Coupon, Price and Yield to Maturity in Depth
Price a bond from its cash flows, distinguish coupon, current yield and yield to maturity, and estimate a yield by hand.
What you will learn
- How a bond price is calculated as the present value of its future cash flows.
- The difference between coupon rate, current yield and yield to maturity (YTM).
- Why bonds trade at a premium or a discount, and what "pull to par" means.
- How to estimate a yield to maturity by hand and check it.
Explanation
The inverse relationship between bond prices and yields was introduced in the macro course (lesson "Bonds and Yields Simply Explained"). Here we go one level deeper and calculate it.
A bond is a loan you give to an issuer. The issuer promises fixed payments: a coupon (the yearly interest, stated as a percentage of the face value) and the face value (also called par or nominal value, the amount repaid at maturity, the final repayment date). German federal bonds and most euro bonds pay the coupon once a year; US Treasury notes and bonds pay half the coupon every six months.
Prices are quoted as a percentage of face value. A price of 96.50 means €965 for each €1,000 of face value.
The price formula
The fair price is the sum of all future cash flows, each discounted at the market yield y:
Price = C/(1+y) + C/(1+y)^2 + … + (C+F)/(1+y)^n
Here C is the coupon in euros, F the face value and n the number of years. Discounting means dividing by (1+y) once for every year you must wait, because money received later is worth less today.
Three different "yields"
- Coupon rate: fixed at issue. It never changes and says nothing about today's return.
- Current yield: coupon ÷ price. It ignores the gain or loss at maturity.
- Yield to maturity (YTM): the single discount rate that makes the formula above equal the market price. It is the annual return you earn if you buy now, hold to maturity, the issuer pays everything on time, and every coupon is reinvested at the same YTM.
That last condition, the reinvestment assumption, is often forgotten. If rates fall, coupons are reinvested at lower rates and the realised return ends slightly below the YTM.
Premium, discount and pull to par
- Coupon rate below the market yield → price below 100 (discount bond).
- Coupon rate above the market yield → price above 100 (premium bond).
- Coupon rate equal to the market yield → price at 100 (par).
Because the issuer repays exactly 100 at maturity, the price moves towards 100 as maturity approaches, even if yields do not change. This is called pull to par.
Exchange prices are usually clean prices (without accrued interest). The interest built up since the last coupon is added when you buy; lesson 8 covers this in detail.
Worked example
Hypothetical example: a 5-year bond, face value €1,000, coupon 3% (€30 per year, paid annually). The market yield for similar bonds is 4%. We assume the purchase is on a coupon date (no accrued interest) and ignore fees, spreads and taxes.
| Year | Cash flow | Discount factor (1.04^t) | Present value |
|---|---|---|---|
| 1 | €30 | 1.0400 | €28.85 |
| 2 | €30 | 1.0816 | €27.74 |
| 3 | €30 | 1.1249 | €26.67 |
| 4 | €30 | 1.1699 | €25.64 |
| 5 | €1,030 | 1.2167 | €846.58 |
| Price | €955.48 (quote 95.55) |
Current yield = €30 ÷ €955.48 = 3.14%. The YTM is 4%, higher than the current yield, because the buyer also gains €44.52 when the bond is repaid at €1,000.
Pull to par: if the market yield is still 4% one year later, the same bond (now 4 years left) is worth €963.70. The price rose by €8.22 without any change in yields.
Estimating a YTM. A second bond: 4 years left, coupon 2.5% (€25), price €965. A common approximation is:
YTM ≈ (C + (F − P)/n) ÷ ((F + P)/2)
= (€25 + €35/4) ÷ €982.50 = €33.75 ÷ €982.50 ≈ 3.44%. Solving the full formula by trial and error (or with a spreadsheet function such as RATE or YIELD) gives 3.45%. The shortcut is close for small discounts and short maturities but becomes less accurate for long or deep-discount bonds.
Common mistake
Practice activity
On paper: (a) Price a hypothetical 3-year bond, face €1,000, annual coupon 2%, market yield 3%. (b) Estimate the YTM of a hypothetical 4-year bond, coupon 5%, price €1,040, using the approximation formula. Ignore fees and taxes.
Show solution
(a) €20/1.03 = €19.42; €20/1.0609 = €18.85; €1,020/1.092727 = €933.44. Price = €971.71 (quote 97.17), a discount bond because the coupon is below the yield.
(b) (€50 + (€1,000 − €1,040)/4) ÷ ((€1,000 + €1,040)/2) = (€50 − €10) ÷ €1,020 ≈ 3.92%. The exact YTM is about 3.90%. The premium of €40 is lost by maturity, so the YTM is below the 5% coupon and below the current yield (€50/€1,040 = 4.81%).
Limitations: YTM assumes no default, holding to maturity and reinvestment at the same rate; it excludes costs and taxes.
Key takeaways
- A bond's price is the present value of its coupons and face value, discounted at the market yield.
- Compare bonds by yield to maturity, not by coupon or current yield.
- Discount bonds rise towards par and premium bonds fall towards par as maturity approaches.
- YTM is a promise only if the issuer pays, you hold to maturity and coupons are reinvested at the same rate.
Next step
In the next lesson you will measure how strongly a bond price reacts when yields change, using duration and convexity.
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