This project is related to the investor's portfolio allocation problem, considering (a) the empirical properties of the expected rates of return of assets and their covariance, and (b) the investor's preference for risk tolerance, investment horizon, and early resolution of uncertainty. The project's goal is to identify the percentage of wealth invested in the major categories of stocks and bonds, in order to maximize the investor's lifetime benefit.

I contribute to the implementation of interest rate term structure models. Our group's first goal is to compare different classes of affine (linear) models. We have created a suite of Nelson - Siegel affine models, and we aim to compare these models with corresponding ones from three different affine model classes. We are constructing a collection of state space models with an arbitrary number of time-invariant components for the state vector. There are two types of Nelson - Siegel models, the arbitrage-free and the dynamic ones that do not impose a "no-arbitrage" relationship. Our goal is to identify an optimal scheme that combines the benefits of these two types of models. These are the simplicity and stability of the estimation calculations from the dynamic models, and the inclusion of economically desirable properties such as the absence of arbitrage opportunities.

We use the Kalman filter algorithm to fit the models to the data, and maximize the relevant log-likelihood function to estimate the optimal model parameters. We benchmark various optimization algorithms to determine the time-invariant state space parameters that best represent the data trends. The current data set that we use is the Fama - Bliss compilation on zero-coupon bond yields during the period 1987-2002. The optimization algorithms include quasi-Newton, gradient descent, and Nelder-Mead schemes. A comparison of yield curves for unconstrained and arbitrage-free Nelson - Siegel models is shown in the movie at the top-left panel of the table below. The rest of the panels on the table show the yield to maturity surfaces of the Nelson - Siegel models for the period 1987-2002 and 16 maturities ranging from 3 months to 30 years.

Yield curves for all models on November 30, 2002.
yield curves
AFNS uncorrelated
Yield surface for the arbitrage-free Nelson Siegel (AFNS) uncorrelated model.
AFNS correlated
Yield surface for the arbitrage-free Nelson Siegel (AFNS) correlated model.
AFGNS
Yield surface for the arbitrage-free generalized Nelson Siegel (AFGNS) model.
DNSS
Yield surface for the dynamic Nelson Siegel Svensson (DNSS) model.
DNS uncorrelated
Yield surface for the dynamic Nelson Siegel (DNS) uncorrelated model.
DNS correlated
Yield surface for the dynamic Nelson Siegel (DNS) correlated model.
DGNS
Yield surface for the dynamic generalized Nelson Siegel (DGNS) model.


The major difficulty with the in-sample analysis is to capture the dynamics of the 15-, 20-, and 30-year maturities. The data indicate an average "concave downwards" structure for these maturities. A straightforward way to reproduce this trend is to introduce additional parameters to the state space, creating thus a new model that captures the trends of short and long maturities slightly better. However, the interpretation of the additional state parameters becomes more cumbersome, because they are unobserved and lack economic content.

The top-left panel of the table below shows the data time-series for 4 different maturities. The rest of the panels demonstrate the evolution in time for some of the state variables and models associated with them. In general, the model variables are divided in three categories: level, slope, and curvature of the yield curve. Every model has usually one level variable that determines the average "height" of the yield curve, and one or multiple slope and curvature variables that attribute the structure to the yield curve.

data time series
Yield data time series for 3-month, 2-year, 10-year and 30-year maturities.

DNS and AFNS model factors
First slope and curvature factors for the dynamic Nelson Siegel (DNS) and arbitrage-free Nelson Siegel (AFNS) models.
DNSS model yield surface.
DNSS and AFGNS model 1st slope curvature
First slope and curvature factors for the dynamic Nelson Siegel Svensson (DNSS), the dynamic generalized Nelson Siegel (DGNS), and the arbitrage-free generalized Nelson Siegel (AFGNS) models.

DNSS and AFGNS model 2nd slope curvature
Second slope and curvature factors for the dynamic Nelson Siegel Svensson (DNSS), the dynamic generalized Nelson Siegel (DGNS), and the arbitrage-free generalized Nelson Siegel (AFGNS) models.


Depending on the level of success in maximizing the Gaussian log-likelihood function, we may demonstrate for the in-sample estimation that arbitrage-free models are equally successful to models that do not obey this important condition. The out-of-sample prediction analysis for the aforementioned models is work in progress. Our efforts are focused on exploring the impact of a potential non-normality for the innovations. In general, the estimation of covariance matrices that may reasonably account for cases of non-normality involves the calculation of derivatives to the log-likelihood function and similar functions related to it. We currently use the outer product of gradients (OPG) estimator and the Hessian matrix combined, to form the White covariance as a "sandwich" operator. In addition, we are in the process of implementing the exponential family of distributions, as an alternative approach for non-normal innovations.

A visualization of the log-likelihood function along certain important variables (such as the parameters of the state transition matrix) may provide an insight on the slopes of the functions that need to be calculated. The plots below show changes to the log-likelihood function for varying certain model parameters, while fixing the rest of the parameters at their optimal values. The overlaid points indicate the parameter values that maximize the log-likelihood function. The dark planes on the surface plots indicate the boundary limits for the specific parameters. Beyond these limits the eigenvalues of the state transition matrix exceed the unit circle region, and the stability condition for the time-invariant Kalman filter is violated. The model shown is the dynamic Nelson Siegel (DNS) uncorrelated model.


logL line DNS uncorrelated level slope curvature
Log-likelihood as a function of the slope and curvature factor loadings for the dynamic Nelson Siegel (DNS) uncorrelated model. The dark dots indicate the optimal values along the specific dimension.

logL surface DNS uncorrelated slope curvature
Log-likelihood surface along the slope and curvature factor loadings for the dynamic Nelson Siegel (DNS) uncorrelated model. The magenta dot indicates the maximum log-likelihood value.
DNSS model yield surface.
logL surface DNS uncorrelated level slope
Log-likelihood surface along the level and slope factor loadings for the dynamic Nelson Siegel (DNS) uncorrelated model. The magenta dot indicates the maximum log-likelihood value.

logL surface DNS uncorrelated level curvature
Log-likelihood surface along the level and curvature factor loadings for the dynamic Nelson Siegel (DNS) uncorrelated model. The magenta dot indicates the maximum log-likelihood value.


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