Der Zusammenhang zwischen Deckungskapital und Sterblichkeit im Jecklinschen Reservemodell

Abstract

The author determines the decrement tables leading, for endowment assurances, to the hyperbolic reserve function ofJecklin’s reserve model. He shows in particular that, for a given maturity age (s), a necessary and sufficient condition for the reserve to follow a rectangular hyperbola with asymptotes parallel to the co-ordinate axes is given by the following mortality laws a $$\begin{gathered} x) = D_x = v^x l(x) = e^{ - \delta x} l(x): \hfill \bar f(x) = k\left( {l - \frac{x}{c}} \right)\left( {l - \frac{x}{s}} \right)^{\lambda (c - s) - 1} e^{ - \lambda x} \hfill \end{gathered} $$ and b $$f(x) = K(c - x)\frac{{\Gamma (s - x)}}{{\Gamma (\sigma - x)}}(1 - \lambda )^x ,$$ according to whether the reserve is defined in (a) a continuous or (b) a discontinuous way. If the above rectangular hyperbola is replaced by a non-rectangular hyperbola with one asymptote parallel to the ordinate then the functions (a) and (b) are replaced by a $$f^ * (x) = k\left( {l - \frac{x}{c}} \right)\left( {l - \frac{x}{s}} \right)^{\lambda A - 1} \left( {l - \frac{x}{{s + a}}} \right)^{\lambda B - 1} $$ and b $$\begin{gathered} f^ * (x) = K(c - x)\frac{{\Gamma (s - x \Gamma (s + a - x)}}{{\Gamma (s - \alpha + 1 - x) \cdot \Gamma (s - \beta + l - x)}} \hfill = K(c - x)(s - 1 - x)^{(\lambda A - 1)} (s + a - 1 - x)^{(\lambda B - 1)} \hfill \end{gathered} $$ These functions include de Moivre’s law as a special case, leading to a hyperbolic reserve curve for every maturity age. The reserves defined by these functions also include the special cases of rectangular hyperbolic and linear reserves.