The Application Of The Relative Nelson-siegel Class Of Models As Predictive Models For Exchange Rate Changes In The Philippines.pdf

An Undergraduate Thesis Presented to the Financial Management Department Ramon V. Del Rosario – College of Business De La Salle University - Manila

The Application of the Relative Nelson-Siegel Class of Models as Predictive Models for Exchange Rate Changes in the Philippines

In partial fulfillment of the Course Requirement in THSEFIN Term 2, A.Y. 2018-2019

Cruz, Zurex Carlo E. Feldia, Agatha Francesca B. Reyes, Paul Michel D.

March 2019

ACKNOWLEDGEMENTS First, Ms. Elvira P. De Lara-Tuprio, a published author and professor that has written an earlier study regarding the Forecasting of Term Structure of Philippine Interest Rates Using the Dynamic Nelson-Siegel Model. She has provided knowledge to her fullest extent regarding the subject matter. Without her input on the fundamentals of the Nelson-Siegel model, the researchers would not have begun studying the said predictive model. Second, Mr. Dioscoro P. Baylon Jr., a finance professor turned thesis advisor. He has provided guidance and encouragement throughout the challenges and triumphs of completing the study. Without him giving the patience and time to the researchers, the researches would falter to the forecasted difficulties regarding the what and how of the Nelson-Siegel model. Third, Mr. Tyrone Panzer L. Chan Pao, a finance professor with an expertise in macroeconomics, business economics, and financial economics. He was the first professor that the researchers encountered regarding the feasibility and creativity of the topic during the introductory process of the Nelson-Siegel model and the researchers. Without his enthusiasm at the topic at hand, the researchers will not be able to be motivated towards a probable successful outcome. Fourth, Mr. Tomas S. Tiu and Mr. Edralin C. Lim, the panelists that have provided the researchers with hard-hitting questions that have made the researchers question the entire study. Without the difficulty that was put through by said panelists, the researchers would have not be been more determined in acquiring the knowledge and skill to defend the study in at the final defense. Most importantly, the researchers would like to take God and our parents. God, for guidance and direction given during disarray through the process of completing the study. Parents, for the continuous support and guidance.

ABSTRACT

Exchange rates are vital to the economy. It regulates local and overseas markets and determines inflation and has an impact on future price movements. Empirical studies have been conducted for the predictive ability of exchange rates. However, despite past studies, due to the volatility of exchange rates, the method for its prediction requires such an instrument containing future economic conditions. With this, the application of the yield curve is introduced as it provides information concerning future economic conditions that can affect the exchange rate based on previous bodies of literature.

The study aimed to examine the yield curve`s predictive ability by utilizing the Relative Nelson-Siegel class of models through a regression analysis of the slope, level, and curvature factors derived from the Relative Nelson-Siegel three and four-factor model against the USD/PHP Exchange Rate. In addition, to further determine the predictive ability of the models, a comparative analysis against the Random Walk model was conducted. With this, the study concluded that the Relative Nelson-Siegel factors of the United States and the Philippines exhibit potential predictive ability for the USD/PHP exchange rate changes in both in-sample and out-of-sample prediction.

TABLE OF CONTENTS CHAPTER ONE - INTRODUCTION ................................................................................ 1 1.1 Background of the Study ........................................................................................... 1 1.2 Statement of the Problem .......................................................................................... 4 1.3 Objectives of the Study ............................................................................................. 4 1.4 Hypotheses ................................................................................................................ 5 1.4.1 Relative three-factor Nelson Siegel model (TF) ................................................ 5 1.4.2 Relative four-factor Nelson Siegel model (SV) ................................................. 6 1.5 Significance of the Study .......................................................................................... 7 1.6 Scope and Limitations ............................................................................................. 10 CHAPTER TWO - REVIEW OF RELATED LITERATURE ......................................... 11 2.1 Exchange Rate Studies ............................................................................................ 11 2.1.1 Derivations of the Monetary Approach ............................................................ 12 2.1.2 Portfolio Balance Approach ............................................................................. 15 2.1.3 Taylor Rule Model ........................................................................................... 16 2.2 Previous Yield Curve Models ................................................................................. 18 2.2.1 Durand’s Basic Yields ...................................................................................... 18 2.2.2 McCulloch’s Discount Function ...................................................................... 19 2.2.3 Exponential Splines Model .............................................................................. 20

2.2.4 Shortcomings of Previous Models ................................................................... 21 2.3 Nelson-Siegel Model ............................................................................................... 23 2.3.1 Original Model ................................................................................................. 24 2.3.2 Svensson four-factor Model ............................................................................. 26 2.3.3 Dynamic three-factor Model ............................................................................ 27 2.3.4 Five factor Dynamic Model ............................................................................. 29 2.3.5 Relative Model ................................................................................................. 31 2.3.6 Application of the Class of Models in Exchange Rates ................................... 32 2.4 Yield Curve Macro Linkage .................................................................................... 33 2.5 Previous Studies in the Philippine Setting .............................................................. 35 2.5.1 Exchange Rate .................................................................................................. 35 2.6 Research Gap ........................................................................................................... 36 2.7 Literature Map ......................................................................................................... 37 CHAPTER THREE - FRAMEWORK ............................................................................. 38 3.1 Theoretical Framework ........................................................................................... 38 Uncovered Interest Rate Parity Theory ..................................................................... 38 3.2 Conceptual Framework ........................................................................................... 40 3.3 Operational Definition............................................................................................. 42 CHAPTER FOUR - METHODOLOGY........................................................................... 44

4.1 Research Design ...................................................................................................... 44 4.2 Data Description and Collection ............................................................................. 44 4.3 Data Analysis .......................................................................................................... 45 4.3.1 Nelson Siegel three-factor and four-factor ....................................................... 45 4.3.2 Regression Analysis ......................................................................................... 48 4.3.3 Tests of Regression .......................................................................................... 49 4.3.4 Root Mean Squared Error (RMSE) .................................................................. 51 4.3.5 Random Walk (without drift) Comparative Analysis ...................................... 52 CHAPTER FIVE – RESULTS AND ANALYSES .......................................................... 53 5.1 Descriptive Statistics ............................................................................................... 53 5.2 Estimating the Dynamic Factors of Relative Nelson-Siegel Class of Mmodels ..... 56 5.3 Predicting Exchange Rate Changes using Relative Nelson-Siegel Factors ............ 60 5.3.1 Parameter Estimates ......................................................................................... 60 5.3.2 Regression Diagnostics .................................................................................... 61 5.4 Maximum Likelihood Estimation ........................................................................... 63 5.5 Prediction Accuracies for Within-Sample and Out-of-Sample Prediction ............. 64 CHAPTER SIX – SUMMARY, CONCLUSION, AND RECOMMENDATIONS ........ 66 6.1 Summary ................................................................................................................. 66 6.2 Conclusion ............................................................................................................... 68

6.3 Recommendations ................................................................................................... 69 REFERENCES .................................................................................................................. 73 APPENDIX A: Statistical Codes ...................................................................................... 81 APPENDIX B: Data (Sample) .......................................................................................... 86 APPENDIX C: Turnitin Receipt/Report ........................................................................... 89 LIST OF FIGURES

Figure 2. 1. Plotted Exponential Spline, Polynomial Spline and Asymptotic Forward Rate ........................................................................................................................................... 22 Figure 2. 2 Flexibility of the Yield Curve ......................................................................... 24 Figure 2. 3. Interpretation of the Yield Curve Factors ...................................................... 25 Figure 2. 4. Literature Map ............................................................................................... 37

Figure 3. 1. Conceptual Framework for three-factor Nelson-Siegel model ...................... 40 Figure 3. 2. Conceptual Framework for four-factor Nelson-Siegel model ....................... 41

Figure 5. 1. USD-PHP Exchange Rates, 1 January 2010 to 31 December 2018 .............. 55 Figure 5. 2. Dynamic factors of the Nelson-Siegel three factor model ............................. 56 Figure 5. 3. Dynamic factors of the Nelson-Siegel four factor model .............................. 58

LIST OF TABLES

Table 5. 1. Philippines Zero-coupon Bonds Interest Rates (in %), 1 January 2010 to 31 December 2018 ................................................................................................................. 53 Table 5. 2. US Zero-coupon Bonds Interest Rates (in %), 1 January 2010 to 31 December 2018 ................................................................................................................................... 54 Table 5. 3. USD-PHP Exchange Rates, 1 January 2010 to 31 December 2018 ............... 55 Table 5. 4. Parameter Estimates of the Nelson Siegel three and four-factor model ......... 60 Table 5. 5. Results of Breusch-Pagan test ......................................................................... 61 Table 5. 6. Results of Durbin-Watson test ........................................................................ 61 Table 5. 7. Results of Variance Inflation Factors test ....................................................... 62 Table 5. 8. Results of Jarque-Bera test .............................................................................. 63 Table 5. 9. Maximum Likelihood Estimation Results....................................................... 63 Table 5. 10. Root Mean Squared Error (RMSE) of the Nelson Siegel three factor, Nelson Siegel four factor, and Random Walk ............................................................................... 64

Table 6. 1. Predictive Performance Summary ................................................................... 67

CHAPTER ONE INTRODUCTION 1.1 Background of the Study For the most part of 2018, the Philippine peso (PHP) continuously depreciated against the United States dollar (USD). In fact, the United States dollar to Philippine peso exchange rate reached USD/PHP 54.13 last September 12, 2018, representing a new 12year low for the Philippine local currency. The last time the Philippine peso depreciated against the dollar at the PHP 54 level was nearly 13 years ago when the Philippine peso was at PHP 54.15. Given this, the continuous depreciation of the Philippine peso presented potential economic challenges for the Philippines as exchange rates are considered vital to the economy. Exchange rates allow the Philippines to: (1) monitor and regulate the local and overseas market for goods and services, (2) determine actual inflation and future price movements, (3) impact foreign trade through price competitiveness, and (4) estimate the cost of servicing on its foreign debt (Bangko Sentral ng Pilipinas, 2018). Moreover, the foreign exchange market growth, as of 2016, had an average total daily trading of $5.1 trillion (Bank of International Settlements, 2016).

Given the economic relevance and impact of exchange rates, addressing exchange rates’ predictability had been the subject of several empirical studies, providing various perspectives as to how exchange rates are treated before the determination of prediction 1

models. Despite the existence of numerous empirical studies, most studies have only succeeded in identifying well-known empirical puzzles. The failures of empirical studies such as Meese & Rogoff (1983) led to the belief that economic fundamentals such as national income, trade balance, and money supply are not important factors in exchange rate prediction. This belief would later be challenged by the Asset Pricing approach, a method that states that the nominal exchange rates are influenced by economic fundamentals. Engel & West (2005) argue that existing models could be transcribed as present-value asset-pricing format which allows for the possibility of providing information about future fundamentals while also noting that their empirical evidence is not uniformly strong. Moreover, succeeding studies have found evidence for the asset behavior of exchange rates and its ability to predict economic fundamentals such as world commodity prices (Chen et al., 2008; Lustig et a.l, 2008). However, Sarno & Taylor (2002) observed, “Overall, the conclusion emerges that, although the theory of exchange rate determination has produced a number of plausible models, empirical work on exchange rates still has not produced models that are sufficiently statistically satisfactory to be considered reliable and robust...In particular, although empirical exchange rate models occasionally generate apparently satisfactory explanatory power in-sample, they generally fail badly in out-of-sample forecasting tests in the sense that they fail to outperform a random walk.”

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Given the impact of economic conditions on exchange rates, the method for its prediction requires an accurate instrument containing future economic conditions such as the yield curve. Furthermore, the yield curve can predict GDP growth, inflation, possible recessions, among others through its three factors: inflation, level, and slope (Mishkin, 1990a, 1990b; Barr and Campbell, 1997; Estrella and Mishkin, 1998; Hamilton and Kim 2002; Rudebusch and Wu, 2007, 2008). With this, empirical evidence entails that an accurate yield curve model is needed for accurate exchange rate predictions.

This leads to the Nelson-Siegel model, a parsimonious yield curve model shows the shapes associated with yield curves (S-shaped, humped, and monotonic) which entails the adaptability and flexibility of the model. Moreover, the above-mentioned three unobservable factors of the yield curve which contain information about future economic development can be extracted from the Nelson-Siegel model to determine the information on expected economic conditions contained within a yield curve.

With this, the researchers recognize the urgency of the Philippine exchange rate condition and finds importance in addressing the predictability of exchange rate changes in the Philippines through a comprehensive comparison of the empirical methods prescribed by previous literature. Thus, the researchers believe that determining a

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feasible model for exchange rate prediction can benefit several stakeholders in the Philippines.

1.2 Statement of the Problem The presence of an alarming exchange rate condition in the Philippines and the lack of studies on exchange rate predictability using yield curve models interests scholarly attention from the researchers. Therefore, the study aims to answer the following questions: ● Can the Relative Nelson-Siegel class of models predict the exchange rate changes in the Philippines? ● Which Relative Nelson-Siegel model is the most significant in predicting exchange rate changes in the Philippines? ● Which factor of the most significant Relative Nelson-Siegel model is the most significant in predicting exchange rate changes in the Philippines? ● Does the Relative Nelson-Siegel class of models outperform the Random Walk model? 1.3 Objectives of the Study As the study aims to address the predictability of exchange rate changes in the Philippines through the use of the Nelson-Siegel Relative Factor model, the study aims to achieve the following research objectives:

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● To determine if the Relative Nelson-Siegel class of models is significant in predicting the future exchange rate changes in the Philippines ● To determine the most significant Relative Nelson-Siegel model in predicting exchange rate changes in the Philippines ● To determine which factor of the most significant Relative Nelson-Siegel class of models is the most significant in predicting exchange rate changes in the Philippines. ● To determine the if the Relative Nelson-Siegel class of models is more significant in predicting exchange rate changes in the Philippines than the Random Walk model. 1.4 Hypotheses The hypotheses to be validated by the study through extensive data analysis are as follows: 1.4.1 Relative three-factor Nelson Siegel model (TF) 1.4.1.1 Level Factor H0: The level factor of the Relative three-factor Nelson Siegel model is not significant in predicting the exchange rate changes in the Philippines Ha: The level factor of the Relative three-factor Nelson Siegel model is significant in predicting the exchange rate changes in the Philippines

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1.4.1.2 Slope Factor H0: The slope factor of the Relative three-factor Nelson Siegel model is not significant in predicting the exchange rate changes changes in the Philippines Ha: The slope factor of the Relative three-factor Nelson Siegel model is significant in predicting the exchange rate changes in the Philippines 1.4.1.3 Curvature Factor H0: The curvature factor of the Relative three-factor Nelson Siegel model is not significant in predicting the exchange rate changes in the Philippines Ha: The curvature factor of the Relative three-factor Nelson Siegel model is significant in predicting the exchange rate changes in the Philippines

1.4.2 Relative four-factor Nelson Siegel model (SV) 1.4.2.1 Level Factor H0: The level factor of the Relative four-factor Nelson Siegel model is not significant in predicting the exchange rate changes in the Philippines Ha: The level factor of the Relative four-factor Nelson Siegel model is significant in predicting the exchange rate changes in the Philippines 1.4.2.2 Slope Factor H0: The slope factor of the Relative four-factor Nelson Siegel model is not significant in predicting the exchange rate changes changes in the Philippines

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Ha: The slope factor of the Relative four-factor Nelson Siegel model is significant in predicting the exchange rate changes in the Philippines 1.4.2.3 First Curvature Factor H0: The first curvature factor of the Relative three-factor Nelson Siegel model is not significant in predicting the exchange rate changes in the Philippines Ha: The first curvature factor of the Relative three-factor Nelson Siegel model is significant in predicting the exchange rate changes in the Philippines 1.4.2.4 Second Curvature Factor H0: The second curvature factor of the Relative four-factor Nelson Siegel model is not significant in predicting the exchange rate changes in the Philippines Ha: The second curvature factor of the Relative four-factor Nelson Siegel model is significant in predicting the exchange rate changes in the Philippines

1.5 Significance of the Study Exchange rates hold significance, mainly due to its relationship with inflation rates, which in turn defines a country’s economic standing which is seen in cost of living, personal and corporate loans, corporate and government bond yields, etc. Inflation rates and prices of said goods and services have a direct relationship. Hence, controlling inflation rates is a must by regulatory institutions such as the government and Central Bank, in order for the success and growth of the country’s economy. Thus, there is significance in predicting exchange rates changes. 7

Government institutions and Policy makers Government institutions and policy-making bodies are confronted with economic decision making regularly. For instance, one of the main responsibility of Bangko Sentral ng Pilipinas is determination of exchange rate policy through adherence to a marketoriented foreign exchange rate policy to foster orderly conditions in the market (http://www.bsp.gov.ph/about/functions.asp). A more thorough appreciation of the relationship between yield curves and exchange rates will lead to the better promulgation and implementation of the monetary policy that fosters a healthier and more robust Philippine economy. If the proposed model is significant enough to estimate exchange rate changes, the BSP may use it to ensure growth and price stability of the Philippine economy.

Financial institutions that deal with foreign currencies For financial institutions like banks that deal with foreign currency, the relationship with foreign exchange involves risks that occurs when (1) a change in domestic and/or foreign currency affects investment’s value and (2) a decision in closing out a currency with either a long position or a short position. The foreign exchange risk is unpredictable for the reason that exchange risk is also unpredictable, If the proposed models are succesful, they would be able to mitigate the the foreign exchange risk that the respective bank occurs.

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Active traders in the Forex market For participants in the forex (FX) market, the forward rate transaction is how the foreign exchange dealer earns profit wherein it holds a private contract with a locked-inrate currency rate and date of transaction, between them and a respective company or client. If the model proposed is successful in prediction, FX traders may use the model to make more educated market decisions.

Businesses engaged in imports and exports For businesses, specifically those engaged in the importation and exportation of goods and services in and out of the country, in gaining capability in predicting through the models, their expected profit or loss would not be far from the computed profit or loss. Moreover, it is an advantage in making business decision with foreign exchange as its basis. If the domestic currency depreciates against the foreign currency, meaning paying less for the same good or service, would be best to purchase. If the domestic currency appreciates, meaning paying more for the same good or services, would be best to sell.

Academe / Researchers For researchers, it is not gaining capability in prediction through the models that holds significance, but it is the likely the shortcomings that can be proved upon by future

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researchers to contribute a less-flawed, more accurate predictive model of exchange rate changes.

1.6 Scope and Limitations The study used only the three-factor and four-factor model of the Nelson-Siegel class of models to predict the USD/PHP exchange rate changes given that the Yield Curve Package in the R Statistical Program only contains the three-factor and the fourfactor. Two (2) datasets were required: (1) the monthly USD/PHP exchange rate and (2) the monthly zero coupon bond yields of the Philippines and the United States for tenors 3-month , 6-month, 1-year, 3-year, 4-year 5-year, 6-year, 7-year, 8-year, 9-year, 10-year, 15-year, 20-year, and 30-year from 2010-2018. All datasets were taken from the Bloomberg Terminal. In addition, the study was limited only to the use of the mentioned models as the main models for the prediction and the use of the Random Walk model as a predictive performance comparison test.

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CHAPTER TWO REVIEW OF RELATED LITERATURE 2.1 Exchange Rate Studies The asset market approach pertains to determining exchange rates, thus portraying nominal exchange rates with similar nature as the discounted present value of expected future fundamentals, which holds resemblance in inflation, output, and cross country differences in monetary policy (Chen & Tsang, 2013). These nominal exchange rates are equivalent to an asset price due to the dependability on expectation of future variables (Obstfeld & Rogoff, 1996, 529). These asset prices are determined through current prices reflecting on markets’ expectations on present and future economic conditions (Frenkel & Mussa, 1985) and are processed arbitrarily close to random walk that requires (a) at least a forcing variable containing a unit autoregressive root and (b) for the discount factor to hold precision with unity. To provide emphasis, there holds a relationship between economic fundamentals and expected future exchange rates. A variety of models indicates the present value relationship which is expressed as: 𝑠𝑡 = (1 − 𝑏) ∑∞ 𝑗=0

𝑏 𝑗 𝐸𝑡 (𝑓1𝑡+𝑗 + 𝑧1𝑡+𝑗 ) + 𝑏 ∑∞ 𝑗=0

𝑏 𝑗 𝐸𝑡 (𝑓2𝑡+𝑗 +

𝑧2𝑡+𝑗 ) (2.1) where exchange rate st is defined by drivers of economic fundamentals being money supplies, money demand shocks, productivity shocks, etc., fit is a measured fundamental and zit is a measurement error and an unobserved shock (i = p 1, 2) ,b jE,st+j imposes the 11

no-bubbles condition (Engel & West, 2005). This shows the linear combination of future fundamentals in a discounted sum is the exchange rate (Chinn & Meese, 1995).

Furthermore, the asset market approach contains branches describing domestic and foreign money with perfect capital substitutability or imperfect capital substitutability, namely the monetary approach or the portfolio balance approach, respectively (Levich, 2001). The monetary approach is derived from (a) domestic and foreign economies with the functions of conventional money demand (Chinn & Meese, 1995) and (b) purchasing power parity (Meese & Rogoff, 1983b). The portfolio balance approach, on the other hand, assumes cumulative current account balances, inflation rates, and domestic-foreign differentials which serve as the variables contributing to nominal exchange rate (Faust, Rogers, & Wright, 2003).

2.1.1 Derivations of the Monetary Approach Conventional money demand functions are identical in domestic and foreign countries. It is expressed as 𝑚𝑖 = 𝑝𝑖 + 𝑎𝑦𝑖 − 𝛿𝑟𝑖

(2.2)

𝑚𝑗 = 𝑝𝑗 + 𝑎𝑦𝑗 − 𝛿𝑟𝑗

(2.3)

where mi and mj are nominal money, pi and pj are the price level, yi and yj are real income, ri and rj are the nominal interest rate, and (α) and (δ) represent parameters (Joseph & 12

Larrain, 2012). The equations assume equality across countries due to the income elasticity α interest rate semi elasticity δ (Frankel, 1984).

Purchasing power parity (PPP) theoretical model is the change of price level between home and foreign currencies. Equilibrium between the two economies will result from the exchange rate changes with goods-market arbitrage mechanism. It is expressed as: 𝒍𝒏 𝒆𝒕 = 𝒍𝒏 𝒑𝒕 − 𝒍𝒏 𝒑∗𝒕 (2.4) where et is the nominal exchange rate, pt and pt∗ are domestic and foreign prices respectively. The equation show the inclusion of price indexes in estimations as the relative version of the theoretical model (Lam, Fung, & Yu, 2008).

The monetary approach contains three exchange rate determinative models which predict and explain exchange rate out-of-sample using structural models with constrained coefficients, namely flexible-price monetary model, sticky-price monetary model, and the sticky-price asset model. The said models hypothesize on the homogeneity in the first-degree which is relative to money supply, a1≃1 (Meese & Rogoff, 1983 a, b). The combined models can be shown as a general specification through its quasi-reduced form specifications expressed as 𝑠 = 𝑎0 + 𝑎1 (𝑚 − 𝑚∗ ) + 𝑎2 (𝑦 − 𝑦 ∗ ) + 𝑎3 (𝑟𝑠 − 𝑟 𝑎5 𝑇𝐵 + 𝑎6 𝑇𝐵

∗

+𝑢

∗ 𝑠)

+ 𝑎4 (𝜋 𝑒 − 𝜋 ∗𝑒 ) +

(2.5) 13

The flexible price monetary model of Frenkel-Bilson holds equivalence to the difference of two identical money demand specifications and imposes PPP (Meese and Rogoff, 1983b) due to the flexibly perfect prices of goods (Frankel, 1984). The imposition of PPP mediums extensively to an exogenous real exchange rate shock (Chinn & Meese, 1995) with variables such as output, yv and the real exchange rate qo (Engel & West, 2005).

The sticky price monetary model of Dornbusch-Frankel takes the assumption of flexibly perfect prices with shorter horizons of one to three months and PPP takes hold of measures with longer horizons of nine to twelve months (Frankel, 1984) but short horizon deviations permit PPP due to sticky domestic prices (Meese & Rogoff, 1983b). Consequently, the assumption leads to consequent deviations (Meese & Rogoff, 1983a) and the slow correlative adjustment of goods prices and assets prices (Chinn & Meese, 1995). Endogenous variables such as short-term differential, relative incomes, and relative money supplies apply to the stick price monetary model (Meese & Rogoff, 1983a; Engel & West, 2005).

The sticky price asset monetary model of Hooper-Moton reveals similarity with the sticky price monetary model of Dornbusch-Frankel. The difference lies in the incorporation of real exchange rate changes in the long run, thus resulting to the unforeseen impact to trade balance deficits or surplus (Meese & Rogoff, 1983a,b). The 14

failure of the PPP in holding broad price indices stimulates said model, like consumer price index and GNP deflators (Chinn & Meese, 1995).

2.1.2 Portfolio Balance Approach Past literature describes the portfolio balance approach as; (1) The illustrative model (2) The analytical model (3) The extended model. Literature developments on the approach held more complexity considering the redistribution of world wealth wherein imbalances in current accounts alter to the bonds demanded in different currency denominations, the adjusted interest differentials, resulting to the adjustment of relative money demands. Additionally, current account is deemed important in determining the exchange rate due to the wealth transfer effect where there is a higher probability of domestic residents holding more proportion of wealth (domestics currency denominated assets) than foreign travelers. The argument to the statement is that expected real returns depend on the consumption basket which can be held parallel to favor the holding of bonds denominated in foreign currency (Krugman, 1981).

To attain portfolio equilibrium, the price of exchange rate is equilibrium to the balance of payments flows is what needs to be taken in consideration. This is in respect to the continuous portfolio equilibrium assumption wherein the market demand separates itself from changes in asset holdings (Kouri, 1976). The balance of payments flow holds an ex-post (realized change) identity which is expressed as where IMD is import 15

deliveries, EXD is export deliveries, EXO is order for exports, IMO is orders for imports (Black, 1973). 𝐵 − 𝐵𝑡−1 = 𝐼𝑀𝐷𝑡 − 𝐸𝑋𝐷𝑡 = 𝐼𝑀𝑂𝑡−1 − 𝐸𝑋𝑂𝑡−1 (2.6)

2.1.3 Taylor Rule Model Monetary policy rule is the fundamental idea of the Taylor rule model. Taylor (1993) describes policy rule to be an inclusion of a nominal income rule which targets nominal income and a contingency plan for a rational amount of time. The adjustment of the monetary policy, or rather interest rates, can be caused by (a) disparity of money supply and target, (b) disparity of exchange rate and target, or (c) weighted deviations of inflation rate (or the price level) and real output from target. Therefore, Taylor constructed a policy rule expressed as 𝑟 = 𝑝 + .5𝑦 + 5(𝑝 − 2) + 2 (2.7) where r is federal funds rate, p is the rate of inflation over the previous four quarters, and y is the percent deviation of real GDP from a target. The stated model contains assumptions of equal weight of 0.5 between the output and inflation gaps and real interest rate and inflation target are held at an equilibrium level of 2%.

Following Molodtsova & Papell (2008), the modified monetary policy rule characterizes flexibility on inflation rates and output gaps which can be expressed as 𝑖𝑡∗ = 𝜋𝑡 + 𝜙(𝜋𝑡 − 𝜋 ∗ ) + 𝛾𝑦𝑡 + 𝑟 ∗ (2.8) 16

where it* is the target for the short-term nominal interest rate, 𝝅t is the inflation rate, 𝝅t* is the target level of inflation, yt is the output gap, and r* is the equilibrium level of real interest rate. The similarity between two of the models explain the increase of short-term nominal interest rate considering the surge of inflation from the output. In addition to the technicalities of the equation, (a) the output cannot exceed potential output for yt=0 due to the natural rate hypothesis and (b) a positive target inflation is the ideal. Furthermore, constant increase of the exchange rate can precipitate the short-term nominal interest rate, if these short-term nominal interest rate were underestimated for its persistent shocks (Gourinchas & Tornell, 2004). The forecasting equation of the Taylor Rule model holds the equivalence between interest rate differential and the difference between interest rate reaction function for the foreign and home interest rate, which can be expressed as 𝑔

∗𝑔

𝑖𝑡 − 𝑖𝑡∗ = 𝛽0 (𝑠 − 𝑠𝑖∗ ) + 𝛽1 (𝑦𝑡 − 𝑦𝑡 ) + 𝛽2 (𝜋𝑡 − 𝜋𝑡∗ ) + 𝜐𝑡 − 𝜐𝑡∗

(2.9)

where 𝝅t = pt - pt-1 is the inflation rate, ygt is the output gap, vt is the shock containing omitted terms, st* is the exchange rate target. The model contains the following assumptions 𝞫1 > 0, 𝞫z > 1 and 0 < 𝞫0 < 1. It explains the derivations of the exchange rate as a present value model and a positive relationship between the model based and actual real exchange rate between the foreign and home countries (Engel & West, 2005).

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Given the previous empirical studies on exchange rate determination models, exchange rates should be valued or treated as the net present value of future fundamentals. Thus, in the latter part of this study, the researchers will be utilizing the information on future fundamentals which are shown in the yield curve in determining the exchange rate changes.

2.2 Previous Yield Curve Models 2.2.1 Durand’s Basic Yields David Durand (1942) was the first researcher who attempted to fit yield curve. The study revolved around corporate debt and focused on 1900 to 1942’s actively markettraded bonds, but the study’s aim was not to create an estimated representation of the yield curve. Moreover, the study used a monotonic envelope approach which utilized a procedure using free-hand curve fitting that presented a curve of minimum yields by term. As a result, the curves produced in this procedure were constrained to a few shapes exhibited by a typical yield curve.

However, the procedure of curve fitting used by Durand (1942) revealed several errors. Meiselman (1962) mentioned states “these measurement problems… introduce some lack of precision”. Also, Sargent (1972) mentioned “(the basic Yields of Durand) are subject to substantial error”. However, Durand (1958) stated himself that “Basic yield curves are designed to create a quick and crude structure of the term structure of high 18

grade bond yields… They are not adequate, however, to support certain types of refined analysis”. Thus, this concludes that Durand’s model for fitting yield curves were insufficient in estimating the yield curve.

2.2.2 McCulloch’s Discount Function McCulloch (1971) developed the yield curve fitting technique through using the “discount function”. McCulloch (1971) derived the yield curve, instantaneous forward interest rates, mean forward interest rates, and consistent values for securities. Specifically, the study (McCulloch, 1971) utilizes a cubic splines method to divide the zero-coupon yield curve into distinct intervals that make up a curve when connected. Moreover, the discount function D(t) is simply the present value of $200 repayable in m years represented by Equation (2.10): 𝐷(𝑡) = 1 + ∑𝑘𝑗=1

𝑓𝑗 (𝑡) ⋅ 𝑎𝑗 (2.10)

where aj are the estimated paramenters and j>k , fi(t) is a cubic polynomial.

However, the study (McCulloch, 1971) discovered that the D(t) is a kth-degree polynomial with unity for its constant term. McCulloch (1971) stated that “a polynomial is straight-forward, but it has no theoretical motivation”. This means that the formulation of D(t) indoes dependent on the distribution of t. As a result of its unity, when D(t) is used to fit a discount function, it will exhibit a finely defined and relatively smooth shape in the first 1-2% of its length, but it will either ignore the short end and conform to the 19

remaining part or vice versa given that there are many bill observations within the regression.

Thus, this presents an inherent limitation contained in McCulloch’s

polynomial splines approach. 2.2.3 Exponential Splines Model Following the study of McCulloch (1971), Vasicek & Fong (1982) presents an approach using the s of the exponential spline in contrary to McCulloch’s polynomial spline fitting. Given this, the difference that the exponential approach exhibits are (1) desirability of asymptotic properties for long-term bonds, (2) flexible, allowing it to produce the different yield curve shapes, and (3) sufficient robustness for stability of forward rate curves

Moreover, the model was introduced due to the shortcomings of using polynomial splines in fiting the discount functions. The argument was that the discount function is in exponential form/shape. Polynomials produce different curves when compared to exponentials. Also, the local fit of polynomial splines is not adequate even with choosing sufficiently large knot points to be close to an exponential curve. Furthermore, when viewed practically, polynomial splines exhibit the tendency to weave around the exponential spline resulting to high levels of unstable forward rates.

20

2.2.4 Shortcomings of Previous Models Polynomial spline models have drawbacks as term structure due to their tendency to yield forward interest rates estimates that exhibit instability, high levels of fluctuations, and drifting to large and negative values. With this, Vasicek & Fong (1982) followed it up with the introduction of exponential splines as a better model for approximation due to its asymptotic properties, flexibility, and stability which will be now referred to as VF model. Given this, Shea (1985) proceeded to producing empirical applications and comparisons of exponential and polynomial splines to verify the claims of Vasicek & Fong (1982) specifically: (1) the model’s robustness to produce stable forward rates and (2) the model’s asymptotic properties. As evidenced by past studies, polynomial spline approximations are rarely linear or nearly linear. This observation would later be the same case for exponential spline approximations in contrary to Vasicek and Fong’s claim of desirable asymptotic properties.

Furthermore, the study (Shea, 1985) shows that there are wide changes in the short term together with the little influence that the asymptotic forward rate brings to the estimated forward yield curve. Also, it illustrates the instability of the VF model approximations in response to previously stated claims of stability due to the data conditioning properties that model imposes. This leads to the unrealistic and unstable resulting asymptotic rates.

21

Figure 2. 1. Plotted Exponential Spline, Polynomial Spline and Asymptotic Forward Rate. Adapted from "Interest Rate Term Structure Estimation with Exponential Splines: A Note" by G.S, Shea. 1985, March. The Journal of Finance, Vol. 40, pp. 319-325..

Moreover, Vasecik and Fong’s claim of the VS model’s ability to produce more stable rates as compared to the polynomial spline approximations are also debunked. Figure 2.1 illustrates the similar estimates obtained by the exponential spline as compared to the polynomial spline. This result is a result of a technicality that Vasecik and Fong failed to recognize: it is difficult for polynomial splines to model exponential functions, but this fact does not extend to local approximations. Thus, polynomial splines produce nearly the same estimates and curve fitting as exponential splines.

Furthermore, Shea (1984) investigated the shortcomings of the equilibrium and spline models in smoothing the yield curve. The study focused on comparing McCulloch’s spline approximation model to the developed piecewise polynomial yield curve model in the paper. Spline approximation techniques allow for the solution to 22

sparse data due to its ability to control the length of constituent polynomial pieces at long terms to maturity, but this was not the case when empirical applications were established as data was found to be too widely spaced (Shea, 1984). However, a solution was suggested which was to add an additional constraint (Schaefer, 1973). Moreover, this suggestion yields unsatisfactory yield estimates that would dive to zero or near-zero interest rate levels. Also, the suggestion is theoretically invalid as polynomial approximations are to be independently determined. Providing a negative slope constraint would therefore violate this. Thus, the approximations by exponential and polynomials splines are not to be considered perfect as evidenced by previous literature discussed above.

2.3 Nelson-Siegel Model The development of a parsimonious model for the yield curve started when Friedman (1977) stated in a money demand study that “Students of statistical demand functions might find it more productive to examine how the whole term structure of yields can be described more compactly by a few parameters”. In addition, Friedman (1977) recommended the development of a parameterized statistical procedure that could describe the height and slope of the yield curve to investigate if money demand will increase or decrease. Nelson and Siegel (1987) stated that “Potential applications of parsimonious yield curve models include demand functions, testing of theories of the term structure of interest rates, and graphic display for informative purposes” 23

2.3.1 Original Model Nelson & Siegel (1987) stated that “A class of functions that readily generates the typical yield curve shape is associated with solutions to differential equations”.The primary objective of the paper was to present simple and parametric models that exhibit influence

but not dependence on the expectations theory, that is sufficiently

parsimonious to present the shapes associated with yield curves. Overall, as evidenced by the empirical results, the model exhibits the necessary smoothness to present a maturity-specific pattern. 𝑅(𝑚) = 𝛽0 + (𝛽1 + 𝛽2) ∗ [1 − 𝑒𝑥𝑝(−𝑚/𝜏)]/(𝑚/𝜏) − 𝛽2 ∗ 𝑒𝑥𝑝(−𝑚/𝜏) (2.11)

Figure 2. 2 Flexibility of the Yield Curve.. Adapted from "Parsimonious Modeling of Yield Curves" by Nelson, C., & Siegel, A. (1987). The Journal Of Business

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As seen on Equation 2.12 and Figure 2.2, parameter a being able to use values from minus six to 12 in equal increments allows the generation of different yield curve shapes (humps, monotonic curves, and S-shapes). According to Nelson and Siegel (1987), the loading levels of the short-term, medium-term, and long-term aspects of the yield curve are contained in the coefficients of the model. The long-term in B0, the shortterm in B1, and the medium term in B2. Among the components, the short-term monotonically decays to zero. On the other hand, the long term is referenced as a constant which exhibits no decay to zero while the medium term is the only component that starts at zero and ends at zero as seen in Figure 2.3.

Figure 2. 3Interpretation of the Yield Curve Factors. Adapted from "Parsimonious Modeling of Yield Curves" by Nelson, C., & Siegel, A. (1987). The Journal Of Business

The primary objective of the model is to investigate its sufficiency in producing any possible relationship between the term to maturity and the yields for U.S. Treasury bills. The study (Nelson & Siegel, 1987) gathered 30 yield and term-to-maturity pairs For the purpose of fitting yield curves the model was converted to a parametric form: 25

𝑚

𝑚

𝑚

𝑡

𝑡

𝑡

𝑅(𝑚) = 𝑎 + 𝑏 ⋅ [1 − 𝑒𝑥𝑝(− )]/( ) + c (exp(- ) (2.13) Nelson and Siegel (1987) also stated that “Another criteria for a satisfactory parsimonious model is that it be able to predict yields beyond the maturity range of the sample.” With this, Nelson & Siegel (1987) determines through their empirical results that there is high correlation between the estimated and actual values.

2.3.2 Svensson four-factor Model The study’s (Svensson, 1994) major contribution is the extension of the NelsonSiegel model through the addition of a fourth, U-shaped, term with two additional parameters 𝛽3and 𝜆2 (second exponential decay parameter). 𝑚

𝑚

𝜆1

𝜆1

𝑓(𝑚; 𝑏) = 𝛽0 + 𝛽1 𝑒𝑥𝑝(− ) + 𝛽2

𝑚

𝑚

𝜆1

𝜆 2

𝑒𝑥𝑝(− ) + 𝛽3

𝑚

𝑒𝑥𝑝(− ) 𝜆2

(2.14) where b = 𝛽0,𝛽1,𝛽2, 𝜆1 𝛽3 , 𝜆1

Svensson (1994) emphasized on the importance of minimizing price errors to result to good fits of yield especially for short maturities as observed in previous literature. Moreover, the empirical results reveal that the four-factor Nelson-Siegel model exhibits better predictive performance when compared to the original Nelson-Siegel model when yield curves become more complex. Thus, the addition of the fourth term (later defined as the second curvature factor) shows improvement in goodness of fit.

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2.3.3 Dynamic three-factor Model Diebold & Li (2006) presented the original model as a three-factor dynamic model of level, curvature, and slope 𝑦𝑡 = 𝛽1𝑡 + 𝛽2𝑡 (

1 − 𝑒 −𝜆𝑚 ) + 𝛽3𝑡 𝜆𝑚

(

1 − 𝑒 −𝜆𝑚 − 𝑒 −𝜆𝑚 ) 𝜆𝑚

(2.14)

The study proceeds to interpret the several parameters in Equation (2.14). The lambda represents the rate of exponential decay. Lambdas with small values result to slower decay and better fit for maturities that are long. This means that the lambda is important to the determination of the maturity where the medium-term loading reaches the maximum. However, in the study, Diebold & Li (2006) determines the lambda value as 0.0609, the 30-month average of the 2 or 3 year maturities are commonly used in the medium-term. Moreover, one of the vital contributions of the study is the interpretation of the factors (long, short, and medium term) found in Nelson & Siegel (1987) as seen in Equation (2.14) as the yield curve’s slope, level, and curvature ( 𝛽2𝑡 ,𝛽1𝑡 t,, and 𝛽3𝑡 ) Furthermore, the study (Diebold & Li, 2006) provides a five-point comparison of the model’s framework and the facts related to the average represented shape of a yield curve to theoretically assess the model’s ability to present a smooth and accurate yield curve. First, previous literature has provided that the average yield curve is concave and increasing which is also matched by the three latent dynamic factors, 𝛽1𝑡 ,𝛽2𝑡 and ,𝛽3𝑡 .

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Second, the average yield curve exhibits different shapes through time which can be matched by the study’s yield curve due to the factor’s association with a yield curve’s slope, curvature, and level. Third, yield curve dynamics are said to be persistent and spread dynamics are less persistent, and this is again replicated by the model’s heavy loading on 𝛽1𝑡 and light loading on 𝛽2𝑡 . Fourth, yield curve’s short end is found to be much more volatile in the long end which is represented by the model’s short end depending on both 𝛽1𝑡 and 𝛽2𝑡 and the long end depending only on 𝛽1𝑡 Finally, previous yield curve literature state that long rates are said to be more persistent than short rate which can be potentially represented by the model’s long term factor, 𝛽1𝑡 , if it is the most persistent factor after data analysis. Thus, the study’s framework is very much consistent in theory.

Also, the study (Diebold & Li, 2006) plotted the the observed yield curve factors (B1t,B2t ,B3t) against the actual/empirical level, curvature, and slope. The correlations for the factors are as follows: p(B1t, lt) = 0.97, p(B2t, st) = -0.99, and p(B3t, ct) = 0.99. Moreover, the results of the descriptive statistics of the factors empirically confirm that the long-term factor, B1t, is the most persistent factor of the yield curve. Thus, the results confirm the theoretical assertions by the study.

In addition, Diebold & Li (2006) conducted in-sample and out-of-sample forecasting tests. First, in the in-sample tests, there appears to be a low degree of cross28

factor interaction and autocorrelation between the estimated factors. Also, when the estimated factors are plotted against the autocorrelations of the AR(1) models, the results show that there is a low level of residual correlation; entailing that the AR(1) models describe the conditional means of the three factors. Lastly, in the out-of-sample tests, the RMSE results show that the Nelson-Siegel dynamic model performs well in the 1 and 6month ahead, but does not outperform other predictive models such as the Random Walk. This outcome may be a result of forecast errors being serially correlated as evidenced in previous literature (Bliss, 1997; de Jong, 2000). However, the Nelson-Siegel model outperforms the other predictive models in the 12-month ahead forecasting performance comparison. Thus, the framework of the study succeeds in simplifying the Nelson-Siegel model interpretation and forecasting the term structure of interest rates at some levels.

2.3.4 Five factor Dynamic Model The study compares the in- and out-of-sample prediction of the Nelson-Siegel class of models with those of a Random Walk. In addition, the study utilizes Quantile Autoregression (QAR), Vector Autoregression (VAR), Autoregression (AR), and Random Walk (RW) to further evaluate predictive performance. However, the study introduces a five-factor model that includes a fifth term or second slope factor to enhance the fitting flexibility of the previous variations of the Nelson-Siegel model allowing for more complex and twisted curves.

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𝑦(𝜏) = 𝛽1𝑡 + 𝛽2𝑡 (

1 − 𝑒_− 𝜆1 𝜏 1 − 𝑒_− 𝜆1 𝜏 1 − 𝑒_− 𝜆1 𝜏 ) + 𝛽3𝑡 ( ) + 𝛽4 ( − 𝑒 −𝜆1 𝜏 ) 𝜆1 𝜏 𝜆1 𝜏 𝜆1 𝜏 1 − 𝑒_− 𝜆2 𝜏 + 𝛽5𝑡 ( − 𝑒 −𝜆2 𝜏 ) 𝜆2 𝜏

(2.15)

Moreover, the study (De Rezende & Ferreira, 2013) utilizes two lambda or exponential decay parameters through an optimization problem that provides the the lambda with the lowest Root Mean Squared Error at each period t: 𝑁 1 (𝜆̂1, 𝜆̂2 ) = 𝑎𝑟𝑔 𝑚𝑖𝑛{ ∑ 𝑁 𝑛=1

𝑇 1 √ ∑ 𝑇 𝑡=1

(𝑦𝑡 (𝜏𝑛 ) − 𝑦̂𝑛 (𝜏𝑛 , 𝜆1 , 𝜆2 , 𝛽𝑡 )2 }

(2.16)

Furthermore, the empirical results of the study show that the five-factor NelsonSiegel model is the best model for in-sample fitting based on RMSE and BIC statistics. However, the five-factor model delivers poor results in out-of-sample forecasting which may be attributed to a case of overparameterization as evidenced by Diebold & Li (2006). Overall, the Nelson-Siegel models can outperform the random walk forecasts depending on prediction horizons such as 1 month and 3 months ahead, but the common denominator among them is that these models perform better when utilizing QAR.

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2.3.5 Relative Model The study focuses on the yield curve by extracting relative factors from crosscountry differences in the yields to gather information on future exchange rate fundamentals. The study shows that the factors contained in any two countries’ yield curves can exhibit predictive performance. Also, the study reiterated the dominant performance of the model in fitting the yield curves through references to papers such as Nelson & Siegel (1987) and Diebold & Li (2003). However, the vital contribution of the study (Chen & Tsang, 2013) is the process created for the purpose of extracting Nelson Siegel factors that are relative for two countries. Given the previous literature that exchange rates contain information about future fundamentals such as interest rate differentials, Chen & Tsang (2013) were able to change the left side of the formula into that of the interest rate differential of the two countries to transform the Nelson-Siegel model relative to two countries which would allow for the prediction of exchange rate changes as shown in Equation 2.17 and 2.18: 𝑖𝑚 𝑐 − 𝑖𝑚 𝑐∗ = 𝐿𝑚 𝑅 + 𝑆𝑚 𝑅 (

1−𝑒 −𝜆𝑚 𝜆𝑚

) + 𝐶𝑚 𝑅 (

1−𝑒 −𝜆𝑚 𝜆𝑚

− 𝑒 −𝜆𝑚 ) + 𝜀𝑚 𝑐 (2.17)

𝛥𝑠𝑡+𝑚 = 𝐵𝑚,0 + 𝐵𝑚,1 𝐿𝑡 𝑅 + 𝐵𝑚,2 𝑆𝑡 𝑅 + 𝐵𝑚,3 𝐶𝑡 𝑅 + 𝑢𝑡+𝑚 (2.18) Furthermore, the study shows that the relative factors can estimate exchange rate movements. The study finds that an increase in the level or slope factors entails an annual 3-4% appreciation of its currency, while changes caused by the curvature factor tend to have a much smaller effect. Results also offer insight into the UIP. According to Chen & Tsang (2013), “the study finds that the deviations from UIP respond to the shape of the 31

yield curves, which in turn capture market perception of future inflation, output, and other macro indicators.”. Thus, the yield curve exhibits predictive ability, but it is limited to certain time horizons.

2.3.6 Application of the Class of Models in Exchange Rates Exchange rate studies have accepted the superiority of the three-factor regression model with a three-month horizon, but have not proven the accuracy and precision in exchange rate changes determination of the four- and five-factor regression models in any horizon. The study Ishii (2018) remodels the three-factor regression equation by Chen & Tsang (2013) to include the Nelson-Siegel four and five-factor extensions. The following remodeled Extended Dynamic model Regression Equations are expressed as:

The study runs an out-of-sample prediction, using statistical tests such as RMSPE and Clark and West to compare results on accuracy performance while also applying the random walk as the benchmark model comparison. The Nelson-Siegel class of models

holds superior to the random walk with a 1% significance level. Based on the empirical results, the three-factor model can predict at a three-month horizon. The four-factor model can predict at a 6- and 12-month horizon. Moreover, the five factor is inferior to

32

the three-factor model at a three-month horizon and is equal to the performance of the four-factor model in six- and twelve-month horizons.

2.4 Yield Curve Macro Linkage As observed by previous bodies of literature cited in Section 2.1, the nominal exchange rate is a result of economic fundamentals, such as cross-country monetary policy differences, GDP, and inflation. This entails that a proper exchange rate determination model must be one that contains information about future macroeconomic fundamentals. Moreover, Mishkin (1988) investigated the relationship of the yield curve and inflation. Generally, the results show that the yield curves do exhibit information of the future path of inflation at the longer-end, specifically at maturities nine months or greater. However, given this finding, Mishkin (1988) recommends that future researchers should practice caution when using the yield curve to predict future inflation changes as the empirical results show that external factors may contribute a drastic change in predictive performance such as major monetary policy changes. Also, Barr & Campbell (1997) conducted a similar study with similar results to Mishkin (1988) of the yield curve’s short end containing little to no information about future inflation. With this, it can be concluded the yield curve has the potential to determine future inflation at certain levels.

33

In addition, given the evidence provided by previous literature, Hamilton & Kim (2000) reinvestigated the capability of the yield curve’s spread to estimate future real GDP growth. The study (Hamilton & Kim, 2000) confirms that both effects are statistically significant: (1) a forecast of lower short-term interest rates resulting to slower GDP growth and (2) an increase in the expected returns from rolling over one-period bonds relative to an n-period bond denotes a slower GDP growth. However, the expectations effect proves to be more statistically and quantitatively significant. Thus, the study (Hamilton & Kim, 2000) confirms that there lies information on future GDP growth within the yield curve

Also, Rudebusch and Wu (2008) observed the lack of macro-finance interpretation of the term structure of interests. The study (Rudebusch & Wu, 2008) focuses on the established macroeconomic and finance facts of the term structure. Moreover, the study (Rudebusch & Wu, 2008) concludes through its empirical results that the level factor of the yield curve exhibits information on the perceived mediumterm central bank inflation target while the slope factor exhibits information related to the cyclical variation in inflation and output gaps when the central bank adjusts the short rate accordingly to policy. Given the previous bodies of literature on the macroeconomic explanatory capability of the yield curve, it can be concluded that the yield curve contains information on future fundamentals which can potentially be applied to estimate movements in exchange rates. 34

However, Meese & Rogoff (1983a, 1983b) compared the predictive performance of models of the 1970s against the Random Walk model. The study (Meese & Rogoff, 1983a, 1983b) empirically determined that no fundamentals, including commodity prices, can predict better than the Random Walk model.

2.5 Previous Studies in the Philippine Setting 2.5.1 Exchange Rate The study (Yu, 2011) considers the application of previous determination models such as the Structural and Time-Series to the USD/PHP Exchange Rate. The study aims to qualify and confirm the results of the Meese-Rogoff (MR) Experiment (refer to Section 2.) through a predictive performance comparison of the mentioned models and Random Walk model during stable and turbulent time periods. To analyze predictive performance, the study employed Mean Average Error, Mean Squared Error, Mean Average Percentage Error, and Diebold-Mariano Statistic. Furthermore, the results of the study confirms the conclusion of the MR Experiment evidenced by the predictive performance of the Random Walk model against the structural models when applied in the Philippine setting. However, the study finds that the parsimonious AR(1) model from the Time-Series category is able to dominate the Random Walk during the crisis, thus indicating the ability of the model to produce accurate forecasts amidst instability.

35

2.6 Research Gap Given the previous bodies of literature discussed in this section, this study will focus on using the three and four factor model of the Relative Nelson-Siegel class of models prescribed by Ishii (2018) as the predictive models for the Exchange Rate Changes in the Philippines. However, for the 𝜆, exponential decay factor, of the Relative Nelson-Siegel class of Models will be determined through the same optimization process utilized by De Rezende & Ferreira (2013) and Ishii (2018). The study will also make use of the Regression Analysis equations for the threefactor and four-factor Models prescribed by Ishii (2018). The Regression Analysis equations will produce values for the p-value, t-statistic, and pearson’s correlation which will be the bases for analyzing in-sample fit of the model. The significance level of 5% will be used as a basis for the significance of variables (Nelson-Siegel Factors). Also, despite numerous out-sample tests used in previous literature, the study will only use the Root Mean Squared Error (RMSE) as an additional measure of predictive performance. However, to comprehensively determine the predictive performance of the Relative Nelson-Siegel class of models, a Random Walk model for predicting USD/PHP Exchange Rate changes will be employed to confirm the validity of the Random Walk superiority in exchange rate determination as stated by Meese & Rogoff (1983). With this, the RMSEs of the Relative Nelson-Siegel class of models and Random Walk model will be compared to assess the more significant model in predicting USD/PHP exchange rate changes. 36

2.7 Literature Map

Figure 2. 4. Literature Map

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CHAPTER THREE FRAMEWORK 3.1 Theoretical Framework Uncovered Interest Rate Parity Theory The Uncovered Interest Rate Parity (UIP) is a theory that states that high yield currencies are expected to depreciate. An increase in the real interest rate should appreciate the currency. The UIP is one most important and prominent concepts in international finance which has affected previous exchange rate literature. According to Fung & Yu (2008) “the UIP gives an arbitrage mechanism that drives the exchange rate to a value that equalizes on holding both domestic and foreign assets.”

We let it be the interest rate on bonds on home currency at time t , and 𝑖𝑡∗ be the interest rate on foreign-currency bond. The UIP holds when: 1 + 𝑖𝑡 = (1 + 𝑖𝑡∗ )𝐸𝑡 {

𝑆𝑡 +1 𝑆𝑡

} (3.3)

where St and St+1 is the nominal exchange rate in period t and t+1 , respectively.

Moreover, it is also a fundamental concept in foreign exchange that implies same deposit placed whether in a local or foreign currency should yield the equal returns. According to Ullenes (2012), “… any returns from interest differentials should be equalized through exchange rate movements.” By writing the UIP in log form, we introduce an approximation under uncertainty: 38

𝑖𝑡 = 𝑖𝑡∗ + 𝐸𝑡 (𝑠𝑡 + 1) − 𝑠𝑡

(3.4)

where the logs of St and St+1 , are St+1 and St , respectively.

Furthermore, Madura (2007) stated that “The country with the higher nominal interest rate is expected to depreciate against the low interest currency, because higher nominal interest rates reflect the expectation of inflation.” The relation predicts that countries with high interest rates, should on average, have depreciating currencies. However according to Flood & Rose (2001), “such currencies tend to have appreciated.” As the demand for high-yield interest currencies attract more foreign investors than lowyield currencies, so does its price, and hence, it will appreciate. Empirical findings reveal that high-yield currencies tend to appreciate, which provide information regarding violations of the UIP. Deviations from UIP may indicate an existence of currency risk premium.

39

3.2 Conceptual Framework The two main objectives to answer in this study are : (1) to determine if the Relative Nelson-Siegel class of models is significant enough to predict USD/PHP Exchange Rate Changes and (2) to determine if the Nelson-Siegel class of models is more significant than the Random-Walk model in predicting USD/PHP Exchange Rate Changes. Moreover, to accomplish the objectives, a comparison between the predictive performances of the three-factor, four-factor, and five-Factor Relative Nelson-Siegel model must be done.

Figure 3. 1. Conceptual Framework for three-factor Nelson-Siegel model (Ishii, 2018)

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Figure 3. 2. Conceptual Framework for four-factor Nelson-Siegel model (Ishii, 2018) Furthermore, the study will start with the extraction of the corresponding exponential decay factor, 𝜆, for each Nelson-Siegel model (three, four, and five-factor) through an optimization process. The study will then proceed with the extraction of the monthly Relative Nelson-Siegel Factors for each variation of each Nelson-Siegel model through an Ordinary Least Squares Regression and AR(1) Time-Series. By obtaining the monthly Relative Nelson-Siegel Factors for each Nelson-Siegel model, the study can proceed to analyzing the predictive performance of the Relative-Nelson-Siegel class of models for exchange rate changes through a Regression Analysis that will produce values for p-value and t-statistics. These produced values will be used to analyze in-sample predictive performance while the RMSE of both the Nelson-Siegel Relative Factors and the Random Walk model will be used to analyze out-sample predictive performance. However, a more comprehensive explanation of the statistical processes is provided in the next chapter.

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3.3 Operational Definition a. Nelson Siegel Predictive exchange rate model with the use of term structure of interest rates b. Relative Nelson-Siegel factors Factors of the yield curve that are extracted from the Nelson-Siegel model through time-series relative to the interest rate differentials of two countries c. Exchange Rate Price of one currency equivalent to another currency

d. Treasury Bond Long term fixed interest rate debt security issued by a respective government e. Bond Yield Realized return gained from a bond f. Yield Curve Also called to be the term structure of interest rates, presenting a graphic illustration of yields of bonds of varying maturities g. Ordinary Least Square (OLS) Regression

42

Statistical method estimating the correlations between independent variables and a dependent variable through the reduction of the sum of squares h. Time Series Chronologically sequenced numerical data points or variables in successive order i. RMSE Used in measuring the spread of prediction errors through resulting to standard deviation of the prediction errors j. Random walk Stochastic process applying random random variables consisting of a succession of random steps on a mathematical space k. Regression Analysis Statistical approach in forecasting change in a dependent variable, if there is a change in the independent variables

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CHAPTER FOUR METHODOLOGY 4.1 Research Design The study utilizes a Descriptive-Comparative research design to determine the predictive performance of the Relative Nelson-Siegel class of models. The study will consist of the statistical tests to be utilized such as Ordinary Least Squares Regression, AR(1) model, Regression Analysis, Maximum Likelihood Estimation. Moreover, after obtaining the results from statistical tests, a comparative analysis between the NelsonSiegel class of models and the Random Walk model will be conducted to assess the predictive performance evidenced by Pearson’s Correlation Coefficient and Root Mean Squared Error (RMSE).

4.2 Data Description and Collection The data to analyzed in the study consists of the daily US Zero Coupon Bond Yields, PH Zero Coupon Bond Yields, and the USD/PHP Exchange Rate from 2010-2018. Moreover, although there is a bootstrapping method for obtaining zero coupon bond yields, the data for the study will utilize the zero coupon bond yield data from the Bloomberg Terminal. This sample period was chosen to thoroughly analyze the predictive performance of the Nelson-Siegel class of models factors.

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4.3 Data Analysis The study mainly revolves around the estimation of the Nelson-Siegel factors and the regression of the Nelson-Siegel factors and the USD/PHP Exchange Rates. However, the researchers would like to note the study’s usage of the Yield Curve Package in the R Statistical Program to streamline the data analysis. With this, the study will be utilizing these statistical processes: 4.3.1 Nelson Siegel three-factor and four-factor 4.3.1.1 Estimation of the Exponential Decay/Lambda Parameter (𝜆)

(4.2)

(4.3) where: 𝑦𝑡 (𝑚𝑛 ) ≡relative zero coupon yields

𝑅,𝐶𝐻 𝑆𝑚 = the Relative Slope of the

n = index of the number of maturities

three-factor model at m

𝑋̂𝑡𝑅 = [𝐿𝑅,𝑆𝑉 , 𝑆𝑡𝑅,𝑆𝑉 , 𝐶1 𝑅,𝑆𝑉 , 𝐶2 𝑅,𝑆𝑉 ]𝑋̂𝑡𝑅 ] 𝑡 𝑡 𝑡

𝑅,𝐶𝐻 𝐶𝑚

𝐿𝑅,𝐶𝐻 𝑚 = the Relative Level of the

three-factor model at m

= the Relative Curvature of the

three-factor model at m 45

First, the study will proceed to the estimation of the exponential decay/lambda parameter that minimizes the RMSE to provide for better fitting accuracy of the treasury bond yields. As stated in previous literature, the three-factor (CH) model utilizes only one lambda parameter while the four (SV) utilizes two distinct lambda parameters. However, the lambda parameters for all models will differ due to the existence of the varying amount of factors in the all models.

Moreover, this is estimation is done through an optimization problem as evidenced by Equation 4.1 and 4.2. Simply put, the optimization problems are mathematical representations of the process. In actuality, the estimation of the lambda parameters that minimizes RMSE is done through rigorously estimating the Nelson-Siegel factors using Ordinary Least Squares with different lambda parameters, then the lambda parameter that minimizes RMSE is chosen.

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4.3.1.2 Estimating the Relative NS Model Factors - Ordinary Least Squares Regression 𝑖𝑚 𝑐 − 𝑖𝑚 𝑐∗ =

𝑅,𝐶𝐻 𝐿𝑅,𝐶𝐻 + 𝑆𝑚 ( 𝑚

1−𝑒 −𝜆𝑚

𝜀𝑚 𝑐 𝑖𝑚 𝑐 − 𝑖𝑚 𝑐∗ =

𝑅,𝑆𝑉 𝐿𝑅,𝑆𝑉 ( 𝑚 + 𝑆𝑚

𝑅,𝐹𝐹 +𝐶2,𝑚 (

(

1−𝑒 −𝜆𝑚 𝜆1 𝑚

− 𝑒 −𝜆𝑚 ) +

(4.4)

1−𝑒 −𝜆1 𝑚

1−𝑒 −𝜆2 𝑚 𝜆2 𝑚

𝜆1 𝑚

𝑅,𝐶𝐻 ) + 𝐶𝑚

𝜆1 𝑚

𝑅,𝑆𝑉 ) + 𝐶1,𝑚

(

1−𝑒 −𝜆1 𝑚 𝜆1 𝑚

− 𝑒 −𝜆𝑚 )

− 𝑒 −𝜆𝑚 ) + 𝜀𝑚 𝑐 (4.5)

Where: m = Bond tenor

𝑅,𝐶𝐻 𝐶𝑚 = the Relative Curvature

imc = the US Treasury Bond Yield

of the three-factor model at m

imc* = the PH Treasury Bond

𝐿𝑅,𝑆𝑉 𝑚 = the Relative Level of the

Yield

four-factor model at m

𝜆=

lambda

parameter

that

controls the speed of exponential decay 𝐿𝑅,𝐶𝐻 𝑚 =

𝑅,𝑆𝑉 𝑆𝑚 = the Relative Slope of the

four-factor model at m 𝑅,𝑆𝑉 𝐶𝑚

= the Relative Curvature

the Relative Level of the of the four-factor model at m

three-factor model at m 𝑅,𝐶𝐻 𝑆𝑚 = the Relative Slope of the

three-factor model at m

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Second, the study requires the extraction of the Nelson-Siegel factors of for each Treasury Bond Tenor (m) through an Ordinary Least Squares Regression of Equation 4.1 presented above. For every tenor, m, (1) the level, slope, and curvature factors are extracted for the CH model and (2) the level, slope, first curvature, and second curvature factors are extracted for the SV model.

4.3.1.3 Autoregressive (1) Model Xt=δ+ϕ1Xt-1+w

(4.7)

Third, with the Time-Series of the Factors of the Relative Nelson Siegel class of models for each time period, the study can now proceed to analyze if the factors can predict the exchange rate changes through a regression analysis which will be discussed in the next section. 4.3.2 Regression Analysis 𝛥𝑠𝑡+𝑚 = 𝐵𝑚,0 + 𝐵𝑚,1 𝐿𝑡 𝑅,𝐶𝐻 + 𝐵𝑚,2 𝑆𝑡 𝑅,𝐶𝐻 + 𝐵𝑚,3 𝐶𝑡 𝑅,𝐶𝐻 + 𝑢𝑡+𝑚 (4.8) 𝛥𝑠𝑡+𝑚 = 𝐵𝑚,0 + 𝐵𝑚,1 𝐿𝑡 𝑅,𝑆𝑉 + 𝐵𝑚,2 𝑆𝑡 𝑅,𝑆𝑉 + 𝐵𝑚,3 𝐶1,𝑡 𝑅,𝑆𝑉 + 𝐵𝑚,4 𝐶2,𝑡 𝑅,𝑆𝑉 + 𝑢𝑡+𝑚 (4.9) Where: m = number of months ahead st+m = USD/PHP Exchange Rate at t+m

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The study proceeds with the Regression Analyses of the USD/PHP Exchange Rates and the Nelson-Siegel factors of each Nelson-Siegel model (CH and SV) to determine the performance of the Nelson-Siegel class of models in predicting exchange rate changes. Moreover, the study will be conducting in-sample tests from 2010-2018 and a out-ofsample test for January 2019 Exchange Rates. The p-value, t-statistic, and Pearson’s correlation coefficient produced by the Regression Analyses will be serve as bases for assessing predictive performance.

4.3.3 Tests of Regression The following procedures are tests for the assumptions of linear regression. Assumptions of the linear regression suggests that if any of the assumptions are violated, the model may be inefficient or biased. Assuming that the model violates the OLS assumptions, the study will proceed to the Maximum Likelihood Estimation Method for Regression Analysis. 4.3.3.1 Homoscedasticity Tests of homoscedasticity may be found using the Breusch-Pagan test. The process allows the heteroscedasticity be a function of one or more of the independent variables by assuming that heteroscedasticity may be a part of the linear function of all the independent variables of the regression model. Results of the Breusch-Pagan test may be interpreted by its p-value.

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4.3.3.2 Autocorrelation Tests of autocorrelation is found using the Durbin-Watson test. The test gives a number from esiduals of a regression analysis. The Durbin-Watson statistic is a value between 0 and 4, Values from 0 to less than 2 indicate a positive autocorrelation and values ranging from more than 2 to 4 indicate a negative autocorrelation, while 2 indicates that there is no autocorrelation present in the regression model. 4.3.3.3 Multicollinearity Multicollinearity is present when there is a significant correlation between independent variables. Detecting multicollinearity is done by measuring the Variance Inflation Factors (VIF). VIFs can be computed for each independent variable, a value of 1 suggests that a variable is not correlated with the other variables. Generally, the higher the value of the VIF the greater the correlation present. Values of 4 to 5 suggests moderate to high multicollinearity, while values of 10 or more suggests very high multicollinearity.

4.3.3.4 Normality of the error distribution Tests for normality are used to determine whether or not a data set is well-modeled by a normal distribution. Normality test is done using the Jarque-Bera test. The test is determined by its p-value at 5% significance level. Values of more than 0.05 suggests that the null hypothesis that the model is normally distributed should be rejected. Values of less than 0.05 on the other hand, suggest that the null hypothesis should not be rejected therefore implying that the model is normally distributed. 50

4.3.4 Root Mean Squared Error (RMSE) 𝜮 𝑹𝑴𝑺𝑬 = √ 𝒊=𝟏

𝒏

= (𝑿𝒐𝒃𝒔,𝒊 − 𝑿𝒎𝒐𝒅𝒆𝒍,𝒊 )𝟐 𝒏

(4.10)

Where: Xobs = observed values Xmodel is modelled values at time/place i.

The RMSE is the standard deviation of the residuals (prediction errors). The residuals are a measure that shows how far from the regression line the data points are. The RMSE measures the spread of these residuals. This entails that the RMSE shows how concentrated the area is around the best-fit line.

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4.3.5 Random Walk (without drift) Comparative Analysis Xt=Xt-1+et

(4.12)

Xt-Xt-1=et

(4.13)

where: Xt = the value in time period t Xt-1= the value in time period t−1 plus a random shock et (value of error term in time period t).

The predictive performance of the Regression Analysis between the Relative Nelson-Siegel Factors and the USD/PHP Exchange Rates will be compared to the predictive performance of a Random Walk without Drift. A Random Walk TimeSeries assumes that changes in data are independent which entails that historical trends of the data do not produce any predictive ability for its future movement. Furthermore, this predictive performance comparison between the Nelson-Siegel model and the Random Walk model will allow us to determine if the Meese-Rogoff puzzle holds true with the Nelson-Siegel model.

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CHAPTER FIVE RESULTS AND ANALYSES 5.1 Descriptive Statistics Table 5. 1. Philippines Zero-coupon Bonds Interest Rates (in %), 1 January 2010 to 31 December 2018 Time to Maturity 3 months 6 months 1 year 4 years 5 years 6 years 7 years 8 years 9 years 10 years 15 years 20 years 30 years

Mean 3.35 3.43 3.59 4.37 4.62 4.68 4.77 4.89 5.02 5.15 5.8 6.49 6.56

Std. Dev 1.8 1.81 1.75 1.55 1.84 1.37 1.18 1.23 1.29 1.35 1.55 1.97 1.99

Minimum 0.75 0.13 0.96 2.27 2.1 2.4 2.57 2.67 2.77 2.81 3.1 3.4 3.62

Maximum 12.43 12.43 12.43 15.17 14.3 9.72 8 8.05 8.4 8.78 9.68 12.72 13.09

Table 5.1 presents the descriptive statistics of interest rates of zero-coupon bonds in the Philippines where the mean interest rate increases drastically along with its time to maturity. On the average, the deviations of the coupons’ interest rates from the mean interest rate is not relatively large. This suggests that the interest rates of the Philippine zero-coupon bonds are stable and not erratic. Also, although the interest rates seem to be increasing along with time to maturity, it appears that the maximum interest rates of coupon bonds with time to maturity between 4 to 8 years show a gradually decreasing trend. This

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could also be indicative that there are short-term coupon bonds with high interest rates. The maximum interest rates of long-term coupon bonds (greater than 10 years) increase along with its time to maturity.

Table 5. 2. US Zero-coupon Bonds Interest Rates (in %), 1 January 2010 to 31 December 2018 Time to Maturity 3 months 6 months 1 year 4 years 5 years 6 years 7 years 8 years 9 years 10 years 15 years 20 years 30 years

Mean 0.45 0.51 0.61 1.39 1.65 1.87 2.09 2.x26 2.42 2.56 3.03 3.24 3.51

Std. Dev 0.64 0.67 0.69 0.63 0.6 0.59 0.59 0.6 0.6 0.59 0.72 0.74 0.7

Minimum

Maximum

0 0.01 0.04 0.44 0.6 0.76 0.91 1.1 1.3 1.41 1.63 1.85 2.33

2.5 2.58 2.72 3.06 3.09 3.23 3.63 3.95 4.17 4.24 5.09 5.21 5.23

As presented in Table 5.2, on the average, the interest rates of US zero-coupon bonds drastically increase with its time to maturity. Compared to Philippines zero-coupon bond interest rates, the average interest rates in US zero-coupon bonds are relatively lower. In addition, the interest rates for US coupon bonds are also more stable. Lastly, there are some short-term US coupon bonds (less than 1 year) that can be acquired at very low interest rates or no interest at all.

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Figure 5. 1. USD-PHP Exchange Rates, 1 January 2010 to 31 December 2018

Table 5. 3. USD-PHP Exchange Rates, 1 January 2010 to 31 December 2018 Mean Nominal Exchange Rate Daily Percentage Change

46.05 0.00575

Std. Dev 3.61 0.33

Minimum

Maximum

40.55 -1.32%

54.31 1.67%

Table 5.3 showed that from 2010 until 2018, the average USD-PHP exchange rate is at 46.05 pesos per US dollar. On the other hand, Figure 5.1 shows the time plot of USD-PHP exchange rate showed a decreasing trend from 2010 until early 2013. This is indicative of an increasing value of the Philippine Peso in an economic standpoint within the same

55

period. The lowest recorded USD-PHP exchange rate is at 40.55 pesos per US dollar on 2013. However, starting from mid-2013, the trend shifted in favor of the US dollar, as evidenced by the increasing amount of USD-PHP exchange prices. The largest recorded USD-PHP exchange rate is at 54.31 pesos per US dollar recorded on the 2nd half of 2018. The lowest recorded daily change from 2010 until 2018 is a 1.32 percent drop in the USDPHP exchange rate while the highest is a 1.67 percent increase in the USD-PHP exchange rate.

5.2 Estimating the Dynamic Factors of Relative Nelson-Siegel class of models

Figure 5. 2. Dynamic Factors of the Nelson-Siegel three Factor model

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As shown in Figure 5.2, the level factor of the three-factor Relative Nelson-Siegel model showed a steady increasing trend from 2010 until end of 2013, with presence of occasional, small to moderate shocks. The level factor remained stable at a slightly lower value than 0 on 2014, where only few shocks have been recorded. However, starting from 2015, the level factor showed multiple large shocks in a daily basis, but stabilizing its value closer to 0 from time to time.

Moreover, the slope factor exhibited a short-term decreasing behavior from 2010 until halfway of 2011. The slope factors displayed a drastic increase starting from the second half of 2011 until end of 2013. During this period, shocks to the slope factor are of low magnitude and low occurrence. However, starting from 2014, the slope factors showed a gradual decrease, approaching 0. There is a period of high volatility on the slope factors from 2015 to 2016 and resurfaced again on 2018.

Furthermore, the values of the curvature factors are usually below zero from 2010 until end of 2013 except during periods with moderate to large spikes. Starting from 2015, there is a gradual decreasing trend in the curvature factors but large spikes tend to be more noticeable within the period, more specifically on years 2016 and 2018.

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Figure 5. 3. Dynamic Factors of the Nelson-Siegel four Factor model

Based on the Figure 5.3, the level factors of the four-factor model showed an increasing trend from 2010 until the first half of 2011, characterized with large spikes. Halfway through 2011, the trend shifted downward, which lasted for a year. A large downward spike occurred midway through 2013 and followed by a steadily increasing trend which lasted until end of the following year. Starting from 2015, the level factor exhibited a gradually decreasing trend until end of 2018, with a high occurrence of moderate to large shocks.

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Furthermore, the slope factors of the four-factor model exhibited an increasing trend from 2011 until mid-2013 along with a large spike on 2013. Halfway through 2013, the trend shifted downwards until end of 2014. Starting from 2015, the slope factor displayed an increasing trend but with a regular occurrence of moderate to large shocks.

Moreover, contrary to other dynamic factors, the first curvature factor is relatively stable, and with values that are close to 0. Some noticeable massive downward spikes from 2010 until 2018 occurred on the 2nd half of 2010, near the end of 2012, and last quarter of 2013 which lasted until early 2014. Aside from these large shocks, the first curvature factors have relatively low variability.

Lastly, the second curvature factor of the four-factor model showed an increasing trend from 2011 until end of 2013, with large spikes that occurred nearing the end of 2012 and end of 2013. Afterwards, the trend shifted downward for a year. The factor approached closer to 0 from 2015 until 2018, with low occurrence of moderate shocks.

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5.3 Predicting Exchange Rate Changes using Relative Nelson-Siegel Factors 5.3.1 Parameter Estimates Table 5. 4. Parameter Estimates of the Nelson Siegel three and four-factor model Intercept Level Slope Curvature Second Curvature Adj. R square

Three-factor model −8.36 × 10−4 (0.496) −2.61 × 10−4 (0.481) −7.04 × 10−5 (0.757) −5.6 × 10−5 (0.487) --0.005432

Four-factor model −2.32 × 10−3 (0.0613)* −4.68 × 10−4 (0.0995)* 1.53 × 10−5 (0.8781) −2.48 × 10−5 (0.6734) −1.79 × 10−4 (0.1052) 0.02407

*-denotes statistical significance at 10% level

Table 5.4 presents the parameter estimates for the Nelson-Siegel three and fourfactor model. As presented, there is no significant parameters estimates for the Relative Nelson-Siegel three factor model. This result is in-line with the results of Ishii (2018) where the three-factor model factors did not show any statistical significance. However, in the four-factor model, the level-factor is statistically significant at the 10% level. This outcome is also in-line with several Nelson-Siegel exchange rate application literature (Ishii, 2018; Chen & Tsang, 2013). Although, the researchers would like to note that the statistical significance of the relative factors vary from a country-pair basis as evidenced in previous literature (Ishii, 2018; Chen & Tsang, 2013). Lastly, the parameter estimates show that the adjusted R squared of the four-factor model is higher than that of the three-

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factor model, making it more statistically significant in the within-sample prediction which is again in-line with a previous study on the exchange rate application of the Nelson-Siegel class of models (Ishii, 2018). 5.3.2 Regression Diagnostics 5.3.2.1 Homoscedasticity -Breusch-Pagan test Table 5. 5. Results of Breusch-Pagan test

BP test statistic p-value

Three-factor model 1.1419 0.7670

Four-factor model 3.6610 0.4538

As shown in Table 5.5. both models do not exhibit heteroscedasticity (nonconstancy of variance). 5.3.2.2 Autocorrelation—Durbin-Watson Test Table 5. 6. Results of Durbin-Watson test

DW test statistic p-value

Three-factor model 2.1507 0.8619

Four-factor model 2.2125 0.9832

As presented in Table 5.6, both models have no evidence of serial correlation of error terms

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5.3.2.3 Multicollinearity—Variance Inflation Factors Table 5. 7. Results of Variance Inflation Factors test Level Slope Curvature Second Curvature

Three-factor model 23.2 9.10 28.8 --

Four-factor model 215 29.4 6.13 244

Table 5.7 presents that the three-factor model somewhat exhibits a mild case of multicollinearity between the Level and Curvature factor. However, in the case of the fourfactor model, there is a clear evidence of strong multicollinearity between the Level and the Second Curvature factors. The case of multicollinearity of the Nelson-Siegel class of models has long been recognized by previous studies as this is a result of the factor-loadings in the lambda parameters represented by 𝜆1 and 𝜆2 (Cabrera et. al, 2014; Annaert et. al, 2013; Gilli et. al, 2010). According to Gilli et.al (2010), “Correlated regressors are not necessarily a problem in forecasting. We are often not interested in disentangling the effects of two single factors as long as we can assess their combined effect. The problem changes if we want to predict the regression coefficients themselves”. Given this, the study can ignore the case of multicollinearity as its aim is to predict exchange rate changes and not to predict the values of the Nelson-Siegel dynamic factors.

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5.3.2.4 Normality of Error Terms—Jarque-Bera Test Table 5. 8. Results of Jarque-Bera test Three-factor model 6.5576 0.0377

JB test statistic p-value

Four-factor model 3.4943 0.1743

Table 5.8 shows that there is evidence of non-normality of residuals from the threefactor model. However, there is no evidence of non-normality of residuals for the fourfactor model 5.4 Maximum Likelihood Estimation Table 5. 9. Maximum Likelihood Estimation Results Variable Constant Level Slope Curvature Null deviance Residual deviance

Estimate -8.36E-04 -2.61E-04 -7.04E-05 -5.60E-05

Std. Error 1.23E-03 3.70E-04 2.28E-04 8.05E-05

zstatistic -0.681 -0.705 -0.309 -0.696

pvalue 0.496 0.481 0.757 0.487

0.0021535 0.0021400

df df

260 257

Although, there is a result of non-normality of residuals in the three-factor model as evidenced in Table 5.8, this can be remedied through shifting the parameter estimation model from Ordinary Least Squares (OLS) to Maximum Likelihood Estimation as it does not need to satisfy the normality of errors assumption. Table 5.9 shows that: (1) a unit increase in the level factor in effect will transmit an average of 0.000261 percentage points

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decrease in the current day exchange rate differentials, (2) a unit increase in the slope factor will transmit an expected 0.0000704 percentage points decrease in the current day exchange rate differentials, and (3) a unit increase in the curvature factor will transmit an expected 0.000056 percentage points decrease in the present day exchange rate differentials. Moreover, these parameter results are similar to those of the OLS regression.

5.5 Prediction Accuracies for Within-Sample and Out-of-Sample Prediction Table 5. 10. Root Mean Squared Error (RMSE) of the Nelson Siegel three factor, Nelson Siegel four factor, and random walk model Nelson-Siegel 3-Factor Nelson-Siegel 4-Factor Random Walk

Within Sample 0.002864321 0.002815731 0.004198101

Out-of-sample 0.00099207 0.001071148 0.003692655

Table 5.10 shows the RMSE values of the three competing models for the within sample (training sample) and the out-of-sample (test data) forecast. The Relative NelsonSiegel 4-Factor model produce the least varied prediction error for the training sample (2018 data) with an RMSE of about 0.00282. The 3-Factor model has an RMSE that is almost 2% higher than the 4-factor model which indicates that the 3-factor model performs just as good as the 4-factor model. In contrast, the variability of the Random Walk prediction errors for the training sample are about 50% more varied than the two previous models. Thus, the two Nelson-Siegel models perform 33% better in predicting exchange rate changes than the random walk model.

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Moving on to the out-of-sample forecasts, the Nelson-Siegel 3-Factor model has the least RMSE value among the three competing models. The RMSE of the 4-factor model is about 8% higher than the 3-factor model which implies that the performance of the two models are just similar. Moreover, the random walk prediction error variation is 3.7 times higher than the 3-factor model and about 3.4 times higher than the 4-factor model making the random walk model the least superior (most inferior) model among the three. This huge discrepancy of prediction accuracy of the Nelson-Siegel models from the random walk model suggests a potential application of the Relative Nelson-Siegel models in predicting future exchange rate changes in the country.

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CHAPTER SIX SUMMARY, CONCLUSION, AND RECOMMENDATIONS 6.1 Summary With the recently rising and volatile behavior of the USD/PHP Exchange Rate, this specific study set out to apply one of the most recent prediction models in exchange rate literature. As discussed in previous chapters, the Nelson-Siegel class of models show reliable results for prediction of yield curves. In addition, several literatures have linked the possible use of the information contained in the yield curve in predicting exchange rates. With this, the study focused on applying the three and four-factor model of the Nelson-Siegel class of models in the prediction of Exchange Rate Changes

Through the statistical analyses as presented in the previous chapter, in terms of statistical significance, the Nelson-Siegel class of models failed to fully exhibit statistical significance in its parameter estimates with only the level factor of the four-factor model being statistically significant. Moreover, the two models violate two regression assumptions, where the three-factor violates normality of errors and the four-factor violates multicollinearity. However, the violation of the normality of errors may be avoided through the use of Maximum Likelihood Estimation as an alternative parameter estimation model. On the other hand, the multicollinearity issue can be disregarded as the aim of study is not to predict the regression coefficients or the Nelson-Siegel dynamic factors as evidenced by

66

previous literature. With this, the study proceeds to focusing on the practical and predictive significance of the Nelson-Siegel class of models. Table 6. 1. Predictive Performance Summary Model Nelson-Siegel 3-Factor Nelson-Siegel 4-Factor Random Walk

Within Sample 0.002864321 0.002815731 0.004198101

Out-of-sample 0.00099207 0.001071148 0.003692655

*lower value denotes better predictive performance

Lastly, acknowledging the empirical determination of the Meese-Rogoff puzzle on the inability of any fundamental to predict better than a Random Walk, the study aims to further examine the prediction capability of the Nelson-Siegel class of models through undergoing both within-sample and out-of-sample prediction tests between the three-factor model, four-factor model, and the Random Walk model. With this, the Root Mean Squared Error tests show that the Nelson-Siegel class of models outperform the Random Walk model in both within-sample and out-of-sample prediction in the 1-month horizon.

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6.2 Conclusion The four objectives of the study revolved around investigating the capability of the Nelson-Siegel class of models in predicting the exchange rate changes in the Philippines. First, the study aims to investigate the Nelson-Siegel class of models’ significance in predicting exchange rate changes. Second, it sought to determine the most significant Nelson-Siegel model among the three and the four-factor. Third, it aims to examine which Nelson-Siegel Relative Dynamic Factor is the most significant in prediction. Fourth, the study aims to establish the predictive performance of the Nelson-Siegel class of models when compared to the Random Walk.

Given the statistical analyses of the Nelson-Siegel class of models, the study confirms that the Relative Nelson-Siegel class of models, through its relative dynamic factors, exhibits predictive ability for exchange rate changes in the Philippines. A result that is consistent with previous Nelson-Siegel exchange rate application studies in different countries (Ishii, 2018; Kurti & Vasstrand, 2018; Chen & Tsang, 2013). Moreover, the Relative Nelson Siegel four-factor model is the most significant in predicting exchange rate changes with an adjusted R squared value higher than that of the three-factor model. In addition, the level-factor of the four-factor model was the only factor to show statistical significance, but statistical significance of the relative dynamic factors vary by countrypair (Chen & Tsang, 2013). Lastly, the statistical test for RMSE both within-sample and

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out-of-sample determines that the Nelson-Siegel class of models exhibits substantially better predictive performance than the Random Walk model in both within-sample and out-of-sample prediction which is consistent the results of the study by Ishii (2018).

Overall, the results of the study show that the relative dynamic Nelson-Siegel factors of the United States and the Philippines contain information on the future changes of the USD/PHP Exchange Rate. In line with previous studies, the use of relative yield curves is, in fact, reliable in predicting exchange rate changes. Thus, relevant stakeholders should consider the use of yield curve prediction through the Relative Dynamic NelsonSiegel model to aid and develop better decision-making in foreign exchange transactions.

6.3 Recommendations Despite the outperforming the Random Walk model, the results of the do not suggest an absolute utilization of the Nelson-Siegel three and four-factor model as sole exchange rate predictive models for any stakeholder. Moreover, the importance of solving the multicollinearity issue is reiterated through ridge regression for any stakeholder that will consider the use of the Nelson-Siegel class of models. This problem was not addressed in the study given that it focuses on assessing the overall impact of the Nelson-Siegel factors on predicting exchange rate changes. Addressing the problem only becomes an issue when the factors themselves are being predicted which is the exact practical

69

application that our study suggests. Assuming that multicollinearity is solved,

it is

recommended that:

For Bangko Sentral ng Pilipinas, the study recommends considering the use of the models to potentially improve the implementation of monetary policies, or rather interest rates, in stabilizing inflation rates to favor the Philippine economy. With this, there will be a moderate economic growth and a reduction in inflationary pressure.

For banks, the study recommends the use of the model as additional tools to mitigate the foreign exchange risk that occurs in the respective bank. Therefore, banks will have the ability to (1) forecast the change in domestic and/or foreign currency and (2) managing risk during the closing out of a currency.

For participants in the force (FX) market, the proponents recommend using the models as additional indicators to make more educated market decisions. Hence, the improved ability to profit from forward rate transactions.

For businesses, the researches recommend the use of the models as instruments to heighten the advantage in making business decisions with foreign exchange as a basis. Those particularly engaged in the importation and exportation of foods and services will be those who profit. 70

For future researchers, despite the positive results of the study, the proponents would like to note that the study still has several areas for further improvement and research. A number of considerations that future researchers should account for would be: (1) study the rest of the Nelson-Siegel class of models, (2) include longer time horizons for predictive performance analysis, (3) fix the model’s violations of regression assumptions, (4) add other exchange rate prediction models for comparison.

The Nelson-Siegel five and six-factor As stated in the scope and limitations, the proponents only conducted a study on the three and four-factor Nelson-Siegel model due to time and technical constraints. During the writing of this study, there is no statistical package that can automate the process of obtaining the optimal lambda parameters together with the estimation of the relative dynamic Nelson-Siegel factors. However, this can be solved through customizing and hardcoding the existing statistical package which would require a lot of processing time. Also, the researchers would recommend the application of the remaining models of the Nelson-Siegel class of models due to the models’ ability to further smooth the estimation of the yield curve which, in theory, will produce better estimates of the factors.

Longer Time Horizons for Predictive Performance The proponents of the study only focused on the short-term application of the NelsonSiegel three and four-factor. With this, there is much left to be examined on the long-term 71

prediction capability of the Nelson-Siegel class of models. This recommendation would contribute greatly to future research since it would determine the time horizon that the models are most effective in.

Solving violations of Regression Assumptions As stated in previous sections of the study, the Nelson-Siegel three-factor model violated normality of errors while the four-factor model violated multicollinearity. Moreover, solving the violations would allow the future Philippine application research to be more in-line with the methodology used in previous literature and regression standards. Additionally, the problem of multicollinearity is a well-recognized problem of the NelsonSiegel class of models which can be resolved through ridge regression (Annaert et. al, 2013).

Comparison with other exchange rate predictive models The study mainly focused on addressing the Meese-Rogoff puzzle which determined the Random Walk as the most superior exchange rate determination model. However, it is believed that there are other exchange rate determination models that can be compared with the Nelson-Siegel class of models. With this, future researchers on the Nelson-Siegel class of models may consider interviewing industry experts to determine which additional models to add for comparison.

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APPENDIX A: Statistical Codes #load required packages library(YieldCurve) library(readxl) library(ggplot2) library(DescTools) library(tseries) library(openxlsx) library(psych) library(gridExtra) library(olsrr) library(Hmisc) #read the datasets ph_zero_coupon<-read_xlsx("Zero-Coupon-Data.xlsx",sheet = "ph_zero_coupon") us_zero_coupon<-read_xlsx("Zero-Coupon-Data.xlsx",sheet = "us_zero_coupon") ph_usd<-read_xlsx("USD_PHP-Historical-Data.xlsx","forex") Interest Rate Differentials #compute the interest rate differentials mydata<-cbind.data.frame("Date"=ph_zero_coupon$Date,us_zero_coupon[,2:14]ph_zero_coupon[,2:14]) delta<-diff(log(ph_usd$Price),differences = 1) ph_usd$delta<-c(delta,NA) ph_usd<-xts(ph_usd[,2:7],order.by = ph_usd$Date) Descriptive Statistics #plot the exhange rate data p1<-ggplot(ph_usd,aes(x=Index,y=Price))+geom_line(col="blue")+ xlab("Year")+ylab("USD-PHP Exchange Rate") p2<-ggplot(ph_usd,aes(x=Index,y=`Change %`))+geom_line(col="red")+ scale_y_continuous(labels=scales::percent)+ xlab("Year")+ylab("Change (in %)") 81

grid.arrange(p1,p2)

in_data<-xts(mydata[,2:14],order.by = mydata$Date)

maturity_a<-c(3,6,12,48,60,72,84,96,108,120,180,240,360) #set the maturity tenor in months Estimating Nelson-Siegel Factors #compute for the NS dynamic factors ph_ns<-Nelson.Siegel(rate=in_data,maturity=maturity_a) ph_sv<-Svensson(rate=in_data,maturity=maturity_a) #merge with the exchange rate data in_factors_ns<-merge.xts(ph_usd,xts(ph_ns[,2:5],order.by = ph_ns$Date),join = "inner") in_factors_sv<-merge.xts(ph_usd,xts(ph_sv[,2:7],order.by = ph_sv$Date),join = "inner") #plot the NS dynamic factors #3-factor model p1<-ggplot(in_factors_ns,aes(x=Index,y=beta_0))+geom_line(col="#F8766D")+ xlab("Year")+ylab("Level Factors") p3<-ggplot(in_factors_ns,aes(x=Index,y=beta_1))+geom_line(col="#00BFC4")+ xlab("Year")+ylab("Slope Factors") p4<-ggplot(in_factors_ns,aes(x=Index,y=beta_2))+geom_line(col="#7CAE00")+ xlab("Year")+ylab("Curvature Factors") p2<-ggplot(in_factors_ns,aes(x=Index,y=Change..))+geom_line(col="#C77CFF")+ xlab("Year")+ylab("Exchange Rate Change")+ scale_y_continuous(labels=scales::percent) grid.arrange(p1,p3,p4) #4-factor model p1<-ggplot(in_factors_sv,aes(x=Index,y=beta_0))+geom_line(col="#F8766D")+ xlab("Year")+ylab("Level Factors") p3<-ggplot(in_factors_sv,aes(x=Index,y=beta_1))+geom_line(col="#00BFC4")+ xlab("Year")+ylab("Slope Factors")

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p4<-ggplot(in_factors_sv,aes(x=Index,y=beta_2))+geom_line(col="#7CAE00")+ xlab("Year")+ylab("First Curvature Factors") p2<-ggplot(in_factors_sv,aes(x=Index,y=beta_3))+geom_line(col="#C77CFF")+ xlab("Year")+ylab("Second Curvature Factors") grid.arrange(p1,p3,p4,p2) Regression Analysis #set the training data for the regression train1<-in_factors_ns["2018"] train2<-in_factors_sv["2018"] #Regression Models #NS 3-factor model model1<-lm(delta~beta_0+beta_1+beta_2,data=train1) summary(model1) Regression Diagnostics #diagnostics ols_plot_diagnostics(model1) #diagnostic plots ols_coll_diag(model1) #multicollinearity DurbinWatsonTest(delta~beta_0+beta_1+beta_2,data=train1) #autocorrelation shapiro.test(model1$residuals) #normality test jarque.bera.test(model1$residuals) #normality test ols_test_breusch_pagan(model1,rhs = T) #heteroscedasticity test #NS 4-factor model model2<-lm(delta~beta_0+beta_1+beta_2+beta_3,data=train2) summary(model2) #Model 2 diagnostics ols_plot_diagnostics(model2) #diagnostic plots ols_coll_diag(model2) #multicollinearity DurbinWatsonTest(delta~beta_0+beta_1+beta_2+beta_3,data=train2) #autocorrelation shapiro.test(model2$residuals) #normality test

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jarque.bera.test(model2$residuals) #normality test ols_test_breusch_pagan(model2,rhs = T) #heteroscedasticity test Prediction Accuracy Tests #Prediction Accuracy sqrt(sum(model1$residuals^2)/NROW(model1$residuals)) #NS 3 factor model sqrt(sum(model2$residuals^2)/NROW(model2$residuals)) #NS 4 factor model z<-na.omit(train1$delta-lag(train1$delta,k=1)) sqrt(sum(z^2)/NROW(z)) #Random Walk model #out-of-sample predictions #load the test datasets jan2019_ph_rates<-read_xlsx("Zero-Coupon-Data-Jan-2019.xlsx",sheet "ph_zero_coupon") jan2019_us_rates<-read_xlsx("Zero-Coupon-Data-Jan-2019.xlsx",sheet "us_zero_coupon") jan2019_ph_usd<-read_xlsx("USD_PHP-Historical-Data-January-2019.xlsx",sheet "data")

= = =

delta<-diff(log(jan2019_ph_usd$Price),differences = 1) jan2019_ph_usd$delta<-c(delta,NA) jan2019_ph_usd<-xts(jan2019_ph_usd[,2:7],order.by = jan2019_ph_usd$Date) jan2019_rates
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#compute for the Jan 2019 forecasts yhat<-predict(object = model1,jan2019_ns) yhat_sv<-predict(object = model2,jan2019_sv) #out-of-sample prediction accuracies sqrt(sum((jan2019_ph_usd[-1,]$delta-yhat[-1]))^2/(NROW(yhat)-1)) #NS 3 factor model sqrt(sum((jan2019_ph_usd[-1,]$delta-yhat_sv[-1]))^2/(NROW(yhat_sv)-1)) #NS 4 factor model z<-na.omit(jan2019_ph_usd$delta-lag(jan2019_ph_usd$delta,k=1)) sqrt(sum(z^2)/NROW(z))

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APPENDIX B: Data (Sample) Philippine Zero Coupon

Start Date 1/1/2010 End Date 12/31/2018 3-Month I10503M Index Last Price Dates PX_LAST 1/1/2010 10.546 1/4/2010 12.426 1/5/2010 12.425 1/6/2010 12.426 1/7/2010 12.426 1/8/2010 12.426 1/11/2010 12.426 1/12/2010 12.426 1/13/2010 12.426 1/14/2010 12.426 1/15/2010 12.426 1/18/2010 12.426 1/19/2010 12.426 1/20/2010 12.426 1/21/2010 12.426 1/22/2010 12.427 1/25/2010 6.262 1/26/2010 6.255 1/27/2010 6.34 1/28/2010 11.799 1/29/2010 11.799 2/1/2010 11.8 2/2/2010 11.799 2/3/2010 11.799 2/4/2010 11.799 2/5/2010 11.799 2/8/2010 11.799 2/9/2010 11.799 2/10/2010 11.799 2/11/2010 11.799 2/12/2010 11.8 2/15/2010 11.799 2/16/2010 11.799 2/17/2010 11.799 2/18/2010 11.799 2/19/2010 4.947 2/22/2010 4.946 2/23/2010 4.945 2/24/2010 6.322 2/25/2010 6.268 2/26/2010 6.325 6-Month I10506M Index Last Price Dates PX_LAST 1/1/2010 10.546 1/4/2010 12.426 1/5/2010 12.425 1/6/2010 12.426 1/7/2010 12.426 1/8/2010 12.426 1/11/2010 12.426 1/12/2010 12.426 1/13/2010 12.426 1/14/2010 12.426 1/15/2010 12.426 1/18/2010 12.426 1/19/2010 12.426 1/20/2010 12.426 1/21/2010 12.426 1/22/2010 12.427 1/25/2010 6.262 1/26/2010 6.255 1/27/2010 6.34 1/28/2010 11.799 1/29/2010 11.799 2/1/2010 11.8 2/2/2010 11.799 2/3/2010 11.799 2/4/2010 11.799 2/5/2010 11.799 2/8/2010 11.799 2/9/2010 11.799 2/10/2010 11.799 2/11/2010 11.799 2/12/2010 11.8 2/15/2010 11.799 2/16/2010 11.799 2/17/2010 11.799 2/18/2010 11.799 2/19/2010 4.947 2/22/2010 4.946 2/23/2010 4.945 2/24/2010 6.322 2/25/2010 6.268 2/26/2010 6.325 1-Year I10501Y Index Last Price Dates PX_LAST 1/1/2010 10.546 1/4/2010 12.426 1/5/2010 12.425 1/6/2010 12.426 1/7/2010 12.426 1/8/2010 12.426 1/11/2010 12.426 1/12/2010 12.426 1/13/2010 12.426 1/14/2010 12.426 1/15/2010 12.426 1/18/2010 12.426 1/19/2010 12.426 1/20/2010 12.426 1/21/2010 12.426 1/22/2010 12.427 1/25/2010 6.262 1/26/2010 6.255 1/27/2010 6.34 1/28/2010 11.799 1/29/2010 11.799 2/1/2010 11.8 2/2/2010 11.799 2/3/2010 11.799 2/4/2010 11.799 2/5/2010 11.799 2/8/2010 11.799 2/9/2010 11.799 2/10/2010 11.799 2/11/2010 11.799 2/12/2010 11.8 2/15/2010 11.799 2/16/2010 11.799 2/17/2010 11.799 2/18/2010 11.799 2/19/2010 4.947 2/22/2010 4.946 2/23/2010 4.945 2/24/2010 6.322 2/25/2010 6.268 2/26/2010 6.325

4-Year I10504Y Index Last Price Dates PX_LAST 1/1/2010 11.861 1/4/2010 12.426 1/5/2010 12.425 1/6/2010 12.426 1/7/2010 12.426 1/8/2010 12.426 1/11/2010 12.426 1/12/2010 12.426 1/13/2010 12.426 1/14/2010 12.426 1/15/2010 12.426 1/18/2010 12.426 1/19/2010 12.426 1/20/2010 12.426 1/21/2010 12.426 1/22/2010 12.427 1/25/2010 6.262 1/26/2010 6.255 1/27/2010 6.34 1/28/2010 5.926 1/29/2010 5.949 2/1/2010 6.023 2/2/2010 6.046 2/3/2010 6.044 2/4/2010 6.055 2/5/2010 6.096 2/8/2010 6.179 2/9/2010 6.265 2/10/2010 6.25 2/11/2010 6.313 2/12/2010 6.251 2/15/2010 6.376 2/16/2010 6.438 2/17/2010 6.375 2/18/2010 6.402 2/19/2010 6.979 2/22/2010 6.826 2/23/2010 6.895 2/24/2010 6.794 2/25/2010 6.761 2/26/2010 6.831

5-Year I10505Y Index Last Price Dates PX_LAST 1/1/2010 12.106 1/4/2010 11.705 1/5/2010 11.687 1/6/2010 11.673 1/7/2010 11.67 1/8/2010 11.669 1/11/2010 11.645 1/12/2010 11.641 1/13/2010 11.631 1/14/2010 11.636 1/15/2010 11.619 1/18/2010 11.578 1/19/2010 11.594 1/20/2010 11.573 1/21/2010 11.565 1/22/2010 11.554 1/25/2010 13.303 1/26/2010 13.275 1/27/2010 13.262 1/28/2010 12.467 1/29/2010 12.451 2/1/2010 12.414 2/2/2010 12.381 2/3/2010 12.392 2/4/2010 12.383 2/5/2010 12.37 2/8/2010 12.328 2/9/2010 12.312 2/10/2010 12.307 2/11/2010 12.286 2/12/2010 12.288 2/15/2010 12.239 2/16/2010 12.236 2/17/2010 12.24 2/18/2010 12.213 2/19/2010 13.151 2/22/2010 14.29 2/23/2010 14.301 2/24/2010 14.082 2/25/2010 14.087 2/26/2010 14.052

6-Year I10506Y Index Last Price Dates PX_LAST 1/1/2010 8.939 1/4/2010 8.699 1/5/2010 8.644 1/6/2010 8.613 1/7/2010 8.63 1/8/2010 8.66 1/11/2010 8.648 1/12/2010 8.659 1/13/2010 8.648 1/14/2010 8.697 1/15/2010 8.654 1/18/2010 8.567 1/19/2010 8.661 1/20/2010 8.602 1/21/2010 8.595 1/22/2010 8.578 1/25/2010 9.554 1/26/2010 9.478 1/27/2010 9.555 1/28/2010 9.069 1/29/2010 9.043 2/1/2010 9.009 2/2/2010 8.92 2/3/2010 8.978 2/4/2010 8.97 2/5/2010 8.965 2/8/2010 8.916 2/9/2010 8.915 2/10/2010 8.911 2/11/2010 8.885 2/12/2010 8.884 2/15/2010 8.829 2/16/2010 8.865 2/17/2010 8.868 2/18/2010 8.807 2/19/2010 9.363 2/22/2010 9.676 2/23/2010 9.72 2/24/2010 9.591 2/25/2010 9.602 2/26/2010 9.582

7-Year I10507Y Index Last Price Dates PX_LAST 1/1/2010 6.565 1/4/2010 6.445 1/5/2010 6.369 1/6/2010 6.332 1/7/2010 6.374 1/8/2010 6.422 1/11/2010 6.452 1/12/2010 6.478 1/13/2010 6.473 1/14/2010 6.555 1/15/2010 6.512 1/18/2010 6.428 1/19/2010 6.572 1/20/2010 6.498 1/21/2010 6.501 1/22/2010 6.478 1/25/2010 7.081 1/26/2010 7 1/27/2010 7.119 1/28/2010 6.797 1/29/2010 6.774 2/1/2010 6.763 2/2/2010 6.683 2/3/2010 6.764 2/4/2010 6.747 2/5/2010 6.752 2/8/2010 6.727 2/9/2010 6.774 2/10/2010 6.779 2/11/2010 6.739 2/12/2010 6.753 2/15/2010 6.718 2/16/2010 6.768 2/17/2010 6.77 2/18/2010 6.698 2/19/2010 7.092 2/22/2010 6.993 2/23/2010 7.08 2/24/2010 7.013 2/25/2010 7.031 2/26/2010 7.017

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Date 1/ 1/ 1/ 1/ 1/ 1/ 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/1 2/1 2/1 2/1 2/1 2/1 2/1 2/1 2/2 2/2 2/2 2/2 2/2

United States Zero Coupon

Start Date End Date

Dates 1/1/2010 1/4/2010 1/5/2010 1/6/2010 1/7/2010 1/8/2010 1/11/2010 1/12/2010 1/13/2010 1/14/2010 1/15/2010 1/18/2010 1/19/2010 1/20/2010 1/21/2010 1/22/2010 1/25/2010 1/26/2010 1/27/2010 1/28/2010 1/29/2010 2/1/2010 2/2/2010 2/3/2010 2/4/2010 2/5/2010 2/8/2010 2/9/2010 2/10/2010 2/11/2010 2/12/2010 2/15/2010 2/16/2010 2/17/2010 2/18/2010 2/19/2010 2/22/2010 2/23/2010 2/24/2010 2/25/2010 2/26/2010

12/31/2009 2/6/2018 3-Month F08203M Index Last Price PX_LAST 0.06 0.069 0.071 0.053 0.052 0.055 0.027 0.045 0.053 0.056 0.04 0.045 0.062 0.058 0.054 0.054 0.048 0.061 0.081 0.094 0.051 0.085 0.097 0.097 0.094 0.051 0.064 0.084 0.076 0.066 0.056 0.059 0.096 0.087 0.11 0.058 0.091 0.108 0.108 0.11 0.11

Dates 1/1/2010 1/4/2010 1/5/2010 1/6/2010 1/7/2010 1/8/2010 1/11/2010 1/12/2010 1/13/2010 1/14/2010 1/15/2010 1/18/2010 1/19/2010 1/20/2010 1/21/2010 1/22/2010 1/25/2010 1/26/2010 1/27/2010 1/28/2010 1/29/2010 2/1/2010 2/2/2010 2/3/2010 2/4/2010 2/5/2010 2/8/2010 2/9/2010 2/10/2010 2/11/2010 2/12/2010 2/15/2010 2/16/2010 2/17/2010 2/18/2010 2/19/2010 2/22/2010 2/23/2010 2/24/2010 2/25/2010 2/26/2010

6-Month F08206M Index Last Price PX_LAST 0.167 0.16 0.137 0.121 0.134 0.136 0.105 0.114 0.12 0.117 0.101 0.105 0.126 0.117 0.116 0.116 0.109 0.113 0.165 0.163 0.125 0.156 0.164 0.166 0.151 0.123 0.144 0.157 0.098 0.142 0.133 0.137 0.172 0.174 0.198 0.145 0.181 0.181 0.172 0.169 0.172

Dates 1/1/2010 1/4/2010 1/5/2010 1/6/2010 1/7/2010 1/8/2010 1/11/2010 1/12/2010 1/13/2010 1/14/2010 1/15/2010 1/18/2010 1/19/2010 1/20/2010 1/21/2010 1/22/2010 1/25/2010 1/26/2010 1/27/2010 1/28/2010 1/29/2010 2/1/2010 2/2/2010 2/3/2010 2/4/2010 2/5/2010 2/8/2010 2/9/2010 2/10/2010 2/11/2010 2/12/2010 2/15/2010 2/16/2010 2/17/2010 2/18/2010 2/19/2010 2/22/2010 2/23/2010 2/24/2010 2/25/2010 2/26/2010

1-Year F08201Y Index Last Price PX_LAST 0.496 0.47 0.441 0.424 0.438 0.406 0.371 0.371 0.386 0.367 0.331 0.333 0.356 0.356 0.335 0.324 0.328 0.332 0.337 0.313 0.291 0.31 0.302 0.336 0.301 0.287 0.301 0.336 0.303 0.346 0.322 0.327 0.327 0.346 0.398 0.336 0.357 0.342 0.324 0.301 0.309

Dates 1/1/2010 1/4/2010 1/5/2010 1/6/2010 1/7/2010 1/8/2010 1/11/2010 1/12/2010 1/13/2010 1/14/2010 1/15/2010 1/18/2010 1/19/2010 1/20/2010 1/21/2010 1/22/2010 1/25/2010 1/26/2010 1/27/2010 1/28/2010 1/29/2010 2/1/2010 2/2/2010 2/3/2010 2/4/2010 2/5/2010 2/8/2010 2/9/2010 2/10/2010 2/11/2010 2/12/2010 2/15/2010 2/16/2010 2/17/2010 2/18/2010 2/19/2010 2/22/2010 2/23/2010 2/24/2010 2/25/2010 2/26/2010

F08203Y Index Last Price PX_LAST 1.682 1.632 1.573 1.576 1.603 1.574 1.528 1.471 1.527 1.483 1.411 1.415 1.466 1.455 1.404 1.378 1.389 1.385 1.465 1.396 1.354 1.419 1.419 1.423 1.329 1.306 1.311 1.407 1.405 1.44 1.399 1.403 1.364 1.441 1.534 1.483 1.512 1.437 1.434 1.376 1.36

Dates

1/1/2010 1/4/2010 1/5/2010 1/6/2010 1/7/2010 1/8/2010 1/11/2010 1/12/2010 1/13/2010 1/14/2010 1/15/2010 1/18/2010 1/19/2010 1/20/2010 1/21/2010 1/22/2010 1/25/2010 1/26/2010 1/27/2010 1/28/2010 1/29/2010 2/1/2010 2/2/2010 2/3/2010 2/4/2010 2/5/2010 2/8/2010 2/9/2010 2/10/2010 2/11/2010 2/12/2010 2/15/2010 2/16/2010 2/17/2010 2/18/2010 2/19/2010 2/22/2010 2/23/2010 2/24/2010 2/25/2010 2/26/2010

4-Year F08204Y Index Last Price PX_LAST

2.222 2.186 2.124 2.147 2.178 2.161 2.12 2.039 2.107 2.063 1.99 1.994 2.053 2.027 1.966 1.959 1.966 1.947 2.016 1.942 1.904 1.985 1.934 1.978 1.863 1.833 1.835 1.936 1.924 1.97 1.932 1.936 1.905 2.001 2.088 2.03 2.072 1.973 1.988 1.918 1.896

87

Bloomberg Terminal Screenshots

88

APPENDIX C: Turnitin Receipt/Report

89

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