14.6 Bonds with Other Than Annual Interest Payments

Question: In the previous examples, both the interest rates and payments always covered a full year. How is this process affected if interest payments are made at other time intervals such as each quarter or semiannually?

As an illustration, assume that on January 1, Year One, an entity issues term bonds with a face value of $500,000 that will come due in six years. Cash interest payments at a 6 percent annual rate are required by the contract. However, the actual disbursements are made every six months on June 30 and December 31. In setting a price for these bonds, the debtor and the creditor negotiate an effective interest rate of 8 percent per year. How is the price of a bond determined and the debt reported if interest payments occur more often than once each year?

Answer: None of the five basic steps for issuing and reporting a bond is affected by a change in the frequency of interest payments. However, both the stated cash rate and the effective rate must be set to agree with the time interval between payment dates. The number of time periods used in the present value computation also varies based on the time that passes from one payment to the next.

In this current example, interest is paid semiannually, so each time period is only six months in length. The stated cash interest rate to be paid during that period is 3 percent or 6/12 of the annual 6 percent rate listed in the bond contract. Similarly, the effective interest rate is 4 percent or 6/12 of the annual 8 percent negotiated rate. Both of these interest rates must align with the specific amount of time between payments. Over the six years until maturity, the bond is outstanding for twelve of these six-month periods of time.

Thus, for this bond, the cash flows will be the interest payments followed by settlement of the face value.

• Interest Payments: $500,000 face value times 3 percent stated rate for a $15,000 interest payment every six months during these twelve periods. Equal payments are made at equal time intervals making this an annuity. Because payments are made at the end of each period, these payments constitute an ordinary annuity.
• Face value: $500,000 is paid at the end of these same twelve periods. This cash payment is a single amount.

As indicated, the effective rate to be used in determining the present value of these cash payments is 4 percent per period or 6/12 times 8 percent.

• The present value of $1 in twelve periods at an effective rate of 4 percent per period is $0.62460.
• The present value of an ordinary annuity of $1 for twelve periods at an effective rate of 4 percent per period is $9.38507.
• The present value of the face value cash payment is $500,000 times $0.62460 or $312,300.
• The present value of the cash interest payments every six months is $15,000 times $9.38507 or $140,776 (rounded).
• The total present value of the cash flows established by this contract is $312,300 plus $140,776 or $453,076. As shown in Figure 14.24 "January 1, Year One—Issuance of $500,000 Bond with a 3 Percent Stated Rate to Yield Effective Rate of 4 Percent Semiannually", the bond is issued for this present value amount. With this payment, the agreed-upon effective rate of interest (8 percent for a full year or 4 percent for each six-month period) will be earned by the investor over the entire life of the bond.

Figure 14.24 January 1, Year One—Issuance of $500,000 Bond with a 3 Percent Stated Rate to Yield Effective Rate of 4 Percent Semiannually

On June 30, Year One, the first $15,000 interest payment is made as reported in Figure 14.25 "June 30, Year One—Cash Interest Paid on Bond for Six-Month Period". However, the effective interest rate for that period is the principal of $453,076 times the six-month negotiated rate of 4 percent or $18,123 (rounded). Therefore, the interest to be compounded for this first six-month period is $3,123 ($18,123 interest less $15,000 payment). That is the amount of interest recognized but not yet paid that is added to the liability as shown in Figure 14.26 "June 30, Year One—Interest on Bond Adjusted to Effective Rate".

Figure 14.25 June 30, Year One—Cash Interest Paid on Bond for Six-Month Period

Figure 14.26 June 30, Year One—Interest on Bond Adjusted to Effective Rate

The compound interest recorded previously raises the bond’s principal to $456,199 ($453,076 principal plus $3,123 in compound interest). The principal is gradually moving to the $500,000 face value. Another $15,000 in cash interest is paid on December 31, Year One (Figure 14.27 "December 31, Year One—Cash Interest Paid on Bond for Six-Month Period"). The effective interest for this second six-month period is $18,248 (rounded) or $456,199 times 4 percent interest. The compound interest recognized on December 31, Year One, is $3,248 ($18,248 less $15,000), a balance that is recognized by the entry presented in Figure 14.28 "December 31, Year One—Interest on Bond Adjusted to Effective Rate".

Figure 14.27 December 31, Year One—Cash Interest Paid on Bond for Six-Month Period

Figure 14.28 December 31, Year One—Interest on Bond Adjusted to Effective Rate

The Year One income statement will report interest expense of $18,123 for the first six months and $18,248 for the second, giving a total for that year of $36,371. The second amount is larger than the first because of compounding.

The December 31, Year One, balance sheet reports the bond payable as a noncurrent liability of $459,447. That is the original principal (present value) of $453,076 plus compound interest of $3,123 (first six months) and $3,248 (second six months). Once again, interest each period has been adjusted from the cash rate stated in the bond contract to the effective rate negotiated by the two parties. Here, the annual rates had to be halved because payments were made semiannually.