What is Bootstrapping Yield Curve?
Bootstrapping is a method to construct a zero-coupon yield curve. The following bootstrapping examples provide an overview of how a yield curve is constructed. Although not every variation can be explained as there are many methods in bootstrapping because of differences in conventions used.
Top 3 Examples of Bootstrapping Yield Curve in Excel
The following are examples of bootstrapping yield curve in excel.
Example #1
Consider different bonds with a face value of $ 100, with the yield to maturity equal to the coupon rate. The coupon details are as below:
Maturity | 0.5 Year | 1 Year | 1.5 Year | 2 Year |
Yield to Maturity | 3% | 3.50% | 4.50% | 6% |
Solution:
Now, for a zero-coupon with a maturity of 6 months, it will receive a single coupon equivalent to the bond yield. Hence, the spot rate for the 6-month zero-coupon bond will be 3%.
For a 1-year bond, there will be two cash flows, at 6 months and at 1 year.
The cash flow at 6 months will be (3.5%/2 * 100 = $ 1.75) and cash flow at 1 year will be (100 + 1.75 = $ 101.75) i.e. principal payment plus the coupon payment.
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From the 0.5-year maturity the spot rate or the discount rate is 3% and let us assume the discount rate for 1-year maturity be x%, then
- 100 = 1.75/(1+3%/2)^1 + 101.75/(1+x/2)^2
- 100-1.75/(1+3%/2)=101.75/(1+x%/2)^2
- 98.2758 = 101.75/(1+x%/2)^2
- (1+x%/2)^2 =101.75/ 98.2758
- (1+x%/2)^2 = 1.0353
- 1+x%/2 = (1.0353)^(1/2)
- 1+x%/2 = 1.0175
- x% = (1.0175-1)*2
- x% = 3.504%
Solving the above equation, we get x = 3.504%
Now, again for a 2 year bond maturity,
- 100 = 3/(1+3%/2)^1 + 3/(1+3.504%/2)^2 + 3/(1+4.526%/2)^3 + 103/(1+x/2)^4
- 100 = 2.955665025 + 2.897579405 + 2.805211867 + 103/(1+x/2)^4
- 100-8.658456297 = 103/(1+x/2)^4
- 91.3415437 = 103/(1+x/2)^4
- (1+x/2)^4 = 103//91.3415437
- (1+x/2)^4 = 1.127635858
- (1+x/2) = 1.127635858^(1/4)
- (1+x/2) = 1.030486293
- x = 1.030486293-1
- x = 0.030486293*2
- x = 6.097%
Solving for x we get, x = 6.097%
Similarly, for a 1.5-year bond maturity
100 = 2.25/(1+3%/2)^1 + 2.25/(1+3.504/2)^2 + 102.25/(1+x/2)^3
Solving the above equation, we get x = 4.526%
Thus, the bootstrapped zero yield curves will be:
Maturity | Zero Rates |
0.5 Year | 3% |
1 Year | 3.50% |
1.5 Year | 4.53% |
2 Year | 6.10% |
Example #2
Let us consider a set of zero-coupon bonds of face value $ 100, with maturity 6 months, 9 months and 1 year. The bonds are zero-coupon i.e. they do not pay any coupon during the tenure. The prices of the bonds are as below:
Maturity | Price ($) | |
Months | 6 | 99 |
Months | 9 | 98.5 |
Year | 1 | 97.35 |
Solution:
Considering a linear rate convention,
Where r is the zero-coupon rate, t is the time
Thus, for 6-month tenure:
- 100 = 99*(1 + R_{6}*6/12)
- R_{6} = (100/99 – 1)*12/6
- R_{6 }= 2.0202%
For 9-month Tenure:
- 100 = 99*(1 + R_{9}*6/12)
- R_{9} = (100/98.5 – 1)*12/9
- R_{9 }= 2.0305%
For 1-year Tenure:
- 100 = 97.35*(1 + R_{12}*6/12)
- R_{12} = (100/97.35 – 1)*12/12
- R_{12 }= 2.7221%
Hence, the bootstrapped zero-coupon yield rates will be:
Maturity | Zero Coupon (Rates) |
6 Months | 2.02% |
9 Months | 2.03% |
1 Year | 2.72% |
Note that the difference between the first and second example is that we have considered the zero-coupon rates to be linear in example 2 whereas they are compounding in example 1.
Example #3
Although this is not a direct example of a bootstrapping yield curve, sometimes one needs to find the rate between two maturities. Consider the zero-rate curve for the following maturities.
Maturity | Zero Coupon (Rates) |
6 | 2.50% |
1 Year | 3.50% |
3 Year | 5% |
4 Year | 5.50% |
Now, if one needs the zero-coupon rate for 2-year maturity, he needs to linearly interpolate the zero rates between 1 year and 3 years.
Solution:
Calculation of zero-coupon discount rate for 2 year –
Zero-coupon rate for 2 year = 3.5% + (5% – 3.5%)*(2- 1)/(3 – 1) = 3.5% + 0.75%
Zero-Coupon Rate for 2 Years = 4.25%
Hence, the zero-coupon discount rate to be used for the 2-year bond will be 4.25%
Conclusion
The bootstrap examples give an insight into how zero rates are calculated for the pricing of bonds and other financial products. One must correctly look at the market conventions for proper calculation of the zero rates.
Recommended Articles
This has been a guide to Bootstrapping Yield Curve. Here we discuss how to construct a zero-coupon yield curve using bootstrapping excel examples along with explanations. You can learn more about fixed income from the following articles –