We saw that exact estimation methods can lead to ragged forward curves. This is because discount factors of similar maturity can be very different. Moreover, the discount curve is sensitive to small changes in the cash flow matrix. This may be critical because market rate data can contain outliers. If our object of interest is a smooth forward curve we should apply a smoothing estimation method. That is, we should directly estimate a smooth forward curve for market rates at the cost of possibly not matching the data exactly. Let us directly formalize what the smoothing estimation method does. For simplicity of notation, we set the spot date T-0 equal to 0. That helps us to avoid writing 2 time arguments all the time. We consider market data consisting of N yields which we denote by Y-i with maturities T-i. The aim is then to find a smooth forward curve f(T) that matches the yields optimally For pricing errors epsilon which are subject to be minimized. There are several good reasons for considering pricing errors. Market micro structure effects could lead to outliers in the data and in general, there are bid-ask spreads which can be wide in times of illiquidity so that prices are not uniquely given. smoothing methods are mainly used buy central banks and any institution that is primarily interested in the qualitative shape of the forward curve. Is it upward sloping? Is it downward sloping? Is it steep in the beginning? or flat? The following 3 are key criteria for smoothing methods. First is smoothness. The estimation method should supply a market expectation for monetary policy purposes rather than precisely pricing all the bonds in the market. Flexibility. The method should be sufficiently flexible to capture movements to capture movements in the underlying term structure. And stability. Small changes in the data at one maturity must not have disproportionate effects on forward rates at other maturities. This is an overview from the Bank of International Settlements in 2005. We see that 9 out of 13 central banks use methods called Nelson-Siegel and Svensson. While the remaining banks use smoothing splines. We will in the following see what Nelson-Siegel and Svensson and smoothing splines are. Let's also look at the third coloumn which indicates what type of error is being minimized. We see that we can minimize the matching error to yields as we saw on our former slide. Alternatively One can also match prices directly and weight the pricing errors and minimize those. The simple smoothing methods consist of given parametric families of forward curves. The 2 major examples are the Nelson-Siegel and the Svensson curves. We see that the Nelson-Siegel curves are forward curves of polynomial exponential type with 4 parameters β-0, β-1, β-2 and A where A is the shape parameter for these exponential functions, and the Betas are linear loadings on these basis functions. The Svensson family extends the Nelson-Siegel family by adding 2 additional parameters expressed in this term. And adding thus small flexiblity to these curve families. We obtain the yield curves in the Nelson-Siegel family by elementary integration as shown here. So that the Nelson-Siegel yield curves linear combinations with parameters β-0, β-1, β-2 of the basis functions i-0, i-1, i-2. The figure on the right hand side shows us these 3 basis functions for the choice of the shape parameter A = 0.3 and we see that these basis functions from MVC. MVC, that is, 3 basis functions are the level of the yield curve, the slope of the yield curve, and the curvature of the yield curve. We will come back to these 3 basis shapes but from a different angle in the section on Principal Component Analysis. The Nelson-Siegel yield curves are clearly smooth. The issue with this family of curves is their lack of flexibility. There are only 3 or 4 parameters. But also the stability meaning that changing a yield data point especially at a longer maturity will lead to a dramatic change of the Nelson-Siegel yield curve at the short end. Smoothing splines can overcome the issues of flexibility and stability of the Nleson-Siegel and Svensson curves. The idea is to find the forward curve matching the data in the space of all possible forward curves. We formalize this by fixing a finite time horizon T* and look at the Hilbert space H consisting of absolutely continuous functions on this time horizon. And we endow this Hilbert space with a scalar product as shown here. Recall that absolutely continuous real functions have a first order derivative which is integrable so that this integral is well defined. We then look at all forward curves in this Hilbert space that optimally match the data, the yields Y-i; and at the same time are smooth in the sense that the square of the first derivative integrated over the time horizon is being minimal. The parameter α shown here tunes the trade off between the smoothness and the correctness of the fit. It turns out that this is a convex optimization problem in the Hilbert space H which is infinite dimensional. The more remarkable it is that there is a unique solution which is given in closed form as shown here. As a linear combination of quadratic basis splines H-i which are defined as piece wise 2nd order polynomials with these characteristics. The right hand side shows quadratic basis splines H-1 and H-2, for the maturities T-1 = 1 and T-2 = 2. We see that H-1 is 2nd order polynomial with slope starting at 1 and then linearly decreasing to 0. And then as of T-1 being constant. In red we see the basis spline for T-2 = 2. The parameters β-0, β-1, up to β-n solve a linear system which is shown on the next slide. Here is the linear system that is solved by the N+1 parameters β-0 up to β-n. It can be shown that this linear system has a unique solution. This theorem is proved using elementary functional analysis. Proof is provided to you in a separate document. This result is a fork theorem in the theory of smoothing splines. I call it Lorimier's theorem for this course because in her thesis she used this result to estimate the forward curve. It can be verified that smoothing splines satisfy the 3 criteria of smoothness, flexibility, and stability. In fact the parameter α tunes the trade off between the smoothness and the correctness of the fit. When alpha (α) turns to 0, in the limit we get the maximally smooth forward curve namely, a constant curve taking value β-0. On the other hand, when tends to infinity we obtain a perfect fit to the yields subject to having a minimal matrix for the smoothness of the forward curve. As an example, we see here on the right hand side the curves we get from applying this smoothing spline theorem to yield data that we get from the Swiss Government bond market in August 2011. In red we see the 8 data points and then we have 3 curves for various values of α ranging from 0.01 to 10. We see when α is close to 0, we get a curve which is closer to a constant than the other 2 curves but it obviously has a bad fit to the market data. On the other hand when alpha(α) is large, that is, when it is 10 we get the dotted curve which basically, perfectly fits the yields. We conclude that the choice of α is critical For a value of α which is too small we can get a very smooth curve which however does a very bad job in fitting the market data. The right choice of alpha(α) has to be done case by case.