APPENDIX TO CHAPTER XIII 383

tant from the present instant by the intervals #, ,, #;, ete., which
may be equal or unequal, whole or fractional, or even positive
or negative according as the income is in the future or in the
past. Let 7 represent the rate of interest. The present value
of such an income stream will be,—
i a ay ag
P= A+)" = (1+) =+ (A+) +5 ete.
Or, in briefer notation,

a
V=2 Thi
where 3 is taken in its usual sense of the summation of the
series of terms of the type of that following it.

 

 

§12 (ro Cm. XIII, § 8)
Diagram and Formula for deriving Capital-value from a given Continuous
Income Stream
For any ineome stream flowing continuously, and represented
in Figure 43 by the area below MN, the capital-value will
be represented by the curve NO. The ordinate of the income
stream at any point, as RS, represents the rate of its flow at that
point, and any area, as RSS" R", represents its total flow through
the period RR". The ordinates of the curve NO will represent
 the capital-value of this income stream. NO is constructed
 from N backward as follows: the curve begins at N
on the income stream and is generated by a point moving in
such a manner that at any position O its direction of motion
is the resultant of two tendencies. To represent these
two tendencies we draw through O the discount curve OP.
Tangent to this curve we draw OH so as to meet the line QH
drawn vertically and distant to the left one unit from OA.
OH represents one of the two tendencies mentioned, that due
to discounting the future. OK drawn vertically up from
and equal to RS, the rate of income at that time, represents
the other tendency. The resultant, OQ, drawn according to
the principle of the parallelogram of forces, will represent the
actual direction in which the curve will be moving at the
point O. In other words, a point moving under the influence
 of two forces, OK and OH, will generate the required

curve NO.