Updated June 3, 2023

## 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.

### 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 + 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 1-year 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 6-month period is 6.03%, and the forward rate for the second 6-month period is 6.80%

#### 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:

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 S_{1y} can be calculated as,

$100 = $104 / (1 + S_{1y})

- S
_{1y}= 4.00%

2. At the end of the 2^{nd} 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 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 the 3^{rd} 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 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 the 4^{th} 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 + 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 the 5^{th} 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 + 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%

### 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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