Techniques for predicting the behaviour of natural events, and the losses that they can cause, have developed into a fine science. Markus Aichinger looks at the use of catastrophe models in setting prices for tropical cyclone exposures.
Fair pricing and accumulation control for natural catastrophe risk require an accurate quantification of the potential exposure. Standard approaches using `as if' adjusted historical loss information have significant drawbacks. Changes in geographical portfolio distribution cannot be considered, and calculations are always based on very limited sets of figures. A more sophisticated approach, based on reconstruction of historical catastrophic events, could additionally take into account a client's current portfolio, as well as prevailing insurance conditions. However, the problem of having to rely on short observation periods remains.
Probabilistic catastrophe models are the accepted solution to this problem. They serve not only to extend the event set, but also provide a complete pricing tool. Like others, PartnerRe's solution for tropical cyclones is composed of three distinct modules. The hazard module calculates the frequency, severity and location of a particular loss event. The engineering module converts the event parameters from the hazard model into a ground-up loss for risks at each location, and the insurance/actuarial module takes into account the programme structure and re/insurance conditions of the particular insured risks and/or portfolio.
Building an accurate model requires a reliable set of meteorological data for research, testing and calibration. It must incorporate all the elements that combine to produce the final characteristics of the overall event. An in-depth understanding of these elements, and of how they interact with one another, is therefore critical to the production of realistic simulations, incorporating the immense variability in these natural phenomena
`Best Track' files containing best-estimates of position and intensity for north-west Pacific cyclones are recorded by the Japanese Meteorological Agency and the Joint Typhoon Warning Centre (JTWC) at six-hourly intervals after each tropical cyclone event.The position of the cyclone centre is given in latitude and longitude. Intensity is defined as the maximum sustained one-minute or ten-minute mean surface wind speed. These files are sufficient to represent the smoothed path of an actual storm, but do not describe the actual size and shape of its wind field. For this, data is also required of the radius of maximum wind (RMW), which is located near the eye wall, and the radius of gale force wind (Rgale), which is a measure of the size of the cyclone's circulation. Contrary to what is often assumed, the RMW alone cannot be relied upon to give a good indication either of the wind profile or of the radial extent of the circulation.
Our analysis of data from tropical cyclone basins reveals a dependency between a cyclone's particular region of origin within each basin, and its characteristic behaviour. Each region of origin provides particular initial conditions for storms originating within it.
Pacific region divide
The north-west Pacific basin can basically be divided into two major regions of origin: the South China Sea and the region east of the Philippines. In both regions, tropical cyclone development is mainly influenced by the large-scale monsoon trough, which triggers around 85% of tropical cyclones within the basin. The Mei-Yu front, however, often also spawns small, fast moving and rapidly intensifying storms over the South China Sea and near Taiwan. Because of the monsoon regime and high sea surface temperatures over large areas, the north-west Pacific basin provides favourable conditions for tropical cyclogenesis almost throughout the year. That makes it the most active tropical cyclone basin in the world - it produces three times more hurricanes and typhoons than the Atlantic basin. Monsoon gyres, which are enormously large low pressure systems, are also quite common features, often responsible for multiple cyclone outbreaks.
Activity is better correlated with the sea surface temperature (SST), and shows a less pronounced seasonality compared to the Atlantic basin. In the Atlantic, almost 60% of tropical storms and 85% of major hurricanes are triggered by smaller-scale convective disturbances called `easterly waves'. These show a definite seasonality - mainly starting in May - and usually spawn smaller cyclones than those observed in the north-west Pacific. The size and shape of a cyclone's wind field is not only dependant on its original source, but also latitude and the cyclone's current stage of development.
Intensity also influences movement pattern, with weak systems often displaying relatively large random changes in track compared to mature typhoons. This is explained by the fact that weak atmospheric vortexes can be influenced by other vortexes, as could be produced by a strong convective outbreak within the primary circulation of the cyclone. Such interaction would not easily occur if the vorticity of the primary circulation of the cyclone was significantly higher in magnitude than that of a secondary vortex.
Simulated cyclones in the PartnerRe model are driven by matrices containing information on climatological mean values, together with the variations of the parameters to which the storm is exposed once it is generated. The track model calculates position, intensity, RMW and Rgale over the entire life cycle of each event, including formation, mature stage, and period of decay over land or open seas. Both radii serve as shape parameters for the wind field calculation. Random variations for all parameters are built in, as are any systematic differences in these parameters within and between different tropical cyclone basins. Each event is treated as an autoregressive time series.
The size and the shape of the wind field are needed to determine how far from the centre damage is likely to occur. A modified Rankine vortex equation is used to approximate the radial shape of the wind field outside the RMW at any given point in time:
(equation 1) (r): the wind speed experienced at a given distance, r, from the storm centre. : the maximum wind speed.
An exponent of x = 1 would signify the assumption of `conservation of angular momentum' for the inflowing air. This would describe solid body rotation and result in very steep wind profiles. In reality, much flatter profiles are often observed (x < 1), owing to the fact that inflowing air loses a significant amount of angular momentum on its way to the centre, due to friction and convectional processes. The wind profile can be dynamically adapted by making RMW and Rgale dependent also on the actual stage of development. On average, simulated category 2 and 3 cyclones show the steepest profiles (x ( 0.6)).
In contrast, flatter wind profiles are modelled for very low and very high intensity events. This is explained by the fact that the circulation in weak tropical storms is not, in fact, confined to the centre. The profile becomes steeper as the cyclone evolves and gains strength, and the eye wall tends to contract. On the other hand, `eye wall cycles' are a common feature of very intense hurricanes and typhoons. They generally lead to a progressive flattening profile and fluctuation in intensity for these storms, as the secondary wind maximum evolves beyond the RMW, gradually moving inward to finally replace the original eye wall. To better represent real conditions at a given location exposed to a tropical cyclone, the hazard model also takes into account asymmetries in the wind field due to forward translation, surface roughness, and sheltering effects at and after landfall.
The engineering module determines the ground-up loss accumulated for each tropical cyclone event in the event set. Construction type and coverage are used to identify the relevant vulnerability functions which, put simply, quantify the expected damage for a given wind speed at any location. This basic description hides a host of research effort into the production of these functions.
Many studies on vulnerability have been undertaken by engineers, including wind tunnel experiments, mechanically stress testing real buildings, and post-event, on-site analyses (in support of developing building codes, their purpose was to find better means of constructing safer buildings, rather than describing insured losses). Many of the widely used damage assessment tools use distributions from these studies. Quite often, however, results were based on a single event or idealised conditions that did not allow for the impact of airborne debris or `gustiness' during a storm. Critically, it is not primarily the static wind load that causes buildings to fail. The dynamic forces of gusty wind and the deconstruction of non-engineered parts, which then serve as airborne missiles, are in fact responsible for much of the damage (as seen in Hong Kong during typhoon York, with mean winds of just around 70kt).
Another crucial factor in mitigating damage is the maintenance of the buildings. Additionally, especially in Japan, Taiwan and the Philippines, the enormous amount of precipitation associated with tropical cyclones, combined with the mountainous landscape and incohesive volcanic soils, lead to faster run-off and landslides, further increasing damage levels.
Clearly it is extremely important to take all these features into account within a CAT model, and not just to assume that all losses are average. We strongly believe that it is absolutely necessary to continuously collect real insurance claim data from our clients. With such data, purely engineering-based loss distributions can be calibrated against real insurance losses, and can facilitate calculation of the whole probability distribution, providing a full description of the loss at each hazard intensity.
In order to turn the calculated ground-up losses into a risk premium, calculations within the model modify the individual loss distributions to reflect all relevant insurance and reinsurance conditions. Multiple locations, involving separate site and policy deductibles and limits, require especially sophisticated treatment. In such cases, the modified individual site loss distributions are subsequently combined to create the overall policy level loss distributions.
The appropriate policy level insurance conditions are then applied. Finally, the effects of inuring reinsurance are calculated to arrive at an estimate of the overall loss and its distribution. Different statistical functions are then fitted to the simulated losses to arrive at a pure risk price for a specific cover.
The Concentric Eyewall Cycle of Hurricane Gilbert, Mon. Wea. Rev, Vol. 120, 947-957, USA.
By Markus Aichinger
Markus Aichinger is a meteorologist with the PartnerRe CAT Research Team in Zurich.
This article is based on PartnerRe's September 2002 Publication, CAT Reinsurance Pricing - Modelling Tropical Cyclone Hazard, which is available for download at www.partnerre.com .