The sources of return from a fixed-rate bond

Where does a bond investor’s return come from?

Holding a bond pays you in three ways. You receive the promised coupon income and principal. You reinvest each coupon you receive, which earns reinvestment income : the interest on interest that makes a bond’s realized return differ from its coupon rate. And if you sell before maturity, you realize a capital gain or loss against what the bond was worth on the path you bought it on.

Total return on a fixed-rate bondmeasured over the investor’s holding period

• Coupon and principal payments Know the idea the promised cash flows, received on schedule
• Reinvestment of the coupons Use it interest on interest, at whatever rate is available
• Capital gain or loss on sale Use it only if the bond is sold before maturity

Interest rates touch the second and third sources. A change in rates changes the rate at which coupons are reinvested, and it changes the price at which the bond sells. The first source, the promised cash flows, does not move unless the issuer defaults.

How do you calculate the horizon yield?

The horizon yield is the annualized return over your holding period. It treats everything you end up with, the reinvested coupons plus the sale price (or the redemption amount), as one future value, and compares it with the price you paid.

First the coupons. Each is reinvested until the horizon, so together they form an ordinary annuity that grows to a future value.

$$FV_{coupons} = C \times \frac{(1 + r)^{H} - 1}{r}$$

Here $C$ is the annual coupon, $r$ the reinvestment rate and $H$ the number of years to the horizon. Everything above $C \times H$ is interest on interest.

Then the horizon yield itself.

$$r = \left(\frac{FV_{coupons} + P_{sale}}{P_0}\right)^{1/H} - 1$$

$P_{sale}$ is the sale price at the horizon, which is 100 if the bond is held to maturity. $P_0$ is the price paid. The $1/H$ root turns the total growth into a per-year rate.

What happens when rates move right after you buy?

Suppose the 8-year, 5% bond is bought at par and yields jump to 6%, or fall to 4%, immediately and permanently. Both effects now show up: the coupons are reinvested at the new rate, and a bond sold early is priced at the new yield with the years it has left.

Horizon	Rates move to 6%	Rates move to 4%
3 years (sell a 5-year bond)	Coupons 15.918, price 95.788: 3.76%	Coupons 15.608, price 104.452: 6.28%
8 years (hold to maturity)	Coupons 49.487, redemption 100: 5.15%	Coupons 46.071, redemption 100: 4.85%

The table shows the pattern the exam tests. Over a short horizon the price effect dominates, so rising rates hurt. Over a long horizon reinvestment dominates, so rising rates help. These are the two interest rate risks, and the next lesson shows where one hands over to the other.

Interest rate risk on a fixed-rate bondtwo effects that pull in opposite directions

• Reinvestment risk Memorize rates fall, so coupons are reinvested at less; dominates over a long horizon
• Price risk Memorize rates rise, so the sale price falls; dominates over a short horizon

Two edge cases follow directly. A buy-and-hold investor sells nothing, so faces reinvestment risk only. An investor who sells before the first coupon is paid reinvests nothing, so faces price risk only. Two investors in the same bond can carry opposite risks if their horizons differ.

Exam tip Rates up: coupons grow, price falls. Rates down: coupons shrink, price rises. Then ask which one has had time to win.

When does the horizon yield equal the yield to maturity?

Only when nothing has gone differently from the assumptions behind the YTM. The base case above is exactly that situation.

Horizon yield = yield to maturity at purchase

• Paid in full, on schedule Memorize no default
• Coupons reinvested at the original YTM Memorize the rate does not change
• Sold on the constant-yield price trajectory Memorize no capital gain or loss (or held to maturity)

Break either of the last two and the realized return departs from the YTM, in a direction set by what happened to rates. This is why the YTM is a promised return, not a forecast.

Go deeper Why the analysis uses a one-time, parallel rate change

The scenarios assume one instantaneous, parallel shift right after purchase: every yield moves by the same amount, once, and stays there. That keeps the arithmetic clean (a single reinvestment rate and a single sale yield) and isolates the horizon effect. Real rates wander, and curves twist, so the offsets described in the next lesson are approximate guides, not guarantees.

How do you split interest income from capital gain?

A bond bought at a discount drifts up toward par as time passes, even if its yield never changes. That drift is not a market gain, because the original yield already priced it in. The curriculum separates the two.

The constant-yield price trajectory is the path the price follows if the yield stays at its purchase level. A point on it is the bond’s carrying value : the purchase price plus the amortized discount (or minus the amortized premium).

$$CV = \text{PV of the remaining cash flows at the original YTM}$$

The interest income part is the return from the passage of time: coupons, reinvestment income and the amortization of the discount or premium. The capital gain or loss is the return from a change in yield, measured against the carrying value.

$$G = P_{sale} - CV$$

Return over the holding periodsplit by what caused it

• Interest income Use it the return from the passage of time
  • Coupons received the cash paid out
  • Reinvestment income interest on interest
  • Amortization of the discount or premium the pull of the price toward par at the original yield
• Capital gain or loss Use it the return from a change in the yield
  • Sale price above carrying value yield fell: a gain
  • Sale price below carrying value yield rose: a loss

Sell exactly at the carrying value and the capital gain is zero. Then, with coupons reinvested at the original yield, the horizon yield is the original YTM again: for this bond, coupons of 5 reinvested at 6% for 3 years are worth 15.918, and (15.918 + 97.327) ÷ 95.083 = 1.1910, whose cube root is 1.0600: a 6.00% return.

Go deeper Taxes and total return

Many tax systems treat interest income and capital gains differently, so the split matters beyond the exam. Check the rules that apply to the account you are analyzing. The horizon yield is also a total return over the holding period; compare it with the single-period holding period return in the rates and returns lessons, which has no reinvestment assumption.