# Yield curve modeling

### Konstantinos Pappas

#### 15/02/2023

The financial industry is constantly searching for models that can help accurately predict the behavior of interest rates. In this article we will explore one of the most widely used models for this purpose, the Nelson-Siegel (NS) model.

## What is the Nelson-Siegel Model?

The NS model is a yield curve factor model. Yield curve factor models are based on the idea that the yield curve can be decomposed into several components, each of which describes a different aspect of the yield curve’s behavior. Yield factor models are a parallel to equity factor models. They can provide a highly accurate representation of the yield curve compressing a lot of information in a low dimension factor structure. As in the equity space these models have an economic theory underpinning as only a small number of systematic risks underlying the pricing of bonds.

The NS model essentially uses 3 smoothing parameters, to describe the shape of the yield curve. The first parameter, $$\beta_0$$, describes the long-term level of interest rates. The second parameter, $$\beta_1$$, describes the slope, while the third parameter,$$\beta_2$$ describes the curvature of the yield curve. These parameters are used to generate a mathematical representation of the yield curve that can be used to make predictions about future interest rate movements.

$$y(\tau)=\beta_0+\beta_1\left(\frac{1-e^{-\lambda \tau}}{\lambda \tau}\right)+\beta_2\left(\frac{1-e^{-\lambda \tau}}{\lambda \tau}-e^{-\lambda \tau}\right)$$

• $$\beta_0$$: the long run levels of interest rates (the loading is 1, it is a constant that does not decay).
• $$\beta_1$$: the short-term component (it starts at 1, and decays quickly to 0).
• $$\beta_2$$: the medium-term component (it starts at 0, increases, then decays to zero)
• $$\lambda$$: the decay factor (small values result in slow decay – better fit the curve at long. maturities, while large values produce fast decay – better fit the curve at short maturities)
• $$t$$: maturity,
• $$y(t)$$:the yield of the curve at maturity $$t$$.

## Extensions of the Nelson-Siegel Model

There have been many advancements in the yield curve modeling space since the publication of NS model in 1987. Since then many extensions have been proposed addressing constraints and weakness of the NS model. For the purpose of this article we will focus on 2 versions that had the biggest impact in the progress of yield curve modeling the Dynamic Nelson-Siegel model(DNS) and Svensson extension (NSS).

### Dynamic Nelson-Siegel

Diebold and Li (2006), modified the the Nelson-Siegel model making the apparent observation that the parameters must be time-varying as the yield curve is also time-varying. Instead of the cross-sectional linear projection of $$y(t)$$ on variables with parameters $$\beta_0$$,$$\beta_1$$,$$\beta_2$$, we now have a time-series linear projection of $$y_t$$ on variables $$\beta_{0 t}$$,$$\beta_{1 t}$$,$$\beta_{2 t}$$,
Hence from a cross-sectional perspective the βs are parameters, but from a time-series perspective the βs are variables. Combining the spatial and temporal perspectives produces the dynamic Nelson-Siegel (DNS) model:

$$y_t(\tau)=\beta_{0 t}+\beta_{1 t}\left(\frac{1-e^{-\lambda \tau}}{\lambda \tau}\right)+\beta_{2t}\left(\frac{1-e^{-\lambda \tau}}{\lambda \tau}-e^{-\lambda \tau}\right)$$

### Nelson-Siegel Svensson model

The Nelson-Siegel Svensson model is an extension of the Nelson-Siegel model, which provides sufficient precision adjustment by including the parameter $$\beta_3$$:a “second hump” in the model.

$$+\beta_3\left(\frac{\left[1-\exp \left(-m / \tau_2\right)\right]}{m / \tau_2}-\exp \left(-m / \tau_2\right)\right)$$

## Applications of the NS and NSS Model

In this quick example we calibrate the models on the U.S. yield curve (date:08.02.2023) and present their factor decomposition. We see that the 2 versions do not deviate almost at all. Furthermore the proposed rates in the table below show that the fourth factor added does not enhance our initial model.

Using the NS and the NSS model we can obtain the rates for any maturity, as well as, calculate the implied forward rate.