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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.

  1. 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.

  1. 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.

  1. 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

  1. 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

  1. 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

  1. 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.

  1. 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

  1. 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

  1. 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.

  2. 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.

  3. 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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