Variation in Ultimates based on the paper by François Morin of Towers Perrin

In the presentation, "Integrating Reserve Risk Models into Economic Capital Models" by François Morin of Towers Perrin, the use of variation in ultimates is introduced to provide a one-year risk horizon measure required by Solvency II. Specifically:

"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)

The paper provides several methods for calculating "the change in the estimate, one year hence".

We have already mentioned here that these methods are unsound, inconsistent on updating, and provide false indications of the change in estimates of the ultimates.

Deterministic link ratio methods have nothing to do with the volatility in the actual data even if they can be also formulated as stochastic methods. For example, the Mack method is the regression (stochastic) formulation of volume weighted average (Chain Ladder) link ratios.

The Mack, Murphy, related methods, and associated Bootstrapping

One approach proposed in the paper involves the Mack, Murphy and related methods, and associated Bootstrapping.

According to Francois Morin:

"Bootstrapping utilizes the sampling-with-replacement technique on the residuals of the historical data",

and

"Each simulated sampling scenario produces a new realization of "triangular data" that has the same statistical characteristics as the actual data." (Emphasis added)

This is worthwhile repeating.

"...that has the same statistical characteristics as the actual data."

Indeed, the above statement is of paramount importance! Bootstrap samples have the same statistical characteristics as the actual data

only

if the model has the same statistical characteristics as the actual (real) data.

The Mack method, the Murphy method and related methods produce Bootstrap (pseudo) samples that have statistical characteristics that are not even remotely related to the actual data because they have nothing to do with the true volatility in the actual data.

The following links demonstrate, inter alia, that the Bootstrap technique (even when used correctly) does not measure variability if the bootstrap samples do not have "the same statistical characteristics as the actual data."

It is also of utmost importance to examine the weighted standardized residuals not only in the development period direction but also in the accident period and calendar period directions.

View a video series explaining the bootstrap technique here, an article detailing the bootstrap, and advanced statistical treatment of the bootstrap here.

Moreover, the one year risk horizon measure (a la François Morrin) if based on the Mack, Murphy or related link ratio methods will not produce consistent estimates of prior year ultimates on updating for most real (actual) loss development arrays. This is due to the fact that forecasting assumptions are not consistent on updating. See this page for more details.

The Solvency Capital Requirement over a one year horizon is the "Value-at-Risk of the basic own funds of an insurance or reinsurance undertaking subject to a confidence level of 99.5% over a one-year period" (CEIOPS-CP-71-09). The Solvency Capital Requirement for the Ultimate (99.5% quantile for the total liability distribution) encompasses all liability stream periods. Both of these calculations require a detailed analysis of the tail end of the liability distributions. The variation in the change in ultimate does not provide analysis of the tail of the liability distribution and alternative calculations should be pursued.

Risk margin and cost of capital

Variation in the change in the ultimate does not adequately encapsulate the risk horizon - the risk margin must also be included. As seen in the pages discussing the calculation of Market Value Margin, the distribution of the data, which is unique to each company, is the most important factor when controlling risk. The distribution of a forecast statistic does not provide a good measure of risk. In order to determine an appropriate risk margin which is completely relatable to the business the distribution of the liability stream must be considered.

In ICRFS-Plus™distribution of the total liabilities is most important. Summary statistics for the entire distribution(s) are computed - mean, standard deviation, with other quantile statistics being available after simulation. The quantile statistics provide the means of determining the risk margin at a level of risk that the company requires. By contrast, the conditional statistics from deterministic methods, do not add any additional information to the original estimations in terms of understanding risk; especially because distributional assumptions about the estimates have to be made which are not linked to the data.

In order to calculate a one-year horizon statistic, the distributions by calendar year are required since it is necessarily to know what the distribution of the liabilities are for the next year. In the case of multiple lines of business, the distributions of the aggregates of the liabilities by calendar year are required. Only probabilistic models which fit distributions to the data are capable of producing these figures.

Since the fair value of liabilities is applicable to the aggregate of multiple lines of business, forecasts are necessary for each line of business and the aggregate of each line. ICRFS-Plus™ calculates the liability stream for multiple lines of business and the aggregate of the lines providing all the distributions required in order to calculate mean and appropriate risk margin in order to meet solvency II requirements. An example of such tables is provided below with the aggregate of 15 lines along with an of the individual line:

Some examples of Market Value Margin calculations where a model is fitted to multiple lines of business and the correlations between them, are available here.