Optimized Yield Curve Matching Mark Davenport 3 December 2007

Outline Motivation Financial Background Black Box Optimizer Alternative Objective Functions Simulations Results Conclusion

Motivation Employee pension benefits provide flows of income to retired or disable employees  Pension funds provide the necessary cash flows to cover these pension obligations The goal of managing these funds is to ensure sufficient cash flows  The focus of this project will be how to best manage the risks associated with attaining this goal

Financial Background Price-yield relationship of a bond  Shifts in the yield curve cause PV of bond to change Duration  Measure of sensitivity of the price of a bond to changes in interest rate Portfolio Immunization  Practice of minimizing amount of interest rate risk to a portfolio  This project explores a duration matching strategy

Black Box Optimizer Optimization routine using Excel Solver  Allocates plan across grouping of assets Initial point  Random proportion of plan size assigned to each bond  Total amount allocated equals plan size Test of assumption  6 Yield Curves, 3 random starting points  Average difference between points $28,969 – trivial considering over $1 billion plan size

Optimization Routine C: difference in contribution to duration T: total plan size B: amount invested D: duration w: maximum relative $ duration v, u: minimum amount invested

Alternative Objective Functions Twofold Process  Five objective function structures  Four weighting strategies  Total of 20 objective functions tested Benchmark Objective Function  NISA’s Current Strategy Subtle implicit weighting Questionable:  Not enough emphasis on short term  Single year diluted across entire investment horizon

Alternative Objective Functions

Weighting Schemes Implicit Weighting  Apply weight = 1 for each observation Simple Short-Term Weighting  Apply weight = 1- Power Weighting  Apply power of six to first ten years, power of four to the next twenty years, power of two for the remaining Simple Weight-to-End  Apply weight =

Simulation  As input, used single upward sloping yield curve Created portfolio for each objective function

Volatility Three volatility environments  Moderate  Extreme  NISA

Moderate Volatility Two shift scenarios  Short-Term & Long-Term For each, 3 types of shifts  100 basis point shift  200 basis point shift  50 basis point bend

Extreme Volatility Examined Short-Term, Long-Term, and Constant volatility scenarios

NISA Volatility Generated in-house at NISA Assuming to be closest proxy for true Yield Curve volatility Used as benchmark for my volatility

Determining Results Results based off of 50,000 simulated yield curve shifts observed for each objective function Difference in PV of assets and liabilities recorded Standard deviation of differences determined the tracking error for each environment  Reported as raw number and in % of liability

Results – Objective Functions Best for Short-Term Volatility in all environments  Least Squares Method  Two-Period Lagged Method (only power-weighted and simple weight-to-end strategies) Best for Constant and Long-Term Extreme Volatility  Five Period Lagged Method  Change in Summation Method

Results – Weighting Strategies No impact on Least Squares Method Short-Term Weighting Strategy in general  Encouraging results for power weighted benchmark and two-period lagged methods  In general, best for extreme short-term and extreme constant volatility

Results – Weighting Strategies Long-Term Weighting  Not a well designed test because cash flows resulting past 40 year mark are essentially zero  However, still positive results Simple weight-to-end best for moderate long-term volatility, also for moderate short-term volatility Oddly, generated better results as compared to short-term weighting strategy

Discussion Least Squares and Two-Period Lagged (power weighted and simple weight-to-end) Methods

Discussion Five-Period Lagged and Change in Summation Methods

Discussion – Which is “best”? Different yield curve environments require different objectives Similarly, different money managing styles require different objectives  Reallocate once a day? Month? Decade? “Best” objective?  Moderate short term volatility most realistic Least Squares or Two-Period Lagged

Extensions Interface Excel and Matlab  Take advantage of strength of Matlab’s NLP capability Introduce knowledge of convexity of assets and liability Explore stronger short term weighting strategies Introduce more asset classes into the problem  Become more interesting  Eliminate skewing effect of large 30 year cash flow Explore different inputs

Accomplishments Optimizer can identify global minimum Simulated results determined better optimization routine  Given the constraints of the problem Demonstrated importance of design in solving this problem  Knowledge of external factors

Questions?