What is Bootstrapping?
The term bootstrapping refers to the technique of carving out a zerocoupon yield curve from the market prices of a set of a coupon paying bonds. The bootstrapping technique is primarily used to make up Treasury bill yield which are offered by government and as such are not always available at every time period. In other words, the bootstrapping technique is used to interpolate the yields for Treasury zerocoupon securities with various maturities. Treasury bills are considered to riskfree and hence are used to derive the yield curve. In this article, we will discuss Bootstrapping Examples.
Examples of Bootstrapping
some of the examples of bootstrapping are given below:
Example 1:
Let us take the example of two 5% coupon paying the bond with zero creditdefault risks and a par value of $100 with the clean market prices (exclusive of accrued interest) of $99.50 and $98.30 respectively and having time for the maturity of 6 months and 1 year respectively. Determine the spot rate for the 6month and 1year bond. Please note that this a par curve where the coupon rate is equal to the yield to maturity.
At the end of 6 months the bond will pay a coupon of $2.5 (= $100 * 5% / 2) plus the principal amount (= $100) which sums up to $102.50. The bond is trading at $99.50. Therefore, the 6month spot rate S_{0.5y} can be calculated as,
$99.50 = $102.50 / (1 + S_{0.5y}/2)
 S_{5y} = 6.03%
At the end of another 6 months the bond will pay another coupon of $2.5 (= $100 * 5% / 2) plus the principal amount (= $100) which sums up to $102.50. The bond is trading at $98.30. Therefore, the 1year spot rate S_{1y} can be calculated using S_{0.5y} as,
$99.50 = $2.50 / (1 + S_{0.5y}/2) + $102.50 / (1 + S_{1y}/2)^{2}
 $99.50 = $2.50 / (1 + 6.03%/2) + $102.50 / (1 + S_{1y}/2)^{2}
 S_{1y} = 6.80%
So, as per the market prices, the spot rate for the first 6month period is 6.03% and the forward rate for the second 6month period is 6.80%
Example 2:
Let us take the another example of some coupon paying bond with zero creditdefault risk with each having a par value of $100 and trading at par value. However, each of them have varying maturity period that ranges from 1 year to 5 years. Determine the spot rate for all the bonds. Please note that this a par curve where the coupon rate is equal to the yield to maturity. The detail is given in the table below:
1. At the end of 1 year the bond will pay a coupon of $4 (= $100 * 4%) plus the principal amount (= $100) which sums up to $104 while the bond is trading at $100. Therefore, the 1year spot rate S_{1y} can be calculated as,
$100 = $104 / (1 + S_{1y})
 S_{1y} = 4.00%
2. At the end of 2^{nd} year the bond will pay coupon of $5 (= $100 * 5%) plus the principal amount (= $100) which sums up to $105 while the bond is trading at $100. Therefore, the 2year spot rate S_{2y} can be calculated using S_{1y} as,
$100 = $4 / (1 + S_{1y}) + $105 / (1 + S_{2y})^{2}
 $100 = $4 / (1 + 4.00%) + $105 / (1 + S_{2y})^{2}
 S_{1y} = 5.03%
3. At the end of 3^{rd} year the bond will pay coupon of $6 (= $100 * 6%) plus the principal amount (= $100) which sums up to $106 while the bond is trading at $100. Therefore, the 3year spot rate S_{3y} can be calculated using S_{1y} and S_{2y} as,
$100 = $4 / (1 + S_{1y}) + $5 / (1 + S_{2y})^{2} + $106 / (1 + S_{3y})^{3}
 $100 = $4 / (1 + 4.00%) + $5 / (1 + 5.03%)^{2} + $106 / (1 + S_{3y})^{3}
 S_{3y} = 6.08%
4. At the end of 4^{th} year the bond will pay coupon of $7 (= $100 * 7%) plus the principal amount (= $100) which sums up to $107 while the bond is trading at $100. Therefore, the 4year spot rate S_{4y} can be calculated using S_{1y}, S_{2y} and S_{3y} as,
$100 = $4 / (1 + S_{1y}) + $5 / (1 + S_{2y})^{2} + $6 / (1 + S_{3y})^{3} + $107 / (1 + S_{4y})^{4}
 $100 = $4 / (1 + 4.00%) + $5 / (1 + 5.03%)^{2} + $6 / (1 + 6.08%)^{3} + $107 / (1 + S_{4y})^{4}
 S_{4y} = 7.19%
5. At the end of 5^{th} year the bond will pay coupon of $8 (= $100 * 8%) plus the principal amount (= $100) which sums up to $108 while the bond is trading at $100. Therefore, the 5year spot rate S_{5y} can be calculated using S_{1y}, S_{2y,} S_{3y} and S_{4y} as,
$100 = $4 / (1 + S_{1y}) + $5 / (1 + S_{2y})^{2} + $6 / (1 + S_{3y})^{3} + $7 / (1 + S_{4y})^{4} + $108 / (1 + S_{5y})^{5}
 $100 = $4 / (1 + 4.00%) + $5 / (1 + 5.03%)^{2} + $6 / (1 + 6.08%)^{3} + $7 / (1 + 7.19%)^{4} + $108 / (1 + S_{5y})^{5}
 S_{5y} = 8.36%
ConclusionBootstrapping Examples
The technique of bootstrapping may be a simple one, but determining the real yield curve and then smoothening it out can be a very tedious and complicated activity which involves lengthy mathematics primarily using bond prices, coupon rates, par value and the number of compounding per year.
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This has been a guide to Bootstrapping Examples. Here we discussed the calculation of Bootstrapping with practical examples. You can also go through our other suggested articles to learn more –
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