Bond pricing and yield to maturity: how a bond is valued
A bond is a loan that you, the investor, make. The issuer promises regular interest payments called coupons and the return of the face value, or principal, at maturity. Pricing a bond means working out what that stream of promised cash is worth today, which is a time-value-of-money problem. If that idea is new, start with the time value of money guide.
The price is a present value
Discount every cash flow at the market yield for a bond of that risk and maturity, and add them up:
Price = Σ coupon ÷ (1 + y)t + face value ÷ (1 + y)N
For a bond paying annually, y is the annual yield and N the number of years. For a semiannual bond, halve the yield and the coupon and double the number of periods.
Take a three-year bond with a 5% annual coupon and a face value of 100, when the market yield is 6%:
- Year 1 coupon: 5 ÷ 1.06 = 4.72
- Year 2 coupon: 5 ÷ 1.062 = 4.45
- Year 3 coupon and principal: 105 ÷ 1.063 = 88.16
The price is 97.33. You can check any bond like this with the bond price, yield and duration calculator.
Why price and yield move in opposite directions
The coupon is fixed when the bond is issued, but the yield investors demand changes every day. If yields rise, the fixed coupons are less attractive, so the price must fall until the bond offers the new yield. If yields fall, the price rises.
That gives three cases, all visible in the example:
- Yield above the coupon rate: the bond sells at a discount. At a 6% yield the 5% bond is worth 97.33, less than par.
- Yield equal to the coupon rate: the bond sells at par. At a 5% yield the price is exactly 100.
- Yield below the coupon rate: the bond sells at a premium. At a 4% yield the price is 102.78.
As a bond approaches maturity, a premium or a discount shrinks toward par, because the principal repayment is getting closer.
Yield to maturity
The price is what you can observe in the market; the yield is the thing you solve for. The yield to maturity (YTM) is the single discount rate that makes the present value of the promised cash flows equal to the price. There is no formula that isolates it, so you find it by trial and refinement, or with a financial calculator.
It is a useful summary, but it rests on assumptions. Yield to maturity is the return you earn only if:
- you hold the bond to maturity,
- the issuer pays every coupon and the principal in full, and
- you reinvest every coupon at the same yield.
The third assumption is the least realistic. If rates fall, the coupons you receive are reinvested at less than the YTM, and your realised return is lower.
Current yield is not yield to maturity
The current yield is simply the annual coupon divided by the price. For the 97.33 bond it is 5 ÷ 97.33 = 5.14%. It ignores the gain you will make as the price returns to par, and ignores the time value of money, so it is a rough income measure rather than a return.
Clean price, dirty price and accrued interest
Bonds are usually priced on days between coupon dates, and the buyer must compensate the seller for the interest earned since the last coupon. That is accrued interest. The dirty (full) price is what the buyer actually pays; the clean price is the dirty price minus accrued interest, and is the figure that is quoted. A bond's quoted price therefore does not jump on coupon dates.
Mistakes to avoid
- Not matching the period to the coupon. For semiannual coupons, use the half-year yield and the number of half-years.
- Treating the coupon rate as the return. The coupon is fixed by the bond's terms; the return depends on the price you pay.
- Mixing up clean and full prices when a question gives a settlement date between coupons.
- Forgetting direction. A rise in yield always lowers the price. How much it moves depends on the bond's duration, covered in the duration and convexity guide.
Where this is tested
Bond valuation is the core of the fixed income topic and returns in derivatives, portfolio management and the ethics of fair valuation. The learning outcomes below set out what you need to be able to do with it.