Variation in estimates of Ultimates and Solvency II

In the presentation, "Integrating Reserve Risk Models into Economic Capital Models" by François Morin of Towers Perrin, the Solvency II one-year risk horizon is measured as the sample variance of the difference between estimates of ultimates today and one year from now. François Morin of Towers Perrin provides several methods of calculating the variance of the change in estimate.

"Under Solvency II the one-year risk horizon is defined as the change in the estimate, one year hence, generally measured as the sample variance of the difference between the deterministic ultimate's one year from now and the deterministic ultimate's today." (see page 20
http://www.casact.org/education/clrs/2008/handouts/erm2-1.pdf)

We discuss the consistent and sound statistical methodology for calculating the variation in estimates of ultimates conditional on the next calendar years' data, the relationship between these estimates, consistent estimates of prior year ultimates on updating (next valuation period), and loss reserve increases on updating.

Calculating conditional statistics and variation in ultimates by using the Mack and related methods including Bootstrapping are inherently flawed. For a more detailed discussion of the paper by François Morin of Towers Perrin please click here.

Reserve increases from year to year

Consider a company that writes the same mix of risks each year with the same exposure level each accident year. Suppose calendar year inflation in the paid losses is stable and is 5% every year.

For reserves to be computed accurately the company should be assuming 5% inflation going forward at each valuation period.

In this scenario, the following applies:

• Each year the company needs to increase its total reserve by at least 5%.
• Each year the company needs to increase its premium (price) by at least 5%.
• The ultimates for prior accident years will remain consistent with each increase in total reserves.
• Ultimates increase by at least 5% from one accident year to the next.

The principal reason for these facts is that calendar year inflation projects both onto the accident years and development years - it also impacts all prior accident years.

The increases in total reserves each year should not be regarded as an upgrade. If the costs in running the business increase by 5% each year then the company should at least increase the price of the product by 5% to stay even. If the total reserves are not increased by at least 5% each year the company will be under reserved!

Estimates of prior ultimates remain consistent on updating from year to year only if the forecast assumptions remain consistent. For further information see this article.

Consistent estimates of prior year ultimates on updating

Consistency of prior (accident) year ultimates on updating is related to the discussion in the previous section on loss reserve increases on updating. Ultimates remain consistent from year to year if forecasting assumptions remain consistent. For example, if a forecast scenario assumes a calendar trend of 10% +- 2% for next year followed by 5% +- 1% thereafter, then, for an accident year ultimate to be consistent from this year to next year requires that the next year's data fall within the 10% +- 2% trend line and that the subsequent calendar trend at this year end continue to be set to 5% +- 1%.

It is only in the PTF modelling framework and the MPTF modelling framework that calendar year trend assumptions are explicit, transparent and can be monitored on updating.

Consistency of Solvency II Risk measures

Most importantly Solvency II capital risk measures and MVM calculations remain consistent on updating only in the PTF and MPTF modelling frameworks. In these probabilistic frameworks (transparent) forecast assumptions remain consistent on updating and trend assumptions are monitored.

For example, a PTF model was applied to a WCom dataset in 1986 and udpated in 1987. The updated model remained consistent in the parameters (the 1987 data followed the projections made in 1986), and the subsequent forecasts maintained the same forecast assumptions.

The calendar year parameters were:

The sample statistics are consistent (the mean increases by around 10% which is consistent with the calendar year trend).

The calculation of risk capital is also consistent between years; note that the VaR_99.5 has decreased slightly in 1988 due to the reduction in parameter uncertainty - we have a new year's worth of data.

Calculating variation in ultimates conditional on next calendar years' data (Conditional Statistics)

Let Ult. denote the Ultimate and CY represent the next calendar year's paid losses.

Step 1:	Using the model obtain the expectation of the ultimate E[Ult.] and variance of ultimate Var[Ult.],
Step 2:	Using the model simulate the next CY's losses.
Step 3:	Re-estimate the model using the array with the additional CY's data and compute accident year ultimates.
Step 4:	Repeat Step 2 and Step 3 N times.

For each simulation Step 3 computes E[Ult.|CY]] and Var[Ult.|CY]] using the model on the updated array.

The N simulations give us estimates of E[E[Ult.|CY]], E[Var[Ult.|CY]] and Var[E[Ult.|CY]].

Provided forecast assumptions (based on the model) are consistent on updating (see previous sections), we also have

Var[Ult.]	=	E[Var[Ult.|CY]] + Var[E[Ult.|CY]], and
E[E[Ult.|CY]]	=	E[Ult.],

François Morin of Towers Perrin mentions "the change in the estimate one year hence" which is equivalent to:

Var[E[Ult.|CY]]

=

Var[Ult.]- E[Var[Ult.|CY]].

In respect of the identified models in the PTF and MPTF modelling frameworks there is no need to conduct simulations as the above mentioned formulae can be computed analytically using the identified parametric models. The statistics include both process variability and parameter uncertainty.

We include two Forecast Summary tables from ICRFS-Plus, one from the aggregate of multiple long tail LOBs and the other Private Passenger Automobile (PPA).

+-Ult|Data refers to SD[E[Ult.|CY]] and Std.Dev|Data to Sqrt(E[Var[Ult.|CY]]).

As mentioned above, consistency depends on whether the predicted forecast assumptions for the next year and following a) occur for the next year and b) that the same forecasting assumptions used previously for the following years are applied in the subsequent forecast. Any change in assumptions will make prior year ultimates on updating inconsistent.

Variation in Ultimates and the associated conditional statistics do not provide sufficient descriptors of the risk margin or cost of capital as required by Solvency II. Cost of capital and risk margins are discussed further here. Excess capital requirements if the first calendar year is in distress is discussed here.

See here for a demonstration video illustrating Solvency II metrics and consistency on updating.