bond to which this value applies we will find it to be as follows on January 1, 1908:

$1030. due Jan. 1, 1910

30. due July 1, 1909

30. due Jan. 1, 1909

30. due July 1, 1908

Let us now suppose this bond to be altered to an annual bond, with all other conditions remaining unchanged. It would then assume the form:

$1060. due Jan. 1, 1910

60. due Jan. 1, 1909

Now it is certainly not taking very much for granted to premise that the value of 101.88 is no longer applicable * to this bond after having made such a radical change in its make-up as deferring for six months the collection of each alternate coupon, and that the value, or worth of the bond to an investor has to a greater or less extent been changed; i. e., enhanced or impaired. Suppose this very simple proposition were submitted to 100 ordinary men accustomed to lending money or buying bonds; who make no pretensions to actuarial skill, but have a general common sense understanding of the principles of banking; know that money is worth money, etc., how many of them would adjudge that the value (i. e., actual intrinsic worth) had been impaired, and how many that it had been enhanced?

No one would go to the labor and expense of preparing and publishing a complete set of bond value tables unless he expected to find a market for same; hence, inasmuch as at least three different editions for quarterly bonds have appeared between 1885 and 1905, the inference is that both computers of tables as well as the users thereof realized that the ordinary semi-annual tables were not available for quarterly bonds, and that special tables were necessary. Inasmuch as the respective contents of all three tally throughout, it is immaterial which one we refer to. One of them, however, which contains both quarterly and semi-annual values side by side affords the following:

3% BOND-25 Years. Coupons maturing. 22% basis

3% basis

42% basis

*One of the authors of the annual tables hereinafter referred to, states in a circular: "There is quite a little inaccuracy in using the semi-annual interest bond value tables to compute the return from bonds with interest payable annually."

Now if words in the English language mean anything, a bond value is the value of a bond, and if these values above quoted are correct, then they mean:

If we are considering the purchase of a 3% bond running 25 years, and have both quarterly and semi-annual bonds to choose from;

If we are willing to accept 22% for our money, the quarterly, bond is worth a trifle * more than the semi-annual bond;

If we are unwilling to accept a less rate than 42%, then the semi-annual bond is worth a trifle* more than the quarterly bond;

And finally if we value our money at 3%, then it is totally immaterial how frequently the coupons mature.

There are also at least two editions of tables for annual bonds, and the same relationship as shown above would have been obtained had we used them instead of quarterly values for comparison with the standard semi-annual values. In fact they might all three have been compared at one time and the general results would have been the same; viz., when the rate to be realized coincided with the nominal rate of the bond it made no difference how frequently the coupons matured, and when the effective rate exceeded the nominal rate, then the less frequently the coupons matured the greater the value of the bond.

I have merely quoted the values of the bonds as they appear in these different publications. Deductions as to the value of the values are left to the judgment of the reader.

One would think that the computers of these tables for annual and quarterly bonds must have observed these absurd and inconsistent properties, and would either have refrained from putting out such calculations, or at least have indulged in some apology or explanation in a preface, but such does not appear to be the case. The author of the quarterly values just quoted does state, however:

"The table of values for Government bonds, interest being paid quarterly, has been computed on the usual assumption that the interest so received constitutes a sinking fund, each installment of which is invested to yield the same rate of interest as the principal."

In the article by Mr. Rollins which appeared in the November number of this magazine he said, quoting from another article of

*It will be noted that the difference is very slight, although as a banking proposition there is considerable difference between .75 every three months and 1.5 every six months, for 25 years.

† My italics.

his own that had appeared shortly theretofore in another publication:

the fundamental principles upon which they (bond values) 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."

It is certainly remarkable, in view of the fact that there are at least four totally different methods of regarding the subject of bond values (as well as making calculations thereof), all equally sound, and giving identical results, that so many authors of tables should have invented a fifth one that is unsound from start to finish, and the absurdities that have just been pointed out in the values that were based on these "usual assumptions" and "fundamental principles" furnish all the evidence that is necessary to indicate how thoroughly mischievous and fallacious these two utterances just quoted are, as well as showing the fatal consequences of pushing too far the luck that attends the "usual assumption" plan.

If the purpose of this article were merely to establish the fact that a fallacy in bond values exists we could rest our case; nevertheless it is proper to go further into the matter and discuss at length and in detail one of the pitfalls that lies in the "reinvestment" path.

There is a more or less well known and perfectly sound expedient for obtaining values for bonds not comprised in tables (such as a 34% or a 10% bond) by ascertaining the constant difference between a 4%, 5%, 6%, etc., bond and making the proper correction. Taking any edition of the semi-annual tables and turning to the 25 year page, we find in the 22% income line: 4% bond 3% bond

...

....127.76

109.25

Their difference is 18.51. Subtracting successively we obtain: 2% bond 90.74; 1% bond 72.23, and finally an 0% bond 53.72.

Taking the tables for quarterly bonds and proceeding in the same manner we obtain 53.62 as the value for an o% bond on a 22% basis, 25 years to run. If these two 0% bonds were closely examined they would not be found to differ from each other in a

*"The two plans would give precisely the same result numerically if we assumed that the rate of re-investment was exactly that of the net income, which is the usual assumption." (From the article: A Fallacy in Bond Values, by Chas. E. Sprague, in the November issu● of THE JOURNAL OF ACCOUNTANCY.)

manner to indicate a difference of .10 in their respective worths; whence more evidence that we are still on the trail of a fallacy.

Mr. C. E. Sprague has, however, recently published a table of values for quarterly bonds whose contents do not tally in a single instance with the other quarterly tables that are under consideration, but on the contrary differ widely. Employing his tables we find: 4% bond, 127.990269; 3% bond, 109.426178; difference 18,564091, and working out an 0% bond in the same manner we obtain 53-733905, or the same as the 53.72 first obtained from the semi-annual tables.

*

The fact that these totally different values for 4%, 3%, 2%, etc., bonds should reduce to a common value for an o% bond is, instead of being a matter to marvel at, merely as it should be, for when the time to run and income basis are fixed, all o% bonds are worth the same, regardless of whether their coupons mature every 100 years or every five minutes. Now if an investor buys an 0% bond running 25 years to yield him 22% income and pays 53.73 therefor, if this price is a correct one it is certainly very clear that the 100. that he will collect in a lump at the end of 25 years will represent the return of his principal, together with interest thereon, and since the problem is not complicated by his having betaken himself to the Knickerbocker every three or every six months with "a sufficient portion of the money received" from his coupons for $0.00, this 53.73 must have been the PRESENT WORTH of 100. for 25 years at 22%, and upon making the calculation or consulting suitable compound discount tables this will be found to be the case.

The 53.62 may also be found in the compound discount tables, but in an entirely different place, and not close by the 53-73 as might be supposed from the fact that in each case the time was 25 years; the amount 100. and the rate 22%. The 53.73 will be found in the 14% rate for 50 periods, and the 53.62 in the 5-8% rate for 100 periods.

This lets the cat out of the bag. In all the tables for semiannual bonds the interest has been compounded semi-annually; in the annual tables it has been compounded annually, and in all the quarterly tables (except those of C. E. Sprague) it has been compounded quarterly. In this connection it is not improper to state that to prepare tables for quarterly bonds and compound the

*This discrepancy of .01 does not signify an error in either tables, but is due to there not having been more decimals in the short tables.

interest quarterly is child's play, using any one of the five methods of procedure, while to construct tables for quarterly bonds and compound the interest semi-annually is a task that is appalling in its difficulties; even if rough accuracy only is desired. Without making any insinuations, however, and taking the reinvestment votaries at their word, it will be plainly seen how utterly impossible it would be to do what they claim is necessary in order for an investor to realize the rate of income the tables had told him he would receive, for according to the "usual assumption" man the investor would not only have to continue buying bonds for small and fractional amounts on the same income basis as his original purchase, but would also have to restrict himself to quarterly bonds, while according to Mr. Rollins,* when a quarterly bond was bought a savings bank would have to be found that would consent to upset its system and compound interest on balances quarterly (with full interest allowances on odd cents), and if the bond happened to be annual, then, unless the bank could be prevailed upon to compound the interest annually, the balance would ultimately be too great, thus causing the whole system to fall to the ground.

The process by which the foregoing analysis was made was simplified by the expedient which I designate as "clearing the bond of its coupons," but at the same time the proposition of an 0% bond is not an absurdity at all, or even far-fetched, and such a bond could easily be conceived to exist. e. g. a village wishes to issue some 25-year bonds to build a city hall; the understanding being that the bonds will be taken by the citizens themselves. A public meeting is held, and the said citizens, actuated by public spirited motives, proceed to "bid down" the rate of interest among themselves. One man offers to furnish the money at 3%, another at 2%, another offers to take a certain amount of the bonds at 1%, etc., until finally a philanthropic merchant (who possibly also figures that the notoriety involved might be worth something to him) creates a sensation by offering to furnish the money at 0%, which means that he pays par for bonds that have no coupons attached. Later on it becomes necessary for him to turn one of these bonds into cash and he is unable to find any one who will buy it on the original basis, but really does find a pur

"In a bond with the interest payable but once a year, this money can only be reinvested and compounded once a year. Likewise in a quarterly table, it will be four times a year." (Annals Am. Acad., September, 1907, p. 55.)