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? The Volatility Glare Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 2 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 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 Economic Balance Sheet in QIS5 Own Funds (Net Asset Value): equity capital Available Solvency Margin / own funds present value future earnings German specialty: going-concernreserve Liabilities: riskmargin cost for options and guarantees Technical Provisions PV of future pol.-holder participation best estimate pv of guaranteed benefits 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 Solvency II – Transition to Economic Balance Sheet - Overview Economic Balance Sheet Local GAAP / BEL EC PVFP SH + PH Market Value Assets ASM PVFPSh RM O&G PVFPPh Best Estimate Guaranteed Benefits Techn. Provision 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 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 Derivation of Solvency Ratio 1 Obtain Own Funds from Economic Balance Sheet Own Funds 2 Evaluation of required capital on given safety level for different risk types Market Risks Underwriting Risks Market value assets Market value Liabilities Operational Risks Default Risks Intangibles Assets Liabilities 3 Solvency Ratio: Available Capital over Required Capital Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 8 Required Capital Standard Formula Concept How 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 Standard Formula with Modular Approach Obtaining each Required Capital per Risk Economic Capital Market Value Assets • 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 Probability SCR = ∆ Economic Capital 0.5% Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 10 Expected Value Market Risks - Overview Interest rate risk Equity risk Property risk Spread risk Concentration risk Currency risk Illiquidity premium risk interest rates up / down scenario Market value reduction of 30% for „equities global“ and 40% for „other“ market value reduction of 25% split of spread risk in five submodules capital requirement depends on rating and duration Aggregation to market risk concentration Rrsk derived based on threshold and rating classes ± 25% change in value of asset and liability in foreign currency Reduction of observable market illiquidity premium by 65% Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 11 vereinfachte Darstellung 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 lapse cost catastrophy revision permanent increase of disability rates by 25%; first year by 35% additional reduction of reactivation rates. by 20% max over mass lapse in first year of projection, increase of lapse rates by 50%, decrease of lapse rates by 50% Aggregation to life underwriting risk permanent increase of costs by 10%; absolute increase in inflation by 1% absolute increase of mortality rates in first year of projection by 1,5‰ for policies with mortality risk irrelevant in Germany Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 12 vereinfachte Darstellung Risk Categories - Overview Market Default 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. 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 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 Risk neutral measures vs. real world explained by a bookmaker Price: 10 € per bet placement. Who wins? Bookmaker sets risk neutral rates. 1,5 3,0 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 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 The problem of arbitrage… What is a fair price of a call option? Z (T ) max S (T ) K ;0 at time T 1 with Strike K 2 The underlying can reach two values with probabilities P ( A ) S ( 0 ) : 2 1 2 and P ( B ) S (1; A ) : 4 1 , 2 S (1; B ):1 According to game theory, the fair price amounts to Z ( 0 ) E P Z (1) 2 1 2 0 1 2 1 Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 17 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 V H ( 0 ) : H 0 H 1S ( 0 ) V H 1, : H 0 H 1S 1, Replication of option payout: find strategy so that ! V H 1, Z 1, H 0 H1 0 H 0 4H 1 2 Value of replicating portofolio at time 0 is fair price: Z ( 0 ) : V H ( 0 ) 2 3 2 3 2 2 3 Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 18 H0 2 3 ; H1 2 3 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) Q 1 (2) Risk neutral price equals game theoretical price V H 0 E Q V H 1 S 0 E Q S 1 ■ For the example this leads to Q A Q B 1 4 Q A Q B 2 Q A 1 3 ; Q B 2 3 Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 19 Replicating portfolio is martingale according to equivalent martingalemeasure Q. Fair price can be calculated without knowing explicit trading strategy! 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 measure is unique equivalent martingale ■ Next steps □ Cox-Ross-Rubinstein, Cox-Ingerson-Ross, Itô, Feynman-Kacformula, Black-Scholes Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 20 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 q=½ 150 100 90 q=¼ 150 100 p=½ 90 D1=10/21 q=½ 100 p=¾ 90 risk neutral: transformation of probabilities D2=30/21 discounting by stochastic deflators 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 Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 21 p=½ 150 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] WBA SA N z1 SB N z2 S 1 ln A 2 t S 2 z1 B t , with z2 z1 t A2 B2 2 AB A B difficulty here: finding the „right“ volatilities Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 22 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 Life Insurance is a Complex Business long term contracts P&L statement does not show Lifetime annuity up to 70 years profitability of life insurance business 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 Liabilities can be of any complexity floor + floater examples of options and guarantees in life insurance guarantees Policyholder options Counteroptions insurer ■ Guaranteed rate □ Guaranteed over entire term □ Has to be earned by undertaking on financial market ■ lapse □ Policyholders can withdraw contract at any time □ Possible realisation of hidden losses ■ Discretionary benefits □ reversionary bonus raises guarantee □ Terminal bonus not guaranteed predictability ■ Guarantee of premiumsof cashflows □ annuities: death benefit is paid up premiums. □ Due to acquisition cost not available in the beginning ■ 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.) Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 25 uncertain cash out Model undertaking and guide it through 1,000 – 10,000 scenarios assets start of projection prices market model asset model and rates liabilities sh equity assets net asset return technical provisions mgmt model total return 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 liability model Modelling needs a ton of information asset manager actuary ■ cash flow modelling ■ cash flow modelling ■ asset classes ■ products □ type □ Line of business □ region □ tariff □ structure □ Age / sex □ maturity □ Term structure □ direct investments □ private / pension scheme Court of Justice EU ■ reference indices ■ mapping of unmodeled products ■ fungibility ■ policy holder behavior ■ distribution on classes ■ bonus participation rules ■ rebalancing ■ solvency ratio ■ economic evaluation ■ liability reserves („RfB“, „estate“) ■ accounting environment ■ accounting environment ■ hidden reserves ■ profit sources Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 27 EIOPA Some product parameters Parameters for technical provisions and pricing additional forecast parameters ■ guaranteed rate ■ net asset return per scenario ■ mortality tables ■ observed mortality ■ occupational disability and other tables ■ observed other termination rates ■ cost parameters (which reference, which rate,…) ■ realistic costs ■ discretionary bonus parameters (which scheme, which reference, which rate, reversionary / discretionary,…) ■ 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 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 Which quality level of answers does a model produce? data storage findings undertakings software needs to store historic data relative comparisons valuable quality of cahs flows stochastic simulations? volatility of results Steering possibilities risk exposure Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 30 Deep and liquid markets – dispelling the myths long term EUR-market shows vacancies… 200.000 90% 95% 180.000 160.000 98% 99% 174 179 Euro market 150 Outstanding Amount (millions) 139 140.000 120.000 100.000 100 90 80.000 79 60.000 53 49 40.000 19 16 9 13 6 614 7 12 1 1 13 2 20.000 20 0 1 3 5 Source: Barrie & Hibbert 7 1 1 31 2 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 time to maturity in years Labels denote number of bonds Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 31 Deep and liquid markets – dispelling the myths different market – different deepness 25.000 98% 90% 95% 99% GBP market, amounts denoted in EUR 45 43 20.000 Outstanding Amount (millions) 43 35 33 15.000 39 23 28 29 24 25 21 20 10.000 18 22 23 98 14 5.000 11 9 13 10 10 58 6 10 4 6 4 4 2 2 1 1 11 1 0 1 3 5 7 Source: Barrie & Hibbert 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 time to maturity in years Labels denote number of bonds Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 32 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 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) Danish hedge (mandatory) Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 34 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 Shareholders take a simple point of view risk is change of undertaking‘s embedded value What are key drivers of risk? Insurer A Insurer B How sensitive is the MCEV*? Which scenarios are hazardous? Value Added Analysis Sensitivity Analysis Aggregation of sub risks? mean 50% of all values in this area 90% of all values in this area 98% of all values in this area RoE spells RARoRAC nowadays! Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 36 *MCEV: Market Consistent Embedded Value What do you really want to know? ■ one quantile? ■ the expected loss on one quantile? ■ the loss distribution? Probability Expected Value 0.5% Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 37 What is „the right“ risk measure? VaR TVaR / CTE stresstests Depends! Certain characteristics are important, c.f. David Blake, „After VaR“ Coherent risk measures: monotonicity: V Y V X X Y subadditivity: X Y X Y positive homogeneity: hX h X h 0 translational invariance: X n X n Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 38 for some certain amount n 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 Tails are of interest for a life insurer Few scenarios render extraordinary profits and the same goes for losses V e r te ilu n g P V F P s Illustrative disitribution of PVFPs* 3 000 000 000 ,00 3,000 Mio. 2 000 000 000 ,00 2,000 Mio. 1,000 Mio. 1 000 000 000 ,00 0 2004 0 ,00 1 44 87 130 173 216 259 302 345 388 431 474 -1,000 Mio. -1 0 0 0 0 0 0 0 0 0 , 0 0 -2.000 Mio. -2 0 0 0 0 0 0 0 0 0 , 0 0 -3,000 Mio. -3 0 0 0 0 0 0 0 0 0 , 0 0 *PVFP: present value of future profits Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 40 517 560 603 646 689 732 775 818 861 904 947 990 2005 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 The volatility glare: asset-liability position looses value for both up and down change of interest rates 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 Reduce risk without eliminating franchise value Receiver swaptions reduce risk Reduced guarantee risk Policyholder protected liabilities Reduced bonus participation Shareholders interest rate risk Raising franchise value assets discretionary benefits but as an option change risk-return profile of asset-liability position Crucial Paths in Stochastic Modelling for Life Insurers – Spring School Jena 2011 /page 43 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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# Crucial Paths in Stochastic Modelling for Life Insurers