5.7 Bonds
Learning Outcomes
Calculate the coupon payment size, market value or gain/loss for bonds.
If a corporation or government is looking to raise money (capital), they can issue bonds. The issuer (the company or government) must pay the bond holder (owner of the bond) a series of equal-sized regular interest payments for a fixed period of time. The size of these payments is determined by the agreed-upon interest rate (coupon rate). At the end of the fixed period of time, or maturity date, the issuer must repay the bond holder the principal (the amount of money the bond issuer borrowed).
We consider the bond is a debt owed by the issuer to the bond holder. The amount owed never increases because the issuer pays the interest owed each period to the holder in the form of a coupon payment. This is why the final amount owed by the issuer to the bond holder (the face value) will be the principal (amount borrowed).
There are several different types of calculations for bonds — see the sections below for the key formulas, tips and examples related to bond calculations.
The coupon payment (PMT) is the interest earned in one period (6 months):
where the principal is the initial purchase price of the bond (when the bond is purchased from the issuer) and i is the periodic rate. The periodic rate is the interest rate for one period.
Example 5.7.1
Amir purchased a $3,000 bond that has a 10-year term (will mature in 10 years). The bond has a coupon rate of 5%, compounded semi-annually. What is the size of the semi-annual coupon payment?
Check your Knowledge for Example 1
Often, the bond holder can sell the bond (transfer the bond) to a secondary bond holder before the end of the term. Because interest rates change throughout the term of the bond, the bond can be worth different amounts at different times throughout its term. The amount the purchaser (secondary bond holder) is willing to pay is called the fair market value of the bond. The fair market value is determined by the current interest rate, the size of the coupon payment, the number of years remaining in the term (number of years before the maturity date) and the face value of the bond.
We will use the BAII Plus to calculate the fair market value of a bond:
| PV | Interest | PMT | FV |
|---|---|---|---|
| Fair Market Value | + % Market Rate | = Coupon Payments | + Face Value |
| − | − | + | + |
Bonds are the ONLY type of investment in this text where we will have a negative value for PV and a positive value for FV. To make sense of this, imagine the bond from the perspective of the bond holder. The bond holder initially ‘lends’ money to the bond issuer when purchasing a bond from the issuer. We consider this amount lent (PV) to be negative because this is the amount the bond holder must outlay to purchase the bond.
Each period, the issuer will owe the bond holder interest on that loan. The issuer pays this interest (coupon payment) to the holder each period. The issuer also repays the face value of the bond (FV) at the end to the holder[1]. For this reason, we consider PMT and FV as positive because they are money that the bond holder receives.
Example 5.7.2
Francis purchases a $4,000 bond that has a 30-year term. The bond has a coupon rate of 5.25%, compounded semi-annually and a semi-annual coupon payment of $65. If Francis chooses to sell the bond after 10 years when the current interest rate is 3%, compounded semi-annually, what is the fair market value of the bond?
Check your Knowledge for Example 2
Conclusion: Francis can sell the bond for $4,149.58 in 10 years.
If the interest rate (coupon rate) on a bond is above the market rate then the bond increases in value (sells at a premium). If the interest rate (coupon rate) on a bond is below the market rate, then the bond goes down in value (sells at a discount). The gain (or loss) is calculated by:
Let us use this formula to determine Francis’ gain or loss in Example 2.
Example 5.7.3: Selling at a Premium
What is Francis’ gain or loss on the sale of his bond in Example 2?
Check your Knowledge for Example 2b
Conclusion: Francis will make a $149.58 gain when he sells the bond in 10 years.
Example 5.7.4 — Selling at a Discount
Redo Example 2 with an interest rate (market rate) of 6% when Francis chooses to sell the bond in 10 years. What is Francis’ gain or loss on the sale of the bond?
Check your Knowledge for Example 2c
First, we need to calculate the fair market value of the bond in 10 years:
The fair market value of the bond will be $2,728.69 when Francis sells the bond in 10 years. We can use this value to now calculate Francis’ gain or loss:
Conclusion: Francis will lose $1,271.31 on the sale of the bond.
Key Takeaways for Bonds
- Coupon Payment (PMT) = PV × i = Principal × Periodic Rate
- Periodic Rate = Nominal Rate ÷ 2 (Bonds are always semi-annual)
- For bonds, PV will be negative and PMT and FV are positive.
- Bonds are the ONLY type of investment in this text where PV will be negative.
- Bonds are the ONLY type of investment in this text where FV will be positive.
- If the market rate rises above the coupon rate, the bond decreases in value.
- If the market rate drops below the coupon rate, the bond increases in value.
- The gain or loss on the sale of bonds is given by:
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- Information thanks to https://www.investopedia.com/terms/b/bond.asp ; https://en.wikipedia.org/wiki/Bond_(finance) and https://www.investopedia.com/articles/investing/062813/why-companies-issue-bonds.asp ↵
In a bond, this is the initial interest rate used to calculate the coupon payment
The termination or ending date for which a loan, bond, or any amount borrowed must be paid back in full.
The original amount of money invested or borrowed.
The regular (usually semi-annual) payment from a coupon bond.
Periodic Compound interest rate
A bond holder who has purchased a bond(s) from either the original bond holder or another secondary bond holder.
The amount of time between when the bond is issued and when the bond matures.
The amount the purchaser (secondary bond holder) is willing to pay for a bond.
In a bond, this is the original purchase price, as well as final payment for the bond.
Sells for a higher selling price than the original purchase price.
Sells for a price lower than the original purchase price.
