Common Practice for Bootstrapping!

[JuniorASA]

The key concept for bootstrapping is to use X+1 bond yield and X year spot rate to solve for X+1 spot rate.

In common US treasury bond, there are 1,2,3,4,5,7,10,20,30 bond yield. Certainly, I could use above logic to solve for 1-5 spot rate.

But without 6 year bond yield. how could I derive the 6 year spot rate?

Furthermore, without 6 year spot rate, I could not solve for spot rate for 7 years upwards.

What is the common industrial approach to solve for this issue?

[NoName]

I'm not a practitioner but ...

The yield curve is an artificial construct anyway. If you're trying to put together an arbitrage trading model and have billions of dollars on the line you probably have more input information than a single yield curve. Presumably you're just doing normal stuff where you just need a number that sort of makes sense. In that case I would say assume the 6 year yield is halfway between the 5 and 7 and be done with it.

[exactuary]

The one-year-old 7-year is now a six year; the one-and-two-year-old 10-years are now the eight- and nine-years, etc.

[RobertNelson]

I am a practioner, and I can outline how it is commonly done.

Convert the yield curve into spot rates, or actually discount factors. Since the yield curve is typically quoted in cash rates for the first year, and in semi annual bond yield curve equivalents for terms greater than one year, this requires two algorithms to decompose the YC. You should end up with discount factors for every 1/2 year point.

To get the points in between the initial curve, common practice is to use an interpolation routine on those discount factors. Cubic spline is perhaps the most common of these. You can find that routine in freeware: either C++, Java, or Excel. I copied mine from "Numerical Recipes in C". I usually do monthly points. Be warned that the forward rates produced in this fashion will have a slight saw-toothed pattern to them. This shouldn't worry you unless the instruments you are pricing are greatly affected by intermediate forward rates.

From the discount curve, you can easily form the forward discounts, and then convert them back into either semi annual bond equivalents, or leave them as spot rates.

Its a bit laborious the first time around - writing the code to go back and forth between discount rates and bond yield equivalents. But once done, you never have to do it again.

[E]

Has anyone written this routine in Excel? I'm trying to develop an Excel-based approached to interpolating between the standard quoted constant maturities to get rates for all monthly maturities from 1 to 360. Thanks.

[DW Simpson]

Has anyone written this routine in Excel? I'm trying to develop an Excel-based approached to interpolating between the standard quoted constant maturities to get rates for all monthly maturities from 1 to 360. Thanks.

http://www.google.com/search?hl=en&lr=&as_qdr=all&q=maturities+filetype%3Axls&btnG=Search

[JuniorASA]

Hi, E,

I found that there is an excel for bootstrapping process in Hong Kong's actuarial site. You may take a look here. See if it could help!

http://www.actuaries.org.hk/

HKD Yield Curve Excel Software

[hunting]

Just a comment ... I mean no disrespect by this, but I am a swaps trader and so I know this area pretty well.

It is not prudent to take spot rates and compute discount factors, and then connect them with a spline. Forward rates should not have a saw-toothed pattern. They should be smooth and differentiable. If they are jagged, the model is wrong. In fact the whole reason for the introduction of splines was to make the forward curve smooth, so that people weren't using models that projected forward rates to bounce all over the place in an unrealistic fashion.

The situation has progressed now to the point that even cubic splines are being replaced by quartic splines to maximize smoothness. Although I do agree with the comment earlier that it definitely depends on what you are doing. Connecting the dots might be adequate depending on your purpose.

Hope it helps ...

[E]

Hi, E,

I found that there is an excel for bootstrapping process in Hong Kong's actuarial site. You may take a look here. See if it could help!

http://www.actuaries.org.hk/

HKD Yield Curve Excel Software

Thanks. This looks helpful.

[RobertNelson]

Just a comment ... I mean no disrespect by this, but I am a swaps trader and so I know this area pretty well.

It is not prudent to take spot rates and compute discount factors, and then connect them with a spline. Forward rates should not have a saw-toothed pattern. They should be smooth and differentiable. If they are jagged, the model is wrong. In fact the whole reason for the introduction of splines was to make the forward curve smooth, so that people weren't using models that projected forward rates to bounce all over the place in an unrealistic fashion.

The situation has progressed now to the point that even cubic splines are being replaced by quartic splines to maximize smoothness. Although I do agree with the comment earlier that it definitely depends on what you are doing. Connecting the dots might be adequate depending on your purpose.

Hope it helps ...
Agree 100%, hunting. It is always hard to respond on this board, since the audience covers such a spectrum of experiences and needs. Without question, if you are trading in fixed cash flows, use a method that results in continuous forwards. There's a number of them.

From the OP, I assumed he was interested in actuarial work. Sadly, that means less than perfect cash flow modeling. Virtually all of the actuarial models I've seen use discrete cash flows to represent continuous flows. Sometimes its calendar year, sometimes policy year. But the work usually results in a cash flow that represents the amount paid/received between two dates. This is not easily translated into a swap-desk type pricing model. Accurate fixed income pricing requires a date with the cash flow.

I wonder how many others have had to translate between the coarse grain actuarial models, and the precision of the capital markets. What do others use to bridge this gap? Mid period flows? end of period flows? Some sort of algorithm to translate the actuarial flows into dates?