On the construction of tables for the calculation of certain survivorship benefits

Abstract

1. Benefits depending on two lives often involve the necessity for calculating tables with two arguments. There are, however, cases where this can be avoided, the chief example being the joint-life annuity when the mortality table is graduated by Makeham's formula. The question therefore arises whether there are other cases where a suitable graduation of the mortality table may render similar services.     

Über das Makeham'sche Gesetz und das Zinsfussproblem bei der Leibrente

Abstract

In dem Folgenden werde ieh einige praktisehe Folgen einigen der vielen bis heute ersehienenen Arbeiten ü das MAKEHAM'sche Sterblichkeitsgesetz und das versicherungstechnische Problem bei Zinsänderungen andeuten. Es handelt sich um einige Zusammenstellungen für die praktische Verwendung einigen der vielen schönen theoretischen Resultaten in diesem Gebiete der Versicherungsmathematik; praktische Zusammenstellungen, die mir bekannt nicht früher klar ausgesprochen worden sind.     

Remarks on iteration.

1. Iteration is an operation with many aspects; we shall here only occupy ourselves with iteration as a means of numerical determination of a real root of an equation. Nor shall we go into the whole of the theory of this subject ² See, for instance, Whittaker & Robinson: The Calculus of Observations, or Runge & König: Vorlesungen über numerisches Rechnen. , our principal object being to show how the process of iteration which is often futile in its primitive form may be improved by a suitable combination of three consecutive values. But before doing this, it will be convenient to recall briefly some of the principles involved in iteration.     

Summation-formulas of Lubbock's type

Abstract

If it is required to calculate a sum consisting of a great many terms, it is natural to ask oneself whether an approximation might not be obtained by adding up every mth term and multiplying the result by m. If the approximation obtained in this way is not considered sufficient, certain supplementary terms must be added to the result, and these may either be expressed as differences or as differential coefficients of the function under consideration. In the former case we have formulas of Lubbock's, in the latter of Woolhouse's type. We here confine our attention to the formulas of Lubbock's type.     

On the remainder form of certain formulas of mechanical quadrature

Abstract

The class of formulas considered in this paper belong to a type where the required integral is approximately represented by a linear function of a certain number of equidistant values of the integrand. Formulas of this type may, for instance, be obtained by integrating Lagrange's interpolation formula between finite limits, as is well known. As regards the function to be integrated we assume only, that it possesses, throughout the interval of integration, a continuous differential coefficient of the highest order of which use is made in deriving the particular formula under consideration. It is not necessary, then, that the function should possess differential coefficients of every order, much less, that it should be a polynomial.     

Consumption-portfolio optimization with recursive utility in incomplete markets

In an incomplete market, we study the optimal consumption-portfolio decision of an investor with recursive preferences of Epstein?CZin type. Applying a classical dynamic programming approach, we formulate the associated Hamilton?CJacobi?CBellman equation and provide a suitable verification theorem. The proof of this verification theorem is complicated by the fact that the Epstein?CZin aggregator is non-Lipschitz, so standard verification results (e.g. in Duffie and Epstein, Econometrica 60, 393?C394, 1992) are not applicable. We provide new explicit solutions to the Bellman equation with Epstein?CZin preferences in an incomplete market for non-unit elasticity of intertemporal substitution (EIS) and apply our verification result to prove that they solve the consumption-investment problem. We also compare our exact solutions to the Campbell?CShiller approximation and assess its accuracy.     

Equilibrium investment with random risk aversion

We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by its expectation.