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Bonds at a Premium or Discount

Lesson
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It takes a lot of time to get a bond ready to be issued. Because of that, it's nearly impossible to know what interest rate a bond should offer.

Often the market interest rate is higher or lower than it was when the bond offering was originally planned.

For this reason, bonds are often sold at a:

  • Premium - A bond is sold for more than its face value when the market rate is lower.
  • Discount - A bond is sold for less than its face value when the market rate is higher.

Often bonds will be assigned a number representing how much of a percent discount or premium its price.

For instance, a bond at 90 is sold at a 10% discount, and a bond at 110 is sold a 10 percent premium.

Buyers and sellers in the bond market need to be able to calculate what a premium or discount should look like in order to make a deal attractive to both parties.

Question
Pepper corp., the infamous cape importer, wants to sell a 3-year $46,000.00 bond. The stated rate is 8.00% and the market rate is 9.00%.
  • The present value factor for an ordinary annuity at 9.00% for 3 years is 2.53.
  • The present value factor for an ordinary annuity at 8.00% for 3 years is 2.58.
Will this bond sell at a discount or a premium - and by how much?
Answer
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👉 Answer:

  • The bond will sell at a discount of $1,164.40.

👨‍🎓 This is how we solve it:

  1. First, remember the formula to find the present value of the principal.
    PRESENT VALUE OF PRINCIPAL = BOND FACE VALUE / ((1 + MARKET INTEREST RATE) ^ YEARS)
  2. Now let's plug in the numbers
    $35,520.44 = $46,000.00 / ((1 + 0.09) ^ 3)
  3. Next, remember the dollar value of an end-of-period payment.
    PAYMENT = BOND FACE VALUE * (1 + STATED INTEREST RATE)
  4. Let's turn the variables into numbers
    $3,680 = $46,000.00 * (1 + 0.08)
  5. Next, get the present value of the periodic payments.
    PRESENT VALUE OF INTEREST PAYMENTS = PAYMENT * PRESENT VALUE FACTOR AT MARKET RATE
  6. Plugging in the numbers, we get
    $9,315.16 = $3,680 * 2.53
  7. Next, add the present value of the principal to the present value of all of the interst payments.
    PRESENT VALUE OF BOND = PRESENT VALUE OF PRINCIPAL + PRESENT VALUE OF INTEREST PAYMENTS
  8. Plugging in the numbers, we get
    $44,835.60 = $35,520.44 + 9,315.16
  9. Finally, Find the difference between the present value of the bond, and its initial cost. If the present value is higher, then people will pay a premium for it. Otherwise, people will demand a discount.
    DIFFERENCE = PRESENT VALUE OF BOND - BOND FACE
  10. And we see the following:
    $-1,164.40 = $44,835.60 - 46,000.00
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