Essays on Foreign Exchange and Credit Risk

Nielsen, Andreas Bang Doctoral Thesis Essays on Foreign Exchange and Credit Risk PhD Series, No. 26.2018 Provided in Cooperation with: Copenhagen Business School (CBS) Suggested Citation: Nielsen, Andreas Bang (2018) : Essays on Foreign Exchange and Credit Risk, PhD Series, No. 26.2018, ISBN 9788793579996, Copenhagen Business School (CBS), Frederiksberg, https://hdl.handle.net/10398/9644 This Version is available at: https://hdl.handle.net/10419/209073 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/ ESSAYS ON FOREIGN EXCHANGE AND CREDIT RISK Andreas Bang Nielsen PhD School in Economics and Management PhD Series 26.2018 PhD Series 26-2018ESSAYS ON FOREIGN EXCHANGE AND CREDIT RISK COPENHAGEN BUSINESS SCHOOL SOLBJERG PLADS 3 DK-2000 FREDERIKSBERG DANMARK WWW.CBS.DK ISSN 0906-6934 Print ISBN: 978-87-93579-98-9 Online ISBN: 978-87-93579-99-6 Essays on Foreign Exchange and Credit Risk Andreas Bang Nielsen Supervisor: David Lando PhD School in Economics and Management Copenhagen Business School Andreas Bang Nielsen Essays on Foreign Exchange and Credit Risk 1st edition 2018 PhD Series 26.2018 © Andreas Bang Nielsen ISSN 0906-6934 Print ISBN: 978-87-93579-98-9 Online ISBN: 978-87-93579-99-6 The PhD School in Economics and Management is an active national and international research environment at CBS for research degree students who deal with economics and management at business, industry and country level in a theoretical and empirical manner. All rights reserved. No parts of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without permission in writing from the publisher. Foreword This thesis is the result of my PhD studies at the Department of Finance at Copenhagen Business School, and it consists of summaries in English, Danish, and three self-contained essays on foreign exchange and credit risk which can be read independently. I gratefully acknowledge the financial support of the Center for Financial Frictions (FRIC), grant no. DNRF102. I have benefited greatly from advice, discussions, and suggestions from a number of people over the years. In particular, I would like to thank my supervisors David Lando and Christian Wagner. I am indebted to David Lando for being a great supervisor and mentor that helped me grow as an academic. I truly appreciate his encouragement and support over the years. Also, I would like to thank him for a great collaboration on the first essay—I have learned a lot from the process. I would like to give special thanks to Christian Wagner for the support and detailed and honest feedback. His reflections and great comments on my research helped me to sharpen and clarify my ideas. The quality of this thesis has benefited greatly from his help. A number of people deserve a special acknowledgement. Mike Chernov for sponsoring my visit at UCLA Anderson School of Management and for taking his time to discuss my research; Peter Christoffersen for much appreciated help and feedback; my fellow PhD students and colleagues that made work a pleasant and joyful experience. Finally, but not least, I would like to thank friends and family for their endless support and for bearing with me in difficult and stressful times. Andreas Bang Nielsen Copenhagen, April 2018 i ii Summary Summary in English Essay 1: Quanto CDS Spreads (co-authored with David Lando) We investigate how currency denomination affects the price of credit risky securities of the same issuer. We focus on eurozone sovereign quanto spreads, i.e., differences in credit default swap (CDS) premiums denominated in U.S. dollar and Euro of the same reference entity. Quanto spreads of eurozone sovereigns reached unprecedented levels during the European debt crisis and have remained significant ever since. Quanto spreads do not simply reflect differences in contractual terms linked to currency denomination, because CDS contracts trade under the same standardized terms independent of currency denomination, including credit events and recovery rates. In order to understand which factors drive quanto spreads, we propose a no-arbitrage model that shows in a simple and rigorous manner that quanto spreads arise without any market frictions through two risk channels. The first channel, currency crash risk, reflects the risk of an adverse jump in domestic versus foreign currency triggered by default of the reference entity. Intuitively, currency crash risk causes the expected recovery payment to be relatively smaller on the domestic CDS compared to the foreign CDS, because the recovery payment on the domestic contract is received in the ’crashed’ currency. The second channel, covariance risk, contributes to quanto spreads through covariance between the exchange rate and default risk of the reference entity. The intuition for how this channel works is as follows. If default risk rises (falls) CDS premiums increase (decrease) in both foreign and domestic currency, i.e., there is a gain (loss) on a long CDS position iii in the relevant currency. However, if foreign currency tends to appreciate (depreciate) versus domestic currency when credit risk increases (decreases) then the gain (loss) is largest (smallest) on the foreign CDS. Foreign CDS protection is therefore more valuable than domestic protection since it has larger expected gains and smaller expected losses, implying a positive quanto spread caused by covariance risk. Guided by the insights of our simple model, we propose an affine term structure model that captures both crash risk and covariance risk. We estimate the model to quanto CDS data for Italy, Spain, Portugal, and Ireland. Our estimations show that the EURUSD is expected to jump more if Spain and Italy were to default compared to if Portugal and Ireland were to default. We document that crash risk accounts for most of the quanto spreads at shorter maturities and that the covariance risk component embedded in quanto spreads increases in maturity. Covariance risk is particularly important in times of distress, when credit risk and exchange rate risk are volatile and co-vary strongly, while crash risk is important throughout the sample period. Finally, we document that yield spreads between bonds denominated in U.S. dollar and Euro issued by eurozone sovereigns are significantly related to our estimated model-implied quanto yield spreads, especially during the peak of the European debt crisis. Our results indicate that a large portion of the differences in bond yields across currency denominations is caused by crash and covariance risk, and thus not solely by market imperfections, as previous research suggests. Essay 2: Forward-Looking Currency Betas This paper proposes a model-free method that uses currency option prices to compute risk exposures (betas) with respect to any currency factor. While traditional currency betas are based on exchange rate covariances estimated from historical data, the option-implied betas that I propose are based on exchange rate covariances derived from the most recent crosssection of currency option prices, without assuming any parametric structure on correlations. Typically, betas are estimated by means of rolling window regressions that are backwardlooking, adjust slowly to new information, and the econometrician has to decide on which subset of the data to use for the estimation. In contrast, since the option-implied betas are inferred from the latest cross-section of option prices, they require neither historical data iv nor choices of estimation window and frequency—they are a market-based measure of betas. I calculate currency betas by inferring the covariances between exchange rates from options on cross-pair exchange rates. For example, consider three currencies: the Euro, the British pound, and the U.S. dollar. Options exist on each pair-wise combination of these currencies. Specifically, the options on the Euro versus the British pound allow me to pin down the covariance between the Euro versus U.S. dollar and the British pound versus U.S. dollar, without assuming any parametric structure on their covariance. Using the same procedure for any other pair of currencies against the U.S. dollar, I calculate the full exchange rate covariance matrix from which betas with respect to currency portfolios can be derived. In order to test the empirical properties of the option-implied betas compared to traditional rolling window betas, I use the dollar factor—an equally weighted portfolio of the G10 currencies against the U.S. dollar—as the systematic factor driving currency excess returns. I use the dollar factor because it captures the aggregate level of foreign currencies versus the U.S. dollar, i.e., it is essentially the market portfolio of foreign currencies from the perspective of a U.S. investor and, more importantly, because it has been documented by Lustig, Roussanov, and Verdelhan (2011, 2014) to carry a significant risk premium. For both types of betas, I separately construct portfolios of currencies sorted by their dollar factor betas. I identify a significant positive relation between option-implied portfolio betas and ex-post portfolio returns, whereas there is an insignificant relation when using rolling window betas. Interestingly, this is because the option-implied betas predict currency spot changes and not because of the interest rate component of the portfolio returns, which is the most typical source of excess returns for currency strategies. Furthermore, I provide evidence that the model prediction errors of portfolio excess returns are significantly smaller when using option-implied betas as inputs in the model compared to using rolling window betas. Finally, I find that option-implied betas are significantly better predictors of realized betas than rolling window betas at all horizons, both for portfolios and individual currencies. This finding strikes as a likely explanation for why option-implied betas are better in predicting currency excess returns than rolling window betas. v xii Introduction In the first essay, we investigate how currency denomination affects the pricing of credit risky securities by studying the case of eurozone sovereign quanto CDS spreads, that is, differences in credit default swap (CDS) premiums denominated in USD and EUR of the same issuer. Since the EUR and USD-denominated CDS contracts are issued under the same standardized terms—including identical recovery rates and trigger events—the quanto CDS spread is not due to contractual differences. Quanto CDS spreads therefore represent a clean way to study how currency denomination affects the pricing of credit risky securities and the interaction between foreign exchange rate risk and credit risk. We develop a no-arbitrage discrete-time model that rationalizes quanto CDS spreads as compensation for risk through two channels. The first channel is currency crash risk, which reflects the risk of a jump in foreign currency (e.g., USD) versus domestic currency (e.g., EUR) in the event of a default. Intuitively, currency crash risk is priced in the quanto CDS spread because the expected recovery payment is larger on the foreign CDS compared to the domestic CDS since the domestic currency is expected to drop at default. The second channel, covariance risk, reflects compensation for taking exposure to negative correlation between credit risk and foreign exchange rate risk. If credit risk rises (falls), it causes both domestic and foreign CDS premiums to go up (down), that is, a gain (loss) for the protection buyer of CDS in either currency. However, since domestic currency simultaneously tends to decrease (increase) relative to foreign currency when credit risk rises (falls), the gain (loss) is larger (smaller) on the foreign CDS. Therefore, the expected gains are smaller, and the expected losses are greater on the domestic CDS for a protection buyer, implying a positive quanto CDS spread. Moreover, we show that this channel has a larger effect on quanto CDS spreads the larger the expected volatility of currency risk and credit risk are. Our model shows that quanto CDS spreads at shorter maturities are xiii primarily driven by crash risk, while the impact of covariance risk increases in maturity. We can therefore disentangle crash risk from covariance risk using the term structure of quanto CDS spreads. Guided by the insights of the discrete-time model, we propose an affine term structure model that encompasses crash risk and covariance risk. We estimate the model to sovereign quanto CDS for Spain, Italy, Portugal, and Ireland, at maturities of 1-10 years. Furthermore, to get accurate assessments of the covariance risk components embedded in quanto CDS spreads, we use currency options to estimate forward-looking currency volatility risk. We find that both covariance and currency crash risk are important contributors to quanto CDS spreads. We estimate the (risk-neutral) expected percent-wise jump in the EURUSD at sovereign default for Spain and Italy to 15.6% and 9.6%, significantly larger than the currency jump size of about 5% in the event of a Portuguese or Irish default. Our estimations show that covariance risk is most pronounced in times of financial distress, i.e., when the exchange rate and credit spreads are volatile and highly correlated. During the most severe period of the European debt crisis, we estimate the covariance components at the 5-year maturity to range from 18.4 bps to 35.6 bps, corresponding to 25%-58% of the average quanto CDS spreads. Without accounting for covariance risk, we would erroneously overestimate the implied jump size in the EURUSD upon sovereign default. Furthermore, consistent with our intuition from the discrete-time model, we find that crash risk accounts for a larger part of quanto CDS spreads at shorter maturities and that the contribution from covariance risk increases in maturity. Finally, we use our estimated model to explain quanto bond yield spreads for Italy, Spain, and Portugal, which are differences in yields on USD and EUR-denominated bonds. From 2010-2013, i.e., at the peak of the European debt crisis, we provide evidence that our model-implied quanto bond yield spreads co-vary significantly with the observed quanto bond yield spreads, while in the post-crisis period they seem unrelated. Our results suggest that in times of market turmoil, crash risk and covariance risk are important determinants of yield spreads between EUR and USD-denominated eurozone sovereign bonds, implying that quanto bond yield spreads, at least partly, are attributable to risk and that they do not necessarily reflect market mispricings. xiv In the second essay, I propose a method for calculating forward-looking betas (risk exposures) with respect to factors constructed from currencies. I make use of a unique feature of currency option markets that allows me compute forward-looking covariances/variances for currencies. In particular, I exploit that there are options traded on each pair-wise combination of the G10 currencies, which I use to infer currency variances and correlations from which I derive currency betas. The option-implied betas that I propose are inherently forward-looking and measured in real time. Whenever option prices change, the option-implied betas adjust immediately, and since the option prices are forward-looking, the option-implied betas are forward-looking as well. In contrast, betas calculated based on rolling window regressions (which is the most commonly used approach to calculate betas) are slow-moving and may not reflect current expectations about future betas over, say, the next month. Purely forward-looking betas cannot be obtained in other major asset classes, for example for stocks, since there is no (liquid) market for options that depend on the price of two stocks. My contribution is important because asset prices reflect compensation based on expected future risk exposures, and not historical realizations of risk exposures that traditional methods offer. In order to test the empirical properties of the option-implied betas compared to traditional rolling window betas, I use the dollar factor—which is an equally weighted portfolio of G10 currencies versus the U.S. dollar—as the systematic factor in currency excess returns. I use the dollar factor because it is well-documented that it carries a significant risk premium and because it reflects the aggregate level of foreign currencies from the perspective of a U.S. investor (Lustig, Roussanov, and Verdelhan (2011, 2014)). However, my methodology can be applied to any currency factor model. I provide evidence that the option-implied dollar factor betas are significantly better predictors of realized dollar factor betas than rolling window dollar factor betas, both for betas of portfolios and for betas of individual currencies. Having established this fact, we would expect that option-implied betas are better in predicting currency returns, which is indeed what I find support for in the data. In order to compare the cross-sectional properties of the two types of betas, I construct monthly rebalanced portfolios of currencies sorted on betas, for each type of beta separately. xv Lustig, Roussanov, and Verdelhan (2014) show that the dollar factor tends to appreciate (depreciate) whenever the average of short-term foreign interest rates is above (below) the short-term U.S. interest rate. Therefore, I construct the portfolios such that the investor goes long (short) in each portfolio whenever the average foreign interest rate is above (below) the U.S. interest rate. When sorting on the basis of option-implied betas, I find a significantly positive relation between ex-ante betas and ex-post portfolio returns, whereas there is an insignificant relation when the rolling window betas are used. Using the option-implied betas, a long-short portfolio that buys the upper tertile beta currencies and shorts the lower tertile beta currencies gives a significant annualized mean excess return of 3.35% (Sharpe ratio of 0.41), whereas it has an insignificant annualized mean excess return of 0.95% (Sharpe ratio 0.11) when sorting on rolling window betas. Interestingly, the difference in mean excess returns on the long-short portfolio for the two types of beta stems from the spot component and not from the carry component (interest rate differential) of the portfolio excess returns, which is in contrast to the currency carry trade, where the excess returns primarily come from the interest rate component. This implies that option-implied betas outperform the rolling window betas for portfolio construction because they are better predictors of currency spot changes. Furthermore, I show that the model time-series prediction errors are smallest, on average, when using option-implied betas and that rolling window betas tend to underestimate low-beta portfolio returns and overestimate high-beta portfolio returns, while option-implied betas deliver unbiased predictions. I provide evidence suggesting that a reasonable explanation for why the option-implied betas are better predictors of currency excess returns is because they are better in predicting realized betas, both for portfolios and individual currencies. Moreover, rolling window betas deliver biased forecasts; they underestimate (overestimate) betas for low-beta (high-beta) portfolios, while the option-implied betas deliver virtually unbiased predictions. In the third essay, I study if risk premia associated with currency volatility risk are attributable to exposure to systematic variance risk. The main objective of the study is to investigate if the large volatility excess returns for individual currencies that have been documented in previous research are driven primarily by systematic variance risk. To this end, I propose a simple method for decomposing variances of exchange rates into systematic xvi and idiosyncratic variance risk, which I use to empirically investigate the relation between systematic variance risk and volatility excess returns. Specifically, I assume that currency excess returns are driven by exposure to the dollar factor which implies that currency variances consist of a variance component stemming from exposure to dollar factor variance risk (systematic variance risk) and an idiosyncratic variance risk component. Because exposure to dollar factor variance risk is the source of volatility excess returns under my hypothesis, I begin the empirical analysis by establishing a number of stylized facts about the volatility risk premium on the dollar factor. The dollar factor volatility risk premium is, on average, negative and tends to have an upward sloping and concave term structure, i.e., it is steep at the short end and virtually flat at longer maturities. This pattern indicates that investors are willing to pay for hedging systematic volatility risk but that they are more concerned with short-term systematic volatility risk relative to long-term systematic volatility risk. The factor structure in currency excess returns allows me calculate forward-looking measures of the systematic variance components by using the option-implied dollar factor betas and variances that I proposed in the second essay. Using this methodology for calculating systematic variance risk, I find a negative relation between the (expected) share of systematic variance and realized volatility excess returns, i.e., excess returns on volatility swaps and forward volatility agreements (FVAs). As a consequence, it has been profitable for investors to sell volatility protection on currencies with a high share of systematic variance and buy volatility protection on currencies with a low share of systematic variance. For example, the monthly mean excess return of a long-short portfolio of 1-month volatility swaps based on the share of systematic variance is 4.47% with an annualized Sharpe ratio of 0.71, and for FVAs, in which the forward contract and volatility have a 1-month maturity, the monthly mean excess return is 2.73% with an annualized Sharpe ratio of 0.95. At shorter maturities, the excess returns of the long-short systematic variance risk portfolios cannot be explained by exposure to traditional currency factors, equity factors, or the volatility carry factor proposed by Della Corte, Kozhan, and Neuberger (2017). xvii Contents Summary in English iii Summary in Danish viii Introduction xiii 1 Quanto CDS Spreads 5 1.1 Introduction.................................... 7 1.2 Literature ..................................... 10 1.3 Default and Recovery in Different Currencies . . . . . . . . . . . . . . . . . . 13 1.4 The Quanto Spread in a Discrete Model . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Model Assumptions and Definitions . . . . . . . . . . . . . . . . . . . 15 1.4.2 Pricing the Domestic and Foreign CDS . . . . . . . . . . . . . . . . . 17 1.4.3 Quanto CDS Spreads Comparative Statics . . . . . . . . . . . . . . . 19 1.4.4 Calibrating the Quanto CDS Term Structure . . . . . . . . . . . . . . 21 1.4.5 Bond Pricing in Different Currencies . . . . . . . . . . . . . . . . . . 23 1.5 A Term Structure Model of Quanto CDS Spreads . . . . . . . . . . . . . . . 26 1.5.1 The Risk-Neutral Dynamics of the Model . . . . . . . . . . . . . . . . 26 1.5.2 Specification of Pricing Kernels . . . . . . . . . . . . . . . . . . . . . 28 1.5.3 CDS Premiums in Domestic Currency . . . . . . . . . . . . . . . . . 29 1.5.4 CDS premiums in Foreign Currency . . . . . . . . . . . . . . . . . . . 30 1.6 Data and Descriptive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6.1 Credit Default Swap Data . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6.2 Currency Options Data . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6.3 InterestRateData ............................ 35 1.6.4 Descriptive Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 35 1.7 Model Results and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.7.1 Estimation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.7.2 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.7.3 Quanto Effects on Bond Yields . . . . . . . . . . . . . . . . . . . . . 46 1.8 Conclusion..................................... 52 1.9 Figures....................................... 54 1.10Tables ....................................... 65 1.11 Appendix: Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . 77 1.11.1 Crash Risk Consistent with No-Arbitrage . . . . . . . . . . . . . . . . 77 1.11.2 Proofs in the Discrete-Time Model . . . . . . . . . . . . . . . . . . . 78 1.12Appendix:AffineModel ............................. 87 1.12.1 Market Price of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 1.12.2 Pricing of CDS in Affine Framework . . . . . . . . . . . . . . . . . . 88 1.13 Appendix: Estimation Approach . . . . . . . . . . . . . . . . . . . . . . . . . 90 1.13.1 The Unscented Kalman Filter . . . . . . . . . . . . . . . . . . . . . . 93 2 Forward-Looking Currency Betas 97 2.1 Introduction.................................... 99 2.2 RelatedLiterature ................................102 2.3 Option-Implied Risk Exposures . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.3.1 ModelSetup................................105 2.3.2 Option-Implied Currency Betas . . . . . . . . . . . . . . . . . . . . . 107 2.3.3 Dollar Factor Betas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.4 TheData .....................................112 2.4.1 Currency Spot and Forward Data . . . . . . . . . . . . . . . . . . . . 112 2.4.2 Currency Options Data . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.5 EmpiricalResults.................................114 2.5.1 The Dollar Carry Trade . . . . . . . . . . . . . . . . . . . . . . . . . 114 2.5.2 Measuring Dollar Factor Betas . . . . . . . . . . . . . . . . . . . . . . 116 2.5.3 Dollar Factor Beta-Sorted Portfolios . . . . . . . . . . . . . . . . . . 118 2.5.4 Evaluation of Model Predictions . . . . . . . . . . . . . . . . . . . . . 122 2 2.5.5 Predicting Dollar Factor Betas for Portfolios . . . . . . . . . . . . . . 125 2.5.6 Predicting Dollar Factor Betas for Individual Currencies . . . . . . . 128 2.6 Conclusion.....................................130 2.7 Figures.......................................132 2.8 Tables .......................................139 2.9 AdditionalTables.................................151 3 Systematic Currency Volatility Risk Premia 154 3.1 Introduction....................................156 3.2 Systematic Variance Risk Premia . . . . . . . . . . . . . . . . . . . . . . . . 160 3.2.1 VarianceSwaps..............................161 3.2.2 Forward Variance Agreements . . . . . . . . . . . . . . . . . . . . . . 162 3.2.3 Forward Variance Price . . . . . . . . . . . . . . . . . . . . . . . . . . 163 3.2.4 Dollar Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.2.5 Forward-Looking Dollar Factor Betas . . . . . . . . . . . . . . . . . . 165 3.2.6 The Share of Systematic Variance . . . . . . . . . . . . . . . . . . . . 166 3.2.7 Systematic Variance Risk Premia in Cross-Currencies . . . . . . . . . 167 3.2.8 The Forward Share of Systematic Variance . . . . . . . . . . . . . . . 168 3.2.9 Calculating Spot and Forward Variances . . . . . . . . . . . . . . . . 170 3.3 Data........................................172 3.3.1 Currency Options Data . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.3.2 Spot and Forward Data . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.4 EmpiricalResults.................................173 3.4.1 Currency Volatility Risk Premia . . . . . . . . . . . . . . . . . . . . . 173 3.4.2 Share of Systematic Variance . . . . . . . . . . . . . . . . . . . . . . 176 3.4.3 Portfolios Sorted by Share of Systematic Variance . . . . . . . . . . . 178 3.5 Explaining SYS-Sorted Portfolio Returns . . . . . . . . . . . . . . . . . . . . 181 3.5.1 CurrencyFactors .............................181 3.5.2 EquityFactors ..............................184 3.5.3 Currency Volatility Factors . . . . . . . . . . . . . . . . . . . . . . . 185 3.6 Dollar Factor Beta-Sorted Portfolios . . . . . . . . . . . . . . . . . . . . . . . 188 3.7 Conclusion.....................................190 3 3.8 Figures.......................................192 3.9 Tables .......................................199 3.10 Appendix: Supplementary Tables and Figures . . . . . . . . . . . . . . . . . 212 3.11Bibliography....................................222 4 for eurozone sovereigns to estimate redenomination risk, that is, compensation for risk that EUR-denominated securities are redenominated into a new devalued currency. The latter focuses on developing a pricing model for quanto CDS spreads and calibrate it to Italian quanto CDS spreads. Carr and Wu (2007b) provide evidence that sovereign credit risk is priced in the currency option markets for Brazil and Mexico. They obtain inference on the (risk-neutral) jump size in local currency upon sovereign default by estimating a joint model for options and sovereign CDS. Since option prices are driven by numerous factors apart from sovereign credit risk, e.g., macroeconomic news (Chernov et al., 2016), this approach makes it difficult to quantify the effect of sovereign default on local currency. Since the payoff on a quanto CDS is directly linked to currency jump risk at default, we contribute by providing a clean method for estimating the crash risk upon default. Our paper is related to the vast literature that studies sovereign credit risk through the lens of CDS premiums, e.g., Longstaff, Pan, Pedersen, and Singleton (2011), A¨ıt-Sahalia, Laeven, and Pelizzon (2014), Pan and Singleton (2008), Benzoni, Collin-Dufresne, Goldstein, and Helwege (2015), and Della Corte, Sarno, Schmeling, and Wagner (2016). The latter is, perhaps, the closest related to this paper. They document empirically a significant relationship between sovereign credit risk and returns on currencies and currency option strategies. While their paper is purely empirical, our objective is to develop models that allow us to quantify and understand the interconnection between credit and currency risk. We contribute to the literature that studies pricing of similar credit risky securities across currency denominations, in particular bonds. There is a growing literature that analyzes deviations in yields for sovereign bonds across currency denominations (Buraschi et al., 2014; Corradin and Rodriguez-Moreno, 2016; Du and Schreger, 2016). In these papers, the objective is to use the so-called ”yield basis”, defined as the difference between yields on a domestic and a synthetic domestic bond (which is constructed from foreign currency denominated bonds using FX forwards), to measure violations of the law of one price. Corradin and Rodriguez-Moreno (2016) show that the yield basis for eurozone sovereigns is large and volatile, and they attribute it to differences in collateral value and ECB purchases of EUR-denominated bonds. Buraschi, Menguturk, and Sener (2014) find a substantial yield basis for emerging market bonds during the 2007-2008 crisis and explain it 11 by frictions in banking capital structure and non-conventional policy interventions. However, our theory shows that a yield basis may arise because of crash risk and covariance risk. Our empirical results suggest that this not only a theoretical concern. We provide evidence that indicates that the yield spread between EUR and USD-denominated bonds for eurozone sovereigns reflects compensation for risk related to covariance and crash risk. 1.3 Default and Recovery in Different Currencies CDS contracts on the same reference entity but denominated in different currencies share a number of characteristics that are important to understand before setting up a model. A Credit Default Swap (CDS) is an insurance against default on debt of an underlying reference entity. The contract involves two parties: a protection buyer and a protection seller. Every period, if no credit event has occurred of the reference entity, the buyer pays a percent-wise premium (often quarterly) of an agreed notional amount to the seller. If a credit event occurs, the buyer receives a recovery of the notional protected. Credit events are defined by the International Swaps and Derivatives Association (ISDA) and involves different scenarios, including outright bankruptcy, restructuring of debt, or deferred interest payments. If a credit event occurs, an auction is held to determine the recovery rate based on a pool of bonds delivered into the auction. Importantly, the recovery rate is the same for all CDS contracts, independently of the currency denomination (see below for more details). The auction is typically conducted between 30-35 days following the event determination date. Once an event has occurred, protection buyers are entitled to settle by physically delivering any of the specified deliverable obligations to settle the contract. According to the standardized ISDA terms, the deliverable bonds are subject to a number of requirements. The payments of the obligation must be made in one of the specified currencies which for reference entities of Western Sovereigns are CAD, CHF, EUR, GBP, JPY, or USD. This means, for example, that a holder of a CDS contract denominated in EUR on Germany can choose to deliver German sovereign bonds denominated in USD. The relevant exchange rates for delivering obligations in a different currency to the CDS contract are fixed the day before the auction at 4pm at the WM/Reuters 4pm London mid-point 12 rate. 1.4 The Quanto Spread in a Discrete Model The option to choose in which currency to deliver bonds of the defaulted issuer means that the currency denomination becomes important. This can be seen through a very simple example: Consider two CDS contracts on Germany: One EUR-denominated with a notional amount of 1 EUR and one USD-denominated with a notional amount of 1 USD. Imagine for simplicity that the exchange rate is 1 at the initiation of the contract. If a default occurs before maturity, and at the same time the EUR drops to, say, a value of 0.5 USD, then the scale of protection offered by the two contracts differs. The holder of the EUR-denominated CDS can deliver 1 EUR notional and receive 1 EUR, whereas the holder of the USD protection can deliver a notional amount of 2 EUR, since the USD equivalent notional of 2 EUR is now only 1 USD because of the ’crash’ of the EUR. Hence the amount of notional protected becomes effectively larger for the USD contract. A similar mechanism is at play when currency depreciation has a positive correlation with a decrease in credit quality. Again, a simple example can provide the intuition. Imagine, as above, that the time 0 exchange rate is 1, and that the value of 1 USD can become 1.2 Euro or 0.8 Euro with equal probabilities 0.5 (under the USD risk-neutral measure) in the period 1, and that the exchange rate stays put in the second period until the CDS matures at time 2. Assume also for simplicity that the default probability of the reference entity is perfectly correlated with the exchange rate and becomes 3 percent in the state where the exchange rate is 1.2 and 1 percent in the other state. Assume zero interest rate in both currencies, and zero recovery in default. In this case, the USD value of protection of the CDS contract in two states is summarized in the following table: State/denomination USD EUR 1.2/3% 0.03 0.03 1.2= 0.025 0.8/1% 0.01 0.01 0.8= 0.0125 Since 0.5·0.025 + 0.5·0.0125 = 0.0187 <0.5·0.01 + 0.5·0.03 = 0.02, we see that the value of the protection leg at time 1 is smaller for the EUR-denominated contract. If we assume (again for simplicity) that default risk is 0 between time 0 and time 1, then we have shown 13 that the effect also applies for correlated default probability and FX-rate. 1.4.1 Model Assumptions and Definitions We now build a simple discrete-time model that makes these observations rigorous. The model allows us to derive comparative statics and to analyze term structure effects. For the remainder of the paper, we define the exchange rate at time t,Xt, as units of domestic currency per unit of foreign currency, i.e., an increase in Xtimplies that the foreign currency has appreciated against the domestic currency. Furthermore, we assume the existence of fixed riskless interest rates in both foreign and domestic currency, which we denote rdand rf, and we let Pi(t, T) = e−ri(T−t)denote the price at time tof a zero-coupon bond paying one unit of currency i=d, f at time T. In a no-arbitrage setting, we can then express the time tforward exchange rate with maturity T,F(t, T), in terms of the foreign and domestic bond prices and the spot exchange rate as F(t, T) = Xt Pf(t, T) Pd(t, T) Our model has a time horizon of ¯ tand we subdivide the time horizon into Nequidistant time points which we label t0= 0, t1= 1, . . . , tN=¯ t. In each time period tthere is a probability λtthat the reference entity will default between time tand time t+ 1. We model FX crash risk upon default of the reference entity by assuming that the exchange rate drops by a fixed fraction of δof the (risk-neutral) unconditional expectation of the exchange rate. Specifically, conditional on default between tand t+ 1, the exchange rate takes two possible values at t+ 1: δ·uXtand δ·u−1Xtwith probabilities qand 1 −q, respectively. Conditional on no default, the exchange rate takes the values C(λt)·uand C(λt)·u−1with respective probabilities qand 1 −q, where C(λt) is a compensating factor C(λt) defined as C(λt) = 1−δλt 1−λt and it is needed to ensure no-arbitrage by compensating the exchange rate movement for crash risk. Had there been no crash risk, the exchange rate would either move up by a factor of uor down by a factor of u−1. We show formally in Appendix 1.11.1 that this model is consistent with no-arbitrage. For tractability, we choose to do the compensation 14 of crash risk through the jump size rather than through the martingale probabilities, which is an alternative option. We assume that the default probability can assume two values (λU, λD) in each period, and for simplicity we assume that the respective probabilities qλ and (1 −qλ) do not depend on the current state. To capture the joint dynamics of default risk and exchange rates, we introduce correlation between the movements in the exchange rate and the default probability. Let Qij denote the one-step probability of the exchange rate to reach state iand the default probability to reach state j(conditional on survival), where i= 1/j= 1 correspond to an up move, and i= 0/j= 0 to a down move. At any point in time, we specify the joint distribution of the exchange rate and default probability as Q11 =q(qλ+A1), Q10 =q(1 −qλ−A1) (1.1) Q01 = (1 −q)(qλ−A0), Q00 = (1 −q)(1 −qλ+A0) (1.2) where, A1=ρqqλ q(1 −q)(1 −qλ) and A0=ρqqλ 1−qq(1 −qλ). The important parameter here is ρ, which is the correlation between the Bernoulli variables controlling the up and down moves of the exchange rate and default probability. Clearly, if ρ < 0, then A1<0 and A0<0, which implies that the exchange rate and the default probability tend to move in the opposite direction compared to the uncorrelated case (ρ= 0). Note that it only takes a specification of the unconditional probabilities qand qλand the correlation parameter to specify all the relevant quantities. qλand ρcan be chosen freely in (0,1) and (−1,1), respectively, but qis endogenously determined through the no-arbitrage condition for the currency movement which can be expressed simply in terms of the one-period forward rate F=F(t, t + 1) as q=F/Xt−u−1 u−u−1(1.3) See Appendix 1.11.1 for the derivation. Figure 1.1 illustrates the joint dynamics of the exchange rate and the default probability over two periods. The multi-period dynamics are obtained by repeating this tree from each individual node. After default of the reference entity, the tree terminates. 15 1.4.2 Pricing the Domestic and Foreign CDS We model a Credit Default Swap (CDS) contract focusing on the ’fair running premium’ that the buyer of protection should pay to obtain credit protection. For a contract with maturity T, we assume that no payment is exchanged at time 0 and that at every period ti≤tN≡T, the buyer of the CDS contract pays a premium if the reference issuer has not defaulted at this time. If default occurs in the time interval (ti−1, ti], the seller of insurance pays 1 −Rper unit face value—which we without loss of generality assume to be 1. In this setting, the CDS premium in domestic currency with maturity T,Sd(0, T), is given by Sd(0, T) = (1 −R)PN i=1 Pd(0, ti)Q(τ=ti) PN i=1 Pd(0, ti)Q(τ > ti)(1.4) According to the standardized rules of ISDA, the foreign CDS contract is subject to the exact same contractual terms as the domestic contract, apart from currency denomination (CDS premiums are paid in foreign currency, and in the event of default, the recovery is received in foreign currency). The rules imply that the recovery rate is the same regardless of currency denomination of the contract. Recall, that Qis the risk-neutral pricing measure when using the domestic bank account as numeraire. Defining Qfas the risk-neutral measure corresponding to having the foreign account as numeraire, we can now express the premium of the same CDS contract denominated in the foreign currency as Sf(0, T) = (1 −R)PN i=1 Pf(0, ti)Qf(τ=ti) PN i=1 Pf(0, ti)Qf(τ > ti)(1.5) where Pf(0, t) denotes the discount factor corresponding to the foreign interest rate. To compare the two expressions we will need to understand the relationship between Qand Qf. Let Mi tdenote the pricing kernel for currency denomination i=d, f. Starting with the objective measure, P, we can price any foreign-denominated security with a price, Zf t, using 16 the foreign pricing kernel: 1 = EP t Mf T Mf t Zf T Zf t!=EQf t Pf(t, T)Zf T Zf t!(1.6) As in, e.g., Backus, Foresi, and Telmer (2001), we construct a domestic security from the foreign security using the exchange rate: XtZf t. Since this claim is denominated in domestic currency, we can price it using the domestic pricing kernel: 1 = EP t Md T Md t XTZf T XtZf t!=EQ t Pd(t, T)XTZf T XtZf t!(1.7) Equations (1.6) and (1.7) hold for any security which implies that there is the following relationship between the domestic and foreign pricing kernels, the exchange rate, and the foreign and domestic risk-neutral measures: Mf T Md T Md t Mf t =XT Xt , MT=XT Xt Pd(t, T) Pf(t, T)(1.8) where MTchanges measure from the foreign to the domestic risk-neutral measure (i.e., MT= dQf dQ (T)). We refer to Appendix 1.11.2 and 1.11.2 for the closed-form model expressions of the domestic and foreign CDS premiums as well as their derivation. 1.4.3 Quanto CDS Spreads Comparative Statics We now discuss how each parameter of the model impacts the quanto spread. First, we show that the quanto spread widens in the expected severity of the crash in foreign currency upon default. Proposition 1. The quanto spread, QS(0, T), is decreasing in δfor all T Proof. See Appendix 1.11.2 To gain some intuition on Proposition 1, we propose a stylized example with a fixed default probability (implying independence between the default probability and the exchange rate), and a crash risk premium of δ. In Appendix 1.11.2, we show that in this case, the 17 CDS premiums in domestic and foreign currency, of any maturity, are given by Sd= (1 −R)λ (1 −λ)(1.9) Sf= (1 −R)λδ (1 −λδ)(1.10) In the case of a fixed default probability, the riskless interest rates do not affect CDS premiums, i.e., the expressions for the CDS premiums in (1.9) and (1.10) hold for any choice of foreign and domestic interest rates. Assume δ < 1, which implies that foreign currency depreciates upon default. Under this assumption, the recovery payment on the foreign CDS, (1 −R)δ, is strictly smaller compared to the domestic CDS. The net present value of the premium leg payments, on the other hand, is larger than on the domestic CDS, because the foreign currency is expected to appreciate vs. domestic currency conditional on survival. Therefore, when δ < 1, the value of the premium leg is greater and the value of the protection leg is smaller than for the domestic CDS, implying a positive quanto spread. Figure 1.2 shows the CDS premiums denominated in foreign and domestic currency plotted against the expected depreciation upon default. The foreign CDS premium decreases as the risk-neutral expected crash in the currency increases, while the domestic CDS premium is fixed for a given level of the default probability, implying that the quanto spread increases in the severity of the crash. Proposition 2. The quanto CDS spread, QS(0, T), is decreasing in ρfor all T≥2. Furthermore, if ρ < 0(ρ > 0) then QS(0, T)is increasing (decreasing) in uand λU−λD. Proof. See Appendix 1.11.2 The intuition behind Proposition 2 is that if there is negative correlation between the exchange rate and default risk, it is more likely that default occurs in states in which foreign currency has depreciated relative to its unconditional expectation. This effectively causes the foreign contract (converted into domestic currency) to deliver a smaller expected recovery payment, in the event of a default, compared to the domestic contract. The value of the premium leg, on the other hand, is largest on the foreign contract. This is because the risk-neutral expectation of the exchange rate conditional on survival must be larger than its unconditional expectation, otherwise, the currency forward is not priced consistently with 18 no-arbitrage. The exchange rate thus tends to move unfavourably in both default and nondefault states for the buyer of foreign CDS, implying that the fair foreign CDS premium must be smaller than the domestic CDS premium, i.e., a positive quanto spread. An increase in the volatility of the exchange rate or the default probability, measured by the spread between up and down states (i.e., uand λU−λD), causes the quanto CDS spread to widen. An intuitive explanation for this is as follows. When credit risk goes up (down), then there are gains (losses) on both the foreign and the domestic CDS in the respective currencies. However, if the exchange rate tends to simultaneously decrease (increase), then the gain (loss) is smaller (larger) on the foreign CDS compared to the domestic CDS. Thus, the larger the moves in the credit risk and the exchange rate, the smaller (greater) the expected gains (losses) on the foreign CDS versus the domestic CDS, causing the quanto CDS spread to widen. Finally, an important aspect of Proposition 2 is that the one-period quanto CDS spread is exclusively driven by crash risk, while the quanto CDS spread of two periods or more are impacted by both crash risk and covariance risk. Crash risk and covariance risk thus affect the term structure of quanto CDS spreads differently which allows us to distinguish between them by using data for quanto CDS spreads at different horizons. 1.4.4 Calibrating the Quanto CDS Term Structure In the following, we use the discrete-time model to get a grasp of the magnitude of the crash and covariance risk embedded in quanto CDS spreads. The purpose is to gain intuition on how crash and covariance risk affect quanto spreads and to get an approximate estimate of their effect on observed quanto CDS spreads. Although the model is static, the central intuition gained from the model carries over to a richer dynamic term structure model, which we will analyze further in section 1.7. We calibrate the model using CDS premiums for Spain, Italy, Portugal, and Ireland over the period August 2010-August 2012, i.e., at the height of the European debt crisis where CDS and quanto CDS spreads peaked. More specifically, the model parameters are calibrated such that they match the average observed 5-year quanto CDS spread, the 5-year CDS spread volatility, the EURUSD FX volatility, and the realized correlation between FX spot and 5-year USD CDS spread changes. We proxy ρ, the default probability/exchange 19 rate correlation, with the correlation between daily percent-wise changes in the 5-year USDdenominated CDS premium and the EURUSD exchange rate. The parameter uis chosen such that the model’s FX volatility matches the average 1-year risk-neutral volatility1. We compute the risk-neutral volatility from EURUSD currency options using the ”model-free” methodology of Bakshi, Kapadia, and Madan (2003) (see section 1.6 for further details on the data). The empirical moments used for the calibration are reported in Table 1.1. Fixing ρand uas described above, we calibrate the default probability parameters, (λD, λU, qλ), and the currency crash risk parameter, δ, such that the model exactly matches the average 5-year CDS premiums denominated in USD and EUR. The calibration shows that the risk-neutral expected crash in the EURUSD in the event of a default is substantially larger for Spain and Italy relative to Portugal and Ireland. In particular, in the event of default of Spain and Italy, we estimate the risk-neutral expected depreciation in the EURUSD to 16% and 15%, respectively, while for Portugal and Ireland we estimate it to 5% and 7%, respectively. The results seem reasonable; the Euro is expected to take a much larger hit in the event of a Spanish or Italian default as these countries are more important economies for the eurozone. If we were to ignore covariance risk (ρ= 0), the impact of a sovereign default on the EURUSD exchange rate would have been overestimated. In this case, for Spain and Italy, we estimate the crash risk to 21% and 19%, and 7% and 9% for Portugal and Ireland, underlining the importance of including covariance risk in the model to get an accurate assessment of the implied effect of a sovereign default on the exchange rate. In Figure 1.3, we show the calibrated term structure of quanto spreads for Portugal, Ireland, Italy, and Spain. We see that the quanto spread increases in time to maturity. In the model—as shown explicitly in equations (1.9) and (1.10)—the term structures of foreign and domestic CDS premiums are flat when there is no covariance risk. Hence, the upward sloping quanto CDS curve is caused by covariance risk. The orange graph shows the quanto CDS spread in the case of no crash risk, i.e., the case where the entire quanto spread stems from covariance risk. We see that the curve is upward sloping in maturity, implying that covariance risk accounts for larger share of the quanto spread at longer maturities. Therefore, consistent with the intuition discussed previously, we can infer the magnitude of 1Since the one-year FX volatility, σF X , and the size of the up step, u, in a Cox, Ross, and Rubinstein (1979) tree are related as u=eσF X . 20 entails that we can compute the following transforms as ψ(xi,t, t, T)≡EQ te−RT tλi,sds=eαi(t,T )+βi(t,T )·xi,t (1.17) φ(xi,t, t, T)≡EQ tλi,T e−RT tλi,sds=ψ(xi,t, t, T )Ai(t, T) + Bi(t, T)·xi,t(1.18) where αi(t, T), βi(t, T), Ai(t, T), and Bi(t, T) solve a set of ordinary differential equations (see, e.g., Duffie, Pan, and Singleton (2000)). The exact specification of the ODEs are reported in Appendix 1.12.2. Given a quarterly payment scheme for the premium leg and a fixed recovery rate on the protection leg, we have that their present values are given by Πprem(t, T) = Sd(t, T)1 4 4T X j=1 Pdt, t +j 4ψxi,t, t, t +j 4(1.19) Πprot(t, T) = (1 −R)Zt+T t Pd(t, t +u)φ(xi,t, t, u)du (1.20) The domestic CDS premium, which is consistent with no arbitrage, is then determined such that the present values of the premium leg and the protection leg are equal: Sd(t, T) = Πprot(t, T) Πprem(t, T)(1.21) 1.5.4 CDS premiums in Foreign Currency In the discrete-time model, we derive the foreign CDS premium directly by using Mt= Xt X0 Pd(0,t) Pf(0,t)to convert each foreign-denominated payment into a domestic payment. In the affine model, this is rather cumbersome. We take a more convenient approach and price the foreign-denominated CDS contract using a change of numeraire technique. Formally, Mt=dQf dQ , is the Radon-Nikodym derivative that changes measure from the domestic to the foreign risk-neutral measure. To apply the change of numeraire technique, we need the dynamics of the Radon-Nikodym derivative between Qand Qf, which is given by: dMt=Mt√vtρWsys,t +p1−ρ2dWx,t+Mt K X i=1 (ζidNi,t +ζiλi,tdt) (1.22) 27 By using Lemma 1 with Mtas the pricing kernel, the default intensity under the foreign risk-neutral measure is given by: λf i,t =λi,t(1 + ζi) (1.23) dvt=κf v(θf v−vt)dt +σv√vtdWf sys,t (1.24) dli,t =κl,i(θl,i −li,t) + σl,iρpli,tvtdt +σl,ipli,tdWf sys,t (1.25) where κf v= (κv−σvρ), θf v=κvθv κv−σvρ, and λi,t is the domestic default intensity. Lemma 1 states that the ratio between the default intensity under the foreign measure and domestic measure equals the jump size conditional on sovereign default: λf t=λt(1+ζ). For this reason, very short-term quanto CDS spreads are exclusively driven by crash risk because Sd(t, T)≈(1−R)λtand Sf(t, T)≈(1−R)(1+ζ)λt, when tapproaches T. Even in the case of a purely idiosyncratic default intensity (i.e., no covariance risk), a quanto CDS spread emerges solely through the crash risk channel. This is consistent with our intuition from the discrete-time model, where we showed that a quanto CDS spread arises in the case of a constant default probability through crash risk. Under the foreign measure, each process that is exposed to Wsys,t is drift-adjusted via the pricing kernel (1.22). For lt, the drift adjustment is σlρ√ltvt, i.e., it depends on the instantaneous volatility of the exchange rate, the systematic default component, and their correlation. If there is negative correlation between the exchange rate and the default intensity, then the drift correction is negative which causes the expected default risk to be smaller under the foreign measure than under the domestic measure, implying a positive quanto CDS spread. The covariance adjustment has less impact at shorter horizons, because the drift adjustment does not affect the instantaneous default risk. An implication of the model is therefore that quanto CDS spreads tend to widen in maturity if there is negative covariance between default and exchange rate risk. This is consistent with the results of our calibration exercise based on the discrete-time model, where we showed that the quanto CDS spread widens in maturity because of covariance risk. To summarize, crash and covariance risk affect the foreign default intensity through different channels; crash risk scales and covariance risk drift-adjusts the default intensity, and this distinction is what allows us to separate the two 28 effects using the term structure of quanto CDS spreads. In order to fit the model into the affine framework, we approximate the term, √ltvt, in the systematic default risk’s drift with a first-order Taylor expansion around the respective processes’ mean reversion levels 3. The foreign transforms are then computed as in the domestic setting ψf(xi,t, t, T) = eαf,i(t,T )+βf,i(t,T )·xi,t (1.26) φf(xi,t, t, T) = ψf(xi,t, t, T)Af,i(t, T) + Bf,i(t, T)·xi,t(1.27) and the foreign premium and protection legs are given by Πprem f(t, T) = Sf(t, T)1 4 4T X j=1 Pft, t +j 4ψfxi,t, t, t +j 4(1.28) Πprot f(t, T) = (1 −R)Zt+T t Pf(t, t +u)φf(xi,t, t, u)du (1.29) From the dynamics of the foreign state variables, i.e., equation (1.25), we see that the currency/default covariance risk introduces vtas an additional state variable compared to the domestic case, that is, xi,t ≡li,t zi,t mi,t vtT. The exact specification of the ODEs which αf,i, βf,i, Af,i, and Bf,i solve are provided in Appendix 1.12.2. 1.6 Data and Descriptive Analysis 1.6.1 Credit Default Swap Data We collect CDS premiums from Markit on eurozone sovereign bonds issued by Austria, Belgium, Germany, Finland, Ireland, France, Italy, Netherlands, Portugal, and Spain denominated in EUR and USD. Markit provides us with daily quotes at maturities of 1, 3, 5, 7, and 10 years. We use the complete restructuring clause on the CDS contracts which allows the protection buyer to deliver bonds of any maturity (and currency denomination) into the CDS auction. Markit performs a number of data cleaning procedures on the CDS data that they receive from their contributors, e.g., to avoid stale quotes and outliers, and 3The exact form of the Taylor approximation is given by: √ltvt= 1/2vtθl θv1/2+ltθv θl1/2. 29 they only report quotes if there are at least three quotes from different contributors. Before August 2010, Markit aggregated quotes across currency denominations into one quote. As our focus is on the impact of currency denomination on the pricing of CDS contracts, we initiate our analysis in August 2010, and our sample ends in April 2016. 1.6.2 Currency Options Data One of our main objectives is to estimate the contribution of covariance risk to quanto spreads which essentially depends on three factors: risk-neutral exchange rate volatility, volatility of systematic default risk, and the correlation between credit risk and the exchange rate. The latter two factors can be identified from USD-denominated CDS premiums and quanto CDS spreads, but CDS data are not particularly informative about the first factor. Therefore, in order to pin down the risk-neutral distribution of exchange rate volatility, we include currency options data in our estimation, as in, e.g., Bates (1996); Carr and Wu (2007a,b). We collect EURUSD currency options data from Bloomberg from August 2010 to April 2016. The data consist of Garman and Kohlhagen (1983) implied volatilities of deltaneutral straddles, 10, 25-delta risk reversals, and 10, 25-delta butterfly spreads which are the common quoting conventions in currency option markets. The maturities are fixed and are 1, 2, 3, 6, 9, and 12 months. A straddle is a portfolio which is long a call and a put option with the same strike and maturity. The payoff of a straddle is directionless and the buyer of the straddle is long at-the-money volatility. A risk reversal consists of a long position in an out-of-the money (OTM) put option and a short position in an OTM money call option with symmetric deltas4. The long position in the OTM put protects against large depreciations in foreign currency (EUR), and in contrast, the short OTM call loses money when large depreciations in the USD occur. Risk reversals therefore measure the slope of the implied volatility curve against moneyness, also called the skew of the implied volatility curve. A butterfly spread is the difference between the average IV of and OTM call and an OTM 4Sometimes the risk reversal is quoted conversely as a long position in a call option and a short position in a put. 30 put and the IV of the delta-neutral straddle. If the butterfly spread is positive, it reflects that the market price of hedging large FX movements (in either direction) is more expensive compared to the case in which returns are log-normal, i.e., the risk-neutral distribution of exchange rate changes is fat tailed. Using the Garman and Kohlhagen (1983) formula for the IVs derived from the straddles, risk reversals, and butterflies, we recover five different strikes, spanning from the strike of a put with a delta of −10 percent to the strike of a call option with a delta of 10 percent. We skip the details on how this procedure works and refer to Della Corte, Sarno, Schmeling, and Wagner (2016) and Jurek (2014) for an elaborate explanation. 1.6.3 Interest Rate Data For the pricing of CDS denominated in Euro and U.S. dollar, we need to compute discount curves in both currencies. We take the most common approach and build discount curves from overnight index swap rates, OIS for U.S. dollar, and EONIA for Euro. We use overnight index swap rates rather than LIBOR swap rates because it is well-documented that they contain a default risk component. Since 2010, maturities of up to 10 years of overnight index swaps have been traded. We therefore exclusively use overnight index swap rates as proxies for riskless interest rates, since the longest maturity in our CDS data is 10 years. Based on the overnight index swap interest rates, we construct zero-coupon curves in Euro and U.S. dollar using a standard bootstrapping procedure. We collect the data on overnight index swap rates from Bloomberg, and the maturities are 3, 6, 9 months, and 1-10 years, and the data start in August 2010 and end in April 2016. 1.6.4 Descriptive Data Analysis Table 1.4 reports the averages and standard deviations of eurozone sovereign CDS premiums denominated in EUR and USD, spanning maturities from 1-10 years, over the period August 2010 to April 2016. First, we note that the USD CDS premium is, on average, unambiguously higher than the corresponding EUR CDS premium for all sovereigns. In absolute terms, the average quanto CDS spreads, e.g., at the 5-year maturity, are largest for Ireland, Italy, Portugal, and Spain, ranging from 36-48 bps, while they are the smallest for Finland, Germany, Netherlands, and Austria, ranging from 8-22 bps. In general, the 31 non-GIIPS countries have much smaller average CDS premiums, indicating that the market deemed it unlikely that sovereign defaults would occur for these sovereigns. As an example, the average 5-year USD CDS premium for Portugal is more than ten times larger than for Germany. In Figures 1.4-1.6, we show the time series of quanto CDS spreads and USD-denominated CDS premiums for all sovereigns at maturities ranging from 1-10 years. The quanto CDS spreads are positive in the entire sample period for all sovereigns. As is the case for the USD CDS premiums, the quanto CDS spreads peak for all sovereigns between the last quarter of 2011 and the Summer of 2012. During this period, the 5-year quanto CDS spreads exceed 100 bps for Spain and Portugal, and almost reach 100 bps for Italy and Ireland as well. From July 2012, in the wake of Mario Draghi’s speech in which he insured that the ECB would do whatever it takes to preserve the Euro, the quanto CDS spreads gradually decline, but they stay positive throughout the sample period. Table 1.5 reports the averages and standard deviations for implied volatilities of straddles, risk reversals, and butterflies for each maturity. The implied volatility for both the 10 and 25-delta risk reversals are, on average, negative, in fact, they are negative throughout our sample period at all maturities. This shows that large downside risk in the Euro has historically been more expensive to insure relative to symmetric downside risk in the U.S. dollar. The focus of our analysis is the relation between currency risk and credit risk. As a first step in exploring this relation, we proxy aggregate eurozone credit risk by the first principal component of eurozone 5-year USD CDS premiums and investigate its relation to EURUSD implied volatility and spot changes. The principal component analysis shows that there is a strong commonality in CDS premiums for eurozone sovereigns. The first principal component of weekly changes in 5-year USD CDS premiums explains 77% of the common variation of the changes in 5-year USD CDS premiums5, consistent with Longstaff, Pan, Pedersen, and Singleton (2011), who document strong commonality in global CDS premiums. Table 1.6 shows results from regressions of weekly innovations in the EURUSD spot exchange rate and the delta-neutral straddle implied volatility on the first principal component 5Similar results are obtained when using EUR-denominated CDS. 32 of the eurozone CDS premiums. Over the entire sample period, there is a significantly negative relation between changes EURUSD spot rate and eurozone credit risk, with a t-statistic of −3.69 and an R2of 8.1%. This result suggest that the Euro tends to depreciate when eurozone credit risk rises. Most of the significance, however, stems from the European debt crisis period, i.e., from August 2010 to December 2012. In the post-crisis period (January 2013 to April 2016), there is a negative, but insignificant, relation (t-statistic of −1.34), and a miniscule part of the variation in spot exchange rates is explained by exposure to sovereign credit risk. The at-the-money implied volatility and eurozone credit risk are significantly positively related over the entire sample period (t-statistic of 3.84), with an R2of 12.1%, i.e., increasing forward-looking EURUSD volatility tends to be associated with increasing eurozone credit risk. Our results are consistent with those of Della Corte, Sarno, Schmeling, and Wagner (2016), who document, for a large sample of countries, that exchange rate spot movements and implied volatilities of options are tightly related to sovereign credit risk. The positive relation between EURUSD implied volatility and eurozone credit risk is highly significant in the crisis period, with a t-statistic of 7.70 and an R2= 27.2%, but their relation is barely significant in the post-crisis period (t-statistic of 2.17, R2= 3.2%). Consequently, the results of our regression analysis indicate that eurozone sovereign credit risk and the currency spot rate and implied volatility primarily co-vary in times of distress. According to our discrete-time model, the significant covariance between exchange rate risk and sovereign credit risk implies a positive quanto CDS spread for eurozone sovereigns, even without any exchange rate crash risk at default. Moreover, the results of the regressions suggest that the covariance risk components embedded in quanto CDS spreads are most pronounced during the crisis period from 2010-2012. In the next section, we analyze these conjectures using the proposed affine term structure model to decompose quanto CDS spreads into a covariance risk component and a crash risk component. 33 1.7 Model Results and Estimation 1.7.1 Estimation Approach We focus on estimating the model for the GIIPS countries: Portugal, Ireland, Italy, and Spain, excluding Greece. We exclude Greece from the analysis because Breuer and Sauter (2012) document that there was virtually no trading activity in the Greek CDS from early 2011, as the market anticipated a Greek default, which, in fact, occurred on March 9, 2012. CDS markets also reflected that a Greek default was anticipated, with elevated CDS premiums on Greek government bonds reaching several thousand bps by the last of quarter of 2011. We focus on the GIIPS countries (excluding Greece) because they are the least creditworthy in our sample and, arguably, the focal point of the European debt crisis. For example, the 5-year CDS premiums (in USD) for the GIIPS all reached levels exceeding 600 bps, with Portugal and Ireland being the most extreme cases with CDS premiums exceeding 1000 bps. In comparison, the German 5-year CDS barely touched 100 bps, and the French 5-year CDS spiked at about 200 bps. In the estimation, we use weekly data (each Wednesday) of quanto CDS spreads, USDdenominated CDS premiums, and currency option implied volatilities. Each week, we have 30 option prices (five strikes at six maturities), five CDS premiums denominated in USD, and five quanto CDS spreads at maturities of 1, 3, 5, 7, and 10 years. If we were to estimate the model in one joint estimation, we would have an unmanageably large set of parameters and a high dimensional state variable vector. For instance, in the case of four sovereigns, the model has 12 state variables and a very large parameter vector containing systematic, country-specific, and measurement error parameters. One approach to reduce the dimension of the state vector is to introduce common factors or to use just one state variable to capture country-specific default risk. However, since we are interested in making accurate assessments of the magnitude of the quanto spreads driven by crash and covariance risk, we need precise estimations. Our estimations suggest that at least two country-specific factors are necessary for the model to accurately fit the cross-section and time-series dynamics of USD CDS and quanto CDS premiums simultaneously. For this reason, we estimate the model stepwise. In the first step, we estimate a time 34 series of the instantaneous currency volatility, vt, and its objective and risk-neutral parameters from currency option implied volatilities. We estimate the model using maximum likelihood estimation in conjunction with the unscented Kalman filter. In the next step, now treating vtas observable and its parameters as fixed, we estimate the parameters for the idiosyncratic and systematic default intensity components, i.e., lt, zt, and mt, using data for USD CDS and quanto CDS spreads for one country at the time. The estimation procedure is described in detail in Appendix 1.13. 1.7.2 Estimation Results Table 1.7 presents the maximum likelihood estimates of the model, and Figure 1.7 illustrates the estimated state variables lt, zt, and mtfor each sovereign. For all sovereigns, the idiosyncratic component of the default intensity, zt, spikes between the last quarter of 2011 and the Summer of 2012. In the wake of Mario Draghi’s (president of the ECB) famous speech in July 2012, in which it was announced that the ECB would do whatever it takes to preserve the Euro within its mandate, the EURUSD exchange rate and the eurozone sovereign credit markets stabilized, which caused both ztand ltto decrease rapidly, for all sovereigns. The systematic component, which captures the part of the default intensity correlated with the foreign exchange rate, lt, exhibits two peaks (with the exception of Portugal), in early 2011 and by mid-2012. The systematic default component has a more stable path over the sample period compared to the idiosyncratic components that have stronger mean reversion and seem to capture transient credit risk shocks. Clearly, for all the sovereigns, mt, is highly time-varying, indicating that it is an important feature of our model to allow the mean-reversion level of ztto be stochastic. Consistent with this, we find considerable improvements in model fits when using a three-factor model instead of a two-factor model. For example, we find that a model in which zthas a constant mean-reversion level is not sufficiently rich to provide reasonable fits of the USD CDS term structure and the quanto CDS term structure. Using the estimated parameters and the filtered state variables, we compute modelimplied USD CDS premiums and quanto CDS spreads and compare them to their observed counterparts. We show in table 1.8 the summary statistics for the model pricing errors, both 35 in terms of root mean squared errors (RMSEs) and mean absolute pricing errors (APEs) in bps. The time-series fits are illustrated in Figures 1.8-1.9 at maturities of 1, 5, and 10 years. The average RMSE across the 1-10 years maturities for the USD CDS range from 23.2126.68 bps for Italy, Spain, and Ireland. The average RMSEs for Portugal, however, are significantly larger at 37.92 bps, especially the 1-year RMSE is comparatively large. Using the APE metric, the Portuguese fit is better, which indicates that large outliers are important contributors to its RMSEs. For all sovereigns, the general pattern is that the pricing errors decline in maturity, i.e., the shorter maturities are the most difficult to capture for the model. A likely explanation for this is that the short end is more volatile/noisy than the long end of the term structure, as shown in Table 1.4. The model seems to fit the quanto CDS premiums reasonably well, as seen from Figures 1.8-1.9. This is also reflected by relatively small average RMSEs for all sovereigns, with the lowest being 0.98 bps for Ireland and the largest being 4.90 bps for Spain. The RMSEs tend to increase in the maturity of the quanto CDS spread, most notably for Spain. From Figure 1.9, we see that for Spain, the model tends to underestimate the 10-year quanto CDS premium and overestimate the 10-year USD CDS premium. Such a bias, however, is not present for the other sovereigns and does not seem to be a general issue with the model. Overall, considering the large fluctuations in the CDS premiums over a relatively short sample period, we believe that the model performs well in capturing both the USD CDS and the quanto CDS dynamics across all tenors. As an example, to underline the strong time-variation of the CDS premiums over our sample period, the 1-year USD CDS premium for Portugal and Ireland range between 0.23%-23% and 0.07%-14.5%, respectively. Next, we use the model estimates to decompose quanto CDS spreads for Italy, Spain, Ireland, and Portugal into a currency/default covariance component and a crash risk component. We compute the covariance and crash risk component of the quanto spread as: FX/default covariance risk component = Sd ζ=1(t, T)−Sf ζ=1(t, T) (1.30) FX crash risk component = Sd(t, T)−Sf(t, T)−Sd ζ=1(t, T)−Sf ζ=1(t, T)(1.31) where Sd ζ=1(t, T)−Sf ζ=1(t, T) denotes the model-implied quanto spread assuming no currency crash at default. Hence, if crash risk accounts for the entire quanto spread, the covariance 36 months to 10 years. The riskless zero-coupon prices in EUR and USD are bootstrapped from their respective overnight index swap rates. For Italy, we study a USD-denominated bond that matures in February 2017, and for Spain we study two USD-denominated bonds with maturities in June 2013 and March 2018, respectively, i.e., the entire sample period from 2010-2016 is covered by a USD-denominated bond for both countries. Portugal, however, only has one USD-denominated bond traded in our sample period with maturity in March 2015. We calculate the observed quanto yield spread as in (1.35), which we refer to as the ”synthetic” quanto yield spread. Besides this, we compute a bond quanto yield spread, defined as the yield spread between a USD bond and a EUR bond with similar maturities corrected for the riskless interest rate differential: QYbond(t, T)≡yU obs(t, T)−yE obs(t, T)−¯rU(t, T)−¯rE(t, T)(1.38) The bonds that we use are specified in the footnote 6. One advantage with the measure specified in (1.38) is that it does not involve the extraction of a full term structure of zero-coupon prices and credit spreads. This spread, however, is a cruder measure than the synthetic quanto yield spread, since it does not take into account the term structure of the risky zero coupon prices, differences in coupon schemes, or maturity mismatch. The justification for this measure is that if there were no quanto effects, or other frictions, only the riskless interest rate differential drives the yield spreads across currency denominations. We would thus expect (1.38) to be close to zero if there are no quanto effects. In the presence of no frictions, the bond quanto yield spread is exactly zero for zero-coupon bonds 7, but it is not necessarily zero for coupon bonds. However, if the duration of the bond is short, the yield spread between a coupon bond and zero-coupon bond is close to zero, which is the case in our sample, where we consider 6The Italian EUR-denominated government bond matures on 1st of February 2017, ISIN: IT0004164775. 4% coupon semi-annual. The USD-denominated Italian government bond has maturity on 12th of June 2017. ISIN: US465410BS63, 5.375% coupon semi-annual. First Spanish bond couple: EUR-denominated government bond matures on January 31 th 2014, ISIN: ES00000121H0, 4.25% coupon semi-annual, and the USD-denominated June 17th 2013. ISIN: XS0363874081, 3.625% coupon semi-annual. Spain bond couple for latter period: EUR bond: 30th of July 2018 4.1% semi-annual coupon rate, and USD bond: maturity 6th of March 2018, 4% semi-annual coupon rate. Portugal bond couple: maturity EUR bond 15th oct 2014 PTOTEOOE0017 and 3.6% coupon rate semi-annual, USD bond maturity 25th march 2015 XS0497536598 and 3.5% coupon rate semi-annual. 7To see this, consider two risky zero-coupon bonds in EUR and USD: PE(t, T) and PU(t, T) and assume 43 only bond maturities of less than seven years 8. In Table 1.11, we report summary statistics of the observed quanto yield spreads. We divide the sample into a crisis period, from August 2010 to March 2013, and a post-crisis period March 2013 to April 2016. For Italy and Spain, the crisis period is characterized by positive and highly significant quanto yield spreads (i.e., t-statistics exceeding >5.42), with respective averages of 40.8 bps (59.7 bps) and 62.7 bps (99.0 bps) of the synthetic (bond) quanto yield spread. For these countries, the corresponding average model-implied quanto yield spreads are in the same order of magnitude of 61 bps and 59 bps. The Portuguese observed quanto yield spreads based on the synthetic and the bond method have respective means of 4.3 bps and 28.6 bps, which are both insignificantly different from zero. However, if restrict the sample period to August 2010 to July 2012, i.e., we consider the sample period prior to Draghi’s speech, then the synthetic quanto yield spread is significant for Portugal as well. In general, the Portuguese quanto yield spread is more noisy than for Spain and Italy and exhibits larger positive and negative swings. In the post-crisis period, we only study bonds issued by Italy and Spain, since there are no USD-denominated bond data for Portugal. In this period, the quanto yield spreads are much smaller (albeit still positive) and less significant compared to the crisis period. For Italy, the synthetic bond yield spread has en insignificant average of just 14.0 bps, and the spread is contained within a more narrow range compared to the crisis period, with a 95% percentile of 56 bps relative to 123 bps in the crisis period. Likewise is the average Spanish synthetic quanto yield spread smaller (33.3 bps) in the post-crisis period compared to the crisis period. The corresponding means of the model-implied quanto yield spreads are about 25 bps and 41 bps for Italy and Spain, our model thus captures the falling trend independence between the exchange rate and the default event (1τ>T ): PE(t, T ) = EQE texp −ZT t rE(s)ds1τ>T =EQU tXT Xt exp −ZT t rU(s)ds1τ>T  =EQU tXT XtEQU texp −ZT t rU(s)ds1τ>T ⇔PE(t, T ) PU(t, T )= exp RT trU(s)ds exp RT trE(s)ds ⇔yU(t, T )−yE(t, T )−¯rU(t, T )−¯rE(t, T )=0 8For example, for Italy, the 7-year EUR spread between the coupon bond and zero-coupon is always negative, with a minimum of −15 bps (3% in relative terms), and since the USD-bond is subject to the same bias, we presume that the bias’ affect in the quanto yield spread is small. 44 in the quanto yield spreads. Next, we test if the observed quanto yield spreads are explained by their model-implied counterparts. We find a significant and positive relation between observed quanto yield spreads and their model-implied counterparts during the peak of the European debt crisis, while they are insignificantly related in the post-crisis period. Our results indicate that a significant portion of the observed yield deviations between EUR and USD-denominated eurozone sovereign bonds is attributable to quanto risk and that quanto yield spreads do not necessarily reflect mispricings. Positive quanto yield spreads persist post-crisis, although much smaller compared to the crisis period, but they are seemingly caused by other factors, such as differences in liquidity and specialness associated with currency denomination (Corradin and Rodriguez-Moreno, 2016). Table 1.12 shows results from regressions of the observed quanto yield spreads, using both the synthetic (1.36) and the bond method (1.38), on the model-implied counterparts. We also include the 5-year quanto CDS spread in the regressions as an alternative measure for quanto effects. In the crisis period, for Spain and Italy, there is in general a significant positive relationship between the observed quanto yield spreads and the model-implied quanto yield spreads and the quanto CDS spreads. In particular for Spain, the slope coefficients of the model-implied quanto yield spread and the 5-year quanto CDS spread range between 1.24-1.93, with t-statistics between 2.99 and 6.60, and R2s ranging from 20.75%-30.54%. Likewise for Italy, there is a positive relation between the synthetic quanto yield spread and its model-implied counterpart and the 5-year quanto CDS spread both with slope coefficients close to unity, with respective t-statistics and R2s of: 1.43, R2= 8.66% and 2.07, R2= 17.06%. Using the bond method to derive the observed quanto yield spreads, we also find significant positive slope coefficients near unity. In this case, a substantial part of the variation in the observed quanto yield spreads are explained by quanto effects, with R2s between 15.90% and 29.19%. In the post-crisis period, we find an insignificant relation between observed and model-implied quanto yield spreads and quanto CDS spreads for both Spain and Italy. To conclude, our findings suggest that joint modeling of credit risk and currency risk is a key ingredient in understanding bond yields across currency denominations and that it becomes increasingly important when sovereign bond markets are under distress. In 45 accordance with our above findings for the eurozone, Du and Schreger (2016) construct a quanto yield spread for emerging market sovereign bonds and find that the covariance between currency and credit risk explains a significant part of the quanto yield spread. These findings suggest that our model could be useful for understanding the variation in yield spreads across currency denominations in emerging bond markets, we leave this topic for future research. 1.8 Conclusion In this paper we analyze quanto spreads in the context of eurozone sovereign CDS contracts. We develop a discrete-time no-arbitrage model, which illustrates how, even in a frictionless setting, quanto CDS spreads arise as a compensation for exposure to two risk factors. The first risk factor is an FX crash risk factor, which captures the market’s (risk-neutral) anticipation of a large adverse jump in foreign currency (EUR) against domestic currency (USD) in the event of a sovereign default. The second factor, the currency/default risk covariance factor, captures the propensity for the EUR to depreciate (appreciate) against the U.S. dollar when eurozone sovereign credit risk rises (declines). Our simple model allows for simple comparative statics. To estimate the relative importance of these factors, we propose an affine term structure model that allows us to distinguish between the two effects and capture their time-variation. We use our model to decompose the quanto spreads for Spain, Italy, Portugal, and Ireland, and find that both covariance and currency crash risk contribute substantially to quanto CDS spreads. The covariance risk factor is highly time-varying and increases in times of distress, when the currency and credit markets are volatile and co-move. However, the implied currency crash risk from sovereign defaults differ greatly in the four cases. We estimate the (risk-neutral) expected jump in the EURUSD conditional on sovereign default for Spain and Italy to 15.6% and 9.6% which, consistent with our intuition, is significantly larger than the estimated currency jump size of about 5% in the event of a Portuguese or Irish default. We document a significant risk premium associated with currency/default covariance risk and currency crash risk, i.e., a risk premium associated with selling protection in the ’expensive’ currency (USD) and buying protection in the 46 ’cheap’ currency (EUR). This risk premium is especially large for Spain and Italy, where it accounts for most of the quanto CDS spread. Finally, we provide evidence that quanto yield spreads, which are differences in yields on USD and EUR-denominated bonds, are significantly related to quanto effects estimated based on our model. This highlights the importance of taking into account currency crash risk and covariance risk when assessing the relative pricing of bonds across currency denominations. 47 1.9 Figures [1, λ0] t= 0 C(λ0)u, λU C(λ0)u, λD δu C(λ0)u−1, λU C(λ0)u−1, λD δu−1 t= 1 C(λ0)C(λU)u2, λU C(λ0)C(λU)u2, λD C(λ0)δu2 C(λ0)C(λU), λU C(λ0)C(λU), λD C(λ0)δ t= 2 1−λUQ11 1−λUQ10 λUq 1−λUQ01 1−λUQ00 λU(1 −q) (1 −λ0)Q11 (1 −λ0)Q10 λ0q (1 −λ0)Q01 (1 −λ0)Q00 λ0(1 −q) Figure 1.1: Two-period model of the default probability and the exchange rate. This figure illustrates the joint dynamics of the default probability and the exchange rate over two periods. At time 0, the exchange rate is 1 and default occurs with a probability of λ0. If default occurs, the exchange rate is adjusted by δrelative to the state of the exchange rate if there were no crash risk. Conditional on survival, which occurs with probability 1 −˜ λ, the exchange rate is adjusted by the compensating factor C(˜ λ), where ˜ λ=λUor ˜ λ=λD. Simultaneously, if survival occurs, a new one-period default probability is drawn which takes either a high value λUor a low value λD, and a relative one-period change of the exchange rate is realized taking two possible values (u, u−1). That is, in total there are four possible outcomes for the default probability and the exchange rate change at each node. The joint probability distribution for reaching each of those four possible states are specified in equations (1.1)-(1.2). There are the same possible states in each survival node. Due to space constraints, we only show the possible states at time 2 starting from the survival node in which the default probability and the exchange rate went up ((λU, u)). 48 0 20 40 60 80 100 Expected Depreciation δ (%) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 CDS premium (%) Domestic CDS Foreign CDS Figure 1.2: Currency crash risk induced quanto CDS spreads. This figure illustrates the impact of an expected depreciation upon default, δ, on the premiums of CDS contracts denominated in foreign and domestic currency. The blue graph is the CDS premium in domestic currency, and the red graph is the CDS premium in foreign currency on the same underlying reference entity. The CDS premiums are computed based on a model with fixed default probability and a fixed risk-neutral expected depreciation upon default. Interest rates do not affect CDS premiums in the model when the default probability is constant. 49 1 2 3 4 5 6 7 8 9 10 Maturity 0 10 20 30 40 50 60 70 80 Bps Portugal Quanto CDS Spread, δ=0.95 Quanto CDS Spread, δ=1 1 2 3 4 5 6 7 8 9 10 Maturity 0 10 20 30 40 50 60 70 80 Bps Ireland Quanto CDS Spread, δ=0.93 Quanto CDS Spread, δ=1 1 2 3 4 5 6 7 8 9 10 Maturity 0 10 20 30 40 50 60 70 80 Bps Italy Quanto CDS Spread, δ=0.85 Quanto CDS Spread, δ=1 1 2 3 4 5 6 7 8 9 10 Maturity 0 10 20 30 40 50 60 70 80 90 Bps Spain Quanto CDS Spread, δ=0.84 Quanto CDS Spread, δ=1 Figure 1.3: Term structures of calibrated quanto CDS spreads. This figure illustrates the term structure of model-generated quanto CDS spreads at maturities of one to ten years. The quanto spread is the difference between the CDS premiums on the same reference entity denominated in USD and EUR. The parameters are calibrated to match the empirical average 5-year EUR and USD CDS premiums, the 1-year EURUSD risk-neutral volatility, and the correlation between the 5-year USD CDS premium and the EURUSD spot exchange rate. All model parameters are assumed fixed, and the calibration period is August 2010 to August 2012. The blue graph illustrates the quanto spread at different maturities. The orange graph is the share of the quanto spread stemming from default/currency covariance risk, i.e., the case of δ= 1. The recovery rate is assumed to be 40%, and the choice of foreign and domestic interest rates has no impact on the quanto spread in the model. 50 Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 50 100 150 200 250 Bps Austria USD CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 0 20 40 60 80 Bps Austria Quanto CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 100 200 300 Bps Belgium USD CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 0 20 40 60 80 Bps Belgium Quanto CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 20 40 60 80 100 120 140 Bps Germany USD CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 0 20 40 60 80 Bps Germany Quanto CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 20 40 60 80 100 Bps Finland USD CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 0 10 20 30 Bps Finland Quanto CDS 1y 3y 5y 7y 10y Figure 1.4: USD CDS and quanto CDS spreads for Austria, Belgium, Germany, and Finland. This figure shows USD CDS premiums and quanto CDS spreads–defined as the difference between USD and EUR-denominated CDS premiums of the same underlying reference entity–for Austria, Belgium, Germany, and Finland. The sample period is August 2010 to April 2016 and comprises 1402 daily observations obtained from Markit. 51 Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 50 100 150 200 250 Bps France USD CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 0 20 40 60 80 100 Bps France Quanto CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 200 400 600 800 1000 1200 1400 Bps Ireland USD CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 0 20 40 60 80 100 Bps Ireland Quanto CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 100 200 300 400 500 600 Bps Italy USD CDS 1y 3y 5y 7y 10y Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16 0 20 40 60 80 100 Bps Italy Quanto CDS 1y 3y 5y 7y 10y Figure 1.5: USD CDS and quanto CDS spreads for France, Ireland, and Italy. This figure shows USD CDS premiums and quanto CDS spreads–defined as the difference between USD and EURdenominated CDS premiums of the same underlying reference entity–for France, Ireland, and Italy. The sample period is August 2010 to April 2016 and comprises 1402 daily observations obtained from Markit. 52 1.10 Tables Table 1.1: Model parameters calibrated to moments for the EURUSD and CDS premiums. This table shows parameter values for the model calibrated to average 5-year quanto CDS spreads for Portugal, Ireland, Italy, and Spain. First column reports the calibrated values for ρ, which is estimated for each sovereign as the correlation between percent-wise changes in the 5-year USD-denominated CDS premium and the EURUSD exchange rate. The second column reports the value for euwhich equals the average risk-neutral volatility derived from EURUSD options maturing in one year. The third column shows annualized standard deviations of daily percent-wise changes in the USD-denominated CDS premiums, and the fourth and fifth columns report the average 5-year CDS premiums of the USD and EUR-denominated contracts, respectively. All moments are estimated over the period August 2010 to August 2012. ρ= Corr∆SU(t, 5y),∆Xt)eu=σF X Std∆SU(t, 5y)meanSU(t, 5y)meanSE(t, 5y) Portugal −36% 14.6% 57% 8.51% 7.81% Ireland −38% —— 51% 6.48% 5.84% Italy −56% —— 68% 3.30% 2.67% Spain −57% —— 73% 3.44% 2.68% 59 Table 1.2: One-period example with crash risk in synthetic bond price. This table shows the payoffs for a long position in a USD zero-coupon bond and a short position in a synthetic USD zero-coupon bond—which is short a EUR zero-coupon bond and long a forward contract. There are no recovery payments on the bonds. All contracts are initiated at time 0 and expire at time 1. The riskless interest rates are 0, the exchange rate is 1 at time 0, and the forward exchange rate is 1. The default states are assumed to be associated with a 50% depreciation in the EUR against the USD. t= 0 No default at t= 1 Default at t= 1 Long USD Bond −PUSD 1 USD 0 USD Short Synthetic USD Bond PUSD,synth −1 EUR + 1 EUR - 1 USD 1 EUR -1 USD Cash Flow L/S in USD 0 0 −0.5 USD 60 Table 1.3: Crash risk and currency/default covariance risk in bond yields. This table compares yields on domestic and synthetic domestic coupon bonds derived via the discrete-time model. The synthetic domestic bond consists of a long position in a foreign bond that pays 1 at t= 1, . . . 5 and 100 at maturity in foreign currency, and a short position in currency forward contracts that match those payments. The yield of the synthetic bond is reported in the first row with crash risk, and in the second row under the assumption of no crash risk. The third row shows the yield on a domestic coupon bond which pays 1 at t= 1, . . . 5 and 100 at maturity. Rows 4-6 show the corresponding prices of the coupon bonds and the prices for each of the coupon payments. All bond payments are conditional on no default, and there are no recovery payments. The parameters used in the model are calibrated to 5-year EUR and USD CDS data for Spain and EURUSD moments (as reported in Table 1.1). t= 1 t= 2 t= 3 t= 4 t= 5 Yield Synthetic Coupon Bond, (δ= 0.84) 4.46 % Yield Synthetic Coupon Bond, (δ= 1) 5.37 % Yield Domestic Coupon Bond 5.73 % Price Synthetic Domestic Coupon, (δ= 0.84) 0.95 0.91 0.88 0.84 80.41 Price Synthetic Domestic Coupon, (δ= 1) 0.95 0.90 0.85 0.81 76.99 Price Domestic Coupon 0.95 0.89 0.85 0.80 75.67 61 Table 1.4: Summary statistics for USD CDS premiums and quanto CDS premiums. This table reports sample estimates of the means and standard deviations of the USD-denominated CDS premiums and quanto CDS premiums for Austria, Belgium, Germany, Finland, France, Ireland, Italy, Netherlands, Portugal, and Spain. For each sovereign, the quanto CDS premium is defined as the difference in premiums on a USD and a EUR-denominated CDS contract at the same maturity. Panel A reports the time-series means of the premiums of the USD-denominated CDS contracts and the quanto CDS contracts in basis points at maturities of 1-10 years. Panel B reports the standard deviations of the premiums on the USDdenominated CDS contracts and the quanto CDS contracts in percentages at maturities of 1-10 years. The sample consists of daily quotes obtained from Markit from August 2010 to April 2016 (1402 observations for each series). Panel A: Mean in bps USD CDS Quanto CDS 1 yr 3 yrs 5 yrs 7 yrs 10 yrs 1 yr 3 yrs 5 yrs 7 yrs 10 yrs AUS 26.74 43.96 65.09 78.38 89.88 8.75 14.64 22.20 27.07 30.79 BEL 53.78 82.84 108.98 124.08 136.27 12.12 21.96 30.55 35.24 39.25 GER 11.26 21.72 39.26 51.72 63.08 3.70 9.07 16.90 21.48 25.51 FIN 12.87 21.05 34.27 43.98 52.93 2.54 5.00 8.49 11.10 12.96 FRA 29.01 54.02 82.49 100.45 115.90 8.39 18.60 28.90 34.17 38.42 IRE 271.14 303.88 297.09 298.27 291.82 22.59 31.35 36.46 38.64 40.55 ITA 132.99 193.37 224.06 240.09 250.72 21.84 32.45 38.39 41.01 43.11 NET 17.50 29.83 48.56 61.00 72.15 5.06 10.24 17.51 22.31 25.97 POR 437.00 489.10 478.21 471.72 456.50 34.43 37.71 41.04 42.72 44.56 SPA 142.89 198.90 225.70 239.26 247.96 30.16 41.60 47.68 50.36 52.84 Panel B: Standard Deviation in % USD CDS Quanto CDS 1 yr 3 yrs 5 yrs 7 yrs 10 yrs 1 yr 3 yrs 5 yrs 7 yrs 10 yrs AUS 0.35 0.45 0.53 0.50 0.49 0.12 0.15 0.17 0.17 0.17 BEL 0.71 0.85 0.85 0.78 0.70 0.17 0.21 0.23 0.22 0.21 GER 0.13 0.19 0.27 0.27 0.27 0.05 0.08 0.13 0.14 0.15 FIN 0.13 0.16 0.19 0.18 0.17 0.03 0.03 0.03 0.04 0.05 FRA 0.35 0.47 0.55 0.52 0.50 0.12 0.18 0.23 0.24 0.24 IRE 3.48 3.37 2.82 2.52 2.20 0.25 0.27 0.26 0.25 0.24 ITA 1.34 1.39 1.33 1.22 1.11 0.20 0.24 0.25 0.25 0.24 NET 0.20 0.25 0.31 0.31 0.30 0.07 0.09 0.11 0.13 0.14 POR 5.03 4.55 3.59 3.08 2.61 0.39 0.29 0.25 0.23 0.22 SPA 1.27 1.45 1.41 1.30 1.17 0.28 0.33 0.34 0.33 0.32 62 Table 1.5: Summary statistics for currency options data. This table reports the means and standard deviations of implied volatilities for EURUSD delta-neutral straddles (STR), EURUSD 10 and 25-delta risk reversals (RR10 and RR25, respectively), and EURUSD 10 and 25-delta butterfly spreads (BF10 and BF25, respectively). All quantities are reported in percentages. The data are obtained from Bloomberg and the sample consists of daily quotes from August 2010 to April 2016 (1402 observations for each series). Mean (%) Std (%) 1 mo 2 mo 3 mo 6 mo 9 mo 1 yr 1 mo 2 mo 3 mo 6 mo 9 mo 1 yr STR 9.83 9.93 10.01 10.25 10.42 10.56 2.74 2.68 2.64 2.57 2.52 2.47 RR10 -1.70 -2.26 -2.70 -3.20 -3.47 -3.61 1.41 1.52 1.62 1.60 1.62 1.61 RR25 -1.02 -1.30 -1.51 -1.78 -1.90 -1.97 0.82 0.85 0.87 0.85 0.84 0.83 BF10 11.08 11.54 11.93 12.66 13.10 13.42 3.19 3.26 3.33 3.41 3.42 3.41 BF25 10.27 10.47 10.62 10.97 11.20 11.36 2.85 2.82 2.80 2.77 2.72 2.68 63 Table 1.6: Regressions of FX spot and implied volatility changes on eurozone sovereign credit risk. This table presents estimates from regressions of contemporaneous weekly changes in the EURUSD spot exchange rate and the EURUSD implied volatility on eurozone sovereign credit risk: ∆Xt=α+β∆PC1CDS t+εt,∆IVt=α+β∆PC1CDS t+εt EURUSD volatility is proxied by the 1-month implied volatility of a delta-neutral straddle, and eurozone credit risk is measured as the first principal component of weekly 5-year CDS premiums for 10 eurozone sovereigns. Columns 1-2 show the results of the regressions using the full sample, columns 3-4 show the results from the crisis period (August 2010 to December 2012), and columns 5-6 show the results for the post-crisis period (January 2013 to April 2016). Newey and West (1987) t-statistics are reported in brackets, and the superscripts *, **, and *** indicate statistical significance at 10 %, 5 %, and 1 %, respectively. The currency spot and implied volatility data are from Bloomberg, and the CDS data are from Markit. The sample period is from August 2010 to April 2016 (281 weekly observations). 2010-2016 2010-2012 2013-2016 ∆X∆IV ∆X∆IV ∆X∆IV α-0.0001 -0.0000 0.0005 0.0007 -0.0003 -0.0003 [-0.27] [-0.06] [0.64] [0.97] [-1.38] [-0.80] β−0.0988∗∗∗ 0.2330∗∗∗ −0.1720∗∗∗ 0.5046∗∗∗ -0.0191 0.0672∗∗ [-3.69] [3.84] [-4.01] [7.70] [-1.34] [2.17] R20.081 0.121 0.162 0.272 0.002 0.032 64 Table 1.7: Parameter estimates for the proposed affine model. This table reports parameter estimates of the affine model specified in equation (1.16). The numbers in parentheses are standard errors of the estimates. The parameters are estimated using maximum likelihood estimation in conjunction with the unscented Kalman filter, using premiums on USD-denominated CDS and quanto CDS contracts with maturities of 1-10 years, and EURUSD option-implied volatilities at five strikes and six maturities spanning 1-12 months. Each time series consists of 281 weekly observations (each Wednesday) from August 2010 to April 2016. Intensity Parameters FX Parameters Ireland Italy Portugal Spain κl0.0326 0.2533 0.0308 0.0907 κv1.2129 (0.0027) (0.0367) (0.0133) (0.0597) (0.5890 ·10−3) θl0.1760 0.0018 0.0608 0.0188 θv0.0183 (0.0051) (0.0011) (0.0019) (0.0069) (0.9021 ·10−4) σl0.4392 0.2892 0.3484 0.4525 σv0.1452 (0.0099) (0.0154) (0.0081) (2.56 ·10−5) (0.3740·10−3) κP l0.0469 0.0005 0.0007 0.0001 κP v1.5935 (0.0136) (0.8737) (0.0101) (0.0119) (0.1305 ·10−3) θP l0.1295 0.0092 (0.0059) 0.0336 θP v0.0174 (0.0086) (0.0068) (0.0007) (0.0267) (0.9532 ·10−2) κz0.2620 0.2460 0.2450 0.1283 ρ-0.6817 (0.0073) (0.0298) (0.0053) (0.0208) (0.9032 ·10−3) σz0.0000 0.0013 0.3660 0.0445 σO0.8512 ·10−4 (0.0093) 0.0025 0.0031 (0.0617) (0.5815 ·10−3) κP z0.0037 0.0041 0.0000 0.0010 (0.0159) (0.1480) (0.0060) (0.1539) κm0.0035 0.0012 0.0241 0.0023 (0.0164) (0.0027) (0.0057) (0.0035) θm0.0000 0.0000 0.0043 0.0116 (0.0071) (0.0092) (0.0139) (0.0052) σm0.2200 0.1099 0.2059 0.1201 (0.0051) (0.0100) (0.0015) (0.0063) κP m0.0010 0.1641 0.0009 0.0639 (0.1690) (0.0967) (0.0037) (0.4256) θP m0.0000 0.0000 0.2615 0.1002 (0.0027) (0.0300) (0.0123) (0.6966) ζ-0.0502 -0.0960 -0.0543 -0.1559 (0.0041) (0.0050) (0.0018) (0.0012) l0(0.0098) (0.0105) (0.0042) (0.0033) (0.0003) (0.0035) (0.0008) (0.0032) z00.0015 0.0029 0.0272 0.0000 (0.002) (0.0107) (0.0040) (0.1965) m00.0435 0.0459 (0.0022) (0.0793) (0.0023) (0.0093) (0.0040) (0.0120) σU3.24 ·10−61.06 ·10−61.91·10−65.16·10−6 (4.40·10−6) (3.53 ·10−6) (2.91 ·10−6) (2.51 ·10−6) σUE 4.12·10−52.47·10−50.74·10−50.45·10−5 (4.97·10−6) (1.549 ·10−6) (2.22 ·10−6) (6.22 ·10−6) 65 Table 1.8: Summary statistics of model pricing errors for USD CDS and quanto CDS premiums. This table reports the root mean squared errors and mean absolute pricing errors for model-implied USD-denominated CDS premiums and quanto CDS premiums at maturities from 1-10 years. Both are reported in basis points (bps). The pricing error is defined as the difference between the observed CDS premium/quanto CDS premium and the model-implied CDS premium/quanto CDS premium (using the updated state variable). The model is estimated using maximum likelihood estimation in conjunction with the unscented Kalman filter using USD CDS data, quanto CDS data (both from Markit), and currency options data from Bloomberg. The sample consists of 281 weekly observations from August 2010 to April 2016. Panel A: Root Mean Squared Errors (bps) USD CDS Quanto CDS 1 yr 3 yrs 5 yrs 7 yrs 10 yrs Mean 1 yr 3 yrs 5 yrs 7 yrs 10 yrs Mean IRE 37.61 36.93 24.55 18.93 15.35 26.68 0.65 0.67 0.44 1.04 2.09 0.98 ITA 26.65 25.23 25.18 21.58 17.41 23.21 3.89 2.16 1.48 2.41 5.36 3.06 POR 50.50 45.82 36.81 28.35 28.12 37.92 3.48 1.77 0.43 1.42 4.70 2.36 SPA 26.27 23.61 20.42 18.70 34.04 24.61 0.15 1.92 5.79 7.66 8.99 4.90 Panel B: Mean Absolute Pricing Errors (bps) USD CDS Quanto CDS 1 yr 3 yrs 5 yrs 7 yrs 10 yrs Mean 1 yr 3 yrs 5 yrs 7 yrs 10 yrs Mean IRE 28.44 32.48 18.92 13.52 9.07 20.49 7.24 8.06 9.86 11.01 11.27 9.49 ITA 19.44 18.72 18.76 15.58 13.97 17.29 6.82 6.31 6.90 7.31 8.80 7.23 POR 30.93 36.70 25.75 15.71 18.12 25.44 11.02 7.22 6.74 7.61 9.43 8.40 SPA 19.81 17.18 15.11 13.58 26.70 18.48 3.81 5.13 7.20 8.29 9.13 6.71 66 Table 1.9: Summary statistics for decompositions of quanto CDS spreads. This table reports summary statistics for model decompositions of quanto CDS spreads into a covariance risk component and a crash risk component. Panel A reports the mean and the maximum of the covariance component in basis points (bps) over the full sample period. Panel B reports the mean and maximum share for the covariance component of the total quanto CDS spread over the full sample. Panel C and D report the same quantities but for the debt crisis period (August 2010 to December 2012). The model is estimated using maximum likelihood estimation in cojunction with the unscented Kalman filter based on USD CDS data, quanto CDS data (both from Markit), and currency options data from Bloomberg. The sample consists of 281 weekly observations from August 2010 to April 2016. Panel A: Full sample (August 2010 - April 2016) Mean covariance component (bps) Max covariance component (bps) 1 yr 3 yrs 5 yrs 7 yrs 10 yrs 1 yr 3 yrs 5 yrs 7 yrs 10 yrs IRE 8.09 18.95 23.49 25.55 27.25 31.26 59.04 60.60 57.23 54.62 ITA 3.98 12.08 16.35 16.84 15.40 14.37 42.19 55.23 55.15 48.39 POR 5.98 13.02 15.16 15.54 15.51 32.24 64.36 70.97 69.32 64.56 SPA 5.66 9.86 9.24 8.03 6.77 24.06 43.61 38.51 30.23 21.89 Share of spread from covariance risk Max share of spread from covariance risk 1 yr 3 yrs 5 yrs 7 yrs 10 yrs 1 yr 3 yrs 5 yrs 7 yrs 10 yrs IRE 0.54 0.72 0.75 0.77 0.79 0.82 0.96 0.98 0.98 0.99 ITA 0.20 0.34 0.38 0.38 0.37 0.39 0.60 0.65 0.65 0.62 POR 0.21 0.31 0.35 0.37 0.40 0.56 0.72 0.76 0.78 0.78 SPA 0.13 0.17 0.20 0.21 0.21 0.35 0.46 0.40 0.31 0.21 Panel B: Debt Crisis Period (August 2010 - December 2012) Mean covariance component (bps) Max covariance component (bps) 1 yr 3 yrs 5 yrs 7 yrs 10 yrs 1 yr 3 yrs 5 yrs 7 yrs 10 yrs IRE 15.78 32.27 35.81 36.04 36.02 31.26 59.04 60.60 57.23 54.62 ITA 7.77 23.14 30.70 30.92 27.34 14.37 42.19 55.23 55.15 48.39 POR 11.95 24.62 27.18 26.57 25.11 32.24 64.36 70.97 69.32 64.56 SPA 12.49 22.00 18.35 13.23 8.18 24.06 43.61 38.51 30.23 21.89 Share of spread from covariance risk Max share of spread from covariance risk 1 yr 3 yrs 5 yrs 7 yrs 10 yrs 1 yr 3 yrs 5 yrs 7 yrs 10 yrs IRE 0.37 0.54 0.58 0.60 0.62 0.61 0.74 0.76 0.77 0.78 ITA 0.27 0.48 0.53 0.53 0.50 0.39 0.60 0.65 0.65 0.62 POR 0.23 0.35 0.38 0.39 0.40 0.55 0.72 0.76 0.78 0.78 SPA 0.22 0.30 0.25 0.18 0.11 0.35 0.46 0.40 0.31 0.21 67 Table 1.10: Summary statistics for risk premiums of USD CDS and quanto CDS. This table shows risk premiums associated with holding USD CDS and quanto CDS for Ireland, Italy, Portugal, and Spain. Panel A reports the mean risk premiums for holding USD CDS and quanto CDS in basis points at maturities of 1-10 years. Panel B reports the average risk premiums for USD CDS and quanto CDS as a fraction of total spreads. The model is estimated using maximum likelihood estimation in conjunction with the unscented Kalman filter based on USD CDS data, quanto CDS data (both from Markit), and currency options data from Bloomberg. The sample consists of 281 weekly observations from August 2010 to April 2016. Panel A: Mean risk premium in bps USD CDS Quanto CDS 1 yr 3 yrs 5 yrs 7 yrs 10 yrs 1 yr 3 yrs 5 yrs 7 yrs 10 yrs IRE 0.66 1.90 2.34 1.68 -3.18 0.92 2.36 3.14 3.44 3.33 ITA 36.60 96.01 134.81 156.92 171.41 3.76 11.46 15.94 16.88 15.81 POR 55.96 146.97 211.29 251.03 278.33 3.00 7.96 11.18 12.83 13.76 SPA 28.84 76.59 114.31 144.67 177.44 2.85 6.64 8.48 9.25 8.56 Panel B: Mean risk premium as a fraction of spread USD CDS Quanto CDS 1 yr 3 yrs 5 yrs 7 yrs 10 yrs 1 yr 3 yrs 5 yrs 7 yrs 10 yrs IRE 0.03 0.05 0.06 0.06 0.05 0.25 0.17 0.15 0.14 0.12 ITA 0.38 0.58 0.66 0.69 0.72 0.63 0.69 0.73 0.75 0.77 POR 0.42 0.57 0.64 0.69 0.75 0.25 0.42 0.40 0.39 0.38 SPA 0.28 0.48 0.59 0.65 0.71 0.49 0.51 0.61 0.45 0.67 68 Pt fEQf 0(1τ>t) = Pt dEQ 0(Xt1τ>t) =Pt dEQ 0 t−1 Y k=0 (1 −λk)C(λk)Xk+1 Xk! =Pt dEQ 0 t−1 Y k=0 (1 −δλk)Xk+1 Xk! =Pt d(1 −δλ0)EQ 0X1 X0t−1 Y i=1 EQ 0(1 −δλk)Xk+1 Xk =Pt d(1 −δλ0)F t−1 Y i=1 X i,j=0,1 Qij(1 −δ˜ λj)u2i−1! =Pt d(1 −δλ0)F X i,j=0,1 Qij(1 −δ˜ λj)u2i−1!t−1 (1.47) Next, we calculate an expression for the last term in equation (1.47) by plugging in the Qijs: X i,j=0,1 Qij(1 −δ˜ λj)u2i−1!=quqλ1−δλU+ (1 −qλ)1−δλD + (1 −q)u−1qλ1−δλU+ (1 −qλ)1−δλD +quA11−δλU−1−δλD + (1 −q)u−1A01−δλD−1−δλU =qu + (1 −q)u−1qλ1−δλU+ (1 −qλ)1−δλD +quA1δ(λD−λU)−(1 −q)u−1A0δ(λD−λU) =F(1 −δ¯ λ)−Kδρ u−u−1λU−λD≡L(1.48) where K=pqqλ(1 −q)(1 −qλ). In the last equal sign, we use the no-arbitrage condition of a one-period forward contract, F=qu + (1 −q)u−1, and the fact that qA1= (1 −q)A0= ρpqqλ(1 −q)(1 −qλ). Next step is to express EQf 0(1τ=t) in terms of EQf 0(1τ>t) . First, from the derivations above, we can express the premium payments on the compact form: 75 Pt fEQf 0(1τ>t) =      Pt dL0if t= 1 Pt dL0Lt−1if t≥2 where L0=F(1 −δλ0) and Land Kare defined above. In order to compute EQf 0(1τ=t) for t≥2, in terms of domestic currency, we express it in terms of differences between defaultable zero-coupon bonds in foreign currency: Pt fEQf 0(1τ=t) = Pt dEQ 0Xt X0 1τ=t =Pt dEQ 0Xt X0 1τ>t−1−Pt dEQ 0Xt X0 1τ>t =Pt dL0F·Lt−2−Lt−1=Pt dL0Lt−2F−L Above, we have used EQ 0Xt X01τ>t−1=FEQ 0Xt−1 X01τ>t−1. Thus, we can express the protection leg payments on the following compact form: Pt fEQf 0(1τ=t) =      Pt dδλ0Fif t= 1 Pt dL0Lt−2(F−L) if t≥2 (1.49) We then obtain the expression for the foreign CDS premium in (1.46) by plugging in the compact form expressions for the premium and protection leg payments, and make use of the expression for a geometric series: Sf(0, t) = (1 −R)PT t=2 Pt dL0Lt−2(F−L) + Pdδλ0F PT t=1 Pt dL0Lt−1 = (1 −R)P2 d(F−L)L0PT t=2(PdL)t−2+Pd(F−L0) L0PdPT t=1(PdL)t−1 = (1 −R)P2 d(F−L)L0PT t=2(PdL)t−2+Pd(F−L0) L0PdPT t=1(PdL)t−1 = (1 −R)P2 d(F−L)L01−(LPd)T−1 1−LPd+Pd(F−L0) PdL01−(LPd)T 1−LPd 76 Proof of Proposition 1 and 2 The domestic CDS premium is unaffected by changes in the severity in foreign currency at default, δ, hence all we need to show is that the foreign CDS premium in (1.46) is increasing in δsuch that the quanto spread, QS(0, T) = Sd(0, T)−Sf(0, T), is decreasing in δ. Evidently both L0and Lare decreasing functions in δ(holding any other parameters fixed), so if we can show that the CDS premium is decreasing in L0and L, we are done. First, we split the CDS premium up in two expressions: Sf(0, T) = (1 −R)P2 d(F−L)L01−(LPd)T−1 1−LPd+Pd(F−L0) L0Pd1−(LPd)T 1−LPd =(1 −R) Pd       (F−L) P2 d1−(LPd)T−1 1−(LPd)T | {z } A + PdF L0−1 1−(LPd)T 1−LPd | {z } B       Next, we show that both Aand Bare decreasing in δ. Consider the expression A. Since the riskless bond is assumed to be more expensive than a risky bond, we have F−L > 0 and 1−LPd>0. This implies that Ais decreasing in Lif and only if (1−(LPd)T−1) 1−(LPd)Tis decreasing in L, since F−Lobviously is decreasing in L. We show that (1−(LPd)T−1) 1−(LPd)Tis indeed decreasing in Lby defining the function: f(m) = 1−(mPd)T−1 1−(mPd)T Differentiating fwith respect to myields f0(m) = − (Pdm)t(Pdm)t−tPdm+t−1 Pdm2(Pdm)t−12 From this expression, we see that f0is negative if and only if (Pdm)t−tPdm+t−1 is positive, which is indeed the case, since this function is strictly convex with a minimum of 0 at m=1 Pd. Hence, we showed that the expression Ais decreasing in Mand hereby in δ 77 as well. An analogue argument can be used to show that 1−(LPd)T 1−LPd−1>0 is decreasing in L and hence in δ. Likewise is F L0−1>0 and decreasing in L0and therefore in δ. Hence, the expression B is decreasing in δas a product of two positive monotonically decreasing functions in δ. The proof for Proposition 2 is conducted in an analogous manner to the proof of Proposition 1. In Proposition 1, we show that the quanto spread is decreasing in L, and since L=F(1−δλ)−Kρ (u−u−1)δλU−λDis decreasing in ρ, then the foreign CDS premium increases in ρ. Evidently from the expression of L,Lis increasing in λU−λDif ρ < 0. The foreign CDS premium is therefore decreasing (increasing) in λU−λDwhen ρ < 0 (ρ > 0). 78 Derivation of the Expressions (1.9)-(1.10) First, the expression (1.9) follows immediately from (1.45) with λU=λD=λand ¯ λ=λ, where λis the fixed probability of default. Hence it follows that for any maturity T: Sd(0, T) = (1 −R)λ 1−λ In order to derive the foreign-denominated CDS premium in the presence of crash risk and fixed default risk, we first notice that L0=L= (1 −δλ)F. Inserting this into (1.46) gives Sf(0, T) = (1 −R) P2 dF1−1−δλF1−δλ1−F(1−δλ)PdT−1 1−F(1−δλ)Pd+PdF−F(1 −δλ) F1−δλ1−F(1−δλ)PdT−1 1−F(1−δλ)Pd = (1 −R) P2 fδλ (1 −δλ)1−((1−δλ)Pf)T−1 1−(1−δλ)Pf+Pfδλ (1 −δλ)Pf((1−δλ)Pf)T 1−(1−δλ)Pf= (1 −R)δλ 1−δλ The last equal sign follows from repeating the exact same calculations that led us to equation (1.45), with Pdreplaced with Pfand ¯ λreplaced with δλ. 79 1.12 Appendix: Affine Model 1.12.1 Market Price of Risk In this section we provide a proposition which help us specify the pricing kernel between the data-generating measure, the domestic measure, and the foreign measure. Cheridito et al. (2007) show that if an affine diffusion exists under a measure M0, and it does not hit the boundary of the state space, then there also exists an affine diffusion under a measure M1 which does not hit the boundary of the state space. More formally, they show (in the case of affine models without jumps) that if the drift and diffusion functions under M0and M1 both satisfy the boundary non-attainment condition and the existence condition9, then a true martingale exists defining the measure change from M0to M1. Lemma 1. Assume that µM0, σand µM1, σsatisfy the boundary non-attainment condition and the existence condition. Define the Radon-Nikodym derivative from M0to M1: Lt=−Lt−γtdWM0 t+Lt− K X i=1 (dZM0 i,t +λM0 i,t ζidt) Where dZM0 i,t is a pure jump process with intensity λiand the jump size distribution with mean jump size ζi. The jump times for Ziare serially and cross-sectionally independent. Define for j= 0,1: µMj:D→Rn, µMj(y) = aMj+bMjy, σ :D→Rn×n, σ(y)σT(y) = aij +bijy LMj:D→Rn, LMj(y) = lMj 0+lMj 1y Then the following three statements hold: 9The existence criterium is a necessary restriction on µ, σ, λ and Din order for an SDE to have a solution. Essentially the matrix σ(Yt)σT(Yt) has to be positive definite on the interior of the state space and positive semi-definite on the closure of state space. In order for the latter to be fulfilled the drift term has to be positive on the closure of Dand σ(Yt)σT(Yt) has to approach the 0-matrix. These two requirements make sure that σ(Yt)σT(Yt) is positive definite on Dand does not fail to be positive semi-definite on the closure of D. The boundary non attainment condition makes sure that the volatility for each coordinate in Ytremains strictly positive. For a detailed discussion of the existence of a solution to SDEs, see Duffie and Kan (1996) and Cheridito et al. (2007). 80 1. There exists a stochastic process Ytthat solves the SDE: Yt=Y0+µM0(Yt)dt +σ(Yt)dWM0 t 2. There exists a measure M1equivalent to M0such that: Yt=Y0+µM1(Yt)dt +σ(Yt)dWM1 t 3. The jump intensities and drifts under M0and M1are related as: λM1 i(Yt) = (1 + ζi)λM0 i(Yt), µM1(Yt) = µM0(Yt)−σ(Yt)γt Proof. Cheridito et al. (2007) show that continuous process: dLC t=−γtdWM0 tis indeed a true martingale with EM0 t(LC T) = 1, provided that the existence and boundary nonattainment condition holds under both M0and M1. The compensated jump process Zi,t + λi,tζi,t is also a true M0martingale, since the mean jump size for each Zi,t is bounded and only exhibits a finite number of jumps. Hence the process PK i=1(dZM0 t,i +λM0 i,t ζidt) is a true M0-martingale since it is a finite sum of martingales. The process Ltis therefore a true martingale and hence 1.-3. follows from Girsanov’s theorem for jump processes. 1.12.2 Pricing of CDS in Affine Framework Pricing of Domestic CDS All the state-variables that are used to price the domestic CDS premium are independent. This makes the expressions for the ordinary differential much more simple, since the variance-covariance structure of the state-variables is a diagonal matrix. Therefore, we can represent the system of ordinary differential equations used for computing (1.17)-(1.18) for the domestic denominated CDS as ∂β(t, T) ∂t =ω−KT 1β(t, T)−1 2Hβ(t, T)◦β(t, T),α(t, T) ∂t =−KT 0β(t, T) (1.50) ∂B(t, T) ∂t =−KT 1B(t, T)−1 2Hβ(t, T)◦B(t, T),A(t, T) ∂t =−KT 0B(t, T) (1.51) Where ◦is the Hadamard product, and 81 ω=     1 1 0      , K0=     κlθl 0 κmθm      , K1=     −κl0 0 0−κzκz 0 0 −κm      , H =     σ2 l0 0 0σ2 z0 0 0 σ2 m      The boundary conditions are α(T, T) = 0, β(T, T) = [0,0,0], A(T, T) = 0 and B(T, T) = [1,1,0] Pricing of Foreign CDS The foreign CDS premium is a bit more involved than the domestic CDS premiums but also fits into the affine framework. Define the vector βj(t, T) = [βv(t, T), βl(t, T), βz(t, T), βm(t, T)], where βj(t, T) corresponds to the beta for state variable j, then the ordinary differential equation for state variable jis given by: ∂βj(t, T) ∂t =ω−KT 1βj(t, T)−1 2βj(t, T)Hjβj(t, T),αj(t, T) ∂t =−KT 0βj(t, T) (1.52) ∂Bj(t, T) ∂t =−KT 1Bj(t, T)−1 2βj(t, T)HjBj(t, T),Aj(t, T) ∂t =−KT 0Bj(t, T) (1.53) where: ω=         0 (1 + ζ) (1 + ζ) 0         , K0=         κf vθf v κlθl 0 κmθm         , K1=          −κf v0 0 0 1 2σlρθl θv1 21 2σlρθv θl1 2−κl0 0 0 0 −κzκz 0 0 0 −κm          Hv=          σ2 v1 2σlσvθl θv1 20 0 1 2σlσvθl θv1 20 0 0 0 0 0 0 0 0 0 0          , Hl=          01 2σlσvθv θl1 20 0 1 2σlσvθv θl1 2σ2 l0 0 0 0 0 0 0 0 0 0          Hz=         0 0 0 0 0 0 0 0 0 0 σ2 z0 0 0 0 0         , Hm=         0 0 0 0 0 0 0 0 0 0 0 0 000σ2 m         82 The boundary conditions are α(T, T ) = 0, β(T, T) = [0,0,0,0], A(T, T) = 0, and B(T, T) = [0,(1 + ζ),(1 + ζ),0,0] 1.13 Appendix: Estimation Approach We estimate the model in two steps. In the first step, we apply maximum likelihood estimation (MLE) in conjunction with the unscented Kalman filter to infer a time-series of the instantaneous currency volatility process vtand estimates of its risk-neutral and objective parameters ([κv, θv, σv, κP v, θP v]). We refer to section 1.13.1 for details on the Unscented Kalman filter and why we use this estimation approach. In this step, we only have one state variable, and the measurements consist of currency implied volatilities. We use a stochastic volatility model a la Heston (1993) as the currency options model, i.e, we assume that instantaneous currency volatility dynamics are unaffected by the jump components in the exchange rate arising from sovereign defaults specified in (1.15). Importantly, this does not mean that we ignore the correlation between sovereign credit and currency risk or the jump risk when pricing the sovereign CDS contracts, which is the focus of the analysis. For pricing the currency options and CDS premiums, the discount factors in Euro and U.S. dollar are needed, which we bootstrap from their respective overnight index swap rates. The model-implied option prices are derived using the Fast Fourier Transform of Carr and Madan (1999) which we then transform into implied volatilities using the Garman and Kohlhagen (1983) formula such that they are comparable to the observables. We use implied volatilities rather than option prices since these are more stable than option prices along the moneyness and maturity dimension (see e.g., Schwartz and Trolle (2009)). Denoting xtthe time tstate variable vector, then the measurement equation in the Kalman filter is given by yt=h(xt) + et(1.54) where ytis the vector of observables, h(xt) is the pricing function at state xt, and etis the vector of measurement errors. In this particular case: xt=vt,ytis the vector of observed implied volatilities, h(xt) is the vector of corresponding Heston (1993) implied volatilities, and etis a vector of IID Gaussian measurement errors with covariance matrix R. To reduce the number of parameters, we make the common assumption that the measurement errors 83 are cross-sectionally uncorrelated (i.e., Ris a diagonal matrix), and furthermore, we assume that the standard deviations of the measurement errors are identical for all options, σO. We approximate the distribution of vtwith a Gaussian distribution such that the moments of the Gaussian distribution match the first two moments of vt. All moments are computed by means of an Euler discretization, and we then cast the model into state space form xt=A+φxt−1+pQt−1εt, εt∼N(0,I) (1.55) where in this particular case A=κP vθP v·dt, φ =e−κPdt, Qt=σ2 vvt·dt (1.56) Through the UKF iterations, we obtain t−1 predictions of the observables at time t, ¯yt, and the corresponding prediction error covariance matrix ¯ Σyy,t. With those at hand, we can then express the log-likelihood function using the prediction error decomposition l(Θ) = N X t=1 −1 2log|¯ Σyy,t|− 1 2(yt−¯yt)T¯ Σ−1 yy,t(yt−¯yt) (1.57) where Nis the number of observations, using weekly sampling we have N= 281 observations. We then find the maximum likelihood estimate of the parameters by maximizing (1.57). In the second step, we estimate the parameters of the default intensities for one sovereign at the time using CDS premiums denominated in EUR and USD, now treating vtas observable and its parameters as given. In this step, we use MLE in conjunction with the UKF to filter out the default intensity state variables, [lt, zt, mt], and to estimate their objective and risk-neutral parameters. The measurements are the CDS premiums denominated in USD and the quanto CDS spread. In the pricing model, the USD contract is taken to be the domestic CDS contract, and the EUR contract is considered to be the foreign-denominated CDS contract. Their respective model-implied CDS premiums are henceforth derived according to (1.20), with 84 2.1 Introduction This paper proposes a novel method to compute currency factor exposures (betas) that are purely forward-looking and adjust immediately to new information. The currency market provides a unique opportunity to calculate forward-looking betas because covariances can be retrieved from cross-pair currency option prices without assuming any parametric structure on variances and correlations. In particular, the covariance between any pair of currencies against, say, the U.S. dollar, can be expressed in terms of their U.S. dollar variances and their cross-pair variance. In this manner, the exchange rate covariance structure can be constructed from option-based variances, and as a result, forward-looking betas of currency portfolios can be calculated. Forward-looking betas are unique to currencies because covariances in other major asset classes, such as stocks, cannot be derived from options, since there is no (liquid) market for options for which the payoff depends on the price evolution of two securities. Factor models have most commonly been used for stocks and bonds, but a growing literature, pioneered by Lustig, Roussanov, and Verdelhan (2011), has emerged, which explains currency risk premiums as exposures to factors built from currencies. In this literature, betas are estimated by means of rolling window regressions of realized currency returns on the realized systematic factors. The betas estimated using this method, however, suffer from a number of caveats. They are backward-looking, adjust slowly to new information, and the econometrician has to decide on which particular subset of the data to use for the estimation. In contrast, since the option-implied betas are inferred from the latest cross-section of option prices, they require neither historical data nor choices of estimation window and frequency. In order to compare the empirical properties of the option-implied betas to the rolling window betas, I use the dollar factor of Lustig, Roussanov, and Verdelhan (2011)—which is an equally weighted portfolio of all foreign currencies taking the perspective as a U.S. investor—as the systematic factor in currency excess returns. However, the methodology that I propose can be used for any currency risk factor. The excess return on the dollar factor is the excess return a U.S. investor receives from borrowing money at home and investing in all (developed) foreign currencies equally weighted, and it carries a significant risk premium and explains a large share of the time-series variation in exchange rates (Lustig, Roussanov, 91 and Verdelhan, 2011; Verdelhan, 2017). Lustig, Roussanov, and Verdelhan (2014) show that the dollar factor tends to appreciate (depreciate) whenever the average short-term foreign interest rates is above (below) the short-term U.S. interest rate. As a result, a conditional dollar factor, which is long the dollar factor whenever the average forward discount (U.S. minus average foreign interest rates) is negative and short otherwise, has collected a larger excess return than the (unconditional) dollar factor (Lustig, Roussanov, and Verdelhan, 2014; Verdelhan, 2017). Conditional on the average foreign discount, I find a significantly positive relation between ex-ante option-implied dollar factor betas and ex-post portfolio excess returns, while there is an insignificant relation when using rolling window betas. Interestingly, the optionimplied betas are strong predictors of portfolio excess returns, because they predict spot exchange rate changes of the portfolios, while rolling window betas exhibit no predictability of spot exchange rate changes. I provide evidence that this is because option-implied betas are more powerful and less biased predictors of realized betas than rolling window betas. Specifically, conditional on the average foreign discount, I sort currencies into portfolios on the basis of dollar factor betas, for each type of beta separately, and construct an HML dollar factor which dynamically buys high-beta and shorts low-beta currencies. At a 1month holding period, when using the option-implied betas, the HML dollar factor has a significant mean annualized excess return of 3.35 percent (Sharpe ratio of 0.41), where 2.35% stems from the spot change component. On the other hand, constructing the HML dollar factor based on 252-day rolling window betas leads to an insignificant mean excess return of 0.95 percent (Sharpe ratio of 0.11), with a spot change component of −0.18%. Thus, the difference in the HML dollar excess returns is entirely due to the fact that option-implied betas are stronger predictors of currency spot changes. The results for the beta-sorted portfolios do not necessarily imply that the option-implied betas are more accurate forecasters of realized currency returns. Betas could be inaccurately measured that would cause large model prediction errors in the time series and still properly rank currencies on betas. The option-implied betas, however, are not only better at ranking currencies on betas, they are also significantly better predictors of portfolio returns in the time series, with smaller mean squared model prediction errors across all portfolios and forecast horizons. Furthermore, using the rolling window betas to forecast portfolio 92 returns delivers biased predictions, whereas the option-implied beta predictions are virtually unbiased. The expected low-beta portfolios, based on rolling window betas, tend to exhibit larger realized returns than the expectation, and vice versa, the expected high-beta portfolios exhibit lower returns than expected. I show that this superior model performance when using option-implied betas is because they are stronger and less biased predictors of realized betas. At any forecast horizon, for both portfolios and individual currencies, the option-implied betas provide significantly smaller prediction errors than the rolling window betas. Furthermore, consistent with the prediction bias for portfolio excess returns, when using rolling window betas, the expected beta of (high) low-beta portfolios tends to be (smaller) larger ex-post than the expectation, while the option-implied betas deliver virtually unbiased predictions. 2.2 Related Literature There are, to the best of my knowledge, no papers that have studied option-implied betas in currencies, while there are several papers that use options to estimate betas in the equity literature, for example: French, Groth, and Kolari (1983); Siegel (1995); and more recently, Buss and Vilkov (2012); Chang, Christoffersen, Jacobs, and Vainberg (2011); Christoffersen, Fournier, and Jacobs (2017). Since there is not (yet) a liquid market for options that depend on the price evolution of two stocks, stock correlations cannot be implied out from options without assumptions. French, Groth, and Kolari (1983) suggest computing betas using a mixture of option-implied volatilities and correlations estimated from historical data. Chang, Christoffersen, Jacobs, and Vainberg (2011) compute purely forward-looking betas under the assumption that stock returns follow a linear factor model where idiosyncratic shocks have no skew, and Buss and Vilkov (2012) derive option-implied betas by parametrically linking risk-neutral and objective correlations (estimated from past returns). I add to this literature by computing betas in currency markets which only use option market information and require no distributional assumptions. This paper is related to the literature that uses option-based information to predict realized returns and moments. Busch, Christensen, and Nielsen (2011) find that implied volatilities are stronger predictors of realized volatilities than historical volatility estimates in 93 fixed income, equity, and foreign exchange. Jorion (1995) finds consistent results in currency markets for a different sample period. Buss and Vilkov (2012) show that option-implied betas are significantly better at explaining the cross-section of stock returns and predicting realized CAPM betas compared to rolling window betas. I contribute to this literature by showing that purely option-implied betas in currency markets are strong predictors of future realized currency returns and betas. The option-implied betas proposed in this paper can be used to estimate risk exposures in any currency factor model, for instance, in the models suggested by Lustig, Roussanov, and Verdelhan (2011), Lustig, Roussanov, and Verdelhan (2014), and Verdelhan (2017), who focus on the carry factor (long high interest rate currencies and short low interest rate currencies), and the dollar factor (long equal-weighted basket of all foreign currencies). Other notable currency factors include the momentum factor of Menkhoff, Sarno, Schmeling, and Schrimpf (2012b) and Asness, Moskowitz, and Pedersen (2013), the global volatility factor of Menkhoff, Sarno, Schmeling, and Schrimpf (2012a), and the international correlation dispersion factor of Mueller, Stathopoulos, and Vedolin (2017). The work of Verdelhan (2017) is perhaps closest to this paper. He documents that timevarying exposure to the dollar factor is of key importance in explaining the cross-section of currency returns and the time-series variation in currencies. I contribute to this paper by documenting that for the G10 currencies, option-implied betas better explain the crosssection of currency returns and exhibit smaller time-series predictions errors for realized returns and betas than the historical rolling window betas. More generally, this paper is related to the literature on time-varying currency risk premiums (Lustig, Roussanov, and Verdelhan, 2014; Mueller, Stathopoulos, and Vedolin, 2017; Sarno, Schneider, and Wagner, 2012). For instance, Lustig, Roussanov, and Verdelhan (2014) show that a static carry trade, which is long currencies with the highest average interest rates and short those with the lowest, only explains about one third of the returns to a dynamic carry trade, which is long-short based on time-varying betas to the carry factor. Therefore, a central theme in this literature is time-varying betas, which I show can be measured in real time using currency options for any given currency factor. This paper is related to the relatively scarce literature which uses cross-pair currency options to study currency risk premiums. Two notable papers are Mueller, Stathopoulos, 94 and Vedolin (2017) and Jurek and Xu (2014). Mueller, Stathopoulos, and Vedolin (2017) find that currency correlations are counter-cyclical and they construct a correlation dispersion measure that explains the cross-section of currency excess returns. More importantly for this paper, they show how to compute risk-neutral covariances between exchange rates by using model-free cross-pair variances derived from options using, e.g., using Britten-Jones and Neuberger (2000). However, rather than constructing a factor from the covariances between currencies, I use them to measure forward-looking risk exposures to currency portfolios. Jurek and Xu (2014) estimate risk premiums using currency options in a latent factor model in which the common factor follows a sufficiently rich dynamic structure that captures the most salient features of currency returns. In contrast, I specify exactly what the systematic factor is and estimate risk premiums without imposing specific distributional assumptions on the common factor. One of the key strengths of the option-implied betas suggested in this paper is that no parameters have to be estimated, which makes them easy to implement and computationally efficient. 2.3 Option-Implied Risk Exposures In the equity literature, there has been a long tradition of modeling expected returns to individual stocks and portfolios as their covariation with a set of systematic factors, but the popular factors used in the equity literature, e.g., the three factors of Fama and French (1993), have little explanatory power for currency returns (Burnside, Eichenbaum, and Rebelo, 2011). Likewise, macro-based models have failed to ”beat” the random walk in predicting currency returns (Cheung, Chinn, and Pascual, 2005; Meese and Rogoff, 1983). Recently, a new stream of literature has emerged which has found that the cross-section and time-variation of currency returns appear to be well-explained by exposure to portfolios of currencies. Arguably, the most notable currency factors are the carry and dollar factors introduced in Lustig, Roussanov, and Verdelhan (2011) 1. A central element in this research is time-varying betas, especially for cross-sectional analysis, which critically relies on accurate measurements of betas. Common to this liter1Other notable examples of papers that use factor models to model currency returns: Menkhoff, Sarno, Schmeling, and Schrimpf, 2012a,b; Mueller, Stathopoulos, and Vedolin, 2017; Ready, Roussanov, and Ward, 2017. 95 ature is that the betas are estimated using rolling window regressions of currency returns on the proposed factors, which implicitly assumes that historical realizations reflect future outcomes. There are a number of caveats with historical betas. First, they do not adjust immediately to structural changes in the currency market conditions, for instance due to unforeseen changes in a country’s monetary policy, that is, they are slow-moving. Second, the econometrician has to decide on a particular time frame and data frequency used for the estimation, both of which are subjective decisions. The betas derived from currency option markets do not suffer from any of these issues; the option market provides the betas in real time, and only numerical implementation errors affect their measurements, which tend to be of minor impact (Della Corte, Ramadorai, and Sarno, 2016; Mueller, Stathopoulos, and Vedolin, 2017). Arguably, the forward-looking nature of the option-implied betas is especially valuable in periods in which future expectations of exchange rates deviate substantially from the past, as was the case, for example, during the financial crisis, the European debt crisis, or the Asian crisis. 2.3.1 Model Setup Define the exchange rate Sji as units of currency iper 1 unit of currency j, that is, an appreciation in the exchange rate corresponds to an increase in currency jrelative to currency i. Moreover, define the log change over [t, t +m] as ∆sji t,t+m≡log Sji t+m−log Sji t. I assume that the log currency dynamics is governed by a single-factor model: ∆sji t,t+m=ii t,t+m−ij t,t+m+βji t,t+mGt,t+m+εji t+m(2.1) where ii t,t+m−ij t,t+mis the interest rate differential between currency iand jover a horizon of length m,Gt,t+mare shocks in the systematic factor, βji t,t+mmeasures the sensitivity of currency jto shocks in the systematic factor, and εji t+mis idiosyncratic risk (non-priced risk). The conditional expected excess return for holding currency jis then given by Etrxji t,t+m=βji t,t+m·λG t,t+m(2.2) where rxji t,t+m= ∆sji t,t+m−ii t,t+m−ij t,t+m, and λG t,t+mis the price of risk for the systematic 96 factor. rxji t,t+mis the excess return from borrowing money in currency iand investing in currency j, measured in terms of currency i. The risk premium defined in (2.2) is based on a log approximation, which has been used in the majority of papers studying currency risk premiums, dating back to Bilson (1981) and Fama (1984). In the actual empirical implementation of currency excess returns, I use discrete returns, rather than log returns, but in general, the difference is miniscule and does not alter the main conclusions of the paper. The model is able to capture time-varying risk premiums through time-dependent risk exposures (and prices of risk), which has been documented in the literature as an important salient feature of exchange rates, e.g., in order to match the failure of the uncovered interest rate parity (Sarno, Schneider, and Wagner, 2012). Formally, a factor structure in the log exchange rates can be constructed in an international complete market model by imposing a factor structure in the law of motion of each country’s log pricing kernel. This is because a standard no-arbitrage argument shows that the difference in log changes of each country’s pricing kernels governs the law of motion of their bilateral log exchange rate. In this manner, heterogeneity in exposures to shocks in the systematic factors drives the cross-section of currency excess returns. This modeling approach has been taken by several papers (e.g., Lustig, Roussanov, and Verdelhan, 2014; Menkhoff, Sarno, Schmeling, and Schrimpf, 2012b; Mueller, Stathopoulos, and Vedolin, 2017; Verdelhan, 2017). However, the approach that I use does not necessarily assume complete markets but only that there is a factor structure in currency excess returns. 2.3.2 Option-Implied Currency Betas In the following, I will suppress the base currency index whenever it is clear from the context that the method applies to any base currency. The time tconditional beta for currency j, βj t,t+mover [t, t +m] in the model (2.1), is given by: βj t,t+m=Covt∆sj t,t+m, Gt,t+m Vt(Gt,t+m)(2.3) 97 If Gt,t+mis the m-period innovation in a portfolio consisting of Ncurrencies with weights wk, that is, Gt,t+m=PN k=1 wk∆sk t,t+m2, then the beta in (2.3) can be expressed as βj t,t+m=PN k=1 wkCovt∆sj t,t+m,∆sk t,t+m PN k=1 PN l=1 wkwlCovt∆sk t,t+m,∆sl t,t+m(2.4) The numerator in (2.4) contains the covariances between currency jand all the constituents of the systematic factor portfolio. The denominator contains all the variances and covariances of the constituents of the factor portfolio. These moments can be computed using traditional rolling window estimates, or they can be implied out from options. In the next section, I show how to compute risk-neutral covariances and variances of exchange rates by means of currency options, and hence how to compute option-implied model-free betas. The model-free measure of the covariance between two currencies, against a given base currency, can be constructed from their respective exchange rates versus the base currency and their cross-pair exchange rate. Denote the cross-pair exchange rate between two foreign currencies kand j,Skj, and the respective base exchange rates Skand Sj. Let the base currency be USD, then one unit of jequals Sjunits of USD, which can be converted into Sj·(Sk)−1units of k. Hence, in the absence of triangular arbitrage, then Sj·(Sk)−1= (Skj)−1 3. Assuming the absence of triangular arbitrage at time tand t+mand taking logs then gives ∆skj t,t+m= ∆sk t,t+m−∆sj t,t+m(2.5) Taking risk-neutral variance on both sides of (2.5), and rearranging, gives: CovQ t∆sk t,t+m,∆sj t,t+m=1 2VQ t∆sk t,t+m+VQ t∆sj t,t+m−VQ t∆skj t,t+m(2.6) Equation (2.6) expresses the risk-neutral covariance between two exchange rates against the same base currency, say the U.S dollar, in terms of the risk-neutral variances of the 2Since interest rates over [t, t +m] are known at time t, they do not impact variances and covariances, therefore we may equivalently think of Gt+mas the log excess return on the dollar factor. 3The data used in Mueller, Stathopoulos, and Vedolin (2017) show that triangular arbitrage spreads on average are below 1 basis point and last for less than a second (Fenn, Howison, McDonald, Williams, and Johnson (2009) report similar quantities). Consequently, the spreads from triangular arbitrage are so small that they do not significantly affect the option-implied moments. 98 respective exchange rates against the U.S. dollar and their risk-neutral cross-pair variance. By using the expression of Britten-Jones and Neuberger (2000), I compute each risk-neutral variances in (2.6) by integrating over a continuum of put and call prices: VQ t∆st,t+m= 2eit,t+mZSt 0 1 K2P(K, t, t +m)dK +Z∞ St 1 K2C(K, t, t +m)dK(2.7) where it,t+mis the riskless interest rate of the base currency over horizon m, and P(K, t, t+m) and C(K, t, t +m) are put and call prices, respectively, with maturity mand strike K. From the expressions (2.6)-(2.7), the entire covariance matrix for all exchange rates can be constructed, and hence option-implied betas to currency portfolios. Deriving the model-free variance from expression (2.7) requires a continuum of put and call prices at different strikes. Options in currency markets are, in general, quoted in terms of Garman and Kohlhagen (1983) implied volatilities at five different strikes, spread evenly across moneyness (see section 2.4 for details on the options data). I interpolate between those available strike/implied volatility pairs using a cubic spline, as in Della Corte, Ramadorai, and Sarno (2016)4and use the Garman and Kohlhagen (1983) formula to convert each strike/volatility pair into put and call prices, which are then used to calculate the integral in (2.7). Jiang and Tian (2005) point out that discretization errors arise from performing the numerical integration of the integral in (2.7), however, Mueller, Stathopoulos, and Vedolin (2017) report that the discretization errors do not exceed 0.5 percentage points of the implied volatilities in currency markets. In the literature, there are different variations on how to compute the model-free moments. As a robustness check, I derived the variances and covariances using the expressions of Martin (2017) and Bakshi, Kapadia, and Madan (2003) and the differences were negligible 5. The historical moments are computed using daily log changes in the exchange rates. Specifically, the annualized realized variance of exchange 4Della Corte, Ramadorai, and Sarno (2016) analyze different interpolation schemes, including the noarbitrage vanna-volga method of Castagna and Mercurio (2007), and find virtually no differences in the derived variances. 5Mueller, Stathopoulos, and Vedolin (2017) find a statistically insignificant difference between option implied currency correlations using the model-free variances of Martin (2017) and Britten-Jones and Neuberger (2000). 99 rate iand its covariance with currency k, using a window of length L, are calculated as RV i t,t−L=252 L L−1 X j=0 ∆si t−j2(2.8) RCOV ik t,t−L=252 L L−1 X j=0 ∆si t−j∆sk t−j(2.9) where ∆stdenote daily log changes in the exchange rate. 2.3.3 Dollar Factor Betas Let the m-period innovation to the dollar factor, defined as an equal-weighted portfolio of foreign currencies against the U.S. dollar, be denoted: ∆Dolt,t+m≡1 N N X i=1 ∆si t,t+m(2.10) The time tvariance over [t, t +m] of the dollar factor and its covariance with exchange rate junder measure M, which may either be the objective measure Por the risk-neutral measure Q, are given by VM t(∆Dolt,t+m) = 1 N2 N X i=1 VM t∆si t,t+m+1 N2 N X i=1 N X k6=i CovM t(∆si t,t+m,∆sk t,t+m) (2.11) CovM t∆sj t,t+m,∆Dolt,t+m=1 N N X i=1 CovM t∆sj t,t+m,∆si t,t+m,(2.12) Following Verdelhan (2017), I exclude the relevant currency from the dollar factor when computing betas to avoid a mechanical relation, i.e., the time tdollar beta for currency j under measure Mis defined as βM jt,t+m=CovM t∆sj t,t+m,∆Dolt,t+m|j VM t∆Dolt,t+m|j= 1 N−1Pi∈N|jCovM t∆sj t,t+m,∆si t,t+m, VM t∆Dolt,t+m|j(2.13) where Dolt,t+m|jdenotes the dollar factor excluding currency j. The m-month risk-neutral dollar factor beta is then derived from (2.11)-(2.12), excluding currency jfrom the dollar factor, by plugging into (2.13). Variances/covariances are computed from the expressions 100 the other hand, it may also raise the concern that they are substitutes for one another. The Qand P-betas, however, are far from perfectly correlated, as shown in Table 2.3, which reports their contemporaneous time-series correlations. The correlation between the 1-month Q-beta and the P-beta is lowest for the NOK at 34%, and it is largest for the CAD at 83%. The average correlation across all currencies is 44%. The empirical results presented below corroborate that the two types of betas are not interchangeable, as they produce vastly different predictions of realized betas and returns. 2.5.3 Dollar Factor Beta-Sorted Portfolios If the conditional dollar factor model is an appropriate model, the expected excess return of a portfolio should increase monotonically in the portfolio’s expected conditional dollar factor exposure, entailing that a high minus low conditional dollar factor portfolio delivers a positive expected excess return. In practice, identifying such a risk-return relation relies critically on accurate measurements of dollar factor exposures. The slow-moving nature of the rolling window betas may not be very informative of the realized risk exposures over the course of, say, the next month, and especially not if there are rapid changes in the factors that drive exchange rates. Naturally, we may then ask if the ex-ante nature of option-implied betas, and their ability to instantaneously incorporate new information, make them better at anticipating future returns than historical betas. As a first step in the comparative analysis of the betas, I construct beta-sorted portfolios using both methodologies. I follow the portfolio construction procedure of Verdelhan (2017). Specifically, each month, for each type of beta separately, I allocate the currencies into three equal-weighted portfolios from low to high based on their dollar factor betas. I then construct three portfolios P1, P2, and P3which are long the respective beta-sorted portfolios whenever the average foreign discount is negative (i.e., average foreign interest rate is larger than the U.S. dollar interest rate) and short otherwise. In other words, the portfolios are constructed based on their exposure to the conditional dollar factor. Table 2.4 shows mean excess returns, standard deviations, and Sharpe ratios for each of the portfolios at horizons of 1-12 months. The brackets below the mean excess returns are t-statistics based on Newey and West (1987), with the automatic lag selection of Newey and West (1994). In the construction 107 of the Q-beta-sorted portfolios, I use options with the same time to expiry as the holding period of the forward contracts. The P-beta used for portfolio construction is calculated on the basis of overlapping daily rolling window regressions with a length of 252, i.e., the same beta is used for each holding period. Both shorter (126 days) and longer (504 days) rolling windows produce similar results, therefore I only report results for the 252-day P-betas. Table 2.4 reveals a clear pattern: at any holding period, the mean excess portfolio returns increase in the ex-ante Q-beta, while a more dispersed pattern is to be found when P-betas are used for portfolio construction. E.g., for the 1-month holding period, a high minus low (HML) factor based on Q-betas, which buys portfolio P3and shorts portfolio P1, gives a significant (t-statistic of 2.58) mean annualized excess return of 3.35 percent (Sharpe ratio 0.41), while the HML factor based on P-betas has an insignificant (t-statistic of 0.57) mean excess return of 0.95 percent (Sharpe ratio 0.11). The mean excess returns to the Q-beta HML factors are positive at longer horizons as well (albeit only significant at the 2-month holding period), and larger than for the corresponding P-beta HML factors. Figure 2.5 illustrates the cumulative returns to monthly rebalanced HML dollar factors, for both types of betas, along with the annualized 1-month Q-volatility and the 252-day rolling volatility of the dollar factor. We see that implementing an HML dollar strategy based on Q-betas, rather than P-betas, gives larger returns throughout the sample period. Interestingly, while the dollar carry trade performs poorly from 2010-2016 (Figure 2.1), the HML dollar strategy continues to deliver high positive excess returns. In this period, the AFD is negative, therefore, the dollar carry trade is short the U.S. dollar and long the equal-weighted basket of foreign currencies. Thus, the dollar carry trade is exposed to an upwards shift in the level of the U.S. dollar relative to all foreign currencies, which in fact occurred over this period. The HML dollar factor, on the other hand, is immune to level shifts in the U.S. dollar, since the long and short side of the portfolio are affected equally. Notably, there is no obvious link between the volatility of the dollar factor and the returns to the HML dollar strategy. For instance, the HML factor does not crash during the financial turmoil in 2008, as the HML carry trade (Jurek, 2014; Menkhoff, Sarno, Schmeling, and Schrimpf, 2012a). This highlights that the HML factor and the HML carry trade appear to be driven by different risk factors (in this sample, their correlation is ∼19%). Table 2.5 shows the mean excess returns of each beta-sorted portfolio decomposed into 108 a spot and interest rate component. Interestingly, at any holding period, the larger mean excess returns on the Q-beta HML factors compared to the P-beta HML factors stem entirely from the spot component. For example, at the 1-month holding period, the spot components are 2.35% and −0.18%, and the interest rate components are 1.00% and 1.12%, for the Q-beta and P-beta HML factors, respectively. Thus, for the Q-beta HML factor the largest proportion of the excess return is due to spot changes, which is in contrast to the HML carry trade, where the return is primarily driven by the interest rate differential (see Table 2.1). For both types of beta, the annualized interest rate components are virtually the same for the HML factor portfolios across different holding periods, whereas the spot components decrease, i.e., the Q-beta predictability of currency spot changes is confined to shorter horizons. One potential explanation for why the Q-betas are better at explaining the cross-section of currency returns, relative to the P-betas, is that they are better at predicting realized volatility and to a lesser extent because they more accurately forecast correlations with the dollar factor. For instance, Jorion (1995) and Busch, Christensen, and Nielsen (2011) provide evidence that implied volatilities from currency options are better predictors of realized currency volatility than historical volatility measures. I examine if this is the case by constructing portfolios on the basis of betas which are built from a mixture of Qand P-moments. Specifically, I follow the method suggested by French, Groth, and Kolari (1983), in which betas are constructed from historical correlations and option-based variances. Supposedly, if the Q-correlations are good predictors of realized correlations, this beta method will be less successful at identifying high and low-beta currencies ex-ante. In the same spirit, I also construct mixed betas based on Q-correlations in conjunction with P-variances. Following the exact same procedure as previously, portfolios are constructed based on both types of mixed betas. Table 2.6 shows the results. The HML factor constructed from betas combining P-correlations and Q-variances gives a lower mean excess return, at any horizon, compared to the Q-beta HML factor. For instance, at the 1-month horizon, the mean annualized excess return is 2.53% (t-statistic 1.53), compared to 3.35% when Q-correlations are used (Table 2.4). Furthermore, the Sharpe ratio declines to 0.30 (11 percentage points), suggesting that the Q-correlations are more effective at constructing 109 betas for cross-sectional analysis than P-correlations. Using betas built from Q-correlations and P-variances to construct portfolios reaffirms that Q-correlations are useful for computing betas. Using this mixed dollar factor beta, the HML factors have larger mean excess returns and Sharpe ratios relative to the corresponding P-beta HML factors at all horizons. There is a monotonic relation between ex-ante portfolio betas and ex-post portfolio excess returns when sorting on the mixed beta, albeit the HML factor excess returns are insignificant. Among all types of betas, the pure Q-betas perform best for portfolio construction, which corroborates that both Q-correlations and Q-variances contain useful information for the computation of betas. 2.5.4 Evaluation of Model Predictions Another important aspect of betas is how strong predictors they are of realized returns when used as inputs in the factor model. We may erroneously reject an accurate model due to poorly measured betas. Although the Q-betas identify a monotonic relation between ex-ante portfolio betas and ex-post returns, it is not certain that they perform well in the context of the model. Betas could be flawed and still capable of properly ranking currencies on ex-ante betas, and thus performing well in a portfolio sorting exercise. In the following, I examine in greater detail how the two types of betas perform in the context of the model. Using the conditional dollar factor model implies a linear relation between expected portfolios excess returns and betas. As a first step to examine this, I plot in Figure 2.6, at the 1-month horizon, the average realized excess returns against the average model predicted excess returns for the dollar beta-sorted portfolios, for each type of beta. Table 2.7 reports the results at horizons of 1-12 months. The model predicted returns are computed assuming a fixed price of conditional dollar risk equal to its unconditional mean over the entire sample (this follows from the Euler equation since the beta of a tradable systematic factor is 1). Since the price of risk is fixed, the model predicted excess return for each portfolio is the average portfolio beta times the conditional dollar price of risk. From Figure 2.6, we see that using the Q-betas leads to a relation between portfolio betas and returns that is too steep. The (low) high-beta portfolio has a (smaller) larger mean excess return than predicted by the model, whereas, when using P-betas, the opposite 110 is the case. As a result, the realized risk-return relation is too flat for the P-betas, whereas for Q-betas it is too steep—but the model predictions and realized returns appear to be better aligned. These results should be considered as indicative and must be interpreted with caution. The exercise assumes a fixed price of risk for the conditional dollar factor (estimated over the entire sample period), and the prediction errors are uninformative of the model’s performance in the time series. The takeaway from Figure 2.6, however, is consistent with the results obtained via dynamic time-series predictions (which I will discuss further below): when using Q-betas rather than the P-betas, the factor model exhibits less biased predictions of returns and the mean time-series prediction errors are smaller. The results for time-series prediction errors are reported in Table 2.8 for both types of betas. For each portfolio, and beta type, the prediction error is computed as the realized excess return less the model predicted excess return. This is done monthly, and thus a time series of prediction errors is generated. The model excess return is computed as the portfolio’s ex-ante beta times the realization of the conditional dollar factor. Columns 1-4, Panels A and B, show the mean squared prediction errors (MSEs) in basis points for portfolios sorted on Q-betas and P-betas, respectively. In general, the MSEs are larger, for both types of betas, for high and low-beta portfolios. But at any horizon, the mean MSEs across all portfolios are smaller when using the Q-betas compared to the Pbetas. For instance, at the 1-month horizon, the mean P-beta MSE is 15.41 bps compared to a mean Q-beta MSE of 13.42 bps, and their difference is statistically significant with a Newey and West (1987) t-statistic of 2.18 (reported in the ninth column). For comparison, I report in Panels C and D the MSEs for a random walk forecast, which predicts that the future spot exchange rates are equal to their current values. For all portfolios, at all horizons, the random walk forecasts have larger mean MSEs relative to both the Pand Q-beta predictions. The t-statistics for the difference in mean MSEs between the random walk forecast and the beta forecasts are reported in the column furthest to the right, and are at all horizons statistically significant at the 1% level. This is consistent with Lustig, Roussanov, and Verdelhan (2014), who document that the average forward discount (U.S. minus the average foreign interest rate) is a strong predictor of aggregate foreign currency changes versus the U.S. dollar—foreign currencies tend to appreciate (depreciate) versus the U.S. dollar when the AFD is negative (positive). 111 Columns 5-8 report the mean prediction errors (ME), in percentages, for the beta-sorted portfolios, and the difference in MEs between the high and low-beta portfolios. The model forecasts based on P-betas appear to have a systematic bias, while there is no notable bias when using Q-betas. At the 1-month horizon, the P-betas tend to underestimate returns to the low-beta portfolio, with an average of 1.17%, and on the other hand, they tend to overestimate returns to the high-beta portfolio, with a mean prediction error of −1.20%. Consequently, the difference in mean prediction errors between the high-beta and the lowbeta portfolio is −2.38%, whereas the bias is slightly positive, 0.44%, for the Q-beta-sorted portfolios. This difference in the bias between the Pand Q-betas is consistent with that the HML dollar factor portfolio based on Q-betas delivers substantially larger excess returns than the HML dollar factor portfolio constructed based on P-betas. To summarize, the model performs better when using the Q-betas—the mean squared prediction errors are smaller at all horizons, and there are no notable prediction biases across the beta-sorted portfolios. Interestingly, a similar prediction bias for the rolling window betas also appears in the CAPM in which (high) low-beta stocks tend to have (smaller) larger returns than predicted by the CAPM—the low-risk anomaly (Frazzini and Pedersen, 2014). Consistent with my findings, Buss and Vilkov (2012) find that when betas are computed based on options, there are no notable biases between low and high-beta portfolios, whereas rolling window betas tend to underestimate returns (ex-ante) of low-beta stocks, and vice versa for highbeta stocks. Thus, this suggests that rolling window betas generally induce biased model predictions. One potential explanation for the better performance of the model when using Q-betas is that they are more powerful and less biased predictors of realized betas. In the following, I investigate this for both portfolios and individual currencies. 2.5.5 Predicting Dollar Factor Betas for Portfolios In a factor pricing model, the expected excess return on a security is given by the expected beta times the price of risk of the factor. Therefore, accurate predictions of future betas are crucial for empirically identifying a monotonic relation between ex-post returns and betas. A true model may erroneously be rejected if the betas are ranked in the wrong 112 order, and furthermore, if the level of betas is inaccurately measured, this may cause large model prediction errors. The superior model performance when using the option-implied betas may thus be due to them being better predictors of ex-post betas. In the following, I investigate this hypothesis by comparing the predictive power of the two types of betas for portfolio betas. Realized beta is not observable and needs to be proxied by a measurable quantity. Following Andersen, Bollerslev, Diebold, and Wu (2006), Buss and Vilkov (2012), and Chang, Christoffersen, Jacobs, and Vainberg (2011), I measure realized betas using daily rolling window regressions, where the length of the window matches the forecast horizon. That is, for forecasts at time tat horizon τ, the realized beta at time t+τis estimated using daily data from (t, t +τ]. I calculate the P-beta predictors of realized betas at horizon τ using a window of length τ, i.e., the time t P-beta forecasts are computed based on daily currency returns from (t−τ, t]. At forecast horizons of up to one year, the maturity that I use to derive the Q-beta predictors matches the forecast horizon, and for forecast horizons exceeding one year, the 1-year options are used. I calculate the portfolio beta prediction errors as follows for each forecast horizon: every day, for each type of beta, three portfolios are constructed based on their expected betas. For each portfolio, I calculate the daily model prediction error as the difference between the realized and the expected portfolio beta. It is important to note that the ex-ante portfolio of currencies is compared to the exact same set of currencies ex-post. I conduct daily forecasts to increase the power of the predictive tests, as in, e.g., Della Corte, Sarno, and Tsiakas (2011), Chang, Christoffersen, Jacobs, and Vainberg (2011), and Jorion (1995). Due to the overlapping data used to derive the rolling window betas, which causes autocorrelation in the prediction errors, I use t-statistics based on Newey and West (1987), with number of lags that matches the forecast horizon, as in, e.g., Della Corte, Sarno, and Tsiakas (2011) and Chang, Christoffersen, Jacobs, and Vainberg (2011). I run the forecasts at horizons that span 6-18 months; i.e., the 1-3 month horizons are not included in the forecasts as was the case for the portfolio return predictions. This mismatch is because the forecast horizon has to be sufficiently long in order to get reliable beta estimates while at the same time avoiding overlapping the data used to estimate the ex-ante and ex-post rolling window betas. 113 Table 2.9 shows the prediction error results for the daily beta-sorted portfolios. First, I compare the biases for the forecasts. The left panel (ME) reports the mean of the timeseries prediction errors for each portfolio, for both types of beta. At all horizons, the P-beta forecasts have a propensity to (overestimate) underestimate betas of the (high) low-beta portfolios. The mean prediction bias between the high and low-beta portfolios is highly significant at all horizons, with t-statistics ranging from −4.02 to −2.92. At the 6-month horizon, as an example, the P-beta forecasts of the low-beta portfolios underestimate realized betas with an average of 7.44% (in beta units). On the other hand, high-beta portfolios are on average overestimated by 8.29%, resulting in a highly significant (t-statistic -2.92) high minus low beta bias of −15.73% (−33.47% relative bias). In comparison, the 6-month mean prediction errors for low and high Q-beta-sorted portfolios are −0.26% and −3.46%, respectively, resulting in an insignificant (t-statistic −1.42) high minus low mean prediction error of −3.20% (−5.92% relative bias). This significant P-beta prediction bias makes them prone to misidentifying low-beta currencies as high-beta (and vice versa), which strikes as being a plausible explanation for why there is no monotonic relation between ex-ante betas and ex-post portfolio excess returns. In contrast, since there is no notable prediction bias for the Q-betas—with a relative bias that is nearly six times smaller—they are to a lesser extent subject to this issue. Furthermore, the difference in the bias between the Q-beta and P-beta forecasts of realized betas is consistent with that the P-betas have a tendency to (overestimate) underestimate the (high) low-beta portfolio excess returns, while there is no noticeable bias for the Q-betas. The high minus low-beta bias increases in the length of the forecast horizon, albeit it is much smaller for the Q-betas at any horizon. The growing high minus low beta bias at longer horizons appears to be a likely explanation for the diminishing returns to the HML dollar portfolios for longer holding periods (see Table 2.4). The panel on the right shows the mean squared prediction errors (MSEs) in percentages. We see that for all portfolios, at all horizons, the Q-beta forecasts have the smallest MSE. Notably, the Q-betas have much stronger predictive power for the low and high-beta portfolios. The average MSE across all three portfolios is smallest for the Q-beta forecasts at any forecast horizon. For example, at the 6-month horizon, the average MSE is 1.34% for the 114 Q-beta forecasts, while it is 1.90% based on the P-beta forecasts. The t-statistics for the difference in the average MSEs are reported for each forecast horizon in the column furthest to the right. We see that the Q-beta average MSEs are statistically significantly smaller at all forecast horizons, with t-statistics ranging from 2.30 −3.56. The Q-betas therefore deliver not only less biased predictions of betas, but also smaller prediction errors. To conclude, the findings that I provide in this section suggest that the stronger predictive power of the Q-betas compared to the P-betas explain why the Q-betas correctly identify a monotonic relation between ex-ante betas and ex-post returns and why they exhibit the smallest model prediction errors of portfolio excess returns in the time series. 2.5.6 Predicting Dollar Factor Betas for Individual Currencies As a final comparison between the predictive power of the two beta types, I conduct predictive regressions of realized betas for individual currencies on ex-ante Q-betas and P-betas: βP it+τ=γQ i0+γQ i1βQ it +εQ it+τ(2.16) βP it+τ=γP i0+γP i1βP it +εP it+τ(2.17) Ideally, the intercept is zero and the slope coefficient unity. As for the portfolio forecasts, the ex-post betas are computed using daily data over the forecast horizon (t, t +τ], and the P-beta predictors are computed using daily data from (t−τ, t]. The Q-beta predictors are based on option prices at time tfor which the time to expiry matches the length of the forecast horizon as closely as possible. The results for the Q-beta and P-beta predictive regressions are reported in Tables 2.10 and 2.11, respectively. As for the portfolio beta predictions, the predictive regressions are conducted on daily data to increase the power of the tests, and t-statistics (reported in brackets under the relevant coefficients) are adjusted for autocorrelation using Newey and West (1987) with number of lags that match the forecast horizon, as in, e.g., Della Corte, Sarno, and Tsiakas (2011) and Chang, Christoffersen, Jacobs, and Vainberg (2011). The results for the individual predictive regressions corroborate that Q-betas are better predictors of ex-post betas than P-betas. At the 5% significance level, 30 out of 45 predictive 115 regressions (nine exchange rates at five horizons) have a significant slope coefficient, while it is only significant in four cases when using the P-betas. In all the predictive regressions, for both beta types, the slope coefficient is less than unity and has a positive intercept, i.e., the beta predictors tend to be biased predictors of ex-post betas. The slope coefficient, however, is closer to unity when using the Q-betas in 30 out of 45 cases, and in the same regressions the intercept is closer to 0. Likewise, Q-betas have a higher explanatory power in 29 of the cases relative to the P-betas when using the R2-metric. It is important to note that there are a few currencies for which the P-betas have notably stronger explanatory power, namely, the GBP, NOK, and SEK. According to a central bank survey conducted by Bank for International Settlements (Bank of International Settlements, 2016), the latter two currencies are among the most illiquid currencies of the G10, e.g., they both account for less than 1% of the overall currency market turnover (other liquidity metrics, such as the volume in OTC currency interest rate derivatives, are consistent with this picture). Thus, one plausible explanation for the weak forecast performance for the SEK and NOK is that the cross-pair options are illiquid, and perhaps especially so for longer maturities. In fact, using 1-month option-implied betas improves the explanatory power for these currencies, most notably at longer forecast horizons where longer-dated option maturities were used to construct predictors (similar improvements are obtained by using 2-month and 3month maturities). Chang, Christoffersen, Jacobs, and Vainberg (2011) find a similar result in predictions of realized CAPM betas using option-implied CAPM betas, where optionimplied betas based on short-term liquid options have a stronger predictive ability than betas based on longer-dated illiquid options, even at longer horizons. If the long maturity options are in fact less liquid, there appears to be an important trade-off between applying the more liquid short-term maturities vs. longer-dated maturities that match the forecast horizon better. The overall predictive performance is better for Q-betas compared to the P-betas according to any metric considered in the analysis, both for portfolios and individual currencies. However, there are a few exceptions, as mentioned above, in which the forecasts would likely benefit from incorporating historical information. As a small step in this direction, I performed multivariate regressions using both types of beta as regressors, and I indeed found 116 99 02 05 07 10 13 15 -0.5 0 0.5 AUD 99 02 05 07 10 13 15 -0.5 0 0.5 CAD 99 02 05 07 10 13 15 -0.5 0 0.5 CHF 99 02 05 07 10 13 15 -0.5 0 0.5 EUR 99 02 05 07 10 13 15 -0.5 0 0.5 GBP 99 02 05 07 10 13 15 -0.5 0 0.5 JPY 99 02 05 07 10 13 15 -0.5 0 0.5 NOK 99 02 05 07 10 13 15 -0.5 0 0.5 NZD 99 02 05 07 10 13 15 -0.5 0 0.5 SEK Figure 2.4: Time series of the 12-1 month Q-beta spread for the G10 currencies. This figure illustrates the time series of the difference between the 12-month and 1-month option-implied dollar factor betas for each currency. The dollar factor beta for a currency iis computed as the covariance between innovations in currency iand the dollar factor normalized by the variance of the dollar factor, which is defined as an equally weighted basket of all foreign currencies vs. U.S. dollar (excluding currency i). The m-month Q-betas are computed using the model-free measures of covariance and variance implied out from currency options with m-month maturity (expressions: (2.6)-(2.7)). The P-betas are computed from 252day rolling window regressions of daily innovations in the exchange iagainst daily innovations in the dollar factor (excluding currency i). The options data are from JP Morgan Dataquery and the exchange rate data are obtained from Reuters through Datastream. The sample period is from January 1998 to August 2016 and the data comprise 4659 daily observations. 123 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 0.6 0.8 1 1.2 1.4 1.6 1.8 Cumulative Value, G10 HML Q-Betas Cumulative Value, G10 HML P-Betas 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 0 0.05 0.1 0.15 0.2 0.25 Annualized Q-Volatility of Dollar Factor Annualized P-Volatility of Dollar Factor Figure 2.5: Cumulative returns of HML dollar factor and dollar factor volatility. The upper panel shows the cumulative return, for Pand Q-betas for the HML dollar factor. Each month, and for each type of beta separately, the currencies are ranked in ascending order based on their dollar factor betas and allocated into three equal-weighted portfolios: P1,P2, and P3. Each month, the HML dollar factor buys (sells) P3and sells (buys) P1when the average forward discount is negative (positive). The Q-betas are derived from 1-month maturity options using the expression in (2.13)—i.e. as the covariance between the relevant currency and a equal-weighted portfolio of all foreign currencies vs. U.S. dollar (the dollar factor) normalized by the risk-neutral variance of the dollar factor. The P-betas are computed using 252-day rolling windows and updated every month. The lower panel shows the annualized 252-day rolling window volatility of the dollar factor along with its risk-neutral annualized 1-month volatility. The options data are from JP Morgan Dataquery and the exchange rate data are obtained from Reuters through Datastream. The sample period is from January 1998 to August 2016 and the data comprise 216 monthly observations. 124 0123456 Predicted Mean Excess Return (%) 0 1 2 3 4 5 6 Realized Mean Excess Return (%) Option-Implied Beta P1 P2 P3 0123456 Predicted Mean Excess Return (%) 0 1 2 3 4 5 6 Realized Mean Excess Return (%) 252-day Rolling Window Beta P1 P2 P3 Figure 2.6: Mean realized vs. mean predicted excess returns for Qand P-betas. This figure shows the scatterplot of mean realized portfolio excess returns plotted against model predicted mean excess returns at a holding period of one month. For each type of beta, the model predicted excess return is computed as mean beta of each portfolio times the unconditional mean excess return of the conditional dollar factor. The conditional dollar factor is long the dollar factor—equal-weighted basket of all foreign currencies vs. U.S. dollar—if the average foreign discount is negative and short this portfolio otherwise. The dollar factor beta for currency iis computed as the covariance between innovations in currency iand the dollar factor normalized by the variance of the dollar factor. The option-implied beta portfolios are constructed based on the 1-month option-implied betas, using expression (2.13). The P-betas are computed from 252-day rolling window regressions of daily innovations in the exchange iagainst daily innovations in the dollar factor (excluding currency i). The options data are from JP Morgan Dataquery and the exchange rate data are obtained from Reuters through Datastream. The sample period is from January 1998 to August 2016 and the data comprise 216 monthly observations. 125 126 2.8 Tables Table 2.1: Excess returns on the dollar carry trade and HML carry trade. This table shows annualized excess mean returns, the spot exchange rate component, forward discounts and standard deviations of the dollar carry trade and the HML carry trade. The dollar carry trade is long an equal-weighted basket of all foreign currencies if the AFD is negative and short the same set of currencies otherwise. The HML carry trade is long the upper tertile interest rate currencies and short the lower decile interest rate currencies. The excess returns for each strategy are reported at six different maturities: 1, 2, 3, 6, 9 and 12 months, and the portfolios are rebalanced at the end of each month. Newey and West (1987) t-statistics are reported in brackets and ***, **, and * indicate significance at a 1%, 5%, and 10% level, respectively. The exchange rate data are from Reuters through Datastream and the sample period is from January 1998 to August 2016 and comprise 216 monthly observations. Panel A: Dollar Carry Trade Horizon Mean ∆St+m St∆it+mStd Sharpe Ratio 1 mo 3.45∗2.32 1.32 8.32 0.41 [1.89] 2 mo 2.71 1.58 1.31 8.57 0.32 [1.24] 3 mo 2.16 1.04 1.31 8.79 0.25 [0.87] 6 mo 1.72 0.63 1.33 9.51 0.18 [0.42] 9 mo 1.80 0.73 1.34 9.67 0.19 [0.49] 12 mo 1.80 0.74 1.35 9.76 0.18 [0.54] Panel B: HML Carry Trade Horizon Mean ∆St+m St∆it+mStd Sharpe Ratio 1 mo 3.27 -0.67 3.93 8.56 0.38 [1.58] 2 mo 3.35 -0.53 3.88 8.65 0.39 [1.24] 3 mo 3.13 -0.70 3.84 9.01 0.35 [1.28] 6 mo 2.98 -0.82 3.80 8.91 0.33 [0.82] 9 mo 2.93 -0.85 3.77 8.51 0.34 [0.98] 12 mo 2.93 -0.84 3.76 7.95 0.37 [1.09] 127 Table 2.2: Descriptive statistics for dollar factor betas. This table shows descriptive statistics for the dollar factor betas. The dollar factor beta for currency iis computed as the covariance between innovations in currency iand the dollar factor normalized by the variance of the dollar factor, which is defined as an equally weighted basket of all currencies vs. U.S. dollar (excluding currency i). The m-month Q-betas are computed using the model-free measures of covariance and variance implied out from currency options with m-month maturity (expressions: (2.6)-(2.7)). The P-betas are computed from 252-day rolling window regressions of daily innovations in the exchange iagainst daily innovations in the dollar factor (excluding currency i). The options data are from JP Morgan Dataquery and the exchange rate data are obtained from Reuters through Datastream. The sample period is from January 1998 to August 2016 and the data comprise 4659 daily observations. Panel A: Q-betas 1-month: AUD CAD CHF EUR GBP JPY NOK NZD SEK Mean 0.95 0.51 1.05 1.12 0.73 0.55 1.16 0.98 1.15 Std 0.23 0.27 0.20 0.17 0.13 0.39 0.13 0.25 0.14 2-month: Mean 0.94 0.50 1.06 1.13 0.75 0.54 1.16 0.97 1.15 Std 0.22 0.26 0.21 0.17 0.12 0.38 0.12 0.24 0.12 3-month: Mean 0.93 0.50 1.05 1.14 0.75 0.55 1.16 0.97 1.16 Std 0.22 0.26 0.20 0.17 0.11 0.39 0.11 0.25 0.11 6-month Mean 0.93 0.50 1.06 1.15 0.76 0.55 1.16 0.96 1.16 Std 0.23 0.26 0.22 0.18 0.10 0.40 0.10 0.26 0.11 9-month: Mean 0.92 0.50 1.05 1.15 0.76 0.54 1.16 0.96 1.16 Std 0.24 0.26 0.22 0.18 0.10 0.41 0.10 0.26 0.10 12-month: Mean 0.92 0.50 1.05 1.15 0.77 0.54 1.17 0.96 1.16 Std 0.24 0.26 0.23 0.18 0.09 0.42 0.11 0.27 0.10 Panel B: P-betas AUD CAD CHF EUR GBP JPY NOK NZD SEK Mean 1.00 0.49 1.02 1.08 0.69 0.43 1.17 1.02 1.17 Std 0.25 0.25 0.23 0.19 0.14 0.41 0.15 0.22 0.14 128 Table 2.3: Contemporaneous correlations between dollar factor betas. This table reports the contemporaneous correlations between dollar factor betas. Panel A reports the contemporaneous correlations between the 252-day rolling window beta and the Q-betas at maturities from 1-12 months. Panel B reports the contemporaneous correlations between the 1-month beta and 2-12 month betas. The dollar factor beta for a currency iis computed as the covariance between innovations in currency iand the dollar factor normalized by the variance of the dollar factor, which is defined as an equally weighted basket of all foreign currencies vs. U.S. dollar (excluding currency i). The m-month Q-betas are computed using the model-free measures of covariance and variance implied out from currency options with m-month maturity (expressions: (2.6)-(2.7)). The P-betas are computed using 252-day rolling window regressions of daily innovations in the exchange iagainst daily innovations in the dollar factor (excluding currency i). The options data are from JP Morgan Dataquery and the exchange rate data are obtained from Reuters through Datastream. The sample period is from January 1998 to August 2016 and the data comprise 4659 daily observations. Panel A: Correlations between Pand Q-betas Maturity AUD CAD CHF EUR GBP JPY NOK NZD SEK 1 mo 0.54 0.83 0.70 0.60 0.54 0.81 0.34 0.45 0.48 2 mo 0.66 0.89 0.77 0.60 0.53 0.83 0.40 0.52 0.45 3 mo 0.67 0.90 0.77 0.61 0.50 0.83 0.37 0.54 0.43 6 mo 0.67 0.91 0.80 0.59 0.46 0.83 0.46 0.57 0.43 9 mo 0.67 0.92 0.80 0.57 0.45 0.83 0.46 0.59 0.40 12 mo 0.65 0.91 0.77 0.55 0.40 0.81 0.43 0.58 0.40 Panel B: Correlations between Q-betas Maturity AUD CAD CHF EUR GBP JPY NOK NZD SEK 1/2 mo 0.92 0.97 0.98 0.98 0.96 0.99 0.96 0.97 0.97 1/3 mo 0.86 0.97 0.95 0.95 0.93 0.97 0.92 0.95 0.93 1/6 mo 0.76 0.95 0.89 0.90 0.87 0.93 0.83 0.88 0.85 1/9 mo 0.70 0.94 0.86 0.86 0.83 0.90 0.77 0.84 0.78 1/12 mo 0.67 0.94 0.82 0.83 0.80 0.88 0.72 0.82 0.73 129 Table 2.4: Descriptive statistics of portfolios sorted on dollar factor betas. This table reports the means, standard deviations and Sharpe ratios of excess returns on monthly rebalanced portfolios sorted on Qand Pdollar factor betas at horizons of 1-12 months. Each month, for each type of beta separately, the currencies are ranked in ascending order based on their dollar factor betas and allocated into three equal-weighted portfolios P1,P2, and P3. The investor buys (sells) each portfolio when the average forward discount is negative (positive). For Q-beta-sorted portfolios, the length of the holding period and the maturity of the options are the same. The dollar factor beta for a currency iis computed according to (2.13)—i.e. as the covariance between innovations in currency iand the dollar factor normalized by the variance of the dollar factor, defined as an equally weighted basket of all foreign currencies vs. U.S. dollar (excluding currency i).The m-month Q-betas are computed using the model-free measures of covariance and variance implied out from currency options with m-month maturity (expressions: (2.6)-(2.7)). The P-beta is computed using a 252-day rolling window regressions of daily innovations in the exchange iagainst daily innovations in the dollar factor (excluding currency i). The options data are from JP Morgan Dataquery and the exchange rate data are obtained from Reuters through Datastream. The sample period is from January 1998 to August 2016 and the data comprise 4659 daily observations. Panel A: Q-betas Mean Std Sharpe Ratio Horizon P1P2P3P3−P1P1P2P3P3−P1P1P2P3P3−P1 1 mo 1.88 3.24 5.23∗∗ 3.35∗∗ 6.68 9.53 10.96 8.20 0.28 0.34 0.48 0.41 [1.08] [1.41] [2.31] [2.58] 2 mo 1.57 2.03 4.53∗∗ 2.96∗7.08 9.86 11.03 8.02 0.22 0.21 0.41 0.37 [0.86] [0.87] [1.97] [1.93] 3 mo 1.13 2.02 3.32 2.18 7.28 10.10 11.24 7.82 0.16 0.20 0.30 0.28 [0.62] [0.75] 1.3] [1.26] 6 mo 0.89 2.22 2.05 1.16 7.91 11.02 11.39 7.54 0.11 0.20 0.18 0.15 [0.49] [0.84] [0.84] [0.70] 9 mo 1.00 2.42 1.99 0.99 8.05 10.80 11.67 7.03 0.12 0.22 0.17 0.14 [0.61] [1.06] [0.88] [0.70] 12 mo 1.08 2.34 1.97 0.89 8.28 10.63 11.81 6.96 0.13 0.22 0.17 0.13 [0.70] [1.08] [0.93] [0.72] Panel B: P-betas Mean Std Sharpe Ratio Horizon P1P2P3P3−P1P1P2P3P3−P1P1P2P3P3−P1 1 mo 2.59 4.23∗3.53 0.95 6.89 9.74 10.80 8.27 0.38 0.44 0.33 0.11 [1.40] [1.93] [1.52] [0.57] 2 mo 1.76 3.65∗2.72 0.97 7.00 9.91 10.98 7.91 0.25 0.37 0.25 0.12 [0.95] [1.71] [1.07] [0.53] 3 mo 1.44 3.09 1.93 0.49 7.19 9.94 11.47 8.29 0.20 0.31 0.17 0.06 [0.72] [1.29] [0.71] [0.30] 6 mo 1.14 2.89 1.14 0.00 7.96 10.33 12.14 8.27 0.14 0.28 0.09 0.00 [0.58] [1.21] [0.42 [-0.01] 9 mo 1.27 2.81 1.33 0.06 8.17 10.63 12.02 7.88 0.16 0.26 0.11 0.00 [0.71] [1.26] [0.58] [0.07] 12 mo 1.39 2.46 1.54 0.15 8.45 10.94 11.78 7.63 0.16 0.23 0.13 0.02 [0.79] [1.17] [ 0.74] [0.20] 130 Table 2.5: Spot and interest rate components for portfolios sorted on dollar factor betas. This table reports spot and interest rate components of monthly rebalanced portfolios sorted on Qand P dollar factor betas at horizons of 1-12 months. Each month, for each type of beta separately, the currencies are ranked in ascending order based on their dollar factor betas and allocated into three equal-weighted portfolios P1,P2, and P3. The investor buys (sells) each portfolio when the average forward discount is negative (positive). For Q-beta sorted portfolios, the length of the holding period and the maturity of the options are the same. The dollar factor beta for a currency iis computed according to (2.13)—i.e. as the covariance between innovations in currency iand the dollar portfolio normalized by the variance of the dollar portfolio, defined as an equally weighted basket of all foreign currencies vs. U.S. dollar (excluding currency i). The m-month Q-betas are computed using the model-free measures of covariance and variance implied out from currency options with m-month maturity (expressions: (2.6)-(2.7)). The P-beta is computed using a 252-day rolling window regressions of daily innovations in the exchange iagainst daily innovations in the dollar factor (excluding currency i). The options data are from JP Morgan Dataquery and the exchange rate data are obtained from Reuters through Datastream. The sample period is from January 1998 to August 2016 and the data comprise 4659 daily observations. Panel A: Q-betas ∆St+m St∆it+m Horizon P1P2P3P3−P1P1P2P3P3−P1 1 mo 1.27 2.08 3.62 2.35∗0.61 1.17 1.61 1.00 [0.78] [0.94] [1.65] [1.78] 2 mo 1.01 0.79 2.94 1.93 0.56 1.24 1.59 1.03 [0.58] [0.37] [1.33] [1.19] 3 mo 0.60 0.86 1.66 1.06 0.53 1.16 1.65 1.12 [0.34] [0.36] [0.69] [0.58] 6 mo 0.47 1.13 0.29 -0.18 0.43 1.09 1.77 1.34 [0.26] [0.47] [0.12] [0.10] 9 mo 0.62 1.46 0.10 -0.51 0.38 0.96 1.89 1.50 [0.36] [0.66] [0.04] [-0.34] 12 mo 0.76 1.48 -0.03 -0.79 0.32 0.86 2.00 1.67 [ 0.46] [0.73] [-0.01] [-0.67] Panel B: P-betas ∆St+m St∆it+m Horizon P1P2P3P3−P1P1P2P3P3−P1 1 mo 1.97 3.21 1.79 -0.18 0.62 1.03 1.74 1.12 [1.15] [1.48] [0.79] [-0.11] 2 mo 1.14 2.62 0.98 -0.16 0.62 1.03 1.74 1.12 [0.70] [1.28] [0.41] [-0.08] 3 mo 0.83 2.07 0.23 -0.60 0.61 1.02 1.71 1.09 [0.49] [0.91] [0.09] [-0.29] 6 mo 0.57 1.87 -0.56 -1.14 0.57 1.01 1.71 1.14 [0.32] [0.85] [-0.22] [-0.44] 9 mo 0.73 1.81 -0.36 -1.08 0.54 1.00 1.68 1.14 [0.42] [0.84] [-0.15] [-0.48] 12 mo 0.86 1.47 -0.13 -0.99 0.52 0.99 1.67 1.14 [0.52] [0.71] [-0.05] [-0.48] 131 Table 2.6: Portfolio sorts on dollar factor exposure with mixed betas. This table reports annualized means, standard deviations and Sharpe ratios of excess returns of monthly rebalanced portfolios sorted on mixed dollar factor betas for horizons of 1-12 months. Each month, for each type of beta separately, the currencies are ranked in ascending order based on their dollar factor betas and allocated into three equal-weighted portfolios P1,P2, and P3. The investor buys (sells) each portfolio when the average forward discount is negative (positive). Panel A reports the results for betas based on correlations estimated using a 252-day rolling window and Q-variances derived via expressions (2.6)-(2.7). Panel B reports results for betas based on Q-correlations and P-variances (252-day rolling window variances). The maturity of the options used to compute the Q-variances matches the holding period of the forward contracts. The dollar factor beta for a currency iis computed according to (2.13)—i.e. as the covariance between innovations in currency iand the dollar factor normalized by the variance of the dollar factor, which is defined as an equally weighted basket of all foreign currencies against the U.S. dollar (excluding currency i). The options data are from JP Morgan Dataquery and the exchange rate data are obtained from Reuters through Datastream. The sample period is from January 1998 to August 2016 and the comprise 216 monthly observations. Panel A: Betas using P-correlations and Q-variances Mean Std Sharpe Ratio Horizon P1P2P3P3−P1P1P2P3P3−P1P1P2P3P3−P1 1 mo 1.97 3.87∗4.50∗∗ 2.53 6.88 9.99 10.70 8.33 0.29 0.39 0.42 0.30 [1.08] [1.71] [2.13] [1.54] 2 mo 1.53 2.56 4.04∗2.51 7.16 10.40 10.47 7.43 0.21 0.25 0.39 0.34 [0.84] [1.17] [1.90] [1.63] 3 mo 1.29 2.20 2.98 1.69 7.33 10.55 10.71 7.45 0.18 0.21 0.28 0.23 [0.70] [0.89] [1.29] [1.07] 6 mo 0.99 2.37 1.81 0.82 7.98 10.99 11.40 7.60 0.12 0.22 0.16 0.11 [0.53] [0.97] [0.75] [0.55] 9 mo 1.17 2.58 1.66 0.49 8.19 10.98 11.52 7.35 0.14 0.24 0.14 0.07 [0.65] [1.12] [0.72] [0.37] 12 mo 1.31 2.45 1.63 0.32 8.45 11.02 11.62 7.54 0.16 0.22 0.14 0.04 [0.77] [1.08] [0.73] [0.25] Panel B: Betas using Q-correlations and P-variances Mean Std Sharpe Ratio Horizon P1P2P3P3−P1P1P2P3P3−P1P1P2P3P3−P1 1 mo 2.62 3.54 4.12∗1.50 6.76 9.77 10.82 8.11 0.39 0.36 0.38 0.19 [1.59] [1.59] [1.81] [1.05] 2 mo 1.52 3.11 3.33 1.81 7.11 9.67 11.03 7.75 0.21 0.32 0.30 0.23 [0.91] [1.49] [1.42] [1.10] 3 mo 1.04 2.66 2.62 1.57 7.32 9.69 11.45 8.02 0.14 0.27 0.23 0.20 [0.57] [1.14] [1.02] [0.92] 6 mo 0.89 2.68 1.57 0.68 7.82 10.41 12.26 8.12 0.11 0.26 0.13 0.08 [0.49] [1.11] [0.60] [0.43] 9 mo 0.79 3.20 1.70 0.91 7.94 10.35 12.32 7.48 0.10 0.31 0.14 0.12 [0.47] [1.43] [0.68] [0.68] 12 mo 0.89 2.99 1.86 0.97 7.95 10.70 11.98 6.90 0.11 0.28 0.16 0.14 [0.56] [1.38] [0.81] [0.85] 132 33. 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