General Insurance Article - Solvency II Reserving Risk and Risk Margins

 By Peter England is global leader of P&C claims reserving at Towers Watson

 Article 101 of the Solvency II Directive states, "The Solvency Capital Requirement (SCR) shall correspond to the Value-at-Risk (VaR) of the basic own funds of an insurance or reinsurance undertaking subject to a confidence level of 99.5% over a one-year period." Essentially, the basic own funds are defined as the excess of assets over liabilities, under specific valuation rules.

 So it seems straightforward to estimate the SCR using a simulation-based model: Simply create a simulated distribution of the basic own funds over a one-year period, then calculate the VaR at the 99.5th percentile. But the thousands of pages of requirements offer little implementation guidance when details matter.

 A principal problem is the capital market-consistent value of insurance liabilities, which requires a best estimate (defined as the expected present value of future cash flows under Solvency II) plus a risk margin calculated using a cost-of-capital approach. So to calculate the SCR using simulation-based internal capital models, an opening balance sheet is required using discounted expected liabilities with a risk margin, along with a simulated balance sheet on the same basis after one year. That one-year balance sheet requires the expected value of liabilities with a risk margin for each simulation, producing a distribution of expected values. Usually we think the other way around — the expected value of a distribution.

 Calculating the risk margin in the opening balance sheet requires an estimate of the notional SCRs to transfer existing liabilities to a third party. So we need two different capital requirements: the overall capital requirement and notional capital requirements for existing liabilities only (including legally incepted business).

 ESTIMATING CAPITAL REQUIREMENTS

 Estimating capital requirements in a theoretically sound way requires a risk profile, risk measure and risk tolerance criterion, and usually a time horizon.

 Article 101 makes it clear that the risk profile is the distribution of basic own funds over a one-year time horizon; the risk measure is VaR, and the risk tolerance criterion is 99.5%. While the use of a one-year horizon only and the definition of the basic own funds is still open for debate as an appropriate basis for protecting policyholders, at least the Solvency II basis for measuring capital requirements contains all the elements to be theoretically sound.

 In practice,insurers calculate the SCR by using either a simulation-based internal model or a standard formula. Internal models produce a distribution of the basic own funds over one year from which the SCR can be calculated. The SCR is calculated at the entity level, although it could be allocated down to risk type or line of business.

 The standard formula, however, attempts to calculate capital requirements in silos by risk type and then aggregate them with a diversification adjustment. The risk profile for each risk type is not clear and neither is whether the standard formula SCR actually satisfies Article 101. In the absence of a distribution of the basic own funds, it is speculation.

 What is clear is that the notional risk profile for each risk type is intended to be some kind of profit/loss distribution. If we can define those profit/loss distributions so that the sum across all risk types equals the change in the basic own funds, then we have a starting point for contrasting the formula-based approach and the simulation-based internal model approach.

 LINE-OF-BUSINESS SCRS

 Following recent changes, the risk margin is now calculated at the entity level. However, a requirement to report technical provisions by line of business means that an allocation of risk margins to lines of business is required. One suggestion for allocating the overall risk margin to lines of business is in proportion to the line-of-business SCRs. Unfortunately, this retains the concept of a line-of-business SCR, which adds to the complexities.

 ENTITY SCR REQUIREMENTS

 To calculate the entity SCR, we need an opening balance sheet on a Solvency II basis, requiring a risk margin at the entity level. That risk margin requires SCRs for existing liabilities only for each future year until the liabilities are run off. Once known, SCRs can be multiplied by the cost-of-capital rate, discounted and summed to form the risk margin. The risk margin calculation is quite complicated: We need many SCRs to be able to complete the calculation.

 Unfortunately, estimating the SCRs for each future year in a theoretically correct way is circular: The SCRs depend on the risk margins, and the risk margins depend on the SCRs. The problem is resolved by starting at the final time period and working backward recursively, requiring repeated simulation on simulation, which is intractable without simplifications.

 One suggested simplification uses a proportional proxy. If the opening SCR can be calculated for existing liabilities, then future SCRs can be approximated by multiplying the opening SCR by the ratio of expected (discounted) outstanding liabilities at each future time period to the opening expected (discounted) outstanding liabilities. Estimating the opening SCR for the existing liabilities simplifies the problem but doesn't account for changing risk characteristics as liabilities run off.

 So even calculating the opening risk margin is complicated. Assuming we can calculate the entity's risk margin for the opening balance sheet, we still need to allocate it to lines of business in proportion to risk.
 Finally, to calculate the overall SCR, we need risk margins on the simulated balance sheet one year ahead, requiring similar calculations for each simulation. Clearly, simplifications are required to make the problem tractable.

 Unfortunately, we don't know what simplifications will be approved, so we asked the U.K. regulator and were told:
 • At present, there is no definitive answer.
 • Do something sensible, and explain why it is sensible.
 • Focus on the best estimate: The risk margin will usually be a lot smaller. Keep proportionality in mind. 

 Assuming we can calculate the risk margin on the opening balance sheet, we could propose two simplifications for approximating the simulated risk margins one year ahead. Option A uses the same (constant) risk margin for each simulation, but doesn't account for new business written in the year or the emergence of prior-year liabilities for each simulation.

 This approach may seem weak, but there is a precedent in the formula-based approach in which it is perhaps surprising that risk margins are not an input to the overall SCR calculation. Risk margins appear only when the calculated SCR is compared against available capital from a Solvency II balance sheet. Since the change in the risk margin is not considered within the standard formula, this is analogous to assuming that the risk margin is constant.

 A more realistic assumption is that the simulated risk margins one year ahead are calculated using SCRs that are proportional to a function of the simulated outstanding liabilities at that point in time — Option B. A good approach recognisesthat:

 • For the opening risk margin calculation, SCRs at each future period should recognisethe changing profile of the liabilities (e.g., short versus long tailed).
 • Opening risk margins by line of business should be additive to the overall risk margin (i.e., the allocation should be automatic).
 • Simulated risk margins one year ahead (and beyond) should recognisenew business written in the intervening periods (and the changing mix by class) and retain their additivity.

 Towers Watson is working on an approach that uses results from simulation-based internal models to calculate the opening risk margin and future risk margins, with these desirable characteristics.

 The opening risk margin calculation requires running the internal model specifically to obtain an opening SCR for existing obligations only (including legally incepted business), making sensible assumptions for catastrophe exposures, reinsurance and expenses. If we also partition the profit/loss distribution of the net technical result by line of business (in a way that sums to the aggregate result), we can simultaneously obtain risk margins by line of business and create a basis for calculating risk margins for each simulation on the one-year balance sheet in a way that preserves line-of-business dependencies.

 IMPLEMENTATION IN INTERNAL MODELS

 When calculating SCRs for the risk margin calculation, there are two implementation options:
 • Put the "plumbing" into the internal model to allow the risk margin calculations to be performed simultaneously with full model runs.
 • Use the existing internal model without modification, but perform additional runs just for the risk margin calculations.

 The first option is preferable since there is a better audit trail and it reduces the possibility of human error — important features because of the many model runs during a model's development and implementation.

 RESERVING RISK UNDER SOLVENCY II

 The traditional actuarial view of reserving risk considers variability of outstanding liabilities over their lifetime. Under Solvency II, reserving risk takes on a different meaning, and ignoring complications like operational risk loadings and credit risk on reinsurance recoveries, the basic risk profile is the profit/loss on reserves held over a one-year period. This is called the claims development result (CDR), or simply the runoff result.

 At its simplest, on an undiscounted basis, the CDR is defined as the opening expected undiscounted reserves, less claims paid (plus expenses) in the year, less closing expected undiscounted reserves after one year. This is simply the difference between the opening expected ultimate cost of claims and the expected ultimate cost of claims after one year. The traditional actuarial view of reserving risk looks at the difference between the opening expected ultimate and the actual ultimate cost of claims after all liabilities have been extinguished. This distinction is easily highlighted with reference to latent claims (like asbestos-related claims): The change in the view of the expected ultimate cost of claims over one year could be negligible, but the actual ultimate cost of claims after all liabilities have been extinguished could be very different.

 Despite the differences in perspective, the two views can be reconciled. Using simulation-based models, it is straightforward to devise an approach that links the traditional view over the lifetime of the liabilities and the one-year view. The approach also enables us to partition the traditional lifetime view into a sequence of one-year views, showing how the risk emerges over time.

 The beauty of a simulation-based approach is that it overcomes the limitations of a purely analytical approach. It also provides a distribution of the CDR, from which the 99.5th percentile can be obtained, a potential starting point for a reserving risk capital requirement.

 Solvency II's lower reserving risk measure has not gone unnoticed. Lloyd's of London requires SCR calculations using the one-year and lifetime views when assessing member capital requirements.

 IMPLICATIONS OF SOLVENCY II FOR CLAIM RESERVING

 Under Solvency II, the best estimate of outstanding liabilities is defined as the expected present value of all future cash flows, implying discounted liabilities. Reserving risk under Solvency II considers the profit/loss on the reserves over a one-year period, a fundamental change in thinking for most actuaries, who commonly estimate outstanding liabilities on an undiscounted basis and are familiar with considering risk over the lifetime of the liabilities. Actuaries and claim-reserving specialists will need to think differently in a Solvency II environment.