Capital Framework Best Practices. IAA Life Section Seminar 19 April 2007 Andrew D Smith AndrewDSmith8@deloitte.co.uk
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
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
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
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
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
Example liability cash flows 600 Cash flow 500 400 300 200 100 0 0 5 10 15 20 25 30 year 7 Capital Framework Best Practices
Example asset cash flows Cash flow 0 5 10 15 20 25 30 year 8 Capital Framework Best Practices
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
Diversification has a large effect for yield curve models Because yields at adjacent terms are highly correlated. Capital required 700 600 500 400 300 200 Diversification benefit Diversified capital 25-30 years 21-25 years 16-20 years 11-15 years 6-10 years 1-5 years 100 0 10 Capital Framework Best Practices
Our example uses this base yield curve. Spot yield 7% 6% 5% 4% 3% 2% 1% 0% 0 5 10 15 20 25 30 Term (years) See CEIOPS (2007) for free yield curve data for European and other economies. 11 Capital Framework Best Practices
Duration and Convexity years 12 10 8 Assets Liabilities 6 4 2 0 Duration Sqr(Convexity) 12 Capital Framework Best Practices
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
We chose the Net Assets equal to the Required Capital 5500 5000 Assets Liabilities Diversified capital 4500 4000 3500 3000 14 Capital Framework Best Practices
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
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
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
, Hi 5 and Lo 15 yield curves Spot yield 7% 6% 5% 4% Hi 5 Lo 15 3% 2% 1% 0% 0 5 10 15 20 25 30 Term (years) 18 Capital Framework Best Practices
Stress Test Effect on Assets and Liabilities 5500 5000 Assets Liabilities 4500 4000 3500 3000 Hi 5 Lo 15 19 Capital Framework Best Practices
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
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
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
The Hi 5 scenario is slightly beneficial for net assets, even though the asset duration exceeds the liability duration. Net assets 150 100 Hi 5 Lo 15 50 0 0% 1% 2% 3% -50 Scenario shift (short rate stress equivalent) 23 Capital Framework Best Practices
We can also express likelihood using a correlation matrix A mathematically equivalent formulation to the SRSE * correlation 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 5 10 15 Yield term 20 25 30 0 5 10 15 20 25 30 Yield term 0.95-1 0.9-0.95 0.85-0.9 0.8-0.85 0.75-0.8 0.7-0.75 0.65-0.7 0.6-0.65 0.55-0.6 0.5-0.55 * 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
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
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% 0 5 10 15 20 25 30 Term (years) 26 Capital Framework Best Practices
Big Bang test results are more extreme than Hi and Lo likely scenarios 5500 5000 Assets Liabilities 4500 4000 3500 3000 Hi 5 Lo 15 BB up BB down 27 Capital Framework Best Practices
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
Big Bang effect on net assets Net assets 150 100 50 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
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
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
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% 0 5 10 15 20 25 30 Term (years) 32 Capital Framework Best Practices
The Big Bang scenario is ruinous, as liabilities > assets BB is unlikely: this does not contradict capital calculation. 5500 5000 Assets Liabilities 4500 4000 3500 3000 Hi 5 Lo 15 BB up BB down BB A-L 33 Capital Framework Best Practices
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
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% 0 5 10 15 20 25 30 35 Capital Framework Best Practices
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
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% 0 5 10 15 20 25 30 * FSA (2005) suggests 94% is a more usual medium bang confidence level, when all risks are taken into account. 37 Capital Framework Best Practices
The scale factor to go from BB to MB is defined from the ratio of diversified capital to undiversified capital. Capital required 700 600 500 400 300 200 Diversification benefit Diversified capital 25-30 years 21-25 years 16-20 years 11-15 years 6-10 years 1-5 years 100 0 38 Capital Framework Best Practices
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% 0 5 10 15 20 25 30 Term (years) 39 Capital Framework Best Practices
If net assets are a linear function of bond prices, then the MB scenario exactly exhausts the required capital 5500 5000 Assets Liabilities 4500 4000 3500 3000 Hi 5 Lo 15 BB up BB down BB A-L MB 40 Capital Framework Best Practices
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
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
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% 0 5 10 15 20 25 30 Term (years) 43 Capital Framework Best Practices
The LSLE also exhausts the required capital exactly. It shares this property with Medium Bang A-L 5500 5000 Assets Liabilities 4500 4000 3500 3000 Hi 5 Lo 15 BB up BB down BB A-L MB LSLE 44 Capital Framework Best Practices
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% 0 5 10 15 20 25 30 45 Capital Framework Best Practices
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
The Case, the LSLE and a straight line between them, represent the least solvent scenario for a given level of SRSE. Net assets 150 100 50 Hi 5 Lo 15 BB up BB down MB A-L LSLE 0-50 0% 2% 3% LSLE: No likely scenario is worse Scenario shift (SRSE) 47 Capital Framework Best Practices
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
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
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 0-471-96207-4. Artzner P, Delbaen F, Eber, J-M and Heath, D. (1998). Coherent measures of risk. http://www.math.ethz.ch/~delbaen/ftp/preprints/coherentmf.pdf Baddon, Richard and Coulthard, Paul (2006). Risk Geographies understanding risk in n dimensions. http://www.actuaries.org.uk/files/pdf/proceedings/erm20060703/coulthard.pdf CEIOPS (2007). Term structures for QIS 3 http://www.ceiops.org/media/files/consultations/qis/qis3/qis3termstructures.xls Cont, Rama (2004) Modelling Term Structure Dynamics: An Infinite Dimensional Approach http://ssrn.com/abstract=148408 Exley, C J & Smith, A D (2006) The Cost of Capital for Financial Firms http://www.actuaries.org.uk/files/pdf/sessional/sm20060123.pdf Financial Services Authority (2005). ICAS one year on. http://www.fsa.gov.uk/pubs/other/isb_icas.pdf Rousseeuw, P.J. and Ruts, I. (1999), The depth function of a population distribution, Metrika, 49, 213-244. 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. http://wwwcfr.jbs.cam.ac.uk/archive/presentations/seminars/lent2001/asmithyield.pdf Or email AndrewDSmith8@deloitte.co.uk 50 Capital Framework Best Practices
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