## What is Bootstrapping?

The term bootstrapping refers to the technique of carving out a zero-coupon 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 yields offered by the government and are not always available at every time period. In other words, the bootstrapping technique is used to interpolate the yields for Treasury zero-coupon securities with various maturities. Treasury bills are considered to risk-free 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% 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 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 6-month 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 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%

4.9 (3,296 ratings)

View Course

#### Example #2

Let us take another example of some coupon paying bond with zero credit-default risks with each having a par value of $100 and trading at par value. However, each of them has a 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 1-year 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 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 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 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 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 4-year 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 5-year 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%

### Conclusion – Bootstrapping 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 that involves lengthy mathematics primarily using bond prices, coupon rates, par value and the number of compounding per year.

### Recommended Articles

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 –