• Solve for the appropriate forward rate that give null NPV to the given swap.. QuantLib: forward curve[r]

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Consistent **interest**-**rate** curve
We need a consistent **interest**-**rate** curve in order to
• Understand the current market conditions (e.g. forward rates) • Compute the at-the-money strikes for Caps, Floor, and Swaptions • Compute the NPV of exotic derivatives

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changes perceptions of economic fundamentals and affects rates at different horizons. Specifically, changes in the **term** **structure** show whether policy actions directly affect distant forward rates or whether the effects die after medium horizons.
An important difference exists between macroeconomic news and monetary policy news. We can quantify economic surprises, but we do not have a measure that satisfactorily captures all aspects of policy surprises. The language of FOMC statements and speeches cannot easily be quantified. To capture the surprise component, researchers have focused on how policy actions affect financial markets (see Kuttner 2001). Thus, to assess the effects of policy actions, I employ the same model-based methodology I used with employment and inflation news, examining changes across the entire **term** **structure** of **interest** rates as policy surprises were made public.

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This Table provides the empirical distributions of various economic statistics for the UIRP regressions under the null
that the exchange **rate** process and the short **rate** processes are random walks with a drift. IMPLIED refers to the implied
regression slope coeÆcients. CORR refers to the correlation statistic. VR refers to the variance ratio statistic. SD refers
to the standard deviation of the risk premium. EVR refers to the Fama excess variance ratio statistic. \Data" refers to

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factor model which can be interpreted as a random volatility specification because the volatility of the instantaneous **interest** **rate** is a function of the two factors. They obtain closed-form formulas for discount bond options which depend on investors' preferences. Chen (1994) develops a three-factor model of the **term** **structure** of **interest** rates. In this model the current short **rate**, the short **term** mean and the current volatility of the short **rate** follow a square-root process. Chen obtains a general formula for valuing **interest** **rate** derivatives that requires the computation of high-dimensional integrals. Two-factor models developed, for instance, by Richard (1978) who argues that the instantaneous **interest** **rate** is the sum of the real **rate** of **interest** and the inflation **rate**. Chen and Scott (1993) decompose the instantaneous **interest** **rate** into two unspecified factors each of which follows a square root process. A common characteristic of these two models is that there is little theoretical support to the choice of the factors 1 . Note that the models of the first approach mentioned above are all preference-dependent. In the framework of the second approach, we derive simple formulas for **interest** **rate** contingent claims.

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It follows that any empirical test of the relationship between the exchange rate and the term structure of interest rates necessitates the utilisation of a measure of the yield curve wh[r]

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United Kingdom and United States. However, United Kingdom used the Svensson model from January of 1982 to April of 1998.
The purpose of the paper is to obtain the **term** structures of **interest** rates with a theNS model to analyze the **term** **structure** of **interest** **rate** of the last decade. The objective is to obtain the parameters (level, slope and curvature) of the model and compare the evolution of theses curves in the Monetary Union. The study include a period of thirteen years, from 1992 to 2004 and we have analyzed the evolution of these **term** structures in six different countries: Spain, France, Germany, Italy, United Kingdom and United States. The first four countries are members of the European Monetary Union (EMU). Germany, France and Italy already participated in the creation of the European Monetary System (EMS). Although Spain didn’t adhere to the EMS up to 198686, it is interesting to see that, as Italy, from an economic situation very different from France and Germany, it was able to reach the approaches settled down by Maastricht. Moreover, both Italy and Spain are able to be part of the Economic and Monetary Union on first ofst January in 1999. The evolution of Germany, France, Italy and Spain allows us to analyze the process of convergence of the single currency countries versus United Kingdom and United States, which are a reference to contrast the differences.

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Theoretically, the expectation theory argues that the shape of the yield can be explained by investors’ expectations about future **interest** rates. The liquidity preference theory states that short **term** bonds are more desirable than long **term** bonds because former are more liquid. The preferred habitat theory elucidates the shape of the **term** **structure** by the assumptions that if an investor is risk averse and such investor can draw out of his preferred habitats only with the promise of a higher yield while market segmentation theory assumes that there are two distinct markets for the short and long **term** bonds. The demand and supply in the long **term** bond market determines the long **term** yield while short **rate** is determined in the short **term** bond market by the forces of demand and supply. This means that the expected future rates have little to do with the shape of the yield curve. Basically, the factors that affect terms of **structure** of **interest** **rate** include the monetary policy, the fiscal policy, taxation and inflation. The monetary policy is used by the government to control the supply of money in the economy. When supply of money in the economy is low then the **interest** rates are expected to be high and vice versa while volatility in money supply growth may lead to higher **interest** rates. Under the fiscal policy, the government hypothetically finance all expenditure for the economy. In cases of budget deficit, the government is forced to borrow from the local markets. This in turn affects the supply of money in the economy which in turn affects the trend of **interest** rates (Olweny, 2011).

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Keywords: **term** **structure** decomposition, optimization, market efficiency 1. Introduction
The use of forward **interest** rates has long been standard in financial analysis, for instance in pricing new financial instruments and in discovering arbitrage possibilities (Svensson, 1994). Bolder and Gusba (2002) note the fundamental aspect and importance of risk-free **interest** rates: In the world of fixed-income, it is difficult to find a more fundamental object than a riskless pure discount bond or, as it is equivalently called, a zero-coupon bond. This is because the price of a pure discount bond represents the current value of one currency paid with complete certainty at some future point in time. Abstracting from the idea of risk premia for longer-**term** bond holdings, it is essentially a representation of the time value of money. A trivial transformation of the bond price is the **rate** of return on this simple instrument or, as it is more commonly termed, the zero-coupon **interest** **rate**. These building blocks of fixed-income finance are tremendously important for a wide array of different purposes, including bond pricing, discounting future cash flows, pricing fixed- income derivative products, constructing forward **interest** rates, and determining risk premia associated with holding bonds of different maturities.

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In chapter 5 we show the Duffle and Kan model is unlikely to be used much in practice because it is exceedingly difficult to specify parameters for the model such that the state variable[r]

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"The Information Content of the Term Structure of Interest Rates: Theory and Practice"', Paper presented at Money Study Group Annual Conference, September 1989.. "The Inter[r]

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We interpret the observed behavior of five-year **interest** rates as the product of short-**term** liquidity effects. This conclusion is based on several findings. First, the predicted relationship between forward rates and spot rates does not persist beyond a few weeks, nor can it be profitably exploited in a systematic way. Both results suggest that short-**term** liquidity forces rather than economic fundamentals are likely to be driving the results. In addition, and in contrast to the behavior of medium-maturity rates, shorter maturity **interest** rates show no evidence of such feedback effects. The ample liquidity of the markets for short-**term** **interest** **rate** products, where market turnover is large relative to hedging demands, makes them an unlikely site for any evidence of positive-feedback effects. Finally, forward rates predict spot rates in the medium-**term** segment of the yield curve only in the weeks when **rate** changes are relatively large. This finding is also consistent with liquidity effects, since large **interest** **rate** changes cause large adjustments to options hedges, which in turn induce trading flows that will be large relative to normal market turnover.

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1. Introduction
As is well known, an **interest** **rate** process describes the profitability of a financial instrument, such as stock, bond or option. Hence if the price change is given by the sequence X = (X n ) n≥0 , then the **interest** **rate** process has in the simplest case the form

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JEL classiﬁcation: C53; E43; E47; G17
Keywords: **Interest** **rate**; Forecast model; Combined forecast Resumo
Problemas como quebras estruturais e vieses causados por má-especificac¸ão dificultam achar um modelo de previsão de estrutura a termo de taxa de juros que domine todos os competidores. Esse artigo tem como objetivo identificar a existência de métodos de combinac¸ão que produzam resultados de previsão superiores a modelos individuais no caso Brasileiro. Resultados empíricos confirmam que não é possível determinar um modelo individual que consistentemente produza previsões superiores. Além disso, o desempenho desses modelos varia temporalmente. Os problemas encontrados nos modelos individuais podem ser reduzidos aplicando esquemas de combinac¸ão de previsão. Os resultados mostram consistentemente ganhos de previsão nos esquemas de combinac¸ão para o período considerado. Em particular, quanto maior o horizonte de previsão, maior a contribuic¸ão do esquema.

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This figure shows that ex post return data are at less than zero percent at the beginning of the implementation of the monetary policies, and thus the zero **interest** **rate** and quantitative easing policies were initially perceived as credible and were expected to last for some considerable time. However, this phenomenon changes over time. Apparently, there is usually an increasing trend in the ex post short-**term** rates during low **interest** **rate** periods, but this trend was absent during the quantitative easing policy. This indicates that, while it is statistically insignificant, some investors anticipated a change in the zero **interest** **rate** policy. This finding is consistent with Marumo et al. (2003), who calculate the probability of the zero **interest** **rate** policy being removed, and conclude that after August 2000 a shift occurred in the distribu- tion of expectations; investors indeed had anticipated a policy change. In contrast, the relatively constant ex post **rate** during the quantitative easing policy would suggest that this measure was expected to last some time. This latter observation is generally consistent with previous research (Okina and Shiratsuka [2004]).

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The absence of arbitrage says that it is not possible to design a risk-free self-financing portfolio that yields more than the instantaneously return of the risk-free (short) **rate** within a time interval. Expected excess returns, then, are the result of explicit risk- taking. This means that arbitrage opportunities exists unless long-**term** bond yields are equal to risk-adjusted expectations of future short-**term** yields. The assumption of ab- sence of arbitrage opportunities seems quite logical in bond markets in which arbitrage opportunities are traded away immediately and markets can be characterized as highly liquid. The so called affine dynamic **term** **structure** models (ATSM) are the most popular among the class of no-arbitrage **term** **structure** models. They are best tractable since they assume bond yields to be affine functions of a set of risk factors driving the whole yield curve. They enable to get closed-form solutions for **interest** rates and such models are maximally flexible to reproduce the moments of bond yields and excess returns. The pioneering work by Vasicek (1977) and Cox et al. (1985) consists of a particular simple form of an affine **term** **structure** model where the short-**term** **interest** **rate** is the single factor that drives the whole yield curve at one moment in time and where it describes comovements of bond yields of different maturities.

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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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After the first question, he states another one: “Even if these two episodes can be seen as reflecting similar correlations between the bond **rate** and the short **rate**, is there any reason to expect the correlation to be stable in the future?” The answer for this question is no. In the low-inflation 1950 and 1969 period, long rates are varied relatively little with short rates. Then inflation expectations were anchored securely, and the range in which the Fed varied short rates to stabilize the economy was smaller in this period than it was in the 1970s, ’80s, and ’90s. Here, if the Fed succeeds in getting full credibility for low inflation in these years, short and long rates should once again co-vary which was in the earlier period. Thus, the late 1980s and mid-1990s can be seen as a transition period because short and long rates continued to exhibit the kind of covariation that observed in the period of high inflation.

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In Table 3, we show the banks’ estimated earnings from **term** transformation normal- ized to total assets (the ratio T M i ( t ) as deﬁned in Equation (13)). We give the results for the median bank and we break down the results into banking groups and years.
Over the whole period 2005-2009 and over all banking groups, the median bank earned 26.3 basis points (in relation to total assets and per annum). There are, however, large diﬀerences across the years and across the banking groups. In 2005, when **term** transfor- mation was quite proﬁtable, the median bank earned more than 56 basis points from **term** transformation, whereas in 2008, when the **term** **structure** was nearly ﬂat, the median bank earned barely more than nine basis points. The results illustrate that earnings from **term** transformation are quite volatile in the course of time, depending on the current and past shape of the **term** **structure**.

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