Excess risk capital requirement conditional on the first calendar year being in distress
The basic idea here is to examine the impact on future calendar years' paid losses (reserve) means, given the first (next) calendar year falls into a particular percentile - e.g.: 99.5%. That is, conditional on the first calendar year losses L(1) being in a distress situation.
Excess risk capital (ERC) is defined as the Value at Risk in the first calendar year (VaR(1)) plus the sum of differences in expected losses minus the mean (unconditional) losses in subsequent calendar years conditional on the first calendar being in distress,
ERC = VaR(1) + Σ{ E[ (L(k) - M(k)) | L(1) in distress ] }, k > 1
where
k - future calendar period,
L(k) - random variable representing total paid loss in the future calendar period k,
M(k) - mean paid loss for future calendar period k.
For a two year run-off
If the first year is in distress, then BEL(1) and VaR99.5(1) are used, ΔBEL(2) is additional the capital required in order to pay out the mean losses in the second calendar year given all risk capital was used in the first year.
For the general case, ΔBEL(k) is computed for each calendar year k . The sum of the ΔBEL(k) and the VaR(1) is the 'excess risk capital'.
ERC represents the risk capital requirement for the first calendar year in distress plus the excess mean capital above the (unconditional) mean for subsequent years. That is, the first calendar year's contribution is risk capital and the subsequent years contributions are the additional mean reserves (conditional on the first calendar year being in distress).
Value at Risk can be replaced by Tail Value at Risk (T-VaR) as an alternative measure.
A single composite model forecasts lognormal distributions and their correlations for each cell for each LOB. By simulating from the correlated lognormals, we can extract information about the samples for which the first calendar year values are in distress. For example, they lie in the 99.5-99.6 percentile.
The first calendar year being in distress could be due to parameter uncertainty, process variability, or both. For the former, the realized parameter is significantly higher than its expected value resulting in all observations being significantly higher than their expected; in the case of process variability only one or two observations would be significantly higher than expected.
The differences between ERC, the VaR for the aggregate reserve distribution VaR(aggregate), and VaR(1) the VaR for the first calendar year is driven by the relationship between parameter uncertainty and process variability.
We have VaR(1) <= ERC <= VaR(aggregate).
If parameter uncertainty is high relative to process variability, then calendar year paid losses distributions are highly correlated (first year in distress implies other years are also in distress) and ERC tends to VaR(aggregate). On the other hand, if parameter uncertainty is zero, the calendar year paid loss distributions are uncorrelated (first year in distress has no impact on the losses in subsequent years) resulting in ERC equal to VaR(1). See the detailed PDF document.
The example below illustrates those facts (at 99.5%),
| Mean Loss | Std Dev of Reserve (or Loss) Distribution |
VaR(1) | ERC | VaR(Aggregate) | |
| Both parameter uncertainty and process variance: |
$1,473M | $169M | $120M | $176M | $496M |
| No parameter uncertainties: | $1,462M | $110M | $111M | $111M | $325M |
| No process variance: | $1,254M | $107M | $26M | $270M | $311M |
Thus, in No parameters uncertainties forecast scenario the ERC $111M is equal to VaR(1) $111M; in No process variance forecast scenario the ERC $270M is close to VaR(aggregate) $311M.
A macro for ICRFS-Plus™ users is available here. This macro operates on a single composite model for multiple LOBs and associated forecast scenarios.
Consistent estimates of ERC on updating
In the section on Variation in Estimates of Ultimates conditions for consistency of prior (accident) year ultimates on updating are discussed. It is mentioned that only models and forecast assumptions based on the PTF and MPTF modelling frameworks achieve the necessary conditions of consistency on updating.
Under the same conditions ERC calculations are also consistent on updating from year to year. For example, if a forecast scenario assumes a calendar trend of 10% +- 2% for next calendar year followed by 5% +- 1% thereafter, then, ERC is consistent on updating one year hence provided the next year's observed paid losses fall on the assumed 10% +- 2% trend line and the subsequent calendar trend is set to (assumed to be) 5% +- 1%.
Note that for a long tail liability LOB the parameter uncertainty reduces on updating as the model is re-estimated with more data.
See also a discussion on Solvency II, variation in ultimates, variation in ultimates according to François Morin, of Towers Perrin, and MVM and Cost of Capital.
