1. A stochastic risk margin will be based on a stochastic model capable of predicting the probability distribution of the total outstanding claims, as the 75th percentile must be estimated.
2. The amount of comfort this gives us about the risk margin will depend on how much of the outstanding claims relates to payments to be made in the next transaction period and how confident we are about the payment pattern.
3. Some of the more common stochastic models used for assessing risk margins are:
(i) Chain ladder bootstrap
(ii) Other non-parametric bootstraps based on different models
(iii) Mack’s model
(iv) Generalised linear models
(V) Adaptive generalised linear models
TESTING THE RISK MARGIN
1. Are the actuals greater than the 75th percentile about 25% of the time? A binomial distribution can be used to quantify whether the result is implausible.
2. A binomial distribution can be used to quantify whether the result is implausible. For example, the probability of getting above the 75th percentile three times out of three valuations is about 0.02 – unlikely to happen by chance, so if it does happen, it suggests that either the risk margin (or the central estimate) is too low.
3. The next simplest test requires an assumption about the form of the probability distribution for the claims in the next transaction period. Whatever assumption was made to determine the 75th percentile of the overall risk margin is likely to be appropriate. Then it is straightforward to estimate the 25th percentile, and we can ask: are the actuals between the 25th and 75th percentiles (the 50% range) about 50% of the time?
4. Particularly for long-tail classes, it would be advisable to also examine the actual versus expected by accident period to see if there are patterns in the differences and, if there are, what the cause might be. If there are no obvious patterns, it would be necessary to review the specific risks incorporated in the risk margin in detail to see whether the risk margin could be reduced.
5. If there is a sufficient number of valuations available, we can quantitatively test whether the actual values have the expected spread. For example, if it is reasonable to assume that the probability distribution is lognormal, we can test whether the log of the actual values appears to be normally distributed with the mean and standard deviation associated with the expected value and the risk margin. A chi-squared test can be performed to test whether the actual standard deviation is consistent (or not) with the expected standard deviation.
CHECKING THE MODEL
1. If the actual values do not seem to be consistent with the expected values and risk margin, the next step is to check whether the model assumptions are reasonable in the light of the most recent history of the particular business being valued. Plots of model residuals may show patterns that indicate some trends or changes have emerged.
2. Are the expected values from the stochastic model materially different to the central estimates? Although not ideal, it is not unusual for a simpler model to be used for determining the risk margin than is used for the central estimate. Any simplifications should be examined carefully to see if they are likely to materially affect the risk margin.
3. Does the amount of process variability follow the pattern implied by the stochastic model? For example, if the variability is higher than expected according to the model in the later development periods, this may cause the variability to be over-estimated in periods where the claim amounts are higher.
4. Is there systematic variation in the calendar direction that is not taken into account? Uncertainty in the future calendar trends can be a major source of risk.
5. Is the development pattern changing over time? For example, in a bootstrapping model, are there any trends in the ratios? Not allowing for changes in development pattern may produce considerable bias. Evidence of changes in the historical data may indicate a source of risk.
SHORT TAIL LINES
1. With short-tail lines, it is often the case that a large part of the outstanding claims will be paid in the next transaction period. Then it is likely that the coefficient of variation (CV) of the payments in the next transaction period will be close to the CV of the outstanding claims.
2. A Short-tail Rule of Thumb for Outstanding Claims is to use the outstanding claims risk margin (as a %) to test whether the actual amount paid in the next transaction period falls within the 50% range about half the time. Some assumption will need to be made about the probability distribution of the outstanding claims to estimate the 25th percentile. Normality or lognormality should be reasonable assumptions for most short-tail lines.
3. A Short-tail Rule of Thumb for Premium Liabilities is to use the premium liabilities risk margin (as a %) to test whether the hindsight estimate after one period falls within the 50% range about half the time.
THOUGHTS
1. Comparing those predictions with outcomes at each valuation will give regular feedback on the validity of at least some of the assumptions behind the risk margin.
2. It may be useful to look at a number of transaction periods combined, to give a better test of some of the assumptions that relate to longer term performance. In that case, care must be taken to not use the same “actuals” more than once.
3. In many cases, a subjective adjustment will be applied to a risk margin that has been calculated from a stochastic model, for example, to allow for model error. Some judgment will need to be made on how much of that adjustment should be applied to the next transaction period. For a long-tail line, much of the model uncertainty might relate to uncertainty in future inflation, which will have a negligible effect on the next transaction period.
4. Using the risk margin to draw confidence limits around expected experience in the presentation of actual versus expected results in insurance liability valuation reports can provide useful information about whether the extent of deviation is within the bounds anticipated by the previous valuation basis.
5. Where the risk margin is based on mainly on judgement or industry standards, the conclusions are less clear-cut, particularly for long-tail lines.
(Source: Institute of Actuaries of Australia)
