Lesson 1 of 5 · All lessons in this module
- The three sources of return on a fixed-rate bondVideo lesson
- Horizon yield: the realized return when rates moveVideo lesson
- Reinvestment risk, price risk and the duration gapVideo lesson
- Calculating Macaulay duration on a coupon dateVideo lesson
- Macaulay duration between coupon dates and the closed-form formula
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Lesson 1 of 5 · 12 min
The three sources of return on a fixed-rate bond
A fixed-rate bond pays you through its coupons and principal, through interest earned on reinvested coupons, and through any gain or loss if you sell before maturity.
In short
- Three sources of return: (1) the promised coupons and principal, (2) reinvestment of the coupons, (3) a capital gain or loss if the bond is sold before maturity.
- Your realized return equals the YTM at purchase only if you hold to maturity, the issuer does not default, and every coupon is reinvested at that same YTM.
- Compounding reinvested coupons adds interest on interest: the future value of the coupons exceeds their simple sum.
- Interest income is the return from the passage of time: coupons, their reinvestment, and amortization of any purchase discount or premium.
- A capital gain or loss comes only from a change in the bond's YTM: a sale above (below) the carrying value on the constant-yield price trajectory.
Three ways a bond pays you
Buy a fixed-rate bond and three things can add to (or subtract from) your wealth. First, the issuer pays the coupons on schedule and repays the principal at maturity. Second, each coupon can be reinvested until your horizon, earning extra interest. Third, if you sell before maturity, the sale price may be above or below what the bond was 'supposed' to be worth, giving a capital gain or loss.
The first source is fixed by the contract (ignoring default). The other two depend on interest rates after you buy. That is why changing rates change your realized return even though the coupons themselves never change.
This module assumes no default, so all risk discussed here is interest rate risk: the risk that rates move after you buy.
| Source | Depends on | Who is exposed |
|---|---|---|
| Coupons and principal | The bond contract (and no default) | Every holder |
| Reinvestment of coupons | Rates at which coupons are reinvested | Anyone who receives coupons before the horizon |
| Gain or loss on sale | The bond's YTM on the sale date | Only investors who sell before maturity |
When your return equals the YTM
The YTM you see at purchase is a promise with conditions attached. Your annualized, compounded return equals it only if (1) you hold the bond to maturity, (2) the issuer pays every cash flow on time, and (3) you reinvest every coupon at that same YTM.
Reinvestment is the condition people forget. The YTM is an internal rate of return, and an IRR quietly assumes interim cash flows earn the IRR. If coupons are reinvested at a lower rate, you end up with less than the YTM promised, even when you hold to maturity.
Compounding matters here. A coupon received in year 1 and reinvested for seven years earns interest, and that interest earns interest. The amount by which the future value of the coupons exceeds their plain sum is called interest on interest. On long bonds it is a large part of total return.
Think of the YTM as a forecast that assumes nothing changes. Once rates move, the forecast is wrong in a predictable direction: reinvestment income moves with rates, the sale price moves against them.
Interest income versus capital gain
The curriculum splits total return into two parts. Interest income is the return earned simply because time passes: the coupons, the reinvestment income on them, and the amortization of any discount (or premium) paid at purchase. A capital gain or loss is the return from a change in the bond's value caused by a change in its YTM.
Why does amortization count as interest? A bond bought below par is pulled up toward par as maturity nears, even if its yield never changes. That climb is part of the yield you locked in, not a market windfall. A premium bond drifts down toward par, which reduces interest income.
The path the price follows when the yield stays at its purchase level is the constant-yield price trajectory. A point on it is the bond's carrying value: the purchase price plus the discount amortized so far (or minus the premium amortized so far). For a bond bought at par, the trajectory is flat at 100.
If you sell exactly at the carrying value, there is no capital gain or loss, and with coupons reinvested at the original YTM your horizon return equals the original YTM. Selling above the carrying value gives a capital gain; below it, a capital loss.
| Component | Classified as | Caused by |
|---|---|---|
| Coupons received | Interest income | Passage of time |
| Interest on reinvested coupons | Interest income | Passage of time (amount depends on reinvestment rate) |
| Discount accretion / premium amortization | Interest income | Pull to par at the original yield |
| Sale price − carrying value | Capital gain or loss | A change in the bond's YTM |
Formulas
- Future value of reinvested coupons
PMT = coupon per period, r = reinvestment rate per period, T = number of coupons received by the horizon. Interest on interest = this FV − T × PMT.
- Carrying value
Equivalently, purchase price + discount amortized to date (or − premium amortized). Capital gain = sale price − carrying value.
Worked examples
Example 1 Interest on interest for a buy-and-hold investor
A Swedish pension fund buys an 8-year bond with a 4% annual coupon at par (SEK 100) and holds it to maturity. If every coupon is reinvested at 4%, how much do the coupons grow to, and how much of that is interest on interest?
Example 2 Splitting a return into interest income and capital gain
An investor buys a 5-year, 3% annual-coupon EUR bond at a YTM of 4% and sells it after 2 years. Coupons are reinvested at 4%. On the sale date the bond's YTM has fallen to 3.4%. Find the horizon yield and split the gain into interest income and capital gain.
On the exam
Expect concept questions on which return sources a given investor faces, the three conditions for realized return = YTM, and the definition of carrying value for a discount or premium bond.
- Forgetting that reinvestment rate matters even for a buy-and-hold investor who never sells.
- Calling the pull to par of a discount bond a capital gain. It is interest income; only a sale above the carrying value is a gain.
- Measuring a capital gain against the purchase price or par instead of the carrying value.
- Subtracting discount amortization from the purchase price. For a discount bond it is added; for a premium bond the premium amortization is subtracted.
Check yourself
Q1A pension fund buys a 6-year bond and will hold it to maturity. Ignoring default, which source of return is uncertain at the time of purchase?
Q2A 5-year, 3.5% annual-coupon bond is bought at par and held to maturity. Coupons are reinvested at 3.5%. Interest on interest per 100 of par is closest to:
Q3A bond bought at a discount is later sold at a price above its carrying value. The difference between the sale price and the carrying value is best described as:
Q4Which of the following is least likely part of a fixed-rate bond investor's interest income?
Q5An investor buys a bond at a discount, holds it to maturity and suffers no default. Her realized return equals the YTM at purchase if coupons are reinvested at:
Answer the 5 questions: +10 XP for each one you get right first time.
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