Good estimates of the term structure of interest rates (also known as the spot rate curve or the zero-bond yield curve) are of the utmost importance to investors and policy makers. There were multiple attempts at bootstrapping discrete spot rates from the market data and then fit a smooth and continuous curve to the data. They all suffered one way or another either from having undesirable economic properties or for being ‘black box’ models. Nelson and Siegel (1987) and Svensson (1994, 1996) suggested parametric curves that are flexible enough to describe a whole family of observed term structure shapes. These models are parsimonious, they are consistent with a factor interpretation of the term structure (Litterman and Scheinkman (1991)) and they have both been widely used in academia and in practice. The Nelson‐Siegel model is extensively used by central banks and monetary policy makers.

In their model Nelson and Siegel (1987) specify the forward rate curve \(f(\tau)\) as follows:

\[f(\tau) = \begin {bmatrix}\beta_0 \\ \beta_1 \\ \beta_2 \end {bmatrix}'\begin {bmatrix} 1 \\ e^{-\lambda\tau} \\ \lambda\tau e^ {-\lambda\tau}\end {bmatrix} = \begin {bmatrix}\beta_0 \\ \beta_1 \\ \beta_2 \end {bmatrix}' \begin {bmatrix} f_0 \\ f_1 \\ f_2 \end {bmatrix}\] where \(\tau\) is time to maturity, \(\beta_0, \beta_1, \beta_2\) and \(\lambda\) are coefficients, with \(\lambda > 0\).

This model consists of three parts reflecting three factors: a constant \((f_0)\), an exponential decay function \((f1)\) and a Laguerre function \((f2)\). The constant represents the (long‐term) interest rate level. The exponential decay function reflects the second factor, a downward (β1 > 0) or upward (β1 < 0) slope. The Laguerre function in the form of \(xe^{-x}\) , is the product of an exponential with a polynomial.

Nelson and Siegel (1987) chose a first-degree polynomial which makes the Laguerre function in the Nelson‐Siegel model generate a hump \((\beta_2 > 0)\) or a trough \((\beta_2 < 0)\). The higher the absolute value of \(\beta_2\), the more pronounced the hump/trough is. The coefficient \(\lambda\), referred to as the shape parameter, determines both the steepness of the slope factor and the location of the maximum (resp. minimum) of the Laguerre function. The spot rate function, which is the average of the forward rate curve up to time to maturity τ, is defined as:

\[r(\tau) = \frac 1 \tau \int_0 ^\tau f(u)du\]

with continuous compounding. Hence, the corresponding spot rate or yield function at time to maturity \(\tau\) reads as following:

\[y(\tau)= \begin {bmatrix}\beta_0 \\ \beta_1 \\ \beta_2 \end {bmatrix}'\begin {bmatrix} 1 \\ (1-e^{-\lambda\tau})/\lambda\tau \\ (1-e^ {-\lambda\tau})/\lambda\tau - e^{-\lambda\tau}\end {bmatrix} = \begin {bmatrix}\beta_0 \\ \beta_1 \\ \beta_2 \end {bmatrix}' \begin {bmatrix} y_0 \\ y_1 \\ y_2 \end {bmatrix}\]

Next slide depicts the three building blocks of the Nleson-Siegel model. The curves \(f_0, f_1\) and \(f_2\) in a Panel A and respectively \(y_0, y_1\) and \(y_2\) in Panel B represent the level, slope and curvature components of the forward rate or yield rate curve.

Note: This figure shows the decomposed components of Nelson-Siegel model for the forward rate curve (Panel A) and the spot rate curve for (Panel B) when the shape parameter is fixed at 1/3.

The Nelson-Siegel model worked well when there were few securities issued with long term maturities. The classical model cannot explain well the “second hump” in the yield curve that may occur for securities with maturities longer than 20 years due to curve convexity. Svenson extended the original model by adding a second curvature term, for a total of 6 parameters: 4 coefficients and two lambdas.

The Svenson model expressed in terms of forward rates has the following functional form:

\[f_t(\tau)=\beta_0 + \beta_1 e^{-\tau/\lambda_1}+\beta_2(\tau/\lambda_1)e^{-\tau/\lambda_1}+\beta_3(\tau/\lambda_2)e^{-\tau/\lambda_2}\]

or in a matrix form :

\[f(\tau) = \begin {bmatrix}\beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \end {bmatrix}'\begin {bmatrix} 1 \\ e^{-\lambda_1\tau} \\ (\tau/\lambda_1) e^ {-\lambda_1\tau}\\ (\tau/\lambda_2) e^ {-\lambda_2\tau} \end {bmatrix} = \begin {bmatrix}\beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3\end {bmatrix}' \begin {bmatrix} f_0 \\ f_1 \\ f_2 \\f_3 \end {bmatrix}\]

This model consists of four components reflecting four factors: a constant \((f_0)\), an exponential decay function \((f1)\) and two Laguerre functions \((f2)\) and \((f3)\). The constant represents the (long‐term) interest rate level. The exponential decay function reflects the second factor, which has influence only the short end of the curve. The third and forth components are responsible for the two “humps/trough” in the yield curve \((\beta_2 >0/\beta_2 < 0)\) and \((\beta_3 >0/\beta_3 < 0)\) respectively.

Obviously, if \(\beta_3=0\) the Svenson model collapses to the classical Nelson-Siegel model, hence this model is an extension of the original one.

The Svenson model can also be presented in terms of zero-coupon yields via integration as follows:

\[y_t(\tau)=\beta_0+\beta_1 \frac{1-e^{(-\tau/\lambda_1)}}{\tau/\lambda_1} + \beta_2 [\frac{1-e^{(-\tau/\lambda_1)}}{\tau/\lambda_1}-e^{(-\tau/\lambda_1)}] + \beta_3 [\frac{1-e^{(-\tau/\lambda_2)}}{\tau/\lambda_2}-e^{(-\tau/\lambda_2)}]\]

If the Treasury issued the full spectrum of debt instruments every day, there would not be any need to estimate the yield curve as we can simply observe the market yields for all maturities. In practice, US debt is issued on an “As Needed” basis therefore on any given day we need to infer the yields and prices of the securities that are not being issued on that day.