# Notes

Question: Although zero-coupon bonds are popular, notes and most bonds actually do pay a stated rate of cash

interest, one that is specified in the contract. If the buyer and the seller negotiate an effective rate of interest that

is the same as this stated rate, an amount equal to face value is paid for the bond. If the stated interest to be paid

is 7 percent each year and a negotiated annual rate of 7 percent is accepted by the parties, the bond is issued for

face value. No discount or premium results; the debtor and creditor are satisfied with the interest being paid. The

effective rate method is not needed because the cash interest and the effective interest are the same—7 percent is

paid and recognized as interest.

However, the negotiated rate often differs from the cash rate stated in a bond contract. Market interest rate

conditions change quickly. The interest that creditors demand will often shift between the printing of the indenture

and the actual issuance day. Or the financial reputation of the company might vary during this time. Information

To illustrate, assume that Smith Corporation decides to issue \$1 million in bonds to the public on January 1, Year

One. These bonds come due in four years. In the interim, interest at a stated cash rate of 5 percent will be paid

each year starting on December 31, Year One. These are term bonds because interest is conveyed periodically by

the debtor but the entire face value is not due until the end of the term.

No investors can be found who want to purchase Smith Corporation bonds with only a 5 percent annual return.

Therefore, in setting an issuance price, annual interest of 6 percent is negotiated. Possibly, interest offered by

other similar companies is 6 percent so that Smith had to match this rate to entice investors to buy its bonds.

Or some event has taken place recently that makes Smith seem slightly more risky causing potential creditors to

demand a higher rate of return. A list of market conditions that can impact the price of a bond would be almost

unlimited. How is the price of a bond calculated when the stated cash rate is different from the effective rate that

is negotiated by the two parties involved?

Answer: The pricing of a bond always begins by identifying the cash flows established by the contract. These

amounts are set and not affected by the eventual sales price. The debtor is legally obligated to make these

payments regardless of whether the bond is sold for \$1 or \$10 million.

Here, Smith Corporation must pay \$50,000 per year in interest (\$1 million × 5 percent) for four years and then the

\$1 million face value:

Cash Flows in Bond Contract

\$50,000 annually for four years

\$1,000,000 in four years

After the cash flows are identified, the present value of each is calculated at the negotiated rate. These present

values are then summed to get the price to be paid for the bond. The \$50,000 interest payments form an annuity

since equal amounts are paid at equal time intervals. Because this interest is paid at the end of each period starting

on December 31, Year One, these payments constitute an ordinary annuity1. As determined by table, formula, or

Excel spreadsheet, the present value of an ordinary annuity of \$1 at an effective annual interest rate of 6 percent

over four years is \$3.465112. Thus, the present value of the four interest payments is \$50,000 times \$3.46511 or

\$173,256 (rounded). Note that the present value computation requires the multiplication of one annuity payment

(\$50,000) rather than the total of the interest payments (\$200,000).

Present Value of an Ordinary Annuity of \$1Present Value of an Ordinary Annuity of \$1

The second part of the cash flows promised by this bond is a single payment of \$1 million in four years. The

present value of \$1 in four years at a 6 percent annual rate is \$0.79209 so the present value of the entire \$1 million

is \$792,090.

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