Bonds and other instruments with payments linked to explicit events are growing in popularity among investors and intermediaries.
Of particular interest to market participants is the pricing of catastrophe event and weather risks. Financial economists find these risks to be unique because their actuarial properties can be objectively modelled with considerable sophistication. The risks are of great interest to practitioners because their prices have fallen dramatically over the past few years, as traditional reinsurance markets have become more competitive. The appropriate level for these prices going forward is a matter of considerable debate among practitioners and academics.

Catastrophe risks need to be priced at a rate that covers the chance of loss, expenses and a profit factor for assuming the risk. If a catastrophic event has a probability of 1%, then the risk will be priced higher than 1%. But how much higher?

Some analysts would argue that the risk premium should be quite small. Cat and weather risks tend to be uncorrelated with the returns on diversified financial portfolios. In addition, each risk is small relative to the risks in financial markets. In most equilibrium models this would imply that event risks should yield an unbiased actuarial estimate of expected loss, implying a profit factor close to zero. This statement arises from well known concepts in financial theory. However, the theory applies only in markets that have reached an equilibrium so that risks are widely shared - a state that cat and weather markets most certainly have not reached. But many observers argue that event risks have not been, and should not be, priced at actuarial levels, even in equilibrium. Many practitioners expect the risks to provide a yield considerably greater than the actuarially expected loss.

There is little agreement, however, on the determinants of this premium spread. For cat-linked instruments, the premium is most commonly determined as a fixed constant times the volatility (or variance) of loss. More sophisticated users add to the premium based on the skew-ness of the loss profile. Others argue that the expected loss conditional on an event effectively explains cat-linked yields. Yet another group, including Warren Buffett argues that the uncertainty associated with actuarial probabilities is an important explanatory factor for high event yields.

Our paper, Issues in the Pricing of Catastrophic Risk, addresses this argument. It begins by asking what the pricing of event bonds ought to be in an equilibrium where actuarial probabilities are uncertain. We first show that uncertainty in actuarial probabilities does not affect the pricing of the event instruments as long as the actuarial probabilities are unbiased and are uncorrelated with the event itself.

Even if the uncertainty about probabilities is correlated with the event outcomes - a relatively unlikely state of affairs - we find that the uncertainty about event probabilities is insufficient to explain the level of cat-event spreads. If the uncertainty about event probabilities is correlated across a number of different events, there is a detectable, though generally small, upward influence on spread. As a result, we argue that parameter uncertainty about the probability of events is unlikely to explain why event spreads are high.

Of course, we cannot rule out the hypothesis that event-risk spreads are high not because of uncertainty per se, but because participants believe that the true average probabilities of event occurrence are higher than those put forth by third-party modellers. In other words, the market believes that the typically measured event probabilities are downward biased. While this hypothesis is possible, it strikes us as unlikely to be true: it would be extraordinary to claim that unsophisticated “gut” views of event probabilities are able to avoid the biases that plague sophisticated models that use historical data, simulation and other techniques. Thus, detectable bias is unlikely to be the reason that event-risk yields are so far in excess of their associated event probabilities.

Why then have yields exceeded actuarially fair levels? For catastrophe event risks in particular, we argue that risk pricing is currently determined by reinsurers. This is inefficient, since reinsurers are equally as informed as modelling firms yet they hold large risk positions (positions that are not diversified, yet benchmarked against absolute returns rather than against the performance of aggregated event risks). As the market learns that it can profitably underbid reinsurers for these risks, future spreads are likely to decline.

  • Professor Kenneth Froot is the Andre R. Jakurski professor of business administration Harvard Business School and special consultant to Marsh McLennan Securities Corporation. This article is excerpted from an in-depth report of the same name prepared by Marsh & McLennan Securities Corporation and published by Guy Carpenter & Company, Inc., in May 2000. The full report is available at