The Reinvestment Process in a Mass Where There is a Probability Law for the Lifetime of Each Capital Object

• Ragnar Frisch

Abstract

In a mass of uniform capital objects such as e.g. telegraph poles, or a specific kind of machine component (cf. the examples given in Chapter 16) it will as a rule be possible to draw up on the basis of statistics a mortality curve (retirement curve) for such objects. It would then be plausible to assume that every new unit which is added to the mass is subjected to a probability distribution of this kind as long as it is in existence. We assume that it is the same curve for all units. This assumption that the durability is subject to a probability law is in one way more general than the assumption that the life-time for each object is a given number (determined by the durability of the old object which the new one is to replace). But on the other hand, we now make the more restrictive assumption that all units are alike. Thus we do not now have a possibility of distinguishing between different typical sorts of objects as we had in Chapter 17. For this reason the analytical schema we are now considering cannot be used for studying the movement in a very complex mass of capital objects (e.g. that of a country); but it provides a useful point of departure for discussing the movement in a fairly uniform mass of capital objects. If a mass with a few different types of object is involved, it is of great assistance to undertake an analysis separately for each part of the total mass.

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References

1. 1.
For a precise interpretation, on the lines of probability theory, of such concepts, see for instance Erling Sverdrup: ‘Basic concepts in life assurance mathematics’ Skandinavisk Aktuarietidsskrift 1952.Google Scholar
2. 1.
If we imagine that departure takes place continuously and commences immediately we shall not accept any limitation such as (18c.1).Google Scholar
3. 1.
This method is used inter alia by Harald Schulthess, Mitth. d. Vereinigung Schweizerischer Versicherungsmathematiker 1937, see especially p. 75, and by Gabriel Preinreich, Econometrica, July 1938, especially p. 223, and The Present Status of Renewal Theory, Baltimore, 1940, especially pp. 17–21.Google Scholar
4. 1.
The curves are constructed [by using various calculations carried out in the work of Preinreich.Google Scholar
5. 1.
By interpreting (18f.3) as a Stieltjes or Lebesgue integral we can formally permit the last term in (18f.3) to be included in the integral, but it is much simpler to keep it separate. If we do, the integral can be assumed to be an ordinary Riemann integral.Google Scholar
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A graphic representation from which the solutions of (18f.13) can be read off approximately, is given by Lotka, Annals of Mathematical Statistics, March 1939.Google Scholar
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The procedure to be considered is a modification of the method of P. Hertz, Mathematische Annalen 1908 and Alfred J. Lotka, The Annals of Mathematical Statistics 1939, especially p. 9. In the latter article the method is used without raising the question of the multiplicity of the roots in the characteristic equation.Google Scholar