Mathematics of investments;

              36 HASKINS & SELLS May
THE accountant sometimes is confronted
with problems concerning compound
interest, annuities, sinking fund con­tributions,
and the like. He may be
asked to find the present value of a lease
calling for a certain number of periodic
payments, or he may want to know what
amount should be set aside each year to
liquidate a debt at some future date. These
situations arise frequently enough so that
some knowledge of the mathematics of
compound interest is considered an essen­tial
part of the technical equipment of the
accountant.
Tables have been published which facili­tate
the computations in the solution of
compound interest problems. These tables
are very helpful when accessible and almost
indispensable in the more complicated in­stances.
However, the accountant should
be sufficiently familiar with the relation­ships
of the several compound interest
functions, so that he can solve elementary
problems when published tables are not
available or when the tables at hand are
incomplete. One should, if necessary, be
able to handle a fair proportion of invest­ment
problems through reasoning rather
than through memorizing complicated
formulas and the attendant algebraic
symbols.
There are six fundamental compound in­terest
functions as follows:
1. Amount of $1 at compound interest.
One dollar invested now at compound in­terest
will amount to how much at the end
of a certain period of time?
2. Present value of $1 at compound in­terest.
What sum of money invested now
at compound interest will amount to $1
at the end of a certain period of time?
3. Amount of $1 per period at compound
interest. A periodic payment of $1 will
amount to how much at compound interest
at the expiration of a certain period of
time?
4. Present value of $1 per period at
compound interest. What sum of money
invested now at compound interest would
be just sufficient to make a periodic pay­ment
of $1 for a certain length of time?
5. Annuity which will amount to $1 at
compound interest. What sum of money
should be regularly set aside at compound
interest to amount to $1 at the end of a
certain period of time?
6. Annuity which has a present value at
compound interest of $1. One dollar set
aside now at compound interest will be
just sufficient to make a regular payment
of how much for a certain length of time?
Computations are made on the basis of
$1 because each dollar in a given trans­action
increases in the same proportion as
every other dollar. The result then is mul­tiplied
by the number of dollars involved.
Generally the dollar sign is omitted be­cause
the tables or computations may be
considered as representing one unit of any
monetary standard—the dollar, the pound,
the franc, and so forth. Time usually is
referred to by periods rather than by years,
a period being the interval between two
successive interest conversion dates. In­terest
rates also are adjusted to the basis of
a period. The amount of 35,000 for three
years at 4% per annum compounded
quarterly would be considered, for purposes
of computation, as the amount of $1 for 12
periods at 1% per period and the result
multiplied by 5,000.
1. Amount of 1
The amount of 1 invested for one period
at 4% is 1 x 1.04, or 1.04. The amount of
1 invested for two periods at 4% is equiva­lent
to the amount of 1.04 invested for one
period at 4% which is 1.04 x 1.04, or 1.0816.
The amount of 1.0816 invested for one
period at 4% is 1.0816 x 1.04, or 1.124864,
which is the same as the amount of 1 in­vested
for three periods at 4%, (1 x 1.04 x
1.04 x 1.04 = 1.124864). The ratio of
the investment value at the end of a period
to the value at the beginning of the period
is known as the ratio of increase. The ratio
Mathematics of Investments