Bootstrapping a Six-Month Spot Rate from Coupon Bonds
Summary
The document examines a bootstrap calculation for a six-month continuously compounded zero rate using bond prices, coupon cash flows, and a one-year zero rate. Its central lesson is that a shorter-maturity spot rate can be higher than a longer-maturity rate; the curve’s ordering is determined by the observed prices and cash flows, not by a rule that rates must decline with maturity.
The answers also expose a convention issue in the calculation. With the six-month time fraction included in the discount exponent, the logarithm gives an annualized continuously compounded rate; interpreting it as the full-period rate would cause a factor-of-two error. The discussion includes inconsistent alternative arithmetic, so the key transferable point is to keep the compounding basis and annualization convention consistent when bootstrapping. The example is limited to its stated bond cash flows and assumptions and does not establish a general shape for the yield curve.
Key ideas
- A bootstrapped short-maturity spot rate can exceed a longer-maturity spot rate.
- Including the maturity fraction in the discount exponent produces an annualized continuously compounded rate.
- Keep the logarithm base and annualization convention consistent throughout the calculation.
- Bond cash flows and prices determine the curve; rate ordering is not fixed in advance.
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Full text
# Basic boostrapping question
# Basic boostrapping question
Suppose I have three bonds:
Coupon bonds are paid semi-annually. Rates are continuous compounding.
I'm trying to bootstrap the zero rates for 0.5 years maturity using the 1 year zero coupon bond and the 1 year fixed rate coupon bond, but my eventual 0.5 year zero rate is higher than my 1 year zero rate.
My calculations are:
- 1-year zero rate: $$95 = 100 \times e^{-r}$$ $$r_{1y} = - \ln (0.95) = 5.129\%$$
- 6M zero rate: $$2.5 \times e^{-0.5 r_{6m}} + 102.5 \times e^{-r_{1y}} = 99.8$$ $$r_{6m} = - 2 \ln \left( \frac{99.8-102.5 e^{-r_{1y}} }{2.5} \right) = 6.118\%$$
I'm not sure if I'm doing it right, don't think a 0.5 year zero rate is supposed to be higher than a 1 year zero rate.
Help please!
## Answer by Niko777 (score 1)
https://quant.stackexchange.com/a/35031
You think you make a mistake where you actually don´t make one. The exercise is just like it is. Resulting in $$r_{6m}>r_{12m}$$
The difference in your both answers, based on the same rounding, lays in the different basis for the logarithm.
$$r_{6m} = - 2 \log_e \left( \frac{99.8-102.5 e^{-r_{1y}} }{2.5} \right) = \textbf{6.118%}$$ $$r_{6m} = - 2 \log_{10} \left( \frac{99.8-102.5 e^{-r_{1y}} }{2.5} \right) = \textbf{2.6571%}$$
Calculation for the last interest rate base on $r_{6m}=0.06118%$ and $r_{12m}=0.05129%$ yields:
$$r_{18m}=-\frac{log_e\left(\frac{102.7-4\times(e^{0.5\times(-0.06118)}+e^{-0.05129})}{104}\right)}{1.5}=0.06019886318=\textbf{6.0199%}$$
## Answer by Christian Hampton (score 1)
https://quant.stackexchange.com/a/41532
Your answer is correct. You included .5 in the exponent and therefore got an annualized result. 6.118% divided by 2 is your bootstrapped 6 month spot rate.
## Answer by JejeBelfort (score -1)
https://quant.stackexchange.com/a/34557
I think your calculation is just wrong.
Starting from the edit, we have for the 6m zero rate:
- 6M zero rate: $$2.5 \times e^{-0.5 r_{6m}} + 102.5 \times e^{-r_{1y}} = 99.8$$ $$r_{6m} = - 2 \ln \left( \frac{99.8-102.5 e^{-r_{1y}} }{2.5} \right) = \textbf{2.6571%}$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.