Bootstrapping Spot Rates and Forward Rates from Coupon Bonds

Summary

The document asks how to infer spot rates at several maturities from three coupon bonds, and how to derive a forward rate from those spot rates. Its central lesson is that a bond’s yield or accrued interest alone does not identify the spot rate for its maturity: each bond price reflects the discounted value of all its coupon and principal cash flows. A curve is typically built by bootstrapping, solving sequentially for discount factors or spot rates using the available bond prices and cash-flow schedules.

The question also proposes a forward-rate relation between the half-year and eighteen-month points, and tests a simple return calculation for the shortest bond. The material provides no accepted answer or worked solution, so it does not establish the correct treatment of the quoted prices, accrued interest, compounding convention, or forward-rate formula. Those conventions and the bonds’ cash-flow timing must be specified before the numerical rates can be trusted.

Key ideas

• Coupon bond prices combine the present values of multiple cash flows, so they do not directly reveal a single maturity spot rate.
• Spot rates can be inferred sequentially by solving for discount factors from bond prices and cash-flow schedules.
• Forward-rate calculations depend on the spot-rate compounding and day-count conventions.
• The document poses the calculation but contains no answer that verifies its proposed rate estimates.

Tags

# How to calculate spot rates using market data of bonds?


# How to calculate spot rates using market data of bonds?












Given 3 Bonds $A$, $B$ and $C$ with \begin{matrix} & \text{Bond } A& \text{Bond } B& \text{Bond } C& \\ \text{Price:}& 101,12\%& 99,03\%& 102,95\%\\ \text{Mat. in years:}& 0,5& 1,5& 2,5\\ \text{Coupon:}& 2\%& 1\%& 3\% \\ \text{Coupon frequency:}& \text{per year}& \text{per year} & \text{per year}\\ \text{Nominal value:}& 1.000& 10.000& 1.000\\ \end{matrix}

How can I determine the spot rates implied by these bonds? I.e. what would be the spot rate for the next 0,5 years, the next 1,5 and 2,5 years?

I assume that it is not correct to just claim that the spot rate for 0,5 years is 1% (the accrued interest from Bond $A$)?

Furthermore, how can I calculate the forward rate from these rates? Is it correct to calculate the 6x18 forward rate $F$ via the equation $(1 + \frac12r_{0,5})(1+F) = (1+\frac32r_{1,5})$ where $r_{0,5}$ and $r_{1,5}$ are the spot rates for the given duration?

Thanks in advance!

Edit: The return I can realise on Bond $A$ would be the following: I can buy at 102,12% (accounting for accrued interest), after 0,5 years, I get 102%. Does solving $$ 102,12 \cdot (1 + \frac12 r_{0,5}) = 102 $$ yield the desired spot interest rate? In this case $r_{0,5} = -0,24\%$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.