Cost of Capital Risk Margins with a One-Year Time Horizon
As we go about building our stochastic loss reserve models and calculating the variability of results implied by these models, we find that there are some different opinions on what to do with the results. There have been some who argue that we should post a range of results. Others, including myself, argue that we should post a single number. This single number should include a risk margin that increases as the uncertainty in the final outcome increases. One such formula that has this property is given in the proposed Solvency II requirements:
where Ct is the capital required at time t, r is the rate of return for a risky asset, and i is the rate of return for a “safe” asset. In the not-too-distant past, I was of the opinion that the Cts could be routinely calculated with one’s favorite stochastic loss reserve model. But at one of the roundtable discussions at the September 2011 CLRS, Ben Zehnwirth convinced me that this opinion was incorrect.
Early on in the International Actuarial Association’s deliberations on the time horizon, a rationale given for the one-year time horizon was that if a “stress event” occurred, an insurer would have time to recover by raising additional capital. The evolving insight that Ben relayed to me at the CLRS was that the risk margin should be sufficient to attract sufficient capital to support the liability after a stress event has occurred. One can calculate C0 with a stochastic loss reserve model. If a stress event occurs in the first year, we can use that information to recalibrate our stochastic loss reserve model, and then calculate C1. The problem is that we have to calculate C1 (and subsequent Cts) before the stress event actually occurs!
Here is one solution to this problem. I will first show this solution for an artificial, but simple, example. Next I will argue that a more realistic solution can be obtained with the right kind of stochastic model.
For the simple example, let’s consider a stochastic loss reserve model with two equally weighted scenarios, each consisting of a two-period runoff. The scenarios, representing the mean of an exponential distribution of reserve outcomes, are given in Table 1.
The initial reserve for this model will be the weighted mean of the two-year payout, 12 = 0.5•(10 + 5) + 0.5•(6 + 3). (To keep things simple I am assuming that the “safe” interest rate, i, is zero.)
Let’s define a “stress event” for each year as a loss equal to the 99.5th percentile of the equally weighted mixed exponential distribution. The result of this calculation is given in third column of Table 2.
For a one-year time horizon, the capital required would be the stress event minus the expected payout in the first year, i.e., 38.49. Prior to September’s CLRS, I would have calculated the risk margin by calculating C0 and C1 and applying the above risk margin formula, with r = 6%, to get a risk margin of 3.46.
Now let’s suppose the stress event happens in the first year. From Table 3, we can see that the stress event is far more likely to come from Scenario 1 than Scenario 2. Using Bayes’ theorem, we can calculate the updated weights, given in the third column of Table 3.
If we want the risk margin to be sufficient to attract investors to support the risky reserve in the second year after a stress event in the first year, I suggest that we should use the weights in Table 3 to calculate the reserve and the capital, C1, for the second year. The results of this reweighting are in Table 4.
Note that the reweighting increases both the capital and the reserve for the second year. The investor who takes on the risky reserve will need both the cost of the updated capital and the increase in the reserve. So I contend that the risk margin should be the sum indicated by the risk margin formula above and the reserve increase. In our example, this risk margin is 4.45.
While I hope the above example simply illustrates the ideas underlying my suggested approach, I will now argue that we can do a similar calculation for real loss reserves. What one needs to do such a calculation is a stochastic model that has a list of scenarios that describe the parameter risk in these models, and also provides the distribution of outcomes for each scenario. It turns out that my paper, “Stochastic Loss Reserving with the Collective Risk Model,” which appears in the 2009 Variance (Volume 3, Issue 2), is such a model. This paper provides two models with 1,000 scenarios generated by a Bayesian MCMC method, with the distribution of the outcomes described by a Tweedie distribution. Table 5 gives the result of my “pre-CLRS” calculation of the risk margin using a model described in that paper. The calculations of the table entries involve some heavy math, so I will not describe them here. I will share them with anybody who is interested, however.
Applying the above risk margin formula with i = 0% and r = 6% to this table, the risk margin is 6% of the sum of the Cts.
Table 6 gives the results of my suggested risk margin calculation reflecting the goal of attracting investors after a stress event. The calculation is complicated by the possibility that after the first year, there may be a stress event in the second year, and in turn there may be a stress event in the third year, and so on. The calculation involves successively updating the scenario weights for each year. In the table, “Total Reserve(1)” is the reserve calculated going into the year, and “Total Reserve(2)” is the updated reserve calculated with the updated weights after a stress event in that year. Other than that, the calculations are similar to those in the simple example above.
It is interesting to note that the indicated risk margin for the second method is almost triple that indicated by the first method, with most of the increase coming from the reserve increase resulting from the stress events.
Hopefully these examples will provoke some discussion about the purpose of reserve risk margins.
Update to November’s Column
While I was writing this quarter’s column, I got a note from Clive Keatinge, who questioned my calculations in Table 1 in last quarter’s column (“Economic Scenario Generators and Correlation,” AR, November 2011). Having worked with Clive in the past, I should not have been surprised to see that he was right. The correct Table 1 is below.
Clive rightfully points out that with the correct calculation, my blanket conclusion about the magnitude of the Solvency II correlations should be called into question. While I still suspect my conclusion is appropriate for most insurers, it is by no means as obvious as I indicated in that column. It depends on the values of the CV and Beta. While I do make frequent mistakes, I am often able to catch them when the calculations give nonintuitive results. Erroneous calculations that agree with one’s intuition are harder to catch. Nobody is immune.