On the decomposition of risk in life insurance

Transcription

1 On the decomposition of risk in life insurance Tom Fischer Heriot-Watt University, Edinburgh April 7, 2005 This work was partly sponsored by the German Federal Ministry of Education and Research (BMBF, 03LEM6DA) 1

2 1 Motivation Situation: Life office/policy obtains gain G s,t during [s, t] Bühlmann (1995): G s,t must be decomposed into financial and biometric (technical) part G s,t = G F s,t + G B s,t (1) G B s,t must be pooled (diversification, Law of Large Numbers) AFIR-problem (Bühlmann, 1995): (G F s,t) + belongs to the insured pricing & hedging? How to decompose? What is pooling? What is the role of hedging? 2

3 2 Model Finite discrete time axis T = {0, 1, 2,..., T } filtered product space (F, (F t ) t T, F) (B, (B t ) t T, B) F T, B T finite d assets with price processes S = ((St 0,..., St d 1 )) t T complete arbitrage-free financial market, unique EMM Q t-portfolio θ = (θ 0,..., θ d 1 ) is vector of integrable F t B t -measurable random variables with t-value θ, S t = d 1 j=0 θ j S j t e.g. zero-coupon bond with maturity 1: θ = (1/S 0 1, 0,..., 0) 3

4 3 Market-based valuation Product measure principle: market value (minimum fair price) π t (θ) of θ at t T given by π t (θ) = S 0 t E Q B [ θ, S T /S 0 T F t B t ]. (2) Q B is one of perhaps many EMM (incomplete market) for t=0: π 0 (θ) = E Q B [ θ, S T /S 0 T ] = E Q[ E B [θ], S T /S 0 T ] History: Brennan and Schwartz (1976), Aase and Persson (1994), and many more Reasons: minimal martingale measure, quadratic approaches, utility approaches, Law of Large Numbers arguments 4

5 4 Life insurance contracts Benefits: t-portfolios γ t premiums: t-portfolios δ t viewpoint of the insurer: company gets δ t γ t at t Important: How are premiums invested? δ r is seen together with the self-financing strategy (δ r,t ) t r starting at r time r r + 1 r T portfolio δ r = δ r,r δ r,r+1 δ r,r+2... δ r,t benefits analogously 5

6 5 Market values and gains Market value of a life insurance contract at time t MV t = r<t π t (δ r,t γ r,t ) } {{ } past stream + r t π t (δ r γ r ) } {{ } future stream (3) company s gain G s,t obtained during [s, t] G s,t := MV t MV s (4) MV t and G s,t in L 0 (F B, F t B t, F B) 6

7 6 One-period decomposition Orthogonal decomposition is proposed Natural properties: G F s,t = E F B [G s,t F t B s ] (5) G B s,t = G s,t G F s,t (6) 1. G F s,t L 0 (F B, F t B s, F B) G F s,t is replicable by a purely financial s.f. strategy starting at s 2. G F s,t closest to G s,t w.r.t E[G B s,t] = 0 4. biometric parts can be pooled (explained below) 5. for s = t 1, biometric parts do not(!) depend on trading strategy 7

8 7 Hedging For a random variable Z in any L 2 (P, P, P) its conditional variance w.r.t. a sub-σ-algebra P P is defined by Var[X P ] = E[(X E[X P ]) 2 P ]. (7) p(s, t s) = price of a ZCB with time to maturity t s at time s PROPOSITION 1 (Locally variance-optimal market value). The locally variance-optimal market value at time t, which can be achieved by a purely financial s.f. strategy of cost 0 starting at s, is MV opt t = p(s, t s) 1 MV s + G B s,t. (8) 8

9 Interpretation Minimization of fluctuation of MV t MV opt t mean like riskless investment (seen from s) develops in the residual risk = biometric part of original gain cost of hedge = (-1) cost of capital MV s at time s Main ingredients of proof: Π t s(g s,t ) = Π t s(g F s,t) = (1 p(s, t s))mv s (9) 9

10 8 Pooling - a convergence property All considered independent individuals i N will have a contract maximum life span, maximum dates of death T i (T = N) bounded, possibly dependent portfolios; A i t := {i signs at t} PROPOSITION 2. Under the above assumptions, 1 m m i=1 T i 1 t=0 T i 1 A i t r=t+1 i G B r 1,r/S 0 r m 0 F B-a.s. (10) The mean aggregated discounted biometric risk contribution per contract converges to zero a.s. for an increasing number of independent policyholders. independent from distribution of contracts on time axis! pooling (10) = core competence of insurance companies 10

11 COROLLARY 1. Assume that (S 0 t ) t N is the price process of the locally riskless money account and that the insurance company sells fairly priced contracts, only, i.e. 1 A i t i MV t = 0 for 0 t < T i when i MV t denotes the present value (cf. (3)) of the i-th (meta-)contract at t. Under the hedge of Proposition 1, started at the beginning of each (sub-)contract for each time period, 1 m m i MV Ti /ST 0 i i=1 m 0 P-a.s. (11) Interpretation. (11) is the mean discounted total gain (= discounted present value at T i ) of the first m contracts that converges to zero almost surely. 11

12 9 Conclusion Natural decomposition of risk by orthogonal projections pooling can/should be considered as a convergence property role of locally variance-optimal hedges explained Consequences for bonus theory? 12

13 References [1] Aase, K.K., Persson, S.-A., Pricing of Unit-linked Life Insurance Policies, Scandinavian Actuarial Journal 1994 (1), [2] Bouleau, N., Lamberton, D., Residual risks and hedging strategies in Markovian markets, Stochastic Processes and their Applications 33, [3] Brennan, M.J., Schwartz, E.S., The pricing of equity-linked life insurance policies with an asset value guarantee, Journal of Financial Economics 3, [4] Bühlmann, H., Editorial. ASTIN Bulletin 17 (2), [5] Bühlmann, H., Stochastic discounting. Insurance: Mathematics and Economics 11 (2),

14 [6] Bühlmann, H., Life Insurance with Stochastic Interest Rates in Ottaviani, G. (Ed.) - Financial Risk in Insurance, Springer [7] Fischer, T., 2005 (2003). A Law of Large Numbers approach to valuation in life insurance. Working Paper fischer/papers/valuation.pdf [8] Fischer, T., On the decomposition of risk in life insurance. Working Paper fischer/papers/decomp.pdf [9] Møller, T., Risk-minimizing hedging strategies for unit-linked life insurance contracts, ASTIN Bulletin 28, [10] Møller, T., Risk-minimizing hedging strategies for insurance payment processes, Finance and Stochastics 5,

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