Risk-Neutral Valuation of Participating Life Insurance Contracts

Transcription

1 Risk-Neutral Valuation of Participating Life Insurance Contracts DANIEL BAUER with R. Kiesel, A. Kling, J. Russ, and K. Zaglauer ULM UNIVERSITY RTG 1100 AND INSTITUT FÜR FINANZ- UND AKTUARWISSENSCHAFTEN APRIA Annual Meeting, Tokyo, August 2006

2 Contents Introduction 1 Introduction 2 Insurance Company and Insurance Contract Development of the Liabilities 3 4 Financial Market Monte Carlo Approach Discretization Approach 5 6 APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 2 / 28

4 Guarantees and Options in Participating Contracts Interest rate guarantee cliquet style. Participation guarantee: Policyholders participate on the company s surplus. Influenced by regulatory prerequisites and insurer s management decisions about surplus distribution. Surrender option. Problems: Plunging stock markets, low interest rates. Insurers are stuck with old, non-hedged guarantees. APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 4 / 28

5 Guarantees and Options in Participating Contracts Interest rate guarantee cliquet style. Participation guarantee: Policyholders participate on the company s surplus. Influenced by regulatory prerequisites and insurer s management decisions about surplus distribution. Surrender option. Problems: Plunging stock markets, low interest rates. Insurers are stuck with old, non-hedged guarantees. APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 4 / 28

6 What s new? Introduction Financial strength is taken into account (via the reserve quota). Surplus distribution model(s) adequate for many markets, in particular the German market. Decomposition of the contract, separation of the embedded options. Allows for the separate securitization/hedging of the embedded options. We can derive an equilibrium condition for a fair contract. APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 5 / 28

7 Contents Introduction Insurance Company and Insurance Contract Development of the Liabilities 1 Introduction 2 Insurance Company and Insurance Contract Development of the Liabilities 3 4 Financial Market Monte Carlo Approach Discretization Approach 5 6 APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 6 / 28

8 of insurer s financial situation Insurance Company and Insurance Contract Development of the Liabilities Assets A t A t Liabilities L t R t A t Insurer s assets A t, where A + t = A t d t is the value right after dividend payments d t. The evolution of the liabilities and the dividends depend on the evolution of the assets and the reserve quota x t = R t L t, i.e. ( L t+1 = L t+1 A t+1, A+ t, L t, x t,... ) and ( d t+1 = d t+1 A t+1, A+ t, L t, x t,... ). APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 7 / 28

9 of insurer s financial situation Insurance Company and Insurance Contract Development of the Liabilities Assets A t A t Liabilities L t R t A t Insurer s assets A t, where A + t = A t d t is the value right after dividend payments d t. The evolution of the liabilities and the dividends depend on the evolution of the assets and the reserve quota x t = R t L t, i.e. ( L t+1 = L t+1 A t+1, A+ t, L t, x t,... ) and ( d t+1 = d t+1 A t+1, A+ t, L t, x t,... ). APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 7 / 28

10 Insurance Contract Introduction Insurance Company and Insurance Contract Development of the Liabilities Assets A t A t Liabilities L t R t A t The insured pays a single premium L 0 at time t = 0 and receives L T at maturity, independent of whether he/she is alive or not. Analyze this contract under different surplus distribution schemes. APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 8 / 28

11 Distribution schemes 1 Insurance Company and Insurance Contract Development of the Liabilities 1 MUST-case (only obligatory payments considered): Year-to-year cliquet style guarantee on the liabilities (minimum rate g). Proportion δ of earnings on book value have to be credited to policyholder s account. "Rest" of the earnings on book values paid out as dividend. "Rest" of the earnings on market values remains in the company (reserves). 2 IS-case (typical behavior of German insurer modeled): Target rate z is credited to the policyholder s account as long as the reserve quota stays within a given range [a, b]. If quota is outside the range this rate is adjusted. Dividends amount to a portion α of any surplus credited to the policy reserves. APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 9 / 28

16 The usual approach: Usual assumptions: There exists a risk-neutral measure Q and a numéraire process (B t ) t [0,T ] (bank account). General [ pricing formula: ] (risk-neutral valuation formula) V 0 = E Q B 1 T L T F 0 Problem 1: Underlying security not traded! Approximate underlying by traded benchmark portfolio. APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 11 / 28

19 Problem 2: Feedback effect Insurer determines price and adjusts his portfolio to hedge the liabilities. Portfolio composition changes and thus underlying changes. Both, price and hedging strategy change since the underlying changes. Insurer determines price and adjusts his portfolio to hedge the liabilities.... Solution: Decomposition of the contract. Securitization of the separated guarantees outside of the insurer s balance sheet. APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 12 / 28

25 Decomposition Introduction Dividends are paid out and reduce the value of the reference portfolio (but not the asset allocation): Cumulated present value: D 0. If the return of the reference portfolio is so poor that granting the minimum interest rate guarantee would result in negative reserves, capital is needed in order to fulfill obligations (capital shots c t ) these increase the value of the reference portfolio; however, the asset allocation stays the same: Cumulated present value: C 0. Value of the surrender option is the supremum of all possible gains from surrendering w t : Cumulated present value: W 0. APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 13 / 28

28 Equilibrium Condition for a fair contract We call a contract fair, if the (cumulated) value of the guarantees/options for the policyholder... (interest rate guarantee, surrender option, participation on initial reserve)...equals the (cumulated) value of the respective guarantees/options for the shareholders/insurer... ( dividend payments, remaining final reserve): [ ]! C 0 + W 0 + R 0 = D 0 + E Q B 1 T R T [ ] [ and since E Q B 1 T L T = L 0 + C 0 + W 0 + R 0 E Q [ ]! L 0 = E Q B 1 T L T B 1 T R T ] D 0 APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 14 / 28

29 Contents Introduction Financial Market Monte Carlo Approach Discretization Approach 1 Introduction 2 Insurance Company and Insurance Contract Development of the Liabilities 3 4 Financial Market Monte Carlo Approach Discretization Approach 5 6 APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 15 / 28

30 Market s Introduction Financial Market Monte Carlo Approach Discretization Approach Assets: Geometric Brownian motion with constant volatility σ A : da t = A t (r t dt + σ A dw t ) Here, r t denotes the short rate of interest models: 1 Classical Black-Scholes model: Constant rate r t = r. 2 Stochastic interest rates: Vasicek : Ornstein-Uhlenbeck process for r t, i.e. dr t = a(b r t ) dt + σ r dz t Cox-Ingersoll-Ross : Square-root process for r t, i.e. dr t = a(b r t ) dt + r t σ r dz t APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 16 / 28

31 Market s Introduction Financial Market Monte Carlo Approach Discretization Approach Assets: Geometric Brownian motion with constant volatility σ A : da t = A t (r t dt + σ A dw t ) Here, r t denotes the short rate of interest models: 1 Classical Black-Scholes model: Constant rate r t = r. 2 Stochastic interest rates: Vasicek : Ornstein-Uhlenbeck process for r t, i.e. dr t = a(b r t ) dt + σ r dz t Cox-Ingersoll-Ross : Square-root process for r t, i.e. dr t = a(b r t ) dt + r t σ r dz t APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 16 / 28

32 Monte Carlo Simulation Financial Market Monte Carlo Approach Discretization Approach For classical Black-Scholes model and Vasicek model: Distributions of random variables that describe the evolution of the contract over one period are known "Exact" simulation possible. CIR model: "Exact" simulation not possible. Integrals are discretized. Problem: Valuation of surrender option! APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 17 / 28

35 Discretization Approach (1) Financial Market Monte Carlo Approach Discretization Approach Between two policy anniversaries, the evolution of the value V t of the insurance contract solely depends on the evolution of the underlying assets r t and A t. Given the state variables (reserve quota, account value) at the beginning of the period, the value of the contract at the beginning of the period can be determined using PDE methods given the value at the end of the period (Feynman-Kac type). For constant short rate: Black-Scholes PDE, integral solution (heat equation). For stochastic short rate: Numerical methods (finite difference scheme). APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 18 / 28

38 Discretization Approach (2) Financial Market Monte Carlo Approach Discretization Approach Arbitrage Arguments Continuity of the contract value on the policy anniversaries. Similar to Tanskanen and Lukkarinen (2004) 1, a backward iteration can be used to compute the value of the contract via a discretization of the state space using interpolation methods at the anniversaries. 1 Fair Valuation of Path-Dependent Participating Life Insurance Contracts. IME, 33: , APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 19 / 28

40 Parameters Introduction Parameter Value Constant interest rate g 3.5% Participation rate δ 90% Book value restriction y 50% Initial reserve quota x 0 10% Target rate IS case z 5% Reserve corridor IS case [a, b] [5%,30%] Dividend rate α Is case 5% Asset Volatility σ A 7.5% Risk-free rate (Black-Scholes) 4% Maturity 10 years APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 21 / 28

41 Influence of the guaranteed rate g on the contract value in the MUST case APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 22 / 28

42 Influence of the guaranteed rate g on the contract value in the IS case APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 23 / 28

43 Contract Values MUST BS MUST OU IS BS IS OU init. inv. 10,000 10,000 10,000 10,000 + C , , , D R E Q Eur. 10, , , , W E Q 10, , , , APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 24 / 28

44 Influence of the initial reserve quota x 0 on the contract value (constant interest rate) APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 25 / 28

46 Future research Include more advanced asset processes (e.g., Lévy processes). Practical Implementation: How to hedge book of liabilities. Empirical Studies: How good is the model, how good is the hedge! Include (stochastic) mortality effects. APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 27 / 28

47 Thank you! APRIA Tokyo, August 2006 D. Bauer RNV of Participating Life Insurance Contracts 28 / 28