A loan you can sell
A bond is a loan split into tradable pieces. The issuer promises a stream of coupon payments and the return of face value at maturity. The holder can sell it before maturity, so its price moves as market interest rates move.
A bond's price is the present value of everything it will pay, discounted at the yield the market demands for that risk and maturity. Nothing about the coupon changes; only the discount rate does.
That is why price and yield move in opposite directions. When market yields rise, the fixed coupons are worth less today, so the price falls. This is interest-rate risk.
Key points
- Coupons and face value are contractual; the yield is set by the market.
- Price is the present value of those cash flows at the market yield.
- Price falls when yield rises, and rises when yield falls.
Pricing and reading a bond
P = Σ [C ÷ (1 + y)ᵗ] + F ÷ (1 + y)ⁿ
The present value of every coupon plus the present value of face value, discounted at the yield to maturity.
- C
- Coupon payment per period
- F
- Face value repaid at maturity
- y
- Yield to maturity
- n
- Number of periods to maturity
Current yield = Annual coupon ÷ Price
The income return, ignoring any gain or loss at maturity.
Coupon = y → par; Coupon > y → premium; Coupon < y → discount
The relationship that tells you which way the bond trades.
ΔP ÷ P ≈ −Modified duration × Δy
A first-order estimate of how much price moves for a small change in yield.
Yield to maturity is the single rate that makes the present value of all cash flows equal the market price. It assumes coupons are reinvested at that same rate, which is an approximation rather than a promise.
A 10-year bond at a 9% yield
Consider a bond with a ৳1,000 face value, a 7% coupon, and 10 years to maturity. The coupon is ৳70 a year; the market yield is 9%.
Discounting the ten coupons and the face value at 9% gives a price of ৳871.65. Because the 7% coupon is below the 9% yield, the bond trades at a discount of ৳128.35. The current yield is ৳70 ÷ ৳871.65 = 8.03%, and the holder also picks up the pull to par as maturity approaches.
If the yield were 7%, the price would be exactly ৳1,000. If it were 5%, the bond would trade at a premium. The coupon sets the cash flows; the yield sets the value.
Key points
- Price ৳871.65, a ৳128.35 discount to face value.
- Current yield 8.03% is below the 9% yield to maturity.
- Coupon below yield always means a discount.
Reading rate expectations through bonds
Two things drive how much a bond's price moves when yields change: maturity and coupon. Longer maturities and lower coupons concentrate value in distant cash flows, which are discounted more heavily, so they are more sensitive. A 20-year low-coupon bond swings far more than a 2-year high-coupon bond.
An investor who expects rates to fall wants high sensitivity, because falling yields produce larger capital gains. An investor who expects rates to rise, or who simply wants to protect capital, wants low sensitivity.
Credit risk is separate from interest-rate risk. A widening credit spread raises the yield the market demands and lowers the price even if the risk-free rate is unchanged. Always ask which yield component moved, and why.
Key points
- Long maturity plus low coupon equals high sensitivity.
- Expecting rate cuts argues for higher duration, and vice versa.
- Separate interest-rate moves from credit-spread moves.
Interactive experiment
Bond pricing and yield
A bond's price is the present value of its coupons and its face value, discounted at the yield the market demands.
P = Σ [C ÷ (1 + y)ᵗ] + F ÷ (1 + y)ⁿ
Your inputs
Amount repaid at maturity.
Annual coupon as a share of face value.
Return the market demands.
Time until the face value is repaid.
Result
- Price
- ৳872
- Current yield
- 8.0%
- Annual coupon
- ৳70
- Face value
- ৳1,000
- Premium / (discount)
- ৳-128
- Capital gain / (loss) at maturity
- ৳128
- Total coupon income
- ৳700
The coupon of 7% is below the 9% yield, so the bond trades at a discount of 128.35.
03 / Apply
Two bonds, one rate view
An investor expects central-bank rates to fall over the next year. Two bonds are available: Bond A matures in 2 years with a 12% coupon, and Bond B matures in 20 years with a 4% coupon. Both have a ৳1,000 face value and currently yield 9%.
The assumptions
- Both bonds carry the same credit risk.
- The investor will sell before maturity.
- The rate view is a forecast, not a certainty.
What to take away
01
Bond B is far more sensitive to a change in yield than Bond A.
02
A rate fall would raise Bond B's price more, which suits the rate view.
03
The same sensitivity means a rate rise would hurt Bond B more.
Your turn
0 of 4 answered
01Question
A 2-year bond has a ৳1,000 face value and a 10% coupon, and the market yield is 10%. What is its price?
02Question
The same 2-year, 10% coupon bond now yields 12%. What is its price? Round to two decimals.
03Question
Using the ৳966.20 price and the ৳100 annual coupon, what is the current yield? Answer as a percentage, rounded to two decimals.
04Question
Rates are expected to fall. Which bond should gain the most in price?
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Open the playgroundSources & further reading
OpenStax · Principles of Finance, Fixed Income chapter ↗Educational sample content. Worked examples use hypothetical figures. Dates refer to the note's latest content update.