Yield Curve Modeling and Forecasting

1 Yield Curve Modeling and Forecasting :The Dynamic Nelson-Siegel ApproachFrancis X. DieboldUniversity of PennsylvaniaGlenn D. RudebuschFederal Reserve Bank of San FranciscoApril 29, 2012iiContentsPrefacexi1 Facts, factors , and Three Interest Rate curves .. Zero-Coupon Yields .. Yield Curve Facts .. Yield Curve factors .. Yield Curve Questions .. use factor models for yields? .. should bond Yield factors and factorloadings be constructed? .. imposition of no arbitrage useful? .. should term premiums be specified? . are Yield factors andmacroeconomic variables related?

2 Onward .. 202 Dynamic Curve Fitting .. Introducing Dynamics .. factorizations .. State-Space Representation .. and transition .. extractions and predictions .. Estimation .. Nelson-Siegel in cross section .. DNS .. DNS .. Multi-Country Modeling .. yields .. yields .. representation .. Risk Management .. DNS Fit and Forecasting .. 423 Arbitrage-Free A Two-Factor Warm-Up .. The Duffie-Kan Framework .. Making DNS Arbitrage-Free .. key result.

3 Yield -adjustment term .. the Bjork-Christensen-Filipovi c critiqueand the Yield -adjustment term .. Workhorse Models .. AFNS Restrictions onA0(3) .. Estimation .. AFNS Fit and Forecasting .. 754 Variations on the Basic Theme .. shrinkage .. factor loadings .. spreads .. Additional Yield factors .. factors :Dynamic Nelson-Siegel-Svensson (DNSS) . factors : arbitrage-freegeneralized Nelson-Siegel (AFGNS) .. Stochastic Volatility .. formulation .. formulation .. and unspanned volatility.

4 Macroeconomic Fundamentals .. DNS approaches .. DNS approaches .. macroeconomy and Yield factors .. nominal vs. real yields .. 1045 Macro-Finance Yield Curve Modeling .. Macro-Finance and AFNS .. expectations and risk .. and interbank lending rates .. Evolving Research Directions .. macroeconomic risks .. zero lower bound .. supply and the risk premium .. 1246 Is Imposition of No-Arbitrage Helpful? .. Is AFNS the Only TractableA0(3) Model? .. Is AFNS Special? .. simple structure facilitates special-izations, extensions, and varied uses.

5 Has strong approximation-theoreticmotivation .. restrictions are not rejected .. 134 Appendices137viCONTENTSA Two-Factor AFNS Risk-Neutral Probability .. Euler Equation .. 140B Details of AFNS Independent-Factor AFNS .. Correlated-Factor AFNS .. 147C The AFGNS Yield Adjustment Term151 Bibliography155 List of Bond Yields in Three Dimensions .. Bond Yields in Two Dimensions .. Bond Yield Principal Components .. Empirical Level, Slope, and Curvature, and FirstThree Principal Components, of Bond Yields .. DNS Factor Loadings.

6 Out-of-Sample Forecasting Performance: DNS Walk .. DNSS Factor Loadings .. DGNS Factor Loadings .. Nominal and Real Yields and BEI Rates .. BEI Rates and Expected Inflation .. Probabilities of Nonpositive Net Inflation .. LIBOR Spreads .. 122viiviiiLIST OF FIGURESList of Bond Yield Statistics .. Yield Spread Statistics .. Yield Principal Components Statistics .. AFNS Parameter Restrictions on the CanonicalA0(3) Model .. Out-of-Sample Forecasting Performance: Four DNSand AFNS Models .. Out-of-Sample Forecasting Performance: RandomWalk,A0(3), and AFNS indep.

7 78ixTo our wivesPrefaceUnderstanding the dynamic evolution of the Yield Curve isimportant for many tasks, including pricing financial assets andtheir derivatives, managing financial risk, allocating portfolios,structuring fiscal debt, conducting monetary policy , and valuingcapital goods. To investigate Yield Curve dynamics, researchershave produced a huge literature with a wide variety of mod-els. In our view it would be neither interesting nor desirable toproduce an extensive survey. Indeed our desire is precisely theopposite: we have worked hard to preserve the sharp focus ofour Econometric Institute and Tinbergen Institute (EITI) Lec-tures, delivered in Rotterdam in June 2010, on which this bookis sharp focus is driven by an important observation: mostyield Curve models tend to be either theoretically rigorous butempirically disappointing, or empirically successful but theo-retically lacking.

8 In contrast, we emphasize in this book twointimately-related extensions of the classic Yield Curve modelof Nelson and Siegel (1987). The first is a dynamized version,which we call dynamic Nelson-Siegel (DNS). The second takesDNS and makes it arbitrage-free; we call it arbitrage-free Nel-son Siegel (AFNS). Indeed the two models are just slightly dif-ferent implementations of a single, unified approach to dynamicyield Curve Modeling and Forecasting . DNS has been highly suc-cessful empirically and can easily be made arbitrage-free ( ,xixiiPREFACE converted to AFNS) if and when that is intended audience is all those concerned with bond mar-kets and their links to the macroeconomy, whether researchers,practitioners or students.

9 It spans academic economics and fi-nance, central banks and NGOs, government, and industry. Ourmethods are of special relevance for those interested in assetpricing, portfolio allocation , and risk use this book, just as we used the EITI Lectures, as anopportunity to step back from the signposts of individual journalarticles and assess the broader landscape where we ve been,where we are, and where we re going as regards the whats andwhys and hows of Yield Curve Modeling , all through a DNS methods and framework have strong grounding in the bestof the past, yet simultaneously they are very much intertwinedwith the current research frontier and actively helping to pushit begin with an overview of Yield Curve facts and quicklymove to the key fact: Beneath the high-dimensional set of ob-served yields, and guiding their evolution, is a much lower-dimensional set of Yield factors .

10 We then motivate DNS as apowerful approximation to that dynamic factor structure. Wetreat DNS Yield Curve Modeling in a variety of contexts, em-phasizing both descriptive aspects (in-sample fit, out-of-sampleforecasting, etc.) and efficient-markets aspects (imposition ofabsence of arbitrage, whether and where one wouldwantto im-pose absence of arbitrage, etc.). We devote special attentionto the links between the Yield Curve and macroeconomic are pleased to have participated in the DNS researchprogram with talented co-authors who have taught us muchenroute: Boragan Aruoba, Lei Ji, Canlin Li, Jens Christensen,Jose Lopez, Monika Piazzesi, Eric Swanson, Tao Wu, and VivianYue.