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Bootstrapping Examples

Madhuri Thakur
Article byMadhuri Thakur

Updated June 3, 2023

Bootstrapping Examples

 

 

Definition of Bootstrapping

The term bootstrapping refers to the technique of carving out a zero-coupon yield curve from the market prices of a set of coupon-paying bonds. Primarily, we use the bootstrapping technique to calculate Treasury bill yields offered by the government, which aren’t always available for every time period.

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Examples of Bootstrapping

Some of the examples of bootstrapping are given below:

Example #1

Let us take the example of two 5% coupons paying the bond with zero credit-default 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. First, determine the spot rate for the 6-month and 1-year bonds. Please note that this is a par curve where the coupon rate equals 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, we can calculate the 6-month spot rate S0.5y as follows,

$99.50 = $102.50 / (1 + S0.5y/2)

  • S5y = 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 1-year spot rate S1y can be calculated using S0.5y as,

$99.50 = $2.50 / (1 + S0.5y/2) + $102.50 / (1 + S1y/2)2

  • $99.50 = $2.50 / (1 + 6.03%/2) + $102.50 / (1 + S1y/2)2
  • S1y = 6.80%

So, as per the market prices, the spot rate for the first 6-month period is 6.03%, and the forward rate for the second 6-month period is 6.80%

bootstapping examples 1

bootstapping examples

Example #2

Let us take another example of some coupon-paying bonds with zero credit-default risks, each with a par value of $100 and trading at par value. However, each has a varying maturity period ranging from 1 year to 5 years. Determine the spot rate for all the bonds. Please note that this is a par curve where the coupon rate equals the yield to maturity. The detail is given in the table below:

coupon rate

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 1-year spot rate S1y can be calculated as,

$100 = $104 / (1 + S1y)

  • S1y = 4.00%

2. At the end of the 2nd year, the bond will pay a coupon of $5 (= $100 * 5%) plus the principal amount (= $100), which is up to $105 while the bond is trading at $100. Therefore, the 2-year spot rate S2y can be calculated using S1y as,

$100 = $4 / (1 + S1y) + $105 / (1 + S2y)2

  • $100 = $4 / (1 + 4.00%) + $105 / (1 + S2y)2
  • S1y = 5.03%

3. At the end of the 3rd year, the bond will pay a coupon of $6 (= $100 * 6%) plus the principal amount (= $100), which is up to $106 while the bond is trading at $100. Therefore, the 3-year spot rate S3y can be calculated using S1y and S2y as,

$100 = $4 / (1 + S1y) + $5 / (1 + S2y)2 + $106 / (1 + S3y)3

  • $100 = $4 / (1 + 4.00%) + $5 / (1 + 5.03%)2 + $106 / (1 + S3y)3
  • S3y = 6.08%

4. At the end of the 4th year, the bond will pay a coupon of $7 (= $100 * 7%) plus the principal amount (= $100), which is up to $107 while the bond is trading at $100. Therefore, we can calculate the 4-year spot rate S4y using S1y, S2y, and S3y as follows,

$100 = $4 / (1 + S1y) + $5 / (1 + S2y)2 + $6 / (1 + S3y)3 + $107 / (1 + S4y)4

  • $100 = $4 / (1 + 4.00%) + $5 / (1 + 5.03%)2 + $6 / (1 + 6.08%)3 + $107 / (1 + S4y)4
  • S4y = 7.19%

5. At the end of the 5th year, the bond will pay a coupon of $8 (= $100 * 8%) plus the principal amount (= $100), which sums up to $108 while the bond is trading at $100. Therefore, we can calculate the 5-year spot rate S5y using S1y, S2y, S3y, and S4y as follows,

$100 = $4 / (1 + S1y) + $5 / (1 + S2y)2 + $6 / (1 + S3y)3 + $7 / (1 + S4y)4 + $108 / (1 + S5y)5

  • $100 = $4 / (1 + 4.00%) + $5 / (1 + 5.03%)2 + $6 / (1 + 6.08%)3 + $7 / (1 + 7.19%)4 + $108 / (1 + S5y)5
  • S5y = 8.36%

5-year spot rate

examples 2

Conclusion – Bootstrapping Examples

The bootstrapping technique may be simple, but determining the real yield curve and then smoothening it out can be a tedious and complicated activity involving lengthy mathematics, primarily using bond prices, coupon rates, par value, and the number of compounding per year.

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