Bond pricing & yield: practice problems with step-by-step solutions

1What a bond is

A bond is a loan that you can buy and sell. A company or government borrows money and promises fixed payments in return.

A typical bond has three numbers printed on it:

• The face value F, usually €1000. It is paid back on the last day, the maturity.
• The coupon rate. Each year the bond pays this percentage of the face value as interest, the coupon C. A 5% coupon on €1000 is €50 a year.
• The maturity n: how many years until the face value is repaid.

The coupons and the face value are fixed. What changes is the price P that investors pay today. That price depends on the return investors want now.

2Price and yield to maturity

The yield to maturity (YTM, written y) is the yearly return the market requires on the bond. It is the discount rate for all its payments.

The price is the present value of everything the bond pays: the coupons form an annuity, and the face value is one amount at the end.

Compare the yield with the coupon rate. This tells you the answer before you calculate:

• Yield equal to the coupon rate: the price is exactly the face value, €1000 (at par).
• Yield above the coupon rate: the price is below face value (a discount).
• Yield below the coupon rate: the price is above face value (a premium).

So bond prices and yields always move in opposite directions. When yields rise, prices fall.

3The method step by step

1. Find the coupon per period C, the yield per period y and the number of periods n.
2. For half-yearly (semi-annual) coupons: use half the coupon, half the yield and twice the number of years.
3. Present value of the coupons: C · [1 − (1 + y)⁻ⁿ] / y.
4. Present value of the face value: F / (1 + y)ⁿ.
5. Add the two. Check the result against the coupon rate: discount or premium?

Example of step 2: a 6% coupon paid twice a year for 5 years, with a yield of 5%, gives C = €30, y = 0.025 and n = 10.

4A worked example

Follow every step. Try to predict the next one before you open it.

5Finding the yield to maturity

Sometimes the price is given and you must find the yield. There is no formula for y, so you try rates.

A bond with a 4% coupon and 3 years to go sells for €960. It sells below €1000, so the yield is above 4%.

1. Try 5%: the price formula gives €972.77. Still above €960, so the yield must be higher.
2. Try 6%: the price is €946.54. Now it is below €960, so the yield lies between 5% and 6%.
3. Interpolate between the two rates.

Do not confuse the YTM with the current yield, C/P = 40/960 = 4.17%. The current yield ignores the €40 gain from €960 up to the €1000 repaid at maturity.

6Common mistakes

• Discounting at the coupon rate instead of the yield. The coupon rate only sets the size of the coupon.
• Forgetting the face value at maturity, or forgetting the coupons.
• Adding the payments without discounting them.
• Mixing annual and semi-annual periods: with half-yearly coupons halve C and y and double n.

7How exam questions look

• Easy: annual coupons. Find the price from the face value, coupon rate, maturity and yield.
• Medium: semi-annual coupons. The same, with half the coupon, half the yield and twice the periods.
• Hard: yield to maturity. The price is given. Try two rates, then interpolate or use a calculator.

Always compare the yield with the coupon rate first. It tells you whether the price must be above or below €1000.