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 timeinvariant components for the state vector. There are two types of Nelson  Siegel models, the arbitragefree and the dynamic ones that do not impose a "noarbitrage" 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 loglikelihood function to estimate the optimal model parameters. We benchmark various optimization algorithms to determine the timeinvariant state space parameters that best represent the data trends. The current data set that we use is the Fama  Bliss compilation on zerocoupon bond yields during the period 19872002. The optimization algorithms include quasiNewton, gradient descent, and NelderMead schemes. A comparison of yield curves for unconstrained and arbitragefree Nelson  Siegel models is shown in the movie at the topleft 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 19872002 and 16 maturities ranging from 3 months to 30 years.


The major difficulty with the insample analysis is to capture the dynamics of the 15, 20, and 30year 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 topleft panel of the table below shows the data timeseries 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.



Depending on the level of success in maximizing the Gaussian loglikelihood function, we may demonstrate for the insample estimation that arbitragefree models are equally successful to models that do not obey this important condition. The outofsample prediction analysis for the aforementioned models is work in progress. Our efforts are focused on exploring the impact of a potential nonnormality for the innovations. In general, the estimation of covariance matrices that may reasonably account for cases of nonnormality involves the calculation of derivatives to the loglikelihood 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 nonnormal innovations.
A visualization of the loglikelihood 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 loglikelihood 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 loglikelihood 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 timeinvariant Kalman filter is violated. The model shown is the dynamic Nelson Siegel (DNS) uncorrelated model.


