Kinds of Bonds.
—Bonds are certificates of indebtedness by means of which the repayment of borrowed money may be spread over a series of years. They are classified as Sinking Fund, Annuity and Serial, depending on their manner of payment.
Sinking fund bonds are paid as a whole at the end of their term, interest being paid annually, or at some other fixed regular period, upon their face value. The name arises because of the custom of establishing a sinking-fund into which a certain proportion of the debt is to be paid annually, and this loaned out so that at the end of the period it will amount to the face of the bonds. Since there is always time lost between the collection and loaning of the sinking fund money the interest derived therefrom will not usually be the same as that of the bonds. For this reason and from the further fact that sinking funds are frequently drawn upon for other purposes than that for which they were created this type of bonds is less economical than either of the other two types.
The sinking fund which must be raised annually to discharge a debt of P dollars in n payments, if it can be loaned at i per cent, is given by the formula:[197]
Sinking fund = i (1 + i)n - 1 . P
To illustrate the use of the formula let the debt be $10,000, the average rate that can be expected from the sinking fund 4 per cent, and the time five years. Substituting in the formula,
S = .04 (1 + .04)5 - 1.$10,000
To solve, the denominator is first evaluated:
| Log (1 + .04)5 | = | 5 log 1.04 |
| = | 5 × 0.017033 | |
| = | 0.085165 |
Taking the antilog,
(1 + .04)5 = 1.21665
and
(1 + .04)5 - 1 = 0.21665
Then
S = 0.4 × $10,000 0.21665 = $1846.27.
Annuity tables, which may be seen at nearly any bank or brokers’ office, or in Bulletin 136, U. S. Department of Agriculture, give the annuity which will amount to 1 in five years at 4 per cent as 0.1846271; this multiplied by $10,000 gives $1846.27.
To the nearest cent the following tabular statement shows the growth of the sinking funds:
| Year | Sinking- fund at Beginning of Year | Interest during Year | Annual Payments into Sinking- Fund | Total Sinking- fund at End of Year | ||||
|---|---|---|---|---|---|---|---|---|
| 1 | 0. | 0. | $1,846. | 27 | $1,846. | 27 | ||
| 2 | $1846. | 27 | $73. | 85 | 1,846. | 27 | 3,766. | 39 |
| 3 | 3766. | 39 | 150. | 66 | 1,846. | 27 | 5,763. | 32 |
| 4 | 5763. | 32 | 230. | 53 | 1,846. | 27 | 7,840. | 12 |
| 5 | 7840. | 12 | 313. | 61 | 1,846. | 27 | 10,000. | 00 |
If this loan, the bonds, bore 5 per cent interest the cost to the borrower would have been the principal plus the interest on principal less the interest on the sinking fund:
$10,000 + $2500 - $768.65 = $11,731.35;
or the interest on the loan plus the sinking-fund payments:
$2500 + $9231.25 = $11,731.35
Serial Bonds are such that a fixed amount of the principal is retired at definite periods of time. Usually the amount retired is an aliquot part of the whole. The payments to be made at any particular time is the fixed portion of the principal plus the interest on the unpaid portion up to that date. The periods of retirement are usually annual or semi-annual.
Assuming the principal to be P and that one nth part of it is paid each year, the formulas are:
Annual payment for the kth year
= P ( 1n + i (1 + 1 - k n)).
Interest for the kth year
= Pi (1 + 1 - k n).
Total amount of interest to the end of the kth year
= Pik (1 + 1 - k 2n).
Total amount of interest and principal paid up to the end of the kth year
= Pk (1 n + i (1 + 1 - k 2n)).
The following table shows how a debt of $10,000 bearing 5 per cent interest would be discharged by equal annual payments in five years:
| Year | Principal at Beginning of Year | Interest for Year | Principal Repaid at end of Year | Total Annual Payment |
|---|---|---|---|---|
| 1 | $10,000 | $500 | $2,000 | $2,500 |
| 2 | 8,000 | 400 | 2,000 | 2,400 |
| 3 | 6,000 | 300 | 2,000 | 2,300 |
| 4 | 4,000 | 200 | 2,000 | 2,200 |
| 5 | 2,000 | 100 | 2,000 | 2,100 |
| Totals | $1,500 | $10,000 | $11,500 |
Annuity Bonds are those wherein a uniform periodic payment is made to discharge the debt in a given time. The formula for the necessary payment to discharge a debt of P, with interest rate i in n years is,
Annual payment = i 1 - (1 - i)-n . P.
Results may be taken from books of tables already referred to or by means of logarithms the formula may be solved. For example let it be required to discharge a debt of $10,000 in five equal payments, the rate of interest being 5 per cent.
Solution:
(1 + i)-n = 1.05-5.
Log 1.05 = -0.021189
-5 Log 1.05 = -0.105945 = 9.894055 - 10
Log-1 (9.894055 - 10) = 0.783529
1 - 0.783529 = 0.216471
Log Annual Payment = Log i - Log 0.216471 + Log P
= Log 0.05 - Log 0.216471 + Log 10,000
= (8.698970 - 10) - (9.335398 - 10) + 4.000,000
= 3.363572
Annual Payment = Log-1 3.363571 = $2309.748.
The following table shows the repayment of the loan by annual payments of $2309.75:
| Year | Principal Owing at Beginning of Year | Interest for Year | Principal Repaid at End of Year | Total Payment for Year |
|---|---|---|---|---|
| 1 | $10,000.00 | $500.00 | $1,809.75 | $2,309.75 |
| 2 | 8,190.25 | 409.51 | 1,900.24 | 2,309.75 |
| 3 | 6,290.01 | 314.50 | 1,995.25 | 2,309.75 |
| 4 | 4,294.76 | 214.74 | 2,095.01 | 2,309.75 |
| 5 | 2,199.85 | 109.99 | 2,199.75 | 2,309.74 |
| Totals | $1548.74 | $10,000.00 | $11,548.74 |
Since it is more convenient to have the bonds in even hundreds of dollars and the interest in dollars some adjustment from the theoretical amounts are usually made but such that the annual payments will be near the theoretical. Sometimes, too, the bonds are made smaller for the first few years then gradually increase so that the natural growth in population and wealth may bear its proportional burden. One adjustment for the example just given is shown:
| Year | Principal Owing at Beginning of Year | Interest for Year | Principal Repaid at end of Year | Total Payment for Year |
|---|---|---|---|---|
| 1 | $10,000 | $500 | $1,800 | $2,300 |
| 2 | 8,200 | 410 | 1,900 | 2,310 |
| 3 | 6,300 | 315 | 2,000 | 2,315 |
| 4 | 4,300 | 215 | 2,100 | 2,315 |
| 5 | 2,000 | 100 | 2,000 | 2,100 |
| Totals | $1,500 | $10,000 | $11,550 |