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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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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∆ s B 6= 0), there is an additional source of risk due to regime shifts and it should also be priced (λ S (z, r t )) in the **term** **structure** model. Introducing the regime switching risk not only can add more flexibilities to the specification of time-varying bond risk premiums, but also can be potentially important in understanding the bond risk premiums over different holding periods. Wu and Zeng [27] use a general equilibrium model to introduce the systematic risk of regime shift in the **term** **structure** of **interest** **rate** and further estimate the model by Efficient Method of Moments. They find that the market price of the regime-switching risk is not only statistically significant, but also economically important, accounting for a significant portion of the **term** premiums for long-**term** bonds. Ignoring the regime- switching risk leads to underestimation of long-**term** **interest** rates and therefore flatter yield curves.

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The **term** **structure** of **interest** rates is of crucial importance for a variety of economic agents. For central banks, the yield curve helps to explain how monetary policy is implemented. For national treasuries, it indicates the yields for which public debt management should pay for issuing fixed **rate** bonds. For financial institutions, the **term** **structure** determines the allocation and exposure limits of different bonds maturities. For macroeconomists, it is an important leading indicator of economic activity and inflation. For firms, the **term** **structure** affects investments decisions. For households, the yield curve influences consuming and savings decisions.

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Contrary to the results of standard **term**-**structure** models with a time-invariant **term** pre- mium, our results show that the expectations hypothesis of the **term** **structure** of **interest** rates is generally supported, even during the QEMP period. Empirical results from a macro-finance model show that the relationship between the macroeconomic and financial variables has changed significantly over time. There is hardly any relationship at the zero lower bound on **interest** rates and deflation, and especially during the quantitative easing period. The structural decomposi- tion of the yield curve into its macroeconomic components shows that, by contrast with conven- tional wisdom, inflation, activity and monetary policy play a less prominent role in explaining the yield curve. They play no role at all particularly during quantitative easing. The variance decomposition of the level factor indicates that the main part of the variation in this factor comes from the yield curve factors, limiting the contribution of macroeconomic variables. Conversely, the relative importance of yield curve factors in the variation of inflation is relatively small and even inexistent during quantitative easing. A more pronounced effect of yield curve factors on the output gap is detected during the period of high **interest** rates, and this effect disappears during quantitative easing. Moreover, the increasing contribution of the level factor in the vari- ation of the call **rate** during quantitative easing reflects the increasing importance attributed by monetary policy to long-**term** **interest** rates. These finding are corroborated by impulse response results.

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Contrary to the results of standard **term**-**structure** models with a time-invariant **term** pre- mium, our results show that the expectations hypothesis of the **term** **structure** of **interest** rates is generally supported, even during the QEMP period. Empirical results from a macro-finance model show that the relationship between the macroeconomic and financial variables has changed significantly over time. There is hardly any relationship at the zero lower bound on **interest** rates and deflation, and especially during the quantitative easing period. The structural decomposi- tion of the yield curve into its macroeconomic components shows that, by contrast with conven- tional wisdom, inflation, activity and monetary policy play a less prominent role in explaining the yield curve. They play no role at all particularly during quantitative easing. The variance decomposition of the level factor indicates that the main part of the variation in this factor comes from the yield curve factors, limiting the contribution of macroeconomic variables. Conversely, the relative importance of yield curve factors in the variation of inflation is relatively small and even inexistent during quantitative easing. A more pronounced effect of yield curve factors on the output gap is detected during the period of high **interest** rates, and this effect disappears during quantitative easing. Moreover, the increasing contribution of the level factor in the vari- ation of the call **rate** during quantitative easing reflects the increasing importance attributed by monetary policy to long-**term** **interest** rates. These finding are corroborated by impulse response results.

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However, there is more to **interest** **rate** changes than direction and magnitude. Bonds that mature at different time horizons do not move in lockstep. Much can be learned by looking at changes in the **term** **structure** of **interest** rates, that is, the entire range of rates from short maturities to long. Sometimes short- **term** **interest** rates move strongly and the long end of the **term** **structure** barely shifts. At other times, the short end remains anchored and the action is only in medium- and long-**term** rates. Identifying changes in **interest** rates across the maturity spectrum can be useful when assessing the impact of news on market expectations about the macroeconomy and future monetary policy. This Economic Letter looks at how different types of news affect **interest** rates across maturities.

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JEL No. F3
ABSTRACT
This paper examines uncovered **interest** **rate** parity (UIRP) and the expectations hypotheses of the **term** **structure** (EHTS) at both short and long horizons. The statistical evidence against UIRP is mixed and is currency- not horizon-dependent. Economically, the deviations from UIRP are less pronounced than previously documented. The evidence against the EHTS is statistically more uniform, but, economically, actual spreads and theoretical spreads (spreads constructed under the null of the EHTS) do not behave very differently, especially at long horizons. Partly because of this, the deviations from the EHTS only play a minor role in explaining deviations from UIRP at long horizons. A random walk model for both exchange rates and **interest** rates fits the data marginally better than the UIRP-EHTS model.

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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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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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Interbank Offered **Rate** (TIBOR) data. 23 This data set covers maturities of one, two,
three, six, nine, and 12 months, and enables us to analyze the performance of the short end of the **term** **structure**.
The results from the TIBOR data are reported in Tables 7–9, where those during transition period A are not included due to the lack of adequate sample size. (The TIBOR data are available from 1995 onward.) Table 7 shows the poor performance of the **term**-**structure** model regardless of the combination of **interest** rates used to calculate the yield spreads. This result is in sharp contrast to that obtained from the long end of the **term**-**structure** model using gensaki rates and JGBs, but is consistent with previous findings obtained elsewhere that the short end of the **term**-**structure** model performs less well than the long-end model.

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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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Nearly all of the banks will gain if the **term** **structure** shifts downward and lose if the **term** **structure** moves upward, because banks tend to grant long-**term** loans and take in short-**term** deposits. For the few banks for which the 190-bp-upward shift is the relevant shock we proceed as follows: Their exposure is multiplied by -130/190 to account for their negative **term** transformation and to rescale their exposure. Observations of parallel shifts of other than 130 basis points are rescaled accordingly. When calculating the eﬀects of the **interest** **rate** shock, banks have to include all on-balance and all oﬀ-balance positions in their banking book.

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ahead). OIS rates, on the other hand, are available at a daily frequency and have a horizon that aligns exactly with those of the zero-coupon nominal government bonds used in the estimation of GADTSMs. The **term** **structure** of OIS rates can therefore be readily added to a GADTSM for nominal government bond yields. I show that augmenting the GADTSM with OIS rates provides additional information, specifically related to future **interest** **rate** expectations, that can help better identify the evolution of these expectations. Using 3 to 24-month OIS rates in an arbitrage-free GADTSM enables the estimation of future short-**term** **interest** **rate** expectations for the whole **term** **structure** — from 3 months to 10 years. Estimates of **interest** **rate** expec- tations from OIS-augmented GADTSMs are superior to those from existing GADTSMs. In particular, short and long-horizon in-sample OIS-augmented risk-neutral yields match patterns in federal funds futures rates and survey expectations. These time series also match qualita- tive daily patterns exhibited by financial market instruments. This implies that OIS-augmented GADTSMs are well suited for daily frequency policy analysis. Thus, OIS-augmented GADTSMs provide reliable and policy-relevant estimates of **interest** **rate** expectations along the whole **term** **structure**.

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Finally, h C t captures uncertainties associated with the curvature of the yield curve, which can vary between convex, linear and concave forms. Obviously, such variations mainly stem from time-varying volatility in bonds with mid-**term** maturities.
An alternative way to capture time-varying volatility in the **term** **structure** of inter- est rates would be to allow Σ itself to be time-varying. However, this would result in an N -dimensional MGARCH or SV model which is not very tractable if the cross-sectional dimension N is high. Therefore, we see our approach as a parsimonious alternative to capture **interest** **rate** risk. Note that the slope and curvature factors can be interpreted as particular (linear) combinations of yields associated with factor portfolios mimick- ing the steepness and convexity of the yield curve. 4 Then, the corresponding slope and

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This paper estimates the **term** **structure** of **interest** rates with the setup of 3-factor no arbitrage model and investigates the trend of **term** premia and the effectiveness of changes in policy **interest** rates. The **term** premia are found to be high in a three-year medium **term** objective, which can be interpreted as reflecting the recognition of investors who expect a higher uncertainty in real activities for the coming three years than for a longer **term**. Then, in order to look into the effect of policy **interest** rates after the recent change of benchmark **interest** **rate**, this paper analyzes the effects of the changes in short-**term** **interest** rates of the financial market on the yield curve of the bond market at time of change. Empirical results show that the discrepancy between call **rate**, short-**term** **rate** in money market, and instantaneous short **rate**, short-**term** **rate** in the bond market, is found to be significantly widened, comparing to the periods before the change in benchmark **interest** **rate**. It is not easy to conclude clearly for now whether such a widening gap is caused by the lack of experiences with managing new benchmark **interest** **rate** or is just an exceptional case due to the recent turmoil in the global financial market. However, monetary policy needs to be operated in a manner that could reduce the gap to enhance its effectiveness.

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of risk, nonparametric estimation allows data to speak for themselves. The model precludes arbitrage opportunities, preserves a simple **structure** and the computational tractability, and at the same time allows for maximal flexibility in fitting into the data.
The paper is organized as follows. Section 2 outlines the spot **rate** approach of mod- eling **term** **structure** dynamics; Section 3 summarizes two well known one-factor models, i.e., the Vasicek (1977) model and the CIR (1985) model, and examines the behavior of these models and their closed form solutions for bond and bond option prices; In Section 4, consistent estimators of the nonparametric drift func- tion, diffusion function and market price of risk are proposed. Procedures to obtain nonparametric prices of **interest** **rate** derivative securities by either solving the PDE numerically or performing Monte Carlo simulations along the risk-neutral process are proposed as well. In Section 5, the nonparametric model is implemented using historical Canadian **interest** **rate** **term** **structure** data. Empirical results not only pro- vide strong evidence that the traditional spot **interest** **rate** models and market price of **interest** **rate** risk are misspecified but also suggest that different model specifications have significant impact on the **term** **structure** dynamics and prices of **interest** **rate** derivative securities. A brief conclusion is contained in Section 6.

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

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