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of the case, premising that the converse reasoning applies to discounts.
The view which I am criticizing as incorrect is the following:
1. That it is "presupposed that the holder of a bond will at the maturity of each one of the coupons, reinvest a sufficient portion of the money received and keep it so invested until the maturity of the bond, so that the face value of the bond, added to the accumulation of reinvested interest will at its maturity be exactly equivalent to the cost of the same.”
The other view, which I consider more correct, is the following:
2. That reinvestment is entirely outside of the problem, which is the division of the amount of each coupon into two portions, the second of which is not income at all, but repayment of principal, and may be reinvested or not.
These two theories may be styled the “reinvestment" and the repayment” plan.
I contend that 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.
The fallacious assumption of the necessity of reinvestment has led logically to a further divergence. The two plans would give precisely the same result numerically, if we assumed that the rate of reinvestment was exactly that of the net income, which is the usual assumption. But it is claimed by the reinvestmentists and conceded by the repaymentists, there is no certainty that reinvestments will be made at that exact rate. Therefore some of the former school have contended that the rate of reinvestment ought to be irrespectiv of the income-rate, but should be assumed at some fair definite rate, say 4 per cent.
This difficulty arises, according to my view, from the forced introduction of the reinvestment idea where it has no natural place. If you do not reinvest, the rate of possible reinvestment is immaterial.
Reinvestment is the purchase of another security; what we are discussing is how much is the percentage of yield of this security. It certainly is held during a time, but not necessarily all of it is held during all that time. Suppose we had a lot of 5 per cent. bonds maturing as follows:
$ 1,000 in 1 year
1,000 in 2 years
10,000 in 4 years These $13,000 bonds are purchased at par, and I think both schools would decide unanimously that the true rate of income was exactly 5 per cent. But no provision has been made for reinvestment. Would not the same reasoning apply here and ought not a portion of each coupon to be reserved in this case also; since it is probable that reinvestment could only be made at a lower rate, say 4 per cent? Our friends would hardly claim this, but it seems to me a logical consequence of their position that the given income is to be earned on the entire initial amount up to maturity.
In the above example the principal invested is not continuously $13,000, but is as follows:
$13,000 for 1 year
10,000 for 1 year And so likewise I claim that a two year 6 per cent. $1,000 bond (semi-annual) bought at 101.881, to pay 5 per cent., does not demand that the entire $1,018.81 should earn at the rate of 5 per cent. for the full time, for the investment is not all for the full time; it varies as follows:
$1,018.81 for 6 months
1,009.64 for 6 months
and 1,004.88 for 6 months In each semi-annual coupon of $30 there is exactly enough to pay 27/2 per cent. on the investment at the beginning of the half year and besides to pay off a part of the investment itself, the repayments being $4.53, $4.64, $4.76, and $4.88. It is a fact that the holder has actually had $25.47, $25,36, $25.24, and $25.12, as income for the several half years, and the previously-named sums in reduction of his principal. And how he has reinvested it of what has become of it has nothing to do with the case. If he hal put it all into mining stocks and lost it, that does not wipe out the fact that he did receive 5 per cent. interest for every day h held the investment. It came out just as he expected; not 5 per
cent. on the $1,018.81, which after six months was non-existent, but 5 per cent. on the successive amounts of his investment; 5 per cent. on what he did have, nothing on what he did not have.
Or looking at it another way; if you had a promissory note for $30, due in two years, which you had bought for $27.18, would you not be getting 5 per cent. compounded semi-annually on your $27.18? for 27.18 X 1.025 X 1.025 X 1.025 X 1.025 = 30. You admit that, do you? Mind, there is no reinvesting in this; you get $27.18 plus interest and it is settled. Now, suppose you discount another note of $30, under the same conditions but payable in 11/2 years, investing $27.86 and receiving 27.86 x 1.025 X 1.025 X 1.025 = 30. Here your money has again earned 5 per cent. of itself, for the whole time. Take another note of $30, payable in one year, and its present worth is $28.55 ; and one of $30 at six months worth $29.27. Finally buy a larger note of $1,000, two years to run; this will cost you $905.95.
Now what is a coupon bond at 6 per cent. for two years but five promissory notes printed on a single sheet of paper; four notes of $30, and one large one of $1,000. The proper prices at which to buy these so as to receive 5 per cent. compound interest are
$1,018.81 The total $1,018.81 is exactly the value of the bond as we have quoted it above; each promise has been fulfilled in cash with interest, and each account closed.
Why lug in reinvestment?
BY MONTGOMERY ROLLINS, Author of “Money and Investments,” etc. The editors of this magazine have requested the writer to present his views as to the somewhat complex theory upon which tables of bond values are based, and have called his attention to a fact, already well known to him, that different financiers or accountants hold antagonistic views upon such subjects.
In an article which the writer contributed to the September number of The Annals of the American Academy of Political and Social Science, Philadelphia, appears the statement that many are at a loss in their understanding of the theory upon which such tables are computed, by their not comprehending the fundamental principles upon which they are based, which presupposes that the holder of a bond will, at the maturity of each one of the coupons, reinvest a sufficient portion of the money received, and keep it so invested until the maturity of the bond, so that the face value of the bond, added to the accumulation of reinvested interest, will, at its maturity, be exactly equivalent to the original cost of the same. In a nutshell, that is part of the point in controversy, but even those who conceive that this theory is correct, still further have divergent views, one class believing that the portion of the coupon money so invested shall be compounded at an interest rate which is equal to the investment rate of the bond; not the rate of interest which the bond bears, but the “net return," so called. The other class believes it an absurdity to suppose that such a reinvestment plan can be followed, but that some definite fixed rate of interest—say from 3 to 4 per cent.—shall be selected, regardless of the net return which the original purchase price of the investment provides. It must be perfectly clear that it is safer to anticipate the future investment rates of money, set well within the limits of safety—say not exceeding 4 per cent.-than at such varying rates, sometimes high and sometimes low, which, at times, would be the equivalent of the net return of the money invested.
Practically all the tables in use are based upon the first plan. Even those who controvert the whole reinvestment theory, and who have themselves computed tables, follow the plan of computation based on the reinvestment at the earning rate. It has been the custom to buy and sell bonds upon such tables for so long a time that the memory of man hardly runneth to the contrary. The writer's own tables are so planned; yet he is free to concede that tables based upon the reinvestment at an average earning rate of 3 to 4 per cent. is much fairer to all parties. But such tables are not likely to find a broad market, owing to the fact that the others have become standard, the same as a yardstick or a pound weight.
With the foregoing principles in mind, it will be easy to approach the objections raised that the reinvestment plan is wrong in its entire; that a creation of a sinking fund, which is what this reinvestment plan amounts to, is neither actually practiced, nor ever can be; or that, if it were, the earning rate would not be what the investor is led to believe.
Let us see now. In the article in The Annals, to which reference has already been made, it is stated that the principle presupposes the holder of a bond, at the maturity of each one of the coupons, will reinvest a sufficient portion of the money received, and keep it so invested, until the maturity of the bond. In actual practice, if he does that, and does it promptly, and as he should do, the face value of the bond, added to the accumulation of the reinvested interest, will, at its maturity, be exactly equivalent to the original cost of the same. That is so easily proved, that it seems almost an absurdity to controvert it.
The ordinary tables of bond values show that a bond having twenty years to run, bearing 5 per cent. interest, nets the investor 4 per cent. at price of 113.68; that is, $1,136.80, for a $1,000 bond, plus the accrued interest, if any. This latter we will consider as a negligible quantity, as we are now dealing with the premium only. The investor is led to believe that he will receive 4 per cent. upon the purchase price of $1,136.80. As a matter of fact, he receives $25.00 each six months, or $50.00 yearly. At the maturity of the bond, he will receive, besides the last interest payment, only the principal sum of $1,000. There must be some manner, therefore, of accounting for the $136.80 premium paid at the time of purchase. It is the sinking fund, already referred to, which provides this. To carry the illustration further, the investor must reckon 4 per cent. upon the total cost price of $1,136.80, which would amount to 22.74 for each six months' period. The semi-annual coupon being $25.00, there is left, therefore, a sum of $2.26, which, if immediately invested upon the day the money is due, will, at the maturity of the bond, added to the principal sum, together with other amounts similarly deposited twice yearly, equal the original purchase price.
In the twenty years which the bond has to run there will be thirty-nine times $2.26 deposited, which will have drawn interest, and one like sum taken at the maturity of the bond, which will have no time to draw interest. Forty times $2.26, however, will be the sum set aside, making a total of $90.40, which is $46.40 less than the actual amount sought. This amount is provided for by the interest-compound-upon the sums set aside.