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

Usage license:: Get license information via the feedback formular.

APPENDIX TO CHAPTER XVI 405 which is the product of two very small quantities. Thus, if q is t}g and ¢ is 144, the value of the fractional term becomes approximately .0005, which is a negligible quantity (compared with 14+7i4+¢=1+4.04 + .01). Hence when ¢ is small, the formula for mathematical value becomes approximately, — o a gy 3 : An Tritg A+itof Arita’ Taritor In other words, when the risk of default is small, its effect is substantially the same as that which follows from a rise in the rate of interest. If the rate of interest when risk is absent is 4%, a risk of 19, will therefore merely increase the “basis” on which the loan can be contracted to about 5%. Thus, if we recur to the so-called 59) ten-year bond, and suppose that the probability of each successive payment is Tos; and the risk of default, ¢, is 114, then the mathematical present value of the bond, when interest is 49, is approximately, — a a4 y hi “Ttite (A+itoP tative 5 5 + (1.05) t m m +, ete., 5 ~1.05 Hogg He In other words, the present value is approximately the same as the present value of a 59, bond on a 59, basis, which is of course par, or 100. But if the risk is great, the approximate formula given will no longer apply. Thus, if the chance of default is 1%; or, in other words, if the chance of payment is only i, the formula for the mathematical value of the property becomes, — ay) | al)’ | a)’ Vi= re ++ ot A +iy +, ete. In this case it is evident that all terms after the first are neg- ligible compared with the first (unless the successive items a, ag, ete., increase with sufficient rapidity to offset the decreasing fractions 11, 1445p, ete.). In the case of a 59 ten-year $100 bond, in which the risk of default is at any moment %, the approxi- mate value of the bond, obtained by omitting all terms after the first, would be ih, or approximately 50 cents! This “ mathematical value ” might be still further reduced by a co- efficient of caution. In other words, the bond is worthless. In

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