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We discuss the Taylor rule near low inflation and interest rates. Using an additional option-like term in the Federal Reserve's loss function (i.e., the “deflation put”) we extend the classic Taylor rule to one with an asymmetric response that is more accommodative when the inflation rate is very low. Once calibrated, this payoff profile gives an exact, and easily communicable prescription for Federal Reserve policy under regimes of low inflation. Simple models of central bank behavior can produce highly complex yield curve shapes. Using the usual Taylor rule and our proposed extension as building blocks, we construct a robust framework for generating realistic yield curves and the evolution of the economy. Our main focus is the impact on the yield curve and the economy of the “deflation put”. We find that for economies like the U.S. the deflation put reduces yields for all maturities. We also find that in highly leveraged economies (such as Japan) the consequence of an asymmetric deflation fighting policy may result in improved economic conditions, but also raises the possibility of higher long-term yields as a consequence.

Traditional models for the yield curve are based on essentially an arbitrage-free statistical approach. After assuming reasonable stochastic processes for the key underlying variables, such models are calibrated using a combination of historical and cross-sectional fits to observed yield curves. While these models provide an elegant and computationally effective way to explain the yield curve, as well as tools to exploit relative value opportunities, they lack in deeper economic underpinnings. On the other hand, pure macroeconomic models attempt to explain the behavior of the yield curve based on macroeconomic aggregates such as GDP, inflation, etc. In doing so, they do not use the information implicit in the prices of traded securities that make up the yield curve, and the forecasts that they make for individual yields are not related by arbitrage constraints. In this paper, we combine the macroeconomic approach and the arbitrage-free approach by going back to the fundamental building block – the short rate, and the behavior of the Central Bank, which drives the short rate via the so-called Taylor rule. In this setting, the short rate is driven by a set of policy rules that link macroeconomic variables and the short rate together. By stitching together the short rate for future periods, we are then able to compute the yield for any maturity. The focus of our paper is to explore the behavior of the yield curve as a consequence of possible asymmetries in the monetary policy rule. This question is especially relevant now, since the current financial crisis has forced central banks to significantly reduce rates. At Jackson Hole in 2003, Federal Reserve Governor Janet Yellen (2003) elegantly stated the need for pre-emptive, asymmetric response when nominal rates and inflation are very low. She argued for a non-linear policy that would call for a central bank to lower its interest rate more rapidly toward zero and hold it at a low level for longer than the classic Taylor (1993) rule would suggest: The research shows that it is important to have a “cushion” in the inflation target to minimize the deterioration in macroeconomic performance due to the “zero bound” problem. For the United States, research suggests that the cushion of at least 1 percent (on top of measurement bias) is needed to avoid significant deterioration in macroeconomic performance, while a 2 percent cushion virtually eradicates economic problems relating to the influence of the zero bound. A larger inflation buffer becomes especially desirable if there is good reason to think that a “neutral real fed funds rate” is particularly low, as it might be in a post bubble economy, so that the odds of hitting the zero bound are high (Janet Yellen, Jackson Hole Meeting, 2003). It has also been pointed out that while linear response rules such as the Taylor rule have worked very well, the Federal Reserve's “risk-management” approach was essential to countering large, negative shocks posing serious asymmetric risks (Yellen, 2003). It is generally accepted that the Federal Reserve has done a very good job of dynamically updating its targets for the structural constants in the Taylor rule – such as the equilibrium real rate and the target inflation rate. But most empirical estimates assume the symmetry of the Taylor rule, and use the simple linear Taylor rule that is obtained by minimizing a quadratic loss function of inflation and output gaps and an inertial term in interest rate changes. The present crisis provides an example of this risk-management approach. During the early phases, the standard Taylor rule called for a Fed funds rate significantly above what the Fed chose to target. Inflation and output had not yet declined significantly, but the Fed estimated that several negative shocks due to the crisis required an asymmetric response. Now nominal funds rates in the U.S. have reached the zero bound, and the Fed has begun to apply other easing techniques. In fact, the standard Taylor rule would have the Fed funds rate significantly negative if that were possible. While it remains to be seen whether the zero bound will be a major problem in this situation, it is clear that as Federal Reserve Chairman Ben Bernanke has remarked “…policymakers are well advised to act pre-emptively and aggressively to a void facing the complications raised by the zero lower bound.” (Bernanke and Reinhart, 2004). Putting the views of Janet Yellen and Ben Bernanke together, it appears logical to explore the deflation avoidance problem with a modification of the Taylor rule that is both pre-emptive and asymmetric. At Jackson Hole in August of 2005, Alan Greenspan explicitly admitted the risk-management nature of policy under his Federal Reserve leadership (Greenspan, 2005): In effect, we strive to construct a spectrum of forecasts from which, at least conceptually, specific policy action is determined through the tradeoffs implied by a loss function. In the summer of 2003, for example, the Federal Open Market Committee viewed as very small the probability that the then-gradual decline in inflation would accelerate into a more consequential deflation. But because the implications for the economy were so dire should that scenario play out, we chose to counter it with unusually low interest rates. The product of a low-probability event and a potentially severe outcome was judged a more serious threat to economic performance than the higher inflation that might ensue in the more probable scenario. Moreover, the risk of a sizable jump in inflation seemed limited at the time, largely because increased productivity growth was resulting in only modest advances in unit labor costs and because heightened competition, driven by globalization, was limiting employers' ability to pass through those cost increases into prices. Given the potentially severe consequences of deflation, the expected benefits of the unusual policy action were judged to outweigh its expected costs (Alan Greenspan, Jackson Hole Meeting, 2005). Under Alan Greenspan's oft-quoted approach of “risk-management” for monetary policy, we are tempted to cast the problem in terms of a monetary policy “put option” (the put option being one where the underlying is inflation rate, and the strike the threshold at which policy becomes asymmetric, e.g. the 1% level of inflation in Janet Yellen's speech). Options markets are well known to provide insurance for risk-managers who are willing to part with some premium to avert the consequences of disastrous yet severe low-probability events. In this paper we show that such a dynamic extension of the Taylor rule can be derived from first principles using an intuitively appealing modification of the usual quadratic central bank loss function. The modified loss function contains an additional term analogous to the price of a put option on future inflation and it is derived using some simple assumptions in Section 3 of this paper. Numerous variations of the Taylor rule have been proposed in the literature to account for country specific monetary policy dynamics (Medina and Valdes, 2002, Siklos et al., 2004 and Plantier and Scrimgeour, 2002). Also, various mechanisms for escaping the zero bound have been proposed (Eggertson and Woodford, 2003, Svensson, 2001 and Benhabib et al., 2001). Econometric estimates have demonstrated that structural shifts in the coefficients of the Taylor rule have occurred in the past, coinciding with the change in the Federal Reserve chairmanship (Clarida et al., 2000 and Judd and Rudebusch, 1998). More recently, it has argued that because of the so-called “Greenspan put”, market participants can benefit disproportionately by availing themselves of cheap insurance embedded in the yield curve (McCulley, 2005). A major goal of this paper is to explore the consequences of our proposed modification of the Taylor rule on the yield curve and the development of the economy. Considerable effort has recently been devoted to including macroeconomic information in term structure modeling. Much of this work has focused on improving our understanding of the economic determinants of the level and shape of the yield curve. Whereas some research has included the effects of macro variables without including macroeconomic structure (Ang and Piazzesi, 2000, Ang et al., 2003, Wu, 2001, Dewachter and Lyrio, 2002, Kozicki and Tinsley,, Piazzesi, 2003, Diebold et al., 2003, Evans and Marshall, 2001, Rudebusch and Wu, 2003 and Diebold et al., 2005), we take a more macroeconomical approach and introduce a simple model of the economy involving the short-term interest rate, inflation, and output gap. This approach is similar to Rudebusch (2002); Hördahl et al. (2002). We us a model for the economy where output gap and inflation evolve forward in time via coupled random walks roughly corresponding to the Phillips curve and the IS equations. Regressions on historical data for inflation, output gap, and the Fed funds rate are used to determine the parameters of these equations. We assume a Federal Reserve policy that chooses the short rate in terms of inflation and the output gap. The solution to the resulting set of equations provides the stochastic evolution of the interest rate for all future times and may be used to compute risk-free yields for any maturity by computing expectations. We consider two different Federal Reserve policies. One is the standard Taylor rule, which gives a simple linear relation between the Fed funds rate, output gap, and inflation. The parameters in the standard Taylor rule are fit to historical data. This Treasury yield model is similar in spirit to some of the recent literature (Rudebusch and Wu, 2003). The other is our proposed modification of the Taylor rule, which has the Federal Reserve responding more aggressively when inflation is unusually low. We find that for the U.S. economy this “deflationary put” decreases longer-term treasury yields for most choices of parameters (note that it always decreases the short rate, given the same inflation and output gap). We also consider an economy where inflation and output gap are more sensitive to the policy rate (i.e., a “leveraged” economy). In such a case the deflationary put term can increase long-term yields. Furthermore, numerical simulations indicate that during periods in which the standard Taylor rule would lead to negative inflation and output gap, the deflationary put term increases these quantities. The difference between the yield curves with the same underlying dynamics points out the importance of estimating the form and correct coefficients for the Taylor rule used by a policy setter. The methods we develop can be extended to the case where the Federal Reserve also responds more aggressively than the Taylor rule to unusually large inflation, i.e., “an inflation call”. We derive a loss function that gives rise to a modified Taylor rule with an inflation call and briefly explore its impact on treasury yields. In the next section, we detail our model for the economy and review the derivation of the standard Taylor rule by minimizing a loss function that is quadratic in inflation and output gaps and contains an inertial term in interest rate changes. The modified Taylor rule is introduced in Section 3. Section 4 details the fits to the parameters from historical data, and in Section 5 we use these results to calculate the yield on Treasury bonds, both for the standard and modified Taylor rules. Section 6 describes the highly leveraged economy and demonstrates the benefits of the deflationary put. In Section 7 we return to the U.S. economy and discuss the impact of an inflation call on treasury yields. Brief concluding remarks are given in Section 8.

We derived a modification of the standard Taylor rule appropriate to a deflation averse Federal Reserve by adding to the loss function an additional term that resembles the payoff on a put option. The modification of the Taylor rule corresponds to a Federal Reserve that is more accommodative when inflation drops below a critically low value than what the standard Taylor rule would imply. Under the assumption that the usual quadratic loss function is supplemented by a term that only contributes for inflation below a critical value, we gave a general argument for its form. Using a simple model of the economy and Taylor-style policy rules for the short rate we calculated Treasury yields. Our macroeconomic model includes the inflationary pressure of a positive output gap and the tendency of positive real short rates to decrease output gap. These standard macroeconomic facts combined with stabilizing mean reversion terms are the basic ingredients of our economy. Taylor rules giving the short rate in terms of output gap and inflation then lead to a complete model for the short rate. The parameters of this model are estimated from historical data from 1954 to 2005. Numerical simulations of our model produce yield curves for a wide range of initial conditions and evolution volatilities. The main purpose of this paper is to explore the implications of an alternative Taylor rule that includes the effect of a “deflationary put option” to manage the risk associated with sustained low inflation. We consider its effects in two different economies, one similar to the U.S. economy before the current crisis and one that responds even more quickly to central bank interest rate changes and has a high risk of deflation. This may be relevant for the U.S. economy during the crisis. In the first economy, we show that the effect of this modified rule on yields can be significant, depending on the current values of inflation and output gap. For some reasonable initial conditions, the modified yield curve can be as much as 50 basis points below the standard one. For the second type of economy, we show that the benefits of the modified Taylor rule can be significant. It does a much better job than the standard rule of warding off deflation, and it keeps output gap closer to zero than it otherwise would be. We also show that in this economy the modified rule can push long-term yields above what the standard Taylor rule would give. This is simply the result of the effectiveness of the modified Taylor rule: it stimulates the economy and therefore allows the future short rate to be higher than with the standard rule. We would like to thank Vadim Yasnov for help with the simulations and Richard Clarida and Paul McCulley for numerous enlightening discussions. We would also like to thank Brian Sack for helpful comments. This article contains the current opinions of the authors and not necessarily Pacific Investment Management Company LLC. These opinions are subject to change without notice. This article is distributed for educational purposes only and does not represent a recommendation of any particular security, strategy or investment product. Information contained herein has been obtained from sources believed reliable, but not guaranteed. No part of this presentation may be reproduced in any form, or referred to in any other publication, without express written permission. Pacific Investment Management Company LLC, 840 Newport Center Drive, Newport Beach, CA 92660, 2005, PIMCO.