1. Luciano Vereda 1 Eduardo Bevilaqua 1 Ana Luiza Abrão 1 Adapting the Diebold and Li Methodology to Deal with Heteroskedasticity in Factor Dynamics 1 IAPUC (Institute for Financial and Actuarial Risk Manegement at PUC-Rio)

2. Introduction Why modeling the term structure of interest rates?  Forecasting the returns of traditional fixed income securities of various maturities, which are in the core of the portfolio of any pension insurance company;  Controlling the risks associated to investing in such assets (specially interest rate and reinvestment risk);

3. Introduction Why modeling the term structure of interest rates?  Calculating the discount factors that are necessary to mark to market assets and liabilities (procedure that is in the heart of solvency analysis);  Important input of any ALM system. These objectives naturally call for realistic econometric models, which should take into account all information available.

4. Introduction This paper builds on three main ideas… Expectations Hypothesis Long rates are risk-adjusted averages of future expected short rates → the former can help predict the later. Risk premium Average of expected future one period rates The yield curve as a whole conveys valuable information. 1

5. Introduction This paper builds on three main ideas… Diebold and Li (2006): trade-off between complexity (which increases the ability to describe observed dynamics) and simplicity (which usually improves forecasting performance). Reccomendation: summarize the information content of the term structure by modeling its driving forces. The Nelson and Siegel framework is suitable for this goal because the yield curve is represented by means of only three factors (level, slope and curvature). 2

6. Introduction This paper builds on three main ideas… Longstaff and Schwartz (1992), Christiansen and Lund (2002), Pérignon and Smith (2004), etc... 3 Financial time series behave in such a way that some periods are more volatile than others. Level effects, GARCH effects and regime shifts are required to adequately model interest rate volatility. These properties are transmitted to the factors that “explain” yield curve dynamics.

7. Introduction Question: is it worth adding these attributes to models which are aimed at forecasting? It is hard to say a priori… SimplicityExplanatory Power Perhaps the answer depends on the economy at hand. Emerging economies usually experience greater volatility levels → Probably these attributes are more important for them! The main purpose of this (working) paper is answering these questions.

8. Outline 1) Discussing the U. S. yield curve → show that interest rate volatilities vary over time; 2) Describing the Nelson and Siegel representation of the yield curve; 3) Discussing the factors → show that their volatilities vary over time; 4) Describing the Diebold and Li methodology; 5) Describing our first attempt to adapt the Diebold and Li methodology; 6) Analyzing the forecasting performance (for three different forecasting horizons) of our variant, comparing it with the performance achieved by the Diebold and Li proposal; 7) Some preliminary conclusions; 8) Future research.

9. The U.S. Term Structure Yields of zero-coupon bonds of several maturities (1, 3 and 6 months; 1, 2, 3, 5, 7, 10, 20 and 30 years); Source → Federal Reserve Economic Data (FRED), St. Louis FED. The yield curve is upward sloping on average; Rates are highly autocorrelated to their past values and to current and past values of other rates; The first order autocorrelation of squared rates is highly significant; Cross-section autocorrelation decreases with the distance between maturities; Their distribution cannot be considered normal (long right tails and excess kurtosis for short rates); Their variance decreases with maturity;

10. The U.S. Term Structure

11. Evidence of heteroskedasticity

12. The U.S. Term Structure

13. Evidence of volatility clustering

14. The Nelson and Siegel Framework 1st factor2nd factor 3rd factor y yy y + +

15. How Factors Look Like?

16. Evidence of heteroskedasticity Cross correlations can be important...

17. How Factors Look Like? Evidence of volatility clustering

18. The Diebold and Li Proposal

19. Our Variant of the Diebold and Li Proposal Mean zero, variance obbeys...

20. Competing Models Random walks: Autoregressive models : VAR models:

21. Empirical Procedure Observations were taken in a monthly frequency. The complete sample is comprised by data coming from September 1982 (after the so called “monetary experience”) until March 2007. Our procedure for examining out-of-sample forecasts is very conventional:  Estimate the models using observations from September 1982 until and including October 2003;  Calculate h -month-ahead forecasts ( h = 1, 6 and 12 months) of all yields.  Repeat the first step using observations from September 1982 until and including November 2003.  Stop when the estimation sample comprises data from July 1999 until March 2006. Observations that are not used during the estimation process are put apart in order to evaluate forecast errors. Estimations were made by applying the OLS technique. The criterion that we use to judge forecasting performance is the mean squared deviation between actual and forecasted rates.

22. Results

23. Final Remarks Results suggest that our very simple way of adding sthocastic volatility to the Diebold and Li proposal enhances forecasting performance for the set of horizons that was considered. This result is valid for our three proxies of short, medium and long term rates. This result holds true even for the U.S. economy, which is a very stable one.

24. Final Remarks It is possible to improve performance further... Refine the strategy adopted to model sthocastic volatility...  GARCH-M.  Volatility as a function of the magnitude of the factors.  Volatility as a function of macroeconomic variables. Testing if sthocastic volatility depends on the characteristics of the economy (emerging vs. developed countries).

25. Final Remarks GARCH-M The values assumed by the factors may depend on prevailing volatility levels.

26. Final Remarks GARCH with level effects white noise

27. Final Remarks Volatility depending on the state of the economy.