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It is not necessary here to show that these sums at compound interest will yield that amount. No "controvertist” will deny that. Why, then, should they claim that it cannot be done; that is, that no such sinking fund ever can be put into practice? Perhaps it never is put in practice, but there must be some theory upon which bond values tables are computed, and, right or wrong, or in want of a better, that is the theory in use, and all tables agree, except where the disagreement arises from different reinvestment rates.
In practice, the holder of a coupon due, say, July 1st, would probably not receive his money, in most cases, on that date, and then, again, it might be, and probably would be, difficult to get it drawing interest promptly upon July 1st, if even so received. But there must be some average rate that money can be so reinvested, even allowing for the delays referred to. That is another reason why, perhaps, the reinvestment plan upon a lower average rate of interest is the more correct. But the theory remains unchanged.
Let us further illustrate with the foregoing example:
When the bond is first purchased, it has exactly twenty years to run. The quotation from The Annals presupposes that it will be held until maturity, that the holder may get his net return of 4 per cent. That, however, does not prevent the holder from selling the bond and still realizing his net rate, provided he sells it upon the same relative basis upon which he purchased it. In this case, let us suppose that he holds it exactly five years, at which time there remains fifteen years to maturity. Having purchased it on a 4 per cent. basis, which is $1,136.80, he now sells it upon the same basis, which, for fifteen years, is $1,112.00, an apparent loss of $24.80. Now, if the “controvertists” are right, that the holder of a bond will not earn what he has been led to expect, then there is some loss here to be explained. We will make this as simple as possible. The original purchaser, when the bond had twenty years to run, at the end of his first six months' period, received a coupon of $25.00. He immediately went down to his savings bank, which paid 4 per cent. interest, and upon which interest was allowed him the moment the money was deposited, and he deposited the money the day the coupon was due. His $2.26 began earning interest at once at 4 per cent. Each six months he betook himself to his savings bank and went through a like process. This process he kept up for five years, until he sold the bond. But he was face to face with the fact that he only obtained $112.00 premium, and that he had paid a greater sum. Where was this money needful to take care of his apparent loss? He suddenly bethinks himself of this savings bank, and, having some knowledge of the theory of bond values, reasons that the sums which he has deposited there have been compounded and that, unless the bond values table had deceived him, there must be in the bank $24.80 awaiting him. In, perhaps, a skeptical frame of mind, having read some of the articles of the “controvertists,” he goes down, presents his book, and demands the full amount there on deposit. Lo and behold! he is presented with exactly $24.80. So, therefore, he finds that he has received 4 per cent. upon the original purchase price of the bond-namely, $1,136.80 for the full time he has held it. He sells it upon the same relative basis upon which he purchased it, and the sinking fund which the “controvertists " claim never is and never can be practiced, does take care of shrinkage in premium.
Bear in mind all the time that the writer of this article is not himself an exponent of the reinvestment plan at the earning rate, but that some rate—say 3 per cent.-should be selected, upon which there can be no question of carrying out this plan.
One word more. The “controvertists” claim that, in actual practice, this sinking fund plan is never carried into effect. Again the writer claims that that does not change the theory. But, whether or no, advanced students in finance and political economists will make this reply: that even if the sinking fund plan is not put into actual practice, it does not change the fundamental principle that money has a value, and that, therefore, even if the sums are not set aside each six months and invested, it does not change the principle that the money must have been used for permanent improvements, living expenses, or what not. If one did not possess the money for such purposes he might have to borrow it and pay at greater or lesser rate of interest for its use. Therefore, money has a value, the same as coal, pig iron or any other commodity, and it is the money value which is taken into consideration.
It may possibly be an opportunity to bring in a point here, not always clear, but, perhaps, somewhat foreign to this discussion, that the holder of this bond, cited above, selling at the end of five years for $1,112.00, made neither a gain nor a loss, but received his 4 per cent. just as expected. Suppose, however, he wished to find out whether he had sold at a profit, or at a loss, he would simply have to find the same relative basis as the purchase price, taking the time that the bond has to run into consideration, which in this case is $1,136.80, and compare that with the price received. If greater than $1,112.00, he has made a profit; if less, he has made a loss.
About Bond Values.
books of reference. The article in this number of THE JOURNAL OF ACCOUNTANCY headed “A Fallacy in Bond Values” is generally, but not always, correct in its statements—although always so in its results.
The reader can refer to the first three paragraphs in that article and also to the paragraph numbered 2.
Following those paragraphs there is a statement that "the plan of using a sinking fund in which to reinvest a portion of the interest periodically received from the bond, for the purpose of wiping out the premium at maturity—which he calls the 'reinvestment plan '-is purely fictitious; that no such creation of a sinking fund for the extinction of a premium is ever practiced or can be practiced, and that if it were put in practice, the holder of the bond would not be earning the rate on his investment which he has been led to believe he would earn."
On the contrary, I believe that the author of the article here commented upon practically, if not consciously, uses the plan himself, although not in the ordinary sinking fund manner of depositing the same amounts each time he makes a deposit.
Now, I contend that if any one is liable for an amount due at some future time, and wishes to decrease that liability by a certain sum, say $5, it may be done in two ways—either by placing in a sinking fund, at the agreed rate, a sum which is equivalent to the present worth of the $5—or, it may be accomplished by discounting (that is, by deducting) now, from the amount of that liability, the present worth of the $5 by which sum it is wished to diminish the future liability. And I contend that the issuer of a bond is liable, not only for the income promised, or implied, but also for the amount which the holder paid for the bond. He pays up these two liabilities by furnishing to the holder, periodically, the interest that the bond promises, and by paying him $1,000 at maturity.
To illustrate practically what I mean, I will use one of the examples used in the article referred to. And it is not necessary to use more than one example—for the principle and operation are the same in all cases.
I buy a two-year $1,000 6 per cent. bond, as a 5 per cent. investment. The proper book value when bought is $1,018.81, which is, practically, owed me by the issuer of the bond; and this debt is to be liquidated by paying to me $30 at the end of each six months and $1,000 at the maturity of the bond. At the maturity, my debtor (the issuer) must, in some way, have paid me back the premium; as he will then have only the $1,000, which the bond will call for.
I receive at the end of each half year $30, the semi-annual interest which the bond pays. The interest on my investment of $1,018.81 for six months at the investment rate is 25.47; and, of course, that is my income for the six months, which I can spend as I please. The other 4.53, equal to 30.00 — 25.47; must, in some way, be devoted to helping pay off the premium. It may be used in either of two ways:
By investing it in an ordinary sinking fund whose rate is the same as the investing rate—or it may be deducted from my original investment, thus reducing my investment to that extentthe issuer of the bond thus, practically, paying me back 4.53 of my investment, and making my investment at the end of the first six months $1,014.28. My income the next six months will be 21/2 per cent. on 1,014.28, that is 25.36, and the sum to be deducted from 1,014.28 will be 30.00 — 25.36, equal to 4.64, leaving my investment and the issuer's liability at end of twelve months, 1,009.64, and so on. To find the sums I have remaining invested at the ends of the next two half-years respectively, the amount which should be successively deducted from the next previous amount found as my investment at those periods, being 4.76 and 4.88 respectively. Thus 1,009.64 — 4.76 equals 1,004.88; and 1,004.88 — 4.88 equals 1,000.00, which the issuer of the bond will then pay me, the bond having matured. There is no necessity of going through all this formula; but, in this case, simply to
deduct 4.53 from the original investment, and then to successively deduct what 4.53 compounded one, two, or three times will amount to, thus: 1,018.81 – 4.53 equals 1,014.28; this, less 4.64, equals 1,009.64; this, less 4.76, equals 1,004.88; and this, less 4.88, equals 1,000.00.
The income from any bond for the first half year equals six months' interest on the original investment; and for each succeeding half year is found by finding the interest on the amount invested at the first of that half year. By this plan I have been practically paid back my premium by varying instalments, amounting in each case to what the first instalment would amount to if interest were compounded on it up to the time it is used.
The results thus obtained are exactly those stated in the article commented upon, whether arrived at by this plan or not.
Now for the “reinvestment” plan.
It assumes that the holder of a bond to maturity is liable, at maturity, for the premium he paid and, in order that he may be able to pay it up then, allows him to collect interest on his whole investment for the whole time for this purpose—and it enables him to do so.
Let us take the same example as we took in the other case. I am not trying to prove that a suitable sinking fund for taking care of the premium could be found-for, as this is purely a theoretical matter, it is immaterial whether such a sinking fund could be found or not. I am only trying to show that if we could find it, everything would be properly provided for, and in order to prove this exactly, it is necessary, as I will do, to use several decimal places.
I will bunch the whole matter and show (as my “Bond and Investment Tables” or my “Building-Loan Interest Tables,” either, enables me to do) that the instalment for such a sinking fund (whose rate is the same as that of our investment), as indicated by deducting the income for the first six months from the interest paid by the bond for that period, is (viz., 30.00 – 25.47025) equal to 4.52975. This sum is to be placed at interest (always at the earning, or investment, rate, is understood) at the end of each half year, for four half years. Now, $1.00 so placed will amount to 4.15252; hence, 4.52975 will amount to 4.52975 times 4.15252, equal to 18.809867—which shows that the holder has wiped out the premium—that he has received an income,