Yield curve bootstrapping

[mihai.bunea]

Could someone please explain the algorithm behind bootstrapping of a yield curve from deposit rates?

I have an example of a yield curve consisting of 2 legs with the following maturities and rates:
1) maturity: 1 year, rate: 10% per year.
2) maturity: 2 years, rate: 20% per year.

How is the zero-rate computed for the second leg?

Legs use simple compounding for interest rate, so the time value of money is F = S (1 + r * t).
F = future sum of money
S = spot (current) sum of money
r = yearly interest rate
t = time, in years

For the first leg: 1$ now becomes 1.1$ in one year. The zero-rate equation is: 1 * e^(r1 * 1) = 1.1, therefore r1 = Ln(1.1) = 0.0953102, which is precisely what Quantlib computes.

For the second leg however, my assumptions on the calculation no longer hold. 1$ becomes (1 + 0.2 * 2) = 1.4$ in 2 years.
The zero-rate equation would be 1 * e^(r2 * 2) = 1.4, so r2 = Ln(1.4) / 2 = 0.168236. Quantlib computes something close, 0.167967 but definitely not exactly the same thing.

Somehow the 1'st leg enters the zero-rate equation of the 2'nd leg, but couldn't find out a documentation on how it's done.

[Luigi Ballabio]

On Thu, 2011-02-24 at 08:16 -0800, mihai.bunea wrote:
> I have an example of a yield curve consisting of 2 legs with the following
> maturities and rates:
> 1) maturity: 1 year, rate: 10% per year.
> 2) maturity: 2 years, rate: 20% per year.

May you post the code?  It might help diagnosing the problem.

Luigi


--

Lubarsky's Law of Cybernetic Entomology:
There is _always_ one more bug.



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[mihai.bunea]

Hello Luigi,

thanks for answering.

Looking over the code again helped and i found the problem: 2012 is a leap year.
I used 1-Mar-2011 as reference date, Unadjusted business day convention and Actual365Fixed day counter. Then added 365 days, respectively 2 * 365 days to obtain maturities of 1, respective 2 years.

The problem is the calendar-based year fraction is not exactly 1 or 2 in this case.
When i replaced the reference date with 1-Mar-2012 (after the leap day of 29-Feb-2012), it worked by the book.

That is, 1y tenor with 10% rate => 1$ after 1y -> 1.1$ => discount =~ 0.909091
           2y tenor with 20% rate => 1$ after 1y -> 1.4$ => discount =~ 0.714286