# Risk Margins and Solvency II Peter England and Richard Millns

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1 Risk Margins and Solvency II Peter England and Richard Millns GIRO conference and exhibition Liverpool, October 2011

2 Agenda Part 1 A quick re-cap from last year (and the year before) Part 2 Further thoughts Part 3 An example model Note: This presentation only considers risk margin calculations in respect of simulation-based internal models

3 A quick recap from last year (and the year before) Czernuszewicz, AJ, and England, PD (2009). The Ultimate and One-Year Views of Reserving Risk with Respect to Solvency and Risk Margins. GIRO presentation. England, PD, and McGuinness, A (2010). Solvency II Balance Sheets in Simulation-Based Capital Models. GIRO presentation.

4 DIRECTIVE OF THE EUROPEAN PARLIAMENT Article 101 The Solvency Capital Requirement shall be calibrated so as to ensure that all quantifiable risks to which an insurance or reinsurance undertaking is exposed are taken into account. With respect to existing business, it shall cover unexpected losses. It shall correspond to 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. So it seems straightforward to estimate the SCR using a simulationbased model: simply create a simulated distribution of the basic own funds over 1 year, than calculate the 99.5%. The devil is in the detail

5 A Projected Balance Sheet View From Article 101, the SCR is calculated from a distribution of net assets over a 1 year time horizon Opening Balance Sheet When projecting Balance Sheets for solvency, we have an opening balance sheet with expected outstanding liabilities Year 1 Balance Sheet The bulk of those liabilities are the reserves (provisions) set aside to pay unsettled claims that have arisen on policies sold in the past We then project one year forwards, simulating the payments that emerge in the year, and require a closing balance sheet, with (simulated) expected outstanding liabilities conditional on the payments in the year, together with the market value of assets at the end of the year A L t=0 A L t=1 Simulations 4

6 Solvency II Balance Sheet In the usual course of business Additional Capital { ORSA Capital Solvency Capital (SCR) { Minimum { Capital (MCR) Assets backing Liabilities } Risk Margin Discounted best estimate Marketconsistent value of Liabilities Assets Liabilities 5

7 Overall SCR GIRO 2009: Simulated Year 1 balance sheet options For each simulation Opening Balance Sheet Economic Basis? Discounted Liabilities (1 Yr View) with Risk Margins A Simulated Year 1 Balance Sheet using: Discounted Liabilities (1 Yr View) without Risk Margins B 99.5% applied to distribution of Net Assets gives excess capital required for the overall SCR, which is then tested against the Opening Capital Discounted Liabilities (Ultimate) without Risk Margins Undiscounted Liabilities (Ultimate) without Risk Margins C D

8 Simulated balance sheet definitions after 1 year? GIRO 2009: A convenient procedure Opening Balance Sheet without Risk Margins in the Liabilities For each simulation Simulated Year 1 Balance Sheet Discounted Liabilities (1 Yr View) without Risk Margin B Excess capital calculated using 99.5% applied to distribution of Net Assets This is used to calculate the overall SCR which is then tested against the opening capital using a Balance Sheet WITH Risk Margins in the Liabilities Under what assumptions can we use a balance sheet definition without risk margins in simulation based internal capital models for calculating the overall SCR?

9 Simulated balance sheet definitions after 1 year? GIRO 2010: Simplification 1 - Constant Risk Margin Opening Balance Sheet with Risk Margin For each simulation Simulated Year 1 Balance Sheet Discounted Liabilities (1 Yr View) with constant Risk Margin Excess capital calculated using 99.5% applied to distribution of Net Assets This is equivalent to Option B, but there is no need to add the opening risk margin back in. Using a constant risk margin appears to be analogous to the assumptions underlying the QIS 5 standard formula (that is, the change in the risk margin is not considered)

10 Simulated balance sheet definitions after 1 year? GIRO 2010: Simplification 2 - Proportional Risk Margin Opening Balance Sheet with Risk Margin For each simulation Simulated Year 1 Balance Sheet Discounted Liabilities (1 Yr View) with proportional Risk Margin GIRO 2010: We could devise more complicated alternatives based on proportions where the risk margin is different for each simulation, giving the appearance of a better solution

11 The Important Question When calculating risk margins, it is impossible to satisfy the Solvency II requirements without simulation on simulation, which is impracticable Simplifications must be made When calculating the opening SCR for the risk margin calculations When calculating future SCRs Simplifications must be made for risk margins for each simulation on the 1 year ahead balance sheet Assume a constant risk margin? Use a simple ratio method? What we don t know is: What methods will be approved? The question can only be answered by the regulators, so we asked the FSA 10

12 What the FSA has said so far * At present there is no definitive answer We don t want to give an answer that turns out to be wrong QIS 5 is not final: it is only a test Do something sensible and explain why it's sensible Worry more about the technical provisions; the risk margin will usually be a lot smaller Proportionality should be borne in mind * Reported at GIRO 2010

13 Do something sensible, and explain why it is sensible So, this presentation considers how to obtain an opening risk margin for a Solvency II balance sheet from a simulation-based internal model, and also how to allocate that to line-of-business at the same time in a consistent way. It also considers how to obtain simulated risk margins for the 1-year ahead balance sheet by line of business and in total, using the same outputs.

14 Solvency Capital Requirements Non-Life Companies Asset Risk: Movement in market value of assets Default Risk on assets, reinsurance and debtors Operational risk Overall Company SCR Premium & Reserve risk on existing obligations Premium risk on new business Catastrophe risk on existing obligations and new business 13

15 The Risk Margin The risk margin is designed to ensure that the value of technical provisions is sufficient for another insurer to take over and meet the insurance obligations It is calculated by determining the cost of providing an amount of eligible own funds equal to the SCR necessary to support the obligations over their lifetime Approach Establish net best estimate technical provisions at each point over the lifetime of the liabilities Estimate the appropriate corresponding SCR at each point Apply the cost-of-capital charge factor Discount and sum It is calculated at the portfolio level, net of reinsurance only The risk margin should take into account underwriting risk, reinsurer default risk, operational risk and unavoidable market risk In practice for most non-life insurers, market risk can be ignored 14

16 Solvency Capital Requirements: Non-Life Companies Asset Risk: Movement in market value of assets Default Risk on assets, reinsurance and debtors Operational risk Default Risk on reinsurance and debtors Operational risk (existing liabilities) Overall Company SCR SCRs for Opening Risk Margin Premium & Reserve risk on existing obligations Premium risk on new business Catastrophe risk on existing and new business Premium & Reserve risk on existing obligations Catastrophe risk on existing obligations only 15

17 Calculating the Risk Margin TP5.9 COCM = COC SCR t 0 RU. t+ 1 (1 + rt + 1) ( t) where COCM = risk margin for the whole business COC = cost of capital rate SCR RU (t) = the SCR for year t as calculated for the reference undertaking r t = risk-free rate for maturity t 16

18 Risk Margin Example Time Best- Estimate Reserve (SCR) Capital Capital Charge (assuming 6% CoC rate) Discounted Capital Charge (assuming flat 2% discount rate) Total (Risk Margin =)

19 Risk-Margin Granularity The risk margin assumes transfer of all business, hence it can allow for diversification between all reserve elements, including lines of business However, technical provisions need to be estimated at class level (own segmentation, or at least no higher than Solvency II LoB level) Hence we need to produce risk margin at this level Sum of class risk margins should equal the total risk margin So we need a way of allocating the overall risk margin to class 18

20 Calculating the Risk Margin by Line of Business Possible Simplification TP 5.28 where COCM lob SCR RU, lob lob =. SCR RU, lob (0) (0) COCM COCM lob = risk margin allocated to line of business SCR RU,lob (0) = SCR of the reference undertaking for line of business at t=0 COCM = risk margin for the whole business Notes: 1) The concept of an SCR by lob is a strange one, and appears to be a relic of QIS 4 2) Although it is not clear from the documentation, the SCRs should be in respect prior year reserves and legally incepted business only (but include an allowance for operational risk and reinsurance default risk). 3) Using this formula, there is no requirement for the sum of capital requirements across lobs to equal the total capital requirement 19

21 Requirements An overall SCR Requires a distribution of the basic own funds after 1 year A risk margin on the opening balance sheet Requires future SCRs in respect of the opening technical provisions only An allocation of the opening risk margin to LoB Requires opening SCRs by LoB (in respect of the opening technical provisions only), at the very least More robust methods require opening and future SCRs by LoB Simulated risk margins for the 1 year ahead balance sheet (for the overall SCR calculation Just use a constant? (analogous to the standard formula approach) Proportional to the simulated expected technical provisions at T=1? Based on a cost-of-capital approach using future SCRs by LoB? 20

22 Desirable characteristics of a suitable method For the opening overall risk margin calculation, SCRs at each future period should recognise the changing profile of the liabilities (i.e. short vs long tailed) Opening risk margins by class should be additive to the overall risk margin (i.e. the allocation should be automatic) Simulated risk margins at t=1, 2, should recognise new business written in the intervening periods (and the changing mix by class), and retain their additivity Simulated risk margins at t=1, 2, should take account the (simulated) yield curve at each period 21

23 The Opening Risk Margin

24 The Opening Risk Margin We require a risk margin for the opening balance sheet This requires SCRs in respect of the opening technical provisions, for all future years How should those be calculated? Using the standard formula? Using output from the internal model? If an internal model is used for the overall SCR then it seems to make sense to use the output available where possible 23

25 The Opening Risk Margin in Internal Models First produce a suitable risk profile in order to calculate the initial SCR This should include the profit(/loss) over one-year arising from: Prior year reserves and expenses Unexpired risk and expenses Legally obliged (but unincepted) business Operational risk, RI default, and unavoidable market risk (not usually material) 99.5% will give the opening SCR amount for the risk margin calculation Then calculate future SCRs: In proportion to the emergence of the (expected) reserves in each future year in aggregate? By LoB? 24

26 Risk Margin Future SCRs QIS 5 Options 1. Make a full calculation of all future SCRs without using simplifications. 2. Approximate the individual risks or sub-risks within some or all modules and submodules to be used for the calculation of future SCRs. 3. Approximate the whole SCR for each future year, e.g. by using a proportional approach. 4. Estimate all future SCRs at once, e.g. by using an approximation based on the duration approach. 5. Approximate the risk margin by calculating it as a percentage of the best estimate. 25

27 Future SCR Calculation An Approach (For the opening risk margin) 1. Take the one-year profit/(loss) distribution of the net technical result by class for the existing business only as described earlier 2. Calculate the expected outstanding cash-flows by class for this business at each year end as the business runs off 3. Approximate the distribution of the profit/loss of the net technical result by class in future year t in proportion to the ratio of the expected outstanding cash-flows by class in future year t divided by the value at t=0* 4. Calculate the 99.5th percentile loss amount of the distributions in 3. to give the standalone SCR run-off by class before diversification. Call this SCR(pre-div) RM (0,class,t) 5. Aggregate the loss distributions in each year t and calculate the 99.5th percentile to give the total SCR run-off over time. Call this SCR RM (0,t) 6. Due to diversification, SCR RM (0,t) will typically be less than the sum of SCR(pre-div) RM (0,class,t) across classes. Hence scale SCR RM (0,class,t), such that Sum class (SCR RM (0,class,t)) = SCR RM (0,t). Note: alternative allocation approaches are possible. 7. Calculate the risk margin using the cost-of-capital approach at the class and total level (it will be additive, and will take account of the changing profile of the liabilities at the total level). * In step 3 we could instead assume that the distribution scales in proportion to a function of the expected outstanding cash-flows e.g. square root Assumptions The coefficient of variation of the one-year distribution around the expected technical provision (or function thereof) is the same at each year in the run-off within a class of business (proportional proxy) The dependency between classes is the same at each point in the run-off 26

28 Example

29 Risk Margins in the 1-Year Ahead Balance Sheets (and beyond?)

30 A Projected Balance Sheet View Remember, from Article 101, the SCR is calculated from a distribution of net assets over a 1 year time horizon So when we project one-year forwards, in addition to the (simulated) expected outstanding liabilities conditional on the payments in the year, we also need a risk margin for each simulation, in respect of the outstanding liabilities at that time. Opening Balance Sheet Year 1 Balance Sheet Simulations A L t=0 A L t=1 29

31 Simulated balance sheet definitions after 1 year? Simplification 1: Constant Risk Margin Opening Balance Sheet with Risk Margin For each simulation Simulated Year 1 Balance Sheet Discounted Liabilities (1 Yr View) with constant Risk Margin Excess capital calculated using 99.5% applied to distribution of Net Assets This is equivalent to excluding the risk margin in the capital model, then adding the opening risk margin back in. Using a constant risk margin appears to be analogous to the assumptions underlying the QIS 5 standard formula (that is, the change in the risk margin is not considered) 30

32 Simulated balance sheet definitions after 1 year? GIRO 2010: Simplification 2 - Proportional Risk Margin Opening Balance Sheet with Risk Margin For each simulation Simulated Year 1 Balance Sheet Discounted Liabilities (1 Yr View) with proportional Risk Margin GIRO 2010: We could devise more complicated alternatives based on proportions where the risk margin is different for each simulation, giving the appearance of a better solution

33 Projection of the Risk Margin Options 1. Fixed risk margin (i.e. set closing risk margin equal to the opening value) 2. Risk margin based on expected best-estimate closing reserves 3. Risk margin based on the simulated best-estimate closing reserves For example, if the risk margin is set as a constant proportion of the reserves we would have: BE t = 0 RM t = 0 BE t = 1 RM t = 1 Opt 1 Opt 2 Opt 3 High Average Low In this presentation, we are proposing a more complicated variation of options 2 and 3. 32

34 Projection of the Risk Margin Project the risk margin to t=1,2,3 etc. by using results from the t=0 calculation and making some further assumptions: 1. From the opening risk margin calculations, express the SCRs by class as a proportion of the expected outstanding cash-flows for each year*, i.e. calculate SCR RM (0,class,t) / Expected Outstanding(0,t) 2. For the one-year ahead risk margins, approximate the SCRs by class by multiplying the expected outstanding cash-flows** at t=1 by the ratio calculated at step 1 3. Sum across classes to give the overall SCR RM (1,t) 4. Calculate the risk margin using the cost-of-capital approach at the class and total level, using the (simulated) risk-free yield curve at t=1 Assumptions The same assumptions as for the t=0 calculations, plus the following: The coefficient of variation of the one-year loss distribution (in each simulation) within a class at t=1 is the same as at t=0 The diversification between classes at each point in the run-off is the same at t=0 and t=1 * Again, in step 1 we could instead assume that the distribution scales in proportion to a function of the expected outstanding cash-flows e.g. square root ** Based on the stochastic or deterministic reserves at t=1 33

35 Example

36 Implementation in Internal Models For the SCRs for the risk margin calculation, there are two implementation options: 1. Put the plumbing into the internal model to allow the risk margin calculations to be performed simultaneously with full model runs Partition the net technical result from the full model into separate components for the existing obligations (including legally obliged business) and new business, making sensible assumptions in respect of catastrophe exposures, reinsurance and expenses 2. Use the existing internal model without modification, but perform additional runs just for the risk margin calculations Modify the inputs so the model relates to the existing obligations only (including legally obliged business) excluding new business, making sensible assumptions in respect of catastrophe exposures, reinsurance and expenses 35

37 Summary There are many complications associated with risk margins under Solvency II: Risk margins are required for the opening balance sheet, and for each simulation at the 1 year ahead position Although a diversified risk margin can now be calculated under QIS 5, there is still a requirement to allocate the risk margin to line of business (TP ) This requires an opening SCR, as well as future SCRs for the cost of capital method This presentation proposes a new approach to calculating the opening risk margin using outputs from an internal model, and also considers how the risk margins might then be calculated for each simulation at the 1 year ahead position. The approach is simple and has many desirable characteristics 36

38 Questions or comments? Expressions of individual views by members of The Actuarial Profession and its staff are encouraged. The views expressed in this presentation are those of the presenter. 37

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