# Bond Curve Bootstrapping

 

Government Bond Curve Bootstrapping Procedure

 

We discuss a method for bootstrapping a set of zero rates from an input set of US government money market securities and bonds. The government bond bootstrapping procedure requires to input a set of financial instruments, of the type below, sorted by order of increasing time to maturity:

 

§  Short term money market instruments (i.e., USD T-Bills with maturity not more than one year),

§  Medium to long term “on the run” US government bonds,

§  Medium to long term “off the run” US government bonds.

 

First of all, we need to have a bond valuation model to calculate bond price or bond yield (see <https://finpricing.com/lib/FiZeroBond.html>)

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To determine discount factors at times intermediate to control points, we can apply a particular interpolation technique. There are three available:

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·         LINEAR

·         LOG\_LINEAR

·         TIME\_WEIGHTED\_LINEAR

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The LINEAR scheme interpolates zero rates linearly between successive control points on the zero curve; that is, if  and  are bootstrapped continuously compounded zero rates at successive control points, then

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The LOG\_LINEAR scheme interpolates linearly between  and , that is,

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The TIME\_WEIGHTED\_LINEAR scheme interpolates between  and ,

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where and . Since t2>t1, the TIME\_WEIGHTED\_LINEAR scheme weights rate information farther in the future more heavily than rate information in the near future.

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Government Bond Bootstrapping proceeds in two phases. The first phase uses short term instruments, which typically mature in one year or less. Consider, for example, a US government money market instrument with

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§      calendar days between settlement date, , and maturity date, , and

§     with a corresponding simple forward interest rate .

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Then

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where the “360” factor arises from the ACT/360 market convention used for US government money market instruments yields, is the forward price at the date, , of a zero coupon bond with maturity date, , and unit face value.

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Let a bond have the cash flow, , at date, , for . Our Benchmark applies a Newton iterative scheme to solve

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for the unknown zero rate at  where  is the settlement date. Here we use the “Time Weighted Linear” interpolation scheme to determine intermediate discount rates.

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