Download PDF by Mario V. Wuthrich, Hans Buhlmann, Hansjorg Furrer: Market-Consistent Actuarial Valuation

By Mario V. Wuthrich, Hans Buhlmann, Hansjorg Furrer

ISBN-10: 1865089117

ISBN-13: 9781865089119

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Example text

T. P ∗ . t. P ∗ (equivalent martingale measure). This implies for the one-period model Q0 [X] = D(F0 ) · E ∗ [ Q1 [X] ] = E [ Y1 · Q1 [X] ] . 69) Qt = ϕt · Qt [X] forms an Ft -martingale under P . 70) E Qt − Qs Fs = 0, which means that there exists no arbitrage strategy (which roots in the idea of risk-neutral valuation). e. 5 Equivalent martingale measures 23 with positive expected gains and without any downside risk. g. 6 in Lamberton–Lapeyre [LL91]), the proof is essentially an exercise in linear algebra.

We assume that the initial sum insured is CHF 1, the age at the entry of the contract is x = 50 and the contract term is n = 5. Moreover we assume that: 30 3 Valuation portfolio in life insurance • The annual premium Πt = Π (t = 50, . . , 54) is due in non-indexed CHF at the beginning of each year. • The benefits are indexed by a well-known index It (t = 50, 51, . . 55) with I50 = 1. – Death benefit is the indexed maximum of It and (1 + i)t−50 for some fixed minimal guarantee i. e. no minimal guarantee in case of survival.

L,L /2) , µ + Σ(h∗ + 1i ) = (δ + σ1,i − σ1,1 /2, . . , δ + σL,i − σL,L /2) . 44) is independent of µ. 45) Eh∗ e−δt · (A1 (t) − A2 (t))+ = A1 (0) · Ph∗ +11 [A1 (t) > A2 (t)] − A2 (0) · Ph∗ +12 [A1 (t) > A2 (t)] = A1 (0) · Ph∗ +11 [W (t) < ζ] − A2 (0) · Ph∗ +12 [W (t) < ζ] , with ζ = log(A1 (0)/A2 (0)) and W (t) = B2 (t) − B1 (t).

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Market-Consistent Actuarial Valuation by Mario V. Wuthrich, Hans Buhlmann, Hansjorg Furrer

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