Essays in Macro-Finance and International Finance

Essays in Macro-Finance and International Finance

Yu Liu

Submitted in partial fulfillment of the

requirements for the degree

of Doctor of Philosophy

in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2015

c©2015

Yu Liu

All Rights Reserved

ABSTRACT

Essays in Macro-Finance and International Finance

Yu Liu

This dissertation contains three essays on macro-finance and international finance.

In Chapter 1, Richard Clarida and I study the term structure of US interest rates using

observable macro factors as inputs to a Taylor-type rule that can account for the time path

of the short term interest rate. Using a standard essentially affine model, we build directly

on the pioneering work of Ang and Piazzesi, Rudebusch and Wu, and others but extend their

analysis to a framework in which all macro factors are observable. We focus on the period

since 1997 when US inflation expectations have been well anchored and inflation indexed

bonds – which provide useful information on expected inflation and expected future real in-

terest rates - have been issued by the US government. In contrast to many previous studies

that – of necessity - focus on earlier periods when low frequency movements in expected infla-

tion appeared to dominate, in our sample variation in expected inflation at longer horizons is

modest and the yield curve is importantly driven by the evolution of the ‘neutral’ real policy

rate as estimated by Laubach and Williams. Deviations of the policy rate from the Taylor

rule path are found to have a marked impact at the front end of the yield curve. In any

factor model yields are linear combinations of factors, and principal components, are linear

combinations of yields. In our model, we can solve explicitly for the mapping from macro

factors to traditional ‘level’ ‘slope’ and ‘curvature’ factors. Our model exhibits surprising

robustness in a post-crisis out-sample study. We also propose a novel, but simple regression

based approach to generate initial values - required to implement the non-linear GMM esti-

mation technique we use - for the affine model’s deep structural parameters.

In both Chapters 2 and 3, I study the portfolio problem associated with currency carry trade.

In Chapter 2 specifically, I analyze the carry trade threshold portfolios. I prove that under

general assumptions, the optimal mean-variance portfolio gives a higher weight to carry trade

having larger forward premium. I then proposes a more robust version of the mean-variance

optimal portfolio: the threshold portfolio, where I construct carry trade threshold portfolios

using thresholds that depend upon forward premium. And I show that empirically, up to

the optimal threshold value, higher-threshold portfolios outperform lower-threshold portfo-

lios. The financial performance then decreases, as the threshold goes higher. I model the

threshold effect in a random-walk model of exchange rates. The model predicts the optimal

threshold value and the relative gain of an optimal threshold portfolio. The model is cali-

brated, and the predictions are tested. I also discuss the threshold effect in a model which

features global risk factor. Following Jurek (2014) and using crash-hedged portfolios, I test

the crash risk explanation for outperformance of threshold portfolio, I show that the crash

risk premium can explain around 25 percent of the excess performance of higher threshold

portfolios.

In Chapter 3, I study the hedging problem associated with currency carry trade. I propose

theoretical frameworks and divide hedging instruments into three categories: insurance, tech-

nical rule, and the market neutral strategy. I then propose and empirically test four hedging

strategies: FX options strategy, VIX future strategy, “Stop-loss” rule and CTA strategy.

Based upon empirical evidence from 2000 to 2012, I find that CTA is the preferred hedging

strategy because it upgrades both return and volatility. The stop-loss strategy reduces risk.

Both the currency options strategy and the VIX future strategy offer good hedges against

tail risk, while also reducing volatility. Unfortunately they are costly to implement. I also

compare the VIX strategy to various currency option strategies, to determine if VIX is a

cheaper form of systematic insurance as compared to the currency options. With respect

to CTA, I study its risk-return aspect, I also provide a new methodology for replicating the

returns of the benchmark CTA index.

Table of Contents

List of Figures v

1 Macro Fundamentals and the Yield Curve: Re-Interpreting Factors in the

Essentially Affine Model 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Modeling Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Stochastic discount factor and the Taylor rule . . . . . . . . . . . . . . 5

1.2.2 Interest rate term structure . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Econometric methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Orthogonality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.2 Exactly identified model . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.3 Over-identified model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.1 Measurement for real neutral interest rate . . . . . . . . . . . . . . . . 11

1.4.2 Measurement for output gap . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.3 Measurement for the expected inflation . . . . . . . . . . . . . . . . . 13

1.4.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Results of exactly identified model . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.1 Macro factors dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.2 Market price of risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5.3 Impulse response function . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5.4 Variance decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 18

i

1.5.5 Robustness checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Extension: Forecasting of Bond Excess Return . . . . . . . . . . . . . . . . . 23

1.6.1 Forecasting using forward yields . . . . . . . . . . . . . . . . . . . . . 23

1.7 Extension: Adding Bonds of Longer Maturities . . . . . . . . . . . . . . . . . 25

1.7.1 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . 26

1.7.2 Impulse response function . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.7.3 Variance decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.7.4 Robustness checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.8 Robustness check and discussions . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.8.1 Testing model on alternative measurements: I . . . . . . . . . . . . . . 33

1.8.2 Testing model on alternative measurements: II . . . . . . . . . . . . . 35

1.8.3 Model using efficient interest rate . . . . . . . . . . . . . . . . . . . . . 37

1.8.4 Taylor rule under zero lower bound regime . . . . . . . . . . . . . . . 39

1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2 Carry Trade Threshold Portfolios 42

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2 Carry Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3 Currency investor’s portfolio problem . . . . . . . . . . . . . . . . . . . . . . . 48

2.3.1 Mean-variance problem . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.3.3 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3.4 Robustness check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.4 Threshold portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.1 Threshold portfolios based on forward premium . . . . . . . . . . . . . 55

2.4.2 Robustness check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.4.3 Comparison with a quantile based sorting rule . . . . . . . . . . . . . 57

2.5 Crash-hedged threshold portfolios . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.5.2 Performance of crash-hedged threshold portfolios . . . . . . . . . . . . 59

2.6 Random-walk model of threshold effects . . . . . . . . . . . . . . . . . . . . . 60

ii

2.6.1 Random-walk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.6.2 Possible variations of the random-walk models . . . . . . . . . . . . . 62

2.6.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.6.4 Model with crash-hedged portfolios . . . . . . . . . . . . . . . . . . . . 63

2.6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.6.6 Threshold portfolios when prices of risks are the same . . . . . . . . . 65

2.6.7 Extending models to include currencies with correlations . . . . . . . . 65

2.7 One-factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.7.1 One-factor model with equal correlations of all currency pairs . . . . . 67

2.7.2 One-factor model with heterogeneous correlations of currency pairs . . 68

2.7.3 Model with increasing downside correlation . . . . . . . . . . . . . . . 70

2.7.4 Implication of random-walk model and one-factor model . . . . . . . . 71

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Currency Carry Trade Hedging 73

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 Framework and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2.2 Hedging performance evaluation . . . . . . . . . . . . . . . . . . . . . 79

3.2.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.3 Hedging using currency options or VIX future . . . . . . . . . . . . . . . . . . 81

3.3.1 Crash-risk hedging strategies . . . . . . . . . . . . . . . . . . . . . . . 82

3.3.2 Financial performances of hedged carry trade portfolios . . . . . . . . 84

3.3.3 Who would buy the insurance? . . . . . . . . . . . . . . . . . . . . . . 86

3.3.4 Hedging effects during crash periods . . . . . . . . . . . . . . . . . . . 87

3.4 Stop-loss Rules Applied to FX Carry Trade . . . . . . . . . . . . . . . . . . . 89

3.4.1 Stop-loss rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.4.2 Financial performances of hedging strategies . . . . . . . . . . . . . . 90

3.4.3 Who would buy the insurance? . . . . . . . . . . . . . . . . . . . . . . 90

3.4.4 Robust check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5 CTA trend following as a hedging strategy . . . . . . . . . . . . . . . . . . . . 92

iii

3.5.1 Constructing CTA portfolio . . . . . . . . . . . . . . . . . . . . . . . . 93

3.5.2 CTA as a hedging strategy . . . . . . . . . . . . . . . . . . . . . . . . 94

3.6 Out-Sample Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.7 Two Issues Related to the Hedging Strategies . . . . . . . . . . . . . . . . . . 97

3.7.1 Compare the price of VIX futures to currency options . . . . . . . . . 98

3.7.2 CTA risk/return analysis . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Bibliography 111

A Appendix for Chapter 1 121

B Appendix for Chapter 2 144

C Appendix for Chapter 3 160

iv

List of Figures

1.1 Price Fitting for Exactly Identified Model . . . . . . . . . . . . . . . . . . . . 16

1.2 Contemporaneous Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Impulse Responses at Future Horizon . . . . . . . . . . . . . . . . . . . . . . . 19

1.4 In-sample and Out-sample Pricing . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5 PCA Factor Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.6 Contemporaneous Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . 27

1.7 Impulse Responses at Future Horizon . . . . . . . . . . . . . . . . . . . . . . . 28

1.8 In-sample and Out-sample Pricing . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1 Cumulative G10 currency carry trade returns . . . . . . . . . . . . . . . . . . 83

3.2 Cumulative hedging strategies returns . . . . . . . . . . . . . . . . . . . . . . 83

3.3 CTA Total Return Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.4 Benchmark Index and Replication Portfolio . . . . . . . . . . . . . . . . . . . 107

3.5 Out-Sample Return Replication . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.1 Macro-factors in Taylor Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A.2 Model Implied Yields and Observed Yields . . . . . . . . . . . . . . . . . . . 123

A.3 Impulse-responses Error Bands . . . . . . . . . . . . . . . . . . . . . . . . . . 125


A.5 Contemporaneous Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . 131

A.6 Impulse Responses at Future Horizon . . . . . . . . . . . . . . . . . . . . . . . 131



v





B.1 Cumulative G10 carry trade portfolio returns . . . . . . . . . . . . . . . . . . 152

B.2 G10/USD currencies’ volatilities . . . . . . . . . . . . . . . . . . . . . . . . . 152

B.3 Threshold portfolios of G10 currencies versus US dollar: Part 1 . . . . . . . . 153


B.5 Mean-variance Optimal Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . 154

B.6 10 Delta Put hedged threshold portfolios of G10 currencies versus US dollar:

Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

B.7 10 Delta Put hedged threshold portfolios of G10 currencies versus US dollar:

Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155



B.10 Unhedged and hedged portfolios of TR0 and TR2.5 . . . . . . . . . . . . . . . 157

B.11 Empirical distribution of forward premiums and fitted exponential distribution 157

B.12 Threshold portfolios’ Sharpe ratios versus random-walk model implied values 158

B.13 Sharpe ratios of crash-hedged threshold portfolio with equal implied volatility 159

B.14 Sharpe ratios of crash-hedged threshold portfolio with variable implied volatility159

C.1 Cumulative return of portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . 164

vi

Acknowledgments

I am deeply indebted to my advisor, Professor Richard Clarida, for his guidance and encour-

agement throughout my years in the Ph.D program. I am very fortunate to have him as my

close mentor, for he has given me a guided tour through the fascinating world of international

finance and macro-finance. I greatly appreciate his generous contributions of time, ideas, and

envouragement, which have helped me to make my own humble contribution to economic

research. I thank Professors Serena Ng and Ricardo Reis for their helpful comments and

suggestions on my research papers. I also wish to thank Professors Jushan Bai and Robert

Hodrick for serving on my thesis committee.

I thank Professors Padma Desai and Edward Lincoln for the great experience I have had

while working with them. I thank Stephane Daul for helpful discussions. I also want to

thank all of the colleagues that I have been so fortnate to have in the Ph.D program: Feiran

Zhang, Shaowen Luo, Lina Lu, Charles Maurin, Zheli He. I am grateful to have had this great

community of smart and motivated people who accompanied me on my journey in pursuit of

economic wisdom. Moreover, they have offered great friendship and shared many important

moments with me.

Finally, this dissertation could never have been written without the support and love of my

wife, Zoey Yi Zhao, and my parents, Hongli Liu and Xiaoneng Jiang. I dedicate it to all of

them.

vii

To my wife, Zoey Yi Zhao, and my parents, Hongli Liu and Xiaoneng Jiang

viii

CHAPTER 1. MACRO FUNDAMENTALS AND THE YIELD CURVE:RE-INTERPRETING FACTORS IN THE ESSENTIALLY AFFINE MODEL 1

Chapter 1

Macro Fundamentals and the Yield

Curve: Re-Interpreting Factors in

the Essentially Affine Model

1.1 Introduction

The Taylor rule (Taylor [1993]) is widely regarded as framework useful for interpreting how

central banks set policy rates - see Clarida et al. [1998a], Clarida et al. [1998b] and Clarida

et al. [1999]. According to the Taylor rule, the central bank policy rate should reflect the real

neutral rate of the economy1, the deviation of the expected inflation vis-a-vis the inflation

target and the GDP gap. It is tempting to investigate the impact of Taylor rule factors on the

bond yield curves. In this chapter, we study the term structure of interest rates using observ-

able macro factors as inputs to a Taylor-type rule. This is an interesting approach, because

the Taylor rule stands right at the intersection of finance and macroeconomics. Its output,

the short-term interest rate, is the building block for longer-term interest rates through the

expectation hypothesis. Its inputs, various macroeconomic fundamentals, reflect the general

states of the economy and are closely tracked by central banks. By using Taylor rule factors

1Real neutral interest rate refers to the real short-term interest rate under which the output converges to

potential, where potential is the level of output consistent with stable inflation.


in a macro-finance model, we set out to explain term structure variations through observable

and interpretable macro factors.

Here we are exploring an arbitrage-free term structure model, where the arbitrage opportu-

nities are eliminated by restrictions on the yields. One approach in finance uses latent factors

to price yields, using an affine stochastic discount factor, see Duffie and Kan [1996]; Dai and

Singleton [2000]. There is also a large amount of literature delving into term structures by

imposing no-arbitrage conditions on yield curves, and the latter are modeled, in some papers,

by using macroeconomic factors. Ang and Piazzesi [2003] is a pioneering paper that models

yield curves using macroeconomic factors in a reduced-form VAR setting. In our model,

all four factors are observable and inputs to a Taylor-type rule: real neutral interest rate,

expected inflation, output gap, and a Taylor rule deviation. We allow for the Taylor rule

deviation to be a priced factor subjected to the restrictions of the essentially affine model.

Previous studies have found that imposing a no-arbitrage condition improves the forecasting

performance of a VAR and that using macro factors improves the forecasting performance,

see Ang and Piazzesi [2003]. Hordahl et al. [2006] models evolution of inflation and output

gap within a New Keynesian framework. They confirm that imposing a no-arbitrage con-

dition, and including macro factors, improve the forecasting performance. Rudebusch and

Wu [2008] combines a term structure model with a New Keynesian model of macro factors.

It shows bi-directional interaction between macro factors and term structure variables. For

more contributions to the macro-finance literature, see for example Piazzesi [2005], De Graeve

et al. [2009], and Bikbov and Chernov [2010]. For testing a general class of affine model, see

Ghysels and Ng [1998].

Our estimation results based on US data suggest that macro variables affect the term struc-

ture in various ways. Yield curve exhibits strong contemporaneous response to macro shocks,

with expected inflation and real neutral rate having the most persistent effect. Real neutral

rate explains most of the price variation at long horizon for all bonds. At short horizon,

expected inflation explains most of the price variation for bonds at belly and long end, and

Taylor rule deviation mainly explains movements of the yield curve on short end. When it

comes to the impact of macro factors on the yield curve, our results are in line with previous

studies. Ang and Piazzesi [2003] finds that macro factors mainly account for movements


at the short and medium ends of yield curves. Ludvigson and Ng [2009] finds that “real”

and “inflation” factors have forecasting power for US bonds’ excess returns. Hordahl et al.

[2006] finds that monetary policy shocks have an impact mainly at the short end of yield

curve. Dewachter and Lyrio [2006] models long-run inflation simultaneously with term struc-

ture. It imposes martingale condition on latent macro factors and finds that for long end

of yield curve, inflation expectation is all-important. For short end, both real rate and in-

flation expectation count. Wu [2006] proposes a DSGE model in an attempt to characterize

the underlying macro factors. It finds that movement of the slope factor can be explained

by monetary policy shocks. And the level factor movement can be explained by technology

shocks. Duffee [2006] offers a model having only observable factors. He finds a positive re-

lation between short-term rate and expected inflation. Bekaert et al. [2010] models macro

factors in a New Keynesian setting. It models pricing kernel consistently with IS curve. The

model shows strong contemporaneous responses of the yield curve to macro shocks. Our

results also feature robustness during the post-crisis period in which the policy rate has been

constrained at the zero lower bound. We use data from 1997 to early 2008 to calibrate the

model parameters. We then use the data from late 2008 to 2012 to do our model testing. For

the out-sample period, our model exhibits good pricing accuracy. The out-sample fitting for

short end curve is, however, worse than medium and long ends. We discuss this issue and find

that this is due to the zero-lower bound regime in the out-sample period which particularly

affects particularly the short end.

After studying an exactly identified model, we extend our analysis to an over-identified model.

We propose a novel method for generating initial parameter values required for our non-linear

GMM estimation framework.

We also discuss the measurement issues of macro factors. In the main text we use Laubach

and Williams [2003]’s measurements of GDP gap and real neutral rate. We test the model

on two alternative measurement specificaions for robustness check.

In addition, we apply our model to an alternative monetary policy rule, referred to as W

rule by Curdia et al. [2014]. For the W rule the output gap doesn’t affect the monetary

policy reaction function. We find similar results regarding the real neutral rate, the expected

inflation, and the rule deviation.


Among the papers studying the Taylor rule the most related work to ours is Smith and Taylor

[2009]. Their interest lies in studying the response of coefficients in the affine equation to the

change in coefficients in the Taylor rule. It also derives the formula linking the coefficients

of the Taylor rule to the coefficients of the implied affine equations for bond yields, but it

doesn’t calibrate the affine term structure model using bond yields data. The interest of the

present chapter is, however, to study the bond pricing using factors derived from Taylor rule.

We now briefly review some other papers that study the interactions between term structure

and macro factors. Among the papers imposing no-arbitrage conditions, some study the im-

plications for macro-economy of term structure variables within this macro-finance setting.

Ang et al. [2006] finds that the short rate has more predictive power than the term spreads

when it comes to GDP forecasting. Rudebusch et al. [2007] studies the effect of bond term

premium on macro variables, in both structural and reduced VAR settings. Another approach

estimates term-structure model without imposing any no-arbitrage conditions. Diebold et al.

[2006], for instance, uses both latent factors and observable macroeconomic factors. It finds

an effect of macro factors on future yield curve, but also an effect in the other direction. It

thus provides evidence of a bidirectional flow of information between the macro-economy and

the financial market.

The standard model can be extended in various ways. Monch [2008] explores the data-rich

environment by extracting factors from a large macro dataset. It improves forecasting per-

formance at the middle and long ends of yield curve. Farka and DaSilva [2011] estimates

the contemporaneous relation between monetary policy shocks and term structures, using

high frequency data around announcements. Following the financial crisis of 2008, some

papers started using non-traditional factors in the standard model to model term structure.

Dewachter et al. [2014] introduces liquidity factor and return-forecasting factors. It finds that

for cross-sectional yield curve fitting, the model has an advantage over most models using

traditional factors. It also finds that financial shocks have an impact on yield curve.

The rest of this chapter is organized as follows. Section 2 introduces a no-arbitrage macro-

finance model so as to study the interest rate term structure dynamics. Section 3 explains

the econometric method. It proposes an initial value generation method for GMM. Section

4 discusses the measurement issues of macro factors. Section 5 explores an exactly identified


model and analyzes the results. Section 6 studies the forecasting of bond excess returns.

Section 7 extends the model to over-identified case. Section 8 tests the model to alternative

measurements of macro factors and alternative monetary policy rule. It also discusses the

mode’s performance under zero-lower regime. Section 9 concludes the chapter.

1.2 Modeling Framework

1.2.1 Stochastic discount factor and the Taylor rule

In the essentially affine model, the short term interest rate is assumed to be an exact linear

function of a small number of ‘factors’. In many papers, the factors are unobserved and must

be inferred from bond yields themselves. We pursue a different course by selecting observable

macro factor as inputs to a Taylor rule. As a matter of accounting, we can always write

it = π∗ + rnrt + 1.5(πt − π∗) + 0.5gt + ut (1.1)

where π∗ is the target inflation rate, it is the nominal short term interest rate, rnrt is an

estimate of the ‘neutral’ real policy rate, πt is an estimate of expected inflation, gt is an

estimate of the output gap, and ut is the deviation of the interest rate from the Taylor rule

path. We assume π∗ equal to 2% for the following. Although in Taylor’s original formulation

the neutral real interest rate was assumed to be constant, the theory of optimal monetary

policy (Clarida et al. [1999]; Woodford [2003]) as well as empirical evidence (Laubach and

Williams [2003]) supports allowing for it to be time varying. With a Taylor rule formulation

and observable macro factors, the affine term structure model’s requirement that the short

rate being an exact linear combination of factors identifies ut – the Taylor rule deviation –

as a fourth factor.

In this chapter we test three measurement-specifications of the Taylor rule’s factors. For

the first specification(Specification I), we use Williams and Laubach’s estimation for the

real neutral rate and the GDP gap. We use Haubrich et al’s estimation for the inflation

expectation. For the second specification(Specification II), we keep the estimation of real

neutral rate and of expected inflation from Specification I. But we estimate the GDP gap

by using Okun’s law. For the third specification(Specification III), we estimate the real


neutral rate by the five-year forward rate five year in the future minus the inflation risk

premium, the expected inflation by the 5-year break-even inflation rate minus the inflation

risk premium, and the GDP-gap by Okun’s law. In the main test, we discuss results of the

first specification. In the section of the robustness check we test the second and the third

specifications.

Crucially the affine term structure model does not require that ut be white noise or that

it be orthogonal to the other factors. It only requires that a first order vector auto regression

in the four factors captures the forecastable variation in the factors. Thus, let the state

Xt = rnrt, gt, πt, ut, where Xt follows a Gaussian vector autoregression process,

Xt = µ+ ΦXt−1 + Σεt (1.2)

where µ is a constant vector, Φ a constant matrix, εt a Gaussian white noise N(0, I4), and

Σ a lower triangular matrix. Thus, as is common in VAR studies, we impose a Cholesky

identification with order rnrt, gt, πt, ut. Assuming the no-arbitrage condition holds, there

will exists a stochastic discount factor that will price bonds conditional on 1.1 and 1.2.

We follow Ang and Piazzesi [2003] (and many others since) and assume that the nominal

stochastic discount factor mt is of the essentially affine class and can be written as

mt+1 = exp

(−rt −

1

2λ∗Tt λt − λ∗Tt εt+1

)(1.3)

where λ∗t is the time-varying market price of risks in the economy, and εt is the innovation to

state variable Xt. Following the affine term structure model literature, we assume that the

market price of risk λ∗t is an affine function of state variable Xt,

λ∗t = λ0 + λ1Xt (1.4)

where λ0 is a vector of constants.

1.2.2 Interest rate term structure

We now model the dynamics for short rate and zero-coupon bond yields, after specifying the

SDF. Since the state variable has been deduced from the Taylor rule equation for short rate,

the affine short rate equation can be specified as

rt = δ0 + δT1 Xt (1.5)


Where by adjusting ut we have δ0 = 0 and δ1 = [1, 1.5, 0.5, 1]′.

Given the no-arbitrage assumption, we have the following relation linking price of the bond

with maturity of n periods at time t to that with maturity of n− 1 periods at time t+ 1

Pnt = E t[mt+1Pn−1t+1 ]

We can recursively solve for the yield of an n-period zero coupon bond, ynt = −log(Pnt ), as a

linear function of the state variable Xt:

ynt = an + bTnXt (1.6)

an and bn are given by an = −An/n and bn = −Bn/n, where An and Bn can be computed

recursively:

An+1 = An +BTn (µ− Σλ0) +

1

2BTnΣΣTBn − δT0 (1.7)

BTn+1 = BT

n (Φ− Σλ1)− δT1 (1.8)

with A1 = −δ0 and B1 = −δ1. From equations 1.7 and 1.8, the time-invariant component,

λ0, affects only the time-invariant component of yield an, while the time-varying parameter

affects the time-varying component of yield bn. Thus, λ0 affects only average bond yield,

and λ1 determines solely the time variation of bond yields. In the following section, we

concentrate on the latter.

1.3 Econometric methods

Since our macro factors are observable, we use GMM to estimate the model. We discuss two

cases: an exactly identified model and an over-identified model.

1.3.1 Orthogonality Conditions

We follow the literature and assume that, in the presence of of measurement error, Eq. 1.6 is

satisfied up to an additive, orthogonal disturbance. Thus we model each yield in the following

way:

ynt = an + bTnXt + ent (1.9)


with the following orthogonality condition:

E [Xtent ] = 0 (1.10)

Given all factors are observable, non-linear GMM which incorporates the model’s cross equa-

tion restrictions is our framework for estimation and testing. In the next subsections we

discuss, respectively, the exactly identified model and the over-identified model.

1.3.2 Exactly identified model

We use Cholesky identification to estimate 1.2, from which we get µ, Σ, and Φ. δ0 and

δ1 are also known. Hence, from equations 1.7 and 1.8, An and Bn are functions of λ0 and

λ1, denoted as An(λ0, λ1) and Bn(λ1). Let us suppose we observe bonds with N different

maturities. For each bond with a specific maturity we have T observations. We then have

the following orthogonality condition:

1

T

T∑t=1

[ynt +

An(λ0, λ1) +BTn (λ1)Xt

n

]= 0 (1.11)

1

T

T∑t=1

[ynt +

An(λ0, λ1) +BTn (λ1)Xt

n

]·Xt = 0 (1.12)

If we have K factors, then we have N(1+K) orthogonality conditions, and we have K(1+K)

unknowns. When K = N , the model is exactly identified.

1.3.3 Over-identified model

Using the framework outlined in subsection 1.3.2, we see that when K < N , the model is

over-identified. Indeed, the number of orthogonality conditions is N(1+K), while the number

of unknowns is K(1 +K). Thus the model is over-identified when K < N . Our motivation is

to study the time variation of bond yield, which is affected solely by λ1 as explained in 1.2.2.

Thus we conduct a two-stage estimation. In the first stage, we estimate λ1 by exploring

demeaned factors and demeaned yields. We then estimate λ0 by using the original factors

and yields.

We have the following moment condition for demeaned factors dXt and demeaned bond yields


dynt :

G(λ1, n) =1

T

T∑t=1

[dynt +

BTn (λ1)dXt

n

]· dXt = 0 (1.13)

Because the model is over-identified, in general we will not find any λ1 that satisfies 1.13 for

all yields to maturity. Let us use MaturitySet to denote the set of bond maturities under

study. In section 1.7 we will estimate a model for bonds with 12 different maturities, such

that MaturitySet = 3M, 6M, 1Y, 2Y, ..., 9Y, 10Y .

Given a set of initial values, we can use the quasi-Newton method to minimize the GMM

objective function:

λ1 = minλ∗1

∑n∈MaturitySet

GT (λ∗1, n)G(λ∗1, n) (1.14)

Since the objective function is highly nonlinear, the performance of a nonlinear optimization

algorithm can depend crucially on the initial values selected for the structural parameters.

One of the contributions made by the present chapter is to innovate a method for obtaining

initial values.

1.3.3.1 Initial value generation

From Eq. 1.8, we see that λ1 is uniquely determined from the following condition, given four

pairs of B whose maturities differ by one month:

BTn+1 = BT

n (Φ− Σλ1)− δT1

For example if we know Bn for n ∈ 1, 2, 4, 5, 6, 7, 9, 10, then BTn+1 = BT

n (Φ− Σλ1) − δT1gives us 16 equations for n ∈ 1, 4, 6, 9. We then can uniquely solve λ1. In order to have

an initial value for λ1, we need an estimate of the reduced form Bn parameters. We obtain

those from a reduced form regression

y(i),nt =

A(U)n

n+B

(U)Tn Xt

n+ νnt (1.15)

Consequently, we solve initial value λInitial1 with the following condition

B(U)Tn+1 = B(U)T

n

(Φ− ΣλInitial1

)− δInitial1 (1.16)

In practice, existing data sets do not include yields to maturity for all tenors between 1 month

and 120 months. As a result, we use an interpolation method to generate approximate yields


to maturity for tenors contiguous to the tenors we study in the exactly identified model. We

follow the method of Nelson-Siegel.

1.3.3.2 Nelson-Siegel Yield Curve Model

Nelson and Siegel [1987] models the forward rate as follows:

f(τ) =[β0 β1 β2

]1

e−τ/λ

(τ/λ)e−τ/λ

where λ is the only model parameter, and τ is the maturity. The spot rate has the following

structure:

r(τ) =[β0 β1 β2

]1

λ/τ(1− e−τ/λ)

λ/τ(1− e−τ/λ)− e−τ/λ

Note that Nelson and Siegel [1987] models the three latent factors of yield curve “level”,

“slope” and “curvature” by using three functions: 1, λ/τ(1−e−τ/λ) and λ/τ(1−e−τ/λ)−e−τ/λ.

1.3.3.3 Generating Hypothetical Yields via the Nelson-Siegel Model

In order to use the Nelson-Siegel Model to generate hypothetic yields, we first fit it to observed

yields in order to estimate λ. We then generate the entire yield curve by computing the

function:

r(τ) =[β0 β1 β2

]1

λ/τ(1− e−τ/λ)

λ/τ(1− e−τ/λ)− e−τ/λ

After we have generated the interpolated yields, we use the methods in subsection 1.3.3.1 to

compute the initial value of λ1.

1.4 Data

We first discuss the measurement issue respectively for the real neutral rate, the output gap,

and the expected inflation. We then detail out three sets of specifications.


1.4.1 Measurement for real neutral interest rate

Real neutral rate can be motivated from the growth theory and related to the long-run

economic equilibrium. This is the method originally used by Taylor in Taylor [1993]. A key

implication of this approach is that real neutral rate is linked to the steady state growth rate

of output because real neutral rate changes over time in responding to technology change

and population growth. Another approach used in Hall [2000] is to characterize the time-

variation of real neutral rate by using the intercept of Taylor rule itself. This is an interesting

approach, especially by treating the real neutral rate as an unobservable variable in a Bayesian

approach.

Here following Archibald et al. [2001] and Laubach and Williams [2003] we focus on the

medium-run concept of real neutral interest rate, which abstracts from the short-run price

and output effects. Real neutral interest rate refers to the real short-term interest rate under

which the output converges to potential, where potential is the level of output consistent with

stable inflation (see Bomfim [1997]). The practical intuition of this assumption is: it may

take several quarters for the effect of the interest rate to take place on inflation. In contrast

to the short horizon, it is this longer horizon that is relevant for our discussion here. Clarida

et al. [1999] and Woodford [2003] have derived the time-varying natural interest rate for the

optimal monetary policy. We review below two methods to measure the real neutral rate.

The first method is based on the decomposition of observed inflation-indexed bond yields.

Note TIPSt the yield of an inflation-indexed bond at time t, rrt the real interest rate at time

t, and πrpt the risk premium of inflation. We then have the following relation

TIPSt = rrt + πrpt (1.17)

which says that the yield of inflation indexed bond is the sum of real rate and inflation risk

premium. We can further decompose the real interest rate into the real neutral interest rate

and a cyclical factor, which represents the monetary policy effect against inflationary pressure

on real rate. Let us denote as mt the monetary policy induced cyclical factor of real rate.

We thus have:

rrt = rnrt +mt (1.18)

TIPSt = rnrt +mt + πrpt (1.19)


Eq. 1.19 offers a way to estimate real neutral rate rnrt = TIPSt − (mt + πrpt ). Among the

three variables on the right-hand side, TIPSt is directly observable and πrpt can be estimated

in a joint nominal and real term-structure model. For mt we note that we can average out

the effects of mt by taking the average of Eq. 1.19. Note F TIPSnt the forward yield of a

one period inflation-indexed bond n periods in the future, rnrnt , mnt , and πrp,nt respectively

the expected real neutral rate, expected cyclical factor and expected inflation risk premium

for n periods in the future. We then have

F TIPSnt = rnrnt +mnt + πrp,nt

We estimate the rnrt by the following approximation:

rnrt =

∑119t=60 F TIPSnt

60− πrpt (1.20)

The second method we discuss is a Bayesian approach. Laubach and Williams [2003] jointly

estimates the output gap and the real neutral rate in a reduced-form equation. In their

model, the output gap is related to its own lags and the lags of real interest rate gap, which

is the difference between the real short-term rate and the real neutral rate. And the real neu-

tral rate is related to the potential output through the trend growth of the latter. Laubach

and Williams [2003] estimates the real neutral rate and output potential together by using

Kalman filter.

1.4.2 Measurement for output gap

We discuss here the output gap measures in the literature. One large class of detrending

method decomposes the log of output, yt into a trend component µt, and a cyclical component,

gaot.

yt = µt + gapt (1.21)

There are several methods to estimate the output trend µt and the output gap gapt. a).

Deterministic trends. This method approximates the output trend as a deterministic

function, with the linear trend and the quadratic trend being the simplest examples. b).

Unobserved component model. This kind of model assumes more complex dynamics


for the trend and the cyclical components. A classical example is HP filter proposed by

Hodrick and Prescott [1997]. For other examples, Watson [1986] models the trend process

to be random walk and cyclical process to be of AR(2). Harvey [1985] and Clark [1987]

both model trend process to be of stochastic trend and cyclical process to be of AR(2). c).

Unobserved component model with a Philips curve. This method adds a Phillips

curve to the univariate model described in b), see for example Kuttner [1994] and Gerlach

and Smets [1999].

The detrending methods have been shown to exhibit poor real time performance. Orphanides

and Van Norden [2002] shows that the reliability of detrending methods for real time output

gap estimation is poor mainly due to unreliability of the end-of-sample estimates of the trend

in output. We discuss next two alternative methods. The first approach is our preferred

approach, which is the joint estimation of output gap and real neutral rate in Laubach and

Williams [2003]. Their method was discussed previously in the subsection 1.4.1. The second

approach is the Okun’s law.

gapt = k(Ut − U∗) (1.22)

where Ut is the unemployment rate at time t, U∗ is the natural rate of unemployment, and

k measures the change in output per unit of change in the unemployment rate. This method

was originally proposed in Okun [1970], for more recent updates see Ball et al. [2013] and

Owyang and Sekhposyan [2012]. Ball et al. [2013] finds that Okun’s law did not change

substantially during the Great Recession. Owyang and Sekhposyan [2012] doesn’t finds

statistical significance of the slope change during the Great Recession for our specification.

Literature estimates k to be between 2 and 3 for US.

1.4.3 Measurement for the expected inflation

A common estimation for the expected inflation is the break-even inflation computed from

nominal and inflation-indexed bond yields. This method, however, doesn’t take into account

the inflation risk premium embedded in inflation-indexed bonds. If one wants to tackle this

issue, one has to model jointly the nominal and real interest rate term structure models.

Haubrich et al. [2012] and Christensen et al. [2010] are two examples. Both studies have

obtained close to zero inflation risk premium for recent years. Both papers also estimate the


expected inflation. We also note that Hilscher, Raviv, and Reis [2014] constructs risk-adjusted

distributions of inflation by using inflation option data.

1.4.4 Data

We use monthly US data from September 1997 to August 2012, which is a total of 180 months.

The 13 zero coupon bond yields of 1, 3, and 6 months as well as of 1, 2, 3, 4, 5, 6, 7, 8, 9 and

10 years all come from the US Treasury Department. For expected inflation we use results

of five-year expected inflation and five-year inflation risk premium in Haubrich et al. [2012],

which covers our whole sample period.

With respect to real neutral rate and GDP gap, our estimations in the main text come from

Laubach and Williams [2003]. As we discussed before the attraction of Laubach and Williams

[2003] is that it models real neutral rate and output gap jointly. As a robustness test we test

our model on two alternative sets of measurements. For the first alternative specification, we

keep the estimation of real neutral rate and of expected inflation from the main text. But

we estimate the GDP gap by using Okun’s law. For the second alternative specification, we

estimate the real neutral rate by the five-year forward rate five years in the future minus the

inflation risk premium, the expected inflation by the 5-year break-even inflation rate minus

the inflation risk premium, and the GDP gap by the Okun’s law.

Lastly the Taylor rule deviation is defined as in equation 1.1 with the short term rate it being

approximated by using the one month maturity bond yield.

1.5 Results of exactly identified model

Following Ang and Piazzesi [2003] and Rudebusch and Wu [2008] we first test the model with

bonds of maturity equal to 1, 3, 12, 36 or 60 months. Under our four factors specification,

the model is exactly identified.

1.5.1 Macro factors dynamics

We plot in Fig. A.1 in the appendix the contribution of each macro factor in Taylor rule

together with the Fed fund rate. Table 1.1 reports the estimates for state dynamics 1.2 and


the corresponding t-ratios. The diagonal of Φ determines the persistence of the macro factors.

We see that the diagonal terms are all higher than 0.8 and all highly statistically significant,

which suggest the strong persistence of macro variables.

Table 1.1: Macro Factors’ Dynamics

µ(×100) Φ Σ

r g π u r g π u

r -0.01 0.99 0.22

(0.00) (66.50)

g -3.39 0.01 0.95 0.02 0.21

(0.78) (0.61) (35.11) (1.40)

π -6.37 0.05 0.02 0.90 0.018 0.006 0.20

(1.33) (1.98) (0.58) (26.89) (1.15) (0.40)

u 52.02 -0.31 0.10 0.32 0.87 -0.27 -0.23 -0.16 0.30

(5.74) (6.09) (2.37) (5.56) (35.29) (11.91) (10.04) (7.24)

The table presents the estimates for the state dynamics following a VAR(1) process. The ab-

solute value of the t-ratio of each estimate is reported. The sample period is from September

1997 to August 2012.

1.5.2 Market price of risk

Using the method detailed above, we compute λ1. We report in Table 1.2 the result. In

Figure 1.1 we plot observed yields and model implied yields. The figure shows that our

model does a superior job of pricing the bonds.

1.5.3 Impulse response function

Here we study, respectively, the contemporaneous response of yields to shocks and responses

at longer horizons. The shock from each factor is assumed to be of one standard deviation

of the factor’s first difference. In Table 1.3 we report the assumed shock size, the unit is

percentage point.


Figure 1.1: Price Fitting for Exactly Identified Model

1997 2000 2002 2005 2007 2010 2012 2015−1

0

1

2

3

4

5

6

7Observed and model implied Yields for 0.25 Years Bond

Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 2015−1

0

1

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 2015−1

0

1

2

3

4

5

6

7Observed and model implied Yields for 1 Years Bond

Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 2015−1

0

1

2

3

4

5

6


Model Implied Yield

Observed Yield

Note: Since this model is exactly identified, the fit of the model simply indicates that our four factors capture

virtually all of the contemporaneous variation in the yields included. We test the over identified version of

the model in Section 7.


Table 1.2: Market Price of Risk

r g π u

r 1.87 -1.65 -0.59 0.28

(0.81) (0.53) (0.71) (0.17)

g 2.65 1.78 0.24 1.61

(2.07) (1.18) (0.21) (1.76)

π 0.17 0.64 -0.33 0.14

(0.47) (1.33) (2.28) (0.56)

u -0.59 -0.15 0.42 -0.37

(2.01) (0.43) (3.13) (1.93)

The table presents the estimates for the market price of essentially affine model. The

absolute value of the t-ratio of each estimate is reported. The sample period is from

September 1997 to August 2012. The standard deviation is corrected following Newey

and McFadden(1994).

Table 1.3: Macro Shock Size

RealRate GDPGap Inflation TYRes

0.22 0.42 0.14 0.52

1.5.3.1 Response of contemporaneous yields

The pricing equation ynt = an+ bTnXt leads us that the effect of each factor on the yield curve

is given by bn, which is in turn determined by the affine asset pricing structure. bn thus

represents the initial response to shocks made by different factors. In Figure 1.2 we plot the

initial responses of bonds having different maturities. The shock from each factor is assumed

to be of one standard deviation of the factor’s first difference.

We look at first the effects of real neutral rate and expected inflation. The responses are

very persistent, and the responses of bonds having different maturities are in the same order.

Thus, shocks to the neutral rate and expected inflation affect mainly the level factor. When

it comes to the other two shocks, the maximum response of yield curve is for bonds having

one month or three months maturity. Thus these shocks are mean reverting.


Figure 1.2: Contemporaneous Impulse Responses

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Bond maturities in Months

Annuliz

ed y

ield

in %

Initial response

RealRate

GDPGap

Inflation

TYRes

1.5.3.2 Response at longer horizon

Our term structure model gives us the yield curve responses to macro factor shocks at long

horizons. We plot in Figure 1.3 impulse responses to yield of maturity 1Y, 3Y and 5Y. We

find that for all three bonds, shock from real neutral rate is most significant at the longer

horizon. At one-month horizon, shock from Taylor residue is most significant for the 1Y

bond. For the 3Y and 5Y bonds the shock from expected inflation is most significant at the

one-month ahead horizon.

1.5.4 Variance decomposition

Based on our VAR dynamics we can decompose the forecasting variance of yields into con-

tributions from four macro shocks. In Table 1.4 we report the results of respectively 1Y,

3Y, and 5Y bonds. For the 1Y bond at one-month ahead horizon Taylor residue contributes

most of the variance. At longer horizons all factors contribute to the variance. For the 3Y

and the 5Y bonds we see results having similar properties. For one-month ahead forecasting,

expected inflation contributes most of the variance. At longer horizons, the real neutral rate

contributes most of the forecasting variance. We thus conclude that at short horizon Taylor

Rule residue chiefly explains the movements of the yield curve at short end. At long horizon

for long maturity bonds, it is real neutral rate that mainly explains the movements of the


Figure 1.3: Impulse Responses at Future Horizon

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Months

Annuliz

ed y

ield

in %

Initial response of 1Y Bond

RealRate

GDPGap

Inflation

TYRes

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes

yield curves.

Our results regarding impulse response analysis and variance decomposition confirm those

gained by many previous studies. The model of Bekaert et al. [2010] also shows strong

contemporaneous responses of the yield curve to macro shocks. Hordahl et al. [2006] finds

that monetary policy shocks impact chiefly at the short end of the yield curve. Dewachter

and Lyrio [2006] finds that at the short end of the yield curve, both real rate and inflation

expectation count.

1.5.5 Robustness checks

In order to test the robustness of our model, we decompose the sample period into two parts.

We then calibrate the model by using in-sample data and test it by using out-sample data.

The in-sample data period is from September 1997 to June 2008, the out-sample data period


Table 1.4: Variance Decomposition

1Y Bond


1Month 0.02 0.03 0.47 0.48

1Y 0.01 0.23 0.60 0.16

5Y 0.30 0.28 0.35 0.07

3Y Bond


1Month 0.04 0.01 0.55 0.41

1Y 0.10 0.11 0.65 0.14

5Y 0.56 0.13 0.27 0.04

5Y Bond


1Month 0.02 0.00 0.93 0.04

1Y 0.16 0.02 0.79 0.02

5Y 0.65 0.03 0.31 0.01

from July 2008 to August 2012. We report in Table 1.5 the price of risk λ1 that we have

obtained by calibrating the in-sample data.

In Figure 1.4 we plot the observed yields and model implied yields. The in-sample period

is from August 1997 to June 2008, the out-sample period from July 2008 to August 2012. In

both cases we use the first ten months’ data in each sample to match the constant term an.

We then measure the accuracy of our model’s pricing by computing the root-mean-square

error for the sample excluding the first ten months.

The figure shows that our model tracks the time variation of bond yields in the out-

sample period. The out-sample performance of 3Y and 5Y bonds are especially good before

September 2011, the onset of Operation Twist. And the model works better for the 3Y

and 5Y bonds than the 3M and 1Y bond. This is confirmed by the root-mean-square error

statistics. In Table 1.6 and Table 1.7 we report the pricing statistics for the in-sample and

the out-sample periods.

We see by looking at Table 1.7 that the pricing error of 5Y bonds increases much less

than for 3M and 1Y bond. In the subsection 1.8.4, when we study an over-identified model,


Table 1.5: Market Price of Risk for In-sample Period

r g π u

r 0.07 -0.20 -3.01 0.11

(0.08) (0.43) (2.15) (0.47)

g -1.10 0.34 1.40 -0.10

(2.07) (1.23) (1.06) (0.60)

π 0.17 0.24 -0.22 0.04

(3.14) (8.20) (1.01) (2.75)

u -0.74 0.02 -0.31 -0.35

(2.20) (0.14) (1.10) (3.88)

The table presents the estimates for the market price of risk of essentially affine model.

The absolute value of the t-ratio of each estimate is reported. The sample period is from

September 1997 to June 2008. The standard deviation is corrected following Newey and

McFadden(1994).

Table 1.6: In-sample Pricing Error

Observed

Mean

RMSE Rlt Error(%)

1M 3.18 0.00 0.00

3M 3.39 0.21 6.14

1Y 3.62 0.20 5.49

3Y 4.05 0.19 4.75

5Y 4.39 0.16 3.54

The table reports the in-sample pricing error. The Observed Mean is the mean of

observed bond yield. RMSE is the root mean square error. Rlt Error is the RMSE

divided by the Observed Mean. The sample period is from September 1997 to June

2008.

we will discuss the reason of the model’s better performance at medium and long ends of the

curve.


Figure 1.4: In-sample and Out-sample Pricing

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014−1

0

1

2

3

4

5

6

7Observed and model implied Yields for 3 Months Bond

← Out−sample starts

Model

Observed

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20140

1

2

3

4

5

6



Model

Observed

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20140

1

2

3

4

5

6



Model

Observed

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20140

1

2

3

4

5

6



Model

Observed

Operation Twist

Table 1.7: Out-sample Pricing Error

Observed

Mean

RMSE Rlt Error(%)

1M 0.07 0.00 0.00

3M 0.10 0.23 228.95

1Y 0.27 0.79 296.09

3Y 0.91 0.81 89.11

5Y 1.67 0.69 41.37

The table reports the out-sample pricing error. The Observed Mean is the mean of


divided by the Observed Mean. The sample period is from July 2008 to August 2012.


1.6 Extension: Forecasting of Bond Excess Return

1.6.1 Forecasting using forward yields

Following Cochrane and Piazzesi [2002], we forecast the excess return of bonds using forward

rates. We find - as did they - a single factor that predicts the excess return of bonds having

maturity between two and five years. We can interpret this factor by using our macro factors.

We note fnt , the forward rate at time t for a loan extended between period t + n − 1 and

t+ n. We also note rxnt+1, the excess return of bond with maturity n years between period t

and period t+ 1.

1.6.1.1 Unrestricted regression

To obtain the unrestricted specification we regress the excess returns on forward rates for

bonds of maturity between two and five years(2 ≤ n ≤ 5):

rxnt+1 = βn0 +5∑i=1

βni fit

Table 1.8 provides the results of that regression:

Table 1.8: Forecasting of Bond Excess Return

β0 β1 β2 β3 β4 β5 R2

2YBond Est -2.82 0.69 -1.01 -1.18 2.46 -0.17 0.20

2YBond Std 2.07 0.64 0.61 1.30 1.57 0.87

3YBond Est -5.36 1.32 -2.37 -2.04 4.96 -0.43 0.18

3YBond Std 3.53 1.26 1.46 2.42 3.40 1.76

4YBond Est -8.00 1.89 -2.80 -4.22 6.93 0.23 0.20

4YBond Std 4.51 1.77 2.25 3.16 4.86 2.61

5YBond Est -10.85 2.37 -2.69 -6.36 7.01 2.28 0.24

5YBond Std 5.14 2.21 2.99 3.80 6.03 3.43

The standard deviation in the table was obtained by doing the Hansen-

Hodrick correction.

It shows that the joint test for all coefficients equal to 0 has p-value below 0.05 for all

bonds except the 4Y bond.


1.6.1.2 Restricted regression

To do a restricted regression we first regress the average excess returns on forward rates:

rxt+1 = γ0 +5∑i=1

γifit

where rxt+1 = 14

∑4i=1 rx

it+1. In Table 1.9 we report the results.

Table 1.9: Bond Excess Return Forecasting Factor

γ0 γ1 γ2 γ3 γ4 γ5 R2

Avg Est -6.75 1.57 -2.22 -3.45 5.34 0.48 0.21

Std 3.79 1.46 1.83 2.62 3.94 2.12

The standard deviation in the table was obtained by doing the

Hansen-Hodrick correction.

In the second stage we regress the individual bond return on the single factor: γ0 +∑5

i=1 γifit .

rxnt+1 = bn(γ0 +5∑i=1

γifit )

In Table 1.10 we report the estimation of bn. We see that b has t-statistic with p-value below

0.001 for all bonds.

Table 1.10: Forecasting of Bond Excess Return

b std R2

2Y Bond 0.43 0.12 0.16

3Y Bond 0.80 0.22 0.17

4Y Bond 1.20 0.30 0.20

5Y Bond 1.57 0.34 0.23

The standard deviation in the table was obtained

by doing the Hansen-Hodrick correction.

In Table 1.11 we represent the factor γ0 +∑5

i=1 γifit by using macro factors. This reveals the

loadings on the macro factors.


Table 1.11: Forecasting Factor Representation using Macro Factors

NEUT GDPGAP EI TAYRES

0.39 -0.63 4.33 -0.09

1.7 Extension: Adding Bonds of Longer Maturities

Here we extend our analysis to bonds of a maturity of up to ten years. More specifically we

look at 13 bonds having maturities of 1 month, 3 months, 6 months and 1 year through 10

years. Our estimation method is similar to that employed in Section 1.3. We report λ1 in

Table 1.12.

Table 1.12: Market Price of Risk

r g π u

r -0.09 -0.30 -0.22 0.10

(0.62) (0.67) (0.71) (0.66)

g 0.04 0.66 0.10 0.34

(0.23) (2.25) (0.42) (2.63)

π 0.29 0.31 -0.42 0.07

(3.98) (1.01) (2.73) (1.54)

u -0.74 0.08 0.49 -0.34

(6.64) (0.34) (2.54) (3.89)

The table presents the estimates for the market price of essentially affine model. The

absolute value of the t-ratio of each estimate is reported. The sample period is from

September 1997 to August 2012. The standard deviation is corrected following Newey

and McFadden(1994).

For the λ1 in Table 1.12, we have∑

1≤n≤13 GT (λ∗1, n)G(λ∗1, n) = 0.013. The J-statistic is

2.282, with a p-value less than 0.01%. Thus we are unable to reject the 32 over identifying

restrictions implied by the theory. We next compare the factor loadings of the over-identified

model with that of the unrestricted regression. In Table 1.13 we report the result. We see

that the factor loadings of the over-identified model are close to that of the unrestricted

2We use the results of the first stage estimation when estimating λ1. Thus the J-statistics is an approxi-

mation.


Figure 1.5: PCA Factor Loadings

1 2 3 4 5 6 7 8 9 10−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8PCA Factor Loadings

Factor 1

Factor 2

Factor 3

regression. In Fig. A.2 in the appendix we plot the observed yields and the model implied

yields for bonds having maturities up to 10 years.

1.7.1 Principal Component Analysis

We first conduct principal component analysis of the yield curves. We then represent the

factors extracted from that analysis by using macro factors. We plot in Figure 1.5 the loadings

on bonds of the first three factors obtained from principal component analysis.

We see that the first factor has almost equal loadings on all bonds. This is the level factor

in the empirical asset pricing literature. The second factor has smaller loadings on bonds

having higher maturities. This is the slope factor. The third factor has first decreasing then

increasing loadings and is thus the curvature factor.

Based on the pricing equation ynt = an + bTnXt we can represent the three factors as a

function of macro factors. In Table 1.14 we report the result.

1.7.2 Impulse response function

Here we study, respectively, the contemporaneous response of yields to shocks and responses

at longer horizons. The shock from each factor is assumed to be of one standard deviation

of the factor’s first difference.


1.7.2.1 Response of contemporaneous yields

The pricing equation ynt = an+ bTnXt leads us that the effect of each factor on the yield curve

is given by bn, which is in turn determined by the affine asset pricing structure. bn thus

represents the initial response to shocks made by different factors. In Figure 1.6 we plot the

initial responses of bonds having different maturities. The shock from each factor is assumed

to be of one standard deviation of the factor’s first difference.

It shows that for all factors other than expected inflation, the response of the bond having

Figure 1.6: Contemporaneous Impulse Responses

0 20 40 60 80 100 120−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6


Annuliz

ed y

ield

in %

Initial response

RealRate

GDPGap

Inflation

TYRes

longer maturity is smaller than that having shorter maturity. The expected inflation has a

hump shape response function. Among all four factors the shocks to expected inflation and

real rate have the most significant response for a longer maturity bond.

1.7.2.2 Response at longer horizon

Our term structure model gives us the yield curve responses to macro factor shocks at long

horizons. We plot in Figure 1.7 impulse responses of bond yields of maturity 1Y, 5Y and

10Y. We find that for all three bonds, shock from real neutral rate is most significant at the

longer horizon. At the one-month horizon, shock from Taylor residue is most significant for

the 1Y bond. For the 5Y and 10Y bonds the shock from expected inflation is most significant

at a one-month ahead horizon.


Figure 1.7: Impulse Responses at Future Horizon

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes

0 10 20 30 40 50 60−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes

In Figure A.3 in Appendix A we report the error bands of impulse responses.

1.7.3 Variance decomposition

Based on our VAR dynamics, we can decompose the forecasting variance of yields into con-

tributions from four macro shocks. In Table 1.15 we report the results of respectively 1Y,

5Y, and 10Y bonds, respectively.

For the 1Y bond at one-month ahead horizon, Taylor residue contributes most of the

variance. At longer horizons, all factors contribute to the variance. For the 5Y and the 10Y

bonds, we see results having similar properties. For one-month ahead forecasting, expected

inflation contributes most of the variance. At longer horizons, the real neutral rate contributes

most of the forecasting variance.


Table 1.13: Model Implied Factor Loadings and Unrestricted Regression Factor Loadings


1M Unres Reg 1.00 1.00 1.00 1.00

1M Over-Iden Model 1.00 1.00 1.00 1.00

3M Unres Reg 0.94 1.04 1.19 0.99

3M Over-Iden Model 0.96 1.03 1.08 0.98

6M Unres Reg 0.84 1.06 1.31 0.94

6M Over-Iden Model 0.90 1.03 1.20 0.94

1Y Unres Reg 0.74 0.94 1.42 0.82

1Y Over-Iden Model 0.79 0.96 1.38 0.84

2Y Unres Reg 0.54 0.67 1.70 0.59

2Y Over-Iden Model 0.61 0.69 1.62 0.62

3Y Unres Reg 0.49 0.44 1.68 0.46

3Y Over-Iden Model 0.47 0.42 1.71 0.45

4Y Unres Reg 0.44 0.25 1.64 0.34

4Y Over-Iden Model 0.38 0.21 1.72 0.31

5Y Unres Reg 0.38 0.07 1.59 0.23

5Y Over-Iden Model 0.31 0.06 1.67 0.22

6Y Unres Reg 0.32 -0.02 1.54 0.18

6Y Over-Iden Model 0.26 -0.03 1.61 0.16

7Y Unres Reg 0.26 -0.11 1.49 0.13

7Y Over-Iden Model 0.22 -0.09 1.53 0.11

8Y Unres Reg 0.24 -0.16 1.41 0.09

8Y Over-Iden Model 0.20 -0.13 1.44 0.07

9Y Unres Reg 0.23 -0.21 1.33 0.05

9Y Over-Iden Model 0.19 -0.16 1.36 0.04

10Y Unres Reg 0.22 -0.25 1.26 0.01

10Y Over-Iden Model 0.19 -0.18 1.27 0.01

The table presents the factor loadings of the over-identified model with that of

the unrestricted regression. For example, 1M Unres Reg is the factor loading

from unrestricted regressions. 1M Over-Iden Model is the factor loading of the

over-identified model. The sample period is from September 1997 to August

2012.


Table 1.14: PCA Factors Representation using Macro Factors

Factor 1 Factor 2 Factor 3

RealRate 1.14 0.60 0.17

GDPGap 0.54 1.16 0.25

Inflation 4.84 0.24 -0.35

TYRes 0.89 0.81 0.17

Table 1.15: Variance Decomposition

1Y Bond


1Month 0.00 0.01 0.31 0.67

1Y 0.01 0.23 0.56 0.21

5Y 0.32 0.28 0.32 0.08

5Y Bond


1Month 0.02 0.01 0.93 0.04

1Y 0.16 0.02 0.80 0.02

5Y 0.65 0.03 0.31 0.01

10Y Bond


1Month 0.06 0.01 0.92 0.00

1Y 0.31 0.01 0.68 0.00

5Y 0.75 0.00 0.24 0.00

In Table A.1 in Appendix A we report the error band of variance

decomposition.


1.7.4 Robustness checks

In order to test the robustness of our model, we decompose the sample period into two parts.

The in-sample period is from August 1997 to June 2008 and the out-sample period from July

2008 to August 2012. We then calibrate the model by using in-sample data and test it by

using out-sample data. We report in Table 1.16 the price of risk λ1 that we have obtained

by calibrating the in-sample data.

Table 1.16: Market Price of Risk for In-sample Period

r g π u

r -1.25 0.56 1.56 0.00

(2.84) (0.95) (1.51) (0.01)

g -1.28 1.02 2.81 -0.07

(1.90) (1.95) (3.07) (0.50)

π 0.14 0.23 -0.40 0.03

(0.92) (1.18) (1.27) (4.08)

u -0.93 0.11 0.59 -0.35

(3.96) (0.38) (1.26) (3.97)

The table presents the estimates for the market price of risk of essentially affine model.

The absolute value of the t-ratio of each estimate is reported. The sample period is from

September 1997 to June 2008. The standard deviation is corrected following Newey and

McFadden(1994).

In Figure 1.8 we plot observed yields and model implied yields for 1Y, 5Y, 7Y and 10Y

bonds. In all cases, we use the first ten months data in each sample to match the constant

term an. We then measure the accuracy of our model’s pricing by computing the root-mean-

square error for the sample, excluding the first ten months.


Figure 1.8: In-sample and Out-sample Pricing

1998 2000 2002 2004 2006 2008 2010 20120

1

2

3

4

5

6

7

Observed and model implied Yields for 24 Months Bond


Model

Observed

1998 2000 2002 2004 2006 2008 2010 20120

1

2

3

4

5

6

7



Model

Observed

Operation Twist

1998 2000 2002 2004 2006 2008 2010 20120

1

2

3

4

5

6

7



Model

Observed

Operation Twist

1998 2000 2002 2004 2006 2008 2010 20120

1

2

3

4

5

6

7



Model

Observed

Operation Twist

The figure shows that our model tracks the time variation of bond yields in the out-sample

period. The out-sample performances of medium and long term bonds are especially good

before September 2011, the onset of Operation Twist. And the model works better for the

medium and long end bonds than the short end bonds. This is confirmed by the root-mean-

square error statistics. In Table 1.17 and Table 1.18 we report the pricing statistics for the

in-sample and the out-sample periods. The Observed Mean is the mean of observed bond

yield. RMSE is the root mean square error. Rlt Error is the RMSE divided by the Observed

Mean.

We see by looking at Table 1.18 that all bonds are priced accurately for out-sample period

except those having maturities less than 2 years. In the subsection 1.8.4 we discuss the reason

of the big pricing error at out-sample period for short end bonds.


Table 1.17: In-sample Pricing Error

Observed

Mean

RMSE Rlt Error(%)

3M 3.39 0.20 5.93

6M 3.52 0.25 7.02

1Y 3.62 0.22 6.10

2Y 3.88 0.21 5.44

3Y 4.05 0.18 4.48

4Y 4.22 0.15 3.66

5Y 4.39 0.15 3.45

6Y 4.48 0.15 3.43

7Y 4.58 0.16 3.48

8Y 4.67 0.17 3.54

9Y 4.77 0.18 3.72

10Y 4.86 0.21 4.38

The table reports the in-sample pricing error. The Observed Mean is the mean of


divided by the Observed Mean. The sample period is from September 1997 to June

2008.

1.8 Robustness check and discussions

1.8.1 Testing model on alternative measurements: I

As a robustness test, we test our model on two alternative sets of measurements for the Taylor

rule factors. We will compare the results to that of the main text. We denote the measurement

in the main text as Williams-Laubach’s data. In the first alternative specification, we estimate

the output gap by using Okun’s law, as in Eq. 1.22, where we have gt = 2.5 ∗ (4.75 − uet).

And we keep the Williams-Laubach’s estimation of real rate and Haudrich et al’s estimation

for the expected inflation. We do all the exercises as in the section 1.7. We report the results

in Appendix A.

We plot in Fig. A.4 the contributions of macro factors in Taylor rule along with the Fed fund


Table 1.18: Out-sample Pricing Error

Observed

Mean

RMSE Rlt Error(%)

3M 0.10 0.25 243.97

6M 0.17 0.60 361.75

1Y 0.27 0.86 320.58

2Y 0.58 0.87 149.80

3Y 0.91 0.86 94.62

4Y 1.29 0.77 60.21

5Y 1.67 0.70 42.17

6Y 2.00 0.61 30.70

7Y 2.34 0.54 23.21

8Y 2.55 0.51 19.92

9Y 2.76 0.48 17.38

10Y 2.96 0.46 15.57

The table reports the out-sample pricing error. The Observed Mean is the mean of


divided by the Observed Mean. The sample period is from July 2008 to August 2012.

rate. In Table A.2 we report the λ1. For the moment condition we have:∑1≤n≤13

GT (λ∗1, n)G(λ∗1, n) = 0.10

The J-statistic is 18.8, with a p-value smaller than 0.03. For this alternative set of measure-

ments we are thus unable to reject the 32 over identifying restrictions implied by the theory.

In Table A.4, we compare the factor loadings of the over-identified model with that of the

unrestricted regression. We see that the factor loadings of the over-identified model are close

to that of the unrestricted regression. In the following of this subsection we analyze the

results of impulse-response analysis, variance decomposition and out-sample test.

1.8.1.1 Impulse-response analysis

Fig. A.5 plots the contemporaneous response of yields to shocks. As in the case of Williams-

Laubach’s data, among all four factors the shocks to expected inflation and real rate have


the most significant response for longer maturity bond. And for all factors other than the

expected inflation, the response of bond having longer maturity is smaller than that having

shorter maturity. The expected inflation has a hump shape response function, which is sim-

ilar to Williams-Laubach’s data.

We plot in Fig. A.6 the responses at longer horizon. We find that for all three bonds, shock

from real neutral rate is most significant at the longer horizon. At the one-month horizon,

shock from Taylor residue is most significant for the 1Y bond. For the 5Y and 10Y bonds

the shock from expected inflation is the most significant at one month ahead horizon. These

properties are similar to that of Williams-Laubach’s data.

1.8.1.2 Variance decomposition

In Table A.5 we report the results of variance decomposition for 1Y, 5Y, and 10Y bonds.

The properties are similar to that of Williams-Laubach’s data.

For the 1Y bond at one-month ahead horizon, Taylor residue contributes most of the variance.

At longer horizons, all factors contribute to the variance. For the 5Y and the 10Y bonds

we see results having similar properties. For one-month ahead forecasting, expected inflation

contributes most of the variance. At longer horizons, the real neutral rate contributes most

of the forecasting variance.

1.8.1.3 Out-sample analysis

We report in Table A.3 the price of risk λ1 that we have obtained by calibrating the in-sample

data. In Table A.6 and Table A.7 we report the pricing statistics for the in-sample and the

out-sample periods. We see by looking at Table A.7 that all bonds are priced accurately for

out-sample period except those having maturities less than 2 years.

1.8.2 Testing model on alternative measurements: II

We now test our model on the second alternative set of measurements. In this specification,

we take the real neutral rate to be the five-year forward rate five year in the future minus the

inflation risk premium as in Eq. 1.20, and the expected inflation to be the 5-year break-even


inflation rate minus the inflation risk premium. And we model the output gap by using the

Okun’s law, as in Eq. 1.22, where we take k equal to 2.75. We do all the exercises as in the

section 1.7. And we report in Appendix A the results.

We plot in Fig. A.7 the contributions of macro factors in Taylor rule along with the Fed fund

rate. In Table A.8 we report the λ1. For the moment condition we have:

∑1≤n≤13

GT (λ∗1, n)G(λ∗1, n) = 0.02

The J-statistic is 2.9, with a p-value smaller than 0.01. We thus see that for this alternative

set of measurements we are again unable to reject the 32 over identifying restrictions implied

by the theory.

In Table A.10, we compare the factor loadings of the over-identified model with that of the





Fig. A.8 plots the contemporaneous response of yields to shocks. As in the case of Williams-

Laubach’s data, among all four factors the shocks to expected inflation and real rate have

the most significant response for longer maturity bond. And for all factors other than real

neutral rate, the response of bond having longer maturity is smaller than that having shorter

maturity.

We plot in Fig. A.9 the responses at longer horizon. We find that for 5Y and 10Y bonds,

shock from real neutral rate is most significant at the longer horizon. For 1Y bond it’s GDP

gap. At the one-month horizon, shock from Taylor residue is most significant for the 1Y

bond. For 5Y bond shock from expected inflation is the most significant at one-month hori-

zon. Most of the properties are similar to that of Williams-Laubach’s data.




The properties are similar to that of Williams-Laubach’s data.



we see results having similar properties. For one-month ahead forecasting, real neutral rate

and expected inflation together contribute most of the variance. At longer horizons, the real

neutral rate contributes most of the forecasting variance.


We report in Table A.9 the price of risk λ1 that we have obtained by calibrating the in-sample

data. In Table A.12 and Table A.13 we report the pricing statistics for the in-sample and the

out-sample periods. We see by looking at Table A.13 that all bonds are priced accurately for

out-sample period except those having maturities less than 2 years.

1.8.3 Model using efficient interest rate

Curdia et al. [2014] studies an alternative specification of monetary policy reaction function,

which the authors refer to as W rules, where the output gap doesn’t affect the reaction

function. The paper argues that W rules fit the US data better than Taylor rule. Here we

test our model on W rules, where we have three factors.

it = rnrt + 1.5(πt − 2%) + wt (1.23)

where rnrt and πt are the same as in Taylor rule, and wt is the W rule deviation. The

model setup and the econometric method are the same as in the case of the Taylor rule. In

Appendix A we report the results.

We plot in Fig. A.10 the contributions of macro factors in W rule along with the Fed fund

rate. In Table A.14 we report the λ1. For the moment condition we have:

∑1≤n≤13

GT (λ∗1, n)G(λ∗1, n) = 0.01


The J-statistic is 2.31, with a p-value smaller than 0.01. We thus see that we are unable to

reject the 27 over identifying restrictions implied by the theory.

In Table A.16,we compare the factor loadings of the over-identified model with that of the





Fig. A.11 plots the contemporaneous response of yields to shocks. Interestingly, as in the

case of Williams-Laubach’s data, among all three factors the shock to expected inflation has

the most significant response for longer maturity bond. And for real neutral rate and W rule

deviation, the response of bond having longer maturity is smaller than that having shorter

maturity. The expected inflation has a hump shape response function, which is the same as

for the four-factor model.

We plot in Fig. A.12 the responses at longer horizon. We find that for all three bonds, shock

from real neutral rate is most significant at the longer horizon. At the one-month horizon,

shock from W rule residue is most significant for the 1Y bond. For the 5Y and 10Y bonds

the shock from expected inflation is the most significant at short horizon. These properties

are similar to that of the four-factor model.



The properties are similar to that of the four-factor model.



we see results having similar properties. For one-month ahead forecasting, expected inflation

contributes most of the variance. At longer horizons, the real neutral rate contributes most

of the forecasting variance.



We report in Table A.15 the price of risk λ1 that we have obtained by calibrating the in-

sample data. In Table A.18 and Table A.19 we report the pricing statistics for the in-sample

and the out-sample periods. We see by looking at Table A.19 that all bonds are priced

accurately for out-sample period except those having maturities less than 2 years.

1.8.3.4 Comparison with the Taylor rule factor model

From the previous analysis we conclude that the W rule factor exhibits similar properties

to the Taylor rule factor. For both models the over identifying restrictions implied by the

theory are not rejected with low p-value. They also feature similar out-sample test results.

We believe it is interesting to further investigate the empirical performance of W rule model

and compare the latter with the Taylor rule model.

1.8.4 Taylor rule under zero lower bound regime

In the subsections 1.5.5 and 1.7.4 we studied, respectively, the out-sample test for the exactly-

identified model and the over-identified model. There we saw that the out-sample perfor-

mance for bonds at short end is much worse than that of bonds at medium and long ends.

Note that our out-sample period extends from June 2008 to June 2012. And for most of the

out-sample period, from December 2008 to August 2012, Fed fixed the short term interest

rate at 0.25%. Thus we are looking at the problem of zero lower bound here. This poses a

problem to our Taylor rule factor model because under the zero lower bound regime the long-

term relation binding bond yields to macro factors may break down. A legitimate question

to raise is: does the underperformance of the out-sample test for both the exactly identified

and over-identified models come from the zero lower bound? We study this question here.

Our strategy is to calibrate the model using the whole sample data from September 1997 to

August 2012. We then look at the fitting performance of the model for two subsamples, with

the first one being from September 1997 to June 2008 and the second one being from July

2008 to August 2012. We already solved the model in the section 1.7. We report in Table 1.19

the fitting results for the second sample. The first column shows the mean of the bond yield

during the period. RMSE 1 is the root mean square error for the baseline model calibrated


using the whole sample data from 1997 to 2012. Rlt Error 1 is the RMSE 1 divided by the

Observed Mean. In Table 1.19 we also recall the results of the out-sample analysis of the

subsection 1.7.4. In Table 1.19, RMSE 2 shows the RMSE of the out-sample test, the same

RMSE as Table 1.18. Rlt Error 2 is the RMSE divided by the Observed Mean. In Table 1.19

the Out-Sample Decay is the ratio between Rlt Error 2 and Rlt Error 1.

By our definition we have the following relation:

RltErrorn2 = RltErrorn1 ×OutSampleDecayn (1.24)

Eq. 1.24 says that the out-sample test’s relative error can be factored into two components:

the relative error of the whole sample model on the same period and the out-sample decay.

Looking at Table 1.19 we see that the out-sample decay is generally decreasing as a function

of the bond’s maturity. We can thus see that the model’s robustness at long end is also better

than that at short end.

Table 1.19: Comparison of Whole-sample Fitting and Out-sample Fitting

Obs Mean RMSE 1 RMSE 2 Rlt Err 1(%) Rlt Err 2(%) OS Decay

3M 0.10 0.14 0.25 133.72 243.97 1.82

6M 0.17 0.14 0.60 84.76 361.75 4.27

1Y 0.27 0.14 0.86 53.19 320.58 6.03

2Y 0.58 0.24 0.87 41.62 149.80 3.60

3Y 0.91 0.26 0.86 28.16 94.62 3.36

4Y 1.29 0.24 0.77 18.32 60.21 3.29

5Y 1.67 0.23 0.70 13.98 42.17 3.02

6Y 2.00 0.24 0.61 12.08 30.70 2.54

7Y 2.34 0.27 0.54 11.62 23.21 2.00

8Y 2.55 0.28 0.51 11.04 19.92 1.80

9Y 2.76 0.30 0.48 10.78 17.38 1.61

10Y 2.96 0.31 0.46 10.63 15.57 1.46

Obs Mean is the mean of the observed yields. OS Decay is the out-sample decay.

We note that the whole sample model shows good fitting for the zero-lower bound period

on the medium and long ends of the curve. The good performance for the long-term bond

maybe reflects the market’s expectation that the Taylor rule will still characterize the short-


term interest rate policy for the long term.

1.9 Conclusion

In this chapter we have interpreted the term structure of US interest rates by using observable

macro factors as inputs to a Taylor-type rule. Consistent with the theory, we allow the

observed deviation from the Taylor-type rule to be a priced factor. We have explained term

structure time variations in terms of observable and interpretable macro factors. We have thus

constructed an arbitrage-free term structure model, one in which the arbitrage opportunities

are eliminated by imposing restrictions on the yields.

Our estimation results of US data suggest that macro variables affect the term structure in

various ways. Yield curve exhibits a strong contemporaneous response to macro shocks, with

expected inflation and real neutral rate having the most persistent effect. The real neutral

rate explains most of the price variation at long horizon for all bonds. At short horizon,

expected inflation explains most of the price variation for bonds at belly and long end. At

short horizon, the Taylor rule deviation, explains most of the movements of the yield curve

at short end. Our results also suggest our estimated model is reasonably robust.

CHAPTER 2. CARRY TRADE THRESHOLD PORTFOLIOS 42

Chapter 2

Carry Trade Threshold Portfolios

2.1 Introduction

Carry trade strategies profit from violations of uncovered interest rate parity. They involve

borrowing low interest rate currency and investing it in high interest rate currency. Carry

trade is related to uncovered interest rate parity (UIP), which states that high interest rate

currency tends to depreciate relative to low interest rate currency such that the expectation

of the carry trade return is zero. A large body of literature rejects UIP, however; see, for ex-

ample, Fama [1984]. Starting with Meese and Rogoff [1983], one view has been that exchange

rates are roughly random walks. Some researchers believe that the random-walk model does

a good job of predicting exchange rates. The predictions of the out-of-sample mean squared

error of the random-walk model are about the same as the models that use fundamental data.

A recent update is Cheung et al. [2005]. More recently Engel et al. [2012] finds that the factor

model improves the forecast for the long horizon (8 and 12 quarters). Overall, the literature

suggests that the random-walk model works well for a short horizon (less than 8 quarters).

A more nuanced view of UIP is related to the forward premium anomaly, which states that

a currency is expected to appreciate when its interest rate is high, see Fama [1984], Hodrick

[1987], and Engel [1996]. Both the random-walk model literature and the forward premium

puzzle literature suggest that currency carry trade strategies are likely to be profitable.

Many papers have studied the carry trade. Empirical research has shown that excess returns

of carry trade portfolios are statistically significant for various sample periods and sets of


currencies. Different theories have been proposed to explain the carry trade excess returns.

According to Brunnermeier et al. [2008], one possible explanation for the excess return of

carry trade is compensation for crash risk. There has been an ongoing debate regarding

whether the classical asset pricing model can explain carry trade return, see Lustig and

Verdelhan [2007], Lustig et al. [2011] and Burnside et al. [2011a]. Daniel, Hodrick, and Lu

[2014] gives a recent summary on carry trade’s risks. This paper contributes to the literature

by studying the carry trade threshold portfolios. It constructs threshold portfolios by using

thresholds that depend upon a forward premium (the interest rate difference between the two

currencies). For example, a threshold portfolio of threshold value equal to 2% includes only

carry trade having a forward premium greater than 2%. We show that from 1990 to 2012

for G10 currencies, higher threshold portfolios outperformed lower ones up to the threshold

value of 2.5%. We explore the connections of the threshold effect to the currency investor’s

portfolio problem. We also study carry trade’s return predictability as a motivation for the

threshold portfolio. To explain the threshold effect, we model the latter in a random walk

currency model. The model predicts the optimal threshold value and the relative gain of the

optimal threshold portfolio. The model is calibrated and the predictions are tested. We also

show the non-existence of the threshold effect in a one-factor currency model.

Related to the recent research on the carry trade, we test the crash risk explanation (Brun-

nermeier et al. [2008]) for the carry trade threshold effect. In order to test the effect of crash

risk, we construct threshold portfolios that are hedged against that risk. Following Jurek

[2014], we construct crash-hedged threshold portfolios using out-of-money put options. We

then use the difference-in-difference method to test the crash risk effect. We first compute

the difference in performance of hedged threshold portfolios versus hedged benchmark equally

weighted portfolios, and the difference in performance of unhedged threshold portfolios ver-

sus unhedged benchmark equally weighted portfolios. We then compare the two differences

in performance. The result suggests that the crash risk premium can explain around 25

percent of the excess performance of the higher threshold portfolios, relative to benchmark

equally weighted portfolios. The remainder of the introduction section explains in detail our

approaches and main results.

Regarding the predictability of the carry trade return, Bakshi and Panayotov [2013] inves-


tigates time series predictability. Lustig et al. [2011], Ang and Chen [2010] and Menkhoff

et al. [2012] identify risk factors that affect the average carry trade return. We motivate

the study of threshold portfolio by looking at the predictability of the carry trade using a

forward premium. A simple regression shows that the loading of the carry trade return on

the forward premium is positive with significant t-statistics. Therefore, a currency investor

allocating more capital on carry trades having a larger forward premium can expect to have

a larger return.

In general, our regression shows a positive loading of carry trade returns on the forward

premiums. The next step is to evaluate the implications of this finding on the perspective of

the currency investor maximizing risk adjusted return. Our setup of the problem has been

conducted in the same spirit as that utilized by Burnside et al. [2008] and Berge et al. [2010].

Burnside et al. [2008] documents the gain achieved by diversifying the carry trade across

different currencies. Berge et al. [2010] discovered that forecasting models that use interest

rates help to improve the returns of currency portfolios. Consistent with the literature of

mean-variance analysis, see Levy and Markowitz [1979], Kroll et al. [1984] and Markowitz

[1991], we approximate the problem of maximizing the investor’s expected utility by solving

a mean-variance problem. When it comes to the literature related to currency portfolios,

Hochradl and Wagner [2010] studies optimal carry-trade portfolios using the Sharpe ratio as

its chief criterion. We study the optimal portfolio that maximizes the expected Sharpe ratio.

Our contribution is to prove that under certain second moment conditions, the investor’s

optimal portfolio always allocates more funds to trades having bigger forward premiums.

We empirically test the model’s implication that the carry trade with larger forward pre-

miums should have a larger weight in the portfolio. We form threshold portfolios, with the

threshold based on the forward premium, and compare their performance. Given the thresh-

old value, our threshold portfolio includes only those carry trades whose forward premiums

are higher than the threshold values. We find empirically that up to the threshold value of

2.5 percent, the threshold portfolio outperforms the benchmark portfolio with respect to the

Sharpe ratio. In addition, the portfolios with higher thresholds beat the portfolios with lower

thresholds. That result is confirmed through the robustness checks of resampling bootstrap

and White’s Reality Check. Recent empirical regime switch studies of the forward premium,


done by Jorda and Taylor [2012], Baillie and Chang [2011], and Christiansen et al. [2011],

also suggest that our threshold portfolios should have different performances.

Garratt and Lee [2010] evaluates the forecast performance of models based upon their eco-

nomic value in the decision making setting of an investor. Model A works better than Model

B, if a portfolio constructed with Model A has a better financial performance than the one

constructed with Model B. In that framework, a test of the threshold model proves that a

model that posits a positive loading of a carry trade return on forward premiums works better

than a model that posits neutral loading. Our threshold approach also allows us to formulate

another portfolio construction rule. Recent studies Lustig et al. [2011] and Menkhoff et al.

[2012] construct portfolios based on forward discounts. Their approaches are similar and yet

different. Menkhoff et al. [2012] constructs portfolios based on the relative order of forward

premium. In their five portfolio example, the first portfolio includes the quintile having the

highest forward premiums. We compare the performance of threshold models based on hard

threshold and relative order. We find that for the sample period 1990-2012, the hard thresh-

old rule led to better financial performance among the G10 currencies.

One recent explanation of carry trade excess return is the crash risk premium. Jurek [2014]

shows that carry trade returns don’t come primarily from the peso problem. This is done

by constructing crash-hedged carry trade portfolios. In the same spirit as Jurek, this paper

tests the crash risk explanation for the superior performance of the high forward premium

carry trade relative to that of the low forward premium. It constructs crash-hedged threshold

portfolios using out-of-money put options. The paper finds persistence of excess return of

hedged higher threshold portfolios, relative to hedged lower threshold portfolios as in the

unhedged case. It compares the relative difference in performance between higher threshold

portfolios and an equally weighted portfolio in hedged and unhedged cases. It shows that

the crash risk premium can explain approximately 25 percent of the excess performance of

higher threshold portfolios, relative to the benchmark equally weighted portfolio.

The paper also finds that the Sharpe ratio of threshold portfolio initially increases, as the

threshold value goes up until it reaches its optimal value. The Sharpe ratio then decreases.

The paper models this threshold effect in a random walk currency model, in keeping with

Meese and Rogoff [1983] and Engel and West [2004]. Using general assumptions on the


volatilities of the currencies and the distribution of the forward premiums, the model pre-

dicts the threshold value most likely to maximize the Sharpe ratio. The model also predicts

the relative gain in the Sharpe ratio of the threshold portfolio as compared to the equally

weighted portfolio. We calibrate the model to G10 currency trades, using data running from

January 1990 to June 2012. We then test the model’s predictions.

Finally the paper extends the random-walk model to a one-factor model where currency’s

innovation has loading on global factor, like in Lustig and Verdelhan [2007] and Jurek and

Xu [2013]. The one-factor model exhibits heterogeneous correlations between currency pairs

and increasing downside correlations of such pairs as in Ang and Bekaert [2002]. The pa-

per analytically solves the one-factor model, and compares the result to that gained via the

random-walk model. It finds that the optimal threshold value doesn’t exist if the currency

returns are correlated and if idiosyncratic currency risks are perfectly diversified across all

threshold portfolios. The nonexistence of optimal threshold value in the one-factor model

tells us that the quasi-independence of currencies movement is a factor that must be taken

into account if we wish to understand the threshold portfolio effect for currency carry trades.

The rest of the paper is organized as follows: The second section studies the prediction power

of the forward premium with respect to the carry trade returns. The third section solves the

currency investor’s mean-variance problem. The fourth section explains the threshold model

and subjects it to empirical tests. The fifth section presents the results of our examination

of the crash-hedged threshold portfolios. The sixth section solves the random-walk model,

with our solution of the one-factor model coming in the seventh section. The eighth section

concludes.

2.2 Carry Trade

This paper studies the portfolio problem of the carry trade investor. It shows a novel stylized

effect, related to the carry trade threshold portfolios. In this section, we introduce the basic

notions of carry trade.

Carry trade tries to profit from the empirical deviation of an exchange rate away from UIP

(uncovered interest rate parity). A carry trade comprises a short position in a low interest


rate currency, and an equivalent long position in a high interest rate currency. Thus, the log

return of a carry trade is

et+1 = it − i∗t − (st+1 − st) (2.1)

where it denotes the log interest rate of the high yield country; i∗t that of the low yield coun-

try; st is the log exchange rate.

Carry trade is related to uncovered interest rate parity (UIP), which states that high interest

rate currency tends to depreciate relative to low interest rate currency such that the expec-

tation of the carry trade return is zero. A large body of literature rejects UIP, however; see,

for example, Fama [1984]. Starting with Meese and Rogoff [1983], one view has been that

exchange rates are roughly random walks. Some individuals believe that the random-walk

model does a good job of predicting exchange rates. The predictions of the out-of-sample

mean squared error of the random-walk model are about the same as the models that use

fundamental data. A recent update is Cheung et al. [2005]. More recently Engel et al. [2012]

finds that the factor model improves the forecast for the long horizon (8 and 12 quarters).

Overall, the literature suggests that the random-walk model works well for a short horizon

(less than 8 quarters). A more nuanced view of UIP is related to the forward premium

anomaly, which states that a currency is expected to appreciate when its interest rate is high,

see Fama [1984], Hodrick [1987], and Engel [1996]. Both the random-walk model literature

and the forward premium puzzle literature suggest that currency carry trade strategies are

likely to be profitable.

We implement the carry trades of the G10 currencies1 versus the US dollar. The sample

period is from January 1990 to June 2012. The carry trades are rebalanced monthly. Table

B.1 details the performance statistics of the carry trades, and also shows the performance

statistics of the equally weighted carry trade portfolio2. In Fig. B.1, we plot the cumulative

return of the equally weighted portfolio. The equally weighted portfolio is our benchmark

portfolio when we study the threshold portfolios.

1These are the G10 currencies: US dollar, Euro, Japanese yen, Canadian dollar, Swiss franc, British pound,

Australian dollar, New Zealand dollar, Norwegian krone, and Swedish krona.

2The equally weighted carry trade portfolio includes nine carry trade of G10 currencies versus USD. It

gives equal allocation to the nine carry trades.


Suppose that a carry trade investor wants to improve the portfolio returns of his carry trades.

What variable should he look at when constructing the portfolio? An intuitive answer is the

forward premium (the interest rate difference), because both the random-walk model of ex-

change rates and the forward premium puzzle suggest that the carry trade return should be

positively related to the interest rate difference at trade initiation. We now test the prediction

power of the interest rate difference with respect to realized carry trade return. Retaining

the notation of Eq. 2.1, we run the following regression:

et+1 = β(it − i∗t ) + εt+1 (2.2)

Doing so brings us this estimation result:

β

Estimation 0.075

Std (0.045*)

N (2430)

* denotes p-value at 10% significance level.

The positive loading of the carry trade return on the yield difference tells us that a currency

investor allocating more capital to carry trades of higher forward premium can expect to see

a higher return. It is this link between forward premium and realized carry trade return

that motivates us to study the threshold portfolios in Section 2.4. Our result also provides

evidence that interest rate term premiums are related to foreign exchange risk premiums, in

keeping with Campbell and Clarida [1987], Clarida and Taylor [1997] and Clarida et al. [2003].

2.3 Currency investor’s portfolio problem

Our simple regression in the last section has revealed that forward premiums predict carry

trade returns. Thus, an investor whose objective is to maximize the expected return would

be wise to allocate more capital to carry trades having higher forward premiums. Does this

maxim also hold true for investors who are maximizing a risk adjusted return? We explore

this question within this section. We also explore the implications of the fact that the forward

premium predicts carry trade returns as they are seen from the perspective of the individual


currency investor. With regard to the portfolio optimization of currency investors, Burnside

et al. [2008] documents the gain that is achieved by diversifying the carry trade across different

currencies. Berge et al. [2010] discovers that forecasting models by using interest rates helps

to augment the returns of currency portfolios. Here we seek to maximize the expected utility

gained by the currency investor by restricting the investor’s portfolio to carry trades. The

investor, therefore, has to determine the allocation weights of carry trades in his portfolio.

2.3.1 Mean-variance problem

Following the literature on mean-variance analysis, see Levy and Markowitz [1979], Kroll

et al. [1984] and Markowitz [1991], we approximate the problem of maximizing the investor’s

expected utility by solving a mean-variance problem. Levy and Markowitz [1979] and Kroll

et al. [1984] showed that for various utility functions and empirical return distributions, the

expected utility maximizing portfolio can be well approximated by constructing an mean-

variance efficient portfolio3.

We first set up the standard mean-variance framework for the carry trade setting. Suppose

the investment universe of carry trades includes N carry trades with expected return RCTi

for carry trade i. The covariance of carry trade retun RCTi and RCTj is Σi,j. The investor’s

mean-variance problem is stated as following

Problem 1 Given the expected return of N carry trades RCT and the covariance of expected

returns Σ, the investor chooses the portfolio weights ω to solve the following problem:

max : ωTRCT

s.t.√ωTΣω = V

where V is the investor’s tolerance of return risk.

The next proposition shows that the solution to the mean-variance problem has a nice struc-

ture.

3In the original model of Levy and Markowitz [1979] short sale is not allowed. Here we relax this constraint

in our currency investment setting, because currency investment by its very nature involves taking a long

position on one currency and a short position on another.


Proposition 2 Regardless of the investors’ risk tolerances V , the investors’ positions on two

carry trades have the same ratio. The investors adjust their total exposure to match their risk

preference. The optimal portfolio maximizes the Sharpe ratio(risk adjusted return) ωTRCT√ωTΣω

.

Proof. Suppose ω∗ is the portfolio that solves Problem 1 for V = 1, then for general V the

solution of Problem 1 is V ω∗. Note that ω∗ also maximizes the Sharpe ratio ωTRCT√ωTΣω

.

Hochradl and Wagner [2010] also seeks to achieve the optimal carry trade portfolio using

the Sharpe ratio as its criterion. We now give the solution to the mean-variance optimal

portfolio.

Lemma 1 Assuming a universe of N carry trades, with expected return RCT = [RCTi ]T and

covariance matrix Σ of expected return, the portfolio that solves the problem 1 has weight

ω = λΣ−1RCT, where ω is such that√ωTΣω = V .

Proof. Our problem can be stated thus:

max :ωTRCT

√ωTΣω

, s.t.√ωTΣω = V (2.3)

Problem 2.3 is equivalent to this one:

max : ωTRCT, s.t.ωTΣω = V 2 (2.4)

The associated Lagrangian of Problem 2.4 is:

L = ωTRCT − λ(ωTΣω − V 2)

The first order conditions give us

ω = RCT ∗ (λΣ)−1

ωTΣω = V 2

Thus we have proved the theorem.

Lemma 1 gives the solution of the mean-variance optimal portfolio. Let us now take a step

back to think our original question: in the mean-variance optimal portfolio, do we place larger

weight on carry trade having larger forward premium? In the same spirit of the G10/USD

carry trades, let’s consider a setting where we implement carry trades of foreign currencies


versus domestic currency. Then the question can be stated as: in the mean-variance optimal

portfolio, do we have ωi > ωj for |rfi − rd| > |rfj − rd|? We study in the next subsection this

question.

2.3.2 Model

Lemma 1 gives the solution of the optimal mean-variance portfolio ω = λΣ−1RCT, do we

have ωi > ωj for |rfi − rd| > |rfj − rd|? In order to compare the weights of two carry trades,

we need to model the first and the second order moments of the expected returns. Before

going into more general discussion, we first state several simple cases. Let’s first suppose that

the carry trade of foreign currency i versus domestic currency has expected return |rfi − rd|,

this property is satisfied under the random walk model of foreign exchange rates. We also

assume that the currencies are not correlated between them and all currencies have equal

volatilities. Thus the covariance matrix of ω is the identity matrix multiplied by a constant,

σI. We thus have that the weight of carry trade k is ωk = λ|rfk−rd|

σ . It then follows ωi > ωj

if |rfi − rd| > |rfj − rd|.

Let’s consider another case where all the previous assumptions are maintained except that

the volatility of currency i is proportional to the forward premium |rfi − rd|: σi = k|rfi − rd|.

Then by simple calculation we have ωi = ωj = λk . Thus all carry trades have the equal

weights in the optimal mean-variance portfolio regardless of the forward premiums. This

simple example shows that in general we need restrictions to have the property ωi > ωj given

rfi > rfj > rd. We now construct our model.

Carry trade return has two parts: et = it − i∗t − (st+1 − st). In line with the literatures

on the random-walk model of exchange rates and the forward premium puzzle, we model

E t[et] = β · (it − i∗t ), where β = 1 for the random-walk model and β > 1 if we assume that

high yield currency tends to appreciate. Thus RCT = [β · |rfi − rd|]Ti , where |rfi − rd| is the

forward premium of the ith carry trade. For the second moment of carry trade returns, we

make the following assumptions: volatility of the carry trade return is the same across all

currencies; covariance of two carry trades’ returns is the same across all currency pairs. In

Fig. B.2 we plot the historical volatilities of G10/USD currencies. We can see that apart

from Canadian dollar other currencies have roughly the same volatility levels. Under these


assumptions, let J be the n by n matrix which has 1 for each of its elements; let I be the

identity matrix, the covariance matrix Σ can be written as aI + bJ. We also assume that

the carry trades are not perfectly correlated, and thus a > 0. For example, the covariance

matrix Σ of RCT for three carry trades has the following form:a+ b b b

b a+ b b

b b a+ b

Now we give a lemma, which will be used in the following theorem:

Lemma 2 If a 6= −nb, then the inverse of aI + bJ is cI + dJ, where c = 1a and d = −b

a(a+nb) .

Proof. We have that J2 = nJ. Thus (aI + bJ)(cI + dJ) = acI + (ad+ bc+ nbd)J. Thus for

c = 1a and d = −b

a(a+nb) we find that (aI + bJ)(cI + dJ) = I.

Now we can give our main result:

Proposition 3 In the optimal Sharpe ratio portfolio, the weight of a higher forward premium

carry trade is higher than that of a lower forward premium carry trade.

Proof. Note that ω = Σ−1RCT. We need to prove that ωj > ωk whenever RCTj > RCT

k.

From the previous lemma we know that Σ−1 = cI + dJ, where c > 0. Thus, ωj = cRCTj +

d∑N

h=1 RCTh, which in turn means that ωj > ωk if RCT

j > RCTk.

Our result implies that a portfolio including carry trades of higher forward premia will have

a higher expected Sharpe ratio than a portfolio containing trades of lower forward premia.

This motivates our study of threshold portfolios in the next section. Lustig et al. [2011] and

Menkhoff et al. [2012] construct portfolios based on forward discounts, and they document

similar results. Menkhoff et al. [2012] constructs portfolios based on relative order of forward

premia. In their five-portfolio example the first portfolio includes the quintile of highest

forward premium, the fifth the quintile of lowest forward premium. The first portfolio has a

higher Sharpe ratio than the fifth.

2.3.3 Backtesting

In this subsection we backtest the historical performance of mean-variance optimal portfolio.

We set the expected return as the forward premium. Because the return from interest rate


difference is risk-free, the volatility of carry trade return is due entirely to the FX position.

Thus we approximate the covariance of two carry trade returns by the historical covariance

of the two exchange rates. Lemma 1 however solves the Sharpe ratio optimization problem

without imposing any restrictions on the weights of carry trades. An investor may not want

to short carry trades in his portfolio. Thus, we consider a long only version of the portfolio

optimization problem, doing so by adding an additional positivity constraint. We define the

long only carry trade mean-variance optimal portfolio as follows:

Definition 1 Using the notation introduced in Lemma 1, the long only carry trade mean-

variance optimal portfolio has the allocation weights that solve the following problem:

max :ωTRCT

√ωTΣω

s.t.

P∑j=1

ωj = 1, ωi ≥ 0,∀i

The positivity constraint deprives the problem of any analytical formulation of its solution.

Numerically, it is a quadratic programming problem and can be solved quickly.

We report two tests with different window lengths when estimating the currency correlations.

In the first test, we use a one-year fixed window length. In the second, we use two-year window

length. The results are the rows of Carry 1Y and Carry 2Y seen in Table B.3. The cumulative

return indices are plotted in Fig. B.5. We see that both mean-variance portfolios have better

Sharpe ratio compared to the benchmark equally weighted portfolio. But the improvements

are not significant. After discussing the robustness of the result in the next subsection, we

will return to discuss the implications of the results.

2.3.4 Robustness check

The literature takes two different approaches to tackle the problem of data snooping bias.

The first approach focuses on data, and tries to avoid reusing the same data set. Testing

a model by using a different but comparable data set may do this. In our case, however,

no such alternative dataset is available. Thus we have used, instead two methods based on

bootstrap: (1) Resampling with bootstrap, see Efron [1979]; (2) White’s Reality Check, see

White [2000] and Politis and Romano [1994]. Here we test equal performance of the long


only mean-variance optimal portfolio against the benchmark portfolio. Our criterion is the

Sharpe ratio. We report the result of Carry 2Y below.

P-value of bootstrap resampling test

Carry 2Y

Benchmark 0.35

P-value of White’s Reality Check resampling test

Carry 2Y

Benchmark 0.40

In neither tests, mean-variance portfolio outperforms sigfinicantly equally weighted portfo-

lio. One concern regarding the mean-variance approach is the estimation error. DeMiguel

et al. [2009] studies this issue. They conclude that ”out of sample, the gain from optimal

diversification is more than offset by estimation error”. In their simple simulation example

of 10 assets, one needs an estimation of 3,000 months in order for the mean-variance portfolio

to outperform the naive equally weighted portfolio. We propose threshold portfolios in the

next section, by drawing on the insights of the mean-variance approach and also retaining

the robustness of the naive equally weighted portfolios.

2.4 Threshold portfolios

A forward premium threshold portfolio includes only carry trades having forward premiums

higher than the threshold value. We now have two motivations for studying threshold port-

folios. In Section 2.2, we saw that a currency investor allocating more capital to carry trades

that have larger forward premiums can expect to have a higher return. In Section 2.3, in a

mean-variance setting, an investor seeking to maximize the risk adjusted return also allocates

a larger weight to a carry trade having a larger forward premium. In addition, recent em-

pirical regime switch studies about the forward premium— Jorda and Taylor [2012], Baillie

and Chang [2011], and Christiansen et al. [2011]— suggest that threshold portfolios based on

forward premiums should have different performances depending on the threshold values.

An equally weighted threshold portfolio gives equal weights to the carry trades in the port-

folio. Compared to the mean-variance optimal portfolio, the equally weighted threshold


portfolio keeps the essential property that carry trades with a larger forward premium have

larger weights. In this case, those carry trades with forward premiums below the threshold

value have weights of zero. The carry trades with forward premiums beyond the threshold

value have positive weights. In contrast to the mean-variance optimal portfolio, the equally

weighted threshold portfolio gives equal weights to all the trades in the portfolio. The equal

diversification among the portfolio is simple, yet robust, as we will see. In the next subsection,

we test threshold portfolios based on the forward premium.

2.4.1 Threshold portfolios based on forward premium

Unlike the the naive portfolio, a portfolio having a forward premium threshold changes its

composition over time. In the same spirit of the G10/USD carry trades setting, we assume

implementing carry trades of foreign currencies versus domestic currency. For a threshold

value γ, the equally weighted threshold portfolio with threshold γ at time period t includes all

carry trades of foreign currency i versus domestic currency such that |rfi − rd| > γ at t. The

intuition driving a threshold portfolio is that the investor wants to collect the maximum yield

given the constraint of capital. He wants to hold more carry trades having larger forward

premiums.

We first test our threshold portfolios for the universe of G10 currencies versus the US dollar.

Table B.2 details the performance statistics of those threshold portfolios. Figure B.3 plots

the historical cumulative returns of the threshold portfolios. We test 13 thresholds, ranging

from 0 to 6 with a step size of 0.5.

The portfolio’s performance improves as the threshold goes up, to 2.5. At that point, the

performance begins to deteriorate. This isn’t surprising if we look at the average number of

trades in the portfolio. When the threshold becomes too large, there are hardly any trades

in the portfolio.

Based on the empirical test, we have identified two threshold effects: 1) Up to the opti-

mal threshold value, the threshold portfolio outperforms the benchmark portfolio (without

threshold) in terms of risk adjusted return; 2) Up to the optimal threshold value, the portfolio

with the higher threshold value outperforms that of the lower threshold value.

Garratt and Lee [2010] evaluates the forecast performance of models based on their eco-


nomic value within the decision making setting of an investor. In their framework, Model

A works better than Model B if the portfolio constructed with Model A has better financial

performance than the portfolio constructed with Model B. Working within that framework,

we tested the threshold model and proved that a model that posits positive loading of the

carry trade return on the forward premium works better than one that posits neutral loading.

The reason is that threshold models beat benchmark equally weighted portfolios in terms of

financial performance.

2.4.2 Robustness check

A common concern vis-a-vis the threshold model is the data snooping issue. In this subsection,

we check the robustness of our previous results. Similar to the robustness check done in

Section 2.3, we use two methods: resampling with a bootstrap (Efron [1979]) and White’s

Reality Check (White [2000]). We test the two threshold effects. For the first threshold

effect test, we test up to the optimal threshold value if a threshold portfolio has a better

Sharpe ratio than does the benchmark equally weighted portfolio. For the second threshold

effect test, we test up to the optimal threshold value if a threshold portfolio having a higher

threshold has a better Sharpe ratio than does a threshold portfolio having a lower threshold.

Our H0 hypothesis is that a threshold portfolio with a threshold value of γj has an equal or

lower Sharpe ratio than does the benchmark portfolio and the threshold portfolio having a

threshold value γk such that γj > γk. First, we report the results of our resampling bootstrap

test.

P-value of bootstrap resampling test

TR 0.5 TR 1 TR 1.5 TR 2 TR 2.5

benchmark 0.17 0.11 0.06∗ 0.03∗∗ 0.01∗∗∗

TR 0.5 0.31 0.13 0.08∗ 0.03∗∗

TR 1 0.15 0.12 0.03∗∗

TR 1.5 0.22 0.05∗∗

TR 2 0.11

(∗) stands for p-value below 0.1, (∗∗) for p-value below 0.05, (∗ ∗ ∗) for p-value below 0.01

The resampling bootstrap statistics support our contention that up to the optimal threshold

value, the threshold portfolio outperforms the benchmark equally weighted portfolio. The


p-values of testing threshold portfolios of threshold values 1.5%, 2%, and 2.5% having the

same performances as the benchmark portfolio are respectively: 0.06, 0.03, and 0.01. This

finding also implies that up to the optimal threshold value, a higher threshold portfolio beats

a lower one. The p-value of testing 2.5% threshold portfolio having equal performance as

0.5%, 1%, and 1.5% threshold portfolios are 0.03, 0.03, and 0.05.

We report next the result of White’s Reality Check. It shows that the interpretations of

p-values are similar to those gained by resampling bootstrap. Namely, up to the optimal

threshold value threshold portfolio outperforms the benchmark portfolio, and higher threshold

portfolios beat the lower ones.

P-value of White’s Reality Check

TR 0.5 TR 1 TR 1.5 TR 2 TR 2.5

benchmark 0.15 0.11 0.03∗∗ 0.03∗∗ 0∗∗∗

TR 0.5 0.31 0.11 0.06∗ 0.02∗∗

TR 1 0.16 0.10∗ 0.03∗∗

TR 1.5 0.22 0.04∗∗

TR 2 0.08∗

(∗) stands for p-value below 0.1, (∗∗) for p-value below 0.05, (∗ ∗ ∗) for p-value below 0.01

2.4.3 Comparison with a quantile based sorting rule

Recent studies — Lustig et al. [2011] and Menkhoff et al. [2012] — construct portfolios based

on forward discounts. Their approach is similar but different. Menkhoff et al. [2012] con-

structs portfolios based on relative order of forward premium. In their five portfolios example,

the first portfolio includes the quintile of highest forward premiums. We note that the objec-

tive of their portfolio construction is not to improve the financial performance but rather to

capture the risk factor of carry trade. It is interesting however, to compare the two methods

of portfolio construction. We name our sorting rule a “value based threshold rule”, while

the rule of Menkhoff et al. [2012] a “quantile based threshold rule”. We then compute the

performances of the constructed portfolios using the “quantile based threshold rule”. The

results are reported in Table B.4. Comparing Table B.2 to Table B.4, we see that for two

portfolios containing roughly the same number of trades, the one constructed on the basis of

a “value based threshold rule” outperforms the one constructed on the basis of a “quantile


based threshold rule”.

2.5 Crash-hedged threshold portfolios

One recent explanation of carry trade excess return relates to crash risk premium, see Brun-

nermeier et al. [2008], and Farhi and Gabaix [2008]. Following Brunnermeier et al. [2008],

we propose a crash-risk explanation for the threshold effects. We then test the crash-risk

explanation following Jurek [2014].

Brunnermeier et al. [2008] showed that FX crash risk increases with the interest rate differ-

ential and the past FX carry trade returns. The paper argues that carry trade is exposed

to crash risk, which limits arbitrage. In the same spirit, we propose a crash-risk explanation

for the threshold effects, which is that the carry trade of larger forward premium is more

exposed to the crash risk. Therefore, in normal times, the carry trade of a larger forward

premium earns a larger return as a compensation for its bigger loss at crash time. Therefore,

up to the optimal threshold value, a portfolio of higher threshold value outperforms that of

a lower threshold value during normal periods.

Jurek [2014] tests the crash risk explanation for carry trade excess return by constructing a

crash hedged carry trade portfolio. By adding an option overlay, the carry trade’s exposure

to exchange rate movements is eliminated beyond the strike price. If the carry trade’s ex-

cess return comes from the crash-risk exposure, then the crash-hedged carry trade portfolio

shouldn’t exhibit an excess return. By examining the return profile of crash-hedged the carry

trade portfolio, Jurek [2014] concludes that crash-risk can explain only part of the carry

trade excess return. In the same spirit as Jurek, we test the crash risk explanation for the

threshold effects by constructing crash-hedged threshold portfolios. We add an option overlay

to carry trades in the threshold portfolio such that the carry trade’s exposure to exchange

rate movements is eliminated beyond the strike price. The crash risk premium explanation

states that the excess return of higher threshold portfolios comes from the crash risk pre-

mium. If this is true, then crash-hedged higher threshold portfolios should perform just as

well as the crash-hedged portfolios at the lower threshold. We use the difference-in-difference


method to test the crash risk effect. First, we compute the difference in the performance of

hedged threshold portfolios versus hedged benchmark equally weighted portfolios, then the

difference in performance of unhedged threshold portfolios versus the unhedged benchmark

equally weighted portfolios. Finally, we compare the two differences in performance.

2.5.1 Data

We use price data on FX options covering G10/USD pairs from January 1999 to June 2012.

The FX option conventions are explained in Section 3 of Jurek [2014]. We use the most

out-of-money put options. In other words, we use the 10δ put options to hedge the crash

risk.

If we let Sf (T ) denote the price of one unit of domestic currency in terms of foreign currency

f , and let Pt denote the price of the out-money put option, then the pay-off of the option

overlaid portfolio is:

ef (t) = erft τ

Sf (t)

Sf (t+ τ)− erdt τ (1 +

erft τPtSf (t)

) + erft τ ·max(

Sf (t)

X−

Sf (t)

Sf (t+ τ), 0) (2.5)

2.5.2 Performance of crash-hedged threshold portfolios

This subsection reports the results of hedged threshold portfolios and compares them to those

of unhedged threshold portfolios. Table B.5 shows the results of hedged threshold portfolios

for the period between January 1999 and June 2012. Table B.6 shows the unhedged results

for the same period. Figures B.6 and B.7 plot the total return indices for hedged threshold

portfolios for the period January 1999 to June 2012. Figures B.8 and B.9 plot the total return

indices for unhedged threshold portfolios for the same period.

We next compare the performances of hedged versus unhedged portfolios. Table B.7 re-

ports summary statistics comparing the return, volatility, and Sharpe ratio of unhedged and

hedged portfolios for the period between January 1999 and June 2012. Hedged threshold

portfolios, in general, have lower returns and lower volatilities compared to unhedged ones,

with the change in return being more pronounced than that in volatility. This is shown by

the lower Sharpe ratio of hedged portfolios compared to unhedged ones. Figure B.10 plots


the unhedged and hedged portfolios of threshold values 0 and 2.5 taken together.

The next step is to see if the performance difference between unhedged threshold portfolios

and unhedged benchmark portfolios persists for hedged portfolios. We report our results in

Table B.8. Hedged Sharpe Diff shows the difference in the Sharpe ratio of the hedged thresh-

old portfolios versus the hedged equally weighted portfolios. Unhedged Sharpe Diff shows

the difference in the Sharpe ratio of the unhedged threshold portfolios versus the unhedged

equally weighted portfolios. Relative change is the relative percentage difference between

Hedged Sharpe Diff and Unhedged Sharpe Diff. We see that the difference in the Sharpe

ratio decreases by around 25% from unhedged to hedged portfolios.

We find here, like in the unhedged case, the persistence of an excess return of higher threshold

portfolios, relative to hedged lower threshold portfolios. This result suggests that the crash

risk premium doesn’t account for all of the outperformance of higher forward premium carry

trades relative to lower forward premium carry trades. The crash risk premium can explain

around 25% of the excess performance of higher threshold portfolios relative to benchmark

equally weighted portfolios.

2.6 Random-walk model of threshold effects

Section 2.4 finds two threshold effects: 1) Up to the optimal threshold value, the threshold

portfolio outperforms the benchmark portfolio (without threshold) in terms of the risk ad-

justed return. 2) Up to the optimal threshold value, the portfolio of the higher threshold

value outperforms that of the lower threshold value. In this section, we propose a model

which exhibits the two threshold effects. In our model, currencies have only idiosyncratic

shocks. Using a parametric probability distribution as an approximation of the distribution

of the forward premiums, the model predicts the best threshold value which is the one that

maximizes the Sharpe ratio. The model also predicts the relative gain in the Sharpe ratio

of the threshold portfolios as compared to the equally weighted portfolio. We calibrate the

model to G10 currency trades, using data ranging from January 1990 to June 2012.


2.6.1 Random-walk model

We denote as Si(t) the exchange rate of foreign currency i at time t. In our model, each

currency has only idiosyncratic risk and equal volatility. Thus,

dSi(t)

Si(t)= σ∗i dWi(t)

Meese and Rogoff [1983] notes that the random-walk model works well for exchange rates.

Engel and West [2004] finds that, under certain conditions, the exchange rate under the

rational expectation present value model, behaves just like random walk. We denote as ri(t)

the short term interest rate of the foreign currency i at time t, and rd(t) the short term interest

rate of the domestic country. We denote as carryi the carry trade of domestic currency and

foreign currency i. According to the random-walk model of currency rates, the expected

return of carryi from time t to time t + 1 is |rd(t) − ri(t)|. In order to model the threshold

effect, we assume that the forward premiums follow a parametric distribution. We also fit

the forward premium by an exponential distribution which has PDF λe−λx.

We next compute the average return and volatility of threshold portfolios. A threshold

portfolio of parameter y includes all those carry trades whose forward premia are greater

than y. We denote that portfolio as Ptfy. We also assume that Ptfy allocates equally to all

the carry trades in the portfolio. We can compute the expected return of Ptfy as

E [rt+1(Ptfy)] =

∫∞y e−λxxλ dx.

e−λy

Let σ20 denote the variance of return for the portfolio having a threshold zero. Now we prove

that the variance of return for the portfolio having threshold y is eλyσ20. Indeed, the mass

of currency having forward premiums higher than y is∫∞y λe−λxdx. = e−λy. The mass of

currency in the portfolio having threshold zero is 1. Both portfolios give equal allocation

to their carry trades. Given that each currency has the same volatility, the ratio of the

variances of the two portfolios should be the inverse of the ratio of the masses of currencies

in the portfolio. Thus we have

Var [rt+1(Ptfy)]

Var [rt+1(Ptf0)]=

1

e−λy= eλy

The volatility is

Vol [rt+1(Ptfy)] = σ0e12λy


And therefore the Sharpe ratio is

Sh[rt+1(Ptfy)] =E [rt+1(Ptfy)]

Vol [rt+1(Ptfy)]=

∫∞y e−λxxλdx.

σ0e− 1

2λy

Sh[rt+1(Ptfy)] is maximized at

y =1

λ(2.6)

For y = 1λ , the Sharpe ratio of Ptfy is 2

e1/2λσ0. Note that the Sharpe ratio of Ptf0 is 1

λσ0.

Thus,Sh[rt+1(Ptf1/λ)]

Sh[rt+1(Ptf0)]= 2e−1/2 = 1.21 (2.7)

Our model makes two key predictions for threshold portfolios:

1. Given λ, the optimal threshold that maximizes the Sharpe ratio is 1λ .

2. The threshold portfolio with threshold value y has Sharpe ratio with the value being the

Sharpe ratio of threshold zero portfolio multiplied byλ2

∫∞y e−λxxdx.

e−12λy

.

2.6.2 Possible variations of the random-walk models

The random-walk model makes two assumptions.

1. The model assumes the currency movement to be random walk. Other studies have found

that currency has momentum and that currency returns are dependent, see Asness et al.

[2013].

2. The currency has only idiosyncratic shocks in the model. Several studies have found that

currencies have loadings on global shock, see Lustig et al. [2011].

Both of those assumptions can be relaxed, as we shall see in Section 2.7 on the one-factor

model.

2.6.3 Calibration

We calibrate the model to G10 currency carry trade using data between January 1990 and

June 2012. In Section 2.4 we computed the financial performances of threshold portfolios,

with those statistics being reported in Table B.2.

We fit the forward premium distribution by exponential distribution. By the maximum

likelihood method, we obtain λ∗ = 0.45. We plot in Fig. B.11 the CDF of both the empirical


distribution and the fitted exponential distribution. The model predicts that the optimal

threshold portfolio will have a threshold value of 1λ∗ = 2.23. In Table B.2, we saw that

the optimal portfolio is that having threshold values of 2.5. Our model, therefore, makes a

reasonable prediction.

When it comes to another prediction, in Fig. B.12 we plot the Sharpe ratios of threshold

portfolios versus the Sharpe ratio implied by the model where λ = 0.45. We see that the

model works well up to a threshold value of 4%, over which a large deviation begins to appear.

This may be due to the relatively small size of the investment universe, which implies a sizable

variance in test results.

2.6.4 Model with crash-hedged portfolios

In this subsection we test the random-walk model on crash-hedged portfolios. Our motivation

is the same as in Section 2.5. Given the domestic and foreign interest rates rdt and rit, Si(t)

denotes the price in domestic currency of one unit of foreign currency i. This means that the

price of a put option, having respectively strike price X and volatility σt(X, i), is :

Pt(Si(t), X, τ, rdt , rit) = e−rdt τ [Fi(t, τ)N(d1)−XN(d2)]

where Fi(t, τ) = Si(t)e(rdt−rit)τ is the forward rate for currency with maturity τ . And

d1 =ln(Fi(t, τ))/X

σt(X, i)√τ

+1

2σt(X, i)

√τ

d2 = d1 − σt(X, i)√τ

The option-overlaid portfolio’s payoff was given in Eq. 2.5. Because of the non-linear pay-off

of the option, we can’t solve analytically the optimal threshold value. Therefore, we study

the threshold effects by doing numerical simulations. We assume that the investor wants to

hedge the downside currency risk beyond 10% loss. Thus, investors buy put options having

strike 0.9Fi(t, τ) so as to hedge their currency crash risk. We simulate two cases of volatility

specifications. In the first case, the implied volatility is constant for all currency pairs. In

the second case, the implied volatility grows as the forward premiums do.


2.6.4.1 Equal implied volatilities for currency options

We assume that the implied volatilities for currency options are the same. Based on empirical

evidence as to the option premium, we assume a volatility premium of implied volatility

compared to historical volatility. We then do a numerical simulation in order to test the

threshold effect. We simulate 1000 carry trades occurring in 500 periods. In each period, the

funding currency interest rate has a uniform distribution on [0, 2%]. The forward premium

has an exponential distribution of mean 2%. We test three cases of currency volatility: 10%,

15% and 20%. We test two cases of implied volatility premium: 5% and 15%. If, for example,

the currency volatility is 10% and the implied volatility premium is 15%, then the implied

volatility is 11.5%. We plot the simulation results in Fig. B.13 and see that the optimal

threshold values fall between 2% and 3%. The Sharpe ratio rises as the threshold value

rises from 0 to between 2% and 3%, after which it declines. What we find is similar to the

result gained in Section 2.5: when currency movement follows random walk, the crash-hedged

threshold portfolios have profiles similar to the non-hedged portfolios.

2.6.4.2 Variable implied volatilities for currency options

We now do another simulation, with all the other assumptions, aside from those relating to

implied volatilities, being the same as those discussed in the previous subsection. We assume

that the option-implied volatility rises along with the forward premium.

σ∗i = (1 + premium) ∗ σ + 0.2 ∗ (1− e−fpi/fp)σ

where σ∗i is the implied volatility for currency i, σ is the historic volatility, and fp is the

mean of the forward premiums. We plot the simulation result in Fig. B.14 and see that the

optimal threshold value in the range of 2% to 3%. The Sharpe ratio rises as the threshold

value rises, from 0 to between 2% and 3%, after which the Sharpe ratio declines. Thus, the

monotonicity property is similar to the case of equal implied volatility that we looked at in

the subsection 2.6.4.1.


2.6.5 Conclusion

Both simulation cases feature two threshold effects: 1) Up to the optimal threshold value,

the threshold portfolio outperforms the benchmark portfolio (without threshold) in terms of

risk adjusted return. 2) Up to the optimal threshold value, the portfolio of higher threshold

value outperforms that of the lower threshold value. Thus, the essential implications of the

crash-hedged models are the same as the unhedged model.

2.6.6 Threshold portfolios when prices of risks are the same

In the previous model, prices of risks are different across different currency markets. This

means that the price of risk λi for currency i is

λi =abs(ri − rd)

σ

Now we prove that if the price of risk is the same across different currency markets, then the

best portfolio’s allocation depends on the volatility of currencies. Therefore the threshold

effect doesn’t exist for the forward premium but does exist for the volatilities.

Proposition 4 In a setting of N currencies of equal volatility, and if abs(ri−rd)σ is the same

for all currencies i, then the portfolio that optimizes the Sharpe ratio gives currency i weight

µ/σi, where µ is a renormalization constant.

Proof. The variance covariance matrix Σ is diagonal that has σ2 on the main diagonal. We

know from mean-variance portfolio theory, that the optimal allocation weight ωi is equal to

ν abs(ri−rd)σ2 , where ν is a renormalization constant. We also know that abs(ri−rd)

σ is the same

for all currencies i. Hence ωi =µ/σi, where µ is a renormalization constant.

Thus in a setting where prices of risk are the same for all currencies, we are well advised to

investigate threshold portfolios with thresholds depending on currency volatilities. Pursuing

that direction of research ever further would be a reasonable extension of the present project.

2.6.7 Extending models to include currencies with correlations

We now extend our baseline model to the case where currencies are correlated. We assume

that all two currency pairs have the same correlation ρ, and that the volatility of each currency


is σ. This means that the variance-covariance matrix of currency returns is

Σ = σ2

1 ρ ... ρ ρ

ρ 1 ... ρ ρ

ρ ρ ... ρ ρ

ρ ρ ... 1 ρ

ρ ρ ... ρ 1

All the other assumptions are the same as in Section 2. Our model suggests that

si(t) =Si(t+ 1)− Si(t)

Si(t)= σi(t)εi(t), 1 ≤ i ≤ N

where:

cov(εi(t), εj(t)) =

1 if i = j

ρ if i 6= j

We assume that the distribution of forward premiums follows the exponential distribution

with parameter λ. When computing the expected return of threshold portfolios, the correla-

tion structure doesn’t change the computation; thus as in the subsection 2.6.1 we have

E [rt+1(Ptfy)] =


e−λy

In the next section, we provide an example in which all currencies have equal correlations.

We will solve the model analytically, just as we did in subsection 2.6.1.

2.7 One-factor model

Here, following Lustig et al. [2011] and Jurek and Xu [2013], we extend our random-walk

model into a one-factor model, where each currency has loadings on the global factor. We

have:

si(t) = wiεg(t) +√

1− w2i εi(t) (2.8)

εg(t) and εi(t) are independent normal variables with zero means and the same standard

deviation: σ, εg(t) ∼ N(0, σ2), εi(t) ∼ N(0, σ2), where εg(t) is the global risk factor and εi(t)

is the idiosyncratic risk factor of currency i.

Our main interest is to study if the one-factor model features the threshold effects, which is


there exists a optimal threshold portfolio in terms of risk adjusted return. And if the optimal

threshold value exists, if the model feature two other effects: 1) Up to the optimal threshold

value, threshold portfolio outperforms benchmark portfolio(without threshold) in terms of

risk adjusted return. 2) Up to the optimal threshold value, portfolio of higher threshold

value outperforms that of lower threshold value.

We first compute the variance and covariance of currencies:

Var (si(t)) = w2i Var (εg(t)) + (1− w2)Var (εi(t)) = σ2

Cov (si(t), sj(t)) = wiwjσ2

If each currency has the same loading on the global risk factor, wi = w, then all currency

pairs will have the same correlations: Corr (si(t), sj(t)) = w2. On the other hand, given ρ,

we can construct a one-factor model in which wi = w =√ρ, such that Corr (si(t), sj(t)) = ρ.

Thus we see that the model posited on an equal correlation of all currency pairs is a special

case of the one-factor model. In the ensuing subsections we study the one factor model while

imposing various restrictions on wi. First, however, we study the case of equal correlations

of all currency pairs.

2.7.1 One-factor model with equal correlations of all currency pairs

Our assumptions with respect to forward premium are the same as those made in Section

2.6. Thus for a threshold portfolio having threshold value y we have

E [rt+1(Ptfy)] =


e−λy

Next we consider the variance of rt+1(Ptfy). Supposing that the portfolio includes N carry

trades with currency movement sij , sij (t) = wεg(t) +√

1− w2εij (t). Thus we have

Var [rt+1(Ptfy)] = w2Var εg(t) +

∑Nj=1(1− w2)Var εij (t)

N2

Var [rt+1(Ptfy)] = (w2 +1− w2

N)σ2

Now we assume that N is sufficiently large relative to 1−ω2

ω2 , such that we could omit the

second term in the first order approximation.


Assumption 1 We assume that N 1−ω2

ω2 for all threshold portfolios, such that the variance

of portfolio return is dominated by the covariance of different trades.

Under this assumption we have

Var [rt+1(Ptfy)] ' w2σ2 (2.9)

In this case the Sharpe ratio of threshold portfolio is

Sh[rt+1(Ptfy)] 'y + 1/λ

wσ(2.10)

Thus under Assumption 1 in the one-factor model, idiosyncratic currency risks are perfectly

diversified within each threshold portfolio. All threshold portfolios therefore have the same

risk, which is the global systematic risk. Hence the Sharpe ratio having a higher threshold

value is always higher than that having a lower threshold value. Different from the random

walk model, in the case of one-factor model there is no optimal threshold portfolio in terms

of risk adjusted return.

In practice Assumption 1 is false, due to the limited number of carry trades. Higher threshold

portfolios, which include a smaller number of carry trades, are more prone to idiosyncratic

risk. In fact the idiosyncratic risk grows so quickly that higher threshold portfolios’ perfor-

mances drop dramatically, just as we found in Section 2.4.

2.7.2 One-factor model with heterogeneous correlations of currency pairs

Some research, for example Brunnermeier et al. [2008], suggest that the currency pairs of

higher forward premiums tend to comove. In this subsection we model this property in our

one-factor framework. We assume that there are in total N currency pairs, and without loss

of generality we assume that fpi < fpj for i < j. In our model the loading of currency pair

on the global risk factor depends upon its forward premium:

si(t) =i

Nεg(t) +

√1− (

i

N)2εi(t) (2.11)

Thus the currency pair of higher forward premiums has a higher loading on the global risk

factor, with Cov (si(t), sj(t)) = ijN2 . The correlation between currency pairs of higher forward

premiums is thus greater than that of lower forward premiums. We next compute the return


volatility of threshold portfolios.

Assume that the portfolio includes N carry trades, with loadings wij (1 ≤ j ≤ N) on the

global risk factor. Thus

Var [rt+1(Ptfy)] =(∑N

j=1wij )2Var εg(t)

N2+

∑Nj=1(1− w2

ij)Var εij (t)

N2

Var [rt+1(Ptfy)] = (wij )2σ2 +

(1− w2ij

)σ2

N

where wij and w2ij

are the means of wij and w2ij

, respectively. We now make the following

assumption, similar to Assumption 1

Assumption 2 We assume that N 1−w2

ij

(wij )2for all threshold portfolios, such that the vari-

ance of portfolio return is dominated by the covariance of different trades.

Under Assumption 2, we have

Var [rt+1(Ptfy)] ' (wij )2σ2

std [rt+1(Ptfy)] ' wijσ

We now compute the return volatility of the threshold portfolio. The portfolio with threshold

value y includes (1 − e−λy)N carry trades having biggest forward premiums , and thus the

average loading of portfolio on global factor is wyij = 1− e−λy

2 . From which we have

std [rt+1(Ptfy)] ' (1− e−λy

2)σ

Sh[rt+1(Ptfy)] 'y + 1/λ

(1− e−λy

2 )σ

It is easy to verify that Sh[rt+1(Ptfy)] is an increasing function for y greater than 1/λ; thus,

the result gained for heterogeneous correlations is similar to that for equal correlations. Under

Assumption 2, idiosyncratic currency risks are perfectly diversified within each threshold

portfolio. When the investor increases the threshold value, the increase in expected return

outdoes the increase in volatility, and therefore the investor always wants to invest in higher

threshold portfolios. In the real world of a small universe of carry trades, Assumption 2 is

also wrong, just as we pointed out about Assumption 1.


2.7.3 Model with increasing downside correlation

It is well known that correlation between assets increases during period of high market volatil-

ity; this phenomenon has been discussed for example in both Ang and Bekaert [2002] and

Bouchaud and Potters [2001]. Now let us show that our one-factor model features precisely

such an increasing downside correlation. We use the following conditional correlation to

measure the downside correlation:

ρ(si(t), sj(t))|εg(t)|>ν =Cov (si(t), sj(t))|εg(t)|>ν√

Var (si(t))|εg(t)|>νVar (sj(t))|εg(t)|>ν

ρ(si(t), sj(t))εg(t)<−ν =Cov (si(t), sj(t))εg(t)<−ν√

Var (si(t))εg(t)<−νVar (sj(t))εg(t)<−ν

where ρ(si(t), sj(t))|εg(t)|>ν is the correlation between si(t) and sj(t), conditional on the global

factor having high volatility. ρ(si(t), sj(t))εg(t)<−ν is conditional on the downside return of

the global factor. We have the following relationship.

Proposition 5 ρ(si(t), sj(t))|εg(t)|>ν and ρ(si(t), sj(t))εg(t)<−ν are increasing functions of ν.

Proof. We prove for ρ(si(t), sj(t))|εg(t)|>ν , and the proof for ρ(si(t), sj(t))εg(t)<−ν is similar.

We have

ρ(si(t), sj(t))|εg(t)|>ν =

wiwjVar (εg(t))|εg(t)|>ν√w2i Var (εg(t))|εg(t)|>ν + (1− w2

i )Var (εi(t))√w2jVar (εg(t))|εg(t)|>ν + (1− w2

j )Var (εj(t))

Note x = Var (εg(t))|εg(t)|>ν , we have

ρ(si(t), sj(t))|εg(t)|>ν =wiwjx

((w2i x+ (1− w2

i )σ2)(w2

jx+ (1− w2j )σ

2))1/2

Thus,

dρ(si(t), sj(t))|εg(t)|>ν

dx=

wiwj(w2i x+ (1− w2

i )σ2)(w2

jx+ (1− w2j )σ

2)− 12wiwjx(2w2

iw2jx+ (w2

i + w2j )σ

2)

((w2i x+ (1− w2

i )σ2)(w2

jx+ (1− w2j )σ

2))3/2


It is easy to verify thatdρ(si(t),sj(t))|εg(t)|>ν

dx > 0, and thus we are done.

The last proposition shows that the one-factor model features increasing downside correla-

tions.

2.7.4 Implication of random-walk model and one-factor model

The random walk model in the section 2.6 features two threshold effects: 1) Up to the optimal

threshold value, threshold portfolio outperforms benchmark portfolio(without threshold) in

terms of risk adjusted return. 2) Up to the optimal threshold value, portfolio of higher

threshold value outperforms that of lower threshold value. We see that these two threshold

effects disappear in the one-factor model, where investor always wants to invest in higher

threshold portfolio. The reason is in a universe of many carry trades, idiosyncratic risks

are perfectly diversified in threshold portfolios. Thus volatility of threshold portfolio return

is determined by the correlation between currencies. As the threshold goes up, the effect

of increase in average return dominates that of increase in volatility. The non-existence of

optimal threshold value in one-factor model shows that quasi-independence of currencies is

important to understand the threshold effects for currency carry trades.

2.8 Conclusion

This paper studies threshold portfolios. It constructs carry trade threshold portfolios using

thresholds depending on the forward premium. It shows that from 1990 to 2012 portfo-

lios with a higher threshold up to the optimal threshold value outperform lower threshold

portfolios. The robustness of the results is tested. Following Jurek [2014] it tests if the out-

performance of higher threshold portfolios is compensation for crash risk. It constructs the

hedged threshold portfolio using currency options and computes the returns of crash-hedged

threshold portfolios. It finds that crash risk could explain around 25% of the excess perfor-

mance of higher threshold portfolios relative to a benchmark equally weighted portfolio.

Empirically, the paper finds that the Sharpe ratio of threshold portfolio initially increases

as the threshold value goes up until the optimal threshold value. The Sharpe ratio then

decreases. The paper models the effects of threshold portfolios in a random walk currency


model. Under general assumptions about the volatilities of the currency and the distribution

of forward premiums, the model predicts the best threshold value that maximizes Sharpe ra-

tio. The model also predicts the relative gain in the Sharpe ratio of the threshold portfolios

compared to the equally weighted portfolio. It calibrates the model to G10 currency trades

using data from January 1990 to June 2012. It finds that both key predictions are verified.

The paper also explores a one-factor model which exhibits rich properties of currencies: het-

erogeneous correlations and increasing downside correlations. It finds that in a one-factor

model, the optimal threshold value doesn’t exist if the currency returns are correlated and id-

iosyncratic currency risks are perfectly diversified in threshold portfolios. The non-existence

of an optimal threshold value in the one-factor model shows that quasi-independence of cur-

rencies is important to understand the threshold effects for currency carry trades.

Related to threshold effects, the paper studies the portfolio problem of a currency investor. It

approximates the problem of maximizing the investor’s expected utility by a mean-variance

problem. It solves the investor’s optimal portfolio given the expected returns and correlations

of currencies. It proves that under certain second moment conditions, the investor’s optimal

portfolio always allocates more funds to trades having larger forward premiums.

CHAPTER 3. CURRENCY CARRY TRADE HEDGING 73

Chapter 3

Currency Carry Trade Hedging

3.1 Introduction

Portfolio hedging1 is an important issue in investment. The available literature has exten-

sively studied hedging options using underlyings2 and hedging spots with futures3, but the

currency hedging problem4 has not been widely studied. Thus, this paper fills that gap in

the literature in seeking to solve the hedging problem for currency investors. It proposes the-

oretical frameworks for currency investment hedging, while then proposing and empirically

testing various hedging strategies.

This paper studies the hedging problems of a popular currency investment strategy - currency

carry trade, which profits from the violations of uncovered interest rate parity. Carry trade

has been pointed out as a popular currency investment strategy: for example Galati et al.

[2007] and Clarida et al. [2009]. Also see Daniel, Hodrick, and Lu [2014] for a summary of

the carry trade’s risks and drawdowns. Carry trade portfolios are prone to large drawdowns

in liquidity crisis periods and times of high currency volatility; see Brunnermeier et al. [2008]

1The term portfolio hedging refers to techniques used by investors to reduce their overall risk exposure

within an investment portfolio. In other words, hedging uses one investment to minimize the negative impact

of adverse price swings in another. The mathematical framework is provided in Section 3.2.1.

2For example, the option sellers hedge exposures to underlying assets by buying or selling the latter.

3For example, airline companies buy oil futures so as to eliminate risks related to future oil prices.

4By currency hedging we mean specifically hedging as part of a deliberate currency investment strategies.


and Clarida et al. [2009]. In Section 2, we study the hedging problem in the context of the

general utility maximization framework of a currency investor. The solution provided by

the model reveals that investors employ hedging instruments for two reasons: either to 1)

boost the portfolio’s expected return or 2) reduce the portfolio’s variance/risk. We refer to

the position taken for reason 1) as the directional position, and for reason 2) as the variance

reduction position. The model’s solution shows that the investor takes the directional posi-

tion if the hedging instrument has a positive expected return. The model also shows that

the investor takes the variance reduction position only if the covariance between the hedging

instrument’s return and the carry trade’s return is negative. We then divide hedging instru-

ments into three categories, according to their expected return rh and their covariance with

the carry trade’s return rc: Cov [rc, rh]. 1) Insurance, which in general reduces the return

and also the risk to the portfolio, rh < 0,Cov [rc, rh] < 0; 2) Technical rule, which similar to

insurance reduces the portfolio’s risk but doesn’t worsen its return, rh ≈ 05, Cov [rc, rh] < 0;

and 3) Market neutral strategy, which boosts the return of the portfolio but doesn’t affect

its risk, rh > 0,Cov [rc, rh] ≈ 0. After setting up the theoretical framework we then propose

various hedging strategies. We also discuss some additional questions related to the hedging

strategies.

We consider four types of hedging strategies: FX options6 strategy, VIX future strategy7,

stop-loss strategy8 and CTA Trend Following strategy9. The first two are examples of insur-

ace, Stop-loss is a technical rule and CTA is a market neutral strategy. We show that based

5For the following we mean the technical rule doesn’t worsen the carry trade portfolio’s return when we

say the technical rule has rh ≈ 0.

6Three kinds of currency options are actively traded: Delta 10, Delta 25, and Delta 50, they are quoted by

the Delta of the option computed in the Garman–Kohlhagen model.

7VIX index tracks market estimate of expected volatility. VIX is calculated by using the real-time prices

of S&P 500 options.

8Stop-loss rules are rules that reduce a portfolio’s position after it has reached a cumulative loss.

9CTA (Commodity trading advisor) is an asset manager trading chiefly in futures. The predominant tactic

of CTAs is the trend-following strategy. CTA tries to profit from the market trends by observing the current

direction and deciding whether to buy or sell based on the momentum. Henceforth we will refer to CTA

trend-following strategy as simply the CTA strategy.


on empirical evidence from 2000 to 2012, CTA is the preferred hedging strategy because

it increases significantly the average portfolio return and decreases dramatically portfolio

volatility and drawdown. Stop-loss strategy reduces portfolio’s risk while decreasing signifi-

cantly minimum drawdown and minimum monthly return. Currency option and VIX future

strategies offer good hedge against tail risk. They also reduce portfolio return volatility.

However they are costly to implement and significantly worsen average return and Sharpe

ratio. Delta 10 is the least expensive, while Delta 25 and VIX are almost equally costly.

Delta 50 is the most expensive.

Our results shed light on the crash risk explanation for the carry trade. According to the

crash risk theory, carry trade portfolio hedged against crash risk shouldn’t have significant

return. In Chapter 2 we saw that a option hedged carry trade portfolio shows significant

return for the period from 1990 to 2012. In the current paper we show that carry trade

portfolio hedged with stop-loss rule also shows significant return. And the stop-loss hedged

portfolio has positive skewness, in contrast to unhedged carry trade portfolio. Our results

thus indicate that crash risk explains only part of the carry trade excess return.

This paper discusses the intuitions and rationales of employing each hedging strategy. Cur-

rency option is the most straightforward hedging candidate because it eliminates the downside

currency risk beyond the option strike price. Previous studies, see Jurek [2014] and Burnside

et al. [2011a], have constructed hedged carry trade portfolios using out-of-money put options.

Actively traded currency options include the Delta 10, Delta 25 and Delta 50 options. We

construct three hedging portfolios using these three options, and find that all three currency

option hedging strategies reduce the volatility of a carry trade portfolio. The hedgings are

costly, however, for they also reduce the average returns. Overall, hedgings worsen the Sharpe

ratio, but they do offer good hedging against tail events because all of them significantly re-

duce the minimum monthly return while also slightly reducing the minimum drawdown.

The second approach we study is the VIX strategy, whereby we take long position in VIX

future as hedging. Clarida et al. [2009] points out that carry trades suffer losses when un-

dertaken within a high volatility regime. Brunnermeier et al. [2008] finds that rising global

volatility, proxied by VIX, predicts an increasing chance of crash. Menkhoff et al. [2012] finds

that the depreciation of high interest rate currency is positively correlated with the increase of


global currency volatility. Their findings suggest that carry trade crash risk could be hedged

by buying global volatility insurance. Thus we propose to hedge carry trade portfolio by

hedging volatility risk. Lacking as we are a traded FX volatility index, we use VIX futures as

a rolling hedging vehicle. We find that VIX hedging exhibits properties similar to those dis-

played by currency option hedgings; namely, it reduces both the return and the volatility of

a carry trade portfolio. Overall, it worsens the Sharpe ratio, but offers good hedging against

tail events like currency options because it reduces significantly the minimum monthly return,

and much less significantly, the minimum drawdown.

The third hedging strategy we consider is the stop-loss rule. Kaminski and Lo [2013] studies

the stop-loss rules, which are rules that reduce a portfolio’s position after reaching a cumu-

lative loss. Kaminski and Lo [2013] define as the ”stopping premium” the marginal impact

of stop-loss rules on a portfolio’s expected return. They show that the stopping premium

can be positive if portfolio returns are characterized by momentum. They also show that

the stop-loss rule reduces portfolio variance if the stop-loss rule involves switching to cash

when the stop-loss threshold is reached. We make the contribution by applying stop-loss

rule to carry trade portfolios. Following our rule, we reduce position in a currency when

the cumulative loss during a month reaches the threshold. We find that the stop-loss rule

improves the average return of carry trade portfolio while reducing the volatility. It is known

that foreign exchange rates exhibit momentum; see Asness et al. [2013]. Burnside et al.

[2011b] documents that carry trade strategies exhibit momentum. Thus our results can be

explained by the model of Kaminski and Lo [2013]. We also find that the stop-loss rule offers

good hedging against tail event. It reduces significantly both the absolute size of minimum

monthly return and the minimal drawdown.

The fourth and last hedging strategy we consider is CTA. As we pointed out previously,

carry trade is prone to volatility risk, and volatility tends to spike after significant market

movement. Brunnermeier et al. [2008] argues that the excess return of carry trade portfolio

is compensation for crash risk, which may correlate with losses in other markets. We thus

suggest to hedge carry trade portfolio with a market neutral strategy. Previous research on

CTA strategies has pointed out the following two stylized facts:

1) CTA payoff is a nonlinear function of market payoff; it’s not correlated with market return.


2) The Beta of CTA strategy is small. The Beta is positive when the market goes up; it is

negative when the market goes down.

These two characteristics make CTA an ideal hedging strategy. Empirically we find that

CTA offers the best hedging by all criteria (detailed below) among the four hedging strate-

gies. We construct an equally volatility weighted portfolio comprising carry trade portfolio

and CTA portfolio. Compared to unhedged carry trade portfolio, it increases the average

monthly return by 32% and reduces monthly return volatility by 31%. Thus, it increases the

Sharpe ratio by 93% and reduces the absolute size of minimum monthly return by 63%, and

the minimal drawdown by 67%.

Besides studying hedging problems the paper also studies two issues related to different hedg-

ing instruments. Related to CTA the paper studies the risk-return aspect of CTA strategy.

In addition we compare VIX strategy to various currency option strategies. The objective

is to determine if VIX is a cheaper form of systematic insurance as compared to currency

options.

The rest of the paper is organized as following: The second section presents the theoretical

framework and the data. The third section studies the currency option and VIX hedging

strategies. The fourth section studies the stop-loss rule. The fifth section presents results

of CTA hedging strategy. The sixth section conducts out-sample test for hedging strategies.

The seventh section explores two issues related to the hedging strategies. The eighth section

summarizes results and concludes.

3.2 Framework and Data

3.2.1 Framework

The purpose of this section is to set up the framework for the complications involved with

carry trade hedging. The literature exploring futures hedging is extensive; see Lien and Tse

[2002] for a literature review, and Cotter and Hanly [2006] for an empirical summary. Within

the setting of futures hedging, the hedger has exposure in the spot market. Thus, carry

trade hedging and futures hedging are similar in that agents want to eliminate risks in both

situations. Nonetheless, there is an essential difference distinguishing one from the other.


Futures generally are regarded as having zero expected return, and generally that is not

true for carry trade hedging instruments. As a result, the minimum variance hedging widely

studied by the analysis of the futures hedging problem, under the assumption that futures

having zero expected return, can be extended to the carry trade setting. To be more specific,

let us suppose that the hedger has utility function U , where U is an increasing function with

negative second-order derivative. We denote as rc the return of carry trade, and as rh the

return of the hedging instrument. Let E denote the expectation with respect to rc and rh.

The optimal hedging ratio αh is determined via this optimization problem:

maximizeα

E [U(rc + αrh)]

subject to 0 ≤ α(3.1)

The first order condition of 3.1 is:

E [U ′(rc + αrh)rh] = 0 (3.2)

which is

Cov [U ′(rc + αrh), rh] + E [rc + αrh]E [rh] = 0 (3.3)

Assume that rc = β0 + β1rh + ε, where rh and ε are independent. Then under the common

assumption E [rh] = 0, where the hedging instrument has zero expected return, the solution

of 3.1 satisfies αh = −Cov [rc,rh]V ar[rh] , which also minimizes the variance within a hedged portfolio.

When E [rh] 6= 0, we follow Lien and Tse [2002] in their way of solving problem 3.1 for mean-

variance utility function. In other words assume that U(r) = E [r] − A2 Var [r] with A

2 the

Arrow-Pratt risk aversion coefficient. The solution of Problem 3.1 then satisfies

αh =E [rh]

A− Cov [rc, rh]

V ar[rh](3.4)

We can interpret αh as being the sum of directional position E [rh]A and minimum variance

position −Cov [rc,rh]V ar[rh] .

Now we look at the possible hedging instruments of carry trade by the light of Eq. 3.4. In

general, we only use a hedging instrument when Cov [rc, rht] ≈ 0 or Cov [rc, rht] < 0.

Insurance : Insurance strategy has return rhi such that E [rhi] < 0, Cov [rc, rhi] < 0. The

examples we study are currency options and VIX futures.


Technical rule : Technical rule strategy has return rht such that E [rht] ≈ 0, Cov [rc, rht] <

0. The example we study is the stop-loss rule.

Market neutral strategy : Market neutral strategy has return rhn such that E [rhn] > 0,

Cov [rc, rht] ≈ 0. The example we study is the CTA.

The above information may become clearer when it is viewed thus:

Cov [rc, rhi] < 0 Cov [rc, rhi] ≈ 0

E [rhi] < 0 Insurance Utility non-improving

E [rhi] ≈ 0 Technical Rule Utility non-improving

E [rhi] > 0 Market Neutral Strategy

In the ensuing sections, we study in detail the performance of these strategies, while also

providing their theoretical motivations. Note, however, that the intuitions of all of them are

well explained by Eq. 3.4. The carry trade investor employs a hedging strategy only when

doing so reduces the variance or enhances the expected return. In the latter case, the hedging

strategy generally should be market neutral, in order to qualify as a hedging strategy. Both

the Insurance and the Technical Rule strategies reduce the variance of the carry portfolio.

The CTA strategy is of positive return, and uncorrelated with the carry strategy. In addition,

CTA is market neutral. After we have introduced the various hedging strategies, we are ready

to discuss the criteria for the evaluation of hedging performance.

3.2.2 Hedging performance evaluation

Markowitz [1952] laid the foundation of modern portfolio analysis in the framework of return

and risk. While there is consensus that the return can be analyzed by expected return,

there is no universally accepted risk measure. The most popular performance metric for

futures hedging is the variance; see, for example, Cotter and Hanly [2006]. Variance is an

intuitive measure employed when the expected return of a hedging strategy is zero, but the

hedging instruments we propose do not all have zero expected returns; insurance strategies

have negative expected returns, whereas the CTA strategy has a positive average return. We

thus propose two categories of performance metrics.


Financial performance indicators : Sharpe ratio; minimum drawdown; minimum monthly

return; moments of return distributions.

Utility effect for trader : We compute the utility gain for a trader who has CRRA utility.

In order to quantify the hedging effect, we compute the minimal amount of risk-aversion

needed in order for the trader to break even. That is the risk-aversion coefficient of the

trader who is indifferent between employing the hedging strategy or not.

Financial performance indicators in the first category are commonly used for characteriz-

ing portfolio performance. Sharpe ratio was proposed in Sharpe [1966], as a measure of

risk-adjusted return. Minimum drawdown and minimum monthly return are both popular

downside risk measures studied, for example, in Young [1998]. Variance is the original risk

measure proposed by Markowitz [1952].

The risk-aversion indicator (Pratt [1964]) in the second category is used specifically for char-

acterizing hedging strategies. Our intuition is that if a trader of low risk-aversion is willing

to take a hedging strategy, then a trader of high risk-aversion will also be. We will test

empirically this proposition.

We use the above two categories of metrics to compare all hedging strategies. We also com-

pare the hedging performances arrived at via currency options and VIX futures. As a preview

of our result, we find that currency options and VIX futures are all expensive as gauged by

the above two performance metrics, but have a strong tail-risk hedging effect. That is to

say, they offer a good hedge during the worst performance months of carry trades. We thus

employ two special metrics to compare currency options to VIX futures. We compute the

utility gain for a trader who is employing currency options or VIX futures and who has CRRA

utility during crisis periods. We also compute the utility cost to the trader during non-crisis

periods. We then look at the following metrics

Insurance Quality : The ratio of utility gain in crisis periods over utility cost in non-crisis

periods.

Tail hedging effect : Returns on currency options and VIX futures strategy during crisis

periods.


3.2.3 Data

We use the following data: FX price data covering G10/USD pairs from January 1990 to

June 2012; price data on FX options covering G10/USD pairs from January 1999 to June

2012; price data on VIX futures from March 2004 to June 2012; price data on 74 liquid

futures from June 2000 to January 2014; Newedge CTA index from June 2000 to January

2014. The Newedge CTA index data comes from Newedge, while the other price data comes

from Datastream. The FX option conventions are explained in Section 3 of Jurek [2014]. We

use on-the-run VIX futures.

3.3 Hedging using currency options or VIX future

In this section we first report the summary statistics of the unhedged G10 carry trade port-

folio. We then implement the currency option hedging strategy. Previous studies, see Jurek

[2014] and Burnside et al. [2011a], have constructed hedged carry trade portfolios using out-

of-money put options. Actively traded currency options include the Delta 10, Delta 25 and

Delta 50 options. We construct three hedging portfolios using these three options, and find

that all three currency option hedging strategies reduce the volatility of a carry trade port-

folio. The hedgings are costly, however, for they also reduce the average returns. Overall,

hedgings worsen the Sharpe ratio, but they do offer good hedging against tail events, because

all of them significantly reduce the minimum monthly return, while also slightly reducing the

minimum drawdown.

The second approach we study is the VIX strategy, whereby we take long positions in VIX

future as hedging. Clarida et al. [2009] points out that carry trades suffer losses when un-

dertaken within a high-volatility regime. Brunnermeier et al. [2008] finds that rising global

volatility, proxied by VIX, predicts an increasing chance of crash. Menkhoff et al. [2012]

finds that the depreciation of high-interest-rate currency is positively correlated with the

increase of global currency volatility. Their findings suggest that carry trade crash risk could

be hedged by buying global volatility insurance. Thus, we propose to hedge carry trade

portfolios by hedging volatility risk. Lacking as we are a traded FX volatility index, we

use VIX futures as a rolling hedging vehicle. We find that VIX hedging exhibits properties


similar to those displayed by currency option hedgings; namely, it reduces both the return

and the volatility of a carry trade portfolio. Overall, it worsens the Sharpe ratio, but offers

good hedging against tail events, like currency options, because it reduces significantly the

minimum monthly return, and much less significantly, the minimum drawdown.

For an interesting question related to currency options we compare in the subsection 3.7.1

VIX strategy to various currency option strategies. Our objective there is to determine if

VIX is a cheaper form of systematic insurance as compared to currency options. We report

next the summary statistics of the equally weighted G10 currency carry trade strategy, as

well as of the option-holding and VIX-holding strategies.

3.3.1 Crash-risk hedging strategies

Carry trades are well known for “going up by stairs and going down by elevators”. Fig. 3.1

illustrates the total return indices for G10 currency carry trades over the period ranging from

January 1990 to June 2012. Table 3.1 reports the summary statistics for the carry trade

portfolio.

Table 3.1: Carry Trade Portfolio Performance Statistics

Mean Std Skewness Kurtosis Minimum Maximum Sharpe N

3.40 5.97 -0.81 5.36 -6.61 5.18 0.57 270.00

Table 1 reports summary statistics for currency carry trades implemented in G10 currencies. Summary

statistics are reported over Jan.1990-Jun. 2012 (N = 270 months). Means, volatilities and Sharpe ratios

(SR) are annualized; Min and Max report the smallest and largest observed monthly return.


Figure 3.1: Cumulative G10 currency carry trade returns

1990 1992 1995 1997 2000 2002 2005 2007 2010 20121

1.5

2

2.5

Figure 3.1 plots total return indices for currency carry trades implemented in G10 currencies. The time period

is from January 1990 to June 2012.

In both places we see that carry trade portfolios tend to achieve positive average return

and a relatively big drawdown. We show in Fig. 3.2 the cumulative returns of currency

option instruments and VIX instrument.

Figure 3.2: Cumulative hedging strategies returns

2004 2005 2006 2007 2008 2009 2010 2011 2012 20130.7

0.75

0.8

0.85

0.9

0.95

1

1.05

50 Delta

25 Delta

10 Delta

VIX

Figure 3.2 plots total return indices for currency option portfolio implemented in G10 currencies. It also plots

total return index for investing in the on-the-run VIX future. The time period is from March 2004 to June

2012. We normalize the VIX strategy so that it has the same volatility as the Delta 25 strategy.


Note that the crash-hedging strategies have average negative returns, reflecting the hedg-

ing cost. In Table 3.2 we show the correlations between the carry trade portfolio and the

various hedging portfolios.

Table 3.2: Correlations between the carry trade portfolio and various hedging instruments

Carry 50 Delta 25 Delta 10 Delta VIX

Carry 1.00 -0.73 -0.62 -0.54 -0.60

50 Delta 1.00 0.95 0.84 0.49

25 Delta 1.00 0.94 0.40

10 Delta 1.00 0.29

VIX 1.00

Table 3.2 reports correlations between carry trade portfolio and different hedging portfolios for the period

from March 2004 to June 2012.

The table confirms the hedging effects of both the currency option and the VIX portfo-

lios. The correlations of carry trade portfolio with the three currency option portfolios are

respectively, -0.73, -0.62, and -0.54. The correlation of the carry trade portfolio with the

VIX portfolio is -0.6. Table 3.2 also reveals high correlations among the currency options

portfolios, with the average correlation being 0.91. Last but not least we see that the VIX

portfolio is positively correlated with all of the currency option portfolios.

3.3.2 Financial performances of hedged carry trade portfolios

Here we compute the average return, volatility, Sharpe ratio, drawdown, and minimum

monthly return of hedged carry trade portfolios versus unhedged portfolios. In Table 3.3

we report the results of a carry trade portfolio hedged with position in the option contract

or the VIX future. The position in hedging instrument is such that its volatility is equal to

10% of that of the carry trade portfolio.

In order to compare currency option strategies and VIX strategy with stop-loss strategy and

CTA strategy later, we report in Table C.1 in appendix the results for the period from June

2000 to June 2012. Because VIX future data is missing from June 2000 to February 2004,


Table 3.3: Carry Trade Hedged with Options or VIX Futures

Sharpe*10 Rlt ∆ Mean Rlt ∆ Std Rlt ∆ DD Rlt ∆ Min Rlt ∆

Carry 0.64 0.04 1.95 -22.48 -6.55

50 Delta 0.44 -30.67 0.03 -30.65 1.95 0.03 -22.87 1.76 -6.50 -0.79

25 Delta 0.52 -19.11 0.03 -19.20 1.95 -0.12 -22.70 0.98 -6.48 -1.18

10 Delta 0.56 -12.13 0.03 -12.29 1.95 -0.18 -22.53 0.25 -6.48 -1.10

VIX 0.54 -15.13 0.03 -15.85 1.94 -0.85 -22.43 -0.22 -6.42 -1.97

Table 3.3 reports summary statistics for unhedged and hedged carry trade portfolios. The construction rule

is detailed in the text. Hedged portfolio includes position in hedging strategy which has 10% of return

volatility of that of unhedged carry trade portfolio. The time period is from March 2004 to June 2012.

Unhedged denotes the unhedged carry trade portfolio. Sharpe is the Sharpe ratio. Mean is the average

monthly return. Std is the volatility of monthly return. DD is the minimum drawdown. Min is the minimum

monthly return. Rlt ∆ denotes the change of the indicator compared to the unhedged carry portfolio.

for that period we replicate VIX strategy using currrency options.

We see that adding an option overlay decreases not only the portfolio’s average return but

also its Sharpe ratio. The volatility of the hedged portfolio is lower than that of the un-

hedged portfolio. The hedged portfolio also has a bigger minimum monthly return than the

unhedged portfolio. Both the Delta 10 option portfolio and the VIX portfolio have smaller

drawdowns than the unhedged portfolio.

Now we propose another measure to evaluate the impact of hedging overlay: the derivative of

the financial performance indicators. Note for example f(ruh) the statistics of unhedged port-

folio, f(ruh + trh) that of a hedged portfolio which includes 1 position in unhedged portfolio

and t position in hedged strategy. We define the derivative f′

of f as following:

f′(ruh)h = lim

t→0

f(ruh + trh)− f(ruh)

t

We report in Table C.2 in the appendix the derivative of financial performance indicators.

We compute f′(ruh)h by numerical methods. We see that all option hedging strategies and

VIX strategy reduce the maximal monthly loss. VIX and Delta 10 improve the minimum

drawdowns.


Next we study the expensiveness of hedging instruments in the framework of investor’s utility

maximization problem.

3.3.3 Who would buy the insurance?

We assume that the trader has CRRA utility function, and he maximizes expected utility

period by period. St is the trader’s wealth at time period t; ωt is the trader’s portfolio at the

beginning of period t; pt is the price of securities. Trader’s problem at period t is:

maxωt+1E t[Uγ(St+1)]

s.t. : ωt · pt = ωt+1 · pt, St+1 = ωt+1 · pt+1

where Uγ(St+1) =S1−γT+1

1− γ

We showed previously that buying currency option or VIX future is on average costly. Adding

currency option overlay or VIX overlay reduces the average return and the Sharpe ratio of

the portfolio. In order to measure the trade-off between cost and payout of hedging overlay,

we compute the break-even risk-aversion coefficient for which trader is indifferent between

taking an additional position in hedging portfolio or not. Empirically, given the return of

unhedged portfolio ruh(t) and that of hedging instrument rh(t), the break-even risk-aversion

coefficient for a trader who takes k position of hedged portfolio is γh such that:∑Tt=1 Uγ(ruh(t))

T=

∑Tt=1 Uγ(ruh(t) + krh(t))

T

Later we will also calculate break-even risk-aversion coefficient for other hedging strategies.

We show in Table 3.4 the results computed for currency option hedging strategies and VIX

future strategy.

Note γh the break-even risk-aversion coefficient for hedging strategy h. Then intuitively

any trader with risk-aversion coefficient γ > γh will also prefer to have the hedging overlay.

That’s what we have verified empirically by running simulations.

Remark 1 Consider two hedging strategies h1 and h2 with break-even risk-aversion coeffi-

cients γh1 and γh2 respectively. We say that h1 is more expensive for CRRA utility trader

than h2 if and only if γh1 > γh2.


Table 3.4: Break-even Risk-aversion Coefficients

k 50 Delta 25 Delta 10 Delta VIX

0.01 26 21 18 14

0.03 26 21 18 14

0.05 27 21 19 14

0.07 27 21 19 14

0.09 27 21 19 14

Table 3.4 computes empirically the break-even risk-aversion coefficient for which trader is indifferent between

taking an additional position in hedging portfolio or not. The time period is from March 2004 to June 2012.

Surprisingly VIX is the best hedging strategy for CRRA utility trader. For example trader

with risk-aversion coefficient of 15 does not gain by buying any currency option but gains by

buying VIX future. Recall that the hedged portfolio overlaid with Delta 10 option delivers

highest Sharpe ratio. We thus find that the cheapest hedging strategy in terms of return

reduction is not the cheapest when we consider the utility of trader. We analyze more in

detail the hedging effects for crash periods next.

3.3.4 Hedging effects during crash periods

In this subsection, we first identify the periods during which carry trade portfolios suffered

huge losses. We then analyze the hedging effects of different hedging strategies during these

periods. In Table C.3, in the appendix, we report the ten months of biggest losses for carry

trade portfolios. For each month, we also report the return of carry portfolios and different

hedging portfolios. We report respectively average portfolio returns for those ten months, as

well as for the six months of biggest carry portfolio losses. On the last row, we report the

number of months on which the hedging portfolio has a negative return, and thus doesn’t

pay off.

Carry portfolios suffered losses greater than 3% during six particular months over the last

several years. Among them, three are associated with the onset of the last financial crisis

(August, September and October of 2008), with losses of -4.46%, -3.17% and -6.55%. Two

ample wreckages take place with the European Debt Crises (September 2011, May 2012),


with losses between 4.3% and 4.5%. The last one is May 2010, with a -5.5% depletion, which

corresponds to the Flash Crash. We now analyze the effects of different hedging strategies;

in particular, we are interested in whether the VIX strategy offers good hedges.

1. Average hedging returns.

All hedging portfolios have, on average, positive returns. The average return of VIX is 0.88%,

between that of Delta 10 at 0.54% and Delta 25 at 1.17%. The Delta 50 strategy, which is

the most expensive among all hedging strategies, also sees the highest payoffs when carry

portfolios suffer considerable losses. In studying average hedging returns, we see that VIX

offers agreeable hedges against currency crash risk.

2. Number of positive hedging returns.

For the aforementioned ten months, all hedging strategies had negative payoffs for certain

periods. Delta 10 had four months of negative returns, while VIX had three months, which

is the same as Delta 25; Delta 50 had only 1 month. By this criterion, VIX again qualifies

as a favorable hedger.

Among the ten months, we study the five months for which Delta 10 strategies showed the

highest returns. We identify these five months as crash periods. They are also the five months

for which carry portfolios suffered the most extensive losses. The five months represent three

events: the onsets of the Financial Crisis, European Debt Crisis and Flash Crash. Specifically,

we study hedging performances during these three periods. In Table 5, we report the gains of

hedging strategies and losses of carry portfolios during the three periods. For the Financial

Crisis period, we include the three-month period from August to October 2008. In European

Debt Crisis, we include September 2011 and May 2012. Flash Crash corresponds to May

2010. We report in the next table hedging portfolio performances during the three major

crisis periods:

We see that all hedging portfolios have, on average, positive returns. The average return

of VIX is 1.42%, between that of Delta 10 at 0.97% and Delta 25 at 1.96%. The Delta 50

strategy, which is the most expensive among all hedging strategies, also has the highest payoff

when carry portfolios suffer from heavy losses. Upon studying average hedging returns, we

see that VIX offers commendable hedges against currency crash risk.

We conclude that currency option strategies and the VIX strategy all offer hedging against


Table 3.5: Hedging Effects During Tail Events


Financial Crisis -14.18 7.94 6.01 3.49 5.05

European Debt Crisis -8.83 5.29 4.07 1.68 1.82

Flash Crash -5.54 2.47 1.71 0.66 1.62

Month Mean -4.76 2.62 1.96 0.97 1.42

Financial Crisis period includes three months from August to October 2008. European Debt Crisis includes

September 2011 and May 2012. Flash Crash corresponds to May 2010.

crash risk.

3.4 Stop-loss Rules Applied to FX Carry Trade

The third hedging strategy we consider is the stop-loss rule. Kaminski and Lo [2013] studies

the stop-loss rules, which are rules that reduce a portfolio’s position after reaching a cumu-

lative loss. Kaminski and Lo [2013] define as the ”stopping premium” the marginal impact

of stop-loss rules on a portfolio’s expected return. They show that the stopping premium

can be positive if portfolio returns are characterized by momentum. They also show that

the stop-loss rule reduces portfolio variance if the stop-loss rule involves switching to cash

when the stop-loss threshold is reached. We make the contribution by applying stop-loss

rules to carry trade portfolios. Following our rule, we reduce position in a currency when the

cumulative loss during a month reaches the threshold. We find that the stop-loss rule im-

proves the average return of carry trade portfolios while reducing the volatility. It is known

that foreign exchange rates exhibit momentum; see Asness et al. [2013]. Burnside et al.

[2011b] documents that carry trade strategies exhibit momentum. Thus, our results can be

explained by the model of Kaminski and Lo [2013]. We also find that the stop-loss rule offers

good hedging against tail events. It reduces significantly both the absolute size of minimum

monthly returns and the minimal drawdown.


3.4.1 Stop-loss rule

Our baseline portfolio comprises carry trades of G10 currency versus USD. It is equally

weighted for nine trades. The portfolio is rebalanced monthly. We form stop-loss portfolios

by adding a stop-loss rule overlay. We apply a stop-loss threshold for each carry trade

position. For each currency pair, we exit the current carry trade position if the month to day

loss is bigger than the stop-loss threshold. We test 10 stop-loss thresholds from 1% to 10%

with step 1%.

3.4.2 Financial performances of hedging strategies

Table 3.6 shows the test result for the period from January 1999 to June 2012. Unhedged is the

equally weighted portfolio comprising G10/USD paris. TR 1 denotes the equally weighted

portfolio overlaid with stop-loss threshold 1%. Similarly for TR 2, etc. Delta 10 is the

hedged portfolio hedged using out-of-money Delta 10 put option, as in Jurek [2014]. Figure

C.1 in appendix plots the cumulative returns for Unhedged, Delta 10 and TR3 portfolio. We

compare stop-loss portfolios respectively to unhedged equally weighted portfolio and hedged

equally weighted portfolio. For simplicity we use the stop-loss portfolio with threshold 3%.

Compared to unhedged equally weighted portfolio, stop-loss portfolio has higher return, lower

volatility, more positive skewness and higher Sharpe ratio. The reduction in loss is achieved

without giving away return. This suggests momentum property of carry trade returns, as

documented in Burnside et al. [2011b].

Compared to 10δ hedged portfolio, the TR3 portfolio has almost the same mean return,

smaller volatility, roughly the same skewness and higher Sharpe ratio.

3.4.3 Who would buy the insurance?

We find that stop-loss overlays of certain threshold values decrease the average monthly

return. Similar to Section 3.5 we consider the utility gain for a trader who has CRRA utility

function. We compute the break-even risk-aversion coefficient for which trader is indifferent

between taking an additional position in hedging portfolio or not. For the setup of the model

please refer to Section 3.3. The minimal risk-aversion coefficient we test is 2. We show in


Table 3.6: Carry Trade Hedged using Stop-loss Rules

Mean Std Skewness Kurtosis Min Max Sharpe

TR 1 2.02 4.87 1.57 7.16 -1.77 7.11 0.41

TR 2 2.72 5.57 0.90 4.66 -3.11 6.83 0.49

TR 3 3.38 5.87 0.43 4.00 -4.14 6.69 0.58

TR 4 3.38 6.01 0.22 4.08 -4.36 6.48 0.56

TR 5 3.43 6.32 -0.03 4.21 -5.58 6.17 0.54

TR 6 3.15 6.45 -0.23 4.67 -6.07 6.41 0.49

Unhedged 3.23 6.64 -0.63 6.51 -8.73 6.47 0.49

10 Delta 3.17 7.17 0.28 5.01 -5.92 7.94 0.44

Table 3.6 reports summary statistics for carry trade portfolios with stop-loss overlay. TR 1 is the stop-loss

portfolio with 1% as stop-loss threshold, similar for TR 2, TR 3, etc. Unhedged is the unhedged equally

weighted G10 carry trade portfolio. Delta 10 is the hedged portfolio hedged using Delta 10 put option. The

time period is from January 1999 to June 2012.

Table 3.7 the results.


TR 1 TR 2 TR 3 TR 4 TR 5 TR 6

0.1 2 2 2 2 2 2

0.3 2 2 2 2 2 2

0.5 2 2 2 2 2 2

0.7 2 2 2 2 2 2

0.9 2 2 2 2 2 2

Table 3.7 computes empirically the break-even risk-aversion coefficient for which trader is indifferent between

including a specified position of stop-loss strategy in the portfolio or not. The time period is from January

1999 to June 2012.

The results show that all traders of risk-aversion coefficients higher than 2 improve their

utility by taking positions in the stop-loss strategy. This is in contrast to currency option

or VIX strategy. For the latter case only traders of high risk-aversion coefficients improve

utility by taking hedging positions. We thus show that stop-loss strategy is a more efficient


hedging strategy.

3.4.4 Robust check

Our test shows that stop-loss portfolio with threshold 3%(TR3) has higher Sharpe ratio and

more postive skewnwess than unhedged portfolio. In this subsection we test if the results

are robust. We use resampling bootstrap, the p-value that TR3 has lower or equal Sharpe

ratio than unhedged portfolio is 0.1. The p-value that TR3 has more negative skewness than

unhedged portfolio is 0. We thus have strong belief that stop-loss rule reduces losses and

improves Sharpe ratio.

We do the same test for TR3 versus hedged Delta 10 portfolio. The p-value that TR3 has

lower or equal Sharpe ratio than hedged Delta 10 portfolio is 0.37. The p-value that TR3 has

more negative skewness than hedged Delta 10 portfolio is 0.39. There is thus no significance

for the comparison of financial performance between stop-loss strategy and Delta 10 hedging

strategy.

3.5 CTA trend following as a hedging strategy

The fourth and last hedging strategy we consider is CTA. As we pointed out previously

carry trade is prone to volatility risk, and volatility tends to spike after significant market

movement. Brunnermeier et al. [2008] argues that the excess return of carry trade portfolio

is compensation for crash risk, which may correlate with losses in other markets. We thus

suggest to hedge carry trade portfolio with a market neutral strategy. Previous research on

CTA strategies has pointed out the following two stylized facts:

1) CTA payoff is a nonlinear function of market payoff; it’s not correlated with market return.

2) The Beta of CTA strategy is small. The Beta is positive when the market goes up; it is

negative when the market goes down.

These two characteristics make CTA an ideal hedging strategy. Empirically, we find that CTA

offers the best hedging by all criteria (detailed below) among the four hedging strategies. We

construct an equally volatility-weighted portfolio comprising carry trade portfolios and CTA

portfolios. Compared to unhedged carry trade portfolios, it increases the average monthly


return by 32% and reduces monthly return volatility by 31%. Thus, it increases the Sharpe

ratio by 93%, and reduces the absolute size of minimum monthly return by 63%, and the

minimal drawdown by 67%.

In the first part of this section, we detail the portfolio construction rule for CTA. In the

second part, we study CTA as a carry trade hedging instrument. For an interesting question

related to CTA, in the subsection 3.7.2 we study the risk/return analysis for CTA strategy.

3.5.1 Constructing CTA portfolio

We construct a CTA trend following strategy which has the following features:

1) Long/Short portfolio: Portfolio takes long or short positions in different assets.

2) Risk management in two levels: a) In the portfolio level: portfolio volatility is controlled.

b) In individual asset level: individual asset volatility is also controlled.

Our investment universe comprises 74 liquid commodity futures. All major asset classes are

included: equity, bond, FX and commodity. We note Pi(t) the price of future i at time t.

We have 32 equity index futures, 11 bond futures, 25 commodity futures and 10 currency

forwards. Equity index futures include benchmark indices of major developed countries.

Bond futures include those of US, UK, Germany, Japan and Australia. Commodities futures

include the most liquid commodity futures. Currency forwards include all G10 currencies

versus the US dollar.

We specify now the portfolio construction rule. The portfolio construction depends on two

hyper parameters: target volatility of portfolio σ∗ and lookback period τ .

1. The lookback period τ is fixed to be 6 months, τ = 6Month. The signal given by the

trend of the lookback period is Si(t) = sign(Pi(t)−Pi(t− τ)), the sign of portfolio’s position

of asset i is determined by Si(t).

2. The absolute size of position of asset i is determined by two criteria: a) The overall

volatility of portfolio is equal to the targeted volatility. b) The volatility of individual asset

is equal across different assets. Note σi(t) the volatility estimated at time t for asset i, Σt the

estimated covariance matrix of all futures at time t. We thus have that the position ωi(t) of

asset i at time t satisfies the following condition:

ωi(t) = k ∗ sign(Pi(t)− Pi(t− τ))/σi(t) (3.5)


Where k is a constant such that: √ω(t)TΣtω(t)T = σ∗ (3.6)

From 3.6 we can solve k and deduce the position of individual assets.

3. The portfolio is rebalanced each week. We set transaction cost to be at 0.05% for all assets,

which is a conservative estimation of transaction cost, as our assets are all liquid futures.

3.5.1.1 Simulation of constructed CTA portfolio

We report in the table below the summary statistics of constructed CTA portfolio from

June 2000 to January 2014. We plot in Figure 3.3 the total return index of simulated CTA

portfolio.

Table 3.8: CTA Performance Statistics

Mean Volatility Skewness Kurtosis Minimum Maximum Sharpe

CTA 7.96 14.94 0.30 2.80 -9.04 12.57 0.53

Table 3.8 reports summary statistics for simulated CTA portfolio. The time period is from June 2000 to

January 2014.

3.5.2 CTA as a hedging strategy

We study in this subsection the hedging performance of CTA. We compute the Sharpe ratio,

drawdown and minimum monthly return of hedging strategies versus unhedged strategies.

We report in the following table financial performance indicators of a portfolio which has

90% position in unhedged strategy and 10% position in CTA strategy.

We see that compared to unhedged carry trade portfolio, CTA hedged portfolio has higher

average return, lower return volatility, higher Sharpe ratio, lower drawdown and lower mini-

mum monthly return.

3.5.2.1 Who would buy the insurance?

Similar to Section 3.3 we consider the utility gain for a trader who has CRRA utility function.

We compute the break-even risk-aversion coefficient for which trader is indifferent between


Table 3.9: Carry Trade Hedged using CTA

Sharpe Rlt ∆ Mean Rlt ∆ Std Rlt ∆ DD Rlt ∆ Min Rlt ∆

Unhedged 0.413 0.224 1.877 -22.763 -5.548

CTA hedged 0.499 20.773 0.246 9.678 1.705 -9.187 -17.453 -23.330 -5.011 -9.681

Table 3.9 reports summary statistics for unhedged and hedged carry trade portfolios. The construction rule

is detailed in the text. Hedged portfolio has 90% position in unhedged strategy and 10% position in CTA

strategy. The time period is from June 2000 to June 2012. Sharpe is the Sharpe ratio. Mean is the average

monthly return. Std is the volatility of monthly return. DD is the minimum drawdown. Min is the minimum

monthly return.

taking an additional position in hedging portfolio or not. For the setup of the model please

refer to Section 3.3. The minimal risk-aversion coefficient we test is 2. We show in Table

3.10 the results. The results show that all traders of risk-aversion coefficients higher than


CTA

0.1 2

0.3 2

0.5 2

0.7 2

0.9 2

Table 3.10 computes empirically the break-even risk-aversion coefficient for which trader is indifferent

between including a specified position of CTA strategy in the portfolio or not. The time period is from June

2000 to January 2014.

2 improve their utility by taking positions in the CTA strategy. This is the same result as

for the stop-loss strategies. Recall that for currency option or VIX strategy, only traders of

high risk-aversion coefficients improve utility by taking hedging positions. We thus show that

CTA strategy is a more efficient hedging strategy than currency options or VIX.


3.6 Out-Sample Analysis

In this section we do an out-sample analysis for all hedging strategies. We divide the whole

sample period into two sub-periods. The in-sample period is from May 2000 to December

2007. The out-sample period is from January 2008 to June 2012. We first calibrate the

optimal hedging ratio using in-sample data. We then test the hedging portfolio using the

out-sample data.

We use Sharpe ratio as the objective function to compute optimal hedging ratio for in-sample

data. We report in Table 3.11 the optimal hedging ratios for the in-sample period.

Table 3.11: Optimal Hedging Ratio

Delta 50 Delta 25 Delta 10 VIX SL 1 SL 2 SL 3 SL 4 SL 5 CTA

OHR 0.00 0.00 0.00 0.00 0.20 0.25 1.00 0.00 0.00 0.12

This table reports the optimal hedging ratios for various hedging strategies.

We report next in Table 3.12 financial performance of hedged portfolio for the in-sample

period.

Table 3.12: In-sample Hedging Performance

Mean Volatility Skew Kurtosis Min Max Sharpe

Unhedged 4.84 4.41 -0.17 2.96 -2.77 3.87 1.10

SL1 4.63 4.18 0.07 3.08 -2.22 4.07 1.11

Gain -4.32 5.15 19.95 5.21 0.88

SL2 4.75 4.30 0.01 2.97 -2.24 4.10 1.10

Gain -1.84 2.48 18.95 5.97 0.66

SL3 5.07 4.44 0.03 3.18 -2.47 4.48 1.14

Gain 4.71 -0.71 120.11 7.38 10.61 15.59 3.97

CTA 7.55 5.80 -0.18 2.41 -3.25 3.84 1.30

Gain 55.83 -31.48 -8.91 -18.55 -17.58 -0.94 18.52

The table presents the in-sample financial performances of hedged portfolios. Gain denotes the improvement

in percentage relative to unhedged portfolio. The in-sample period is from May 2000 to December 2007.


Based on in-sample Sharpe ratio, stop-loss hedging with threshold 3% is chosen among

all stop-loss strategies. We report in Table 3.13 the financial performance of hedged portfolio

for out-sample period.

Table 3.13: Out-sample Hedging Performance

Mean Std Skewness Kurtosis Min Max Sharpe

Unhedged -0.69 9.72 -0.35 3.93 -8.73 6.47 -0.07

SL3 -0.42 8.00 0.76 3.27 -4.14 6.69 -0.05

Gain 39.90 17.71 52.54 3.37 26.96

CTA 0.66 9.68 -0.08 3.11 -6.57 7.49 0.07

Gain 195.79 0.38 24.80 15.69 196.15

The table presents the out-sample financial performances of hedged portfolios. Gain denotes the

improvement in percentage relative to unhedged portfolio. The out-sample period is from January 2008 to

June 2012.

We see that the hedging strategies have robust performance for out-sample period.

3.7 Two Issues Related to the Hedging Strategies

Here we study two issues related to the hedging instruments. In the first part of this section,

we compare the VIX strategy to various currency option strategies. The objective is to de-

termine if VIX is a cheaper form of systematic insurance as compared to currency options.

Caballero and Doyle [2012] studies this question and argues that compared to VIX, currency

options provide a cheap form of systematic insurance. We tackle the issue by comparing

the effects of different hedging strategies on the carry trade portfolio. First we compare the

financial performance of VIX hedging with that of currency options hedging. Secondly, we

look at the utility gain achieved by traders employing different hedging strategies. Lastly, we

study the tail hedging effects.

Overall, our findings, with respect to both financial performance and tail risk hedging quality,

suggest that Delta 10 is the least expensive in terms of hedging, followed by Delta 25 and

VIX, while Delta 50 is the most expensive. This result helps to shed light on the mispricing


issue with vis-a-vis FX options, and makes a contribution to the literature testing crash risk

explanations of carry trade excess returns.

In the second part of this section, we study the risk-return analysis of CTA strategy. We

provide a new methodology for replicating the returns to a benchmark CTA index. We find

that the risks and returns of CTA benchmark index can be matched with simple trend follow-

ing strategy. The trend following strategy provides transparency over its risk management

and trend-picking period. Fung and Hsieh [2001] provides evidence that the non-linear risk

exposure of CTA strategy resembles that of lookback straddle options. However, this paper

doesn’t provide evidence that the CTA funds implement momentum strategy through look-

back straddle options. We construct a CTA index by specifying technical trading rules, and

we find that our CTA index replicates with statistical significance the return of the bench-

mark CTA index. The replication delivers high R2. The constructed CTA index also does a

superior job of matching the drawdown patterns of the benchmark CTA index. Some previ-

ous papers construct passive CTA index by specifying technical trading rules; see Schneeweis

et al. [2001]. Our innovation is to replicate the benchmark CTA index by portfolio generation

using explicit rules.

3.7.1 Compare the price of VIX futures to currency options

After studying the hedging effects of currency options and VIX, we compare now the VIX

strategy to various currency option strategies. The objective is to determine if VIX is a

cheaper systematic insurance compared to currency options. Caballero and Doyle [2012]

studies this question; they argue that, compared to VIX, currency options provide a cheap

form of systematic insurance. We tackle the issue by comparing the hedging effects of different

hedging strategies on the carry trade portfolio. In doing this, we extensively draw on previous

results. First we compare the financial performance statistics in 3.3.2. We look at the hedging

effects on minimum drawdown, minimum monthly return, return volatility, average return

and Sharpe ratio, and we find that currency options and VIX offer the same return smoothing

and tail hedging effect. While Delta 10 is the least expensive, Delta 50 is the most expensive,

with Delta 25 and VIX in between. Secondly, we study in the framework of the investor’s

utility maximization problem, as in 3.3.3. The result shows that currency options are more


expensive than VIX for a currency trader.

Thirdly, we study the tail hedging effects. We compare the hedging returns for the ten months

during which carry trade suffered the most substantial losses, as in 3.3.4. We also propose

a measure called hedging quality. For a CRRA trader, we compute the average utility gain

during a crisis period, thanks to hedging pay-off. We also compute the average utility cost in

non-crisis periods due to constructing the hedging portfolio. We define the ratio of hedging

utility gain and hedging utility cost as hedging quality. We find that Delta 10 has the highest

hedging quality, followed by Delta 25, VIX and Delta 50. We also conduct a robustness test.

3.7.1.1 Financial performance comparison

We have the relevant results in Table C.1 in the appendix. We find that currency options

and VIX reduce, by roughly the same amount, minimum monthly return and monthly return

volatility, ranging respectively between 37% and 40%, and 13% and 15%. They are, however,

all costly to buy, while more out-of-money options are less costly on average. The Delta 10

strategy decreases the average return by 30%. Both Delta 25 and VIX decrease by 37%, while

Delta 50 decreases by 50%. More out-of-money options perform better in terms of Sharpe

ratio. The Delta 10 strategy decreases the Sharpe ratio by 19%. Delta 25 and VIX decrease

between 26% and 27%, and Delta 50 decreases by 41%. As the options are costly to buy,

the effect on minimum drawdown is less significant. The Delta 10 strategy reduces minimum

drawdown by 9%, VIX by 8%, Delta 25 by 4% and Delta 50 by 2%. In general, financial

performance statistics show that currency options and VIX offer the same return smoothing

and tail hedging effect. While Delta 10 is the least expensive, followed by Delta 25 and VIX,

Delta 50 is the most expensive.

3.7.1.2 Utility improvement for traders

We study in the framework of the investor’s utility maximization problem, as in 3.3.3.We

have the results in Table 3.4. We find that VIX delivers the lowest break-even risk-aversion

coefficient, followed by Delta 10, Delta 25 and Delta 50. It suggests that currency options

are more expensive than VIX for the CRRA utility trader.


3.7.1.3 Tail hedging effects

We compare now the tail hedging effects between different option hedging strategies and the

VIX future strategy. We recall the result in 3.3.4, where we compared the hedging returns

for the ten months during which carry trade suffered the largest losses. We found that Delta

50 gives the highest hedging return, followed by Delta 25, VIX and Delta 10.

We now propose a new measure called tail-hedging quality. For a CRRA trader, we compute

the average utility gain during a crisis period, thanks to hedging pay-off. We also compute the

average utility cost in non-crisis periods, due to constructing the hedging portfolio. We define

the ratio of hedging utility gain and hedging utility cost as hedging quality. We find that

Delta 10 has the highest hedging quality, followed by Delta 25, VIX and Delta 50. This result

is robust for traders with different risk-aversion coefficients. Using a resampling bootstrap,

we find it significant that the more out-of-money currency option has higher hedging quality,

and it is not significant that currency option strategies have higher hedging quality than VIX

strategy. We describe in the following the definition of tail hedging quality, the empirical

result and the robustness test.

Definition

We study the hedging gain and cost from the view of currency trader. We model the trader

to have CRRA utility function.

In a crisis period T + 1, assume the carry portfolio suffers loss of −rc, without hedging the

trader’s utility is thus U(ST (1− rc)). With hedging strategy hi the trader’s utility would be

U(ST (1− rc + rhi )), where rhi is the return of hedging strategy i. The utility gain of hedging

strategy i is thus :

GT+1i (rc, r

hi ) =

U(ST (1− rc + rhi ))− U(ST (1− rc))abs(U(ST (1− rc)))

= 1− (1− rc + rhi

1− rc)1−γ (3.7)

Denote by ΩT+1C the set of events corresponding to crisis at period T + 1. Given the joint

distribution FC(rc, rhi ) of rc, rhi conditional on ΩT+1

C , 3.7 computes the expected utility

gain of hedging strategy i conditional on ΩT+1C , which is:

ET [GT+1i |ΩT+1

C ] =

∫GT+1i (rc, r

hi ) dF (rc, r

hi ) =

∫1− (

1− rc + rhi1− rc

)1−γ dFC(rc, rhi ) (3.8)

We now compute hedging gains given the observation on portfolio returns during crisis period.

Suppose we observe M crisis periods, with carry trade portfolio loss rtc, hedging portfolio


return rh,ti for crisis period t, with 1 ≤ t ≤ M . Given the risk-aversion coefficient γ, the

average hedging gain of strategy i for M crisis periods is

Gi(γ) =

∑Mt=1 1− (

1−rtc+rh,ti

1−rtc)1−γ

M(3.9)

Similarly we compute the hedging cost. In a non-crisis period T+1, assume the carry portfolio

gains return of rc, hedging strategy i suffers loss of rhi . The utility cost of hedging strategy i

is thus :

CT+1i (rc, r

hi ) =

U(ST (1− rc))− U(ST (1− rc + rhi ))

abs(U(ST (1− rc)))= (

1 + rc − rhi1 + rc

)1−γ − 1 (3.10)

Denote by ΩT+1NC the set of events corresponding to non-crisis at period T +1. Given the joint

distribution FNC(rc, rhi ) of rc, rhi conditional on ΩT+1

NC . The following equation computes

the expected utility cost of hedging strategy i conditional on ΩT+1NC :

ET [CT+1i |ΩT+1

NC ] =

∫CT+1i (rc, r

hi ) dFNC(rc, r

hi ) =

∫(1 + rc − rhi

1 + rc)1−γ − 1 dFNC(rc, r

hi )

(3.11)

Again suppose we observe K non-crisis periods, with carry trade portfolio return rtc, hedging

portfolio loss rh,ti for non-crisis period t, with 1 ≤ t ≤ K. Given the risk-aversion coefficient

γ the average hedging cost of strategy i for M crisis periods is thus:

Ci(γ) =

∑Kt=1(

1+rtc−rh,ti

1+rtc)1−γ − 1

M(3.12)

We use the ratio of Gi(γ) and Ci(γ) to measure the hedging quality Qi of hedging strategy i.

Qi =Gi(γ)

Ci(γ)(3.13)

Result

Jackwerth [2000] and Bliss and Panigirtzoglou [2004] estimate the CRRA risk aversion coef-

ficient using equity option data. They find γ to be between 2 and 10. We report in Table

C.4 and Table C.5, in the appendix, the average utility gain for crisis periods and average

utility loss for non-crisis periods of different hedging strategies. We take into consideration

six months listed in Table C.3, where carry trade suffered the largest losses as a crisis period.

Another 86 months are considered as a non-crisis period. In computation, we assume that the


hedged portfolio takes one position in unhedged portfolios and one position in currency op-

tions or VIX future portfolios. The VIX future portfolio is normalized such that the volatility

of the VIX portfolio is the same as that of the Delta 25 currency option strategy.

We see that utility gain is an increasing function of risk aversion coefficient. Traders who

are more risk-averse value more the hedging gains. Across the different hedging strategies,

Delta 50 has the highest utility gain, followed by Delta 25, VIX and Delta 10, which has the

least utility gain. Utility cost is also an increasing function of risk aversion coefficient. Better

insurance thus demands higher prices. Across the different hedging strategies, Delta 50 has

the highest utility cost, followed by Delta 25, VIX and Delta 10, which has the lowest utility

cost. We report in Table 3.14 the hedging quality Qi, computed from 3.13.

Table 3.14: Hedging Quality

50 Delta 25 Delta 10 Delta VIX

γ = 2 4.80 6.59 8.30 6.18

γ = 3 4.68 6.48 8.21 6.05

γ = 4 4.57 6.36 8.12 5.93

γ = 5 4.46 6.25 8.04 5.81

γ = 6 4.36 6.14 7.96 5.70

γ = 7 4.26 6.03 7.87 5.59

γ = 8 4.16 5.93 7.79 5.48

γ = 9 4.06 5.83 7.71 5.37

γ = 10 3.97 5.72 7.63 5.27

Mean 4.37 6.15 7.96 5.71

Table 3.14 computes empirically the hedging quality for trader who takes one position in unhedged portfolio

and one position in hedging strategy. The VIX future portfolio is normalized such that the volatility of VIX

portfolio is the same as that of Delta 50 currency option strategy. The time period is from March 2004 to

June 2012.

We see that Delta 10 has the best hedging quality, followed by Delta 25 and VIX. Delta

50 has the lowest quality, which is 55% of that of Delta 10.

As a particularly interesting result, we find from Table 3.14 that a large variation of hedging


quality exists across currency options. The hedging quality increases as the options become

more out-of-money. Delta 50’s hedging quality is 55% of that of Delta 10. Delta 25’s is 79%

of that of Delta 10. VIX’s hedging quality is slightly below Delta 25. From the table, we

conclude that VIX’s hedging quality is better than Delta 50, worse than Delta 10, and almost

the same as Delta 25.

Robustness Test

In the last section, we conclude that hedging quality increases as options become more out-

of-money, and VIX has roughly the same hedging quality as the Delta 25 option. We now

test the robustness of our conclusion relative to varying risk aversion coefficients and different

samples of crisis periods.

From Table 3.14 we see that the relative order of hedging quality is robust to varying

risk-aversion coefficients. To test the robustness of the result relative to selection of crisis

period, we test for a larger sample of crisis period, including all ten months of Table C.3. We

compute the ratio of hedging qualities of all hedging strategies relative to that of Delta 10.

We then compare the ratios to the results of Table 3.14. We report in Table C.6 in appendix

the results. For example, the 1.06 for Delta 50 and γ = 3 is computed from Ratio10Ratio6

, where

Ratio10 = Q50DeltaQ10Delta

is the ratio of hedging qualities considering 10 months of Table 3.14 as

crisis period, and Ratio6 = Q50DeltaQ10Delta

is the ratio of hedging qualities taking 6 months of Table

3.14 where carry trade suffers biggest losses as crisis period. We see that the results are

robust, because the hedging quality ratio doesn’t change after using a larger sample of crisis

period.

Lastly we use bootstrap resampling to test robustness. We test hypothesis that the hedging

quality of one strategy is better than the othee one. We resample 1000 times. We report in

Table 3.15 the p-values. For example, the p-value of the hypothesis QDelta50 >= QDelta10 is

0.02.

From the above table we see that increasing hedging quality as option becomes more out-

of-money is robust. There is no significance for the comparison between VIX strategy and

currency option strategies.


Table 3.15: Bootstrap Resampling

VIX 25 Delta 10 Delta

50 Delta 0.29 0.01** 0.02**

VIX 0.45 0.33

25 Delta 0.10*

Table 3.15 reports the p-value that one strategy has better hedging quality than other one from bootstrap

resampling.

3.7.1.4 Discussions and Implications

Overall, the financial performance and tail risk hedging quality suggest that Delta 10 is the

least expensive in terms of hedging, followed by Delta 25 and VIX, while Delta 50 is the most

expensive. Our result helps to shed light on the mispricing issue of FX options, which is

important for the literature testing crash risk explanation of carry trade excess returns. We

approach the problem by comparing the expense of currency options to VIX futures, which

is a benchmark insurance against systematic risk. Following Brunnermeier et al. [2008], one

possible explanation for the excess return of carry trade portfolios is compensation for crash

risk. In order to test the effect of crash risk, Jurek [2014] constructs crash hedged carry trade

portfolios using out-of-money put options. By comparing returns of hedged and unhedged

carry trade portfolios, it concludes that crash risk could explain only 25% of carry trade excess

returns. One assumption of Jurek’s approach is that currency options are correctly priced. If

currency options were mispriced and implied crash risks were lower than the accurate crash

risk, then hedged carry trade portfolios could earn higher returns. Jurek [2014] discusses

this issue by comparing the option implied variance and skewness with realized variance and

skewness. It finds variance premium and skewness premium for currency options, which

suggests that options are not mispriced. Our computation shows that, overall, VIX hedging

is as expensive as Delta 25 hedging, and more expensive than Delta 10 hedging, while VIX

hedging is less expensive than Delta 50 hedging. Our evidence shows that the more in-the-

money Delta 25 and Delta 50 are not underpriced compared to VIX futures.


3.7.2 CTA risk/return analysis

We constructed a CTA portfolio with explicit rules in Section 3.5. We now show that the

benchmark CTA return index can be replicated by our simulated CTA portfolio. We use the

Newedge CTA Index as the benchmark index, which tracks the largest 20 (by asset under

management) CTAs. This is representative of the managed future funds. Funds selected for

Newedge CTA Index meet the following two criteria: 1) Funds are open to new investment.

2) Funds report returns on a daily basis.

We set the target volatility of the simulated portfolio such that the volatility of the portfolio

for the whole sample is the same as the volatility of the CTA benchmark index. We show in

Table 3.16 the summary statistics of the simulated CTA portfolio and Newedge CTA Index.

We plot in Figure 3.3 the total return indices of our simulated portfolio and the Newedge

CTA Index.

Table 3.16: CTA Performance Statistics

Mean Volatility Skewness Kurtosis Min Max Sharpe

Simulated 7.96 14.94 0.30 2.80 -9.04 12.57 0.53

Newedge 8.84 14.94 0.18 3.62 -13.45 14.61 0.59

Table 3.16 reports summary statistics for the simulated CTA portfolio and Newedge CTA index. The time

period is from June 2000 to January 2014.


Figure 3.3: CTA Total Return Indices

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 20150.5

1

1.5

2

2.5

3

3.5Total return indices

Simulated Portfolio

NewEdge

Figure 3.3 plots the total return indices of the simulated CTA portfolio and Newedge CTA index for the period

from June 2000 to January 2014.

From the graph above we see that the simulated portfolio and the benchmark index

comove. We study next the question of replicating CTA benchmark index using the simulated

CTA portfolio. We first run the following regression:

rrtb = βrrts + εt (3.14)

Where in the equation above rrtb is the return of the benchmark index, rrts is the return of

the simulated portfolio. We have the following result: We see β = 0.7 is significant at 0.01%

Table 3.17: Regression Analysis

value std

α · 100 0.06 0.05

β 0.67 0.02

R2 0.51

level. R-square is 0.52.

We construct a replication portfolio which has return rtrep = βrrts+(1− β)rrs at period t. We

plot next the total return indices of the benchmark portfolio and the replication portfolio.


Figure 3.4: Benchmark Index and Replication Portfolio

2000 2002 2005 2007 2010 2012 20150.5

1

1.5

2

2.5

3

3.5Total return indices

NewEdge

Replication

Figure 3.4 plots the total return indices of Newedge CTA Index and the replication portfolio.

We see from the figure above that the replication portfolio replicates the return of the

benchmark portfolio. In addition the replication portfolio has lower volatility than the bench-

mark portfolio.

We conduct next out-sample analysis.

3.7.2.1 Out-sample Analysis

In this subsection we do the out-sample analysis for CTA benchmark return index replication.

We first look at the out-sample behavior of the return’s replication.

Replication of return

We divide the whole sample to the in-sample and the out-sample periods. The in-sample

period is from Juanary 2000 to July 2009. The out-sample period is from August 20009 to

January 2014. We estimate the β by using Eq. 3.16 for respectively the in-sample period(βIS)

and the out-sample period((βOS)). We then test the H0 hypothesis that βIS 6= βOS . We

report in Table 3.18 the results.

We see from Table 3.18 that the H0 hypothesis βIS 6= βOS is rejected with p-value 0.51. We

next construct the out-sample replication portfolio rtrep,OS = ˆβISrrts,OS + (1− ˆβIS)rrs,OS by

using the βIS estimated using the in-sample data. We plot in Fig. 3.5 the total return indices

of the replication portfolio and Newedge index for the out-sample period.


Table 3.18: Replication Residue

β std R2 t-Stat(H0) p-value(H0)

In Sample 0.69 0.03 0.51

Out Sample 0.66 0.04 0.55 -0.66 0.51

Std is the standard deviation of β. p-value is for the hypothesis that βIS 6= βOS .

Figure 3.5: Out-Sample Return Replication

2009 2010 2011 2012 2013 2014 20150.9

0.95

1

1.05

1.1

1.15

1.2

1.25Out Sample Total return indices

NewEdge

Replication

Figure 3.5 plots the total return indices of Newedge CTA Index and the replication portfolio.

We see in Fig. 3.5 that the out-sample replication portfolio tracks well the benchmark

total return index. We next do similar out-sample analysis for the drawdown time series.

Replication of drawdown

We define the drawdown seris of a return time series as the following:

ddt = min(0, rt) (3.15)

We denote the drawdown series of the benchmark index return as ddtb, the drawdown series

of the simulated portfolio return as ddts. We look at the replication behavior for drawdown

series, where we run the following regression respectively for the in-sample data and the

out-sample data.

ddtb = βddts + εt (3.16)

We then test the H0 hypothesis that βIS 6= βOS . We report in Table 3.19 the results.


Table 3.19: Out-Sample Drawdown Replication

β std R2 t-Stat(H0) p-value(H0)

In Sample 0.59 0.06 0.30

Out Sample 0.62 0.07 0.42 0.44 0.66

Std is the standard deviation of β. p-value is for the hypothesis that βIS 6= βOS .

We see from Table 3.19 that the H0 hypothesis βIS 6= βOS is rejected with p-value 0.66.

The out-sample analysis shows that the replication is robust. We conduct next factor analysis

of CTA risk/return.

3.7.2.2 Explanation using traditional factors

We test the explanation of CTA benchmark index using traditional factors. In Table 3.20 we

report the result for Fama-French’s three factors model and Fung-Hsieh’s five factors model.

We also report the results of combining the above model with our rule based portfolio. We see

that traditional factors models don’t explaint the CTA benchmark index return. The loadings

on the rule based portfolio are significant in models combining traditional factos and rule-

based factor. In the table Mkt-Rf, SMB and HML are Fama-French’s three factors(Fama and

French [1993]). PTFSBD, PTFSFX, PTFSCOM, PTFSIR and PTFSSTK are Fung-Hsieh’s

five factors.


Table 3.20: Explanation Using Traditional Factors and Simulated CTA Index

1 2 3 4 5 6 7

Intercept 0.83 0.70 1.03 0.25 0.22 0.17 0.30

0.34 0.34 0.36 0.26 0.26 0.27 0.28

Mkt-RF -0.14 -0.15 0.04 0.02

0.07 0.08 0.06 0.06

SMB 0.10 0.13

0.13 0.10

HML 0.19 0.00

0.11 0.08

PTFSBD 0.03 0.04

0.04 0.03

PTFSFX -0.01 -0.03

0.03 0.02

PTFSCOM 0.06 0.04

0.04 0.03

PTFSIR -0.00 -0.01

0.02 0.02

PTFSSTK 0.05 -0.02

0.04 0.03

RuleBased 0.65 0.66 0.67 0.66

0.06 0.06 0.06 0.06

R2 0.02 0.04 0.05 0.44 0.44 0.45 0.46

The table reports coefficients from monthly return regressions under several risk models over the period from

Junuary 2000 to June 2014. The dependent variable is the return of Newedge CTA index. Specification 1

corresponds to a CAPM-style model with a single factor computed as the S&P500. Specification 2

corredponds to the Fama-French model. Specification 3 corresponds to Fung-Hsieh’s 5-factor model.

Specification 4 corresponds to the model of rule based CTA. Specification 5 corresponds to Specification 1

plus the rule based CTA model. Specification 6 corresponds to Specification 2 plus the rule based CTA

model. Specification 7 corresponds to Specification 3 plus the rule based CTA model.


3.8 Conclusion

In this chapter, I have studied the hedging problem associated with currency carry trade. I

have proposed theoretical frameworks and divided hedging instruments into three categories

according to the expected return of the hedging instrument and its correlation with the carry

trade: insurance, technical rule, and the market neutral strategy.

I then proposed and empirically tested four hedging strategies: FX options strategy, VIX

future strategy, “Stop-loss” rule and CTA strategy. Based upon empirical evidence from

2000 to 2012, I have found that CTA is the preferred hedging strategy because it upgrades

both return and volatility. The stop-loss strategy reduces risk. Both the currency options

strategy and the VIX future strategy offer good hedges against tail risk, while also reducing

volatility. However they worsen significantly the portfolio’s average return.

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311–344, 2005.

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APPENDIX A. APPENDIX FOR CHAPTER 1 121

Appendix A

Appendix for Chapter 1

Table A.1: Variance Decomposition Error Bands

1Y Bond

rr ErrBand g ErrBand π ErrBand u ErrBand

1Month 0.00 [0.00,0.04] 0.01 [0.00,0.07] 0.31 [0.21,0.41] 0.67 [0.54,0.77]

1Y 0.01 [0.00,0.14] 0.23 [0.04,0.45] 0.56 [0.32,0.71] 0.21 [0.13,0.34]

5Y 0.32 [0.02,0.66] 0.28 [0.02,0.59] 0.32 [0.13,0.64] 0.08 [0.04,0.23]

5Y Bond


1Month 0.02 [0.00,0.07] 0.01 [0.00,0.04] 0.93 [0.87,0.95] 0.04 [0.03,0.06]

1Y 0.16 [0.03,0.41] 0.02 [0.00,0.18] 0.80 [0.51,0.91] 0.02 [0.01,0.04]

5Y 0.65 [0.07,0.79] 0.03 [0.00,0.26] 0.31 [0.17,0.78] 0.01 [0.00,0.03]

10Y Bond


1Month 0.06 [0.02,0.13] 0.01 [0.00,0.06] 0.92 [0.84,0.97] 0.00 [0.00,0.00]

1Y 0.31 [0.11,0.53] 0.01 [0.00,0.10] 0.68 [0.44,0.86] 0.00 [0.00,0.00]

5Y 0.75 [0.17,0.85] 0.00 [0.00,0.13] 0.24 [0.14,0.74] 0.00 [0.00,0.00]

This table shows the error band for variance decomposition. ErrBand is the 90% error band computed

by using bootstrap. rr represents the real neutral rate, g represents the GDP gap, π represents the expected

inflation, and u represents the Taylor rule residue.


Figure A.1: Macro-factors in Taylor Rule


We plot below observed yields and model implied yields obtained in the section 1.7 for

bonds with maturity up to 10 years.

Figure A.2: Model Implied Yields and Observed Yields

1997 2000 2002 2005 2007 2010 2012 20150

1

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 20150

1

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 2015−1

0

1

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 20150

1

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 20150

1

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 20150

1

2

3

4

5

6


Model Implied Yield

Observed Yield


1997 2000 2002 2005 2007 2010 2012 20150

1

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 20150

1

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 20150

1

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 20151

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 20151

2

3

4

5

6


Model Implied Yield

Observed Yield

1997 2000 2002 2005 2007 2010 2012 20151

2

3

4

5

6


Model Implied Yield

Observed Yield


Figure A.3: Impulse-responses Error Bands

This figure plots the 90% error band for impulse-response.


Appendix 2

Table A.2: Market Price of Risk

r g π u

r 0.92 -0.52 -0.16 0.14

(1.82) (1.75) (0.61) (1.22)

g 0.54 0.09 -0.35 0.32

(0.62) (0.20) (1.02) (1.95)

π -0.02 0.23 -0.48 -0.00

(0.11) (2.70) (2.46) (0.08)

u -0.45 -0.35 0.40 -0.33

(2.32) (2.76) (1.47) (4.07)

The table presents the estimates for the price of risk of the essentially affine model. The absolute value

of the t-ratio of each estimate is reported. The sample period is from September 1997 to August 2012. The

standard deviation is corrected following Newey and McFadden(1994).

Table A.3: Market Price of Risk for In-sample Period

r g π u

r 1.31 -0.65 -1.40 0.32

(2.00) (1.44) (1.25) (1.48)

g -0.29 -0.20 0.77 -0.04

(0.27) (0.62) (0.90) (0.16)

π 0.03 0.54 -0.71 -0.01

(0.12) (3.10) (2.03) (0.29)

u -0.44 -0.18 -0.59 -0.39

(1.64) (0.60) (1.05) (2.23)

The table presents the estimates for the market price of risk of essentially affine model. The absolute

value of the t-ratio of each estimate is reported. The sample period is from September 1997 to June 2008.

The standard deviation is corrected following Newey and McFadden(1994).


Table A.4: Model Implied Factor Loadings and Unrestricted Regression Factor Loadings


1M Unres Reg 1.00 1.00 1.00 1.00

1M Over-Iden Model 1.00 1.00 1.00 1.00

3M Unres Reg 0.87 1.01 1.18 0.96

3M Over-Iden Model 0.89 1.01 1.07 0.96

6M Unres Reg 0.68 1.00 1.28 0.89

6M Over-Iden Model 0.77 0.99 1.18 0.90

1Y Unres Reg 0.53 0.88 1.38 0.75

1Y Over-Iden Model 0.63 0.91 1.35 0.79

2Y Unres Reg 0.41 0.63 1.68 0.55

2Y Over-Iden Model 0.50 0.67 1.58 0.59

3Y Unres Reg 0.39 0.47 1.68 0.42

3Y Over-Iden Model 0.45 0.46 1.69 0.44

4Y Unres Reg 0.36 0.33 1.65 0.31

4Y Over-Iden Model 0.42 0.30 1.71 0.33

5Y Unres Reg 0.33 0.19 1.62 0.21

5Y Over-Iden Model 0.39 0.18 1.68 0.25

6Y Unres Reg 0.33 0.12 1.58 0.18

6Y Over-Iden Model 0.36 0.10 1.61 0.18

7Y Unres Reg 0.32 0.05 1.54 0.15

7Y Over-Iden Model 0.33 0.04 1.52 0.13

8Y Unres Reg 0.30 0.01 1.47 0.10

8Y Over-Iden Model 0.30 0.01 1.43 0.09

9Y Unres Reg 0.27 -0.04 1.39 0.05

9Y Over-Iden Model 0.27 -0.02 1.34 0.05

10Y Unres Reg 0.24 -0.08 1.31 0.01

10Y Over-Iden Model 0.24 -0.04 1.24 0.02

The table presents the factor loadings of the over-identified model with that of the unrestricted regression.

For example, 1M Unres Reg is the factor loading from unrestricted regressions. 1M Over-Iden Model is the

factor loading of the over-identified model. The sample period is from September 1997 to August 2012.


Table A.5: Variance Decomposition

1Y Bond


1Month 0.02 0.07 0.28 0.64

1Y 0.02 0.13 0.56 0.29

5Y 0.52 0.12 0.25 0.10

5Y Bond


1Month 0.03 0.01 0.90 0.06

1Y 0.17 0.02 0.78 0.04

5Y 0.69 0.02 0.28 0.01

10Y Bond


1Month 0.09 0.00 0.91 0.00

1Y 0.34 0.00 0.65 0.00

5Y 0.77 0.00 0.23 0.00

Table A.6: In-sample Pricing Error

Observed Mean RMSE Rlt Error(%)

1M 3.18 0.00 0.00

3M 3.39 0.21 6.23

6M 3.52 0.28 7.98

1Y 3.62 0.26 7.20

2Y 3.88 0.24 6.25

3Y 4.05 0.20 4.93

4Y 4.22 0.17 3.94

5Y 4.39 0.16 3.65

6Y 4.48 0.17 3.74

7Y 4.58 0.18 3.89

8Y 4.67 0.18 3.92

9Y 4.77 0.18 3.87

10Y 4.86 0.19 3.99

The table reports the in-sample pricing error. The Observed Mean is the mean of observed bond yield.


RMSE is the root mean square error. Rlt Error is the RMSE divided by the Observed Mean. The sample

period is from September 1997 to June 2008.

Table A.7: Out-sample Pricing Error


1M 0.07 0.00 0.00

3M 0.10 0.13 125.32

6M 0.17 0.20 121.54

1Y 0.27 0.32 120.20

2Y 0.58 0.37 64.91

3Y 0.91 0.46 50.32

4Y 1.29 0.47 36.93

5Y 1.67 0.51 30.54

6Y 2.00 0.51 25.61

7Y 2.34 0.53 22.46

8Y 2.55 0.56 22.09

9Y 2.76 0.63 22.84

10Y 2.96 0.75 25.23

The table reports the out-sample pricing error. The Observed Mean is the mean of observed bond yield.


period is from July 2008 to August 2012.


Figure A.5: Contemporaneous Impulse Responses

0 20 40 60 80 100 120−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6


Annuliz

ed y

ield

in %

Initial response

RealRate

GDPGap

Inflation

TYRes

Figure A.6: Impulse Responses at Future Horizon

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes

0 10 20 30 40 50 60−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes


Appendix 3


r g π u

r 0.30 -0.43 -0.20 0.01

(1.39) (1.83) (1.18) (0.16)

g -0.09 -0.01 0.11 0.01

(0.27) (0.03) (0.31) (0.04)

π -0.65 0.90 0.15 0.03

(1.58) (2.00) (0.43) (0.20)

u 0.14 -0.36 -0.05 -0.07

(0.85) (2.20) (0.28) (0.61)





r g π u

r -0.33 -1.03 -0.80 0.11

(0.97) (2.25) (1.83) (0.45)

g -0.43 0.05 -0.12 0.18

(1.42) (0.07) (0.24) (2.04)

π 0.28 1.74 1.15 -0.11

(0.47) (2.31) (1.55) (0.28)

u -0.32 -0.33 -0.51 -0.33

(1.54) (1.02) (2.02) (1.71)







1M Unres Reg 1.00 1.00 1.00 1.00

1M Over-Iden Model 1.00 1.00 1.00 1.00

3M Unres Reg 1.07 1.03 1.05 1.02

3M Over-Iden Model 1.00 1.01 1.02 0.98

6M Unres Reg 1.05 1.03 1.06 0.99

6M Over-Iden Model 1.01 1.01 1.03 0.95

1Y Unres Reg 1.05 0.96 1.00 0.90

1Y Over-Iden Model 1.04 0.97 1.02 0.89

2Y Unres Reg 1.17 0.83 0.97 0.77

2Y Over-Iden Model 1.11 0.84 0.97 0.77

3Y Unres Reg 1.18 0.70 0.88 0.65

3Y Over-Iden Model 1.16 0.71 0.91 0.65

4Y Unres Reg 1.18 0.58 0.80 0.53

4Y Over-Iden Model 1.18 0.59 0.84 0.55

5Y Unres Reg 1.17 0.47 0.72 0.43

5Y Over-Iden Model 1.18 0.48 0.77 0.46

6Y Unres Reg 1.15 0.40 0.69 0.39

6Y Over-Iden Model 1.16 0.39 0.70 0.39

7Y Unres Reg 1.14 0.32 0.66 0.35

7Y Over-Iden Model 1.12 0.32 0.63 0.32

8Y Unres Reg 1.09 0.28 0.60 0.29

8Y Over-Iden Model 1.08 0.26 0.57 0.26

9Y Unres Reg 1.04 0.23 0.54 0.23

9Y Over-Iden Model 1.02 0.22 0.51 0.21

10Y Unres Reg 0.99 0.18 0.49 0.17

10Y Over-Iden Model 0.96 0.20 0.45 0.17






1Y Bond


1Month 0.00 0.03 0.14 0.83

1Y 0.01 0.05 0.27 0.67

5Y 0.02 0.20 0.30 0.48

5Y Bond


1Month 0.38 0.01 0.29 0.31

1Y 0.44 0.02 0.27 0.26

5Y 0.55 0.06 0.20 0.18

10Y Bond


1Month 0.68 0.01 0.25 0.05

1Y 0.76 0.01 0.17 0.05

5Y 0.86 0.02 0.09 0.03



1M 3.18 0.00 0.00

3M 3.39 0.20 5.81

6M 3.52 0.26 7.41

1Y 3.62 0.32 8.84

2Y 3.88 0.50 12.89

3Y 4.05 0.50 12.29

4Y 4.22 0.48 11.36

5Y 4.39 0.46 10.49

6Y 4.48 0.46 10.31

7Y 4.58 0.46 10.01

8Y 4.67 0.45 9.57

9Y 4.77 0.43 9.01

10Y 4.86 0.42 8.57







1M 0.07 0.00 0.00

3M 0.10 0.11 104.25

6M 0.17 0.23 140.68

1Y 0.27 0.41 154.14

2Y 0.58 0.54 93.46

3Y 0.91 0.65 71.72

4Y 1.29 0.65 50.59

5Y 1.67 0.63 37.66

6Y 2.00 0.56 28.10

7Y 2.34 0.49 21.04

8Y 2.55 0.45 17.62

9Y 2.76 0.40 14.55

10Y 2.96 0.35 11.83






0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Annuliz

ed y

ield

in %

Initial response

RealRate

GDPGap

Inflation

TYRes


0 10 20 30 40 50 60−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

Months

Annuliz

ed y

ield

in %


RealRate

GDPGap

Inflation

TYRes


Appendix 4


r π u

r 0.10 -6.43 -2.15

(0.07) (4.18) (1.47)

π 0.30 0.52 0.38

(1.38) (2.02) (1.67)

u -0.34 -0.49 -0.47

(1.61) (2.88) (3.06)





r π u

r 1.81 -3.85 -0.02

(4.73) (3.36) (0.08)

π -0.13 0.04 0.04

(1.47) (0.26) (1.50)

u -0.20 -0.65 -0.31

(0.81) (2.28) (3.24)






r π u

1M Unres Reg 1.00 1.00 1.00

1M Over-Iden Model 1.00 1.00 1.00

3M Unres Reg 0.97 1.18 1.00


6M Unres Reg 0.91 1.29 0.97


1Y Unres Reg 0.81 1.40 0.84

1Y Over-Iden Model 0.80 1.45 0.84

2Y Unres Reg 0.58 1.69 0.61


3Y Unres Reg 0.48 1.68 0.45


4Y Unres Reg 0.38 1.65 0.32


5Y Unres Reg 0.28 1.62 0.20


6Y Unres Reg 0.20 1.58 0.14


7Y Unres Reg 0.12 1.54 0.08


8Y Unres Reg 0.10 1.46 0.03


9Y Unres Reg 0.09 1.38 -0.01

9Y Over-Iden Model 0.08 1.38 -0.01

10Y Unres Reg 0.07 1.31 -0.05

10Y Over-Iden Model 0.05 1.30 -0.03






1Y Bond

r π u

1Month 0.00 0.29 0.71

1Y 0.02 0.62 0.36

5Y 0.53 0.35 0.13

5Y Bond

r π u

1Month 0.02 0.93 0.05

1Y 0.20 0.77 0.03

5Y 0.71 0.28 0.01

10Y Bond

r π u

1Month 0.04 0.96 0.00

1Y 0.29 0.71 0.00

5Y 0.75 0.25 0.00


Observed Mean Error Std Rlt Error(%)

1M 3.18 0.00 0.00

3M 3.39 0.24 7.17

6M 3.52 0.31 8.93

1Y 3.62 0.28 7.75

2Y 3.88 0.26 6.61

3Y 4.05 0.20 4.95

4Y 4.22 0.16 3.78

5Y 4.39 0.15 3.46

6Y 4.48 0.16 3.48

7Y 4.58 0.17 3.78

8Y 4.67 0.19 4.03

9Y 4.77 0.20 4.20

10Y 4.86 0.22 4.46






Observed Mean Error Std Rlt Error(%)

1M 0.07 0.00 0.00

3M 0.10 0.19 183.21

6M 0.17 0.44 268.27

1Y 0.27 0.63 235.58

2Y 0.58 0.64 110.25

3Y 0.91 0.67 73.95

4Y 1.29 0.63 49.17

5Y 1.67 0.61 36.33

6Y 2.00 0.56 27.86

7Y 2.34 0.53 22.46

8Y 2.55 0.52 20.43

9Y 2.76 0.52 18.84

10Y 2.96 0.52 17.59






0 20 40 60 80 100 120−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6


Annu

lized y

ield

in

%

Initial response

RealRate

Inflation

WRes


0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Months

An

nu

lize

d y

ield

in

%


RealRate

Inflation

WRes

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Months

An

nu

lize

d y

ield

in

%


RealRate

Inflation

WRes

0 10 20 30 40 50 60−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Months

An

nu

lize

d y

ield

in

%


RealRate

Inflation

WRes

APPENDIX B. APPENDIX FOR CHAPTER 2 144

Appendix B


Subsection 2.6.1

Proof for the maximization of Sharpe ratio for threshold portfolios:

E [rt+1(Ptfy)] =


e−λy= (y + 1/λ)

Thus :

Sh[rt+1(Ptfy)] = (y + 1/λ)e−12λy/σ

The derivative of Sh[rt+1(Ptfy)] is 12(1 − λy)e−

12λy/σ. Sh[rt+1(Ptfy)] is thus maximized at

y = 1λ .


Table B.1: Carry Trade Performance Statistics

Table B.1 reports summary statistics for currency carry trades implemented in G10

currencies. Summary statistics are reported over Jan.1990-Jun. 2012 (N = 270 months).

SR represents the Sharpe ratio. Means, volatilities and Sharpe ratios (SR) are annualized;

Min and Max report the smallest and largest observed monthly return.

AUD CAD CHF EUR GBP JPY NOK NZD SEK EQW

Mean 5.98 2.66 -0.61 3.59 0.64 1.22 2.59 4.72 4.65 3.40

Std 11.62 7.91 11.48 10.36 9.59 11.45 11.03 11.61 11.78 5.97

Skewness -0.54 0.74 -0.22 -0.31 -0.61 -0.43 -0.37 -0.22 -0.52 -0.81

Kurtosis 5.49 9.85 3.91 3.99 5.19 4.95 3.98 5.97 4.34 5.36

Minimum -16.12 -7.56 -13.04 -10.03 -11.88 -14.76 -11.91 -12.93 -14.44 -6.61

Maximum 9.59 14.84 11.69 9.74 8.86 11.41 7.99 14.25 9.56 5.18

SR 0.51 0.34 -0.05 0.35 0.07 0.11 0.24 0.41 0.39 0.57

N 270.00 270.00 270.00 270.00 270.00 270.00 270.00 270.00 270.00 270.00


Table B.2: Carry Trade Threshold Portfolios Performance Statistics

Table B.2 reports summary statistics for threshold portfolios of G10 currencies versus US

dollar. NrTr is the average number of trades in the portfolio.

Mean Std Skew Kurt Min Max Sharpe NrTr N

TR 0 3.40 5.97 -0.81 5.36 -6.61 5.18 0.57 9.00 270.00

TR 0.5 4.70 7.67 -0.60 6.78 -8.74 8.54 0.61 7.40 270.00

TR 1 5.27 8.42 -0.48 6.60 -10.41 9.42 0.63 6.10 270.00

TR 1.5 6.09 8.89 -0.66 6.71 -11.95 9.28 0.68 5.10 270.00

TR 2 6.81 9.49 -0.60 6.70 -12.73 11.37 0.72 4.20 270.00

TR 2.5 7.99 10.03 -0.42 5.59 -11.52 11.89 0.80 3.30 270.00

TR 3 7.32 10.39 -0.52 4.75 -10.79 9.59 0.70 2.50 270.00

TR 3.5 6.66 11.22 -0.52 5.00 -11.30 11.23 0.59 2.00 270.00

TR 4 7.31 10.77 -0.18 4.43 -9.57 11.41 0.68 1.60 270.00

TR 4.5 5.27 10.41 -0.58 6.71 -14.76 11.41 0.51 1.20 270.00

TR 5 1.04 8.03 -1.11 9.40 -13.25 7.44 0.13 0.70 270.00

TR 5.5 -0.14 5.93 -0.77 11.20 -8.73 7.53 -0.02 0.50 270.00

TR 6 0.67 5.16 -0.43 16.48 -8.26 7.83 0.13 0.40 270.00

Table B.3: Carry Trade Mean-variance Optimal Portfolios Performance Statistics

Table B.3 reports summary statistics for the long only mean-variance optimal portfolio, and

that of the equally weighted portfolio. Carry 1Y is the mean-variance portfolio using one

year window when estimating the currency correlations, Carry 2Y the one using two years.

The time period is from February 1992 to June 2012.

Mean Std Skewness Minimum Maximum Sharpe Ratio

EqWeighted 2.92 5.89 -0.76 -6.61 5.18 0.50

Carry 1Y 2.60 4.81 -0.57 -6.26 4.14 0.54

Carry 2Y 2.76 4.96 -0.69 -5.62 4.00 0.56


Table B.4: Carry Trade Threshold Portfolios Performance Statistics: Quantile Based Rule

Table B.4 reports summary statistics for threshold portfolios of G10 currencies versus USD.

Portfolios are constructed using ”quantile based threshold rule”. Portfolio1 contains the

carry trade having biggest forward premium. Portfolio2 contains 2 carry trades having

biggest forward premiums, etc.


Portolio 1 5.80 12.46 -0.57 4.85 -14.23 9.79 0.47 1.00 270.00

Portolio 2 5.41 11.23 -0.38 4.96 -12.13 11.64 0.48 2.00 270.00

Portolio 3 5.59 10.14 -0.55 5.66 -13.35 9.28 0.55 3.00 270.00

Portolio 4 4.45 9.37 -0.60 5.96 -12.27 8.38 0.48 4.00 270.00

Portolio 5 4.28 8.74 -0.62 6.28 -11.87 8.48 0.49 5.00 270.00

Portolio 6 4.41 8.28 -0.68 6.76 -11.48 8.12 0.53 6.00 270.00

Portolio 7 4.32 7.62 -0.50 5.84 -7.94 8.17 0.57 7.00 270.00

Portolio 8 3.58 6.68 -0.63 5.67 -7.10 6.71 0.54 8.00 270.00

Portolio 9 3.42 5.98 -0.81 5.32 -6.61 5.18 0.57 9.00 270.00


Table B.5: Carry Trade Crash-Hedged Threshold Portfolios Performance Statistics

Table B.5 reports summary statistics for 10 Delta Put hedged threshold portfolios of G10

currencies versus US dollar for the period from January 1999 to June 2012. NrTr is the

average number of trades in the portfolio.


TR 0 3.17 7.17 0.28 5.01 -5.92 7.94 0.44 9.00 161.00

TR 0.5 4.55 8.68 0.37 6.19 -7.16 10.41 0.52 6.92 161.00

TR 1 5.17 9.31 0.51 6.31 -7.68 11.41 0.56 5.57 161.00

TR 1.5 6.03 9.93 0.52 6.11 -7.93 11.41 0.61 4.63 161.00

TR 2 6.90 10.29 0.22 5.85 -9.25 11.41 0.67 3.73 161.00

TR 2.5 7.64 10.74 0.44 6.46 -9.25 13.25 0.71 2.65 161.00

TR 3 6.30 10.50 0.07 5.50 -9.25 10.81 0.60 1.89 161.00

TR 3.5 4.63 10.27 -0.27 5.83 -10.36 10.19 0.45 1.43 161.00

TR 4 4.87 10.21 -0.61 7.42 -12.49 10.19 0.48 1.05 161.00

TR 4.5 2.38 8.46 -1.45 13.08 -13.01 7.93 0.28 0.63 161.00

TR 5 -1.45 5.16 -5.12 42.94 -13.01 5.15 -0.28 0.25 161.00

TR 5.5 -0.50 2.87 -2.18 32.22 -5.81 5.15 -0.18 0.14 161.00

TR 6 -0.16 1.64 -6.80 60.86 -4.25 1.53 -0.10 0.09 161.00


Table B.6: Unhedged Carry Trade Threshold Portfolios Performance Statistics

Table B.6 reports summary statistics for unhedged threshold portfolios of G10 currencies

versus US dollar for the period from January 1999 to June 2012.

Mean Std Skew Kurt Min Max Sharpe N

TR 0 2.97 6.10 -0.57 5.14 -6.61 5.18 0.49 162.00

TR 0.5 5.03 8.61 -0.47 6.36 -8.74 8.54 0.58 162.00

TR 1 6.05 9.59 -0.40 6.00 -10.41 9.42 0.63 162.00

TR 1.5 7.27 10.19 -0.60 6.03 -11.95 9.28 0.71 162.00

TR 2 8.31 10.82 -0.57 6.19 -12.73 11.37 0.77 162.00

TR 2.5 9.24 11.14 -0.39 5.45 -11.52 11.89 0.83 162.00

TR 3 8.66 11.03 -0.43 4.86 -10.79 9.59 0.78 162.00

TR 3.5 7.29 11.48 -0.54 4.91 -11.30 9.59 0.63 162.00

TR 4 7.91 10.41 -0.11 4.63 -9.57 9.59 0.76 162.00

TR 4.5 4.88 9.04 -0.04 5.83 -8.92 9.59 0.54 162.00

TR 5 0.15 5.71 -0.32 9.58 -7.04 6.24 0.03 162.00

TR 5.5 -0.40 4.37 -0.70 14.92 -6.32 5.72 -0.09 162.00

TR 6 0.43 2.65 2.48 29.27 -3.23 5.72 0.16 162.00


Table B.7: Comparison of Hedged and Unhedged Carry Trade Threshold Portfolios

Table B.7 reports summary statistics for comparison of return, volatility and Sharpe ratio

between unhedged and hedged portfolios for the period from January 1999 to June 2012..

M H denotes the average return of hedged portfolio. M U denotes the average return of

unhedged portfolio. M Dif denotes the relative difference in percentage between M H and M

U. Vol H denotes the return volatility of hedged portfolio. Sh H denotes the Sharpe ratio of

hedged portfolio. Similarly Vol U, Vol Dif, Sh U, and Sh Dif are defined.

M H M U M Dif Vol H Vol U Vol Dif Sh H Sh U Sh Dif

TR 0 3.17 2.97 6.59 7.17 6.10 17.49 0.44 0.49 -9.28

TR 0.5 4.55 5.03 -9.63 8.68 8.61 0.85 0.52 0.58 -10.39

TR 1 5.17 6.05 -14.64 9.31 9.59 -2.97 0.56 0.63 -12.02

TR 1.5 6.03 7.27 -17.04 9.93 10.19 -2.51 0.61 0.71 -14.90

TR 2 6.90 8.31 -16.98 10.29 10.82 -4.89 0.67 0.77 -12.71

TR 2.5 7.64 9.24 -17.32 10.74 11.14 -3.65 0.71 0.83 -14.19

TR 3 6.30 8.66 -27.27 10.50 11.03 -4.89 0.60 0.78 -23.53

TR 3.5 4.63 7.29 -36.46 10.27 11.48 -10.56 0.45 0.63 -28.95

TR 4 4.87 7.91 -38.45 10.21 10.41 -1.98 0.48 0.76 -37.21

TR 4.5 2.38 4.88 -51.25 8.46 9.04 -6.47 0.28 0.54 -47.88

TR 5 -1.45 0.15 -1050.61 5.16 5.71 -9.62 -0.28 0.03 -1151.84

TR 5.5 -0.50 -0.40 26.04 2.87 4.37 -34.35 -0.18 -0.09 91.99

TR 6 -0.16 0.43 -137.83 1.64 2.65 -37.94 -0.10 0.16 -160.96


Table B.8: Comparison of Hedged and Unhedged Carry Trade Threshold Portfolios

Table B.8 reports summary statistics for relative strength of threshold portfolios versus

benchmark equally weighted portfolio in terms of Sharpe ratio for the period from January

1999 to June 2012. Hedged Sharpe Diff shows the difference of Sharpe ratio of hedged

threshold portfolios versus hedged equally weighted portfolio. Unhedged Sharpe Diff shows

the difference of Sharpe ratio of unhedged threshold portfolios versus unhedged equally

weighted portfolio. Relative change is the relative difference between Unhedged Sharpe Diff

and Hedged Sharpe Diff in percentage.

Hedged Sharpe Diff Unhedged Sharpe Diff Relative Change

TR 0.5 0.08 0.10 -15.97

TR 1 0.11 0.14 -21.33

TR 1.5 0.17 0.23 -27.01

TR 2 0.23 0.28 -18.68

TR 2.5 0.27 0.34 -21.19


Figure B.1: Cumulative G10 carry trade portfolio returns

1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 20120.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Figure B.1 illustrates the total return indices for G10 currency carry trade portfolio over the period from

January 1990 to June 2012.

Figure B.2: G10/USD currencies’ volatilities

1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 20120

0.005

0.01

0.015

0.02

0.025

0.03

AUD

CAD

CHF

EUR

GBP

JPY

NOK

NZD

SEK

Figure B.2 illustrates the G10/USD currencies’ volatilities from February 1990 to June 2012. The volatility

index is calculated by exponential weighted moving average formula: EMV 2t = α ∗ EMV 2

t−1 + (1 − α) ∗ r2t ,

where rt is the currency return, EMVt is the exponential weighted moving average volatility, α = 0.98.


Figure B.3: Threshold portfolios of G10 currencies versus US dollar: Part 1

1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 20120

1

2

3

4

5

6Threshold portfolios of G10 currencies versus US dollar

TR 0

TR 0.5

TR 1

TR 1.5

TR 2

TR 2.5


1987 1990 1992 1995 1997 2000 2002 2005 2007 2010 20120

1

2

3

4

5

6Threshold portfolios of G10 currencies versus US dollar

TR 2.5

TR 3

TR 3.5

TR 4

TR 4.5

TR 5

TR 5.5

TR 6

Figure B.3 and Figure B.4 illustrate the total return indices for threshold portfolios of G10 currencies versus

US dollar over the period from January 1990 to June 2012.


Figure B.5: Mean-variance Optimal Portfolio

1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20140.8

1

1.2

1.4

1.6

1.8

2

2.2Mean−variance optimal portfolio

EqWeighted

Carry 1Y

Carry SY

Figure B.5 illustrates the total return indices of two long only mean-variance optimal portfolio, and that of

the equally weighted portfolio. Carry 1Y is the mean-variance portfolio using one year window for estimating

the currency correlations, Carry 2Y the one using two years. The time period is from January 1990 to June

2012.


Figure B.6: 10 Delta Put hedged threshold portfolios of G10 currencies versus US dollar:

Part 1

1997 2000 2002 2005 2007 2010 20120.5

1

1.5

2

2.5

310Delta Put Hedged Threshold portfolios of G10 currencies versus US dollar

TR 0

TR 0.5

TR 1

TR 1.5

TR 2

TR 2.5

Figure B.7: 10 Delta Put hedged threshold portfolios of G10 currencies versus US dollar:

Part 2

1997 2000 2002 2005 2007 2010 20120.5

1

1.5

2

2.5

310Delta Put Hedged Threshold portfolios of G10 currencies versus US dollar

TR 2.5

TR 3

TR 3.5

TR 4

TR 4.5

TR 5

TR 5.5

TR 6

Figure B.6 and Figure B.7 illustrate the total return indices for 10 Delta Put hedged threshold portfolios of

G10 currencies versus US dollar over the period from January 1999 to June 2012.



1997 2000 2002 2005 2007 2010 20121

1.5

2

2.5

3

3.5Threshold portfolios of G10 currencies versus US dollar

TR 0

TR 0.5

TR 1

TR 1.5

TR 2

TR 2.5


1997 2000 2002 2005 2007 2010 20120.5

1

1.5

2

2.5

3

3.5Threshold portfolios of G10 currencies versus US dollar

TR 2.5

TR 3

TR 3.5

TR 4

TR 4.5

TR 5

TR 5.5

TR 6

Figure B.8 and Figure B.9 illustrate the total return indices for threshold portfolios of G10 currencies versus

US dollar over the period from January 1999 to June 2012.


Figure B.10: Unhedged and hedged portfolios of TR0 and TR2.5

1997 2000 2002 2005 2007 2010 20120.5

1

1.5

2

2.5

3

3.5Unhedged and hedged portfolios of TR0 and TR2.5

Hedged TR 0

Hedged TR 2.5

Unhedged TR 0

Unhedged TR 2.5

Figure B.10 illustrate the total return indices for unhedged and hedged threshold portfolios of threshold 0 and

2.5% over the period from January 1999 to June 2012.

Figure B.11: Empirical distribution of forward premiums and fitted exponential distribution

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x

)

Empirical CDF

Exp(2.23)

Forward Premium

Figure B.11 plots the CDF of the empirical distribution of the forward premiums and the fitted exponential

distribution having mean 2.23.


Figure B.12: Threshold portfolios’ Sharpe ratios versus random-walk model implied values

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.8

0.9

1

1.1

1.2

1.3

1.4

1.5Threshold portfolios Sharpe ratios versus model implied values

Threshold value

Sh

arp

e r

atio

Model implied

Empirical test

Figure B.12 plots the Sharpe ratio of G10 currency carry trade threshold portfolios versus the Sharpe ratios

implied by the random-walk model with λ = 0.45. The Sharpe ratios of zero threshold portfolios are normalized

to 1.


Figure B.13: Sharpe ratios of crash-hedged threshold portfolio with equal implied volatility

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Threshold Value

Port

folio

Sharp

e R

atio

Crash Hedged Threshold Portfolio

V10 Pre5

V10 Pre15

V15 Pre5

V15 Pre15

V20 Pre5

V20 Pre15

Figure B.14: Sharpe ratios of crash-hedged threshold portfolio with variable implied volatility

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.051

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Threshold Value

Port

folio

Sharp

e R

atio

Crash Hedged Threshold Portfolio, Variable Volatility

V10 Pre5

V10 Pre15

V15 Pre5

V15 Pre15

V20 Pre5

V20 Pre15

Figure B.13 and B.14 plot the Sharpe ratio of simulated threshold portfolios. We simulate 1000 carry trades

for 500 periods. The funding currency interest rate has uniform distribution of [0, 2%]. The forward premium

has exponential distribution with mean 2%. The currency volatility is 20%. Figure B.13 plots the case for

equal implied option volatility of 24%. Figure B.14 plots variable implied volatility as explained in Section

6.4.2. Vol 10 Pre5 means currency volatility of 10% and implied volatility premium of 5% against historical

volatility. Similarly for others.

APPENDIX C. APPENDIX FOR CHAPTER 3 160

Appendix C


Table C.1: Carry Trade Hedged using Options or VIX Futures

Sharpe Rlt ∆ Mean Rlt ∆ Std Rlt ∆ DD Rlt ∆ Min Rlt ∆

Carry 0.418 0.227 1.883 -23.836 -8.731

Delta 10 0.337 -19.402 0.159 -30.120 1.632 -13.298 -21.729 -8.843 -5.526 -36.714

Delta 25 0.309 -26.177 0.144 -36.613 1.616 -14.136 -22.773 -4.462 -5.416 -37.967

VIX 0.306 -26.790 0.142 -37.376 1.610 -14.460 -21.986 -7.762 -5.263 -39.725

Delta 50 0.249 -40.526 0.114 -49.673 1.593 -15.380 -23.434 -1.690 -5.525 -36.720

Table C.1 reports summary statistics for carry trade portfolio hedged with currency options and VIX future.

The time period is from June 2000 to June 2012. VIX future data is missing from June 2000 to February

2004. For that period VIX future return is replicated from currency options. Carry represents the unhedged

carry trade portfolio.


Table C.2: Carry Trade Hedged using Options or VIX Futures: Change in Performance

Der Sh Der Mean Der Std Der DD ∆dd/∆mean Der Min ∆Min/∆mean

Delta 50 -0.0149 -0.0087 -0.0185 -0.0957 -11.0339 0.1028 11.8509

Delta 25 -0.0103 -0.0061 -0.0165 -0.0296 -4.8939 0.1137 18.7798

Delta 10 -0.0069 -0.0041 -0.0148 0.0748 18.1641 0.1027 24.9451

VIX -0.0097 -0.0057 -0.0165 0.0491 8.5626 0.1290 22.5209

Table C.2 reports estimation for the derivatives of financial performance indicators. Der Sh is the derivative

of Sharpe Ratio. Der Mean is the derivative of average monthly return. Der Std is the derivative of monthly

return volatility. Der DD is the derivative of minimum drawdown. Der Min is the derivative of minimum

monthly return.

Table C.3: Hedging Effects of Options and VIX Futures for Crisis Periods


Oct-2008 -6.55 4.34 3.12 1.41 3.38

May-2010 -5.54 2.47 1.71 0.66 1.62

Sep-2011 -4.48 2.88 2.15 0.63 0.89

Aug-2008 -4.46 3.44 3.08 2.30 -0.39

May-2012 -4.35 2.41 1.92 1.05 0.94

Sep-2008 -3.17 0.16 -0.19 -0.21 2.06

Aug-2010 -2.61 0.26 -0.37 -0.21 -0.28

Jan-2009 -2.43 0.51 0.35 -0.23 0.77

Apr-2006 -2.24 1.51 1.15 0.51 -0.29

Nov-2008 -2.16 -0.39 -1.17 -0.51 0.14

Mean -3.80 1.76 1.17 0.54 0.88

Mean of 6 Months -4.76 2.62 1.96 0.97 1.42

Number of Negative Return 1 3 4 3


Table C.4: Average utility gain in crisis period


γ = 2 2.67 2.02 1.01 1.46

γ = 3 5.25 3.98 2.00 2.88

γ = 4 7.74 5.89 2.97 4.26

γ = 5 10.16 7.75 3.93 5.62

γ = 6 12.49 9.56 4.87 6.94

γ = 7 14.75 11.32 5.80 8.22

γ = 8 16.93 13.04 6.71 9.48

γ = 9 19.04 14.71 7.61 10.71

γ = 10 21.09 16.34 8.49 11.91

Mean 12.24 9.40 4.82 6.83

Table C.4 computes empirically the average utility gain for trader who takes one position in unhedged

portfolio and one position in hedging strategy. The VIX future portfolio is normalized such that the

volatility of VIX portfolio is the same as that of Delta 25 currency option strategy. We take six months

listed in Table 5 as crisis period. The time period is from March 2004 to June 2012.


Table C.5: Average utility cost in non-crisis period


γ = 2 0.56 0.31 0.12 0.24

γ = 3 1.12 0.61 0.24 0.48

γ = 4 1.69 0.93 0.37 0.72

γ = 5 2.27 1.24 0.49 0.97

γ = 6 2.86 1.56 0.61 1.22

γ = 7 3.46 1.88 0.74 1.47

γ = 8 4.07 2.20 0.86 1.73

γ = 9 4.69 2.53 0.99 1.99

γ = 10 5.31 2.85 1.11 2.26

Mean 2.89 1.57 0.61 1.23

Table C.5 computes empirically the average utility loss for trader who takes one position in unhedged

portfolio and one position in hedging strategy. The VIX future portfolio is normalized such that the

volatility of VIX portfolio is the same as that of Delta 25 currency option strategy. The time period is from

March 2004 to June 2012.

Table C.6: Ratios of relative hedging qualities


γ = 2 1.05 0.98 0.95 1.00

γ = 3 1.06 0.97 0.94 1.00

γ = 4 1.06 0.97 0.94 1.00

γ = 5 1.06 0.97 0.94 1.00

γ = 6 1.06 0.97 0.94 1.00

γ = 7 1.06 0.97 0.94 1.00

γ = 8 1.06 0.97 0.94 1.00

γ = 9 1.06 0.97 0.94 1.00

γ = 10 1.06 0.97 0.94 1.00

Table C.6 reports the ratio of hedging quality taking ten months as crisis period and that taking six periods

as crisis period.


Figure C.1: Cumulative return of portfolios

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 20130.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8Stop Loss vs Unhedged vs Hedged portfolio

TR3

Unhedged B

Hedged B

This figure plots the total return indices of unhedged equally weighted portfolio(Unhedged B), hedged equally

weighted portfolio(Hedged B), and stop-loss portfolio with threshold 3%(TR3) for the period from January

1999 to June 2012.

Essays in Macro-Finance and International Finance

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