Open-access content Thursday 31st March 2016 — updated 5.50pm, Wednesday 29th April 2020
Edward Tredger considers whether a Bayesian framework can help actuaries manage pricing in an uncertain environment
The phenomenon of the underwriting cycle is well known, and the prevailing soft conditions are arguably the most significant current issue facing the London Market, manifesting itself in over-capacity, falling rates and a flurry of merger activity.
Understanding the nature and drivers of the underwriting cycle is of real importance to most (re)insurers. The most commonly discussed driver of this cycle is the supply and demand for capital, as investors look to bring it into the insurance market to seek out higher returns.
However, there is another cause of the cycle that relates to updating our view on the true price of risk in an uncertain environment.
This 'pricing' cycle (which will drive the well-known 'reserving' cycle as the true price of risk is recognised) can be at least as significant as the flow of capital for some lines of business.
Here I will explore this part of the cycle from a point of pricing under uncertainty and show how it can be managed quantitatively at both an individual risk and a portfolio level.
In the London Market, premium rating is often based on underwriters' expert judgment and in some cases can be highly uncertain. In many cases, there is a widespread belief among underwriters (if not actuaries) that it is impossible to produce a 'correct' price for certain risks, given the information available, if indeed such a concept is appropriate at all.
In the event of no claims, rate decreases are normally in the region of 5-10% or more. On the other hand, there has been an implicit understanding that, once a large claim occurs, the risk will renew at a significant rate increase, which will offer the insurer some 'pay-back' for the claim, somewhat mitigating the initial uncertainty surrounding the true claims cost.
To some extent, these rate increases are designed to repay an insurer for a recent claim, but also reflect an updated view of the true price for the risk, based on new information. While, of course, there is no deliberate link to Bayes' theorem, this updating of views in light of experience is exactly what Bayesian statistics attempts to formalise.
In a Bayesian framework, we could call the initial price for a risk the 'prior', which can then be updated based on claims experience, to give the 'posterior', which then forms next year's prior and so on. It should be noted that the extent of updating using new information depends on the relative information content of the prior versus the experience. For example, a brand new line of business which experiences a large influx of claims would respond much more heavily than a well-established class with close-to-expectation experience. While this point may seem obvious, the reliability of the prior is often overlooked in practice.
Although Bayes' theorem offers a mathematically precise way of changing rates in the event of claims (or no claims), we are now facing a market where repayment of claims via large rate increases is far from guaranteed. With this in mind there are good grounds to revisit the issue of understating the gap between theory and current market conditions.
An example of a 'pricing' cycle, driven purely by claims experience, is shown in Figure 1 (below).
Consider a single risk, where our initial estimate of claims frequency is 0.05 claims per year, based on a confidence level of 10 years' experience. Since we are uncertain about the true frequency of claims, we will adjust our estimate over time. New claims are then simulated with a probability of 0.05 per year -our initial estimate was in fact precisely correct but we do not KNOW it is correct, and so we will logically refine it over time. The graph demonstrates the cycle that can occur as a result of revising our estimates using a Bayesian approach, and that a pricing cycle is perfectly consistent with a market attempting to revise its judgment in light of uncertainty.
There are times where pricing is more than adequate (>0.05) and times where the rate drops more than 60% below adequacy. This approach is in fact the mathematically optimal way to revise prices year on year, assuming each risk should be rated from its own experience alone.
While a loss in this example should theoretically lead to rate rises of over 100% in some cases (early in the cycle, where less past data is available), such marked increases are increasingly difficult to achieve in a soft market, partially owing to competitors feeling able to undercut without having personally experienced the claim. However, if a portfolio of similar risks is held, a Bayesian approach would suggest sharing the claims experience across the group, resulting in less volatile rate changes at an individual risk level. This is an example of diversification at work to dampen out claims experience, the very basis for insurance itself.
It should be noted that where claims occur owing to market-wide events, which will be highly correlated across a portfolio of risks, diversification of rate changes is necessarily much more limited. These market-wide drivers then become a potential cause of the underwriting cycle, owing to changes in our estimation of risk, not purely the demand for capital. On the other hand, superfluous capital will dampen the upwards impact of claims on rate change, as was seen in the aviation market recently, but this dampening will often go beyond what is actuarially justifiable, given the need for the market to respond to new claims data.
These observations raise some interesting questions for actuaries, such as:
- Are the observed rate decreases we see in the market partially a function of uncertain pricing updating for new information, rather than a market force driven by supply and demand for capital?
- Should actuaries adopt a coherent, Bayesian, framework for updating hazard rates in light of new experience?
- Does the market introduce mis-pricing by restricting rate increases on loss-making risks?
Actuaries need to face the reality that prices are inherently uncertain and that there are consequences to this acceptance. Therefore, pricing views are in constant need of refinement, both technically and also taking into account soft information to help manage the cycle and make actuaries more commercially relevant to employers.