By Mario V. Wuthrich, Hans Buhlmann, Hansjorg Furrer
ISBN-10: 1865089117
ISBN-13: 9781865089119
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VaPo (Xk ) = Xk at time k. 3. A VaPo X(k) = A VaPo X(k+1) + A (VaPo (Xk )) . 11) For the evaluation of the valuation portfolio at time m ≤ k we define Am VaPo(X(k) ) = A VaPo(X(k) ) Fm = Q X(k) Fm . 14) which tells again, that the VaPo for Xk at time k is simply Xk . This observation is fundamental and should hold independently of the value assigned to the VaPo by the accounting principle A. 2. 5 VaPo protected against technical risks So far we have considered an ideal situation which is an important point of reference to measure deviations.
20) is a stochastic vector in RL with distribution F (x, t) = P [Bi (t) ≤ xi , i = 1, . . 21) for all t ≥ 0 and x ∈ RL . We define the moment generating function as follows, for z ∈ RL and t ≥ 0 M (z, t) = E exp zT · B(t) . 22) Assumptions. Assume that the stochastic process {B(t)}t≥0 has stationary, independent increments and that t M (z, t) = [M (z, 1)] . 23) Moreover, we assume that B(t) has a density for t ≥ 0: f (x, t) = ∂L F (x, t) . 25) M (h, t) the corresponding moment generating function is given by M (z, t; h) = M (z + h, t) .
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. The benefits are paid at the end of the period. This means, for example, that the survival benefit is given by a financial instrument I whose price is a stochastic process (It )t . This index can be any financial instrument like a stock, a fund, etc. Hence, to hedge the survival benefit we need to by one unit of index I at the price I50 = 1 and it generates the (random) survival benefit I55 at time t = 55.