The nature of capital and income

Bibliographic data

Monograph

Identifikator:: 102659555X

URN:: urn:nbn:de:zbw-retromon-82920

Document type:: Monograph

Author:: Fisher, Irving http://d-nb.info/gnd/118533541

Title:: The nature of capital and income

Place of publication:: New York

Publisher:: The Macmillan Company

Year of publication:: 1923

Scope:: XXI, 427 Seiten

Digitisation:: 2019

Collection:: Economics Books

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404 NATURE OF CAPITAL AND INCOME evidently not p,, but p,p,; for one of the first principles of the theory of probabilities is that the chance of two successive events is the product of their successive probabilities. Thus, if the chance of heads appearing in coin tossing is 4, the chance of two successive heads is 1X 4, or }, and the chance of three succes- sive heads is, in like manner, 4 X + X 4, or §, etc. Hence the « mathematical value” of the second installment, a, when due is agp ps of which the present value is a In like manner, 2 the mathematical present value of the third installment is rr and so on. The sum of the expressions for present value thus obtained is the total present mathematical value of the property. If we denote this mathematical value by V.,, we have, — _ GUD UaP1Ps | GPrPePs , ... 4 %aP1PePs °° Pa. Va Tita 1 +19)? histk 1+ 9)" If we suppose that all the probabilities are equal, we may denote all the p’s simply by p, and simplify by substituting p* for p, ps, and p® for pp. ps, ete. Since the p’s represent the probability of receiving the in- stallments, it is clear that the chance or risk of not receiving them is the difference between this and unity. This risk of default we shall denote by the letter g¢. Thus, ¢=1—p,, ete, and also py=1—q, ete. If all the ¢’s are equal, we shall de- note them by ¢, and the present value of the property may then evidently be written, — a (1—q) , a(1—gY , a(1—q)* a. (1—g)" Vo ee 1474 = 1417)? ¥ 141) bir A+) In case the risk of default q is very small, it is evident that This may .1—gq. 5 the fraction s approximately equal to ——— I 1 oi PProxir y eq T+ityg be seen by dividing the numerator and denominator of the first fraction by 1 — g, which will give for the new numerator unity, 3s and for the denominator 1 + i+¢q +4, In this expression 4g the fractional term becomes negligible when ¢ is small, because the denominator, 1—gq, is approximately unity, while the numerator, ¢ + ig, is made up of two terms, each of

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