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Nils Dennstedt, Appointed Actuary, march 23

Crucial Paths in Stochastic Modelling for Life Insurers Spring School „Stochastic Models in Finance and Insurance“ Jena March 21 through April 1. Nils Dennstedt, Appointed Actuary, march 23. Agenda. What is the Matter? Big Deal! What is the Complexity? What is the Risk?

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Nils Dennstedt, Appointed Actuary, march 23

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  1. Crucial Paths in Stochastic Modelling for Life InsurersSpring School „Stochastic Models in Finance and Insurance“ Jena March 21 through April 1 Nils Dennstedt, Appointed Actuary, march 23

  2. Agenda What is the Matter? Big Deal! What is the Complexity? What is the Risk? The Volatility Glare  Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 2

  3. Agenda What is the Matter? Solvency II – Quick Overview Big Deal! Simple Approach on Pricing Models Risk Neutral Valuation What is the Complexity? A Typical Insurance Product What is the Risk? Key Risk Drivers The Choice of Risk Measure The Volatility Glare  Steering Mechanisms in Life Insurers Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 3

  4. Agenda What is the Matter? Big Deal! What is the Complexity? What is the Risk? The Volatility Glare  Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 4

  5. Economic Balance Sheet in QIS5 Own Funds (Net Asset Value): equity capital present value future earnings German specialty: going-concern-reserve Available Solvency Margin / own funds Liabilities: riskmargin cost for options and guarantees PV of future pol.-holder participation best estimate pv of guaranteed benefits TechnicalProvisions deferred taxes market value other liabilities Assets: market value assets market value other assets Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 5

  6. Solvency II – Transition to Economic Balance Sheet - Overview Economic Balance Sheet Local GAAP / BEL ASM Market Value Assets EC PVFP SH + PH PVFPSh RM Techn.Provision O&G PVFPPh Best Estimate Guaran-teed Benefits RM=Risikmargin, PVFP= Presetn Value of Future Profits, SH = Shareholder, PH = Policyholder, O&G = Time Value of Options and Guarantees, EC = Economic Capital, ASM = Available Solvency Margin, BEL = Best Estimate Liabilities Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 6

  7. Evaluation of Solvency Capital Required (SCR) SCR (Solveny Capital Required): Required capital due to change of economic capital under stress change in economic capial = Net-SCR Economic capital Market value of assets Market value of insurance liabilities Stress Marktwert KA example: interest rates down (due to higher duration of liabilities market value of liabilities grows more than market value of fixed income assets) Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 7

  8. 2 3 1 Derivation of Solvency Ratio Obtain Own Funds from Economic Balance Sheet Market value assets Own Funds Evaluation of required capital on given safety level for different risk types Market value Liabilities Market Risks Required Capital Underwriting Risks Operational Risks Default Risks Intangibles Assets Liabilities Solvency Ratio: Available Capital over Required Capital Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 8

  9. Standard Formula ConceptHow to Obtain the Solvency Capital Required (SCR) After Risk Mitigation and OpöRisk 2. Level of Aggregation 1. Level of Aggregation Source: QIS 5 Technical Specifications, p. 90 Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 9

  10. Probability ExpectedValue 0.5% Standard Formula with Modular ApproachObtaining each Required Capital per Risk Market Value Assets Economic Capital • Required capital equals change of economic capital in stress scenario • Required Capital is determined on a defined safety level on a one year horizon • Safety level for 1-year-VaR equals 99,5%. Market Value Liabilities SCR = ∆ Economic Capital Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 10

  11. Market Risks - Overview Interest rate risk interest rates up / down scenario Equity risk Market value reduction of 30% for „equities global“ and 40% for „other“ Property risk market value reduction of 25% Spread risk split of spread risk in five submodules capital requirement depends on rating and duration Aggregation to market risk Concentration risk concentration Rrsk derived based on threshold and rating classes Currency risk ± 25% change in value of asset and liability in foreign currency Illiquidity premium risk Reduction of observable market illiquidity premium by 65% Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 11 vereinfachte Darstellung

  12. Underwriting Risks - Overview mortality permanent increase of moartality rates by 15% for policies with mortality risk longevity permanent decrease of moartality rates by 20% for policies with longevity risk disability permanent increase of disability rates by25%; first year by 35% additional reduction of reactivation rates. by 20% Aggregation to life underwriting risk lapse max over mass lapse in first year of projection, increase of lapse rates by 50%, decrease of lapse rates by 50% cost permanent increase of costs by 10%; absolute increase in inflation by 1% catastrophy absolute increase of mortality rates in first year of projection by 1,5‰ for policies with mortality risk revision irrelevant in Germany Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 12 vereinfachte Darstellung

  13. Risk Categories - Overview Market Aggregation of market risks by covariance matrix. The matrix is depends on whether interest rate up or interest rate down is the key risk in the interest rate category. Default Die Kapitalanforderung für das Gegenparteiausfallrisiko hängt von Typ und Rating der Forderung ab. aggregation to BSCR Life Aggregation of Life Underwriting Risks by covariance matrix. Intang For intangible assets a loss in value of 80% is assumed. Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 13

  14. Agenda What is the Matter? Big Deal! What is the Complexity? What is the Risk? The Volatility Glare  Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 14

  15. Who wins? 1,5 3,0 Risk neutral measures vs. real world explained by a bookmaker Price: 10 € per bet placement. Bookmaker sets risk neutral rates. rates get set according to incoming bets. Brazil gets twice the votes since clear favorite. Bookmaker calculates: 2.000 x 10 € x 1,5 = 30.000 € 1.000 x 10 € x 3,0 = 30.000 € Bookmaker is free of risk, so no dependancy on turnout and the real probability of Brazil to win Risk neutral probabilities are (derived from rates and thus equal to market prices): Brazil: 2/3, England 1/3 Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 15

  16. What do customers (fans) think in real life? Fan checks available information and weighs subjectively strengths home / guest current trends injured players etc. Say, (subjective) real world probability turns out to be ¾, thus more than expected by bet rates. • Investment seems attractive • but there is risk: (brazil can still loose) Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 16

  17. The problem of arbitrage… What is a fair price of a call option? at time with Strike The underlying can reach two values with probabilities and , According to game theory, the fair price amounts to Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 17

  18. leads to martingales! A financial investor can also invest directly into the underlying which leads to risk neutral pricing. • A risk neutral fair price is the value of the replicating portfolio • The game theoretical fair price leads to arbitrage • Risk neutral fair prices can be seen as expected value of option payout under the equivalent martingale Portfolio: Invest in Cash and Underlying Replication of option payout: find strategy so that Value of replicating portofolio at time 0 is fair price: Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 18

  19. If price differs from risk neutral price you got arbitrage! • Market price < risk neutral price • Buy option • Sell replicating portfolio for risk neutral option price • Market price > risk neutral price (opposite way around!) • Martingale constraint can be derived! (1) (2) Risk neutral price equals game theoretical price • For the example this leads to Replicating portfolio is martingale according to equivalent martingalemeasure Q. Fair price can be calculated without knowing explicit trading strategy! Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 19

  20. What makes martingale measure unique? • Harrison-Pliska • No arbitrage implies existence of martingale measure • Completeness-Theorem • (For each derivative a replicating portfolio exists) • Assumption (no arbitrage) • Market model is complete equivalent martingale measure is unique • Next steps • Cox-Ross-Rubinstein, Cox-Ingerson-Ross, Itô, Feynman-Kac-formula, Black-Scholes Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 20

  21. Risk neutral scenarios vs. deflators You can use real world scenarios for pricing if you apply stochastic deflators e.g., risk free rate set to 5%, equity can have two states after one year 150 150 150 100 100 100 90 90 90 Deflators contain information on probability transformation explicitly • pro: Same scenario set can be used for pricing and risk assessment • con: • numerically less stable • generating deflators is a complex matter D1=10/21 q = ½ q = ¼ q = ½ D2=30/21 p = ½ p = ¾ p = ½ risk neutral: transformation of probabilities discounting by stochastic deflators Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 21

  22. No matter what – two scenario sets are needed Pricing: risk neutral environment Risk assessment / investment: real world environment Alternatively, evaluation of financial guarantees might work through a closed formula approach: policyholder can „switch“ assets, namely value of guaranteed benefit (A) versus asset value (B = A + future bonuses) [Margrabe-Option] with difficulty here: finding the „right“ volatilities Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 22

  23. Agenda What is the Matter? Big Deal! What is the Complexity? What is the Risk? The Volatility Glare  Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 23

  24. Life Insurance is a Complex Business P&L statement does not show profitability of life insurance business long term contracts Lifetime annuity up to 70 years Embedded Value • Evaluate value of undertaking by future profits and losses • Analysis of undertaking’s profitability Steering by (Market Consistent) Embedded Value Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 24

  25. floor + floater examples of options and guarantees in life insurance Policyholder options Counteroptions insurer guarantees predictability of cashflows uncertain cash out Liabilities can be of any complexity • lapse • Policyholders can withdraw contract at any time • Possible realisation of hidden losses • Discretionary benefits • reversionary bonus raises guarantee • Terminal bonus not guaranteed • Guaranteed rate • Guaranteed over entire term • Has to be earned by undertaking on financial market • Lump sum option • annuities: lump sum payment instead of life time annuity payment • Possible realisation of hidden losses • Change in risk exposure of asstes (e.g. equity or real estate ratio, corporates vs. Financials, etc.) • Guarantee of premiums • annuities: death benefit is paid up premiums. • Due to acquisition cost not available in the beginning Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 25

  26. Model undertaking and guide it through 1,000 – 10,000 scenarios assets liabilities sh equity start of projection assets technical provisions asset model liability model prices mgmt model total return net asset return market model and rates asset allocation cash flows liabilities sh equity assets technical provisions end of projection local gaap and p&l Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 26

  27. Court of Justice EU EIOPA Modelling needs a ton of information • cash flow modelling • asset classes • type • region • structure • maturity • direct investments • reference indices • fungibility • distribution on classes • rebalancing • economic evaluation • accounting environment • hidden reserves asset manager actuary • cash flow modelling • products • Line of business • tariff • Age / sex • Term structure • private / pension scheme • mapping of unmodeled products • policy holder behavior • bonus participation rules • solvency ratio • liability reserves („RfB“, „estate“) • accounting environment • profit sources Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 27

  28. Parameters for technical provisions and pricing additional forecast parameters Some product parameters • guaranteed rate • mortality tables • occupational disability and other tables • cost parameters (which reference, which rate,…) • discretionary bonus parameters (which scheme, which reference, which rate, reversionary / discretionary,…) • net asset return per scenario • observed mortality • observed other termination rates • realistic costs • policholder lapse behavior • lump sum pick up rates • Payment exemption rates • Indexation cancellation rates • … Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 28

  29. Do undertakings know their specific parameters? lump sum pick up in 2035? lapse behavior with 10y gov‘t at 7,5% and total return policyholder at 4,9%? life expectancy of a 65-year-old in 2045? indexation: termination rate in 2025? social security retirement age at 2030? Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 29

  30. Which quality level of answers does a model produce? data storage undertakings software needs to store historic data quality of cahs flows stochastic simulations? findings relative comparisons valuable volatility of results Steering possibilities risk exposure Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 30

  31. Deep and liquid markets – dispelling the mythslong term EUR-market shows vacancies… 98% 99% 90% 95% Euro market Source: Barrie & Hibbert Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 31

  32. Deep and liquid markets – dispelling the mythsdifferent market – different deepness 98% 99% 90% 95% GBP market, amounts denoted in EUR Source: Barrie & Hibbert Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 32

  33. Non liquid markets present significant (model) risk A solvency model which relies significantly on singular data outside a deep and liquid market horizon in order to determine technical provisions is deemed to produce (high) model errors • For the undertaking itself with long term liabilitites • For the supervisor of undertakings with long term liabilities • For markets also! • Strong impact on market prices in nonliquid segment due to minor changes in demand • Possible increase in demand due to supervisory regime (and thus self-energizing!) • Change in demand by explicit speculation against undertaking with long term liabilities Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 33

  34. Danish hedge (mandatory) Influence of mandatory option hedge in danish market on € fixed income market • Danish Supervisor introduces new stress test / solvency system in 2001 • Swaption volatility exploded due to sudden demand by Danish life insurance industry • For the same reason flattening of €-yield curve (falling forward rates) Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 34

  35. Agenda What is the Matter? Big Deal! What is the Complexity? What is the Risk? The Volatility Glare  Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 35

  36. Insurer A Insurer B mean 50% of all values in this area 90% of all values in this area 98% of all values in this area Shareholders take a simple point of view risk is change of undertaking‘s embedded value What are key drivers of risk? How sensitive is the MCEV*? Which scenarios are hazardous? Value Added Analysis Sensitivity Analysis Aggregation of sub risks? RoE spells RARoRAC nowadays! *MCEV: Market Consistent Embedded Value Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 36

  37. Probability ExpectedValue 0.5% What do you really want to know? • one quantile? • the expected loss on one quantile? • the loss distribution? Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 37

  38. What is „the right“ risk measure? VaR TVaR / CTE stresstests Certain characteristics are important, c.f. David Blake, „After VaR“ Coherent risk measures: monotonicity: subadditivity: positive homogeneity: translational invariance: for some certain amount n Depends! Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 38

  39. Agenda What is the Matter? Big Deal! What is the Complexity? What is the Risk? The Volatility Glare  Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 39

  40. Illustrative disitribution of PVFPs* 3,000 Mio. 2,000 Mio. 1,000 Mio. 0 -1,000 Mio. -2.000 Mio. -3,000 Mio. *PVFP: present value of future profits Tails are of interest for a life insurer Few scenarios render extraordinary profits and the same goes for losses Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 40

  41. Duration and Convexity measure price sensitivity of bonds • Duration: linear approximation of change in price • Convexity: quadratic approximation of change in price Current yield Interest rates down Interest rates up Source: J. Willing, MunichRe Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 41

  42. The volatility glare: asset-liability position looses value for both up and down change of interest rates  negative convexity A/L position = asset - liability Negative convexity needs risk management attention Source: J. Willing, MunichRe Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 42

  43. Policyholder protected liabilities Shareholders interest rate risk Raising franchise value assets discretionary benefits Reduce risk without eliminating franchise value Receiver swaptions reduce risk but as an option change risk-return profile of asset-liability position Reduced guarantee risk Reduced bonus participation Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 43

  44. What‘s in the takeaway box? • Definitely crucial paths for life insurers detected • Long term complex path dependant liabilities not easily replicated by financial instruments • Complex parameter structure might not be easy to set and to monitor • Deep and liquid markets might not always exist • Market wide alignment of risk strategies can severely impact markets • Life insurance products might change • Stochastic modelling bears some risk Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 44

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