Ebook The Path of the Ultimate Loss Ratio Estimate
Ultimate loss ratio estimates change over time. The initial loss ratio estimate that emerges from the pricing analysis for a tranche of policies soon gives way to a new estimate as time passes and claims begin to emerge (or not). By the time all claims have been paid, the loss ratio is likely to have been re-estimated many times. The focus of this paper is on how to model the future revisions of these ultimate loss ratio estimates. We illustrate the approach using loss ratio estimates based on chain ladder and Bornhuetter-Ferguson methods underpinned by a simple stochastic model described by Hayne.
There appears to be little, if any, actuarial literature on the subject of behavior of a ultimate loss ratio estimate behveen the time when it is made and the time when its final value becomes known, i.e., the point at which all claims have been paid. Various authors have sought to address uncertainty in the ultimate loss ratio estimate, but generally from the perspective of a single point in time.
For example, Hayne proposed a lognormal model of loss development that supports the construction of confidence intervals around the ultimate loss ratio estimate . Kelly and Kreps also used a lognormal framework to explore issues of parameter estimation and parameter uncertainty, respectively. Hodes, Feldblum and Blumsohn used a slightly different lognormal development model to quantify the uncertainty in workers compensation reserves. Mack, Venter and Zehnwirth have all written extensively about stochastic modeling of the loss development process:. Others, including Van Kampen, Wacek and the American Academy of Actuaries Property & Casualty Risk-Based Capital Task Force, have sought to quantify the uncertainty in the ultimate loss ratio estimate used in pricing and reserving applications direcdy, without reference to the loss development process. The question on which all of these authors focused their attention is the potential variation in the final loss ratio at ultimate compared to the current ultimate loss ratio estimate, with no reference to how the ultimate loss ratio estimate might vary at intermediate points in time.
In contrast, in his acclaimed paper on solvemy measurement Butsic observed that loss estimates change in their march through time. He recognized that they, like stock prices, are governed by a diffusion process, a type of continuous stochastic process with a time-dependent probability structure. However, he did not propose a model of this stochastic process.
How ultimate loss ratio estimates change in the future depends in part on the method used to make the estimates. In this paper we assume that loss ratio estimates are derived from a consistently applied estimation process with minimal subjective overriding of the indicated result. We model the behavior of loss ratio estimates using stochastic versions of two loss development methods: the chain ladder method and the Bornhuetter-Ferguson method, both using paid development data. To model chain ladder estimates, we combine Hayne's and Butsic's ideas to synthesize a lognormal diffusion model for the path of the ultimate loss ratio. Then we adapt that model to the Bornhuetter-Ferguson method.
This conceptual framework, which could easily be adapted to handle other loss development models, provides actuaries with the means to give their clients more information about how much their loss ratio or reserve estimates may fluctuate from period to period. As such, it can be a useful tool for managing expectations about the variability of loss reserve estimates. It also has potential application in a number of other areas of actuarial analysis, as we will discuss later.
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