What you need before you start
Every bond problem gives you the same five pieces: the face value, the coupon rate, the number of years to maturity, how often it pays, and the market yield. Convert all of them to the payment period before you touch a calculator. A bond that pays twice a year has twice as many periods and half the annual coupon and yield in each one.
A worked example
A bond has a $1,000 face value, a 6% annual coupon paid semiannually, and 10 years to maturity. Similar bonds yield 8% a year, compounded semiannually. What is the price?
- Periods: 10 years times 2 payments a year is 20 periods.
- Coupon each period: 6% of $1,000 is $60 a year, so $30 every six months.
- Yield each period: 8% a year divided by 2 is 4%.
- Discount the 20 coupons as an annuity and the $1,000 face value as one lump sum.
| Piece | Calculation | Present value |
|---|---|---|
| 20 coupons of $30 | $30 x (1 - 1.04^-20) / 0.04 | $407.71 |
| Face value of $1,000 | $1,000 / 1.04^20 | $456.39 |
| Price | $864.10 |
The bond sells for $864.10, below its $1,000 face value, because it pays a 6% coupon when the market asks 8%. Buyers will only pay less than face value for a coupon that is smaller than the going rate. Its current yield, annual coupon divided by price, is $60 / $864.10, or about 6.94%.
What happens when the yield moves
| Market yield | Price | The bond sells at |
|---|---|---|
| 10% | $750.76 | A discount |
| 8% | $864.10 | A discount |
| 6% | $1,000.00 | Par |
| 5% | $1,077.95 | A premium |
Price and yield move in opposite directions. When yields rise, existing bonds with lower coupons become less attractive, so their prices fall. That relationship is why a quick sanity check works on exams: if the yield is above the coupon rate your answer must be below $1,000, and if it is below the coupon rate your answer must be above $1,000.
Mistakes that cost points on exams
- Using the annual rate for each six-month period. The rate and the number of periods have to match the payment frequency.
- Pricing a semiannual bond as if it paid once a year. In this example that gives $865.80 instead of $864.10, which is close enough to look right and different enough to miss the answer choice.
- Mixing up the coupon rate and the yield. The coupon sets the payment. The yield sets the discount rate.
- Forgetting the face value, which comes back in the final period along with the last coupon.
- Missing the sign check. A discount bond priced above face value means something went wrong.