurance premium. Additionally, the more geographic regions and potential perils that expose a contract, the higher the price. As a result, most reinsurers employ some form of portfolio theory in pricing catastrophe reinsurance, so the more diversification a catastrophe contract can bring in, the lower the price and vice versa. Moreover, the quality of a cedant s exposure data determines the accuracy of its modeling output. Poor data quality reduces reinsurers confidence in a cedant s modeling result and might prompt them to demand a higher price. If specific exposure characteristics (such as construction and occupancy codes) cannot be captured, models generally will use internal distributions for these parameters, and the cedant will have lost the opportunity to influence price. Data quality also matters because the insured-to-value ratio should be properly coded. This is a crucial element for the calculation of a cedant s real liability. Other approaches in this category include peer comparison: Reinsurers will charge a similar price for layers based on similar risks, territories and Catastrophe reinsurance is priced mainly on exposure rather than experience. Emphasis 2012/2 13
Joseph Qiu reinsurance analytics and catastrophe modeling. Philadelphia Ming Li reinsurance analytics and catastrophe modeling. Philadelphia loss distributions. Finally, many reinsurers have proprietary models or use multiple models for reinsurance pricing. It is worth noting that post-rms v11.0, blending models has become a common practice, and the choice of blending methodology does affect price to a meaningful extent. Non-Modeling Factors Catastrophe modeling is a very important component of the reinsurance price formation process. Unfortunately, no model can accurately estimate a catastrophe loss distribution, and model risk always exists. This makes it impossible to rely solely on catastrophe models to calculate reinsurance premium. For example, the recent U.S. hurricane model change in RMS v11.0 in February 2011 dramatically increased the loss estimate for most (re)insurers portfolios. Even so, the (re)insurance market has not fully factored the model change into pricing practices for the January 2012 renewal. What is being seen in the market is that many other non-modeling factors can also affect price, and sometimes their impact is significant. These factors include territory, non-modeled exposures, cedants loss experience, reinsurance relationships, (re)insurance market cycles and reinsurers underwriting strategy, among other points. The territory of a cedant s exposures provides a good example of how these factors can impact reinsurance price. A catastrophe layer covering U.S. hurricanes will be more expensive than a layer covering Mexico earthquakes, even if the two layers have similar loss distributions, because a reinsurer probably already has a certain level of U.S. hurricane exposures in its portfolio. Additional U.S. hurricane risk presents more correlation, which will be charged at a higher price. But reinsurers portfolios are probably not heavily weighted in Mexican exposures. Since a Mexico earthquake layer can diversify a portfolio, it will be less expensive. Catastrophe models were created only for major territories and perils. Usually, these territories and perils have generated significant insured losses and have a good amount of claim data for scientists to develop models. But the modeling process will not capture the risk from those exposures that cannot be modeled (e.g., a commercial building located in Saudi Arabia, a country that has no model). In such instances, reinsurers will manually factor non-modeled exposures into their pricing. Loss experience may not change the long-term equilibrium price, which should be based on underlying risks. Still, it does matter in the short run, because reinsurers can reasonably charge higher prices for the layers that experienced losses in recent years. Observations suggest that reinsurance prices may increase dramatically after major catastrophe events. This is partially due to cedants loss experience and partially due to the reduced capital base of reinsurers. It is also easy to understand that the reinsurance relationship, market cycle, reinsurers underwriting strategy and capital adequacy will also affect pricing. The common characteristics of these factors are: They are not based on modeling output. How much impact they have on reinsurance premium depends on subjective judgment. They are usually considered in the price negotiation stage rather than the modeling communication stage. 14 towerswatson.com
Academic Approaches Academia has also attempted to implement financial and actuarial theories, including option pricing theory, arbitrage pricing theory and utility optimization theory. Currently, lack of data and unrealistic assumptions make it impossible to directly implement these theories in the catastrophe reinsurance practice. Researchers have argued that a reinsurance contract is essentially a call option where catastrophe losses are part of the security price and the option pricing theory may be applied to calculate catastrophe reinsurance premiums. The challenge of this approach is that no parameterized distribution or stochastic process can properly describe catastrophe losses, which makes the option pricing formula impractical. The second academic approach is based on arbitrage pricing theory. If a portfolio of securities has exactly the same contingent payoff as a catastrophe reinsurance layer, then, theoretically, the layer price can be derived from the prices of these securities. Otherwise, an arbitrage opportunity will exist. But unlike financial markets, the reinsurance market does not have a sufficient quantity of actively traded and catastrophe-related securities with observable prices. So the arbitrage pricing method remains theoretical because it is technically impossible to construct a market portfolio with payoffs that mimic catastrophe reinsurance. Finally, utility optimization theory assumes that buyers and sellers in the marketplace will negotiate an equilibrium price that maximizes both of their expected utilities. (Utility measures one s happiness and is used to study decision making.) This might be the most practical theory for the market to implement. Unfortunately, no utility function is sophisticated enough to consider all pricing factors. Many quantitative and qualitative elements can affect catastrophe pricing, and there is no mathematical model that can include everything in the equation. A Mix of Art and Science As a critical financial management tool for insurance companies, catastrophe reinsurance is priced in a complex way. The complexity is reflected by the fact that many elements, modeling output and nonmodeling factors can affect premiums, and some of these elements cannot be quantified. So it is fair to say that catastrophe reinsurance pricing is a mix of art and science. Modern catastrophe models have significantly improved market participants knowledge of underlying risks. But model risk remains inevitable, which introduces non-modeling factors and contributes to the historical volatility of catastrophe reinsurance prices. Market and academic approaches to catastrophe reinsurance pricing may lead to significantly different pricing. In-depth analytics are required to identify market opportunity and achieve pricing accuracy. For comments or questions, call or e-mail Joseph Qiu at +1 215 246 1767, joseph.qiu@towerswatson.com; Ming Li at +1 215 246 1743, ming.li@towerswatson.com; Qin Wang at +86 27 59901150 ext. 115, qin.wang@towerswatson.com; or Bo Wang at +86 27 59901150 ext. 107, bo.wang@towerswatson.com. Qin Wang research and analytical services. Wuhan, China Bo Wang research and analytical services. Wuhan, China Among these pricing methods, approaches based on catastrophe modeling output require the lowest level of subjective input but are inevitably associated with model risk. Emphasis 2012/2 15