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# Solvency II all downhill from here?

Solvency II is a project to reform the current solvency regime that governs insurance companies in the European Union. It aims at an improved prudential supervisory system that is consistently implemented across all member states with solvency requirements that are better aligned to the risks to which companies are exposed.
The Committee of European Insurance and Occupational Pensions Supervisors (CEIOPS) recently released the second quantitative impact study QIS2. The purpose of QIS2 is to estimate the impact of the proposed specifications on the balance sheet and assist CEIOPS with the calibration of different parameters. Companies that choose to participate are required to provide extensive information of the impact of different prescriptive tests on their capital positions. Information is required by 31 July.
Although QIS2 is not a final piece of regulation, it provides us with some insights of what the regime might look like.

SCR calculation method
As for the current ICA regime, Solvency II focuses on firms’ ability to meet a solvency capital requirement, or SCR. QIS2 defines the SCR using a tree structure, as shown in figure 1.
As with existing statutory solvency requirements, most of these boxes are completed using standard factors applied to measures of exposure, or, in some cases, the results of stress tests to key assumptions. QIS2 suggests an initial calibration for all the necessary parameters.

Using correlation matrices for aggregation
Capital is no longer simply added to produce a total requirement. Instead, a correlation formula is used for capital aggregation. This is the basis on which Solvency II is described as ‘risk-based’. For example, to calculate capital for life insurance risk, the following formula applies:
Life insurance risk capital=

Here, i and j range over the relevant risk types. The correlation matrices are specified. For example, the life insurance risk correlation matrix is shown in table 1 across.

Diversification
The use of correlation matrices is helpful because it allows capital relief for diversification, that is, the spreading of risks across different risk categories. For example, consider a company X which requires £60 of capital for mortality risk and £320 for disability risk. The tabulated correlation is 25%, so the required combined capital is:

This calculation gives an answer £40 lower than simply adding the components, reflecting the diversification of the two risks.
Although diversification usually provides a benefit, it is possible to contrive examples where the reverse occurs. Suppose company X now merges with another company, Y, which has an SCR of £340, due entirely to morbidity risk. The combined required capital is £685, more than the sum of the SCRs for X and Y separately. It is therefore possible that two firms could meet their SCRs when viewed separately, but not when viewed in aggregate.

Nested aggregation
The aggregation calculations are nested, involving several correlation matrices in succession. For example, the non-life capital for premium risk involves aggregation across 11 classes of business, using one correlation matrix. The non-life capital requirement is the aggregation of premium, reserving, and catastrophe risk, also applied using a correlation matrix. Finally, a third correlation matrix defines the total SCR as an aggregate of capital for life, non-life, health, market, credit, and operational risk.

Two liability valuation methods
Any solvency test must compare required capital to available capital. The available capital is defined as assets minus liabilities. The valuation of assets is at market value, but the valuation of liabilities usually requires some kind of model.
The UK realistic balance sheet regime uses one simple approach to liability valuation. Under this regime, any cashflows depending on capital markets are valued using market prices. For more complicated liabilities, this often requires a Monte Carlo approach, using economic scenarios that replicate market prices.
The controversial question is whether this calculation remains appropriate for cashflows subject to non-market risks, for example lapse risk or mortality risk. It could be argued that that no further adjustment is required, as efficient markets put a zero price on diversifiable risks. Others argue that, within a financial institution, there is a cost even to bearing diversifiable risks, for example, the frictional costs of any additional capital that needs to be held for those risks.
QIS2 allows for frictional costs to be incorporated in one of two ways. The first way is to replace best-estimate cashflows with 75th percentiles, and the second way is to discount explicit frictional cost projections as the liabilities run off. This second ‘cost of capital’ approach is based on the notion that any counterparty taking on insurance liabilities will tie up costly capital for which it will be compensated. The cost in QIS2, and in the existing Swiss solvency standard, is set at 6% of regulatory capital.
Both of these approaches are challenging to implement in practice. In either case, the intention is that the margin should reflect only non-market risk. There are several algorithms to exclude market risk from a calculated percentile or capital number; QIS2 does not express a preference. More pragmatic observers have suggested that the cost of capital calculation could be scrapped entirely, perhaps capturing a similar effect by using a 99.6%-ile rather than 99.5%-ile of the net asset distribution.

Use of internal models
Solvency II permits firms to use approved internal models in place of the standard SCR. Your internal model makes sense to you but does it make sense to a regulator? The publication of QIS2 gives us some clues to how regulators might approach the use of internal models.
QIS2 formulas are tantalisingly close to those employed in some existing internal models. For example, some internal models assume that a firm’s net assets respond linearly to each risk factor, and the risk factors follow an elliptically contoured distribution (for example, normal). In this situation, it is correct to combine capital requirements for each risk according to a correlation formula, as QIS2 proposes.
Other internal models may be both complex and sophisticated. Few regulators have the resources or the inclination to scrutinise such models in detail. You cannot expect regulators to validate your scenario generators or check your stochastic integrals. Instead, what regulators can do is to examine model assumptions and output, benchmarking firms against one another or against the standard SCR.
This means that, in practice, firms may choose to re-express internal model output in the format of the SCR, using the same intermediate steps and the same risk classification. Such a presentation allows a firm to discuss the model with its regulator, comparing assumptions line by line to demonstrate that the internal model is sufficiently prudent.

Reasons for differences between SCR and internal models
Despite the use of risk-based formulas in QIS2, there remain several aspects in which internal models could be expected to deviate from QIS2. These include non-linearity, risk direction and nesting.
Consider an annuity fund, matched with respect to interest risk assuming a certain mortality table. If the mortality basis is strengthened, then the liability duration increases, exposing the fund to any fall in yields. More sophisticated internal models expose this double-whammy effect with scenario tests. For example, the ‘least solvent likely event’, or LSLE, is the most painful combined 99.5% test, allowing for the interaction between market and mortality risk. Non-linearity adjustments to each of the single-factor capital calculations provide a reconciliation of an internal model to the standard formula approach in QIS2.
The QIS2 approach requires a 75% correlation between equity and interest rate capital, irrespective of the direction of the interest rate exposure. This creates the challenge of creating a single-probability model for which the QIS2 calculation consistently computes the 99.5%-ile. One possible approach uses LSLE techniques to estimate tail correlations.
Many internal models show a high correlation between equity returns and corporate bond returns, especially if the equity and bonds are issued by the same company. An SCR calculation should therefore recognise a significant correlation between market risk and credit risk capital, at the top level of nesting. However, this marketcredit correlation is only there if the market risk is generated from equity exposure. If the market risk arises from foreign exchange, then the correlation with credit risk would be much smaller. Unfortunately, under QIS2, by the time you get to the top-level aggregation, the nesting approach has discarded the information of where the market risk capital came from. Firms seeking to restate their internal models in QIS2 terms will wish to make full allowance for the constituents of market risks when justifying top level correlation assumptions.

What firms can do now
Many observers expect Solvency II to look similar to what we now know as QIS2. Firms can now start to investigate the capital requirements for their major product lines. In addition, firms can adapt their internal models to present them in a form closer to QIS2. This is likely to facilitate future discussions with regulators.

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