Bootstrapping a Two-Year Spot Rate from Coupon Bonds

Summary

The example shows how to infer a two-year spot rate from prices of two government coupon bonds with different maturities. The one-year bond first provides a one-year spot rate by discounting its coupon and principal. That rate is then used to discount the first coupon of the two-year bond, allowing the remaining cash flow to imply the one-to-two-year forward rate. Combining the one-year spot and forward rates gives the two-year spot rate.

The worked answer reports a one-year spot rate of about 4.0733%, a forward rate of about 8.5927%, and a two-year spot rate of about 6.3090%. These calculations illustrate a basic bootstrapping procedure and assume annual compounding and the stated bond cash flows and prices. The result depends on those conventions and inputs; the post does not discuss alternative compounding rules, curve fitting, or market frictions.

Key ideas

• A short-maturity coupon bond can be used to infer the corresponding spot rate from its price and cash flows.
• The two-year bond’s first coupon is discounted using the previously inferred one-year spot rate.
• The remaining two-year bond cash flow implies the one-to-two-year forward rate.
• The two-year spot rate follows by combining the one-year spot and forward rates under annual compounding.

Tags

# Calculating spot rate of interest


# Calculating spot rate of interest












You are given the following information regarding the domestic government fixed-interest bond market:

- The current price of a one-year bond paying coupons at a rate of $4.5$% per annum and redeemed at par is £100.41 per £100 nominal

- The current price of a two-year bond paying coupons at a rate of $6.5$% per annum and redeemed at par is £100.48 per £100 nominal

Calculate the two-year spot rate of interest, $y_2$

I'm not sure how to start this question. Do we have to work out the first and second year forward rates?

## Answer by user5155 (score 0, accepted)

https://quant.stackexchange.com/a/7776

Solving for annual interest rates:

The one year annual spot rate r1: $$ 1.045/(1+r1)=1.0041 => r1 \approx 4.0733\% $$ The one-two year forward rate r1,2: $$ .065/(1+r1)+1.065/(1+r1)(1+r1,2)=1.0048 => r1,2 \approx 8.5927\% $$ The two year spot rate r2= $$ (1+r2)^{2}=(1+r1)(1+r1,2) => r2 \approx 6.3090\% $$

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