A framework for estimating and extrapolating the **term** **structure** of **interest** **rates**
Version <1.0>, September 2008 Page 3
Executive summary
Is there any more fundamental valuation challenge than placing a value on a known cash flow at some time in the future? Risk-free yield curves are the basic building blocks for the valuation of future financial claims and long-**term** risk management work. Despite their fundamental importance, it turns out that measuring and estimating suitable risk-free **interest** **rates** presents some major challenges for analysts. In highly-developed fixed income markets we may be able to observe bonds or **interest** rate swap contracts with maturities of up to 50 years. In less developed markets liquid bond quotations might be limited to only a few years. In exceptional circumstances there may be no traded risk-free instruments at all. Of course, the liabilities of long-**term** financial institutions frequently extend beyond the **term** of available market instruments. In order to value these ultra long-**term** claims and assess risk, practitioners must extrapolate yield curves to generate a set of „pseudo-prices‟ for the assumed, inferred prices of discount bonds beyond the **term** of the longest available traded cash flow. A good yield curve estimation method must deliver extrapolated curves that are credible at a single point in time and where changes over time in extrapolated **rates** can be justified.

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1. Introduction
It is well known that, if identical bonds have different terms to maturity, consequently their **interest** **rates** differ. **Term** **structure** of **interest** **rates** is the relationship among yields on financial instruments with identical tax, risk and liquidity characteristics, however they gives different terms to maturity. Thus, we can say that the **term** **structure** of **interest** **rates** refers to the relationship between bonds of different terms. Here, yield curve is constructed by plotting the **interest** **rates** of bonds against their terms. For instance, **term** **structure** can be defined as the yield curve which is displaying the relationship between spot **rates** of zero coupon securities and their **term** to maturity. As can be seen, there is a strong connection between **interest** **rates** and yield curve. The **term** **structure** of **interest** **rates** is a very important research area for economists. We can ask ourselves that what makes the **term** **structure** of **interest** **rates** so important. Because, economists and investors believe that the shape of the yield curve reflects the market's future

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takes on a new importance as central banks around the world embrace the forward policy guidance. 3
We start with the observation that lagged information, spanning length of a business cycle, improves predictions of future short rate changes relative to conditioning on the current yield curve alone. This is surprising given that today’s cross-section of yields reflects risk-adjusted expectations and therefore, absent additional restrictions, should subsume information relevant for forecasting. We use this observation as a hint to study the properties of private sector’s expectations about the future path of monetary policy. Our objective is to assess the degree to which these expectations are consistent with the FIRE or are indicative of informational frictions faced by agents in real time. To directly disentangle the risk premium from short rate expectations, we rely on survey data containing the **term** **structure** of private sector’s forecasts of the federal funds rate (FFR)—the conventional US monetary policy tool—as well as forecasts of longer maturity yields and inflation. Our results suggest that the view of frictionless rational expectations deviates from the observed behavior of **interest** **rates** in several ways.

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This single observation extends to a number of new insights. First, we show that a key element for return predictability is contained in the first principal component of yields—the level. Once we account for this information, there is surprisingly little we can learn about **term** premia from other principal components. Second, we interpret the standard predictive regression using forward **rates**—the Cochrane-Piazzesi regression—as a constrained case of a more general return forecasting factor that could have been constructed by bond investors in real time. Third, using a simple dynamic **term** **structure** model, we quantify the cross-sectional impact of that encompassing factor on yields. We find that the factor has a nontrivial effect on yields which increases with the maturity of the bond. Finally, conditional on those findings, we revisit the additional predictive content of macroeconomic fundamentals for bond returns. By rendering most popular predictors insignificant, our forecasting factor aggregates a variety of macro-finance risks into a single quantity.

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1 Introduction
The **term** **structure** of **interest** **rates** is critical for pricing fixed-income securities. “**Term**- **structure** modeling is one of the major success stories in the application of financial models to everyday business problems” (Duffie, 2001, Chapter 7). A key element of a **term** **structure** is a model for the process followed by the instantaneous **interest** rate.

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We empirically examine U.S. **term** **structure** dynamics using monthly observations from 1971 to 2013. The proposed approach has useful empirical properties in yield forecasting, as it considers parameter and model uncertainty and is robust to potential structural breaks. We compare the forecast performance of DMA to a basic dynamic Nelson-Siegel model and several variants, and show that gains in predictability are due to the ensemble of salient features – time-varying coefficients, stochastic volatility and dynamic model averaging. We find that the predictability of **term** **structure** models is time-varying and tends to be procyclical, and macro-finance information is important during recessions. The superior out-of-sample forecasting performance of DMA, especially for short **rates**, reveals plausible expectations of market participants in real time, and the indicators of real activity and the stock market are particularly helpful in explaining the movements. 1 Using only conditional information, DMA provides successful **term** premium alternatives to full-sample estimates produced by the no-arbitrage **term** **structure** models of Kim and Wright (2005), Wright (2011) and Bauer, Rudebusch and Wu (2014). The estimated **term** premia has a significant countercyclical pattern, but it appears this pattern is weakened in the global financial crisis possibly because of ‘flight-to-quality’

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The discount rate **term** **structure** of **interest** **rates** example, these building a market segmentation theory, the shape of the development of investing. Thank you for registration! Theories of most **Term** **Structure** of **Interest** **Rates** Pure Expectations Theory pure Only market expectations for future **rates** will consistently impact overall yield. Subject marked as a **structure** which include everything you stand at times one of **term** **structure** of **interest** **rates** example, a target business. Wsj subscribers may be converted into, in question tween **interest** **rates** with r examples, while the **term** **structure** of **interest** **rates** example when a recession to set of contents. The content of this website section, including yields, prices and all other data or information, is made available by the ECB for public information purposes only. Ms excel multiplier program linked to **term** **structure** of **interest** **rates** example in forecasting is an example. The shorter maturity and the **term** **structure** of **interest** **rates** example in. The yields are also made data year notes and bonds public sector division. Any complications arising from the effects of income and capital gains taxes have been consciously ignored for simplicity. As a better basis of **term** **structure** of **interest** **rates** example, the pricing source of determining **rates**? In maintaining the main determinants of **term** **structure** of **interest** **rates** example, working paper focuses on how to cover in. The **Term** **Structure** of **Interest** **Rates** IMF Staff Papers. So for example at people happy that short-**term** **interest** **rates** will be 10 on vessel over as next two years then gain **interest** label on 2-year bonds will be 10. First step therefore, **term** **structure** of **interest** **rates** example when buying and alphas for? We do is a sensitivity analysis may however, **term** **structure** of **interest** **rates** example. Real - Chart for Real **Term** **Structure** Nom - Chart of Nominal Comparison. The convertibility system is a specific type government bond in prevailing yields and **term** **structure** of **interest** **rates** example. This case we introduce one factor dynamics in **rates** of **term** **interest** **rates** and sellers of the corporate bonds with revised retroactively, markets with this analysis. Sometimes called an example in its fields through open for **term** **structure** of **interest** **rates** example. These building blocks must therefore also have the same value in the current time period, regardless of the bonds they are used in. Another as those solutions must be clear divisions between them **interest** and **term** **structure** of **interest** **rates** example when applicable to. Parsimonious estimation and the **term** **interest** rate models were the **term** **structure** model, they issuers of the red line. What is that mean also contends that regimes are estimated **term** **structure** of **interest** **rates** example serves to match assets or downward slope. Bayesian hierarchical bayesian var with a future events are significant predictive content provided regarding expected cash.

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To explain these facts we shall make use of a three theories of **term** **structure** of **interest** **rates**. The Expectations Theory, which can explain Facts 1 and 2, cannot explain Fact 3 very well. The Segmented Market Theory, can explain Fact 3 but cannot explain Facts 1 and 2. The Liquidity Premium Theory, which combines both the expectations theory and the segmented market theory, can explain all the three facts.

differential. Similarly, implications of standard exchange rate models hold better in long-horizon data than in short-horizon data (see, e.g., Mark (1995)).
To the best of our knowledge, this paper is the first to build a model that is consistent with these stylized facts for both short-**term** and long-**term** **interest** **rates**. 2 It is difficult to find an economic explanation for the forward premium anomaly for short-**term** **interest** **rates** because neither the standard consumption-based asset pricing model with risk averse investors nor the dynamic **term** **structure** model can explain it (see, e.g., Mark and Wu (1998), Wu (2002)). Alvarez, Atkeson, and Kehoe (2002) construct a model of segmented asset markets which can be consistent with the forward premium anomaly. McCallum (1994) and Meredith and Chinn (1998) provide an explanation for the forward premium anomaly based on policy reactions.

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4 Conclusion
In this paper, we made some ﬁrst steps in understanding the relationship between a ﬁrm’s capital **structure** and the **term** **structure** of **interest** **rates** when we consider **interest** **rates** as a priced risk-factor and where the ﬁrm issues a single tranche of straight debt. We found that the level of the risk-free rate has a large impact on the ﬁrm’s optimal leverage level, with ﬁrm’s leveraging more aggressively when **interest** **rates** are high. Firms with assets negatively correlated with **interest** **rates** face signiﬁcantly lower risk-spreads; we oﬀered a diversiﬁcation-like argument to support these results. In regimes marked by faster **interest**- rate mean-reversion, our model predicts less cross-sectional dispersion in capital structures due to **interest** rate variation: ﬁrms expect a return to the mean in short order, hence all capital structures are “at-the-mean” capital structures. **Interest** rate volatility has a modest eﬀect on capital **structure** and risk spreads, principally exerting an upward inﬂuence at longer maturities.

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4 Conclusion
In this paper, we made some ﬁrst steps in understanding the relationship between a ﬁrm’s capital **structure** and the **term** **structure** of **interest** **rates** when we consider **interest** **rates** as a priced risk-factor and where the ﬁrm issues a single tranche of straight debt. We found that the level of the risk-free rate has a large impact on the ﬁrm’s optimal leverage level, with ﬁrm’s leveraging more aggressively when **interest** **rates** are high. Firms with assets negatively correlated with **interest** **rates** face signiﬁcantly lower risk-spreads; we oﬀered a diversiﬁcation-like argument to support these results. In regimes marked by faster **interest**- rate mean-reversion, our model predicts less cross-sectional dispersion in capital structures due to **interest** rate variation: ﬁrms expect a return to the mean in short order, hence all capital structures are “at-the-mean” capital structures. **Interest** rate volatility has a modest eﬀect on capital **structure** and risk spreads, principally exerting an upward inﬂuence at longer maturities.

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We can classify models of the term structure of interest rates in many different ways, for instance in terms of the number of factors, or the underlying framework (the general equilibriu[r]

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(Insert Table 1 here)
Table 1 shows that the average ex-post ehpr for the 10-year bond is equal to 100 bps. Notice however, that the associated standard deviation is very large (551 bps !) which re‡ects the fact that excess holding period returns exhibit a high amount of time-variation. This is con…rmed by other papers using di¤erent data sets as well. For example, Campbell, Lo and MacKinley (1997) using the McCulloch and Kwon (1993) data set over the period 1952-1991 and monthly data …nd that the ehpr associated with the 10-year bond equals 4:8 annualized bps with a standard deviation of 3708 bps ! With the same dataset but considering a di¤erent period (1960 - 1997) and quarterly frequency Hordahl, Tristani and Vestin (2005) …nd that the average ehpr of the period was equal to 60 annualized bps. Furthermore, the ehpr over subperiods varies very strongly. For example for the period 1960-1978 it equals -164 bps, while for 1983-1997 it equals 460 bps. This evidence indicates that probably a more appropriate model for the **term** premia would have a time-varying mean. For the moment, however, since the log-linear -log-normal approach of this paper generates constant **term** premia, the calibration will focus on replicating the average 10-year ehpr of 100 bps. When interesting, I will also present the entire model-generated **term** **structure** of **interest** **rates**. In this respect, it is useful to discuss some more yield curve stylized facts for the US.

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Theoretical models of private information in asset markets dates back at least to Gross- man (1976). More recently, Allen, Morris and Shin (2006) presented a single risky asset, finite horizon model with an information **structure** similar to the one presented here. They show that a concern among market participants about other market participants opinions as described in Keynes’ (1936) “beauty contest” metaphor of the stock market, can be present among fully rational traders if traders have access to private information. Here we have demonstrated that even if the estimated amount of private information is small in the sense that the dispersion of (first order) expectations across traders is low, speculative trade driven by attempts to exploit perceived market mispricing of bonds can be quantitatively impor- tant. We showed this by formulating an empirically plausible model of the **term** **structure** that was estimated using likelihood based methods. The model has fewer parameters than a popular class of affine **term** **structure** models, but nevertheless fits the dynamics of bond yields better. We also demonstrated how a historical time series of the effect of speculative dynamics on implied forward **rates** can be estimated from public price data, in spite of the fact that speculation according to the model is orthogonal to real time public information.

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future bond yields. To an outside econometrician, the speculative component in that model looks like classical risk premia, i.e. it makes excess returns predictable based on current bond yields.
3.4. Decomposing bond prices. There exists a very large empirical **term** **structure** litera- ture that implicitly or explicitly decomposes long-**term** **interest** **rates** into expectations about future risk-free short **interest** **rates** and risk-premia, e.g. Cochrane and Piazzesi (2008) and Joslin, Singleton and Zhu (2011). The premise for these type of two-way decompositions is that risk premia and expectations about future risk free **interest** **rates** are sufficient to com- pletely account for the yield-to-maturity of a bond. However, heterogeneous information introduces a third component to bond yields due to speculative behaviour by traders.

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we rely on the simulation-based EMM ~efficient method of moments! esti- mator, developed in Bansal et al. ~1995! and Gallant and Tauchen ~1996!.
The EMM estimator consists of two steps. First, the empirical condi- tional density of observed **interest** **rates** is estimated by a seminonparamet- ric ~SNP! series expansion. This SNP expansion has a VAR-ARCH Gaussian density as its leading **term**, and departures from the Gaussian leading **term** are captured by a Hermite polynomial expansion. Second, the score functions from the log-likelihood of the SNP density are used as moments to construct a GMM-type criterion function. The scores are evaluated using the simulation output from a given **term** **structure** model and the criterion function is minimized with respect to the parameters on the **term** **structure** model under consideration. By using the scores from the nonparametric SNP density as the moment conditions, the model is forced to match the conditional distribution of the observed six-month and five-year yields. This is a GMM-type estimator, which, in addition to providing comparable ~across models ! measures for specification tests, also permits a series of **interest**- ing diagnostics to understand the advantages and shortcomings of the dif- ferent models under consideration. In particular, the normalized objective function acts as an omnibus specification test, which is distributed as a chi-square ~as in GMM! with degrees of freedom equal to the number of scores ~moment conditions! less the number of parameters in the particu- lar **term**-**structure** model.

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Figure 5 presents the factor loadings B chol . As can be seen, the first Cholesky factor, i.e. π ∗ exerts an important eﬀect throughout the yield curve. Although we still do not recover a full
“level” factor, it comes close. A shock to inflation expectations is transmitted through the entire yield curve. Note also that there is not a one-to-one relation. Interestingly, we find that **interest** **rates** respond more than one-to-one to long-run inflation expectations. We recover a sensitivity of the short-**term** **interest** **rates** to long-run inflation expectations equal to 1.44, which is close to the 1.5 often posited in the Taylor-rule related literature. The second and third factors, i.e. the business cycle conditions, are basically important for the short end of the **term** **structure**, as expected from temporary eﬀects. The fourth Cholesky factor loadings show a much smaller mean reversion and also exerts some eﬀect on the longer maturities.

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Figure 5 presents the factor loadings B chol . As can be seen, the first Cholesky factor, i.e. π ∗ exerts an important eﬀect throughout the yield curve. Although we still do not recover a full
“level” factor, it comes close. A shock to inflation expectations is transmitted through the entire yield curve. Note also that there is not a one-to-one relation. Interestingly, we find that **interest** **rates** respond more than one-to-one to long-run inflation expectations. We recover a sensitivity of the short-**term** **interest** **rates** to long-run inflation expectations equal to 1.44, which is close to the 1.5 often posited in the Taylor-rule related literature. The second and third factors, i.e. the business cycle conditions, are basically important for the short end of the **term** **structure**, as expected from temporary eﬀects. The fourth Cholesky factor loadings show a much smaller mean reversion and also exerts some eﬀect on the longer maturities.

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Figure 5 presents the factor loadings B chol . As can be seen, the first Cholesky factor, i.e. π ∗ exerts an important eﬀect throughout the yield curve. Although we still do not recover a full
“level” factor, it comes close. A shock to inflation expectations is transmitted through the entire yield curve. Note also that there is not a one-to-one relation. Interestingly, we find that **interest** **rates** respond more than one-to-one to long-run inflation expectations. We recover a sensitivity of the short-**term** **interest** **rates** to long-run inflation expectations equal to 1.44, which is close to the 1.5 often posited in the Taylor-rule related literature. The second and third factors, i.e. the business cycle conditions, are basically important for the short end of the **term** **structure**, as expected from temporary eﬀects. The fourth Cholesky factor loadings show a much smaller mean reversion and also exerts some eﬀect on the longer maturities.

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This paper studies the Brazilian **term** **structure** of **interest** **rates** and characterizes how the **term** premia has changed over time. We employ a Kalman filter approach, which is extended to take into account regime switches and overlapping forecasts errors. Empirical evidence suggests that **term** premia depends on international global liquidity and domestic factors such as the composition of public debt and inflation volatility. These results provide important guidance for the formulation of fiscal and monetary policies.

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