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
