Solvency II Economic Balance Sheet: Risk Capital (SCR), One-Year and Ultimate Year Risk Horizons, Technical Provisions, Market Value Margin (MVM) and Cost of Capital (CoC) approach

Economic Balance Sheet showing the components of the liability side

An Economic Balance Sheet is a balance sheet with risk margins.

In order to compute, Technical Provisions (TP), MVMs using the Cost of Capital approach and SCR for both one-year risk horizon and ultimate year risk horizon for the aggregate of all long tail LOBs and each LOB, the following critical information is required:

• Probability distributions of paid losses (liability stream) by calendar year (i=1,..,n) and their correlations, for each LOB and the aggregate of all LOBs.
• Probability distributions of total reserves for each LOB and the aggregate of all LOBs.
• Probability distributions of the aggregate paid losses from calendar year i to calendar year n for each LOB and the aggregate of all LOBs. This is required for each i ranging from 1 to n, where complete run-off is achieved at the ultimate calendar year n.
• Conditional probability distributions conditional on the first (next) calendar year being in "distress".

Armed with the above mentioned distributions any risk measure can be computed for each LOB and the aggregate of all LOBs including:

• VaR(i) for the paid losses (total loss) in calendar year i,
• VaR*(i) for the aggregate paid losses from calendar year i to N, and
• VaR(i |"first year in distress"); the VaR of the distribution of the loss in calendar year i conditional on the first year being in distress for i>2.

Insurance liabilities fair value is defined as a sum of the Best Estimate of Liabilities (BEL) and MVM, where MVM is the (present value) cost of holding (raising) risk capital. This quantity is also referred as technical provision.

For the one-year risk horizon calculations conditional statistics, BEL(i| first year in distress), VaR(i| first year in distress) and MVM(i | first year in distress) for i=2,..,N are also needed in order to compute the TP (see below) and therefore the total SCR. We are conditioning on the first year in distress.

The Solvency Capital Requirement (SCR) is the risk capital needed to cover non-hedgeable (that is, insurance) risk. It is defined from VaR or T-VaR for a particular quantile (e.g. 99.5%) of the predictive aggregate loss distribution. In ICRFS-Plus™ this can be estimated for a single LOB, and the aggregate of multiple LOBs. The cost of holding the SCR is assumed to attract a premium over the risk-free interest rate which is called the Nominal Cost of Capital.

ICRFS-Plus™ is the only product in the world that satisfies these critical requirements with a (single) composite model in a sound and transparent statistical modelling framework that is also consistent on updating.

A single composite model identified (designed) for multiple LOBs describes the trend structure and process variability for each LOB and the correlations between the LOBs. There is one model for the whole company.

With the one-year risk horizon, risk capital RC(i) is raised at the beginning of calendar year i (for i=1,...,n) and released at the end of that calendar year to cover adverse development in that year up to a certain (e.g. 99.5%) quantile of the predictive loss distribution for that year. The amount of risk capital released back to the capital providers at the end of year i depends on RC(i), the mean loss for calendar year i, and the actual loss for calendar year i.

With the ultimate risk horizon, risk capital is sufficient to cover adverse development up to a certain (e.g. 99.5%) quantile in the predictive aggregate loss (reserve) distribution for the whole run-off period. The entire risk capital, that is, Value at Risk for the aggregate reserve (VaR(aggregate) = VaR*(1)) is raised at inception and released back to the capital providers at the end of each calendar year. At the beginning of calendar year i (for i>2) the amount of risk capital retained for the remaining run-off period is VaR*(i). The amount of risk capital released back to the capital providers at the end of calendar year i (for i=1,...,n) depends on the total loss in year i, the mean loss for year i, risk capital VaR*(i) (retained at the beginning of year i), and risk capital VaR*(i+1) (retained at the beginning of year i+1).

Probability distributions, VaRs and T-VaRs

ICRFS-Plus™ incorporates sound statistical modelling frameworks for forecasting the reserve distributions of individual LOBs and the aggregate of all long tail LOBs. Probability distributions by calendar year, accident year and total are obtained for each LOB and the aggregate of all LOBs. The distributions include both process variability and parameter uncertainty. Aggregate distributions across all LOBs incorporate the two types of correlations, namely, parameter and process correlations. Models are transparent and forecasting assumptions are explicit, and can be related to past experience - both are key requirements of Solvency II. Each model for a LOB is represented by four graphs that are easily interpretable.

For example, an extract from a forecast table for the aggregate of 15 lines of business is shown below:

Although we show values for the aggregate of all LOBs, the black and red values in each LOB tab correspond to the mean and standard deviations of the predicted lognormal for that cell. To find the distributions in the cells of the aggregate table displayed above with means and standard deviations calculated, and the distributions of aggregates by calendar year and accident year for each LOB and the aggregate of all LOBs we simulate from the corresponding lognormal distributions in each cell in each LOB (and their correlations).

Additional benefits are percentiles, and VaR and T-VaR tables for each LOB and for the aggregate of all LOBs. Iinteractive graphs for capital allocation by LOB and calendar year are also provided. See video Chapter 4 on Capital Management of multiple LOBs.

The distributions are based on probabilistic models that describe the volatility in the past data. Assumptions for the future are explicit and can be related to past experience.

The Probabilistic Trend Family (PTF) modelling framework is used to design a model for a LOB that describes the trend structure and volatility about the trend structure in the data. The Multiple Probabilistic Trend Family (MPTF) modelling framework is used to design a composite model for the aggregate of all LOBs that in addition to the volatility in each LOB describes the two types of correlations (parameter and process) between them.

In the PTF modelling framework, the Predictive Aggregate Loss Distribution (PALD) module computes probability distributions of reserves (paid losses) by calendar year, accident year, and the aggregate for a LOB.

The PALD module for a single composite MPTF based model calculates the probability distributions of reserves by calendar year, accident year and total for the aggregate of all LOBs.

Iinteractive graphs for capital allocation by LOB and calendar year are also provided. See video Chapter 4 on Capital Management of multiple LOBs.

The calendar year paid loss distributions is also know as the distributions of the liability stream.

The PALD module output also includes percentiles, VaRs and T-VaRs for each calendar year, accident year and total. For a composite model interactive capital allocation graphs by LOB for specified VaRs or T-VaRs are also given.

A risk measure of Excess capital conditional on the first (next) calendar year being in "distress" is discussed here.

A related article on Variation in Estimates of Ultimates (conditional on next years' data) for a one-year risk horizon is available at Variation in estimates of Ultimates and Solvency II. This article is related to the paper by François Morin of Towers Perrin. A more detailed discussion of Morin's paper can be found here.