Bootstrapping zero rates from bond prices
1. The idea
Pricing a bond with zero rates found a bond’s price from the zero rates. Markets work the other way round: bond prices are known, and the zero rates are found from them, one date at a time. This is called bootstrapping.
A zero-coupon bond pays only its face value, $100, at maturity \(T\), so its price is \(P = 100\,e^{-zT}\), and its zero rate is
\[ z = -\frac{\ln(P/100)}{T}, \]
where \(\ln\) is the natural logarithm.
A bond with coupons pays several times. If the zero rates for all its payments but the last are already known, discount those payments and subtract them from the price: what is left is the last payment, the face value plus a coupon, discounted at the one rate still unknown.
An example. A six-month zero-coupon bond costs $98.00, so \(z_{0.5} = -\ln(0.98)/0.5 = 4.04\%\). A one-year bond paying a coupon of $2 every six months costs $99.50. Its first coupon’s present value is \(2\,e^{-0.0404 \times 0.5} = 1.9600\), which leaves \(99.50 - 1.9600 = 97.5400\) for the last payment of $102. So \(102\,e^{-z_1} = 97.5400\), and \(z_1 = -\ln(97.5400/102) = 4.47\%\).
A rate quoted with another compounding frequency, such as a six-month rate with semiannual compounding, is first turned into a continuous rate, as in the lesson on converting rates.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The rate on a chart
First, how a bond’s price and its rate move.
Now move the zero rate. The bond’s price falls as the rate rises, and the rate the market’s price implies is where the two lines cross.
4. One worked example, then your turn
In the worked example, we start from a six-month rate with semiannual compounding.
Now a one-year bond, with hints, one step at a time.
And one on your own, one date further.
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 from a zero-coupon bond, a one-year coupon bond and an 18-month coupon bond. 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: