Optimized Yield Curve Matching
Final Report
Presentation

Author:
Mark Davenport
mark.davenport@wustl.edu

Advisor:
Dan Scholz
Manager, Investment Strategies
NISA Investment Advisors, L.L.C.
Dan.Scholz@nisanet.com

Abstract

The strategy of immunizing a portfolio against changes to the yield curve is a very relevant topic in light of the recent sub prime crisis and its impact on interest rates.  In addressing this topic, this project will create a black box optimizing routine which will input any yield curve and output a portfolio of fixed-income securities based on an assigned objective function.  Using this tool, I investigate different objective functions which place emphasis on distinct periods along the investment horizon.  Monte Carlo simulations are employed to produce 50,000 hypothetical shifts of the inputted yield curve.  Tracking error will be measured to determine the performance of each objective function to these shifts, with results suggesting that the current routine is sub optimal for this simulated environment.  However, these results highlight a deeper problem of determining what it means for an objective function to truly be optimal.

Motivation 

Employee pension benefits provide flows of income to retired or disabled employees.  Using actuarial projections, a firm knows the estimated expected cost to all of these payments.  To manage these flows, sponsors of these pension benefits invest in collections of assets with the money they have set aside for the pension obligations.  These pools of assets are known as pension funds and are managed by professionals like NISA Investment Advisors, L.L.C. (NISA).  In managing these funds, the main objective is to ensure sufficient cash flows from the assets chosen to cover the liabilities for each year.

One of NISA’s primary management services is to devise investment strategies that minimize the amount of deviation between the change in value of the assets and liabilities from changes in the yield curve.  The goal of minimizing this risk will be the focus of my project. 

Financial Background

For the completion of this project, I had to master a number of financial topics.  First of all, I had to understand that for a shift in the yield curve, the price of a bond changes.  However, for different bonds, the change is not parallel.  Additionally, the duration of a bond is a sample statistic of a bond which serves as an approximation of the sensitivity of the price of a bond to changes in the yield.  Finally, these two concepts culminated into the theory of portfolio immunization.  This is the practice of minimizing the amount of interest rate risk to a portfolio.  This project explores a duration matching strategy.

Optimization Strategy

This project expanded upon the current optimization routine in place at NISA. I looked at alternative objective functions to explore different structures and weighting strategies.  This process was twofold.  I examined five objective functions and four different weighting strategies for a total of twenty different objective functions explored.  A more in depth description of this process can be found at the bottom of the page.  

The objective function structures explored were the following:

• Sum of squared sum of differences

• Sum of squared differences

• Two-period lagged function

• Five-period lagged function 

• Change in summation for ten year periods 

For each of these objective functions, I used four different weighting strategies.

• Implicit weighting

• Power short-term weighting

• Simple linear short-term weighting

• Simple linear long-term weighting

Simulation

In order to determine the performance of each objective function, I employed Monte Carlo simulations to generate yield curve shifts of the input yield curve seen in Figure 1. 

Figure 1 – Input Yield Curve

These shifts were generated for four different volatility environments, shown below in Figure 2:

• Moderate Long-Term 

• Moderate Short-Term

• Extreme

• NISA-generated

 
Figure 2 – Generated Volatility Environments

Results

The following figure demonstrates an example of an optimal portfolio allocation, with a graphical representation of contribution to durations of the assets and liability as well as the cumulative difference between the two.  This depicts the contribution to duration across the entire investment horizon for the optimal portfolio under the inputted yield curve.  Performance for each objective function was measured as the standard deviation of the differences between the change in value of the assets and liabilities to simulated shifts in the yield curve. 

Figure 3 – Sample Graphs Measuring Performance

The main findings from this study were as follows:

• The least squares and two period lagged functions are best for a short term volatility case

• The five period lagged and change in summation functions are best for extreme constant and long term volatility

• Short term weighting structures worked for the power weighted squared sum of differences function and two period lag

• Applying more weight to the long-term is very successful for long-term moderate volatility

Extensions

• Interface between Excel and Matlab to take advantage of the strength of Matlab’s nonlinear optimization routines.

• Use knowledge of the convexity of yield curves to construct objective function.

• Test the performance of objective functions for different inputs of yield curves.

• Extend this project to account for multiple asset classes.   

• Explore even stronger short term weighting strategies. 

Last updated by Mark Davenport on 3 December 2007

Washington University in St. Louis
School of Engineering and Applied Sciences
Department of Electrical and Systems Engineering