Property catastrophes represent a significant risk for property owners and insurers. By definition, catastrophes involve large losses and, what is even more fundamental, violate the most basic tenets of insurance – independence and diversifiability of risk.
So far at least, very few CAT events have had worldwide consequences, so that catastrophic losses should be diversifiable internationally through the reinsurance market. Currently the purchase of catastrophe reinsurance costs the Australian insurance industry several hundred million dollars a year. Most of this coverage is for earthquake, severe storm and tropical cyclone losses, and to a lesser extent, bushfire and other flood losses.
One of the primary goals of Risk Frontiers has been to improve the industry’s understanding of catastrophe risk in our region and to ensure that these risks are correctly priced. Our principal tools are simulation modelling and Geographic Information Systems. To date we have mainly concentrated on modelling single categories of CAT events impacting upon a single city or region: flood risk in the Hawkesbury-Nepean River system; earthquake risks in Sydney or Perth; a hail storm in Sydney or Brisbane or volcanoes in Auckland. Important as this information is, companies having national coverage are more interested in their whole of portfolio risk. In what follows, we briefly illustrate this process using a hypothetical residential portfolio concentrated in four make-believe cities.
Figure 1: Event losses versus annual exceedance probabilities for individual cities and the combined national portfolio (labelled “All Hazards”)
Three of these cities are subject to earthquakes(EQ). One of these is also prone to severe storms (Hail) and, in the case of the fourth, its biggest threat arises from tropical cyclones (TC). The cities are well separated so than any particular CAT event impacts upon one city only. Statistical stability is ensured by simulating event losses over a long period, typically 50,000 to 100,000 years, well beyond the 100- to 1000- year time frame of interest for calculating Probable Maximum Losses. Figure 1 shows risk as a function of the probability of exceedance in any one year – most catastrophe reinsurance contracts run for a single year. The national risk is obtained by ranking the entire list of possible losses from the individual cities and from this, calculating return periods and related annual probabilities of exceedance.
Figure 1(front page) illustrates results for individual cities and the national portfolio. For low probabilities of exceedance (long return times) the national exposure is dominated by an earthquake in City 1. This result could stem from this city’s proximity to zones of higher than average seismicity, low attenuation in the bedrock, value of the company’s residential portfolio, dominance of double brick construction and/or prevalence of soft soils, or any combination of these factors. As with some real Sydney portfolios, the curves for severe storms (labelled here as “Hail”) and earthquake cross over for City 1: at higher probabilities (shorter return periods) insured losses from severe storms exceed those from earthquakes; as the probability reduces and time scale increases, we expect earthquake losses to dominate Probable Maximum Losses.
We remind the reader that the event losses plotted in Figure 1 (and in subsequent Figures) are purely fictitious and that Probable Maximum Losses for real company portfolios may differ enormously for the same hazard and for the same city. Neither should the relative ranking of the hazards in Figure 1 be taken as a guide to their actual importance in the Australian context. In particular the relatively low losses accorded a Tropical Cyclone do not imply that cyclone risks to Brisbane or Cairns, for example, are inconsequential in relation to an earthquake in one of the other cities. In fact, recent evidence suggests the reverse and that return periods for tropical cyclones may have been over-estimated in the past (Risk Frontiers Briefing Note 25).
Figure 2 shows the same information but with the axes reversed and using a linear scale. For a given event loss we can add probabilities because these are associated with independent events. Thus the probability of exceeding a certain event loss x is approximately calculated as:
The right hand side of the equation sums the probabilities of single events losses (X) - an earthquake in City 1 or a Tropical Cyclone in City 4 or a hail storm in City 1 - that individually cause insurance losses equal to or greater than x. The equation ignores multiple events whose losses collectively sum to a value greater than or equal to x. To a first approximation, this is justified because their joint probability - calculated by multiplying together their individual probabilities - will generally be several orders of magnitude lower than the sum given in Equation (1).
Figure 2: Cumulative probability distribution of event losses.
Readers of the actuarial persuasion may recognise Figures 2 and 1 as cumulative descending probability distributions. The underlying probability density function (Prob[X = x]) can be extracted by differentiation but therein lies another story.
Reinsurance cover is often sought for the 250-year event loss. This is equivalent to an annual probability of exceedance of 0.4%. For easier extraction of Probable Maximum Losses, Figure 3 plots the same event losses shown previously against recurrence intervals (RI) for each city/hazard combination as well as for the overall national portfolio. Probable Maximum Losses are tabulated below. Both the 250- and 1000-year event losses are given to provide some guidance to how rapidly potential losses are growing as a function of RI.
There are two salient points. Firstly, we note that the national PML (last column) is not just a simple sum of the individual city risks – in fact it is quite a bit less. This difference shows the influence of spatial diversification and results from the non-intuitive manner by which probabilities combine even for these uncorrelated events.
Secondly, for a given recurrence period, reinsurance cover for the maximum likely event amongst the cities when considered individually does not provide the same level of protection for the entire national portfolio. For this hypothetical example, the 1000-year PML for an earthquake in City 1 is roughly the same as the 250-year PML for the national portfolio. This is why whole of portfolio analysis is important, and companies need to understand this or they risk buying either too much or too little cover.
Finally we note that while multiple CAT events in a single year are unlikely, reinsurance treaties actually allow for reinstatement of cover after an event. Thus the companies are buying cover for a CAT event plus the option to reinstate cover for the remainder of the period should an event occur. We will examine that question in a later issue.
losses versus recurrence interval for individual cities and the combined national
portfolio (labelled “All Hazards”). The time scale is restricted
to return periods of primary interest for insurers seeking reinsurance cover
for CAT events.
HailAUS 2.1 is a new desktop version of Risk Frontiers’ probabilistic hail loss model for Sydney and Brisbane. HailAUS 2.1 allows users to analyse their motor and household portfolios to:
Key features of
HailAUS 2.1 include:
Use of HailAUS 2.1
software is based on a 3-year licence. One free upgrade will be included in
that time and e-mail and phone help will be provided. One licence covers multiple
copies of the software within a single company. For further information or
to arrange a demonstration contact Russell Blong or Roy Leigh on 61 2 9850
9683 or firstname.lastname@example.org