Pricing a bond with zero rates
1. The idea
A bond pays its holder fixed amounts on set dates: a coupon, here every six months, and its face value at the end. Money paid later is worth less today, so each payment is discounted with the zero rate for its date: the interest rate on money lent from today until that date, with nothing paid in between. The bond’s price is the sum of these present values:
\[ \text{price} = \sum_i c_i\, e^{-r_i t_i}, \]
where \(c_i\) is the cash flow at time \(t_i\), in years, and \(r_i\) the zero rate for that time, with continuous compounding, as a decimal. The factor \(e^{-r_i t_i}\) is the payment’s discount factor.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The price on a chart
The bond in the question pays $3 in six months, and $103 in one year: the last coupon and the $100 face value. The present value of the $3 paid in six months is \(3 \times e^{-0.04 \times 0.5} = \$2.94\). Discount the $103 in the same way and add the two present values: that sum is the bond’s price.
Now give every date the same zero rate and move it. When rates rise, the price falls.
4. One worked example, then your turn
Now one with hints, one step at a time.
And one on your own.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for a price above the face value and one for a price below. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: