Bond Valuation in 35 Minutes
A bond's price is the present value of its coupon payments plus the face value repaid at maturity, discounted at the market yield. Most confusion comes from mixing up the coupon rate, which fixes the payments, with the yield to maturity, which sets the discount rate.
What you will learn
- How a bond's coupons and face value become cash flows
- Why the coupon rate and yield to maturity are different things
- How to price a bond as an annuity plus a lump sum
- Why bonds trade at a discount or premium to face value
- Why bond prices fall when market yields rise
The formulas
- C
- Coupon payment per period = Coupon rate × Face value
- r
- Yield to maturity per period
- N
- Number of periods to maturity
- F
- Face (par) value, often $1,000
- Annual coupon
- Total coupon paid per year
Worked example
A 5-year bond has a $1,000 face value and a 6% annual coupon. Similar bonds yield 8%. What is its price?
- Coupon C = 6% × 1,000 = 60 per year
- PV of coupons = 60 × [1 − 1 ÷ 1.08⁵] ÷ 0.08 ≈ 239.56
- PV of face value = 1,000 ÷ 1.08⁵ ≈ 680.58
- Price ≈ 239.56 + 680.58 = 920.15 (figures rounded to the cent)
Answer: The bond is worth about $920.15, a discount to face value because its 6% coupon is below the 8% market yield.
Common questions
How do you calculate the price of a bond?
Find the coupon payment, then discount every coupon and the face value back to today at the yield to maturity. The coupons form an annuity, so you can use the annuity formula, and the face value is a single lump sum at maturity. Add the two present values.
What is the difference between coupon rate and yield to maturity?
The coupon rate is fixed when the bond is issued and only sets the size of the interest payments. Yield to maturity is the return the market currently demands, and it is the discount rate you use to price the bond. They are equal only when the bond trades at par.
Why do bond prices go down when interest rates go up?
A bond's coupons are fixed. When new bonds start paying more, the old bond is less attractive, so its price must drop until its yield matches the market. Mathematically, a higher discount rate lowers the present value of the same cash flows.
When does a bond sell at a premium or a discount?
If the coupon rate is above the market yield, the bond pays more than buyers need and sells at a premium, above face value. If the coupon rate is below the yield, it sells at a discount. If they are equal, it sells at par.