# Solvency II: Risk Margins and Technical Provisions Peter England

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1 Solvency II: Risk Margins and Technical Provisions Peter England GIRO conference and exhibition October The Actuarial Profession

2 Agenda The traditional actuarial view of reserving risk vs Solvency II and the one-year view We will look at the two perspectives, and try to reconcile them in some way Risk margins in internal models Practical considerations

3 Lifetime perspective One Yr view The Reserve Risk Puzzle Simulation based Analytic Formula based

4 Reserve Risk: The traditional actuarial view Looking over the lifetime of the liabilities The traditional actuarial view of reserve risk looks at the uncertainty in the outstanding liabilities over their lifetime We have to start talking statistics Given a statistical model, we can derive analytic formulae for the standard deviation of the forecasts Given a statistical model, we can also generate distributions of outstanding liabilities, and their associated cash-flows, using simulation techniques (eg bootstrap or MCMC techniques) We can do this in a way that reconciles the analytic and simulation approaches

5 Simulation vs analytic approaches to reserve risk We can do this the easy way, or we can do it the hard way

6 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 simulation-based model: simply create a simulated distribution of the basic own funds over 1 year, then calculate the 99.5%.

7 A Projected Balance Sheet View When projecting Balance Sheets for solvency, we have an opening balance sheet with expected outstanding liabilities We then project one year forwards, simulating the payments that emerge in the year We then require a closing balance sheet, with (simulated) expected outstanding liabilities conditional on the payments in the year Opening Balance Sheet Year 1 Balance Sheet

8 Solvency II For Solvency II, a 1 year perspective is taken, requiring a distribution of the expected value of the liabilities after 1 year, for the 1 year ahead balance sheet in internal capital models If the standard formula is used, a 1 year-ahead reserve risk standard deviation % is required. This could be: The standard parameter for the line-of-business An undertaking specific parameter The 1 year-ahead reserve risk standard deviation is the SD of the distribution of profit/loss on reserves after 1 year Note: this is a different definition of risk from the traditional actuarial view

9 The one-year run-off result (undiscounted) (The view of profit or loss on reserves after one year) For a particular origin year, let: The opening reserve estimate be The reserve estimate after one year be The payments in the year be The run-off result (claims development result) be Then CDR R 0 R 1 C 1 1 = R0 C1 R1 = U 0 U 1 CDR 1 Where the opening estimate of ultimate claims and the estimate of the ultimate after one year are U,U 0 1

10 The One-year Run-off Result (the view of profit or loss on reserves after one year) Merz & Wuthrich (2008) derived analytic formulae for the standard deviation of the claims development result after one year assuming: The opening reserves were set using the pure chain ladder model (no tail) Claims develop in the year according to the assumptions underlying Mack s model Reserves are set after one year using the pure chain ladder model (no tail) The mathematics is quite challenging. This is the HARD way The M&W method is gaining popularity, but has limitations. What if: We need a tail factor to extrapolate into the future? Mack s model is not used other assumptions are used instead? We want another risk measure, not just a standard deviation (eg 99.5%)? We want a distribution of the CDR?

11 Merz & Wuthrich (2008) Data Triangle Accident Year 12m 24m 36m 48m 60m 72m 84m 96m 108m 0 2,202,584 3,210,449 3,468,122 3,545,070 3,621,627 3,644,636 3,669,012 3,674,511 3,678, ,350,650 3,553,023 3,783,846 3,840,067 3,865,187 3,878,744 3,898,281 3,902, ,321,885 3,424,190 3,700,876 3,798,198 3,854,755 3,878,993 3,898, ,171,487 3,165,274 3,395,841 3,466,453 3,515,703 3,548, ,140,328 3,157,079 3,399,262 3,500,520 3,585, ,290,664 3,338,197 3,550,332 3,641, ,148,216 3,219,775 3,428, ,143,728 3,158, ,144,738

12 Merz & Wuthrich (2008) Prediction errors Analytic Prediction Errors Accident Year 1 Year Ahead CDR Mack Ultimate ,488 1, ,923 4, ,723 10, ,443 30, ,954 35, ,119 45, ,320 69,552 Total 81, ,401 Expressed as a percentage of the opening reserves, this forms a basis of the reserve risk parameter under Solvency II (QIS 5 Technical Specification)

13 The one-year run-off result in a simulation model The EASY way For a particular origin year, let: The opening reserve estimate be The expected reserve estimate after one year be The payments in the year be The run-off result (claims development result) be Then CDR ( i) 1 = R 0 C ( i) 1 R ( i) 1 = U 0 R 0 R ( i) 1 ( i) 1 ( i) 1 C CDR U ( i) 1 Where the opening estimate of ultimate claims and the expected ultimate ( ) after one year are U for each simulation i U, i 0 1

14 The one-year run-off result in a simulation model The EASY way 1. Given the opening reserve triangle, simulate all future claim payments to ultimate using bootstrap (or Bayesian MCMC) techniques. 2. Now forget that we have already simulated what the future holds. 3. Move one year ahead. Augment the opening reserve triangle by one diagonal, that is, by the simulated payments from step 1 in the next calendar year only. An actuary only sees what emerges in the year. 4. For each simulation, estimate the outstanding liabilities, conditional only on what has emerged to date. (The future is still unknown ). 5. A reserving methodology is required for each simulation an actuary-in-thebox is required*. We call this re-reserving. 6. For a one-year model, this will underestimate the true volatility at the end of that year (even if the mean across all simulations is correct). * The term actuary-in-the-box was coined by Esbjörn Ohlsson

15 The standard actuarial perspective: forecasting outcomes over the lifetime of the liabilities, to their ultimate position A single accident year, 4 years developed Actual simulated future amounts

17 Actual simulated future amounts Expected payments conditional on year 1 position

18 Example

19 1 Year ahead Simulation 1

20 1 Year ahead Simulation 2

21 1 Year ahead Simulation 3

22 Merz & Wuthrich (2008) Analytic vs Simulated: Summary Accident Year Analytic Prediction Errors 1 Year Ahead Mack CDR Ultimate Simulated Prediction Errors 1 Year Ahead Mack CDR Ultimate ,488 1,566 1,486 1, ,923 4,157 3,916 4, ,723 10,536 9,745 10, ,443 30,319 28,428 30, ,954 35,967 20,986 35, ,119 45,090 28,110 44, ,320 69,552 53,406 69,713 Total 81, ,401 81, ,992

23 Cascading Bootstrap Run-off Results The input to a Bootstrap Run-off Result can be another Bootstrap Run-off Result. This can be used to give the CDR between the 1 st and 2 nd years ahead, and so on

24 Multiple 1 yr ahead CDRs An interesting result Creating cascading CDRs over all years gives the following results: Accident Number of years ahead Sqrt(Sum of Mack Year 1 Yr 2 Yrs 3 Yrs 4 Yrs 5 Yrs 6 Yrs 7 Yrs 8 Yrs Squares) Ultimate , ,564 1, ,916 1, ,151 4, ,745 3,837 1, ,560 10, ,428 9,679 3,824 1, ,303 30, ,986 27,438 9,343 3,693 1, ,998 35, ,110 20,404 26,922 9,162 3,613 1, ,055 44, ,406 27,798 20,236 26,687 9,111 3,600 1, ,600 69,713 Total 81,226 52,344 38,513 29,010 10,120 3,879 1, , ,992 The sum of the variances of the repeated 1 yr ahead CDRs (over all years) equals the variance over the lifetime of the liabilities Under Mack s assumptions/chain ladder, this can be proved Therefore we expect the risk under the 1 year view to be lower than the standard ultimo perspective

25 Re-reserving in Simulation-based Capital Models The advantage of investigating the claims development result (using rereserving) in a simulation environment is that the procedure can be generalised: Not just the chain ladder model Not just Mack s assumptions Can include curve fitting and extrapolation for tail estimation Can incorporate a Bornhuetter-Ferguson step Can be extended beyond the 1 year horizon to look at multi-year forecasts Provides a distribution of the CDR, not just a standard deviation But it is not without its difficulties, so we need simpler alternatives Simply allow the ultimo variability to emerge steadily over time (but there is the problem of calibration)

26 The Reserve Risk Puzzle Harmony has been restored Lifetime perspective One Yr view Analytic Simulation based Formula based

27 Risk Margins in Internal Models

28 DIRECTIVE OF THE EUROPEAN PARLIAMENT Article 77 Article 77 The risk margin shall be such as to ensure that the value of the technical provisions is equivalent to the amount insurance undertakings would be expected to require in order to take over and meet the insurance obligations the risk margin shall be calculated by determining the cost of providing an amount of eligible own funds equal to the Solvency Capital Requirement necessary to support the insurance obligations over the lifetime thereof. So we need an SCR for each future year as the reserves run-off

29 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 Reserve risk on existing obligations Underwriting risk on new business Catastrophe risk on existing obligations and new business Reserve risk on existing obligations Catastrophe risk on existing obligations only

30 Overall SCR GIRO 2010: Simulated Year 1 balance sheet options? For each simulation Opening Balance Sheet with Risk Margin Simulated Year 1 Balance Sheet using: Discounted Liabilities (1 Yr View) with constant Risk Margin Discounted Liabilities (1 Yr View) with proportional Risk Margin A B Whichever method is used, we still need a risk margin for the opening balance sheet. The opening SCR for the risk margin calculation could be calculated using the standard formula (maybe) or a modified version of the internal model itself. Use a simple approach for the future SCRs for the risk margin calculation.

31 The Opening Risk Margin in Internal Models Using the Standard Formula This has the advantage of appearing to be simple There is no need to justify the assumptions in the standard formula The risk margin method would be standardised across companies Calculate the opening SCR by entering reserve and premium volumes in respect of the (expected) technical provisions (legal obligations basis only) Market risk not required (usually) Calculate future SCRs: In proportion to the emergence of the (expected) reserves in each future year, or By repeatedly calculating the SCR using the standard formula, but adjusting reserve and premium volumes in each future year The capital requirement percentages can be calculated, relative to the opening SCR

32 The Opening Risk Margin in Internal Models Using the Internal Model The internal model basis itself could* be used Assume opening assets = 0** For premium volumes, use legal obligations basis only (no new business in the forthcoming year) Remember to modify assumptions about cat exposures, reinsurance and expenses 99.5% will give the TOTAL capital required, for the SCR calculation Calculate future SCRs: In proportion to the emergence of the (expected) reserves in each future year, or Using the proportions implied by the recursive standard formula method * It is possible that the internal model basis should be used, but given the concept of proportionality, using the standard formula may be sufficient ** Other assumptions could be used

33 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

34 What We Asked the FSA 1. Will it be acceptable to have opening and 1 year ahead balance sheets excluding risk margins, and use the change in the balance sheet on that basis to estimate the overall SCR (after adding the opening risk margin back in)? If that is not acceptable, what simplifications will be approved for calculating risk margins for each simulation in the 1 year ahead balance sheet? 2. If the proposal in (1) is acceptable, will it also be acceptable to use the standard formula for estimating the opening risk margin, even with an internal model? 3. If the standard formula basis is not acceptable for estimating the opening risk margin when using an internal model, what methods will be approved for estimating the initial SCR for the risk margin calculation from the internal model, and what simplifications will be approved for estimating the future SCRs for the risk margin calculation?

35 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 * Thanks to the FSA for clarifying the current position

36 Questions or comments? You ve heard it from the ORSA s mouth 2010 The Actuarial Profession 35

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