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Instantaneous Forward Rates

Note: We enclose a spreadsheet on Nelson-Siegel and Svensson curve fitting to this part of the topic.
Constructing a yield curve is a 2 step process. The first step in most cases is fairly easy and straightforward as it deals with selecting the appropriate bonds.
For example, if we are looking at a highly developed market with a large universe of borrowings, such as the US treasury market, the curve is constructed using on-the-run instruments.
Or if we are looking at a selection of instruments issued by a single entity and wish to build an issuer or borrower curve using this selection, we should use those bonds, that truly represent the credit quality and features that are typical for this entity.
I.e. these securities should not be subordinate or illiquid, and if they carry embedded options, appropriate adjustments should be made to the calculated yields (although this can be argued to subjective reasoning).
The second step is to populate the curve using the selected bond data, which is purely mathematical.

As mentioned above, the yield curve construction can be done either by interpolation or approximation (i.e. regression) to control its shape.
From a quantitative perspective these methods define how a variable changes between two values taking into account time as a third input value.
A simple approach is that there is a sequence of data points and a curve should be constructed with a shape that closely follows a path of this sequence.

The simplest interpolation method is focused between two values and is called Lerp or Linear function.
In order to apply a smooth acceleration/deceleration some smooth interpolation methods can be used as well.

If there is a need to interpolate between a multiple set of values there is a spline method to use.
They are defined as piecewise polynomials where each interval is handled separately. Splines like linear methods can be generally defined as polynomial curves:

Depending on the power of x the polynomial functions can be:
Order 0: Constant, where x is zero;
Order 1: Linear, where b is the highest none zero coefficient;
Order 2: Quadratic, where c is the highest none zero coefficient;
Order 3: Cubic, where d is the highest none zero coefficient. Cubic polynomial is widely used for constructing curves because it supports inflection and is well behaved numerically without jumpy intervals.
Cubic polynomial curves can be defined using 4 points but more complex curves typically built by using constructing of piecewise polynomial curves and parameterization. In order to build such a curve its line should pass through all data points and have the same slope and curvature in these points.

Splines provide an extremely high degree of flexibility in terms of the shapes of the curves.
One of the widely used piecewise polynomial approximation models is the basis splines (or B-spline) to build a yield curve.
B-splines have a good convergence and a high degree of derivative continuity.

Although the spline methods are quite straightforward to construct there are some moments to take into consideration.
This is the quantity of the knot points.
The choice of knot positions defines the explanatory power of the spline interpolation and has to be done in conjunction with choosing the number of knots.
McCulloh defined the number of knots to be calculating using a specific formula:

where K is number of instruments with specified maturities.

The second step is to define location of knot points and adjacent intervals so the points divide the maturity spectrum into segments.
After this, the number of basis functions f(x) calculated which insure continuity and smoothness at the knot points.
The discount function and its parameters are defined using an OLS linear regression model.
The discount function is obtained using parameter values and basis functions.
As a result, the yield curve is obtained from the discount function.

Of course the most precise method of curve fitting is to use linear or polynomial interpolation so that all constituent bonds will be on the curve, but it is a common practice to implement logarithmic or complex parametric models, such as the Nelson-Siegel and Svensson (NSS) approach, which characterizes the curve and discount at all maturities.
The NSS method is given by the following equation:

The parameters β0 and β1 determine the beginning of the yield curve in terms of initial level and slope, whereas β2 and β3 determine the magnitudes and signs of the humps at times τ1 and τ2 of the yield curve, and n represents tenor.
These humps ensure that the curve is convex at the short and long end.
The second hump was introduced by Svensson because the original Nelson Siegel model was inaccurate in fitting the curve at longer maturities.
As the convexity tends to pull the longer term yields down, the curve bends to a negative extent at its long end.
This drawback is eliminated by the Svensson extension, which adds the second hump thus making the curve approach a horizontal asymptote.
This model utilizes maximum likelihood to estimate the six parameters and achieve a minimized sum of squared deviations of the theoretical yields from the actual bond yields. Although NSS is popular, it still has certain drawbacks, like lack of flexibility in comparison with spline based models.

Selection of either curve fitting method affects all spread measures, and first of all the I-spread of the observed risky bonds, which is basically the absolute difference between the risky bond’s yield and the interpolated area of the yield curve. The spread difference arises from the degree of curvature imposed by the curve fitting technique.
For example the logarithmic approximation may result in a higher I-spread on the short end of the yield curve, than the linear interpolation would produce, and ultimately lead to different trading decisions.
These decisions can also be derived from various assumptions, such as identification of mispriced bonds or expectations of a shift in the yield curve, which can be based on simulations carried out by market practitioners.