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Interest Tables
Source: Encyclopedia of Banking & Finance (9h Edition) by Charles
J Woelfel
(We recommend this as work of authority.)
Tables constructed
to show the amount of INTEREST that will accrue on a given convenient
(round number) sum, e.g., $1, $100, or $1,000, at different rates of interest
for various intervals of time, rendering unnecessary separate and independent
computations for each interest transactions.
Interest
tables are prepared in many different forms with varying degrees of detail
and refinement in decimal places, interest rates, and time intervals,
to meet a wide variety of uses.
The following list, illustrated by appended tables, includes the
more important types of interest tables.
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Simple
interest computed on a given principal.
The formula for computing annual simple interest is
I
= Pr
where P
= principal and r = rate of interest per year.
For example, if principal of $1,000 is invested at a 5% annual
rate of interest, the dollar interest per year (assuming that the interest
is not reinvested) is
I = 1,000(0.05)
I = 50
Ordinary
simple interest is computed on the basis of a 360-day year (see Table
1), while exact simple interest is computed on the basis of a 365-day
year (366 days in leap years).
To determine the dollar amount of interest for principal invested
for less than one year (using exact interest basis of 365-day year),
the formula is
I
= (PR)(D /365)
where
D =
the number of days for which the principal is invested.
For example, if principal of $1,000 is invested for 31 days at
an annual rate of interest of 5%, the dollar interest for the 31-day
period is
I
= 1.000(0.05)(31/365)
=
50(0.0849315)
= 4.2466
By
contrast, to determine the dollar amount of interest for principal invested
for less than one year (using ordinary interest basis of 360-day year),
the formula is
I
= (PrV)(D/360)
so
that
I
= 1.000(0.05)(31/360)
= 50(0.086111)
= 4.3056
(1,4.1667 for 30 days plus 0.1389 for 1 day)
Thus
the 360-day year basis, although simpler to calculate, results in higher
interest. Simplicity of
calculation is illustrated by the 60-day, 6% method:
$1,000
for 60 days @ 6% annually equals
$10,00 (simply point off two decimal places)
-8.8333 (1/6th less, or 1%/6%, for
5% rate)
4.1667
for 30 days at 5%
or
1/12th (30/360) of the $50 interest
for one year (1,000 x 0.05) equals $4.1667 for the 30 days.
-
Compound
interest - the amount of interest that a given principal will accumulate
if invested at specified rate, compounded at specified frequency for
specified total number of periods, if the interest generated is reinvested
at the same rate. The
formula for the compound interest as such is
(1
+ r V)n = 1
or
the compound amount of $1 invested at rate r per interest period for a specified number of interest periods n
minus the principal of $1.
As
indicated by Table 3, the compound amount of $1 invested at 5%, compounded
annually for 10 years, is as follows:
(1
+ .05)10 =
1.6289
so
that deducting the $1 of principal implied, the amount of compound interest
is
1.6289
- 1 =
0.6289
The
compound amount of $1,000 invested at 5%, compounded annually for 10
years, therefore is as follows:
C
= P(1 + r)n
= 1000(1 + 0.05)10
= 1000(1.6289)
= 1,628.90
where
C = compound amount of principal.
To
adjust for a specified frequency of compounding, divide the annual rate
of interest by the frequency of compounding per year to obtain the interest
rate per period; multiply the specified number of years by the frequency
of compounding to obtain the total number of interest periods.
For example, if the 5% rate above is compounded quarterly, rate
of interest per interest period is
0.05
(annual rate of interest)
___ = 0.0125(1.25%)
4 (frequency of compounding)
and
the number of interest periods is
10
x 4 =
40
so
that the compound amount of principal and the amount of compound interest
may be determined as above, based on this adjusted interest rate and
adjusted number of interest periods.
-
Future
value of a series of payments - the amount to which a series of payments
at the end of each period will accumulate at compound interest. The basic formula is
S = P1
(1 + r)n-1 + P2(1 + 4)n-2 + .
. . + Pn(1 + r)0
where
S - future value, P1,
P2, . . ., Pn = the payment at
end of each period, r = interest rate, and n
= number of periods. The
value of (1 + r)n-1
can be derived from Table 3. For
example, the future value at the end of two years of payments of $1,000
and $2,000 at the end of the first and second years, respectively, invested
at 5%, will be
S = 1000(1 + 0.05)1
+ 2000(1 + 0.05)0
= 1000(1.05)
+ 2000
= 1050 < + 2000
= 3050
-
Future
value of an annuity. This
is a special case of the future value formula above.
It is the future value of a series of equal future payments
for a given number of periods, at specified interest rate.
Applying the future value formula, the future value of an annuity
is
S = P(1 + r)n-1
+ P(1 + r)n-2
+ . . . + P(1 + r)0
where
S = future value, P
= periodic payment, r = interest
rate, and n = number of periods.
Since P, the payment
for each period, is equal, the formula can be simplified to
S
= P(1 + r)n-1}/r
The
value of { (1 + r)n-1}/r
can be found in Table 4, for the specified interest rate r
and the number of periods n
over which the annuity will extend.
For example, the value at the end of 10 years of an annuity of
$1,000 invested at a 10% interest rate will be
=
1000(1 + 0.01)10 - 1}/r
= 1000(15.9374)
= 15,937.40
-
Sinking
fund accumulations - the amount of installment to be set aside periodically
(annually, semi-annually, or quarterly) that at a specified rate of
compound interest will accumulate to a total sinking fund sufficient
at specified maturity to retire the principal of a given amount of
funds. To determine the
periodic payments, the formula for calculating the future value of
an annuity may be used, as follows:
S = P
(1 + r)n-1}/r
In
the above formula, the periodic payment is known, but the sum of the
payments at the end of the total period at specified interest rate is
unknown. For the sinking
fund accumulation, the sum is known, but the periodic payment is unknown.
Therefore the annuity formula must be solved for P,
the periodic payment, rather than S,
the sum of the accumulation. Therefore,
the formula is
P
= S/{ (1 + r)n
-1}/r
For
example, to determine the amount to be set aside yearly at a 6% annual
rate of interest that will accumulate to $1 million at the end of 10
years, Table 4 provides the value of { (1 + r)n
- 1}/r, with r at 6% and n at 10, as
equal to 13.1803. Therefore:
P =
1,000,000 / 13.1803
= 75,867.93
-
Present
value of one or a series of payments to be received (or paid out)
in the future, discounted at specified discount rate.
The formula for computing present value is
PV
= 1
+ 1
+ . . . + 1
(1 + k) (1 + k)2
(1 + k)a
where
PV = present value, n = year
of last payment received (or paid out), and k = discount rate. For
example, the present value of $1 to be received at the end of the first
and second years from the present time, discounted at 5% is as follows:
PV
= 1
+ 1
(1 + 0.05)
(1 + 0.05)2
= $1 +
$1
1.05 1.1025
= 0.9524
+ 0.9070
= 1.8594
Table
7 provides the present value of $1, received in 1 to 30 years, discounted
at the rate of 1% to 50%. Rather
than divide each numerator by the denominator in the above equations,
multiply the numerator by the present value of $1 discounted at the
specified rate for the specified time period, found in Table 7.
For example, to determine the present value of a $100 inflow
at the end of year one, $200 inflow at the end of year two, and $50
outflow at the end of year three, we may multiply these flows by the
present value of $1 indicated in Table 7 for the respective years, as
follows, at 5% discount rate.
PV
= 100(0.9524) +
200(0.9070) -
50(0.8638)
=
95.24
+ 181.40
- 43.19
=
233.45
In
capital budgeting, one technique of analysis of feasibility of investing
in specific investment proposals is the net present value technique:
cash flows each year anticipated from net income plus depreciation
for the full useful life of the proposed investment in plant ad equipment
are discounted at a selected discount rate; the sum of such discounted
present values is then compared with present investment outlay to show
net excess of sum of discounted present values over the investment outlay.
-
Present
value of an ANNUITY. A
special case of the present value formula above is the present value
of a sum of either equal inflows or equal outflows for a given number
of periods, discounted at specified rate.
Adapting the present value formula in 6 above, since the numerator
($1) is the same for each period, the formula can be simplified to
PV
= 1{1
- (1 + k)-n}/k
the
value of which can be found in Table 8 at the specified discount rate
k and the number of periods
n over which the annuity will
extend. For example, the
present value of an annuity of $1,000 received at the end of each year
for 10 years, discounted at 10%, may be determined by multiplying the
$1,000 by the factor 6.1446, shown in Table 8.
PV
= 1000(6.1446)
= 6.144.60
-
Doubling
of principal. Given the
interest rate, Table 9 will indicate the number of years it will take
a given principal to double in amount.
For example, at 6% compounded annually, it will take 11.896
years to double the principal.
The
formula for determining the number of years in which a given sum will
double at different interest rates may be derived from the compound
interest formula (above), except that the equation is solved for the
number of periods rather than the sum to which the principal will grow.
Thus, the compound interest formula is
S
= P(1 + r)n
but
S, the sum, is specified as equal to twice the principal (S = 2P), so
that substituting for S,
2P
= P(1 + r)n
which
may be simplified to
2
= (1 + r)n
which
by the use of logarithms becomes
n
log(1 + r) = log 2
n
= log 2
log (1 + 0.06)
= 0.301030
0.025306
=
11.896
-
Monthly
savings to attain a specified estate.
See appended Table 10 for the amount to be saved per month,
with interest compounded at 4% semi-annually, to accumulate a specified
sum at age 65.
-
Bond
interest table. See appended
Table 11 for the amount of accrued interest on a $1,000 bond for 1
day to 6 months at coupon rates for every 0.25% from 3.5% to 5%, and
6%. The interest on a
$1,000 bond at 4.5% for 4 months and 23 days would be $15.00 for 4
months and $2.875 for 23 days; total, $17.875.
-
Income
from dividend stocks. See
appended Table 12 for the approximate current return, or YIELD, from
dividend-paying stocks at prices from 20 to 200, having a cash rate
from $2 to $10 annually.
Simple
interest tables for computing interest on short-term loans are based on
a 360-day and 365-day year. Commercial
banks customarily use the 360-day tables, but the Federal Reserve banks
compute their transactions on the 365-day table.
There are a number of published tables showing the amount of interest
on a given sum at various interest rates from 1 to 365 days.
See AMORTIZATION LOANS
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