Fitting Nelson–Siegel–Svensson Curves to Coupon Bond Yields
Summary
The document explains how to fit a Nelson–Siegel–Svensson (NSS) yield curve when zero-coupon yields are not directly available. Its central point is that zero rates can be treated as outputs of the model rather than required inputs: for each trial set of NSS parameters, calculate theoretical bond prices, convert those prices to yields, and minimize the differences from observed market yields.
It also sketches an alternative route using coupon bonds to infer equivalent zero rates sequentially. A small example with bonds of different maturities illustrates how shorter-maturity information can contribute to estimates for longer maturities. The discussion does not compare the accuracy or stability of direct yield fitting with bootstrapping, and the numerical example is too limited to establish how well either method performs in practice. Instrument conventions and optimization choices are not specified.
Key ideas
- NSS parameters can be estimated by minimizing differences between market yields and model-implied bond yields.
- The model can price coupon bonds directly for each trial parameter set.
- Coupon bond prices can also be used to infer zero rates across maturities.
- The document gives a brief example but does not evaluate fitting accuracy or optimization stability.
Tags
Full text
# Legitimate input parameters for Nelson Siegel Svensson model
# Legitimate input parameters for Nelson Siegel Svensson model
I had previously asked this question and have come to better understand the answer with regards to setting the input parameters for the Non-Linear Optimization problem that provides the NSS parameters. However, I realize that the answer assumes that I have zero coupon yields and can thus use the equation (2) on Page 2 of this pdf in order to obtain the input NSS parameters. However, if I don't have the Zero-Coupon Yield data, how may I proceed in order to obtain the input parameters. Would I need to bootstrap these Yields and if so, won't this hamper the whole process of obtaining a satisfactory solution to the Non-Linear Optimization problem.
Thank You
## Answer by Helin (score 1)
https://quant.stackexchange.com/a/19071
Zero coupon rates are outputs, not inputs. As mentioned in the other post, given the parameters (say the initial guesses), you can easily compute the theoretical prices of each bond, which can then be converted into their theoretical yields (standard price to yield conversion). You should minimize the residuals between these theoretical yields and the market yields.
EDIT: I made a pretty crude spreadsheet that illustrates what you need to do: https://app.box.com/s/6i3vae7lb02n6glam7vwts7qpuc44q2w
## Answer by phdstudent (score 1)
https://quant.stackexchange.com/a/19075
A simple example might help. You need to transform your coupon bonds in equivalent zeros. Imagine that you have 5 coupon bonds and you are at the end of 2011:
- Coupon, Maturity, Price = 5.25% 2012 101.69
- Coupon, Maturity, Price = 4.5% 2013 101.52
- Coupon, Maturity, Price = 5.5% 2014 104.49
- Coupon, Maturity, Price = 5% 2015 103.35
From the first one you can get: $R_{0,1}= 3.5\%$, from the second and first ones you get: $R_{0,1}= 3.70\%$, from the third, second and first you get: $R_{0,3}= 3.9\%$, etc.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.