The equitable distribution (technically known as “amortization”) of the discount and premium on bond investments is the essence of the problem of their valuation. There are two methods of making this distribution, termed the “straight line” and the “scientific” method. Under the straight line method the amount of the discount or premium is divided by the interest periods the bond has yet to run, and the quotient is made the amount of periodic amortization. Although not scientifically accurate, the method commends itself because of its simplicity and consequent ease of operation. Its use is allowed by the Department of Banking of the State of New York for valuing the investments of savings banks. The scientific method is based upon compound interest calculations and will be best understood by means of examples. Under this method the discount or premium is looked upon as the amount of an annuity. The portion which must be written off for any period is the present worth of the annuity on that date, taking into consideration the rate of interest and the time in interest periods the bond has still to run.
In practice, however, the amount of amortization is not found in that way. To find the periodic amortization, it is necessary to know the cost of the bond, its coupon interest rate, and the effective rate. By effective rate is meant the real income rate on the basis of the price paid for the bond. Assume that a 3% bond, interest payable January and July, has 3 years (6 periods) to run and is bought for $971.99 so as to yield 4% on the investment. At the end of the first semiannual period the actual interest received will be $15, but the real income on the investment is $19.44 because it was purchased on a 4% basis for $971.99 (2% on $971.99). Of the total discount of $28.01, $4.44 (19.44-15.00) is to be credited to the current period. The adjustment of discount brings about a new valuation of the bond, it being now worth $976.43 (971.99 + 4.44), because nearer by six months to maturity when it will be worth par or $1,000. So, for the next period the coupon is $15 and the effective income, $19.53 (2% on $976.43), hence the amortization is $4.53; and so on for the six periods, at the end of which the discount will have been completely distributed, i.e., amortized.
The following schedule shows the periodic amortization and new values of the bond:
3% bond, par $1,000; bought on a 4% basis for $971.99;
3 years to run; interest January and July.
| Date | Nominal Income | Effective Income | Periodic Amortization | Value of Bond | Discount Adjusted Amounts |
|---|---|---|---|---|---|
| Jan. 1, 1915 | $..... | $..... | $..... | $971.99 | $28.01 |
| July 1, 1915 | 15.00 | 19.44 | 4.44 | 976.43 | 23.57 |
| Jan. 1, 1916 | 15.00 | 19.53 | 4.53 | 980.96 | 19.04 |
| July 1, 1916 | 15.00 | 19.62 | 4.62 | 985.58 | 14.42 |
| Jan. 1, 1917 | 15.00 | 19.71 | 4.71 | 990.29 | 9.71 |
| July 1, 1917 | 15.00 | 19.81 | 4.81 | 995.10 | 4.90 |
| Jan. 1, 1918 | 15.00 | 19.90 | 4.90 | 1,000.00 | ..... |
| Total discount amortized | $28.01 | ||||
A similar schedule for a bond bought at a premium immediately follows:
5% bond, par $1,000; bought on a 4% basis for
$1,028.01; 3 years to run; interest May and November.
| Date | Nominal Income | Effective Income | Periodic Amortization | Value of Bond | Discount Adjusted Amounts |
|---|---|---|---|---|---|
| May 1, 1917 | $..... | $..... | $..... | $1,028.01 | $28.01 |
| Nov. 1, 1917 | 25.00 | 20.56 | 4.44 | 1,023.57 | 23.57 |
| May 1, 1918 | 25.00 | 20.47 | 4.53 | 1,019.04 | 19.04 |
| Nov. 1, 1918 | 25.00 | 20.38 | 4.62 | 1,014.42 | 14.42 |
| May 1, 1919 | 25.00 | 20.29 | 4.71 | 1,009.71 | 9.71 |
| Nov. 1, 1919 | 25.00 | 20.19 | 4.81 | 1,004.90 | 4.90 |
| May 1, 1920 | 25.00 | 20.10 | 4.90 | 1,000.00 | ..... |
| Total premium amortized | $28.01 | ||||
The problem of amortization is thus seen to be comparatively simple when the cost of the bond, its nominal rate, and effective rate are known, and successive valuations of the bond are equally simple. The crux of the whole calculation is thus seen to be the determination of the original purchase price of the bond. At the time of the purchase of the bond the following facts are known: the par of the bond, the time it has to run, its rate of interest, and the rate of earning desired on the investment. There are three methods or formulas by which this price can be determined and they will be explained in turn. However, the development of the formulas will be more easily understood if some points in compound interest and annuity calculations applicable to the three methods of valuing the bond are first explained.
Formulas for Compound Interest