APPENDIX TO CHAPTER XIV 397

case, it is the capital-value of such an income stream which
will vary from time to time. As has been seen, the capitalvalue
 of such an income stream is found by dividing the rate of
income a by the average of the individual rates of interest, such
as, in the above example, 10%, 5%, 6 %, etc., ad inf., the average
 being obtained by the formula given in Appendix to Chap.
XII, § 5. If such average rate of interest be called jj, the
. a .
capital-value will be =. Suppose, for instance, that the person
J
has a uniform perpetual income of $5 per annum. If the rate

of interest to-day, j,, is 5%, the capital-value to-day will be $100.
If next year, j, (the average of the future rates in individual
years, beginning at that time) is 4.99, the capital-value will
be $102. If, two years from date, j, be 5.19, the capital-value
will sink to $98. Adopting such an income stream as a standard,
 the propositions as to impairment or increase will still be
true, provided such impairment or increase is measured with
reference to the variable capital-value just shown. Thus, if at
the end of the first year more income than $5 is received, the
capital-value will be impaired by the difference, this impairment
 to be reckoned with respect, not to $100, but to $102,
which would be the value had the income remained standard.
Thus, the effect of a difference between real and standard
 income may be stated in the same terms, whichever of the
two definitions of standard income is adopted. In the one case
the standard is with reference to constant capital and variable
income; in the other, to variable capital and constant income.
In practical life, the former standard is usually employed,
although for certain purposes the latter would be more suitable.
We all know of cases of investors who, twenty years ago, invested
 at a high rate of interest, and who have taken pains
merely to maintain the value of their capital unimpaired,
although they were well aware that the rate of interest was
constantly sinking. In consequence, these persons are now
forced, when reinvesting, to suffer a large decrease in income,
which could have been avoided had they kept in view the maintenance,
 not of their capital, but of their income, and laid aside
each year a certain sum in order to offset the fall in the rate of
interest. The reason such a procedure is not common is