Das Risikokapitalmodell der Allianz Lebensversicherungs-AG

Transcription

1 Das Risikokapitalmodell der Allianz s-ag Ulm 19. Mai 2003 Dr. Max Happacher Allianz s-ag

2 Table of contents 1. Introduction: Motivation, Group-wide Framework 2. Internal Risk Model: Basics, Life Approach 3. Standalone Risks and Liability Response 4. Simulation of Assets & Fair Value of Liabilities 5. Liability Response: Dynamic Reseves/Benefits 6. Liability Response: Valuation of Liabilities 7. Risk Aggregation, Diversification and Mitigation s-ag 2

3 Table of contents 1. Introduction: Motivation, Group-wide Framework 2. Internal Risk Model: Basics, Life Approach 3. Standalone Risks and Liability Response 4. Simulation of Assets & Fair Value of Liabilities 5. Liability Response: Dynamic Reseves/Benefits 6. Liability Response: Valuation of Liabilities 7. Risk Aggregation, Diversification and Mitigation s-ag 3

4 Main goals for Allianz internal risk analysis Supporting value management Systematic risk assessment based on internal data Improving risk adjusted profitability measurement Portfolio management / asset mix Capital allocation Compare outcome against rating agencies models Meeting legal requirements Risk reporting More detailed quantitative requirements by DRSC or SEC Solvency New solvency requirements on the way; new Basel II-like requirements for insurance companies (Solvency II) and financial conglomerates allow for internal risk analysis models s-ag 4

5 Risk analysis project covering AZ Group flagships PC: SGD, AZRE, Cornhill, AZ Elementar, AZ Suisse, RAS, Lloyd Adriatico, AZ Seguros, AGF, FFIC, AIC, AZ Australia, Hermes/Euler Life: AZ Leben, AZ Elementar, AZ Suisse, RAS, Lloyd Adriatico, AZ Seguros, AGF, AZ Life, AZ First Life Banking: Dreba Group Consistent framework of risk categories for all segments Respect material differences between & within segments by specific risk models s-ag 5

6 Consistent framework for risk analysis, risk capital & capital allocation Insurance Risks Premium Risk Prospective Claims Pricing Non-Cat PC Risk Reserve Risk Retrospective Claims Life Risk Biometric Risks - Mortality - Longevity - Disability - Calamity Counterparty Risk Reinsurer Credit Worthiness (Security) Market Risk Market Risks Credit Risks Other Risks Operating Risk Business Risk - Lapse - Cost Operational Event Risk - IT failure - Litigation - Fraud Cat Risks of AZ Leben Link to S&P standard categories C3 C4 C5 C2 C1 C6 s-ag 6

7 Table of contents 1. Introduction: Motivation, Group-wide Framework 2. Internal Risk Model: Basics, Life Approach 3. Standalone Risks and Liability Response 4. Simulation of Assets & Fair Value of Liabilities 5. Liability Response: Dynamic Reseves/Benefits 6. Liability Response: Valuation of Liabilities 7. Risk Aggregation, Diversification and Mitigation s-ag 7

8 Capital requirement depends on point of view ASSETS Market Value LIABILITIES Best Estimates Required Capital Excess Capital Regulatory Capital Rating Agency Capital (Economic) Risk Capital Net Asset Value = Economic Capital Internal risk approach measures the required risk capital on an economic point of view s-ag 8

9 Economic Capital is volatile due to unexpected developments affecting assets and liabilities ASSETS Market Value LIABILITIES Fair Value Large Claims Economic Capital INSOLVENCY ECONOMIC CAPITAL (NAV) BE NAV 0 Probability Market Crisis All expected profits and losses are included in the best estimate NAV, unexpected developments induce volatility or risk into the business s-ag 9

10 Risk Capital: Potential loss of expected NAV for the next 12 months (VaR approach) SOLVENCY STANDARD AAA 0.01% AA 0.03% A 0.07% Probability Density Key parameters Confidence interval / solvency level Risk distribution Risk correlation Risk concentration Worst case Diversification Worst Case 0 Required Risk Capital Expected NAV NAV Risk capital is the capital required to run the business tied to a given solvency / worst case level s-ag 10

11 Modelling risks - conceptional overview Standalone risks Aggregation Mitigation Equity Interest rate Real estate Credit Life FX Business Operational event Liability response via simulation Risks shareable with policyholder (correlated) Non-shareable risks BE WC IR CV IR WC EQ/RE CV EQ/RE WC Shar. CV Shar. CV Credit CV FX CV Life CV Business CV Op. Event TCV - Total Change in Value CV IR CV EQ Busin. Life Op. Shareholder Policyholder Diversification s-ag 11

12 Risk sharing with policyholder... completely via profit sharing ( liability response ) Equities (EQ) Real estate (RE) Interest rate (IR)... partially via profit sharing: Credit (Default) FX Life (calamity, volatility, trend) Business (lapse, new business, cost inflation)... not sharable via profit sharing: Operational event } via asset returns s-ag 12

13 Table of contents 1. Introduction: Motivation, Group-wide Framework 2. Internal Risk Model: Basics, Life Approach 3. Standalone Risks and Liability Response 4. Simulation of Assets and Fair Value of Liabilities 5. Liability Response: Dynamic Reseves/Benefits 6. Liability Response: Valuation of Liabilities 7. Risk Aggregation, Diversification and Mitigation s-ag 13

14 Modelling life risks: worst case change in value Change in Value (CV) as difference between asset value impact and liability response Value Best Estimate Asset Value Worst Case Asset Value Impact Investment Change in Asset Value (CV) Value Impact Liability Value Impact Worst Case Best Estimate Liability Value Liability Value Impact s-ag 14

15 Standalone risks for life insurance Worst case shock scenarios According to 0.07%-quantile of corresponding distribution Impact of shock in the first year Propagation for next n years necessary for valuation of liabilities Three shocks are considered: Equity: 1st year loss of roundabout 40 % Interest rates: down shift of yield curve by 1,5 % Biometrical, operational and business risks lead to an impairment of gross profits s-ag 15

16 Equity risk... before liability response Market value view (after hedging) Risk driver: volatility of stock returns Equity portfolio modeled by 4 indices (DAX100, MSCI: EMU, EUROPE, WORLD) Lognormal distribution Best estimate (BE) return: 8,5% Total volatility:» historical volatilities of indices» covariance matrix for correlation effects Issues: time frame, ex post / ex ante, private equity, derivatives Probability Worst Case Best Estimate Risk Capital Portfolio Value s-ag 16

17 IR down risk: Risky present value due to volatile discount rate Down-side Risk Down shift of interest rate curve in year 1 Shifted IR curve also used in years 2 to n Lower discount rate leads to higher values of (fixed income) assets and liabilities Present Value t=0 t=1 t=2 t=3 t=4 Yield (%) Time Yield Curve Time Increase of liabilities higher due to higher duration Impact of lower asset returns on the liabilities lessened by guaranteed rate for P/H s-ag 17

18 Other sharable risks... before liability response Risk category Investment (cont d.) Life Subrisks Method Value [Mio. ] Credit FX Calamity Volatility Trend Distribution/ Simulation Analytic GdV-Model GdV-Model GdV-Model > 1000 small < 10* > 10* > 100* Ratio PH:SH Issues 50:50 Huge portfolio 100:0 80:20 80:20 Hedges Availability of internal data Business Lapse New business Cost inflation GdV-Model GdV-Model Proxy > 10 > 100 > 10 80:20 80:20 80:20 Availability of internal data Total aggregated sharable risks: ~ 1 Mrd. * Values after reinsurance s-ag 18

19 Table of contents 1. Introduction: Motivation, Group-wide Framework 2. Internal Risk Model: Basics, Life Approach 3. Standalone Risks and Liability Response 4. Simulation of Assets & Fair Value of Liabilities 5. Liability Response: Dynamic Reseves/Benefits 6. Liability Response: Valuation of Liabilities 7. Risk Aggregation, Diversification and Mitigation s-ag 19

20 Fair value of liabilities is the present value of future cash flows... is influenced by asset movements: Liability cash flows (usually) depend on future portfolio (asset) returns Profit sharing creates a tie between asset return and liability value Management rules to describe the interdependence... depends on interest rates: Present value of future liability cash flows changes with moving interest rates s-ag 20

21 Fair value of liabilities: Models Deterministic world: Future asset returns are assumed to be given Future cash flows on basis of a known profit sharing scheme Options of insurer and PH are negligible Real world: Returns are volatile Profit sharing according to actual return Options of insurer and PH have to be considered s-ag 21

22 Guaranteed rates have negative financial value for the shareholder Investment Return Best Estimate Return Guaranteed rate The risk of investment return below guaranteed rate is pure shareholder risk Time Investment return paths may be below guaranteed rate s-ag 22

23 The cash flows of profit sharing products are only dependent on the investment return above the guaranteed rate Investment Return Guaranteed rate LIABILITY CASH FLOWS (Schematic) Time Thus financially the liability cash flow structure of this product is similar to a floor on the investment return s-ag 23

24 Pricing this floor can easiest be done by simulation 1. Simulate different scenarios of future interest rates and inv. returns Scenario i Year 1 Year 2 Year 3 Year 4 Year 5 Investment Return 7.0% 6,5% 5,5% 2,5% 3,0% Interest Rates 5.5% 4,5% 3,0% 2,0% 2,8% 2.Calculate in each scenario the corresponding future liability cash flows Scenario i Year 1 Year 2 Year 3 Year 4 Year 5 Cash Flow Discount these cash flows at the simulated interest rates 5 1 Value _ Scenario _ i = = Cash _ Flow( k) k 1 (1 + Interest _ Rate( k)) 4.Fair Value: Average value over all scenarios Fair _ Value = ( ) n Value _ Scenario _ i / n i =1 k s-ag 24

25 Portfolio return: Combined return of asset classes FIXED INCOME RET. EQUITY RETURN REAL ESTATE RETURN Return Return Return Time Time Time PORTFOLIO RETURNS Return Time s-ag 25

26 Simulation of equity / real estate Input: Expected asset performance µ 2 Volatility of the portfolio σ (cont. compounded) Output: 1-year performance P Valuet + 1 Model: P = ~ LogN ρ,σ ; Value therefore: t 2 2 σ σ ρ = log(1 + µ ) = i 2 2 E( P) = 1+ µ = e ρ + σ 2 / 2 with i the cont. compounded return expectation s-ag 26

27 Simulation of FI-returns Input: Portfolio statement comprising the cash flows of the FI portfolio (CFs for each investment) Model for simulation of future interest rates Assumptions: Reinvestment in risk free bonds with same duration as initial portfolio Bonds are held to maturity Simulation model for interest rates: Crucial: IR model has to be arbitrage free s-ag 27

28 Table of contents 1. Introduction: Motivation, Group-wide Framework 2. Internal Risk Model: Basics, Life Approach 3. Standalone Risks and Liability Response 4. Simulation of Assets & Fair Value of Liabilities 5. Liability Response: Dynamic Reseves/Benefits 6. Liability Response: Valuation of Liabilities 7. Risk Aggregation, Diversification and Mitigation s-ag 28

29 Dynamic reserves and benefits Problem: For each simulation run a new liability projection necessary Retrospective Approach: Two projections are used: declared profit sharing (BE) and minimum guaranteed (MG) Basis is the minimum guaranteed projection run For each simulation run dynamic reserves are generated retrospectively via interpolation Dynamic benefits are generated via scaling s-ag 29

30 From simulated returns to dynamic reserves and benefits PH credited rate cr t as moving n-year average of simulated returns (after shareholder profit) Initial reserves R 0 and initial cash flows Rim 1 identical for BE and MG Dynamic reserves calculated retrospectively: Dynamic benefits according reserves at b.o.y.: Rim R + t = ( 1 + max(mg, crt )) * Rt 1 Rimt + = MG Rim t Rt R Rt R Rim Valuation reserves are used as buffer ( Rim MG t 1 BE t + 1 MG t + 1 BE t MG t ) s-ag 30

31 Table of contents 1. Introduction: Motivation, Group-wide Framework 2. Internal Risk Model: Basics, Life Approach 3. Standalone Risks and Liability Response 4. Simulation of Assets & Fair Value of Liabilities 5. Liability Response: Dynamic Reseves/Benefits 6. Liability Response: Valuation of Liabilities 7. Risk Aggregation, Diversification and Mitigation s-ag 31

32 Valuation of cash flows Central problem: Appropriate discount factors for cash flows Approach: Simulate asset returns using actual return expectations ( i. e. 5 % on RE, 8,5 % on EQ,...) Discount cash flows at risk free rate Problem: Return expectations are risk adjusted Discounting at risk free rates overstates PV s-ag 32

33 Risk neutral valuation 2 alternatives: a) Adjust discount factors to intrinsic risk of cash flows b) Simulate asset returns using risk free rates and discount liability cash flows at risk free rates No applicable way to obtain option adjusted discount factors for liability cash flows in a). Alternative b): Concept of risk neutral valuation s-ag 33

34 Summary: Liability CF simulation and valuation Liability value: Done 1000 times in each simulation 1 Generate risk free rates 2 Generate risk free asset returns 16% 12% 8% 4% 0% Rate Year % -30% Return Year 5 Average liability value 3 Determine liability cash flows 4 risk free rate liability value = Y Mio. 50 CF's - (50) liability value i = X Mio. (100) (150) Year (200) s-ag 34

35 Simulation used to calculate the liability value in a given shock scenario Portfolio of Assets PV of Assets Shock Scenarios BE Case WC Shocks Simulation of investment returns after next year PV of the Net Position Portfolio of Life Policies Dynamic Liabilities PV of Liabilities s-ag 35

36 Table of contents 1. Introduction: Motivation, Group-wide Framework 2. Internal Risk Model: Basics, Life Approach 3. Standalone Risks and Liability Response 4. Simulation of Assets & Fair Value of Liabilities 5. Liability Response: Dynamic Reseves/Benefits 6. Liability Response: Valuation of Liabilities 7. Risk Aggregation, Diversification and Mitigation s-ag 36

37 Liability response & ALM model Boundary conditions for WC scenarios: WC scenario EQ/RE FI IR curve BE BE BE BE EQ/RE WC BE BE IR down BE WC WC (down) Sharable BE WC Shar. BE BE Scaling of assets and liabilities: A: 100% = capital investments + cash L: 100% = insurance reserves - free RfB Asset mix: constant s-ag 37

38 Standalone risks... before and after liability reponse Liability Value Impact Change in Value Asset Value Scenario Impact EQ RE IR down Shareable Liability response is a 1st order effect s-ag 38

39 Aggregation to total risk: TCV Standalone change in value after liability response: CV i are aggregated to total change in value (TCV) via parametric aggregation: TCV = CVi Cij i, j CV with covariance matrix Cij expressing interdependence between standalone risks. Disaggregation for risk analysis: CV j i, div = TCV CVi CV i Diversification benefits: TCV = CVi, div CV i i i s-ag 39

40 Risk mitigation: Hierarchy of buffers S/H Risk Capital Without risk mitigation TVR: Total valuation reserves fr.rfb: free RfB LTD: Liability for terminal dividends SE: Shareholder Equity With risk mitigation TVR fr. RFB 1/2 LTD 1/2 SE LTD TCV Σ 1 Σ 2 Σ 3 Shareholder part q SH = 0,1 * TVR/ Σ 1 Shareholder risk capital: RC = q SH TCV if TCV < Σ 1 s-ag 40

41 Review: standalone risks... before and after... E.g. credit risk: 100 After 50:50 risk sharing with P/H: 50,0 + 50,0 After liability response: 50,0 + 45,0 = 95,0 After diversification: 65 After risk mitigation: 5 s-ag 41

42 Backup: Interest rate model Y 0 (t): current, continuously comp. zero coupon yield => Discount factor: exp( Y0 ( t) t) => Continuously compounded forward rate: f0( t1, t2) = ( Y0 ( t2) t2 Y0 ( t1) t1 /( t2 t1) For tenor structure T define forward rate processes as 2 σ t (, 1) 0(, 1) X i ft Ti Ti+ = f Ti Ti+ e, where X~ t N, σ i 2 with 2 [ ] [ ] ( σ ) t( i, i+ 1) = 0( i, i+ 1), t( i, i+ 1) = 0( i, i+ 1) e -1 E f T T f T T Var f T T f T T Choose volatility to meet 5-year swap volatility, (5 = average duration of FI portfolio) 2 s-ag 42