Journal of Accounring Educalion Vol. 4, No. 2

Fall, 1986

COMMENT AND REPLY “A THEORETICAL DEFICIENCY IN

ACCOUNTING FOR BONDS.”

Editor’s Note: We are publishing the following as a service to our readers. Professors Myles Stern

and Meryle Hirschland, Wayne State University, have submitted comments concerning an

article, “A Theoretical Deficiency in Accounting for Bonds,” by Serge Matulich, Rollins College,

which originally appeared in the Fall 1984 issue of the Journal. What follows are the comments of

Professor Stern and Professor Hirschland and the replies of Professor Matulich.

Professors Stern and Hirschland

Under compound interest, an investment (or debt) grows along a com- pound interest curve from the issue date or one interest payment date to the next interest payment date or redemption date. For convenience, and because the difference is almost always immaterial, calculations made under the effective interest method within a single interest payment period have tradi- tionally been on a straight-line basis for any partial period.

Consider the simplest case of a six-month note for $10,000, carrying interest of 12 percent (i.e., 6 percent for the six-month period) and issued at par (i.e., $10,000). Only along the compound interest curve, however, does the invest- ment grow at a constant 6 percent rate during the entire period. With the straight-line method, the investment grows at a rate exceeding 6 percent early in the period and at a rate less than 6 percent late in the period.

We suggest that for any investment (debt), the interest income (expense) for a period ending with an interest payment is the difference between the invest- ment’s price at the end (immediately before the interest payment) and the beginning of the period, calculated by strict application of compound interest.

Matulich is inconsistent in his approaches to his two topics. Had he used the same numerical example throughout, this inconsistency might have been more obvious. When he prices a bond issued between interest dates, he correctly applies compound interest theory. However, when he prepares the fiscal year-end adjusting entry to record interest expense and discount amorti- zation, he violates that theory. The corrected entries for the mid period accrual of interest are shown below:

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December 3 1:

Interest expense $4,424.67 Discount on bonds payable Interest payable

Expense: $ 95,899.80 x [( 1.07)x - I] = $4,424.67 Accrual: $100,000.00 x .12 x 4/ 12 = 4,OOO.OO

$ 424.67 $4,000.00

February 28:

Interest expense $2,288.3 1 Interest payable 4,ooo.oo

Discount on bonds payable Cash

Expense: $ 95,899.80 + $4,424.67 = $100,324.47 $100,324.47 x [( 1.07)s - J] = 2,288.31

288.31 6,OOO.OO

An alternative calculation would move the interest liability along a com- pound interest path. The accrual on December 31 would be:

$100,000.00 x [( 1.06)‘/6 - I] = $ 3,961.03

The amount of discount amortized is thus $4,424.67 - $3,961.03 = $463.64. The only difference here is in the classifi:cation of the liability between “dis- count on bonds payable” and “interest payable.”

In advancing the mid-period straight-line proration, Matulich states that “in any compound interest problem with discrete compounding periods, growth of the principal does not occur between periods.” We disagree with this basic premise and argue instead that, as discussed above, under com- pound interest the principal grows at a constant rate. Further, if the principal were not continuously changing, no entry to record amortization should be made at fiscal year-end.

Professor Matulich

Professors Stern and Hirschland are correct that my treatment of the mid-period accrual of interest and bonds issued between interest dates is inconsistent. Dealing with interest accrual at the end of an accounting period between interest payment dates, was intended to illustrate common error that occurs frequently in the classroom. After recording the interest accrual cor- rectly, students often use the end-of-year book value of the bonds to calculate interest expense for the remainder of the six-month period which occurs in the following accounting year. My intent was to show that using the end-of-year book value introduces an additional compounding period; the correct cal-

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culation is based on the book value of the bonds after the previous interest payment, and not after the end-of-year interest accrual.

It was not the intent of my discussion to imply that the recording of accrued

interest at year-end should be done in a theoretically correct fashion. The practical considerations outweigh the theoretical aspects, although I realize that growth takes place continuously. In order to avoid confusing the practi- cal discussion with the subsequent theoretical treatment of the two distinct issues covered in my paper, I deliberately used different bond examples in the mid-period accrual of interest and bonds issued between interest dates.

Professors Stern and Hirschland

Matulich applies compound interest theory correctly in pricing the bonds in his second example. [p. 581 The figure for “present value of $1 for ‘/ period at 5 percent” was apparently calculated by the formula l/ (1 + r)n, where “r” is the interest rate per period and “n” is the number of periods. Using this formula, we obtain the number 0.975900233 rather than the 0.975609756 Matulich uses. Ourcalculationsprice the bondsat $115,715.16ratherthan$115,680.73.

We would make either the first entry (with slight corrections of the amounts) Matulich shows for this case or one based on the computation of the interest liability along a compound interest curve (using our own calcula- tions):

Cash 115,715.16 Interest payable 3,440.80 Bond payable 100,000.00 Bond premium 12,274.36

$100,000 x [( 1.07)x - l] = $3,440.80

Had Matulich applied this same approach in his first example, he would have obtained the amounts we show in our December 3 1 adjusting entry. The present value of $1 for I,$ period at 7 percent is 0.9776996. The present value of the bonds on February 28 of year 2, immediately before the $6,000 interest payment is $96,612.79 (amount given by Matulich on p. 55) + $6,000 q

$102,612.79. Discounting this amount back to December 31 of year 1 gives 0.9776996 x $102,612.79 = $100,324.48. The difference between this last figure and Matulich’s book value for the bonds on August 31 of year 1 is the amount of interest expense recorded on December 31: $100,324.48 - $95,899.80 = $4,424.68. This expense is identical, except for a one-cent rounding error, to the amount in our adjusting entry.

Matulich finds a paradox in the amortization of premium on bonds issued between interest dates. One method [Table 5, p. 601 “violates theory because it assumes the borrowers can obtain a part of the bond issue interest-free during

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the first compounding period.” A second method [Table 3, p. 591 “violates theory by introducing a partial compounding period.”

We also reject the former approach, but contend that Table 3 is, contrary to Matulich’s statement, completely consistent (ignoring the slight discrepancy in the initial price of the bond) with compound interest theory! Since his initial price for the bonds was calculated in a theoretically-proper fashion, there is no “partial compounding period” introduced! To confirm this statement, take the price of the bonds had they been issued at the start of the 4-year period, and move it along the compound interest curve: $112,926.43 x (1.05)‘/6= $115,715.16. This amount is identical to the price calculated earlier for the bonds three months prior to the first interest payment.

Professor Matulich

The second issue raised by Professors Stern and Hirschland is easily dis- posed of: The present value of $1 for ‘/2 period at 5 percent is 0.97590000747 as calculated by Professors Stern and Hirschland. The value used in the paper -0.975609756 - is one-half of the present value of $1 for one period, and was based on the incorrect assumption that growth between interest payments occurs in a straight-line fashion as assumed in practice. But as Professors Stern and Hirschland indicate, growth of an outstanding bond is continuous and hence curvilinear. Using the correct value for the first half period during which the bonds are outstanding, the price of the bonds is $115,7 15. I5 at the time of issue. The correct version of Table 3 in the original paper is shown below, with calculations carried to seven decimal places to show with preci- sion how the premium amortizes from the original issue price to the face value of the bonds.

Interest Period Payment

TABLE 3

Interest Premium Expense Amortization

Book Value of Bonds

.5

$7,000 2,857.5943438 $4,142.4056562

7,000 5,578.6373368 1,421.3626632

7,000 5,507.5692037 1,492.4307963

7,000 5,432.9476639 1,567.0523361

7,000 5,354.5950470 1,645.4049530

7,000 5,272.3247994 1,727.6752006

7,000 5,185.9410394 1,814.0589606

7,000 5,095.2380913 1,904.7619087

115,715.1523927

111,572.7467365

110,151.3840733

108,658.9532770

107,091.9009409

105,446.4959879

103,718.8207873

101,904.7618267

99,999.9999180

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The corrected version of Table 3 addresses the issue of bonds issued between interest dates. It shows that the first cash payment of $7,000 includes $2,857.59 of interest and amortizes $4,142.41 of premium. The beginning book value of the bond includes the bond price plus accrued interest. Conse- quently, to record the bond issue as done in practice the entry is:

Cash 115,715.15 Interest payable 3,500.oo

Bonds payable 100,000.00

Bond premium 12,215.15

On the first interest payment date, the $3,500 of interest payable is written off and $642.41 of premium amortization reduces the book value of the bonds to $111,572.75 as shown in the table above. One of the more interesting issues raised in my paper is the question: Is it possible for companies to receive accrued interest from bond investors at the time the bond is issued and simply return this money on the first interest payment date without paying interest on it? Although Professors Stern and Hirschland do not address the issue directly, the answer to the question is implied in the journal entry they provide. They record the initial bond issue by showing interest payable at only $3,440.80, which is the theoretically correct amount of interest liability accrued by the issue date of the bond. Their entry results in a larger amount of bond premium than my entry shown above.

The implication is that when the first interest payment of $7,000 is made to investors three months after the bonds are issued, it includes $2,857.59 of interest accumulated during the past three months on the entire amount borrowed. The rest of the payment is a return of $3,440.80 plus $701.61 of the borrowed principal, leaving a principal book value of $111,572.75 as shown in the table. (A rounding error of $.Ol occurs when calculations are rounded to two decimal places.)

Interest payable 3,440.80 Bond premium 701.61 Interest expense 2,857.59

Cash 7.000.00

Whether we call part of the principal “interest payable” or “bond pre- mium,” it is still money borrowed from investors at the time the bonds are issued and interest is incurred on the entire debt. To the extent the usual accounting treatment of these transactions deviates from the theoretically correct treatment described by Professors Stern and Hirschland, the error can be justified on grounds of practicality and lack of materiality.

REFERENCE Matulich, Serge (1984), “A Theoretical Deficiency in Accounting for Bonds,” Journal of

Accounfing Educarion (Fall 1984), 53-62.