Parametric Portfolio Policy using Currencies as an Asset Class

Parametric Portfolio Policy using

Currencies as an Asset Class

A dissertation by

Arnar Ingi Einarssoncpr. 221180-3145

submitted to the Faculty of Economics in partial

fulllment of the requirements for the degree of

Master of Science in Advanced Economics and Finance.

Supervisor: Lisbeth la Cour, PhD

Copenhagen Business School

December 6, 2012

CBS-MS-2012-1

Pages:76 Characters: 112.000

Copenhagen Business School

Faculty of Economics

Porcelænshaven 16 A, 1 DK-2000 Frederiksberg

Tlf: +45 3815 2575, Fax: +45 3815 2576

www.cbs.dk

Preface

This thesis was prepared at the Department of Economics, of CopenhagenBusiness School in Denmark in partial fulfillment of the requirements foracquiring the Master of Science degree in Advanced Economics and Finance.The project was carried out over the period from June 1st 2012 to December6th 2012.The subject of the thesis is practical asset allocation with the parametricportfolio policy using currencies as an asset class.

Frederiksberg, Desember 2012Arnar Ingi Einarsson

Summary

A recent innovation in asset allocation was introduced by Brandt et al. [2009]where they propose a dynamic parametric model for asset allocation calledParametric Portfolio Policy (PPP). The method is very adaptable and can beused on different asset classes. The aim of this thesis is to analyse how itwould perform when using currencies as an asset class.The PPP was found to give robust performance in and out of sample. Themodel was benchmarked to the APT model and was found to give superiorperformance. The limitations to the method were found to be that it onlyconsiders asset specific variables and that it requires considerable modeling.In order to pave the way of PPP into practice the modeling code is supplied.

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aim of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Asset Management 4

2.1 Asset Management . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 Risk Management . . . . . . . . . . . . . . . . . . . . . 62.1.2 On Risk in Asset Management . . . . . . . . . . . . . . 9

2.2 Asset Allocation Methods . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Deficiency of Markowitz . . . . . . . . . . . . . . . . . . 102.2.2 Fixes to the Markowitz deficiencies . . . . . . . . . . . 12

CONTENTS vi

2.2.3 Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . 132.3 PPP Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Estimation and Statistical Inference . . . . . . . . . . . 162.3.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Concluding on Asset Management . . . . . . . . . . . . . . . . 183 Monetary Economics 21

3.1 Monetary Economics Theory . . . . . . . . . . . . . . . . . . . . 213.1.1 Exchange Rates . . . . . . . . . . . . . . . . . . . . . . . 223.1.2 Covered Interest Parity, CIP . . . . . . . . . . . . . . . 233.1.3 Uncovered Interest Parity, UIP . . . . . . . . . . . . . . 233.1.4 Purchasing Power Parity, PPP . . . . . . . . . . . . . 243.1.5 Complete Theory of Exchange Rates . . . . . . . . . . 26

3.2 Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 The Toolbox of Central Bankers . . . . . . . . . . . . . 313.2.2 History of Currencies . . . . . . . . . . . . . . . . . . . 33

3.3 Practical Monetary Economics . . . . . . . . . . . . . . . . . . . 363.3.1 The Market for Foreign Exchange . . . . . . . . . . . . 363.3.2 Currency Analysis . . . . . . . . . . . . . . . . . . . . . 383.3.3 Economic factors . . . . . . . . . . . . . . . . . . . . . . 38

CONTENTS vii

3.3.4 Market Sentiment - The Real Market Movers . . . . . 403.4 Technical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Methodology and Data 43

4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.1.1 Different Modeling procedures . . . . . . . . . . . . . . 464.1.2 Statistical Inference . . . . . . . . . . . . . . . . . . . . 484.1.3 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.1 Optimal set of Data . . . . . . . . . . . . . . . . . . . . 534.2.2 Data Resources . . . . . . . . . . . . . . . . . . . . . . . 544.2.3 Limitations of the Analysis . . . . . . . . . . . . . . . . 60

5 Modeling Results 62

5.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.1 Comparison of Modeling Procedures . . . . . . . . . . 645.2.2 Results for P2 . . . . . . . . . . . . . . . . . . . . . . . 655.2.3 Bootstrapping Results - Standard error . . . . . . . . 71

6 Conclusion 73

CONTENTS viii

6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A Data Appendix 77

A.1 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . 77B Programming 88

B.1 The R Language . . . . . . . . . . . . . . . . . . . . . . . . . . . 88B.2 R code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Bibliography 90

List of Tables

5.1 Results for different modeling procedures. . . . . . . . . . . . . 665.2 Results for Quarterly data from modeling process P2. . . . . . 675.3 Results for Monthly data modeling process P2. . . . . . . . . . 695.4 Results for Daily data from modeling process P2. . . . . . . . 705.5 Currency and period specific results from Model (26). . . . . . 71

List of Figures

2.1 Asset Management Process . . . . . . . . . . . . . . . . . . . . 52.2 Asset Management Process that Incorporates Investor Views . 194.1 Simple Regression like Modeling Process P1. . . . . . . . . . 474.2 Stepwise Modeling Process P2. . . . . . . . . . . . . . . . . . . 484.3 One Step Modeling Process P3. . . . . . . . . . . . . . . . . . 484.4 Quarterly Exchange Rates Returns from Q3-1992 to Q2-2012.

Data are centered to the first observation . . . . . . . . . . . . 565.1 Mean Variance Analysis with Efficient Frontiers. . . . . . . . . 635.2 Cumulative Return of Different Modeling Procedures. . . . . . 65A.1 Monthly Exchange Rates Returns from Q3-1992 to Q2-2012.

Data are indexed to the first observation . . . . . . . . . . . . 79

LIST OF FIGURES xi

A.2 Correlation of Monthly Exchange Rates for the period Q3-1992 to Q2-2012. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.3 GDP from Q3-1992 to Q2-2012. . . . . . . . . . . . . . . . . . 81A.4 Correlation of GDP from Q3-1992 to Q2-2012. . . . . . . . . . 82A.5 Foreign Reserves from Q3-1992 to Q2-2012. It is apparent

that smaller economies hold larger proportions of their GDPin foreign reserves. . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.6 Current Account as a ratio of GDP from Q3-1992 to Q2-2012. 84A.7 10 Year Bond Yields from Q3-1992 to Q2-2012. . . . . . . . . 85A.8 Consumer Price Index from Q3-1992 to Q2-2012. . . . . . . . 86A.9 M2 Money Supply from Q3-1992 to Q2-2012. . . . . . . . . . 87

Chapter 1Introduction

1.1 Background

A recent innovation in asset allocation was introduced by Brandt et al. [2009],where they propose a new method for optimizing portfolios. This methodnamed parametric portfolio policy (PPP) by the authors should be of greatinterests to participants in financial markets. As the name implies this is aparametric method, where each asset’s portfolio weight are directly estimatedby using asset specific variables. This computationally simple method caneasily be modified and extended to meet different purposes. A great featureof the PPP is that it can be applied to all asset classes i.e. stocks, bondsand currency portfolios. As currencies are of major importance in financialmarkets, it is chosen as the asset class under consideration. Currencies areprobable the single most important factor in asset allocation as they are anasset type, and act as a denominator in stocks, commodities and bonds.The PPP seems to be well suited for constructing currency portfolio’s as ituses asset-specific variables to optimally select portfolio weights. Currenciesare generally seen as a reflection of the health of the economy where it is

1.2 Aim of thesis 2

used and there are quite many metrics on economic health available.1.2 Aim of thesis

The objective of this thesis is to show the PPP in action, document the mod-elling procedure and evaluate its performance in predicting return from cur-rencies. Furthermore, to show that it can be applied with data that are ofdifferent frequencies, i.e. quarterly, monthly and daily data. This would be ofgreat importance as economic data are released with different frequencies.As the quality of data available to students is not compatible to the qualitythat can be expected to be available to banks. Due to this lack of quality datathere is thus little focus on analysing the effect of each descriptive variableon model performance1. The focus will be on contributing to the theory byproviding a practical perspective on how PPP could be developed to be usedin practice.A comprehensive guide to how to implement PPP will be provided and sug-gestions on how the modeling process could be improved will be provided. AllR code will be made available, which will hopefully ease the way for PPP tobecome used in practice.The performance of the PPP model is benchmarked to the Markowitz modeland the Arbitrage Pricing Theory (APT). The PPP outperformed the bench-mark models but the PPP is a dynamic model whereas the others are static.Modeling issues and limitations to the PPP are documented and discussed.It is concluded that the PPP model is attractive in an economic sense asit seems to be able to be implemented into a profit machine and could assuch be applied by market participant, whether it is central banks, or in highfrequency trading. Furthermore, it is concluded that the PPP could lead toadvances in academic research on how financial markets work in practice, thatcould eventually lead to policy changes for the greater good of the generalpublic.

1Data were collected through Thomson Reuters DataStream database Thomson Reuters[2012] spanning the past twenty years.

1.3 Outline of thesis 3

1.3 Outline of thesis

This thesis is related to two different fields of study, Asset Allocation andMonetary Economics and moreover the practical side of those fields.The structure of the thesis is as follows.Chapter 2: Asset Management. Introduces the asset management processand the PPP methodology along with other asset allocation methods.Chapter 3: Monetary Economics. Gives a brief discussion on theory andapplied aspects of monetary economics.Chapter 4: Methodology and Data. Gives a quite detailed description onthe methodology used and description of the data used in the analysis.Chapter 5: Modeling Results. The main findings are presented.Chapter 6: Conclusion. Concludes on the thesis and includes a sectionabout further work.Appendix A: Data Appendix. Additional description of the data used in thethesis.Appendix B: Programming. An introduction to R, the programming languageused. Includes a description of how the modeling results can be re-generated.

Chapter 2Asset Management

This chapter focuses on the theory of asset management and introduces theasset management process. Fundamental asset allocation methods are pre-sented and discussed.In Section 2.1 the asset management process is discussed and the role ofcurrencies in financial markets is explored. In Section 2.2 fundamental assetallocation methods are presented and discussed. Section 2.3 introduces theParametric Portfolio Policy (PPP) method and Section 2.4 concludes on assetmanagement.2.1 Asset Management

As the main subject of this thesis is practical asset management, using cur-rencies as an asset class, it is an ideal starting point to define what assetmanagement is and which role currencies play in asset management. As-set management can be defined as the practice of managing assets in order

2.1 Asset Management 5

Model Selection

Preferences Decision Model

Portfolio Weights Return

Expected Return & Risk

Measureable Risk

Risk Assessment

Un-measureable Risk

Investor

Asset Manager

Figure 2.1: Asset Management Process

to achieve the greatest historical return as possible. This definition can bedeepened further by defining asset management as the ongoing process ofmaximizing returns while minimizing risk for all imaginable horizons. It isthus necessary to evaluate potential returns and all possible risk factors in-fluencing each asset class as well as individual asset.Considering the asset management process in Figure 2.1, investors reallyface a numerous set of separate problems. First, is the definition of investorpreferences, i.e. the risk that the investor is willing to accept, in order toattain a certain return. Second, is to develop a risk assessment framework,i.e. a methods for measuring and forecasting risk and return of assets. Therisk exposure can be divided into measurable and unmeasurable risk. It isimportant to note that measurable risk is the only risk considered in a riskassessment process. Measurable risk can then be divided further into assetspecific and asset unspecific risk. The framework for risk assessments mustbe intuitively appealing and understandable to the investor.Third, is the model selection process. There is a wide variety of risk as-sessment models within the asset pricing and asset allocation framework, allwith their particular strengths and weaknesses. Finally, there is the issue ofestimation error. The quality of the estimation and forecasting of measurablerisk is difficult to estimate ex ante. Decisions made based on forecasts of


measurable risk, are subjected to measurement error. The realized return isalso subjected to unmeasurable risk. Even ignoring the fact that markets canact quite irrationally, it can be concluded that asset management is not aneasy task.2.1.1 Risk Management

Assets under management can be be divided into groups such as equities,commodities, currencies, fixed income, real estate and other properties, withthe first four asset classes traded on financial markets. From a risk perspectivedifferent asset groups are subjected to different risk factors. Introduction ofrisk factors for the different asset classes that are traded on financial marketsfollows. The discussion on fixed income is based on risk definitions fromAlexander and Sheedy [2005] and Zenios [2005], while the risk factors for theother asset classes were identified by the author. The discussion is by nomean a complete list of risk factors and is supposed to show the importanceof currencies in asset management.Fixed Income

The fixed income market is a decentralized, over-the-counter (OTC) marketsplitted into a primary and secondary markets. The yield of bonds1 reflectsthe riskiness of that particular bond. Fixed income as an asset group is subjectto counter-party or credit risk. i.e. the risk that the issuer of the bond doesnot repay the debt. The holder of the bond is then subjected to maturity ormarket risk, i.e. of change in the value of the bond from issuance to maturity.Convexity risk is the risk of adverse changes in the convexity of the bond yieldcurve. interest rate risk is the risk of changes in interest rates. Convexityrisk and interest rate risk can also be considered as part of the market risk.Currency denomination risk is the risk that the bond denomination currency

1Fixed income securities or bonds can be sectioned into many subgroups, based on theissuers credit-worthiness; investment-grade or junk (high-yield), based on the issuer; govern-ment or corporate and based on duration short-term, intermediate, long-term based on locationdomestic, foreign, emerging markets; or Convertible bond if the debt can be converted into eq-uity. There are also option embedded bonds, most commonly callable bonds where the issuerhas the right to redeem all or part of the debt before the specified maturity date.


depreciates in value. Bonds are often dollar denominated to increase theirliquidity. Then there is liquidity risk, as bond yields are also sensitive tothe level of liquidity. If a call provision exist, the investor faces a call riskthat the bond will be called before maturity, the same investor then faces areinvestment risk, i.e. the risk that she will be unable to purchase anothersecurity of similar return upon the expiration of the current security.Commodities

Commodities are traded by future contracts on regulated commodities ex-changes, in which they are bought and sold in standardized contracts. Com-modities can be divided into the subgrops of metals, energies and soft com-modities2. Commodity risk refers to the uncertainties of future market valuesand can be sectioned into supply and demand risks as there are usually un-certainties on both sides. Resources risk is the supply side risk as there isoften some uncertainty about the volume and cost of the supply. For example;the uncertainty of harvesting volumes of grains and extraction cost of metals.On the demand side there is also some uncertainty on the demand mostly dueto uncertainty of global economic health i.e. economic risk. Then there isspeculative risk, the risk that speculators make up a large demand or sellinginterest based on their speculations. This risk factor probable makes up forlarge price fluctuations in commodities3.Commodities are also subjected to currency denomination risk as they aredenominated in some currency, most often in U.S. dollars. This fact has ledthe U.S. economy to benefit from this dollar denomination of commodities, asmoney creation can be transformed into commodity creation. This fact alsoleads to external intervention risk as the U.S. and to some degree othereconomies have been running wars and overthrowing governments in order togain control of resources4.

2Metals being gold, silver, palladium, copper, etc.. Energies being; WTI crude, heating oil,gasoline, natural gas, etc.. Softs being; sugar, coffee, cocoa, grains, soy beans, livestock, etc..3The topic whether speculation can drive up commodity prices is a debated topic, forarguments fore see [Bos and van der Molen, 2010], [Masters, May 20. 2008], [Masters andWhite, 2008a,b, 2009], [Bank, 2008] and against [Irwin et al., 2009] and [Irwin and Sanders,2010]4There is very limited academic research available on this delicate subject. See [Clark,2005] and references therein.


Equities

The risk factors for equities can be sectioned into two major risk factors,systematic risk and firm-specific risk. Considering first the systematic risk, itis the risk of arbitrary price movements, of stock value, due to external factors.Systematic risk can be divided into sector risk, country risk, index risk andcurrency denomination risk. There can be an arbitrary price movement of aparticular sector, country or market index even though a single firm withinthat segmentation is doing well. As an example of country and sector risk, anItalian bank that has extremely healthy financial statements can be hugelyoversold if there is a certain risk aversion against Italian stock and the bankingsector. If that particular stock would be in some cross European stock index,which were also being sold, that would constitute to being an example of indexrisk. As an example of currency denomination risk a Chinese investor couldbe exposed to currency denomination risk if he was exposed to U.S. stock atthe time of devaluation of the dollar given that the U.S. stock value is fullyindependent or negatively effected by dollar value. Using an APT framework,Gupta and Finnerty [1992], find that currency risk is generally not priced inequity prices and thus not diversifiable.Now considering firm-specific risk, each individual stock is subjected to someoperating risk, which can be explained by uncertainty in revenues and expen-ditures. This uncertainty can be caused by currency, commodity and economi-cal risk factors. Then there is also the risk of poor management or agency costrisk. It is thus clear that equities are both directly and inderectly subjectedto currency risk.Currencies

Currencies are like the other asset classes subjected to a series of risk factors.Economic risk is the risk that the economy is under-performing the generalexpectation of the market, e.g. higher inflation, slower economic growth, etc..There are numerous economic measures published for each economy, thosemeasures will be given a more elaborate discussion in section 3.3.2. Inter-ventional risk is the risk that a central bank takes some action that influencetheir home currency, e.g. lowers interest rates. Banking risk is the risk

2.2 Asset Allocation Methods 9

that there is excessive bank lending in this currency/economy, that leads toa credit crisis that constrain economic activity and thus leads to a weakercurrency. Furthermore, there is the risk of currency crisis following a bankcrisis. Fiscal risk is the risk of politicians running to much budgetary deficitsfor a prolonged period that results in burdensome fiscal situation, where debtpayments weighs so much on the economy that it constrains economic activity.2.1.2 On Risk in Asset Management

All asset classes are then exposed to inflation risk that is the risk that thepurchasing power decreases over the investment period. All economies arethen exposed to Political risk which can be sectioned into political uncer-tainty risk and legal risk. Political uncertainty risk rises when some politicaluncertainty affects the countries economic outlook. Legal risk, is the risk thatsome legislation regarding a specific asset class is changed or that tax rateson a specific asset class is increase. Allocation risk refers to diminishingreturns due to re-allocation of assets into other asset classes, e.g. sale ofstock to buy bonds or commodities. Furthermore, all economies are subjectedto event risk, which can be a natural disaster, terrorist attack or war, etc..It is interesting to evaluate the relationship between the asset groups. Inter-estingly enough is that all asset groups that are traded on financial marketsare subject to currency risk, showing its importance in asset management.Even though it is not a dominating risk effect for all assets it is still importantif maximum return and diversification is the objective.2.2 Asset Allocation Methods

At the heart of asset allocation lies the revolutionary work of Markowitz[1952] on mean-variance analysis. Mean-variance analysis studies the tradeoffbetween portfolio reward, as measured by the portfolio expected return, andportfolio risk, as measured by its variance. The traditional Markowitz model


can be defined as follows,maxω

ω>r − ω>Σ−1ω (2.1)s.t. ω>1 = 1,where ω are the optimal portfolio weights, r are the expected returns and Σis the variance of returns.Even though the revolutionary work of Markowitz has the benefits of beingsimple and provides good intuition about the relation between returns, vari-ance as a measure of risk and correlation of assets, it is quite inefficient inpractice. Following is a discussion of some of the deficiencies.2.2.1 Deficiency of Markowitz

As the expected returns and covariance matrix need to be estimated, the portfo-lio choice problem is subjected to what is known in the literature as estimationrisk. Estimation risk is the effect of estimation error, Best and Grauer [1991]show that mean-variance efficient portfolio’s weights are extremely sensitiveto changes in asset means, even though the portfolio’s returns are virtuallyunaffected. Chopra and Ziemba [1993] confirm their findings and concludethat the impact of estimation error in the mean has up to ten times as strongimpact as an estimation error in the variance. By using a Bayesian approachBengtsson [2003] contradicts the results from Chopra and Ziemba [1993] inthat estimating the covariance matrix correctly is strictly less important thanestimating the mean vector correctly.Furthermore, estimation is conducted using historical data and does thuspresent past dynamics and furthermore depend on the quality and time span ofthe data. Even if the quality of data is perfect their might always be errors inthe estimates, consider two firms in the same sector where one is establishedand the other one has experienced high growth until now. Historically thegrowth firm might have had greater returns and had low correlation withthe other firm, but in the future the returns might be similar and a highercorrelation would be observed.As the number of assets increases the mean-variance analysis gets computa-


tionally heavy, it requires modeling N first and (N2 +N)/2 second momentsof returns. The mean-variance analysis does not consider higher moments ofreturn, the investor may be concerned with the skewness of the return distri-bution or the probability of very large negative shocks measured e.g. by thefourth moment. Furthermore, The mean-variance analysis does not considerany specific sources of risk and it can thus be concluded that it does notcapture the general complexity of risk.It is also necessary to ensure positive definiteness of the covariance matrix,which is not always ensured by just calculating the covariance matrix5. If thenumber of assets is large, it is particularly hard to obtain reliable estimatesof the N covariance matrix. In particular if the covariance matrix, Σ, is closeto singular it’s inverse Σ−1 will be extremely sensitive to uncertainty.Also mean-variance analysis treats gains and losses symmetrically. Varianceis a general measures of risk or uncertainty in statistics but investors do nottreat gains and losses symmetrically. They only want to minimize downsiderisk i.e. losses.Finally, Jorion [1985] and DeMiguel et al. [2009] show that the result of thetraditional Markowitz model result in extreme optimal portfolio weights. Thatis that along the efficient frontier there are large positioning in few assetsonly and substantial changes along the efficient frontier.It is also important to note that the Markowitz model is expected to be a poormodel for forming a currency portfolio. This is due to the close relation ofeconomic fundamentals and political issues that the mean and variance canby no mean capture. In order to state this in a bit more theoretical terms theMarkowitz model only considers a fixed information space, while the PPP canuse the same space and additional space provided by other variables. It is thusnot compatible to compare the Markowitz model with the PPP consideringthe difference in informational spaces. The comparison of model performancecan be seen in the results Chapter 5.

5See [Kwan, 2010]


2.2.2 Fixes to the Markowitz deficiencies

Different approaches have been proposed to solve some or many of those prob-lems, first to mention is putting restrictions on portfolio weights. This is doneto solve the problem of extreme portfolio weights. Short-selling restrictions isthe most commonly applied constrain as it is commonly considered hard forthe private investor to sell short, even though this is becoming increasinglymore accessible to private investors. Furthermore, a maximum exposure re-striction can be applied and a recent innovation by DeMiguel et al. [2009] ofnorm constrained portfolios serves the same purpose. Jagannathan and Ma[2002] show that the introduction of short-selling restriction improve out-of-sample performance substantially. Introduction of these additional constraintsincreases the computational burden of the optimization quite significantly.Second, imposing a factor structure on the covariance matrix. There is a largeempirical literature on this, see e.g. Ledoit and Wolf [2003, 2004], Jagannathanand Ma [2003], Chan et al. [1999] and Kawakatsu [2006]. As and example, if Nis reasonably small, it is possible to estimate the expected mean and varianceby a multivariate autoregressive conditional heteroskedastic (MARCH) model,forecast the time varying conditional mean and variance and plug these intothe Markowitz framework.Third, Jorion [1985] suggests trying to deal with estimation risk by shrinkingthe portfolio weights. The shrinking target has been the key difference whereJorion [1985] shrinks expected returns to a common mean or towards the mini-mum variance portfolio. Wheras Black and Litterman [1992] and Pástor [2000]shrink towards the market portfolio. Pástor [2000] and Pastor and Stambaugh[2000] suggest introducing an asset pricing model into the Markowitz model.In contrast to Pástor [2000], Black and Litterman [1992] allows the investor tohave a view on future returns.Finally, the ground-breaking work of Black and Litterman [1992] providesthe investor the opportunity to have views and to blend the investors viewswith prior information. This is done by using heuristic approaches relying onBayesian Statistics, by providing the Markowitz model with an intuitive prior,the CAPM equilibrium market portfolio, as a starting point for estimation ofasset returns. There is a large number of articles explaining the workings and


implementation process6. It is worth noting that the Black-Litterman modelwas originally proposed for low-dimension global asset allocation and theneed to ensure positive definiteness of the constructed covariance matrix isstill unresolved.It is concluded that even though there are several patches available for thetraditional Markowitz model, most of them only address one of it’s problems.It is thus appropriate to search for other alternatives. The parametric portfoliopolicy (PPP) method introduced in Brandt et al. [2009] (BSCV) might be a rev-olutionary alternative to the traditional mean-variance approach of Markowitz[1952]. The PPP is computationally simple and provides an interesting per-spective in using descriptive variables and thus also giving explanation to whythose particular portfolio weights are being selected.2.2.3 Arbitrage Pricing Theory

The Arbitrage Pricing Theory (APT) in a Markowitz framework Ross [1976a,b]is used as a benchmarking model. The derivation is adapted from Zenios[2005]. By using this model as a benchmark the same information space isused, but the APT is static compared to the PPP model. Now introducing theframework of multi-factor models and following up with the APT.Multi-factor Model

The return of the ith asset is related to the factors fj for j = 1, 2, . . . , K ,through a linear relationri = αi + K∑

j=1 βij fj + εi (2.2)The variance of the jth factor is given by σ 2

j and the security-specific residualterm εi is assumed to be normally distributed with mean 0 and variance σ 2ε . Itis assumed that the asset returns are correlated only through their response to

6See for example [Drobetz, 2001], [He and Litterman, 2002] and [Walters, 2011]


the common factors. There are additionally three assumptions that generallyfollow an introduction to the multi-factor model, they are as follows:1. The covariance of the asset-specific residual term with the factors iszero, Cov(εi, fj ) = 0 for all i,j .2. The covariance of the risk factors is zero, Cov(fj , fj ′ )=0, for all j 6= j ′

3. The covariance of the residuals is zero, Cov(εi, εi′ )=0, for all i 6= i′

The first assumption will hold by construction if the model includes sufficientnumber of predictive factors. The second assumption can be made to hold byselecting factors that are uncorrelated. Factors can be made uncorrelated bye.g. the Principal Component Analysis (PCA). The third assumption will holdif sufficient factors are incorporated so that any residual term is indeed assetspecific, i.e. not systematic.Arbitrage Pricing Theory (APT) in a Markowitz Framework

Now defining the Arbitrage Pricing Theory (APT) in a Markowitz Framework.The expected return of the ith asset is given byri = αi + K∑

j=1 βij f j (2.3)The variance of the ith asset is given by

σ 2i = K∑

j=1 β2ijσ 2

j + σ 2εi (2.4)

2.3 PPP Methodology 15

Now representing the APT in a Markowitz frameworkminω∈Ω φ

K∑j=1 β

2pjσ 2

j + n∑i=1 σ

2εiω2

i

− (1− φ) n∑

i=1 αiωi +K∑j=1 βpj f j

(2.5)s.t. n∑

i=1 ωi = 1 (2.6)βpj = n∑

i=1 βijωi ∀ j = 1, 2, . . . , K . (2.7)

2.3 PPP Methodology

The basic idea behind the PPP is to directly model the portfolio weights ineach asset as a function of each asset’s characteristics. This is done by max-imizing the investor’s average utility of the portfolio’s return over the sampleperiod. The methodology is given in Brandt et al. [2009] as follows; at eachdate t , there are Nt number of assets, each having a return ri,t+1 from datet to t + 1. There are also some asset specific characteristics xi,t observedat date t . The investor’s problem is to choose the portfolio weights wi,t tomaximize the conditional expected utility of the portfolio’s return rp,t+1,

maxwi,tNti=1

Et[u(rp,t+1)] = Et

[u( Nt∑i=1 wi,tri,t+1

)]. (2.8)

The optimal portfolio weights are parametrized, by θ, as a function of assetcharacteristics, xi,t ,wi,t = f (xi,t ; θ). (2.9)Now substituting the parametrization into Equation (2.8) the conditional opti-mization can be written as an unconditional optimization problem with respectto θ: max

θE[u(rp,t+1)] = E

[u( Nt∑i=1 f (xi,t ; θ)ri,t+1

)]. (2.10)


The sample analog can be written as,maxθ

1T

T−1∑t=0 u(rp,t+1) = 1

T

T−1∑t=0 u

( Nt∑i=1 f (xi,t ; θ)ri,t+1

). (2.11)

There are several points to be made about the model at this point. First,(BSCV) propose a linear portfolio weight function that captures the idea ofactive portfolio management relative to some benchmark portfolio:wi,t = w i,t + 1

Ntθ>xi,t , (2.12)

where wi,t is the weight of asset i at date t in some benchmark portfolio, θis a vector of coefficients to be estimated, xi,t are the asset specific variables,standardized cross-sectionally to have mean zero and unit variance across allassets at date t . This standardisation, is a vital part of the PPP as it impliesthat the optimal cross-sectional average of θ>xi,t is zero, meaning that thedeviation of the optimal portfolio weights from the benchmark weights sum tozero, and thus that the benchmark portfolio weights always sum to one.The cross-sectional standardization ensures that the distribution of xi,t isstationary through time, while the raw xi,t may be non-stationary. The 1/Ntterm is a normalization that allows for arbitrary and time-varying number ofassets in the portfolio weight function. Otherwise doubling the number ofassets would result in twice as aggressive allocations.2.3.1 Estimation and Statistical Inference

The standard Markowitz model is an quadratic programming problem makingit easily solvable, by matrix manipulation. The PPP is solved using numericaloptimization, that will require some advanced programming to be implementedin an appropriate manner.In order to estimate the error on each parameter Brandt et al. [2009] suggesteither to estimate the asymptotic covariance matrix or to use bootstrapping.The bootstrapping is easier to implement in practice.A further discussion on the estimation procedure and inference can be seenin Chapter 4.


Refinements and Extensions

One of the strongest components of the PPP is the adaptability of the methodto accommodate a large universe of different methodologies. The objectivefunction can be change to meet an asset managers preferences, any utilityfunction can be used, different performance measures e.g. Sharpe og infor-mation ratios can be used. Furthermore, drawdowns can be controlled andminimizing VaR or CVaR in a scenario generation framework.The PPP framework also allows for portfolio weight constraints, time-varyingcoefficients, shrinkage and easy implementation of transaction costs. Further-more, the portfolio weight function in Equation (2.12) can take any form, solong as the portfolio weights sum to one.2.3.2 Literature Review

Focusing again on the pioneering work of Brandt et al. [2009] and the relatedliterture. BSCV choose to use the well-known Fama-French-Carhart - factorsbook-to-market and size Fama and French [1993] as well as a momentumfactor Carhart [1997] as asset characteristics. Finding that for a CRRA utilityinvestor with risk aversion γ = 5, an annualized certainty equivalent gainof 11.1% in sample and 5.4% out of sample based on monthly U.S. data from1974 to 2002.Also using PPP Hand and Green [2011], who introduce the influence of threesimple, firm specific annual accounting characteristics - accruals, change inearnings, and asset growth. They find that the accounting characteristics yieldan out-of-sample, pre-transactions costs annual information ratio of 1.9 ascompared to 1.5 for the standard price-based characteristics of firm size, book-to-market, and momentum. The performance fails though to be so impressiveafter introducing transaction costs.Castro [2010] also uses the PPP and incorporates unobserved effects into theportfolio policy function. These effects measure the importance of unobservedheterogeneity for exploiting the difference between groups of assets. Wherethe source of the heterogeneity, is locally priced factors, such as industry or

2.4 Concluding on Asset Management 18

country.By sorting currencies on their interest rate, Roussanov et al. [2008] identifya slope factor in currency returns, driven entirely by common exchange ratevariation. The higher the currency’s interest rate, the more the currency isexposed to this slope factor. This suggests that a standard APT can be usedto explaining carry trade returns. The loadings on this slope factor line upwith the average returns on the currency portfolios. They also show that theforward discount, i.e. the differences in interest rates between two currencies,is the single most important risk factor in currencies.Barroso and Santa-Clara [2012] use PPP to find that the interest rate spread,momentum and reversal create economic value for investors. The resultingoptimal portfolio outperforms the carry trade and other naive benchmarks inan extensive 16 year out-of-sample test.2.4 Concluding on Asset Management

“ Wide diversification is only required when investors do not

understand what they are doing.

— Warren Buffett ”To conclude on asset allocation methods it is important to note what assetmanagers really want. It is universally accepted that they want to maximizereturns while minimizing the risk of their investments. Some investors reallywant to diversify their portfolio while others, e.g. Warren Buffett, have pointedout that by greatly diversifying a portfolio they might as well hold the marketportfolio. So in order to beat the market portfolio it must come on the costof diversification. From this perspective, it is easy to conclude that investorsreally want to use their personal view in their investments.From the same perspective it is important to consider the reasoning behind theBlack-Litterman model. It is the introduction of uncertainty on the investorsviews that is of great importance. Furthermore, from the overview of financialoptimization literature Zenios [2005] it is apparent that scenario optimization


Model Selection

Preferences Decision Model

Portfolio Weights Return

Expected distribution ofReturn & Risk

Measureable Risk

Risk Assessment

Un-measureable Risk

InvestorAsset Manager

Investor views

Scenario generation

Figure 2.2: Asset Management Process that Incorporates Investor Views

is the way to proceed. By constructing educated guesses on the distributionof likely outcomes for each variable and to construct an optimal portfoliobased on a random draw from those variables. Each draw will result in aportfolio return and repeating this process will result in a return distribution.Then instead of maximizing the return and minimizing the variance, a betterframework is to maximize the return while minimizing the downside risk of thereturn distribution. This framework is known as scenario optimization, whichfrequently uses the VaR og CVaR risk measures as a benchmark.The PPP can be adapted to accommodated the desirable features just de-scribed and depicted in Figure 2.2. The PPP limits the asset specific risk,but misses like most other methods the asset unspecific risk factors.Another perspective is to limit systematic or asset unspecific risk factors, thisis possible in the Arbitrage Pricing Theory (APT) framework proposed by Ross[1976a] and Ross [1976b]. The real advantage of APT is that it can be usedto minimize specific risk in a Markowitz modeling framework as proposed inRoss [1976c] and Roll and Ross [1980].The author has a certain feeling that there might be a certain lack of identifi-


cation and definition of risk factors, in the sense how they affect asset returns.There are at least four aspects that have to be considered in asset manage-ment, that is whether risk factors are asset specific or unspecific and whetherthey have homogeneous effect on an asset class. Furthermore, is it importantto identify whether risk factors are time dependent. Consider some examples,consider first the systematic risk of oil, it might be included as an risk factorin a currency APT model, but increasing oil prices might have positive ef-fects on some currencies while having negative effect on others, thus havingnon-homogeneous effect. Furthermore, consider that the U.S. is becoming in-creasingly dependent on imported oil, so the effect of oil prices might be timedependent for the U.S. dollar. Failing to identify the different aspects of riskfactors would result in less efficient limitation of specific risk factors.As pointed out in Roll and Ross [1995] APT has not had great influence inpractice, it is easy to hypothesise that it might be due to lack of understandingof the model, or that it is both data and development demanding, but nothingcan be concluded. It will be interesting to see if the PPP will fall into thesame category as APT or if can make an impact on asset management inpractice. The author feels that there are clear academic and practical benefitsto the PPP that are not matched by other modeling frameworks known to theauthor.

Chapter 3Monetary Economics

In this chapter introduces the theory and practical aspects of monetary eco-nomics. Section 3.1 gives a short introduction to the basic monetary theory.Section 3.2 discusses monetary policy and Section 3.3 introduces practicalmonetary economics.3.1 Monetary Economics Theory

In this section a short introduction to monetary economic theory is presented.It is obvious that such a big topic can not be covered in a short text so the textis only meant as an short recap of the theory. The theory is mostly adaptedfrom the comprehensive textbook of Feenstra and Taylor [2011].

3.1 Monetary Economics Theory 22

3.1.1 Exchange Rates

Currencies can be defined to have three key functions in an economy, it is a– store of value, price unit and a medium of exchange. An exchange rate (E )is the price of some foreign currency in terms of a home currency. The pricechanges of one foreign currency can be defined as follows:EH/F ↑ when the home country’s exchange rate EH/F rises, the price of

one foreign currency goes up in home currency terms and the foreigncurrency experiences an appreciation.

EH/F ↓ when the home country’s exchange rate EH/F falls, the price of oneforeign currency goes down in home currency terms and the foreigncurrency experiences an depreciation.

Assuming that that the Marshall-Lerner condition1 hold the exchange rateeffect on trade can be defined as follows:EH/F ↓ when home country’s exchange rate depreciates, home exports

become less expensive to foreigners and foreign imports becomemore expensive.

Export ↑Import ↓

EH/F ↑ when home country’s exchange rate appreciates, home exportsbecome more expensive to foreigners and foreign imports becomeless expensive.

Export ↓Import ↑

Exchange rates obviously play an important role on trade but the relationshipis endogenous i.e. ease of trade also plays an important role in the behaviourof monetary policy makers.1The condition states that, for a currency devaluation to have a positive impact on tradebalance, the sum of price elasticity of exports and imports must be greater than 1.


3.1.2 Covered Interest Parity, CIP

Using no-arbitrage the covered interest rate parity can be derived to be asfollows (1 + i$) = (1 + ie) F$/eE$/e (3.1)

where F$/e is the forward exchange rate, i$ and ie are the nominal interestrates in dollar and euro, respectively. CIP generally holds in normal marketconditions, whereas Baba and Packer [2009] show that during the turmoil ofthe 2007-2008 financial crisis, sharp and persistent deviations from the CIP,and associate it with the US dollar funding shortages of non- US financialinstitutions in wake of the crisis. Furthermore, Obstfeld and Taylor [2004]show how the CIP converges into holding after the abolishment of capitalcontrols in the U.K and Germany in the early 1970s.3.1.3 Uncovered Interest Parity, UIP

Using no-arbitrage, and considering a more risk loving investor,(1 + i$) = (1 + i$) E[E$/e]

E$/e (3.2)where E[E$/e] is the expected future spot rate, i$ and ie are the nominalinterest rates in dollar and Euro, respectively. It is observable from the CIPand UIP equations that the forward rate must equal the expected future spotrate E[E$/e] = F$/e. Manipulating the UIP a useful approximation can bederived, ∆E$/e

E$/e = i$ − ie (3.3)The UIP states that the the dollar depreciation is equal to the interest ratedifferential between the dollar and the Euro, i.e. higher interest rates in theU.S. than in the Eurozone the no-arbitrage implies that the dollar shoulddepreciate against the Euro, by that interest rate differential.


3.1.4 Purchasing Power Parity, PPP

Given the law of one price, applied on a basket of goods founds the theory ofpurchasing power parity (PPP)2. Defining the relative price of a basket ofgoods in Europe versus the U.S., qUS/EUR ;

qUS/EUR = (E$/ePEUR )PUS

(3.4)where (E$/ePEUR ) and PUS are the European and U.S. prices of a basketexpressed in dollars. PPP holds when the price level in two countries, whenexpressed in a common currency, are equal or qUS/EUR = 1.Relative PPP

Considering inflation relative PPP can be defined as follows:∆E$/e,tE$/e,t = πUS,t − πEUR,t (3.5)

the relative PPP implies that the rate of depreciation of the nominal exchangerate equals the inflation differential between the two countries. Dornbusch[1985] surveys the literature and Taylor [2003] picks up from where Dornbuschleft off. The PPP is considered to hold for the long term and to be a useful inexplaining exchange rates as shown by Taylor and Taylor [2004] and Rogoff[1996] presents the PPP puzzle i.e. that deviations from PPP are quitepersistent.Quantity Theory of Money

Introducing the simple model known as the quantity theory of money, Assumethat a rise in nominal income will cause a proportional increase in economic2In order to avoid the mixing the Parametric Portfolio Policy (PPP) and the PurchasingPower Parity (PPP) a different font is used when refering the to the latter one.


activity and, hence, in aggregate money demand.Md︸︷︷︸Moneydemand

= L(i)︸︷︷︸money demandfunction× PY︸︷︷︸Nominalincome

(3.6)where is the money demand decreasing function in i, assumed constant, as L,for simplicity. Where the real money balance can then be defined as,

Md

P = L× Y (3.7)Defining the fundamental equation of the monetary model of the price level,

PUS = MUS

LUS × YUSPEUR = MEUR

LEUR × YEUR. (3.8)

Now using the equation (3.4) for absolute PPP a fundamental equation ofthe monetary approach to exchange rates. Introducing the money growth rate,µUS,t = ∆MUS,t/MUS,t and growth rate of real income gUS,t = ∆YUS,t/YUS,t .It is easily shown that the inflation rate equals the money supply growth rateminus the real income growth rate.

πUS,t = µUS,t − gUS,t πEUR,t = µEUR,t − gEUR,t (3.9)Now introducing the newly defined money growth and real income growthin the relative PPP equation (3.5). Using Money supply-demand equilibriumand PPP the rate of depreciation of a nominal exchange rate can be derivedas follows,∆E$/e,t

E$/e,t = πUS,t − πEUR,t = (µUS,t − gUS,t)− (µEUR,t − gEUR,t)= (µUS,t − µEUR,t)︸︷︷︸Differential in nominalmoney supply growth

− (gUS,t − gEUR,t)︸︷︷︸Differential inreal output growth(3.10)

So the nominal exchange rate can be described as the differential between thedifferentials in nominal money supply growth and real output growth. Rapachand Wohar [2002] find evidence in favour of the monetary model of exchangerate determination in the long run.


The Fisher Effect

Considering that if the UIP approximation in equation (3.3) and the relativePPP in equation (3.5) hold, then the nominal interest differential should equalthe expected inflation differential.

i$,t − ie = πeUS − πeEUR (3.11)This effect named after the prominent American economis Irving Fisher. Asthis result relies on an assumption of PPP, it is therefore likely to hold onlyin the long run. The Fisher effect then states that a rise in the expectedinflation rate in a country will, ceteris paribus, lead to an rise its nominalinterest rate.Real Interest Parity

Considering the Fisher Effect in equation (3.11), it is apparent that subtractingthe inflation rate (π) from the nominal interest rate (i) results in the realinterest rate (r). Thus if PPP and UIP hold, then expected real interest ratesshould be equal across countries. Thus in the long-run all real interest ratesshould converge to the expected world real interest rate, r∗,r∗ = reUS = reEUR (3.12)so interest rates can be considered to be reflecting the world real interestrate and domestic inflation expectations.

i$,t = r∗ + πeUS ie = r∗ + πeEUR (3.13)Vitek [2005] surveys the empirical literature concerning the predictability ofnominal exchange rates using structural macroeconomic models over the recentfloating exchange rate period.3.1.5 Complete Theory of Exchange Rates

In order to give a complete theory of exchange rates the monetary and assetapproaches are unified. Starting with the monetary approach or the long-


run model of the future exchange rate the fundamental equations and thepurchasing power parity are used to predict the future exchange rate.

Short-run moneymarket equilibrium

PeUS = Me

USLUS (ie$)× Y eUS

PeEUR = MeEUR

LEUR (iee)× Y eEURPurchasing powerparity

Ee$/e = PeUS

PeEUR

The Monetary Approach

(3.14)It is assumed that forecasts of future money Me, real income Y e, and nominalinterest rates ie are known. Knowing these the future price levels can becalculated and then the expected future exchange rate. Now considering theasset approach or the short-run model,

Short-run moneymarket equilibrium

PUS = MUS

LUS (i$)× YUSPEUR = MEUR

LEUR (ie)× YEURUncovered interestparity

i$ = ie + Ee$/e − E$/e

E$/e

The Asset Approach

(3.15)where the nominal interest rates and the spot exchange rate are assumedunknown. There are of course some real interest rates and a spot exchangerate available but calculated interest rates and exchange rates will found abasis for any strategy taken. It is easy to imagine that strategic positions andthe size of those positions are formed based on the deviance from the actualinterest rates and spot exchange rate and the calculated ones.Even though the the theory is stated as complete is is actually not thatcomplete as the money demand function has to be estimated. Hetzel [1984]

3.2 Monetary Policy 28

shows some functional form estimation techniques whereas Sala-i Martin andMulligan [1992] gives an econometric approach.3.2 Monetary Policy

The objective of monetary authorities, i.e. central banks, is to control monetaryissues in order to induce economic growth and build the foundation for a stableeconomy. Central banks do this by managing monetary issues based on themonetary policy taken at each period called a monetary regime. There arefew monetary policies available, most are based on the interlinked nominalvariables – the money supply, interest rate, price level, and exchange rate.By constraining one of the nominal variables policy makers hope to achievelong-term objectives while having short-term flexibility of the other nominalvariables.Policy makers generally agree that low levels of inflation is the most novelobjective in order to support economic growth. Feenstra and Taylor [2011]provide a comprehensive introducing the different monetary policies.Fixed Exchange Rate Fixed exchange rate policy is one of the most com-mon policy of choice. Consider rearranging the relative PPP in equa-tion (3.5) and anchoring of the nominal exchange rate,

πUS = ∆E$/eE$/e + πEUR (3.16)

if no depreciation is allowed it is called a peg and the home inflationshould be stable and equal to foreign inflation. Pegs are also labelledas being soft or hard, soft being policies where some fluctuations in thepegged exchange rate are allowed and hard meaning that no or littlefluctuations are allowed. Allowing for some small constant depreciation,called a crawl, home inflation will be higher than foreign inflation.Allowing for some variation around a target in the exchange rate, calleda band, increases the volatility of the home inflation. The home currencycan be fixed to a single currency or a basket of currencies. The largerthe central banks currency reserves the more trustworthy the peg willbe presumed.


Monetary Aggregates Monetary aggregates or money supply targetingimplies holding the money supply growth at a fixed rate, reviewingthe fundamental equation, eq. 3.10,πH︸︷︷︸Inflation

= µH︸︷︷︸Money supplygrowth− gH︸︷︷︸Real outputgrowth

(3.17)fixing the money supply growth, does not fix the inflation rate as thereal output growth can fluctuate. This will also result in low inflationin periods of high growth and high inflation in periods of low growth.It is for sure debatable whether this is a desirable result or not, but itis generally argued that this is undesirable scenario.

Inflation Targeting From the Fisher effect another anchoring variable canbe suggested,πeH = iH − r∗ (3.18)if the world real interest rate is assumed to be constant the averagehome nominal interest rate is kept stable, inflation can also be keptstable. A central bank will adjust a nominal interest rate based oninflation expectations to keep the inflation stable in the long term. Thisis an increasingly popular policy of choice.

Gold Standard The gold standard is a monetary system where monetaryvalue is fixed such that the exchange rate from home currency for goldis fixed.πH = ∆EH/Gold

EH/Gold+ πGold (3.19)

where the gold inflation can be thought of as the supply growth of goldwhich is much slower than most money supply growth rates. The goldstandard thus has a mixed effect of the fixed exchange rates and moneysupply targeting monetary systems. Home inflation should be low andstable, but as for the fixed exchange rate system the larger the reservethe more credible the gold standard will be.Mixed Policy A mixed policy usually refers to central banks that are sup-posed to ensure full employment as well as price stability. It is ob-vious that this is a really difficult task as the tools for ensuring fullemployment are usually to lower interest rates in order to encourageinvestments and public spending. The U.S. central bank or the Fed, isforced to work under this dual mandate or mixed policy.


Discussion

The fixed exchange rate policy and gold standard have the disadvantage thatthe central bank sacrifices monetary policy autonomy, i.e. free control overmonetary aggregates. The other monetary policies can be used to dampenthe business cycle. Those monetary policies are referred to as either beingexpansionary or contractionary. Expansionary policy is traditionally used in arecession by lowering interest rates in the hope that easy credit will increasespending and investment, and hence reduce unemployment. Contractionarypolicy is intended to slow inflation in hopes of avoiding the resulting distor-tions and deterioration of asset values.This monetary policy utilizes the assumption that money supply can influenceeconomic growth it is important to realize that there are a lot of other factors,including money demand, demography, tax- and government spending policies,foreign business cycles etc. that also influence economic growth.Dornbusch [1976] theorized about exchange rate overshooting that is thatpermanently expanding monetary base not only weakens the exchange ratedue to the increased monetary base and the temporarily lower interest rates.The exchange rate then only partially appreciates, when interest rates returnto their initial level, failing to reach its initial level. Bjørnland [2009] providesempirical evidence of Dornbusch’s theory and furthermore shows that the UIPseems to hold for most parts.There is substantial literature on monetary policy, where the survey of thetheory and evidence of monetary policy, by Blinder et al. [2008] is an idealstarting point. Mussa [1986] documents another real-exchange-rate anomalythat industrial countries which switched from fixed to floating exchange rateregimes experienced dramatic and persistent rise in nominal exchange ratevolatility. The volatility increase could not be accounted by changes in do-mestic price levels, building a foundation for reasoning that the increasedvolatility is due to speculation.After the recent crisis some economists have been calling for a review ofthe monetary and financial systems, stating the the current system is notworking. While some economists have lashed out at the inflation targeting andturning their attention to the gold standard, others have been focusing on Irving


Fisher’s ideas of full reserve banking versus the current fractional reservebanking. The focus is on the fact that the current system is unsustainable asmoney is created as interest bearing debt, mostly by privately held banks.The expected world real interest rate should reflect on the expectation ofworld inflation and what is the outlook for world prices. It can be argued thatthere is increased demand from developing nations, there is limited resourcesand there are central banks running expansionary policies. There are manyevidences supporting for increases in prices so that the world inflation andnominal interest rate can be expected to rise in the near future.3.2.1 The Toolbox of Central Bankers

As an insight into the which tools central bankers have in their toolbox andintroduction to them follows. The text is adapted from Sarno and Taylor [2001].Interest Rates Interest rates are determined by central banks. This gener-ally refers to interest rates on funds kept at the central bank overnight.Higher interest rates generally strengthens a currency as funds seektowards higher returns. Higher interest rates, though have the adverseeffect of reduced investments so that investors may withdraw funds fromthe country’s stock market, causing the country’s currency to weaken.Sterilized Intervention Central banks can influence the market in manydifferent ways, they can talk the currency up or down without takingdirect measures. Central banks can also directly intervene on openmarket, by buying or selling, without telling anyone about it. Sterilizedintervention is an intervention by a central bank that does not influencethe money supply, whereas non-sterilized intervention does. Nonsteril-ized intervention affect the exchange rate through purchasing or sellingforeign money or bonds with domestic currency.For example, if the aim is to depreciate the exchange rate of the homecurrency, home central bank could purchase foreign currency bonds.This will cause extra supply of home currency dragging down the homecurrency price, and the increased demand of the foreign currency willpush up foreign currency price. As a result, the exchange rate drops.


This process has been exercised extensively by Asian export economiesto depreciate their home currency value compared to the dollar in orderto induce exports.The evidence of the effectiveness of sterilized intervention is mixed. Thesterilized intervention has little or no effect on home interest rates,since the level of the money supply has remained constant. However,according to Mussa [1981], sterilized intervention can influence the ex-change rate through two channels: the portfolio balance channel3 andthe expectations or signaling channel4.Nonsterilized Intervention Nonsterilized intervention influences the ex-change rate by inducing changes in the stock of the monetary basewhich, in turn, induces changes in broader monetary aggregates, inter-est rates, market expectations and ultimately the exchange rate. Sarnoand Taylor [2001] surveys official intervention and it’s effectivness. Fur-thermore, reporting that non-sterilized intervention is effective.Quantitative Easing At the time of crisis central bankers have engaged inthe unconventional intervention of quantitative easing to stimulate thenational economy. Originated by the Japanese central bank Bank ofJapan (BOJ) to fight domestic deflation in the early 2000s. The BOJhad maintained short-term interest rates at close to zero since 1999,so the impossibility to lower interest rates called for unconventionalmeasures. With quantitative easing, it flooded commercial banks withexcess liquidity to promote private lending, leaving them with largestocks of excess reserves, and therefore little risk of a liquidity short-age. The BOJ accomplished this by buying vast amounts of governmentbonds, asset-backed securities and equities.Similar policies have been used by the United States, the United King-dom and the Eurozone during the Financial crisis of 2007-2012 underdifferent names but with similar purposes. Ugai [2006] and Shiratsuka[2009] examines Japan’s experience of the quantitative easing policy.There is not much literature on the effectiveness of the current QE inthe U.S., U.K., Europe and Japan but Joyce et al. [2010] provides ananalysis on the effect QE had on the U.K. market and Krishnamurthy

3Sterilized intervention will be strengthened by portfolio balancing of other market partic-ipants in the same matter as the central bank.4 Market participants will view exchange rate intervention as a signal about the futurestance of policy.


and Vissing-Jorgensen [2011] evaluate the effect of the Fed’s QE1 andQE2 on interest rates. This will however probably be a popular topicof future research.Reserves Accumulation of foreign currency reserves has risen exponentiallyover the last 15 years, mostly driven by the accumulation of the exportdriven Asian economies, especially China and Japan [ECB, 2006]. Thereare basically two currencies that are kept as reserves and that is thedollar to the greatest extent, accounting for approximately 60%, and theEuro accounting for approximately 25% of known allocation of reserves.5.Gold is also kept in foreign exchange reserves as a hedge against thedollar.Capital Controls Capital controls are an extreme action most often takenby governments in the wake of currency crisis. Capital controls restrictor banish free trading of foreign currencies and thus other financialassets, e.g. equities and bonds. Edwards [2002] gives great introduc-tion to the literature on capital controls and capital flows in emergingeconomies.Edison [1993] surveys the effectiveness of central-bank intervention. Whereas,Blinder et al. [2008] surveys the theory and evidence of central bank commu-nication.3.2.2 History of Currencies

Feenstra and Taylor [2011] provides a long version of the history of currencies,a very short summary follows.The Gold Standard, 1816-1933

The gold standard was a fixed commodity standard, where participating coun-tries fixed their exchange rates to a physical weight of gold or silver. The5For discussion of reform of the international monetary system see [Moghadam", 2010] andhistorical statistics Statistics [2012]


U.S. where a late adopter of the standard and became the standard-bearer,replacing the British pound when Britain and other European countries comeof the system with the outbreak of World War I in 1914. Eventually, the wors-ening international depression led even the dollar off the gold standard by1933, which marked the period of collapse in international trade and financialflows prior to World War II. During the 1920s, the U.S. government "sterilized"gold in-flows from Europe used to purchase products, in an effort to preventthe U.S. dollar from strengthening and hurting the export economy. This setthe stage for the Great Depression, as it caused the money supply of gold inEurope to shrink (deflation). Decreasing monetary base.The Bretton Woods System, 1944-1973

The post-World War II period saw Great Britain’s economy in ruins, its in-frastructure having been bombed. The country’s confidence with its currencywas at a low. By contrast, the U.S., thanks to its physical isolation, was leftrelatively unscathed by the war. Its industrial might was ready to be turned tocivilian purposes. In the aftermath of the World War II 45 countries attended,at the behest of United States, at a resort hotel in New Hampshire, BrettonWoods, to formulate a new international financial framework6. This frame-work was designed to ensure prosperity in the postwar period and preventthe recurrence of the 1930s global depression.The Bretton Woods system formalized the U.S. dollar as the new global re-serve currency, with its value fixed into gold, with other currencies then fixedbut adjustable to the dollar. The other major currencies at that time the Britishpound, the French franc and the German mark were all pegged to the dollar,with other minor currencies frequently pegged to them, making the vast ma-jority of the world’s currencies ending up, directly or indirectly pegged to thedollar. This "dollar standard" led to the dollar’s rise to prominence, becom-ing the reserve currency of choice and staple to the international financialmarkets. Furthermore, this led to dollar denomination of commodities mostimportantly oil. This further strengthened the reserve currency status of theU.S. dollar.

6The meeting also laid the foundations for the formation of The International MonetaryFund and the World Bank.


After the Bretton Woods era

After close to three decades of the Bretton Woods system, the system cameto an end due to growing structural imbalance amongst economies, leading tomounting volatility and speculation. The core of the Bretton Woods’ problemswere deteriorating confidence in the dollar’s ability to maintain it’s converta-bility into gold, due to heavy cost of the Vietnam war. Furthermore, there wasunwillingness of surplus countries to revalue their exchange rate due to theadverse effect on external trade.A group of European economies tried to preserve a fixed exchange rate systemamong themselves by founding the European Monetary System (EMS) ofmonetary cooperation and the pegging mechanism, Exchange Rate Mechanism(ERM) in 1979. In ERM many European currencies were pegged to theGerman Mark. The United Kingdom faced deficit problems, initiating them toallow the Sterling to appreciate while imposing capital control.Still willing to form some kind of currency union Britain joined the ERMagain in 1990, but lasted only for two years as Germany ran tight monetarypolicy due to the large fiscal shock caused by the reunification of Germany.This led to the ERM crisis where the British pound and the Swedish kronaamongst others suffered from an exchange crisis.Euro currency union

The Europeans persisted on unification and finally on January 1, 2002, theEuro became the official currency of 12 European countries7.7Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Nether-lands, Portugal and Spain. Since 2002, 9 more countries have adopted the Euro - Andorra,Cyprus, Malta, Monaco, Montenegro, San Marino, Slovakia, Slovenia and Vatican City

3.3 Practical Monetary Economics 36

3.3 Practical Monetary Economics

“ It is well enough that people of the nation do not understand our

banking and monetary system, for if they did, I believe there would

be a revolution before tomorrow morning.

— Henry Ford ”In this section the focus is shifted from the theory of monetary economic tothe more practical side of monetary economic, introducing the market, actorsand market movers.3.3.1 The Market for Foreign Exchange

The Market

The market for currency exchange also called foreign exchange, forex or FXmarket is not an organized exchange like stock markets, it is an over-the-counter (OTC) market were market participants electronically exchange cur-rencies. The FX market actually consists of two separate markets the spotmarket and the future market. The spot market has no physical location nor acentral exchange, it operates through an electronic network of banks, corpo-rations, and individuals trading one currency for another. The lack of physicalexchange enables the FOREX market to operate 24-hours a day from thebanking opening hours in New Zealand on Monday to Friday closing-hoursin western states of the United States. The future market is a currency futuremarket were standardized contracts of currency exchange are traded. Thevolume of the future market is small compared to the spot market.Carry Trade

There is still another way to profit from the differences in interest rates be-tween currencies, that is by borrow in low yielding currencies to invest in ahigh yielding one Fama [1984]. From UIP, carry trades should not yield a


predictable profit because the difference in interest rates between two coun-tries should equal the rate at which investors expect the high interest ratecurrency to depreciate against the low interest rate one. However, for carrytrading pairs the high interest rate currency is expected to appreciate againstthe low interest rate currency due to the selling of the borrowed low interestrate currency and buying interest of high interest rate currency. This is a self-fulfilling effect as increased demand for carry trades should strengthen thepotential for profit-taking. The reverse process is of course also self-fulfilling,i.e. the closing of carry trades should weaken the high interest rate currencyand strengthen the low interest rate currency.Roussanov et al. [2008] extend the result of Fama [1984] that higher than usualinterest rates lead to further appreciation, by showing that investors earn largeexcess returns simply by holding bonds from currencies with interest rates thatare currently high, not only higher than usual. Comparing the UIP and thedocumented profit opportunities from carry trade shows the heterogeneity ofthe forward rate.Actors

The actors in the currency market can be categorized as follows; Businessand Speculation. Business are basic transactions by the public, corporatesand governments made in order to conduct their business. Speculation is amarket participation in order to profit from price movements. Speculation isconducted by most other market participants from large banks, to small timespeculators. Participant can further be distinguish as being either traders orinvestors, traders trade in assets for shorter periods whereas investors holdtheir investments for longer periods. Traders ride waves while investors sailships.Transaction Cost

Transaction cost is the spread between bid and ask prices. This spread ismeasured in so called pips which is the smallest historically reported decimalpoint of an exchange rate. This is four decimal points for most currencies,


even though one additional decimal point might be reported. There is greatvariability in the spread on currency pairs, the dollar is the most activelytraded currency and thus generally has the lowest spread. The Euro is thesecond most traded pair and thus has the second lowest spread on it’s currencypairs, the spread on the retail market can be as low as one pip for theeurodollar and up to approximately 40 pips for minor currencies such as thescandics and up and above 100 pips for more exotic currencies.Galati [2000] documents that volatility and spreads are positively correlatedand argues that correlation between trading volumes and volatility to be pos-itive during normal times but negative during periods of stress.3.3.2 Currency Analysis

It is commonly accepted that there are two methods for forming trading strate-gies in any market, these being fundamental and technical analysis. Theformer being based on economic factors while the other is based on priceaction. Fundamental analysis are more focused on the long-term, whereasthe technical analysis are about the short run. Considering the fundamentalanalysis, exchange rates are a reflection of the balance between the supplyand demand for a particular currency pair. Monetary policy and the generalstrength of the economy are the two primary factors that affect supply anddemand. The general strength of the economy can be estimated from thefollowing economic factors.3.3.3 Economic factors

There are a series of economic indicators that are of importance to exchangerates and financial markets in general. These economic indicators are re-published in so-called economic calenders available by many sources8. Theseeconomic calenders also publish so-called consensus estimates of the eco-nomic indicators. The following list of indicators are some of the most impor-tant.8See for example http://www.forexfactory.com/ or http://www.tradingfloor.com/


Current Account The current account9 is the sum of balance of trade, factorincome10 and cash transfer. Current account surplus, the country’s netinternational asset position increases correspondingly. Barrios et al.[2009] find an important role for the current account in determining riskpremia.Balance of Trade The trade balance is the net difference between the im-ports and exports of a nation over a certain period. The balance of tradeis sometimes divided into a goods and a services balance.Capital Account The capital account reflects net change in national own-ership of assets. Capital account can be broken further into net valuesof Foreign Direct Investment (FDI), portfolio investments, other invest-ments and reserve account.Gross Domestic Product The gross domestic production (GDP) is the to-tal market value of all goods and services produced within an economy.The growth rate of GDP indicates the pace at which a country’s economyis growing, or shrinking. GDP is generally considered to be backwardlooking, as it reflects on how the economy was doing in the past.Industrial Production Industrial production (IP) is a measure of the changein the production of a nation’s factories, mines, and utilities. IP alsoconsiders capacity utilization that is industrial capacity and availableresources among factories, utilities, and mines.Purchasing Managers Index The purchasing managers index (PMI) is acomposite index of national manufacturing conditions, reflecting pur-chasing managers’ acquisition of goods and services. PMI is dividedinto manufacturing and non-manufacturing sub-indices and is generallyconsidered to be a forward looking measure.Producer Price Index The producer price index (PPI) is a measure of pricechanges in the manufacturing sector. It measures average changes inselling prices that producers receive for their output.Consumer Price Index The consumer price index (CPI) is a measure of theaverage price level paid by consumer for a fixed but adjustable basket

9The balance of payments has two primary components, current account being one and theother being capital account. The two components should sum to zero.10Factor income is the earning on foreign investments minus payments to foreign investors


of goods and services. The CPI basket differs between countries, but issupposed to measure the cost of living and measure changes in pricelevels or simply inflation.Durable Goods The durable goods orders indicator measures new ordersplaced with domestic manufacturers for immediate and future deliveryof durable goods. Durable goods being defined as a good that lasts foran extended period of time.Employment Index Is a measure of employment but also reflects on thespending power in the economy.Retail Sales Retail sales is an standardized indicator of broad spendingpatterns and thus implies the real consumer confidence.Housing Starts Housing starts measures the number of residential con-struction permits issued each month. Housing starts is sensitive to in-terest rates and consumer confidence, and has an effect on the generallevel of employment.Bond yields Government bond yields are also an important measure forrelative strength of currencies, as higher yields will lead to weakercurrencies. Bond yields are an immediate measure on the marketsperspective on bond yields.The producer price index, the consumer price index and the gross domes-tic production are generally considered the indicators that have the biggestimmediate impact.3.3.4 Market Sentiment - The Real Market Movers

Besides the monetary policies and economic factors there are some underlyingeffects that influence pricing of currencies. These effects can be summarizedas follows.Risk On/Off This is a market phenomenon describing the herding be-haviour amongst traders. During a market sentiment of Risk on, the


market participants are optimistic and more willing to take risk in ex-change for possibly better returns. When the market sentiment is Riskoff, there is pessimism in the market and it will favour perceived lowerrisk assets.

External Debt Ownership of debt has a huge effect on currencies. Japan isa great example of locally owned debt whereas US has external debtto a much greater extent. The fact that U.S. debt is to some extentexternal, increases the incentive to devalue the currency as that is alsodevaluing the debt.Gros and Alcidi" [2011] examines the periphery countries and states thatexternal debt is the key factor in whether economies face solvency prob-lems or not, pointing out that Greece and Portugal have high externaldebt.Reserve Currency Effect The effect of being a reserve currency is a mas-sive effect as it founds a huge demand for that particular currency. Ifthe dollar would lose just a partial of their reserve currency status itwill cause a massive depreciation of the dollar.Safe Haven Effect During economic uncertainty or downturns investors movetheir money into assets that are known to preserve their values duringan market turmoil. These investments are also often self-fulfilling asincreased demand will cause prices to rise. The currencies that aregenuinely considered to be safe haven currencies are the Japanese yen,Swiss franc and the U.S. dollar.Portfolio Balancing Effect Major movements to and from particular re-gions or countries can lead to large fluctuations in exchange rates. Thiseffect is also self-fulfilling as a movement out of a particular market cancause the currency of that economy to weaken, due to selling interest,and thus trigger a further selling of that market.Trade Effect The Manufacturing sector in each economy benefit from weakerhome currency and thus it is easy to see that net exporting nations ben-efit from weaker currency. To limit the negative effect on trade by theircurrency appreciating against the dollar Japan and China, along withseveral other developing countries in Asia, sought to peg or control theirexchange rates to limit their appreciation against the dollar.

3.4 Technical Analysis 42

The weakening of the Euro in the summer of 2012 sparked life intothe economic discussion on who was going to lead the world economyout of the current crisis and how. Weaker Euro shifted the attention tothe fact that the U.S. was not particularly up for a prolonged period ofstrong dollar compared to the Euro, so further QE followed. It is easyto suggest that it was done in order to weaken the dollar and hopingthat it would enhance economic growth. It is though difficult to see thatthere is any sustainability in that growth. It is maybe worth referencingBarro [1974] where he argues that every bond-financed deficit must bemet by a future tax increase.Furthermore will the loose monetary policy of the Federal Reserve11force other large central banks to partially loosen up their own monetarypolicy if they do not what their home currency to appreciate against thedollar. It is thus easy to suggest that globally looser monetary policywill eventually lead to a greater world inflation.3.4 Technical Analysis

Technical analysis are mainly used by traders in order to find good entranceand exit points. There are really many different technical analysis availablebut to name a few; Moving Average, Relative Strength Index (RSI) and Mov-ing Average Convergence/Divergence (MACD). Technical measures only usean assets own asset price development to spot changes in the strength, di-rection, momentum, and duration of a trends in asset prices. As they have noeconomic value there will be no further discussion of them, but there is a bulkof textbooks on technical analysis available.

11The Federal Reserve System is the central banking system of the United States, alsoknown as the Fed.

Chapter 4Methodology and Data

In this chapter the methodology of the analysis is given a comprehensivedescription. It shows how the modeling procedures where implemented andshould make the reading of the programming code much easier. Furthermore,the data used in the analysis are presented.Section 4.1 describes the modeling and inference methodologies in an acces-sible manner. Section 4.2 describes the optimal set of data and presents theactual set of data.4.1 Methodology

Now the modeling process will be given a step-by-step description. This willbe a more practical orientated discussion whereas the more elegant mathe-matical description is given in Section 2.3.The whole modeling process can be sectioned into three main processes; dataprocess where the data are handled, modeling process where the modeling

4.1 Methodology 44

steps are taken and the result process where the results are handled. First,is the data process.1. Import Data. The data consist of N asset returns, yi,t , and M descriptivevariables, Xj ,t , spanning time T . Data are described in Section 4.2.2. Data Manipulation, data are manipulated such that they are readyfor modeling. As an example for GDP, there are GDP figures for alleconomies, first difference is calculated in order to make the variableindependent of the economies size. All data manipulations are specifiedin Section 4.2.3. Standardization, each of the M descriptive variable groups, Xj ,t , isstandardized by a non-conventional standardization, Xj ,t being a stan-dardized descriptive variable group. The standardization is done cross-sectionally by subtracting the row mean at time t of that variable groupand then dividing by the row standard deviation.14. Specify initial optimization conditions. These include initial parametervalues, θ0, utility parameter, γ , shrinkage parameter λ and the numberof bootstrapping iterations B.5. Specify initial benchmark weights, w i,t . It is important to note theimportance of the benchmark portfolio in the PPP framework, it could bethe market index if equities where the asset class under consideration.There is no market index in currencies but aggregate market positionsfor large market participants are available once a week and could serveas benchmark weights2.

Second, is the modeling process. There are three different modeling processesconsidered and presented in Section 4.1.1, but they all contain the followingprocesses.6. From the design matrix containing all the standardized variables, Xj ,t ,

1A subtraction by row median could be considered as well as a way to account for potentialoutliers.2Saxobank publishes a great weekly report on the topic on tradingfloor.com. See CFTC’swebsite for the orginal data.

http://www.tradingfloor.com

http://www.cftc.gov/marketreports/commitmentsoftraders/index.htm

http://www.cftc.gov/marketreports/commitmentsoftraders/index.htm

4.1 Methodology 45

an asset specific design matrix, Xi,t is created. Xj ,t is of size (T × i)and Xi,t is of size (T × j)7. Portfolio weights are calculated by,

wi,t = w i,t + 1N Xi,tθ

>, (4.1)where θ> is a (j × 1) vector of parameter estimates. The result fromXi,tθ> is thus a (T × j) × (j × 1) = (T × 1) matrix of variable drivendeviations from the benchmark portfolio.

8. Portfolio returns are calculated by the following formula,rp,t+1 = N∑

iwi,t · ri,t+1

anlong with the standard deviation of the portfolio returns.9. Utility from returns is calculated. The standard CRRA utility function isused along with a shriking of estimated parameters. The utility functionis then as follows

u(rp,t) = (1 + rp,t)1−γ1− γ − λ · 1M ·

√√√√ M∑jθ2j (4.2)

where γ is the utility parameter and λ is the shrinkage parameter. Theshrinkage is done in order to prevent unnecessary variable inflation,especially in the bootstrapping process3.10. Optimization of estimation parameters θ is done with the Nelder-Meadoptimization procedure. The Nelder-Mead does not handle single vari-able optimization, hence the Brent optimization procedure is used forsingle variable models. These methods were chosen as they do notrequire a gradient function, the disadvantage is though that it is notpossible to conclude that the optimum is a global optimum. Differentinitial conditions were tried in order to get some conviction that optimalvalues were indeed a global optimum, with satisfactory results.

3Further discussion in Section 4.1.2

4.1 Methodology 46

Implementing an manual estimation of the gradient and the Hessianmatrix would enhance the modeling procedure and make it possible toaccess whether the optimization results are the global maximum or justa local one. Furthermore it would allow for the asymptotic estimationof standard errors of parameter values. The estimation of gradient andthe Hessian matrix where not conducted in this thesis due to the limitedtime frame.Third, is the result processing

10. Annualization of return and standard deviations are calculated. Theannualization is done by multiplying the returns by the frequency, f ,and the standard deviation by the square root of f . The frequency is4 for quarterly data, 12 for monthly data and 252 for daily data. Fromthe average annualized returns and standard deviation the Sharp ratiois calculated as followsSharp Ratio = rp

σp11. Inference by bootstrapping, the bootstrapping process is given a thor-ough description in section 4.1.2.12. Calculate out-of-sample performance. The out-of-sample performanceis examined by a 10-fold cross-validation.4.1.1 Different Modeling procedures

As economic factors are published with different frequencies it would be ex-tremely practical to be able to utilize them in decision making. It is of coursepossible to implement this in numerous different procedures. The focus willbe on three different modeling procedures that are presented as follows.P1: Simple Regression like Modeling Process

The first modeling process can be seen in Figure 4.1. In this process quarterly

4.1 Methodology 47

Figure 4.1: Simple Regression like Modeling Process P1.

and monthly data are extracted such that they are on the same format asthe daily data. The advantage with this modeling method is its simplicitywhereas it might be possible to get better results by using different modelingprocedures.P2: Stepwise Modeling Process

The stepwise modeling process was inspired by the interesting feature of thebenchmark weights in the weight function, Equation 2.12. With the initialquarterly benchmark weights set as the equally positively weighted portfolio,the output of the quarterly model is then extracted such that it can be used asthe monthly benchmark weights. The same extraction of the monthly weightsto the daily benchmark weights help produce the final daily portfolio weights.It is the hope that the stepwise modeling process would produce better results,than the simple regression like modeling process, P1.P3: One Step Modeling Process.

It is then easy to see that optimizing the same process in one step couldprovide even better results. The gain would come from adjustments in quar-terly parameters resulting in lower quarterly returns but higher daily andthus aggregate return.

4.1 Methodology 48

Results

Data

Process

Data(Q) Data(M) Data(D)

DailyWDMonthlyWMQuarterly

Results

Figure 4.2: Stepwise Modeling Process P2.

Figure 4.3: One Step Modeling Process P3.

4.1.2 Statistical Inference

In order to estimate the error on each parameter Brandt et al. [2009] suggesteither to estimate the asymptotic covariance matrix or to use bootstrapping.The asymptotic covariance matrix is much more evolved as it requires thegradient and the Hessian of the utility function with respect to θ. Statisticalinference is thus conducted by bootstrapping.

4.1 Methodology 49

Bootstrapping

Bootstrapping is a wide field of resampling methods for different purposeswithin statistics and applied mathematics4. Here nonparametric bootstrap-ping is considered, meaning that the obtained data are treated as an accuratereflection of the parent population. Repeated samples are drawn, with re-placement, from a pseudo-population consisting of the obtained data. Dueto special characteristics of time series, they are often subjected to more ad-vanced bootstrapping methods. Goncalves and Politis [2011] reviews thosemethods, but here the focus will be on normal bootstrapping and block boot-strapping. Block bootstrapping is probably the easiest of these methods toimplement, even though there are numerous variations, one selects stretchesof time series, either overlapping or not and of fixed length or random, i.e.whatever that can guarantee stationarity in the samples. The block lengthmust increase with increasing sample size to enable the block bootstrap toachieve asymptotically correct results, according to Härdle et al. [2003].It is also important to note that bootstrapping small samples requires specialattention5. The reason for this is that the inference from bootstrapping isdependent on the number of observations. As an example consider modelingsome landscape first with very few data points and then again with a greatnumber of observations, with the same model. Bootstrapping the samples theresulting picture of how that landscape looks like is by no mean related tothe quality of the model it is only dependent on the number of data.Application

1. The appropriate bootstrapping method, normal or block, is selected andthen whether it is a single- or multi-factor model due to the differentoptimization algorithms.2. Draw B independent bootstrap samples X1, X1, . . . , XB from population3. The optimization is done B number of times and the parameter estimatesare saved for all instances.

4See, e.g. Efron and Tibshirani [1993] for more on bootstrapping5See e.g.Tanaka [1987]

4.1 Methodology 50

4. The standard error of parameter estimates is calculated by the followingequation given in Efron and Tibshirani [1993],seboot(θ) = ( 1

B − 1 B∑b=1(θb − θ∗

))1/2 (4.3)where θ∗ = 1

B∑B

b=1 θb

4.1.3 Benchmark

The Arbitrage Pricing Theory (APT) in a Markowitz framework Ross [1976a,b]is used as a benchmarking model. The mathematical description is given inSection 2.2.3, whereas the following description is a more applied one. Thederivation is adapted from Zenios [2005] and extended to fit the current settingwith asset specific factors.Modified APT Model

In order for the usage of asset specific factors in the APT model the followingmodifications are suggested. The expected return in Equation 2.3 will changedtori = αi + K∑

j=1 βij f ij (4.4)in order to allow for asset specific factors. The variance of the ith asset willthe be given by

σ 2i = K∑

j=1 β2ijσ 2

ij + σ 2εi (4.5)

where the factor variance, σ 2ij , is now also asset specific. The factor variance isnow a matrix but there is actually no fundamental change. The factor variancevector is actually just asset specific just like the parameter estimates. Now

4.1 Methodology 51

restating the modified optimization problemminω∈Ω φ

K∑j=1 β

2pjσ 2

ij + n∑i=1 σ

2εiω2

i

− (1− φ) n∑

i=1 αiωi +K∑j=1 βpj f ij

(4.6)s.t. n∑

i=1 ωi = 1 (4.7)βpj = n∑

i=1 βijωi ∀ j = 1, 2, . . . , K . (4.8)Reviewing the conditions for the first condition is unaffected as it only con-siders if the model fit is good enough and it should actually be better withasset specific variables. The second condition is also unaffected as it is ful-filled by the PCA6. This makes it very easy to calculate the covariance in themean-variance analysis, as all off-diagonal elements are zero. The third andthe last condition is furthermore also unaffected as it states that their canbe no systematic effect left in the residuals. This could be more probable asthere is no modeling of say risk on/off behaviour of investors, but hardly to asignificant degree if numerous factors are considered.7Application

The first two steps of the APT method are the same as for the PPP in Section4.1 so the modeling process can be described as follows.3. Specify initial optimization conditions. These include initial parametervalues, ω0, and a risk aversion parameter, φ.

6Principal components are guaranteed to be independent only if the data set is jointlynormally distributed. There are though numerous ways around this is the data are far frombeing jointly normally distributed7An oversight of a big systematic risk factor would cause a deviation in both the covarianceand return terms in the objective function.

4.2 Data 52

4. From the design matrix containing all the variables an asset specificdesign matrix, Xi,t is created.5. Conduct a Principal Component Analysis (PCA) on the asset specificdescriptive variables, Xi,t to get f i,j . Normally this would just be doneonce, but is done here for each asset specific design matrix.6. Calculate the factor variance,σ 2

ij , of each principal component f i,j for allassets. The factor variance forms the first part of the variance part ofthe objective function, Equation 4.6.7. Regress the asset specific principal components, f i,j , on asset returns,ri,t . Done for each asset.8. Calculate the predicted values, ri,t , from the regression of the principalcomponents on asset returns. The predicted values form the return partof the objective function after being multiplied with the portfolio weights,ωi9. Calculate the error variance, σ 2

εi, from each regression. The error vari-ance forms the second term in the variance part of the objective function.10. Calculate the Objective function by collecting the terms in notes 6.8. and 9. The constraint that portfolio weights should sum to one ismultiplied by a large constant so that it is fullfilled the last constraintis redundant as it is already applied in the objective function.11. Optimization is carried out numerically with the Nelder-Mead algo-rithm.12. Calculate the returns as follows, rp,t =∑Ni wi · ri,t , and it is noticeablethat the APT returns static portfolio weights.

Returns and standard deviation are annualized and the model performance isvalidated by 10-fold cross-validation.4.2 Data

In this section the optimal set of data is discussed and then data are given athorough description.

4.2 Data 53

4.2.1 Optimal set of Data

Defining the optimal set of data requires a definition on what the goal isin the beginning. In order to test and implement the PPP a limited setof variables is sufficient, as can be seen from the fact that both Brandt et al.[2009] and Hand and Green [2011] show superior performance relative to theirbenchmarks using only three variables.The goal of this thesis is to implement and analyse the asset managementprocess using the PPP as a modeling method on a portfolio of currencies.So the data requirements for this objective is just a complete set of data fora number of the most traded currency crosses. An optimal set of descriptivedata would contain a complete set of all variables imaginable that influencecurrencies. The horizon of the analysis would depend on the accessibility ofdata, in the sense that using yearly data spanning a ten year period wouldnot leave many observations. The challenge of having different frequencydata and given the significance of low frequency data play a part in the dataselection.Horizon

The horizon of the analysis also depends on the purpose. If the objective is toinvest or to rebalance a portfolio at a central bank, low frequency data shouldbe sufficient. Estimates of future values of economic indicators could then beused in a strategic evaluation of future profit taking.If the objective is to profit from trading, i.e. short term profit taking, more highfrequency data are required both in the dependent and independent variables.These data are really hard to come by, as those who possess these data knowthat a good modeling framework could extract profits from price movementsleaving less on the table for themselves.It is also worth mentioning that the PPP could be an ideal framework forautomatic trading strategies where the time frame is really short and solelybased on models using technical analysis along with some fundamental anal-ysis. The PPP furthermore uses a cross-sectional standardization, which calls

4.2 Data 54

for attention on behalf of the modeller that the variables are suitable for stan-dardization in the sense that the data have to be economically comparableand in a numerically similar range.Data quality

Data quality is vital for a good analysis, no matter whether the focus is oncausality or model performance. A survey conducted by Barra [2011] showsthat asset managers value data quality as the single most important factor inmodel performance.Potential Variables

There are numerous variable that could be interesting to try in the currentframework. The variables from Section 3.3.3 and the effects from Section 3.3.4are ideal candidates. It would be wise to incorporate both lagged and forwardlooking variables. There is simple plenty of other variables that could becreated, such as Risk On/Off variable, by e.g. high positive correlation of JPYand CHF and negative with the other currencies in a risk off season. It wouldalso be wise to try to determine whether currencies are in an trending phaseor ranging. Creating such variables and using some powerful data miningtool, CART regression should also help create additional value. There arealso few variables in [Barroso and Santa-Clara, 2012] that are not consideredhere.4.2.2 Data Resources

Data are extracted from the Thomson Reuters DataStream database Thom-son Reuters [2012] spanning the twenty year period from 26/06/1992 to27/06/2012. Data where extracted on the 28/06/2012. Data where validatedthrough comparison with other data resources; e.g. central bank webpages,the World Bank database and the external wealth of nations database byLane and Milesi-Ferretti [2001].

4.2 Data 55

Exchange Rates

Currency under consideration will be G10 currencies USD, EUR, JPY, GBP,CHF, CAD, AUD, NZD, SEK, NOK. The reason behind selecting these curren-cies is two-folded. First, these are the ten most liquid currencies on financialmarkets, in the order they are presented8. Currencies that are frequentlytraded also have lower transaction costs. Second, is the fact that these arethe currencies that have the most complete set of descriptive data.Nine currency crosses from the ten currencies form the exchange rates thatare used to calculate currency returns. The exchange rates can be seen inFigure 4.4. The U.S. dollar (USD) is used as the foreign currency i.e. inverseof the U.S. exchange rate are used, e.g. Ee/$. Returns are then continuouslycompounded.Descriptive Variables

There are numerous economic factors that could serve as descriptive variablesand the variables selected here are those who had complete data series forall economies under consideration. The PPP requires that all the descriptivevariables are asset specific i.e. that they are directly related to that specificasset. This creates a problem that an economically meaningful variable cannot be used if there are no observations for one of the economies underconsideration. Graphical representation of the descriptive variables can beseen in Appendix A.Quarterly Data

The main economic indicators Gross Domestic Production (GDP) and CurrentAccount (CA) are presented quarterly. These are also presented with a delayas the figures take considerable time to be estimated. Foreign Reservespositions are also considered as a descriptive variable and they are released8Deutsche Bank has popular currency funds and one focusing on the G10 with 2:1 leverage.See www.dbfunds.db.com

http://www.dbfunds.db.com/dbv/index.aspx

4.2 Data 56

0.6

0.8

1.0

1.2

1.4

1.6

Quarterly Exchange Rates

Year

Qua

rterly

Exc

hang

e R

ates

JPYAUDCADEURNOK.NZDSEKCHFGBP

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure 4.4: Quarterly Exchange Rates Returns from Q3-1992 to Q2-2012. Data arecentered to the first observation

quarterly by central banks. Now considering expected signs of the quarterlyvariables;GDP [+] A positive sign is expected, as rise in GDP is likely to strengthenthe confidence of market participants.Foreign Reserves [+/−] Rising foreign reserves would add to confidence thata central bank can maintain currency stability. This would particularlyapply for fixed exchange rate regime currencies and small floating cur-rencies. Considering the effect on large economies is more difficult topredict, e.g. the foreign currency holding of the U.S. is very minimaland the Euro-area has been decreasing their reserves for quite sometime while Japan has huge reserves. The conclusion can be drawn thatthe effect is maybe not even homogeneous for all the economies under

4.2 Data 57

consideration. It can thus be concluded that it is not sufficient to haveasset specific variables they also need to have homogeneous effects onthe currencies. The effect of increase in foreign reserves is expected tobe positive in general.Current Account [+] A current account surplus for an economy should resultsin a stronger currency as it represents net wealth transfer into theeconomy.Monthly Data

There are numerous economic indicator available on monthly bases. Ten yearbond yields, consumer price Index (CPI) and M2 money supply were chosenas some of the most import ones. There is of course other indicators thatcould have been used as well, but were made redundant due to missing dataor other problems in making them useful. Now considering expected signs ofthe monthly variables;Bond yields [−] A negative effect is expected as higher yields mean that itis more expensive for the economy to finance it’s debt.M2 [+/−] It is not simple to predict the effect of the M2 money supply. In-creasing money supply should induce growth in economic activity mean-while it decreases the value of money if there are greater amounts ofthem available, thus positive and negative effects respectively. Increasein money supply is generally expected to be positive in the short term.Furthermore, there is the Keynesian medicine that at the time of crisisthe monetary base should be expanded in order to dampen the crisis.The increased monetary supply should have an negative effect while ifthere is any positive effect of the monetary expansion on the economythe increased economic activity should have a positive effect. Monetaryexpansions at the time of crisis could decrease the observed positiveeffect, whereas the positive effect might be delayed.CPI [+/−] CPI or the first difference of the CPI should have an inverse Ueffect of currencies. That is increasingly positive effect up to somelevel and then a decreasingly negative effect after that level is reached.

4.2 Data 58

The positive effect comes from the economic conclusion that a growingeconomy should experience some inflation while the inflation is nothingbut a depreciation of currency value and when inflation gets to high itstarts having disruptive effects on the economy, negative real interestrates and an increased risk of a credit crunch. The level can easilybe argued to by changing with time and even different for differenteconomies.Daily Data

For daily data technical data are used as no complete set of high quality,economically meaningful, data were available for the desired modeling period.Hence, the deviations from current exchange rates and the 21 day, 55 dayand 200 day Moving Averages (MA), respectively, where used along with theRelative Strength Index (RSI).Moving Averages [−] The moving averages are all expected to have a nega-tive effect as they are supposed to catch reversals or corrections.RSI [−] The momentum oscillator should have negative effect as high valuesindicate selling might be a good idea and low values buying opportu-nities.Data Manipulation

Now describing the data manipulation process for each variable.GDP For the GDP it is actually the rate of change in the GDP that is used.This is readily available and it is from the GDP estimated using theexpenditure approach.Current Account Raw CA figures are denominated in home currency so CA ismanipulated by dividing by GDP which is also home currency denom-inated. That leaves a usable CA that is independent of the economiessize. Having CA that can be positive and negative for all the ten

4.2 Data 59

economies, creates a problem when the relative ratio of the currentaccounts is desired. Dividing the CA would result in un-meaningfulfigures so absolute differences are used, i.e. the U.S. CA is subtractedfrom the other nine economies CA.Reserves Reserves are denominated in Standard Drawing Rights (SDR) andare divided by GDP in order to make them independent of the sizeof the economies and then multiplied by the exchange rate to makethem all in the same currency SDR/USD. There are few problems withthis variable as the U.S holds a very limited reserves whereas Japanhas huge reserves and Swiss reserves have been piling up after 2008.Whereas it is possible to calculate the relative ratio here it is still optedto use the absolute difference, which could be thought of as the U.S.being a benchmark of how little reserves an economy could hold. Therate of change in the reserves was not tested but it might actually bemore meaningful.Bond yields The ten year bond yield is a good measure of economic healththat is actually available on a daily bases but is not easily availableat that frequency. The absolute difference is used for the bond yieldsas for most of the other variables.CPI The first difference of CPI is calculated and then the absolute differenceinstead of relative ratio.Money Supply As a measure of money supply the M2 money measure isused. The first difference of money supply is used and as before absolutedifference is used instead of a relative ratio.Moving Averages The standard 21 day, 55 day and 200 day moving averagesare calculated for each asset. Then as the PPP requires compatible datathe relative deviation from current exchange rate was used. Movingaverages are calculated as the average of the past n observations.RSI The RSI is calculated for each asset and used as it is. RSI is an extremelypopular momentum indicator. The RSI is calculated as follows,

RS = EMA(U, n)EMA(D, n) (4.9)

4.2 Data 60

where EMA is the exponential moving average and n is the exponentialsmoothing factor. The U and D are calculated from the data as follows,U = ∆yi,t , if ∆yi,t > 00, otherwise (4.10)D = 0, otherwise∆yi,t , if ∆yi,t < 0 (4.11)

The Relative Strength Index can then be calculated as follows,RSI = 100− 1001 + RS . (4.12)

As can be seen the RSI is a scale with the neutral value being 50,so low values are indicate that there might be a buying opportunitywhereas high values signal a potential selling opportunity.From the description of the data and the data manipulation it is clear thatthere is great room from improvement within this area. Reflecting that variableselection was not the main focus of this thesis.4.2.3 Limitations of the Analysis

To conclude on the limitation of the methodology they are quite a few giventhe limited development time for a thesis.Limitations

As the implementation of the PPP took considerable time, some simplificationswere made in the analysis.1. Interest Rate Differential (IRD) is ignored. IRD is the profit/loss from netinterest rate differentials in currency crosses. IRDs can be consideredas dividend like returns whereas they can in fact also be negative.

4.2 Data 61

2. There is no riskless rate implemented, i.e. it is set at zero. There aretwo reasons for this, first the investor can take his desired positions inbonds and on retail accounts. If trading in bonds they are subjected tocurrency risks as bonds are denominated in some currency. Secondly,the investor can choose whether he has a retail account with or withoutspecific calculations of IRDs. If the investor is carry trading he wouldchoose a market maker that returns IRDs.3. Data quality is poor. Better data would provide more intuitive results.Due to this data manipulation and variable selection were given limitedtime frames.4. More advanced optimization framework would provide greater confi-dence in the results.5. Better estimates of standard error is vital in order to conduct reliablevariable selection. Which methodology would be used, the asymptoticvariance estimation or bootstrapping is non-important.6. Transaction cost was not implemented into the model even though itis extremely simple in the PPP model. Barroso and Santa-Clara[2012] actually get better out-of-sample returns after having imple-mented transaction costs.7. The Benchmark model was not modified to by dynamic. That would berequired in order to get an appropriate comparison of the two methods.Furthermore, was their little effort made in validating the modelingprocess of the APT model, i.e. to see if regressions of returns gave goodresults and so forth.

Chapter 5Modeling Results

In this chapter the most important findings of the development process of thePPP are presented. The development process is given a full description inChapter 4. The findings are presented in an order of significancy. Section 5.1summarizes the results whereas the general results are presented in Section5.25.1 Summary of Results

The main results of the analysis are plotted in Figure 5.1. Considering firstthe different return and standard deviations of each currency cross the CHF,NZD, JPY and AUD are all in similar range, they have the highest return andstandard deviations. The Scandics, NOK and SEK have similar volatility butsignificantly lower return. The GBP and EUR have approximately zero returnover the twenty year period. The CAD then has a positive return while havingthe lowest standard deviation.Now considering the model performances, the Markowitz model was used to

5.1 Summary of Results 63

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Figure 5.1: Mean Variance Analysis with Efficient Frontiers.

plot the efficient frontier and was furthermore one of the models under con-sideration along with the APT and PPP. The in-sample(IS) performance ofthe Markowitz model can be quite good and is close to the efficient frontierwhich is constructed using the whole dataset. The out-of-sample (OOS) ofthe Markowitz model is though out of the charts as it either hits or misses i.e.it results in either great or abysmal returns due to its extreme and informa-tionless portfolio weights.The performance of the APT model is much more stable but the return ismaybe not extremely good but still quite good considering that it is a staticmodel, i.e. uses the same portfolio weights in the test sample. The OOS of theAPT is has rather extreme portfolio weights but does not result in as extremeportfolio performance as the Markowitz model does. The PPP has of coursemuch greater return but it has to be considered that it is a dynamic model

5.2 General Results 64

and thus has a great competitive advantage. The OOS of the PPP is worsethan the IS performance An efficient frontier for the PPP was constructed byaltering the utility parameter, γ , values from being very risk averse to veryrisk loving1. The P1 modeling procedure were used for the PPP.5.2 General Results

5.2.1 Comparison of Modeling Procedures

The comparison of the parameter values and model performance of the threemodeling procedures; P1, P2 and P3, that were introduced Section 4.2.1can be seen in Table 5.1. Furthermore, Model (P1* ) uses the P1 modelingprocedure with the extracted monthly benchmark weights from Model (P2) asa benchmark weight. This is done to show the effect of benchmark weightson the model. First, checking on the sign of the parameter values it can beseen that the sign on the foreign reserve (Res) and 10y bond yields (bond )have opposite signs in different models i.e. they are both positive in Models(P1) and (P1* ) and negative in Models (P2) and (P3). This result is maybenot so surprising at least not for the foreign reserves variable as there wasa certain doubt about the sign of the effect. The result for the bond yield ismore surprising and it is hypothesised that it is due to some heterogeneityof the bond yield effect. On the comparison between observed and expectedsigns, Models (P2) and (P3) comply the best to the signs that were expected.The results for Model (P2) are discussed further in Section 5.2.2.On the performance of the different procedures the P1 and P2 perform simi-larly whereas the P3 outperformed them quite significantly. The benchmarkportfolios play a vital role in the performance of the PPP method as can beseen by comparing the model performance for Models (P1) and (P1* ). Sim-ilar hikes in performance where observed for the other models such as theP3, when quality benchmark portfolio weights where used, even though it isnot reported. This has led to the conclusion that the P3 modeling procedureresulted in the best performance and furthermore that the best performance isobserved when extruded modeling results from modeling procedure P2 were

1Utility parameter values, γ , from 0.001 to 200


used as benchmark weights.Further considering the model performance, Figure 5.2 shows the cumulativereturns of the four models under consideration. It is apparent that Models(P1) and (P1* ) perform really well after crisis whereas Models (P2) and (P3)are more stable performers.

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Figure 5.2: Cumulative Return of Different Modeling Procedures.

5.2.2 Results for P2

Considering first the quarterly variables, the results can be seen in Table 5.2.Model (1) is the equally weighted portfolio, where all portfolio weights are setat equal amounts and kept throughout the modeling period. The equal weightswere chosen as positive meaning that the portfolio consist of short postitionin the USD and equally long in all others. The average annualized return is


P1 P2 P3 P1*GDP 3.362 0.669 1.466 7.971st.err. 3.342 0.548 2.369 4.209Res 1.241 -0.401 -0.080 0.760st.err. 2.591 0.644 2.558 5.137CA 3.400 0.889 2.922 3.014st.err. 2.847 0.422 2.875 6.478bond 2.223 -1.450 -1.209 2.301st.err. 6.052 2.692 3.176 4.473m2 4.707 1.948 3.692 3.425st.err. 2.958 2.492 3.049 6.878cpi 2.484 1.036 3.823 1.828st.err. 3.266 2.400 2.179 4.752ma21 -1.446 -1.497 -0.372 -11.785st.err. 3.790 3.162 2.889 5.804ma55 -0.284 0.098 -0.832 2.791st.err. 4.111 3.812 2.424 5.312ma200 -1.911 -0.498 -0.427 -1.951st.err. 4.884 2.294 2.867 5.767RSI -4.683 -1.827 -4.073 -3.143st.err. 5.546 3.144 3.761 6.112Ave. Ret. .0415 .0366 .0373 .0412Sd. .0498 .0439 .0415 .0451Sharpe .8328 .8323 .8982 .9143Note: Model (P1*) is modeling procedure P1 with dif-ferent benchmark weights. The variables under con-sideration are GDP (GDP), Foreign Reserves (Res)and Current Account (CA), 10y Bond yield (Bond ), M2Money Demand (M2) and Current Account (CPI), 21-day moving average (MA21), 55-day moving average(MA55 ), 200-day moving average (MA200) and rela-tive strength index (RSI). Ave Ret is the average annu-alized return, Sd is the annualized standard deviationand Sharpe is the Sharpe Ratio.Table 5.1: Results for different modeling procedures.


positive and serves as a benchmark for the other models as these positivelyequally weighted portfolio weights are the benchmark portfolio weights in theother models.(1) (2) (3) (4) (5) (6) (7) (8)GDP 0.225 0.403 0.406 0.669st.err. 0.282 0.540 0.373 0.538Res -0.077 -0.066 -0.221 -0.401st.err. 0.333 0.518 0.543 0.540CA 0.170 0.441 0.340 0.889st.err. 0.275 0.388 0.618 0.474Ave. Ret. .0028 .0044 .0028 .0042 .0046 .0066 .0042 .0077Sd. .0364 .0258 .0314 .0334 .0261 .0234 .0278 .0210Sharpe .0768 .1699 .0886 .1266 .1770 .2842 .1529 3693Note: The variables under consideration are GDP (GDP), Foreign Reserves (Res)and Current Account (CA). Model (1) is the equally weighted portfolio. Ave Ret isthe average annualized return, Sd is the annualized standard deviation and Sharpeis the Sharpe Ratio. Utility parameter γ = 5 and shrinkage parameter λ = .001.

Table 5.2: Results for Quarterly data from modeling process P2.Models (2) through (4) consider the modeling power of single variables, andeach variable increase the Sharpe ratio whereas the foreign reserves is thevariable which improves the return the least, compared to the benchmark re-turn. Models (5) through (7) then consider pairs of variables. Combiningforeign reserves with GDP and current account, in Model (5) and (7) respec-tively, results in little better results whereas the combination of GDP andcurrent account, in Model (6), boosts the Sharpe ratio significantly. Model(8) considers all variables and results in the best performance.Now considering the sign of parameter values and comparing them with thepredicted effects which can be seen in Section 4.2.1. Starting with the GDPthe realized sign is positive as predicted. Next considering the sign of foreignreserves (Res) it has negative sign whereas the predicted sign was gener-ally positive. Hypothesising why, it might be due to the currencies of thelarge economies, U.S., Japan and the Eurozone, not benefiting from holdingincreased foreign reserves.Considering the standard errors in Table 5.2 it is clear that there is not


a single parameter estimate statistically significant, the parameter estimatewould have to be at least twice as large as the standard error (st.err.) in orderfor the parameter estimates to be significant. This result will be discussedfurther in Section 5.2.3.Now the monthly variables are considered, the results can be seen in Table5.3. Model (9) is a model free of variables but considers the extracted quarterlyweights as the benchmark weights. It is apparent that the positive return fromthe quarterly modeling is transferred through and results in positive monthlyreturns. Models (10) through (12) consider the modeling power of singlevariables, and each variable increases the Sharpe ratio significantly. Models(13) through (15) consider pairs of variables and there is a good harmonybetween the variables in Model (13) whereas there is little gain in Model(14) and worse performance in model (15). Finally, considering all variablestogether in model (16) resulting in the best performance so far.Considering the sign of parameter values, starting with the bond yields theobserved sign is negative which fits the expected negative sign. Next observingthat there is a positive sign for the M2 money supply there is a positivesign indicates that the induced economic growth by increased money supplyoutweighs the negative effect of increased money supply, at least in the shortterm. Finally, considering the first differential of CPI it is generally expectedto be positive which aligns with the observed sign.


(9) (10) (11) (12) (13) (14) (15) (16)Bond -0.145 -0.705 -0.015 -1.450st.err. 1.325 2.244 1.917 3.388M2 0.727 1.242 0.249 1.948st.err. 1.502 2.255 1.558 2.112CPI 0.326 0.650 0.610 1.036st.err. 1.403 2.087 1.789 2.275Ave. Ret. .0131 .0145 .0172 .0169 .0209 .0170 .0179 .0244Sd .0520 .0470 .0395 .0440 .0388 .0440 .0450 .0396Sharpe .2519 .3086 .4354 .3849 .5381 .3851 .3966 .6175Note: The variables under consideration are 10y Bond yield (Bond ), M2 Money De-mand (M2) and Current Account (CPI). Model (1) is the equally weighted portfolio.Ave Ret is the average annualized return, Sd is the annualized standard deviationand Sharpe is the Sharpe Ratio. Utility parameter γ = 5 and shrinkage parameterλ = .0001.

Table 5.3: Results for Monthly data modeling process P2.The results for the daily variables can be seen in Table 5.4. Model (17) isthe benchmark model using the monthly portfolio weights and it is apparentthat most of the performance from the monthly model is transferred though tothe daily returns. Models (18) through (21) consider the modeling power ofsingle variables, and each variable increases the Sharpe ratio significantly.The performance of the shorter horizon moving averages are better than thelonger ones. Models (22) through (25) consider pairs of the moving averagesvariables and it is apparent that two of them would have been sufficientwhereas there is no gain in having them all three, as can be seen from model(25). Model (26) considers all of the daily variables resulting in a goodperformance, but not as good as considering only RSI as in Model (21).Considering the signs of the daily variables a negative sign was expected onall of the variables and negative signs are also observed, except for MA55 inmodel (26) which might indicate that it is over-driven to have three movingaverages variables.


(17) (18) (19) (20) (21) (22) (23) (24) (25) (26)MA21 -0.899 -1.464 -1.521 -2.259 -1.497st.err. 0.805 1.945 1.527 2.834 3.143MA55 -0.630 -0.355 -1.154 -0.292 0.098st.err. 0.852 1.941 1.685 2.538 3.158MA200 -0.458 -0.378 -0.107 -0.457 -0.498st.err. 1.102 2.104 2.153 2.779 2.693RSI -0.946 -1.827st.err. 1.104 3.294Ave. Ret. .0225 .0360 .0314 .0275 .0357 .0361 .0363 .0313 .0364 .0366Sd .0409 .0458 .0454 .0454 .0420 .0458 .0460 .0455 .0463 .0439Sharpe .5508 .7861 .6914 .6052 .8490 .7877 .7891 .6892 .7865 .8323Note: The variables under consideration are 21-day moving average (MA21), 55-daymoving average (MA55 ), 200-day moving average (MA200) and relative strength index(RSI). Model (1) is the equally weighted portfolio. Ave Ret is the average annual-ized return, Sd is the annualized standard deviation and Sharpe is the Sharpe Ratio.Utility parameter γ = 5 and shrikage parameter λ = .00001.Table 5.4: Results for Daily data from modeling process P2.

Currency and Period Specific Results

Table 5.5 presents the modeling results of Model (26) dissected into shorterperiods and currency specific results. Starting at the lower end of table it canbe seen that the Euro (EUR) is the only currency that has negative returnsover the modeling period. This indicates that the Euro-dollar exchange rateis the most difficult currency cross to model and it is generally accepted tobe the currency cross that is most affected by news. The Australian Dollar(AUD) is also has rather poor performance which might be caused by the factthat it is really popular in the carry trade, but this is purely a hypotheticalguess. The British pound (GBP) it the single currency cross that has the bestperformance over the modeling period.Now considering the right most columns in Table 5.5 it seems that the modelperformance is somewhat stable over the whole modeling period with theexceptions of the 1995-1996 and 2005-2006 periods. The best performancesare in the 1993-1994, 2001-2002 and 2009-2012 periods. This is quite aninteresting result as it is not that the model breaks down at the time of crisis


but rather results greatest returns over and after economic crisis2.Periode JPY AUD CAD EUR NOK NZD SEK CHF GBP Portfolio1993-1994 .0125 .0048 .0059 .0033 .0087 .0018 .0216 .0053 .0055 .06951995-1996 -.0329 .0106 .0077 .0026 .0139 -.0041 .0121 .0028 .0016 .01431997-1998 .0079 -.0059 .0053 -.0033 .0004 .0193 .0082 -.0067 .0091 .03441999-2000 .0002 .0193 .0066 -.0155 -.0071 .0218 .0013 .0008 .0045 .03192001-2002 .0143 -.0116 .0010 .0065 .0038 .0149 .0066 .0045 .0158 .05582003-2004 .0102 .0079 -.0062 .0005 -.0173 -.0054 -.0021 .0320 .0192 .03882005-2006 .0108 -.0033 .0055 -.0015 .0120 .0012 -.0121 -.0008 -.0033 .00852007-2008 .0141 -.0085 -.0019 -.0092 -.0138 .0201 .0075 .0237 .0039 .03602009-2010 .0119 .0084 .0038 -.0055 .0179 .0069 -.0101 .0127 .0106 .05672011-2012 .0049 .0008 .0081 -.0025 .0279 -.0088 .0159 -.0273 .0005 .0195Ave. Ret. .0053 .0023 .0034 -.0025 .0040 .0073 .0044 .0056 .0069 .0366St. Dev. .0186 .0155 .0103 .0095 .0158 .0218 .0165 .0166 .0105 .0439Sharp Ratio .2860 .1462 .3322 -.2670 .2505 .3326 .2662 .3353 .6572 .8323Note: Reportet results are the average annualized returns.

Table 5.5: Currency and period specific results from Model (26).5.2.3 Bootstrapping Results - Standard error

Considering the standard error estimates it is easy to see that no variableis significant. The reason for that is the problem with inflation of parameterestimates in the bootstrapping process.In the implementation process of the bootstrapping methods it was observedthat for approximately 30% - 40% of samples, parameter estimates were reallyhigh compared to the general trend of parameter estimates and they evenshifted signs. There are few plausible reasons for this, first, the samples arequite small, especially for the quarterly and monthly data. Then there is thefact that the data are time series meaning that autocorrelation could be anissue. Last but not least their is obvious endogeneity in many of the variables.Barroso and Santa-Clara [2012] conduct a similar analysis as done in thisthesis but get very robust estimates, this might be due to more quality dataor alternative leads to the suggestion that there could be some unobservederror in the bootstrapping procedure.

2The 1992 European currency crisis, 2001 dot-com bubble and financial crisis of 2008.


There is a lot of literature on bootstrapping methods for time series and thereason being that they are quite tricky to bootstrap. Most bootstrappingmethods depend heavily on the assumption that the sampling sample is large.Many of the bootstrapping methods for time series are heavily evolved butthe simple one, block bootstrapping, which is applied here does not seem toproduce stable estimates. There is some literature on small sample estimation,but the samples used here would probable be defined as very small. It is thusnot possible to read much out of the standard errors even though they can beconsidered as somewhat indicative of how stable the estimates are.This leads to the question whether the PPP can be used to say somethingabout causality. Castro [2010] incorporates unobserved effects into the portfo-lio policy function, incorporating unobserved effects could improve the stabilityof parameter estimates. The implementation of parameter shrinkage could notstop the parameter inflation. A better modeling procedure could provide betterresults, so even though the results here might indicate that the PPP is notsuitable for theorizing about causality, this is not the conclusion drawn bythe author due to the fore-mentioned reasons.

Chapter 6Conclusion

This chapter contains a short summary of results found in the thesis, in Section6.1. Suggestions on possible further work related to the work done in thisthesis are discussed in Section 6.2.6.1 Summary of Results

On the asset management subject it has been shown that currencies playa major role on financial markets as it effects all asset classes directly, in-directly or through currency denomination. Furthermore considering all thedeficiencies of the Markowitz model, suggest that maybe it is time to give thetraditional Markowitz model some rest and pursue with other methods.The performance of the PPP model is benchmarked to the Markowitz modeland the Arbitrage Pricing Theory (APT). The PPP outperformed the bench-mark models but the PPP is a dynamic model whereas the others are static.The PPP was documented to have robust in-sample and out-of-sample per-formance. The importance of how to best model data of different frequencies

6.2 Further Work 74

was shown. Furthermore the importance of the benchmark portfolio weightswas shown. Most of the variables under consideration performed as expected,whereas the foreign reserve effect seemed to be heterogeneous and the bondyields seemed to be plagued with heterogeneous effects as well.There is considerable implementation cost to the PPP and the procedure doesalso require complete sets of high quality data. Standard error estimates wereobserved to be great, suggesting that either an improved estimation procedureor better data were required.It is concluded that the PPP model is attractive in an economic sense as wasable to attain a Sharpe ratio of 0.91 with limited focus on variable selection.This is a similar Sharpe ratio as attained in Brandt et al. [2009] where equitiesare the asset class under consideration. Given the fact that retail markets forcurrencies allows for quite significant leverage it seems to be possible toimplement a profit machine1.Furthermore, it is concluded that the PPP could lead to advances in academicresearch on how financial markets work in practice. Increasing the generalknowledge of how currency markets work could result in better monetarypolicies.6.2 Further Work

Variable selection seem as an obvious next step to the analysis done in thisthesis. There are numerous variables that are not considered here that couldresult in an much better results and furthermore provide crucial knowledge onwhat drives currency returns. This would not just serve as profit machine butalso provide fundamental new knowledge of how currency markets work inreality. On the variable selection process the following process is suggested.Use asset specific parameters θj for each descriptive variable and comparethem with the single variable θ for that particular descriptive variable. Thisshould be easy to implement and could help in ensuring that a variable has ahomogeneous effect in explaining returns. Time-varying parameter could justas well be implemented.1Retail accounts provide leverage from 200x up to 500x.

6.2 Further Work 75

Then maybe more crucially is the fact that only asset specific variables canbe considered. This limits the set of variables that can be considered asdescriptive variables. The following process is a suggestion on how unspecificvariables could be included.By regressing both asset specific variables and asset unspecific variables ontothe real effective exchange rate of each currency under consideration. Eachfactor would then be decomposed into real effects. Those effects could thenbe used in the PPP framework. Divide variables into asset specific variablesand asset unspecific variables, and then calculate the asset specific effect thatthe asset unspecific variable has on the asset. Considering for example thecurrency unspecific variable, oil, it would be possible to estimate the assetspecific effect of that risk factor. The decomposed effects would be subjectedto estimation error.In order to improve the performance it would also be wise to use consensus ofeconomic indicators and select a good benchmark. Furthermore, incorporatingviews into the modeling framework, i.e. by conducting scenario optimization.It would also be wise to classify currency pairs into a carry trade portfolio,momentum that is emerging market currencies and value currencies that couldbe benchmarked by the performance of the similar Deutche Bank funds. Itis likely that different factors would explain the returns for different currencyclasses.APT

In order to make the APT competitive with the PPP it would have to be madedynamic. This could probable be done in numerous ways but the followingsolution is suggested. Observations would be given decaying weights suchthat the newest weights would have the highest weights and the older onesdiminishing weights. This should result in dynamic portfolio weights and thedecaying parameter would in turn control how dynamic the model would be.It would also be interesting to see whether the idea behind the PPP couldimprove the APT as the cross-sectional standardization seems to be wellsuited in explaining returns. This can be explained by the negative and

6.2 Further Work 76

positive values that are the result of the standardization and maybe easier tounderstand that strictly positive variables can not result in both positive andnegative returns given that they are multiplied by a constant, θ.An interesting extension to the APT model would be to do the orthogonaltransformation not along the axis that has the greatest variance but ratheralong a predetermined risk factor. Say if an asset manager were to foreseea specific risk, e.g. a burst of an asset bubble, he could adjust his portfolioby minimizing that specific risk out of his portfolio or maximizing his positionagainst the risk event.

Appendix AData Appendix

A.1 Exploratory Data Analysis

In this section the exploratory data analysis is conducted and documented.The following are the most obvious observations. From Figure A.1 it can beseen that the JPY and NZD are quite volatile. The SEK has the greatestdepreciation following the 1992 ERM crisis. From 2002 to 2008 the U.S.dollar experienced quite the depreciation which shows up as an appreciationmor most of the other currencies. It is also interesting to notice the correlationof the JPY and CHF during the financial crisis of 2008 ans all the othercurrencies depreciate they appreciate, most probable due to their safe-havenstatus.Considering next the Quarterly correlations in Figure A.2 it is apparent thatthe JPY has low correlation with most of the other currencies except for theAUD and CHF. The JPYAUD is a popular carry trade pair and the JPY andCHF are both safe-haven currencies. The GBP also has low correlation withthe other currencies with most of the other currencies. The AUD and CADare heavily correlated as both economies are heavily dependent on natural

A.1 Exploratory Data Analysis 78

resources. Economies that are geographically close to each other also usuallyhave higher correlations.Now considering the GDP in Figure A.3 it can be seen that the GDP isgenerally positive with Japan struggling the most to maintain economic growth.The financial crisis of 2008 hit most of the economies quite hard but they alsorecovered quite quickly. Considering the correlations of GDP in Figure A.4 itis apparent that the U.S., U.K and Eurozone economies that have the highestcorrelations. This might be interpreted as the world economy.Now considering the foreign reserves in Figure A.5 it is noticeable that theU.S. reserves is almost non-existence in comparison with the other economies.Furthermore, that the smaller economies hold greater proportion of their GDPas currency reserves. Most noticeable in the great accumulation of foreignreserves by CHF in the 2009-2012 period caused by heavy buy demand ofthe CHF due to uncertainty in financial markets in the aftermath of the 2008crisis.The current account as a ratio of GDP can be seen in Figure A.6. The smallwealthy economies of Swiss, Norway and Sweden have the highest ratios,whereas the Australian and New Zealand have the lowest ratios. Japan hasa pretty stable and positive current account.The 10 year bond yields can be seen in Figure A.7. It is interesting to notehow interest rates have been lowering for the past twenty years. Swedenhas lowered their bond yields from over 10% to under 2%. It is interestingto note the low interest rates for the prolonged period from 2011 to date. Itrepresents the extreme pessimistic prospects of financial markets.The consumer price index can be seen in Figure A.8. It shows how Japan hasbeen struggling with deflation and interestingly shows that Australia and theU.S. have had the greatest inflation of the economies under consideration.Considering the M2 money supply in Figure A.9 shows that the AUD has hadthe greatest increase in money supply over the observation period.


0.6

0.8

1.0

1.2

1.4

1.6

Monthly Exchange Rates

Year

Mon

thly

Exc

hang

e R

ates

JPYAUDCADEURNOK.NZDSEKCHFGBP

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure A.1: Monthly Exchange Rates Returns from Q3-1992 to Q2-2012. Data areindexed to the first observation


JPY

0.5 0.9 1.0 1.4 0.4 0.7 0.6 1.0

0.7

1.1

0.5

0.9

0.71AUD

0.61 0.90CAD

0.7

1.0

1.0

1.4

0.52 0.84 0.83EUR

0.53 0.89 0.87 0.96NOK.

0.12

0.20

0.4

0.7

0.57 0.93 0.82 0.83 0.88NZD

0.38 0.78 0.70 0.88 0.89 0.77SEK

0.10

0.18

0.6

1.0

0.80 0.91 0.86 0.84 0.87 0.85 0.71CHF

0.7 1.1

0.15 0.37

0.7 1.0

0.46 0.56

0.12 0.18

0.58 0.53

0.10 0.18

0.58 0.29

1.4 1.8

1.4

1.8GBP

Monthly Exchange Rates

Figure A.2: Correlation of Monthly Exchange Rates for the period Q3-1992 to Q2-2012.


−50

5

GDP

Year

GD

P

JapanAustraliaCanadaEuroNorwayNew.ZealandSwedenSwitzerlandU.K.U.S.

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure A.3: GDP from Q3-1992 to Q2-2012.


Japan

1 3 5 −4 2 −2 4 −2 2 −4 2

−5

5

13

5

0.27

Australia

0.45 0.45

Canada

−4

26

−4

2

0.62 0.38 0.80Euro

0.35 0.56 0.45 0.52

Norway

−2

4

−2

4

0.53 0.49 0.55 0.43 0.57New.Zealand

0.62 0.34 0.79 0.86 0.37 0.44

Sweden

−6

28

−2

2

0.46 0.17 0.53 0.76 0.27 0.12 0.68Switzerland

0.65 0.63 0.75 0.77 0.61 0.72 0.70 0.46

U.K.

−6

06

−5 5

−4

2

0.58 0.62

−4 2 6

0.81 0.74

−2 4

0.62 0.55

−6 2 8

0.68 0.41

−6 0 6

0.84U.S.

GDP

Figure A.4: Correlation of GDP from Q3-1992 to Q2-2012.


0.0

0.5

1.0

1.5

Foreign Reserves as a Ratio of GDP

Year

Fore

ign

Res

erve

s as

a R

atio

of G

DP

JPYAUDCADEURNOKNZDSEKCHFGBPUSD

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure A.5: Foreign Reserves from Q3-1992 to Q2-2012. It is apparent that smallereconomies hold larger proportions of their GDP in foreign reserves.


−0.1

0−0

.05

0.00

0.05

0.10

0.15

0.20

0.25

Current Account as a Ratio of GDP

Year

Cur

rent

Acc

ount

as

a R

atio

of G

DP

JPYAUDCADEURNOKNZDSEKCHFGBP

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure A.6: Current Account as a ratio of GDP from Q3-1992 to Q2-2012.


24

68

10

10y Bond Yields

Year

10y

Bond

Yie

lds


1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure A.7: 10 Year Bond Yields from Q3-1992 to Q2-2012.


1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Consumer Price Index

Year

Con

sum

er P

rice

Inde

x

JPYAUDCADEURNOK.NZDSEKCHFGBPUSD

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure A.8: Consumer Price Index from Q3-1992 to Q2-2012.


12

34

M2 Money Supply

Year

M2

Mon

ey S

uppl

y


1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Figure A.9: M2 Money Supply from Q3-1992 to Q2-2012.

Appendix BProgramming

In this chapter a brief introduction is made on the programming proceed-ings. All programming was performed in the statistical software R. A briefdiscussion about the software can be seen in Section B.1.B.1 The R Language

R is a programming language and environment for statistical computation andsoftware development. R’s main strengths lie in statistical and time-seriesanalysis, whereas it can also be used in general matrix calculations. R alsohas great data manipulation abilities, and fine graphical facilities.R is a great environment for statistical software development whereas it ishighly flexible in programming new functions. R objects can be manipulatedby the programming language C and for computer intensive tasks C, C++and Fortran code can be linked and called at run time, making R particularlypractical for model development. R can also be used from within Microsoft

B.2 R code 89

Excel. The R language is one of the most widely used statistical softwareamongst statisticians1.R is an open source program made available by the R Development CoreTeam [2007]2. The term open source means that it is free of charge and allprograms are written by active practitioners. In order to give credit to thewriters of the additional packages, used in the programming process of thisthesis, a brief notation about the packages follows, with citations for furtherinformations.B.2 R code

In order to recreate the results presented in this thesis the following procedureshould be executed. Set directory, load data and run functions in QuickStart.rthen run the CombinedModelRun.r and MeanVarPlot.r.Code appendix is omitted but all code is available up on request. For dataand code requests please send e-mail to [email protected].

1Along with it’s commercial twin sister S-PLUS.2For further informations see R’s homepage:http://www.r-project.org/¸

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Parametric Portfolio Policy using Currencies as an Asset Class

Documents

uncovered

purchasing

nominal interest

nominal interest

bretton woods

producer price

gross domestic

real interest