millton 发表于 2011-9-13 12:37 
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APPENDIX TO CHAPTER XII 363
$ 100 will to-day buy the right to $ 4 a year forever, and a
year hence this right to $4 annually may be sold on a 3%
basis. It will therefore fetch $133.33. The investor will
consequently receive in all, at the end of this first year, a total
of $4-}- 1331, or $137i. Hence the rate of interest in the
sense of a premium will be for that year 37^%. In succeed-
ing years the rate of interest, considered either as a premium or
a price, will evidently be 3%.
5 (TO CH. XII, 6)
Mathematical Relations between the Rates of Interest as a Premium and
as a Price
In general, if we let i\ represent the rate of interest in the
premium sense for this year, i 2 for next year, i 3 for the third
year, and so on, whereas ji,j& and j s , etc., represent the
rates of interest in the price sense for the same successive
years, the following relations between the i's and fa may be
proved :
.ad inf.
' ' ' ad mf-
... ad inf.
-M* (1-M*)U + ^) (l + ^Hl + ^Hl-M*)
etc.
These equations may be said to determine j l} as a peculiar
sort of mean of the magnitudes i l} i z , i s , . . . ad inf., and j 2 as a
similar mean of i< 2 , i s , i' 4 , ... ad inf., etc. Their proof,
which is simple, is left to those readers who are interested in
mathematics.
The preceding equations express the values of the fa in
364 NATURE OF CAPITAL AND INCOME
terms of I's. The following equations give the i's in terms of
the fa :
Ji
etc.
Thus if, as in our example, J, = .04 and j 2 .03,
The proof of these formulae is also left to the mathematical
reader. He will observe that the two sets of equations may be
proved independently or either set may be proved and the
other set derived from it. To show that either set may be de-
rived from the other, it will be found useful to substitute for
the left-hand members of the first set their simpler values as
derived by algebra. These are, -,-,-, etc. An easy proof of
ji h h
this is found by actually dividing 1 by j^ etc.
From the formulae it is clear that if ij = * 2 = 3 =, etc., then
j l= j 2 =j s =:, etc., and that then all the i's = the fa. The
converse is also evident.
6 (TO CH. XII, 7)
Mathematical Relations between the Rates of Interest and Discount
Let V, due one year hence, be the equivalent of V available
in the present. Then the rates of interest and discount are
expressed respectively by the formulae :
Whence, by multiplying the two equations together we
derive (1 + i) (1 d) = 1, which reduces to d = i id. That
is, the number representing the rate of discount equals the
number representing the equivalent rate of interest less a
APPENDIX TO CHAPTER XII 365
small correction, equal to the product of the rates of interest
and discount.
The relation between the rates of discount and interest
when semi-annually reckoned is analogous to the relation which
we found between those annually reckoned. We then have
V d'
the equations, = 1 -
V i'
and = l ~
whence
By multiplication and reduction,
d'-i'-^
2 '
We may apply similar reasoning to quarterly-reckoned
interest and discount rates.
It follows that (l + j Vl - ^\ = 1
or
4
The same reasoning may obviously be applied to reckonings
n times a year. We then find,
j(.n)fj (n)
(jfr) -JO*)-* * .
n
It is evident that if n, the number of parts into which the
year is divided, be sufficiently increased, the term - - be-
n
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