Capital Framework Best Practices. IAA Life Section Seminar 19 April 2007 Andrew D Smith

Transcription

1 Capital Framework Best Practices. IAA Life Section Seminar 19 April 2007 Andrew D Smith AndrewDSmith8@deloitte.co.uk

2 Capital Framework Best Practices Presentation Overview Presenting capital and risk information Communication examples based on simple ALM Duration and convexity Capital allocation Stress test likelihood Big bang and medium bang Least solvent likely event Better decisions from capital understanding Conclusions 2 Capital Framework Best Practices

3 Capital and risk reporting structures may be complex Standard Basic Operation Non-life Market Health Default Life Premium FX Expense Mortality Cat Property Claim Lapse Longevity Concentration Interest Equity Spread Accumulation Expense Disability Cat Revision Solvency II QIS 3 Structure 3 Capital Framework Best Practices

4 Calculations may be onerous, so computational efficiency is important. mortality interest Stochastic Model Office response function equity Capital and Risk calculations Probability Stochastically generated drivers Net assets 4 Capital Framework Best Practices

5 Results may be challenging to communicate Graphical techniques address these challenges. Fitness for purpose Capital based on 99.5% policyholder confidence Is this right the right measure to drive product price loading? Or for shareholder value analysis? Model complexity Thousands of assumptions Prevents interactive analysis Can t do the sums in your head Effect of assumptions unclear Capital results change unexpectedly Need to control all inputs Sub-additivity Financial reporting consolidation requires capital to add up Diversification effects are balancing items Remote actions: change in interest rate hedge affects capital allocated for natural catastrophe risk Graphical Communication Best practice adopts stress scenarios to communicate risk exposures 5 Capital Framework Best Practices

6 Capital Calculations have high dimensionality. Today we consider an example based on interest rate risk Loss Ratio: Motor Loss Ratio: Catastrophe Foreign exchange Inflation expectations 1 year yield 30 year yield Credit spreads Bad debts Equity market level Property market level Implied volatility surface: interest rates Implied volatility surface: equities Correlations and forward correlations Mortality: assured lives Mortality: pensioners Expense risks Lapse rates Option take-up rates Operational losses 1 year yield 2 year yield 3 year yield 30 year yield 6 Capital Framework Best Practices

7 Example liability cash flows 600 Cash flow year 7 Capital Framework Best Practices

8 Example asset cash flows Cash flow year 8 Capital Framework Best Practices

9 Our base analysis uses a market yield curve Calibrated to market prices of traded bonds Abbreviation Definition Calibrated to market bond/swap prices of various terms 9 Capital Framework Best Practices

10 Diversification has a large effect for yield curve models Because yields at adjacent terms are highly correlated. Capital required Diversification benefit Diversified capital years years years years 6-10 years 1-5 years Capital Framework Best Practices

11 Our example uses this base yield curve. Spot yield 7% 6% 5% 4% 3% 2% 1% 0% Term (years) See CEIOPS (2007) for free yield curve data for European and other economies. 11 Capital Framework Best Practices

12 Duration and Convexity years Assets Liabilities Duration Sqr(Convexity) 12 Capital Framework Best Practices

13 The Case lies in middle of all likely combinations. 6% 15 year yield 5% Likely locus 4% 3% 2% 2% 3% 4% 5% 6% 5 year yield This an example of the risk geographies approach. See Baddon & Coulthard (2006) for more details. 13 Capital Framework Best Practices

14 We chose the Net Assets equal to the Required Capital Assets Liabilities Diversified capital Capital Framework Best Practices

15 Developing management intuition about threat scenarios Our analysis has shown the traditional capital model outputs: Policyholder perspective How much capital is at risk of loss Sensitivity to parallel moves in interest rates Detail on, potentially, hundreds of other risks Benefits of diversification Other things we would like to know: What has to go wrong to create financial distress? Which risks in combination pose the most significant threats? What are the most likely ways to fail? How to mitigate the risks of failure? What is the shareholder costs of bearing those risks? * * This presentation focuses on the risk to policyholders. See Exley & Smith (2006) for the shareholder perspective. 15 Capital Framework Best Practices

16 Leading firms illustrate their risks by examining a series of stress scenarios Abbreviation Hi 5 Definition Calibrated to market bond/swap prices of various terms Highest likely 5-year yield The Hi 5 Scenario considers the highest likely 5 year yield Other yields are set to their most likely given the 5 year yield 16 Capital Framework Best Practices

17 The Lo 15 Scenario uses the lowest likely 15 year yield Other yields are set to their most likely given 15 year yield Abbreviation Hi 5 Lo 15 Definition Calibrated to market bond/swap prices of various terms Highest likely 5-year yield Lowest likely 15-year yield 17 Capital Framework Best Practices

18 , Hi 5 and Lo 15 yield curves Spot yield 7% 6% 5% 4% Hi 5 Lo 15 3% 2% 1% 0% Term (years) 18 Capital Framework Best Practices

19 Stress Test Effect on Assets and Liabilities Assets Liabilities Hi 5 Lo Capital Framework Best Practices

20 Mathematical Note: Defining Likely Stresses We need to define what we mean by likely and unlikely scenarios Likely yield curve scenarios are Continuous Close to the base case Flat Smooth We define the size of a stress test using the Short Rate Stress Equivalent (SRSE) Let P t base be the time 0 price of a t-maturity zero coupon bond Let P t stress be zero coupon bonds from an alternative yield curve scenario Pick α > 0 (in our example we use α = 15%) Then define SRSE by: α 0 d dt P P 2 2 stress 2 stress 1 d P t t dt dt base + 2 base t α dt P = 0 t flatness SRSE In this presentation, likely is defined as SRSE 2% See Smith & Wilson (2000) for more motivation behind this objective function. smoothness 2 20 Capital Framework Best Practices

21 Likely locus contains scenarios equivalent to short rate stress 2% 6% 15 year yield 5% Likely locus 4% 3% 2% 2% 3% 4% 5% 6% 5 year yield 21 Capital Framework Best Practices

22 All likely tests lie on the Likely Locus 6% 15 year yield 5% 4% Likely locus Hi 5 Lo 15 3% 2% 2% 3% 4% 5% 6% 5 year yield 22 Capital Framework Best Practices

23 The Hi 5 scenario is slightly beneficial for net assets, even though the asset duration exceeds the liability duration. Net assets Hi 5 Lo % 1% 2% 3% -50 Scenario shift (short rate stress equivalent) 23 Capital Framework Best Practices

24 We can also express likelihood using a correlation matrix A mathematically equivalent formulation to the SRSE * correlation Yield term Yield term * Technically, the link to SRSE is the concept of half-space depth. See Rousseeuw & Ruts (1999). The resulting covariance structure is consistent with an infinite dimensional yield model, as described by Cont (2004). 24 Capital Framework Best Practices

25 Big Bang sets all risks to their extreme likely values. Correlations make a Big Bang unlikely (SRSE > 2%) Abbreviation Hi 5 Lo 15 BB down BB up Definition Calibrated to market bond/swap prices of various terms Highest likely 5-year yield Lowest likely 15-year yield Big Bang all yields to their lowest likely value Big Bang all yields to their highest likely value 25 Capital Framework Best Practices

26 The Big Bang up and down scenarios represent the most extreme values for each yield from the Hi and Lo scenarios. Spot yield 7% 6% 5% 4% 3% 2% 1% Hi 5 Lo 15 BB up BB down 0% Term (years) 26 Capital Framework Best Practices

27 Big Bang test results are more extreme than Hi and Lo likely scenarios Assets Liabilities Hi 5 Lo 15 BB up BB down 27 Capital Framework Best Practices

28 Big Bang scenarios lie outside the likely locus, at the most extreme likely values for each yield. 6% 15 year yield 5% 4% Likely locus Limits Hi 5 Lo 15 BB up BB down 3% 2% 2% 3% 4% 5% 6% 5 year yield 28 Capital Framework Best Practices

29 Big Bang effect on net assets Net assets Hi 5 Lo 15 BB up BB down 0 0% 1% 2% 3% -50 Scenario shift (short rate stress equivalent) 29 Capital Framework Best Practices

30 Choosing Combined Scenarios to Illustrate the Effect of Interacting Risks Classical Approach Pre-defined scenario tests One risk at a time Specified at a chosen level of confidence Apply risks at full strength simultaneously Example: Big bang Scenarios Reflecting My Risks Monte Carlo Generate scenarios randomly with equal probability. Advantage: Easy to generate scenarios from a standard model. Big Bang A-L Medium Bang A-L LSLE Disadvantage: Waste time computing scenarios that are not relevant or painful for our firm. 30 Capital Framework Best Practices

31 The Asset-Liability Big Bang moves each yield to its least favourable likely value Abbreviation Hi 5 Lo 15 BB down BB up BB A-L Definition Calibrated to market bond/swap prices of various terms Highest likely 5-year yield Lowest likely 15-year yield Big Bang all yields to their lowest likely value Big Bang all yields to their highest likely value BB Asset Liability all yields to their least solvent likely value 31 Capital Framework Best Practices

32 The Asset-Liability Big Bang does not produce a smooth curve, because big bang ignores correlations. Spot yield 7% 6% 5% 4% 3% 2% Hi 5 Lo 15 BB up BB down BB A-L 1% 0% Term (years) 32 Capital Framework Best Practices

33 The Big Bang scenario is ruinous, as liabilities > assets BB is unlikely: this does not contradict capital calculation Assets Liabilities Hi 5 Lo 15 BB up BB down BB A-L 33 Capital Framework Best Practices

34 The BB A-L scenario has a low 5 year yield and a high 15 year yield, because 15 years is a bond maturity date. 6% 15 year yield 5% 4% 3% Likely locus Limits Hi 5 Lo 15 BB up BB down BB A-L 2% 2% 3% 4% 5% 6% 5 year yield 34 Capital Framework Best Practices

35 Big Bang Scenarios Represent 0.5% / 99.5% confidence For each yield viewed on its own 100% 99.5% percentile 80% 60% 40% BB up BB down BB A-L 20% 0% 0.05% Capital Framework Best Practices

36 Medium Bang A-L takes all yields at a consistent lower confidence level, selected to allow for diversification Abbreviation Hi 5 Lo 15 BB down BB up BB A-L MB A-L Definition Calibrated to market bond/swap prices of various terms Highest likely 5-year yield Lowest likely 15-year yield Big Bang all yields to their lowest likely value Big Bang all yields to their highest likely value BB Asset Liability all yields to their least solvent likely value Medium bang as for BB, but scaled back for diversification 36 Capital Framework Best Practices

37 In our example, 99.5% confidence after diversification implies 62.7% confidence before diversification * 100% 99.5% percentile 80% 60% 40% BB up BB down BB A-L MB A-L 20% 0.05% 0% * FSA (2005) suggests 94% is a more usual medium bang confidence level, when all risks are taken into account. 37 Capital Framework Best Practices

38 The scale factor to go from BB to MB is defined from the ratio of diversified capital to undiversified capital. Capital required Diversification benefit Diversified capital years years years years 6-10 years 1-5 years Capital Framework Best Practices

39 MB A-L is a scaled version of BB A-L. Because of the scaling, the MB A-L yields are close to base case, but less smooth. Spot yield 7% 6% 5% 4% 3% 2% Hi 5 Lo 15 BB up BB down BB A-L MB A-L 1% 0% Term (years) 39 Capital Framework Best Practices

40 If net assets are a linear function of bond prices, then the MB scenario exactly exhausts the required capital Assets Liabilities Hi 5 Lo 15 BB up BB down BB A-L MB 40 Capital Framework Best Practices

41 The Medium Bang scenario is unlikely. Its appearance in this chart is an artefact of the projection into 2-D. 6% 15 year yield 5% 4% 3% Likely locus Limits Hi 5 Lo 15 BB up BB down BB A-L MB A-L 2% 2% 3% 4% 5% 6% 5 year yield 41 Capital Framework Best Practices

42 The Least Solvent Likely Event minimises net assets, over all likely scenarios with SRSE 2% Abbreviation Hi 5 Lo 15 BB down BB up BB A-L MB A-L LSLE Definition Calibrated to market bond/swap prices of various terms Highest likely 5-year yield Lowest likely 15-year yield Big Bang all yields to their lowest likely value Big Bang all yields to their highest likely value BB Asset Liability all yields to their least solvent likely value Medium bang as for BB, but scaled back for diversification Least solvent likely event LSLE is the mean of a generalised scenario considered by Artzner et al (1998). 42 Capital Framework Best Practices

43 LSLE is a smooth curve which finds the vulnerabilities in a firm s net cash flow pattern Spot yield 7% 6% 5% 4% 3% 2% 1% Hi 5 Lo 15 BB up BB down BB A-L MB A-L LSLE 0% Term (years) 43 Capital Framework Best Practices

44 The LSLE also exhausts the required capital exactly. It shares this property with Medium Bang A-L Assets Liabilities Hi 5 Lo 15 BB up BB down BB A-L MB LSLE 44 Capital Framework Best Practices

45 The LSLE represents 99.5% confidence overall, but other levels of confidence for each yield taken individually. 100% percentile 80% 60% 40% 20% Hi 5 Lo 15 BB up BB down BB A-L MB A-L LSLE 0% Capital Framework Best Practices

46 LSLE is a likely test, unlike Medium Bang. Appearances can be deceptive: MB A-L hides behind likely locus 6% 15 year yield 5% 4% 3% Likely locus Limits Hi 5 Lo 15 BB up BB down BB A-L MB A-L LSLE 2% 2% 3% 4% 5% 6% 5 year yield 46 Capital Framework Best Practices

47 The Case, the LSLE and a straight line between them, represent the least solvent scenario for a given level of SRSE. Net assets Hi 5 Lo 15 BB up BB down MB A-L LSLE % 2% 3% LSLE: No likely scenario is worse Scenario shift (SRSE) 47 Capital Framework Best Practices

48 Examples using LSLE for better decision making Interest Rates and Mortality An annuity book matches projected asset and liability cash flows. The LSLE combines increased longevity with lower interest rates. Efficient interest rate hedges focus on the improved longevity scenario. Market / Policyholder Behaviour Savings products with guaranteed surrender values may imply a LSLE with increased lapses and higher interest rates. Policyholder / Operational LSLE revealed a threat from high lapses in particular market stresses. This highlighted the operational challenge of processing surrender payments, which had not previously been identified as critical. Policyholder / Volatility A savings product offers guarantees in exchange for monthly charges. The LSLE shows market falls, lower than expected lapses and a rise in market implied volatilities. 48 Capital Framework Best Practices

49 Conclusions Events are easier to communicate than distributions But you need to choose events that matter for the business Painful enough to exhaust capital Believable enough that they may happen The LSLE is different for different firms It is tempting avoid the best models to make explanation easier Dumbing-down is unnecessary, and self-defeating We need to get better at communicating good models Capital requirements and allocations are of limited use on their own Need to understand effect of key assumptions The most significant threats must be clearly understood And the likely effect of alternative business decisions 49 Capital Framework Best Practices

50 Further Technical Reading Anderson, N, Breedon, F, Deacon, M, Derry, A and Murphy, M (1996). Estimating and interpreting the yield curve. John Wiley & Sons. ISBN Artzner P, Delbaen F, Eber, J-M and Heath, D. (1998). Coherent measures of risk. Baddon, Richard and Coulthard, Paul (2006). Risk Geographies understanding risk in n dimensions. CEIOPS (2007). Term structures for QIS 3 Cont, Rama (2004) Modelling Term Structure Dynamics: An Infinite Dimensional Approach Exley, C J & Smith, A D (2006) The Cost of Capital for Financial Firms Financial Services Authority (2005). ICAS one year on. Rousseeuw, P.J. and Ruts, I. (1999), The depth function of a population distribution, Metrika, 49, ftp://ftp.win.ua.ac.be/pub/preprints/99/depfun99.pdf Smith. A.D. & Wilson, T. (2000). Fitting yield curves with long constraints. Financial Options Research Centre, Warwick. Or AndrewDSmith8@deloitte.co.uk 50 Capital Framework Best Practices

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