Stavros Christofides explains why securitisation of catastrophe risk will become the norm over the next few years

As we get closer to the deadlines for next year's cat programme renewals, the question property underwriters are asking is: what impact the unprecedented occurrence of four hurricanes - Charley, Frances, Ivan and Jeanne - in a six week period will have on their reinsurance costs?

These costs were expected to fall, so increases to what are already high prices are likely to turn attention once again to alternatives such as cat bonds.

Securitisation of catastrophe risk has been around for almost 10 years yet only USAA, out of all the large direct property underwriters, has used this option to help manage its catastrophe exposures. The rest of this market sees securitisation as far too expensive to be worth taking seriously. Recent research now indicates that the capital markets could undercut the specialist catastrophe reinsurers significantly enough to provide the incentive that property insurers need in order to look at securitisation as a real alternative to traditional cover.

So how expensive is traditional catastrophe cover and how much room is there for reducing these costs? Fortunately, as the spreads of cat bonds are currently indistinguishable from the corresponding reinsurance rates on line (ROL), the level of disclosure required by the capital markets for the issue of securities provides us with a wealth of information on how the market is pricing catastrophe risk.

The simplest way to look at this question is to compare expected losses to the layer, as a percentage of the cover amount, against the reinsurance cost as a ROL or as a spread. In the latest issue by USAA, for example, the expected loss was given as 0.48% and the issue spread, on an annualised basis, was just over 500 basis points (or 5%). The price is then no less than 10 times the expected losses (or recoveries).

In a competitive market, such high mark-ups would soon be identified and new players would rush in forcing prices down to more reasonable levels.

This is not happening with catastrophe reinsurance, however, as the main reason for such high prices is the underlying business model rather than lack of competition. At these high levels of catastrophe cover there are simply not enough independent perils or exposures worldwide to enable reinsurers to balance their books sufficiently in order to reduce their year-on-year volatility and the consequent high levels of risk capital required to underwrite such business.

For insurers these high prices are only half the problem. A number of them are now beginning to realise just how critical the amount of catastrophe protection they buy is to their internally estimated risk capital requirements.

For example, in the UK the proposed Enhanced Capital Requirements (ECR) require companies to show the regulators results "based on a confidence level equivalent to 99.5% over a one-year period". Conveniently, most property insurers buy catastrophe cover to around the 200 year return estimate so that catastrophe exposures have little, if any, impact on the proposed ECR calculations. However, their internal models also produce estimates for higher percentiles and these will now include the impact of catastrophe events that exhaust the reinsurance cover with these excess costs adding almost pound for pound to the risk capital estimate. For example, the ECR at 1 in 500 years' level may well turn out to be twice or even three times the amount estimated at the 1 in 200 year level with most of this increase caused by the uncovered catastrophe exposures.

Once this sort of information is understood and relayed to the audit committees of these insurers, things are likely to go from bad to worse.

The only realistic option would be to buy more catastrophe cover. The problem is that such cover is either not available or only available in limited amounts and can be extremely expensive.

The only other source of such capacity has to be the capital markets and the only product that can meet such a demand has to be catastrophe bonds. For this to happen the capital markets have to see a sizeable and viable market with a product that they understand and can price. The obvious comparison is with corporate bonds, but whilst these have identical loss characteristics to cat bonds the comparisons end there.

**Reducing uncertainties**

Corporate bond defaults are triggered by two different types of events: deteriorating market conditions and company specific calamities. Failures from adverse market conditions give these defaults a cyclical look and operational failures provide a steadier stream of defaults. In pricing corporate bonds, therefore, one has to include costs for both the systematic or market risks and the non-systematic or company related risks.

Cat bonds will only default following a specified catastrophic event and such events are not caused by movements in the capital markets. So cat bonds only carry non-systematic risk and pricing them should then be a lot simpler. As there is no possibility of operational or management failure the only remaining uncertainty is that associated with the loss estimates.

The uncertainty in cat model estimates is evident in practice every time we compare results from such models. Figure 1 shows the results for UK storm exposure of a household portfolio from three models. Note also that these estimates extend to the 1000 year return period, even though catastrophe reinsurance may only be available up to 200 or 250 year return levels.

In order to be in a position to price layers of exposure we need an appropriate method and a good understanding of the inherent variability around these loss estimates. Fortunately, this challenge can be met because of the existence of two very powerful results, the first from actuarial risk pricing and the second from extreme value theory (EVT).

The first of these results concerns a pricing principle known as the Wang proportional hazards (or PH) transform. This uniquely satisfies a set of actuarial conditions including that of producing consistent prices for excess of loss layers, making it the only contender for pricing single period non-systematic risk.

In the case of high level catastrophe cover the Wang PH transform price (ROL or spread) can be turned into a simple formula involving only the expected loss (EL) of the layer as a percentage. The derived formula has just one parameter r (rho) for us to calibrate or determine in order to finally calculate the price. The formula is:

Spread = (Expected Loss) (1/r)

The derivation of this powerful formula involves two simplifications but it is still accurate to within one or two per cent. This is an insignificant difference when we recall that current pricing multiples can be as high as ten times the expected loss.

The Wang PH transform now enables us to associate a rho with each securitisation by applying it directly to the exceedence curve produced by the catastrophe model for an accurate estimation or by using the formula given above for a quicker estimate. In the USAA example, the expected loss was 0.48% and the spread 502bp, so we can deduce the implicit rho as 1.78 (5.02% = 0.48% (1/1.78)). The average spread of the 70 or so cat bond issues up to March 2003 was just over 5.5% with an average expected loss of 0.8%, indicating an average rho of 1.66. The individual rhos ranged from around 1.5 to 1.8.

These rhos are merely indices and we need to understand what they represent before we can comment on their appropriateness for setting market prices.

This is where we need to use a result from EVT known as the 'points over threshold' (or POT) theorem. In essence, what this says is that, irrespective of what the underlying loss distribution might be, there is, eventually, some threshold over which the distribution of excess losses tends to a single shape, that of the generalised pareto distribution (GPD).

This result now enables us to associate values of rho with the levels of uncertainty in the expected loss estimate for each bond. These associations will depend on the peril being studied and also the remoteness of the layer being priced. With more reliable historic data and lower (less remote) layers the uncertainty surrounding the estimated losses to the layer will be less and this should be reflected in the market price.

As an example, the estimated loss costs, adjusted for inflation and exposure changes, for the 30 costliest US hurricanes of the last century satisfy the POT conditions. The parameters of the underlying GPD can now be estimated to obtain the base distribution for this peril which in turn enables us to estimate the expected loss to a layer of reinsurance or to the equivalent cat bond.

In order to understand the variability around such a point a Monte Carlo simulation is then performed, drawing random samples of other sets of equally likely sets of events over a 100 year period for further analysis.

This can be done very easily in a spreadsheet and each new set of generated or simulated results enables us to get another estimate of losses to the layer in question. By repeating this process 10,000 times, say, we can estimate the levels of uncertainty in the base estimate for this and for any other layer of interest.

These results can now be used to associate a meaningful statistic with each value of the index parameter rho. For example, let us choose a layer with the average characteristics mentioned earlier, that is an expected loss of 0.8% and a spread of 5.5%. We simply need to find how unlikely a value of 5.5% is in the distribution of possible mean losses around the current estimate of 0.8%. It turns out, in this example, that 5.5% is found at around the 98th percentile. The implicit rho of 1.66 can now be associated with pricing at the 98th percentile. The shape of this relationship between rho and the underlying percentiles is shown in Figure 2. Note that the shape and level of these curves will depend on both the peril and the remoteness of the layer under investigation.

Whilst there is no single correct choice for such a pricing percentile (or rho) and such values will move to reflect market conditions, the approach enables to make informed choices. Selecting the 98th percentile seems penal, however, and produces prices that are five to ten times expected losses. A modest move to the 90th percentile reduces rho to around 1.3 and this indicates a spread or ROL of 2.44%, or a price reduction of over 55%. Moving down to a still conservative 75th percentile, reduces rho to 1.15 and the spread to just 1.5%, a reduction of over 70% in current spreads.

Although significantly lower than existing prices, these new spreads are still 200% and 88% respectively above expected losses and the capital markets do not demand such loadings over their average historic loss rates on their corporate bond spreads. Cat bonds priced between the 75th to the 90th percentile level could thus prove attractive to investors and, once this is appreciated and acted on, the specialist catastrophe reinsurance market stands to lose much of its business particularly at the top end of current catastrophe programmes.

For insurers there are potentially double benefits. They can look forward to significant reductions in existing catastrophe costs and can then use some of these savings to buy more protection by issuing additional cat bonds to much higher return periods. This will provide them with much needed top end risk capital protection for just a fistful of basis points.

Time is on the side of the capital markets and the insurers. Cat bonds are coming.

The paper 'Pricing of Insurance Linked Securities' presented at the 2004 ASTIN Colloquium can be downloaded from: