As longtime readers of this column will recall, I have been exploring the application of Bayesian Markov chain Monte Carlo (MCMC) models to stochastic loss reserving. With the publication of my monograph on the subject,1 I have turned my attention to applications of these models.
In that monograph I focused on predicting the outcomes after the 10 years of development that we could get from successive Schedule Ps. While that timeframe is good for testing models, operationally it is of limited use. Insurance executives and regulators have a much shorter timeframe. We see evidence of that concern in the one-year time horizon specified by Solvency II for calculating the amount of capital to support what we Americans call the loss reserve. One rationale for the one-year timeframe that I heard was that insurers have the opportunity to make adjustments as time passes. One such adjustment could be to add or release capital supporting the reserve. If the amount of capital that might be added down the line is not too burdensome, I contend that the one-year time horizon is a good idea. The purpose in this column is to describe how to check that out using a Bayesian MCMC stochastic loss reserve model with the Schedule P data that is in the CAS Loss Reserve Database.
Given a Schedule P triangle and the assumptions underlying the model, a Bayesian MCMC model produces (say) 10,000 equally likely scenarios consisting of loss outcomes and underlying parameters. One can use these scenarios to represent the possible future developments of the loss triangle.
What gets posted at the end of one year is not the ultimate loss. Instead, one posts the estimate of the ultimate loss given one more year of data. One way to get the distribution of these estimates is to first simulate the next year of losses and then run the MCMC model with that extra year of data to get the loss estimate. Repeating this exercise 10,000 times gives the distribution of loss estimates. Given that it takes a minute or so to get a single estimate, getting 10,000 estimates would take about a week to run on my late-model laptop.
Instead, I implemented a much faster approach. I first simulate the next year of losses. I then reweight the 10,000 equally likely scenarios with Bayes’ theorem and calculate the expected value of the outcome as my ultimate loss estimate. Repeating this calculation 10,000 times, which takes about three and a half minutes on my laptop, gives us a predictive distribution of ultimate loss estimates.2
Let’s look at an example. I ran 10,000 future scenarios with the correlated chain ladder (CCL) model for the incurred data of the illustrative insurer in my monograph (and some of my prior columns) sequentially with one more, then two more, etc., and finally nine more years of data. Figure 1 shows the paths taken by the ultimate loss estimates for a sample of these scenarios. One can see that the paths level off as we include successive years of data in the estimation.
Adopting the Solvency II risk-based capital standard of the 99.5% Value at Risk for a one-year time horizon, I then calculated the capital requirement for each successive calendar year, given the result of the previous calendar years. Consistent with the leveling off that we see in Figure 1, Figure 2 shows that the required capital usually, but not always, decreases as we include successive calendar years of data in the estimation. This makes sense because we expect more precise estimates with the additional data in successive calendar years.
When the required capital (after consideration of the change in the expected loss) decreases, the insurer can release some of the capital that supports the liability. Similarly, when the required capital increases, the insurer will have to add capital.
If the required capital always decreases as we include successive calendar years of data, there would be no problem with a one-year time horizon. But since it increases sometimes, we have to consider how often this happens, and if it does, how much capital the insurer must add.
At the risk of being overly simplistic, I set an upper bound to the amount of capital that must be added to be equal to the 51st lowest increase in capital that arises in our 10,000 simulations. This is one way to recognize that insurers may choose to leave the business if they incur large losses. It turned out for the illustrative insurer that 373 of the 10,000 simulations resulted in an addition to capital. The upper bound was 51.6% of the initial capital.
I did the same calculation for all 200 loss triangles in my monograph. It turned out that the illustrative insurer was one of the nicer examples. Figures 3 and 4 show sample paths for the estimates and the capital requirements in each successive calendar year for a different loss triangle. Additions to capital occurred in 4,672 of the 10,000 simulations and the upper bound of the capital additions was 106.8% of the initial capital.
This was by no means the worst example. The upper bound of the capital addition was over 100% of the initial capital for 88 of the 200 loss triangles.
While some discussion is in order, these examples suggest that the one-year time horizon may not be adequate for going-concern insurers. In addition, it may not be best for insurance company owners. One interesting presentation at last November’s CAS centennial meeting was given by Daniel Bauer and George Zanjani.3 They discussed their work on their research project, “Allocation of Costs of Holding Capital,” which is being sponsored by the CAS Committee on the Theory of Risk. One of the points they made was that the cost of capital under “stressed” circumstances will be higher than the cost of capital under ordinary circumstances. I am still thinking about the implications of that. Stay tuned.
If the 99.5% VaR capital requirement actually goes into effect, this analysis suggests that some insurers should voluntarily carry some “excess” capital. They can get their capital now at a competitive cost, or run the risk of paying dearly for it later.
1 The first CAS Monograph was posted to the CAS website in January 2015.
2 I plan to provide the details of this calculation in a chapter of the book Predictive Modeling Applications in Actuarial Science – Volume II, which is scheduled to be released this fall.
3 https://cas.confex.com/cas/annual14/webprogram/Session7443.html
