A Notes on Compound Interest and Interest Tables Appendix Interest is the cost of using money. It is the rental charge for funds, just as renting a building and equipment entails a rental charge. When the funds are used for a period of time, it is necessary to recognize interest as a cost of using the borrowed (“rented”) funds. This requirement applies even if the funds represent ownership capital and if interest does not entail an outlay of cash. Why must interest be consid- ered? Because the selection of one alternative automatically commits a given amount of funds that could otherwise be invested in some other alternative. Interest is generally important, even when short-term projects are under con- sideration. Interest looms correspondingly larger when long-run plans are studied. The rate of interest has significant enough impact to influence decisions regarding borrowing and investing funds. For example, $100,000 invested now and com- pounded annually for 10 years at 8% will accumulate to $215,900; at 20%, the $100,000 will accumulate to $619,200. Many computer programs and pocket calculators are available that handle computa- tions involving the time value of money. You may also turn to the following four basic tables to compute interest. Table 1—Future Amount of $1 Table 1 shows how much $1 invested now will accumulate in a given number of peri- ods at a given compounded interest rate per period. Consider investing $1,000 now for three years at 8% compound interest. A tabular presentation of how this $1,000 would accumulate to $1,259.70 follows: Cumulative Interest Called Compound Total at End of Year Interest per Year Interest Year 0 $— $ — $1,000.00 1 80.00 80.00 1,080.00 2 86.40 166.40 1,166.40 3 93.30 259.70 1,259.70 interest tables
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ANotes on Compound Interest and Interest Tables
A p p e n d i x
Interest is the cost of using money. It is the rental charge for funds, just as renting abuilding and equipment entails a rental charge. When the funds are used for a periodof time, it is necessary to recognize interest as a cost of using the borrowed(“rented”) funds. This requirement applies even if the funds represent ownershipcapital and if interest does not entail an outlay of cash. Why must interest be consid-ered? Because the selection of one alternative automatically commits a given amountof funds that could otherwise be invested in some other alternative.
Interest is generally important, even when short-term projects are under con-sideration. Interest looms correspondingly larger when long-run plans are studied.The rate of interest has significant enough impact to influence decisions regardingborrowing and investing funds. For example, $100,000 invested now and com-pounded annually for 10 years at 8% will accumulate to $215,900; at 20%, the$100,000 will accumulate to $619,200.
Many computer programs and pocket calculators are available that handle computa-tions involving the time value of money. You may also turn to the following fourbasic tables to compute interest.
Table 1—Future Amount of $1Table 1 shows how much $1 invested now will accumulate in a given number of peri-ods at a given compounded interest rate per period. Consider investing $1,000 nowfor three years at 8% compound interest. A tabular presentation of how this $1,000would accumulate to $1,259.70 follows:
Cumulative InterestCalled Compound Total at End of
This tabular presentation is a series of computations that could appear as follows:
S1 = $1,000(1.08)1
S2 = $1,000(1.08)2
S3 = $1,000(1.08)3
The formula for the “amount of 1,” often called the “future value of $1” or“future amount of $1,” can be written
S = P(1 + r)n
S = $1,000(1 + 0.08)3 = $1,259.70
S is the future value amount; P is the present value, $1,000 in this case; r is the rate ofinterest; and n is the number of time periods.
Fortunately, tables make key computations readily available. A facility in select-ing the proper table will minimize computations. Check the accuracy of the preced-ing answer using Table 1.
Table 2—Present Value of $1In the previous example, if $1,000 compounded at 8% per year will accumulate to$1,259.70 in 3 years, then $1,000 must be the present value of $1,259.70 due at theend of 3 years. The formula for the present value can be derived by reversing theprocess of accumulation (finding the future amount) that we just finished.
If S = P(1 + r)n
then P =
P = = $1,000
Use Table 2 to check this calculation.When accumulating, we advance or roll forward in time. The difference
between our original amount and our accumulated amount is called compound inter-est. When discounting, we retreat or roll back in time. The difference between thefuture amount and the present value is called compound discount. Note the followingformulas (where P = $1,000):
Compound interest = P[(1 + r)n − 1] = $259.70
Compound discount = S[1 − ] = $259.70
Table 3—Amount of Annuity of $1An (ordinary) annuity is a series of equal payments (receipts) to be paid (or received)at the end of successive periods of equal length. Assume that $1,000 is invested at theend of each of 3 years at 8%:
1(1 + r)n
$1,259.70(1.08)3
S(1 + r)n
A-2 APPENDIX A
End of Year Amount
1st payment $1,000.00 $1,080.00 $1,166.40, which is $1,000(1.08)2
2nd payment $1,000.00 1,080.00, which is $1,000(1.08)1
The preceding arithmetic may be expressed algebraically as the amount of anordinary annuity of $1,000 for 3 years = $1,000(1 + r)2 + $1,000(1 + r)1 + $1,000.
We can develop the general formula for Sn, the amount of an ordinary annuityof $1, by using the example above as a basis:
4. Subtract (2) from (3): 1.08Sn − Sn = (1.08)3 − 1Note that all terms onthe right-hand side areremoved except (1.08)3in equation (3) and 1in equation (2).
5. Factor (4): Sn(1.08 −1) = (1.08)3 − 1
6. Divide (5) by (1.08 − 1): Sn = =
7. The general formula forthe amount of an ordinaryannuity of $1 becomes: Sn = or
This formula is the basis for Table 3. Look at Table 3 or use the formula itselfto check the calculations.
Table 4—Present Value of an Ordinary Annuity of $1Using the same example as for Table 3, we can show how the formula of Pn, the pres-ent value of an ordinary annuity, is developed.
End of Year
1st payment = $ 925.93 $1,000
2nd payment = $ 857.34 $1,000
3rd payment = $ 793.83 $1,000
Total present value $2,577.10
For the general case, the present value of an ordinary annuity of $1 may beexpressed:
7. Multiply by : Pn = [1 − ]The general formula for the present value of an annuity of $1.00 is:
Pn = [1 − ] =
Solving,
Pn = = 2.577
The formula is the basis for Table 4. Check the answer in the table. The pres-ent value tables, Tables 2 and 4, are used most frequently in capital budgeting.
The tables for annuities are not essential. With Tables 1 and 2, compoundinterest and compound discount can readily be computed. It is simply a matter ofdividing either of these by the rate to get values equivalent to those shown in Tables3 and 4.