# Bootstrapping Examples: Yield Curve Calculation

## 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 + 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%

#### 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 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%

### 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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