Interest rate changes and stock returns: a

european multi-country study with wavelets

R. Ferrer1, V. J. Bolos2, R. Benıtez3,

1 Dpto. Economıa Financiera y Actuarial, Facultad de Economıa,

Universidad de Valencia. Avda. Tarongers s/n, 46022 Valencia, Spain.

e-mail: roman.ferrer@uv.es

2 Dpto. Matematicas para la Economıa y la Empresa, Facultad de Economıa,

Universidad de Valencia. Avda. Tarongers s/n, 46022 Valencia, Spain.

e-mail: vicente.bolos@uv.es

3 Dpto. Matematicas, Centro Universitario de Plasencia, Universidad de Extremadura.

Avda. Virgen del Puerto 2, 10600 Plasencia (Caceres), Spain.

e-mail: rbenitez@unex.es

July 2014

Abstract

This paper investigates the linkage between changes in 10-year government bondyields and stock returns for the major European countries in the time-frequency domainby using a number of cross-wavelet tools in the framework of the continuous wavelettransform, mainly the wavelet coherence and phase-difference. The results reveal thatthe degree of connection between 10-year bond rate movements and stock returns dif-fers considerably among countries and also varies over time and depending on the timehorizon considered. In particular, the UK shows the greatest interdependence betweenlong-term interest rates and equity returns across time and frequencies, while the re-lationship is much weaker for several peripheral European countries such as Portugal,Ireland and Greece. The highest level of connection is observed for most countries sincethe onset of the recent global financial crisis. In addition, the significant linkage ismainly concentrated at investment horizons from one to two years.

1 Introduction

Understanding interrelations between interest rates and stock prices is important for in-vestors, portfolio managers, corporate managers and policy makers as it has critical impli-cations for different areas of finance such as asset allocation, portfolio management, riskmanagement and monetary policy transmission. The link between changes in interest ratesand stock returns is based on financial theory. Indeed, modern financial theory postulatesthat any firm generates a stream of future cash flows and the stock value of that firm isequal to the present value of all expected future cash flows discounted at the appropriate dis-count rate. In general, interest rates impact stock prices through two basic channels. Firstly,movements in interest rates have a direct effect on the discount rate used in standard equityvaluation models. Secondly, interest rate changes affect firms’ expectations about future cash

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flows by altering the cost of borrowing. Accordingly, it is expected that interest rates will bea significant determinant of stock prices.

An extensive literature has investigated the relationship between interest rate fluctuationsand stock returns. This body of work is principally focused on the time domain and the timeseries methods applied range from standard OLS regression to more sophisticated techniquessuch as cointegration, Granger causality or vector autoregressive (VAR) models. However,little attention has been devoted to the influence of a potentially relevant feature such as theinvestment horizon on the interest rate-stock market nexus. Financial markets like stock andbond markets are complex systems consisting of thousands of heterogeneous agents operatingover different time horizons which range from seconds to years, who collectively determineaggregate market behavior. For instance, agents with very short investment horizons, suchas chartists or day traders, are typically linked to speculative activity and their decisionsare largely based on ephemeral phenomena such as sporadic events, market sentiment orpsychological factors. On the opposite side, agents with long time horizons, such as funda-mentalists or big institutional investors, are more involved in investment activity and followmore closely macroeconomic fundamentals. Therefore, it is not unreasonable to think thatthe degree of connection between interest rates and stock prices may vary across frequenciesassociated with different investment horizons of market participants. In such a case, thewavelet methodology appears as a very attractive alternative that takes into account bothtime and frequency domains simultaneously. Wavelet analysis is a comparatively new, atleast in finance, and powerful tool for signal processing which offers a unique opportunity tostudy the interdependence between economic time series in the time-frequency space, pro-viding a deeper understanding of possible dependencies than time domain methods, whichaggregate all time horizons together.

The aim of this paper is to examine the interactions between changes in 10-year govern-ment bond yields and stock returns for the major European countries over time and acrossfrequencies by using several cross-wavelet tools based on the continuous wavelet transform.The main research question of this study is to determine whether the connection between10-year bond rate fluctuations and equity returns of a set of European countries changessignificantly over time and/or across different investment horizons. A priori we expect thatthis relationship will be stronger at long investment horizons because stock prices are morelikely to better reflect market fundamentals, such as interest rates, on longer time frames.However, at shorter horizons the linkage may be obscured by short-term noise caused bytransitory factors.

This work contributes to the existing literature in several ways. First, to the best of ourknowledge, this study is the first to apply continuous wavelet tools to analyze the interestrate-stock market at multiple investment horizons in a multi-country setting. Second, thisresearch combines standard cross-wavelet techniques such as the wavelet coherence and phase-difference with the development of novel tools based on the wavelet coherence in an attemptto get a better insight into the connection between these variables. Third, by covering arelatively long time period, this study allows us to determine whether major economic eventssuch as the introduction of the euro in 1999 or the international financial crisis which startedin the summer of 2007 have significantly altered the extent of linkage between long-terminterest rates and equity markets.

Our results indicate not only that the connection between movements in yields on 10-year sovereign bonds and stock markets is not uniform across countries, but also that itvaries significantly over time and across time horizons. In particular, the UK is identified asthe country with the strongest interdependence between long-term bond yields and equityreturns, followed by Germany, France, the Netherlands and Spain. In contrast, this linkageis much weaker for several peripheral European countries such as Portugal, Ireland andGreece. For most countries the linkage between government bond yields and equity marketsincreases since the beginning of the recent global financial crisis. The positive sign of thisrelationship suggests that 10-year bond rates and stock prices have tended to move in the

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same direction over the past few years, probably following economic prospects. This positiveassociation may also have been partly driven by flight-to-quality effects from stocks towardslong-term government bonds of European countries with stronger economic fundamentals andhigher credit rating. In addition, the significant interest rate-equity market link is mainlyconcentrated at investment horizons from one to two years.

The rest of the article proceeds as follows. Section 2 briefly reviews the relevant relatedliterature. Section 3 introduces the wavelet methodology and the continuous cross-wavelettools used to characterize the linkage between changes in long-term interest rates and equitymarkets. Section 4 describes the dataset employed. Section 5 reports and discusses the mainempirical findings and Section 6 concludes.

2 Literature review

An important body of research has been developed over the past few decades to investigate therelationship between changes in interest rates and stock returns. The bulk of this literaturehas concentrated on the banking industry due to the particularly interest rate sensitive natureof the financial intermediation business. In fact, a large proportion of income and expensesof banks is directly dependent on interest rates. Specifically, the typical maturity mismatchor duration gap between banks’ financial assets and liabilities resulting from the maturitytransformation function of banking institutions (i.e. the financing of long-term loans withshort-term deposits) is the most commonly cited reason for the interest rate sensitivity ofthese firms [1–3]. This positive duration gap (the average duration of banks’ assets exceedsthe average duration of banks’ liabilities) implies that rises in interest rates have a detrimentaleffect on banks’ value and vice versa. This occurs because when interest rates increase, thepresent value of banks’ assets falls more than that of banks’ liabilities and also the costs ofbanks’ liabilities rise faster than the yields on banks’ assets.

Nevertheless, movements in interest rates may also exert a significant influence on thevalue of non-financial corporations through various channels. Firstly, hikes in interest ratesincrease debt service payments of firms and can also reduce the demand for products byheavily indebted consumers, which means lower corporate profits and has a negative impacton share prices. Secondly, within the framework of dividend discount models rises in interestrates increase the cost of capital. This causes an increase of the discount rate used byinvestors and reduces the present value of future cash flows, thereby adversely affectingcompanies’ equity prices. Thirdly, changes in interest rates alter the market value of financialassets and liabilities held by non-financial firms. Finally, interest rate fluctuations can favourthe implementation of portfolio rebalancing strategies. As yields on fixed-income securitiesdecline, investors may shift into equities in search of higher yields, increasing demand forequities and therefore their prices. Regarding non-financial firms, regulated and/or highlyindebted industries such as Utilities, Real Estate, Telecommunications and Basic Resourcesare typically recognized as the most interest rate sensitive [4–6].

All these reasons suggest an inverse connection between interest rate fluctuations andequity returns. A positive link is, however, also possible because interest rates and stockprices can move in the same direction following changes in macroeconomic factors such asinflationary expectations or economic prospects. The flight-to-quality effects from stocks tobonds in an environment of increased financial uncertainty, such as that in force during therecent global financial crisis, may also contribute to the emergence of a positive relationship.Flight-to-quality occurs in times of financial turmoil as investors move capital away fromstocks toward safer investments such as government bonds. This process leads to a dramaticdecrease in the yield on long-term government bonds because of the big increase in thedemand for this type of securities and generates a positive correlation between changes inyields on sovereign bonds and stock returns.

Early studies in this field provided evidence of a significant negative relationship betweenchanges in interest rates and stock returns of both financial and non-financial firms [7–9].

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However, some more recent works, such as those of [4,5,10]), show that this relationship doesnot remain constant over time. In fact, the interest rate-equity market link seems to showa downward trend over the last years, mainly due to the increased availability of improvedtools for managing interest rate risk. More specifically, the spectacular growth in interestrate derivative markets and the expansion of European corporate bond markets as a resultof the euro’s adoption may have played a key role in this respect. In addition, firms’ stockreturns tend to be more closely linked with movements in long-term interest rates than toshort-term rates [11–14].

So far, most of the empirical research on the interest rate-stock market link has beenconducted in the time domain. Specifically, a wide range of time series methods has beenutilized, including linear regression [4–6], VAR techniques [15–17], cointegration analysis [18–20], Granger causality tests [21–23], nonlinear models [11, 12, 24], and GARCH (generalizedautoregressive conditional heteroscedasticity) models [1, 25, 26]. One obvious limitation ofthese approaches is that they are restricted to one or at most two time scales, i.e. the shortrun and the long run.

Nonetheless, there are a few recent papers investigating the interest rate-stock marketnexus through wavelet methods. For instance, [27], [28] and [29] explore the multi-scalerelationship between movements in interest rates and stock returns by using the maximaloverlap discrete wavelet transform, a variant of the discrete wavelet transform widely uti-lized in economic applications of wavelet analysis. These studies focus on aggregate equitymarkets of different countries and show that the linkage between interest rate fluctuationsand stock market differs from country to country and is dependent on the time horizon con-sidered. In turn, [30] is so far the only work that explores the existence of Granger causalitybetween interest rates and share prices across different frequencies using a continuous waveletapproach. He concludes that causal relations between both variables in the Indian economyvary significantly over time and across frequencies.

It is also possible to find a number of studies examining the relationship between interestrate changes and equity returns from an international perspective [4, 13, 31]. These worksdetect significant variations in interest rate sensitivity across countries and over time, al-though they do not take into account the possible effect of the investment horizon. However,no previous research has used continuous cross-wavelet tools to assess the interest rate-stockmarket nexus in a multi-country framework.

3 Methodology

Wavelet analysis is a powerful mathematical tool for signal processing in the time-frequencydomain that overcomes the main limitations of the Fourier transform [32]. Due to its flexi-bility, wavelet analysis is extensively used in disciplines such as geophysics, medicine, clima-tology or astronomy, although its application in economics and finance is a relatively newphenomenon [33–35].

A wavelet is a small “wave packet” that grows and decays in a limited time period. It isgiven by a function ψ in L2 (R) centered at the origin (more or less), with zero average andnormalized. A family of daughter wavelets ψu,s(t) can be obtained by simply scaling andtranslating ψ:

ψu,s(t) :=1√sψ

(t− us

), (1)

where s is a scaling parameter that controls the length of the wavelet, and u is a locationparameter that indicates where the wavelet is centered. Given a signal x(t) in L2 (R), itscontinuous wavelet transform CWT with respect to the wavelet ψ is a function of two variables

Wx(u, s) :=

∫ +∞

−∞x(t)ψ∗u,s(t) dt, (2)

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where ∗ denotes complex conjugation. It represents the frequency components (or details) ofx(t) corresponding to the scale s and time location u, providing a continuous time-frequencydecomposition of x(t), while the discrete wavelet transform (DWT) uses a specific subset ofdiscrete scale and location values.

By its very nature, the CWT contains a large amount of redundant information on theoriginal signal that makes it much easier to interpret the empirical results as it provides amore visually intuitive output. As argued by [36], the CWT is better for feature extractionpurposes, while the DWT is more useful for multiresolution analysis, particularly for noisereduction and data compression. For a long time, the discrete wavelet analysis has prevailedin economic research [37–41] due to its greater simplicity and more parsimonious nature.However, in recent years the continuous wavelet analysis has also become very popular in theeconomic-finance literature [42–45]. One of the major benefits of the CWT is its ability todescribe localized co-movement between two time series in the time-frequency space throughthe use of cross-wavelet tools.

Several types of wavelet families with different characteristics are available in the liter-ature. The application presented here utilizes the Morlet wavelet because it is the mostcommonly used wavelet and implies a very simple inverse relationship between scale andfrequency. Moreover, the Morlet wavelet is a complex wavelet that can be decomposedinto real and imaginary parts. This feature allows separation of amplitude and phase ofthe signal under study, providing more information about synchronization and delays be-tween two time series. The Morlet wavelet was introduced by [46] and can be defined as

ψ(t) = π−1/4eiω0te−t2/2, where ω0 denotes the central frequency of the wavelet. We set

ω0 = 6 since this choice provides a good balance between time and frequency localizationand it is very often employed in economic and financial applications [42].

In order to detect and quantify relationships between time series, two cross-wavelet tools,introduced by [47] within the framework of the CWT, can be used, namely the waveletcoherence and wavelet phase-difference. According to [48], the wavelet coherence betweentwo time series x(t) and y(t) is defined by

R2(u, s) =|S(s−1Wxy(u, s)

)|2

S (s−1|Wx(u, s)|2)S (s−1|Wy(u, s)|2), (3)

where Wxy(u, s) := Wx(u, s)W ∗y (u, s) is the cross-wavelet spectrum (∗ indicates the complexconjugate), and S is a smoothing operator in both time and frequency. The wavelet coher-ence (3) ranges from 0 (no correlation) to 1 (perfect correlation) and is analogous to thesquared correlation coefficient in linear regression. This concept is particularly useful for de-termining the regions in the time-frequency domain where two time series have a significantco-movement or interdependence.

In spite of its usefulness for measuring the strength of the linkage between any two timeseries in the time-frequency space, the wavelet coherence is able neither to determine thesign of this link nor to identify lead-lag relations between the two series. This problem canbe solved by using the wavelet phase-difference, which characterizes possible delays in theoscillations between the two series, providing information on lead-lag effects as well as thesign of the association. Following [48], the phase-difference is defined by

φxy(u, s) = tan−1

(=(S(s−1Wxy(u, s)

))< (S (s−1Wxy(u, s)))

), (4)

where < and = represent the real and imaginary parts, respectively.The phase information is graphically displayed on the same figure that the wavelet coher-

ence by plotting arrows inside the regions characterized by high coherence. A phase-differenceof zero indicates that the two time series move together at the specified frequency. Arrowspoint to the right (left) when the two time series are in phase (anti-phase). When the twoseries are in phase, they move in the same direction. Anti-phase means that the two series

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move in the opposite direction. Arrows pointing up suggest that the first time series leadsthe second one, while arrows pointing down indicate that the second series leads the firstone.

In addition, three novel cross-wavelet tools which complement the graphic informationprovided by the wavelet coherence and, therefore, allow gaining a more complete insight intothe interconnection between time series are proposed in this study. For defining these tools,it is convenient to re-scale the wavelet coherence using base 2 power scales

R2base2(u, k) := R2(u, 2k), (5)

where k is the binary logarithm of the original scale variable s, called log-scale. The reasonfor this change of variable lies in the fact that the family of dyadic wavelets is an orthonormalbasis of L2 (R) (see [49]) and so, the “natural” variable for the scale should be k instead of s.

Firstly, the percentage of total volume (PTV) of the wavelet coherence summarizes in asimple number the degree of coherence between the time series under study in a given rangeof times and log-scales. It is, in fact, the average of the wavelet coherence (5) re-scaled from0 to 100 and is given by

PTV :=100

(u1 − u0)(k1 − k0)

∫ k1

k0

∫ u1

u0

R2base2(u, k) dudk, (6)

where [u0, u1] × [k0, k1] determines the range of times and log-scales that we are interestedin studying (note that the range of scales is in fact

[2k0 , 2k1

]).

Secondly, as its name suggests, the percentage of significant area (PSA) of the waveletcoherence measures the proportion of the area in [u0, u1] × [k0, k1] which is statisticallysignificant at a given level of significance over the total area, and is given by

PSA :=100

(u1 − u0)(k1 − k0)

∫ k1

k0

∫ u1

u0

χlevel(u, k) dudk, (7)

where χlevel is the characteristic function of the corresponding significance level set.Thirdly, the percentage of significant volume (PSV) of the wavelet coherence is a mixture

of these two concepts, and is given by

PSV :=100

(u1 − u0)(k1 − k0)

∫ k1

k0

∫ u1

u0

R2base2(u, k)χlevel(u, k) dudk. (8)

The CWT applied to finite length time series inevitably suffers from boundary problems asits computation close to the edge of the signal involves nonexistent values. Although there area number of methods for dealing with these edge effects, all of them produce artificial waveletcoefficients at the borders [49]. In this context, the cone of influence delimits the region ofthe wavelet spectrum which is influenced by edge effects. Outside the cone of influence edgeeffects become important and values should be interpreted with great caution [47].

All these three novel measures can be computed inside the cone of influence by simplymultiplying the integrands by the characteristic function of the interior of the cone of influ-ence, χCoI, and substituting the denominator (u1 − u0)(k1 − k0) by the area of the interiorof the cone of influence. For example, the PTV inside the cone of influence is given by

PTVCoI :=100

AreaCoI

∫ k1

k0

∫ u1

u0

R2base2(u, k)χCoI(u, k) dudk, (9)

where AreaCoI :=∫ k1

k0

∫ u1

u0χCoI(u, k) dudk.

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4 Data description

Our dataset consists of yields on 10-year government bonds and stock market indices often European countries (Germany, the UK, France, Italy, Spain, the Netherlands, Finland,Ireland, Portugal and Greece) from January 1993 to December 2012. The starting date of theanalysis is January 1993 for two main reasons. First, this choice enables us to avoid possibledistortions in the results on the interest rate-stock market nexus caused by turbulences instock and bond markets within the framework of the crisis of the Exchange Rate Mechanism(ERM) of the European Monetary System (EMS) during the second half of 1992. In essence,the ERM crisis was linked to strong exchange rate pressures which culminated with the exitof the Italian lira and the British pound from the EMS in September 1992. Thus, given thatthe key role of exchange rate risk during this crisis might lead to a misperception of the realconnection between interest rates and stock returns, we have decided to begin the sampleperiod in 1993. Second, the lack of reliable data or even the non-existence of governmentbond data in the period prior to 1993 for several countries, especially Greece, Portugal andSpain. The stock indices considered are the DAX 30 (Germany), FTSE 100 (UK), CAC 40(France), FTSE MIB (Italy), IBEX 35 (Spain), AEX (Netherlands), OMXH (Finland), ISEQ(Ireland), PSI 20 (Portugal) and FTSE ATHEX 20 (Greece).

The use of the 10-year government bond yield as a proxy of long-term interest rateshas become increasingly popular in the literature on the linkage between interest rates andstock prices [1, 24, 50]. This choice is supported by the following arguments. First, 10-yearinterest rates incorporate market expectations about future prospects for the economy anddetermine to a large extent the cost of borrowing funds. Thus, long-term rates are likely tohave a critical influence on investment decisions and profitability of firms and, hence, on theirstock market performance. Second, long-term government bonds are often viewed as closermaturity substitutes to stocks, which may presumably increase the degree of linkage betweenboth financial assets. All data series are drawn from Thomson Financial Datastream.

In line with, among others, [8], [50] and [14], weekly data series (sampled on Wednesdays)are employed (a total of 1044 observations). Weekly frequency is preferred to daily andmonthly frequencies for several reasons. Firstly, daily data are more contaminated by noiseand anomalies such as day-of-the-week effects or non-synchronous trading bias than weeklyobservations. Secondly, compared to monthly data the weekly frequency provides a numberof observations large enough to yield more reliable results. Stock returns for each country arecalculated as the first log difference of stock price indices. Changes in 10-year sovereign bondyields are obtained as the first difference in the level of bond yields between two consecutiveobservations.

Table 1 presents some common descriptive statistics for the variables. The average weeklystock return over the whole sample period is positive but very small for all equity markets.The average weekly change in 10-year government bond rates is very close to zero in all cases.Standard deviations reveal, as expected, that movements in yields on 10-year bonds havelower volatility than stock returns for all countries. The sign of skewness varies, althoughit tends to be negative for equity returns and positive for 10-year bond rate fluctuations.Furthermore, all the series have positive kurtosis coefficients in excess of three, suggestingthe existence of heavy tails compared to a normal distribution. The Jarque-Bera statisticsconfirm this result, rejecting the null hypothesis of normality for stock returns and changesin 10-year bond rates of all countries at the 1% significance level. In order to determinethe order of integration of the series, the standard Phillips-Perron (PP) unit root test andthe Kwiatkowski-Phillips-Schmidt-Shin (KPSS) stationarity test are conducted. Both testsindicate that all the series of stock returns and movements in yields on 10-year bonds arestationary (integrated of order zero) at the 1% level.

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Variables Mean Median Min. Max. Std. Dev. Skewness Kurtosis JB statistic PP KPSSGerman stock returns (DAX 30) 0.00154 0.00572 -0.16804 0.17155 0.03229 -0.72 6.76 704.48*** -32.63*** 0.15British stock returns (FTSE 100) 0.00073 0.00250 -0.12732 0.13587 0.02393 -0.35 6.37 513.56*** -35.84*** 0.13French stock returns (CAC 40) 0.00067 0.00318 -0.14796 0.16625 0.03067 -0.36 6.14 450.09*** -37.11*** 0.21

Italian stock returns (FTSE MIB) 0.00059 0.00268 -0.15150 0.10625 0.03108 -0.43 4.81 175.08*** -33.15*** 0.32Dutch stock returns (AEX) 0.00094 0.00380 -0.17538 0.20377 0.03145 -0.56 8.42 1330.91*** -36.05*** 0.10

Spanish stock returns (IBEX 35) 0.00120 0.00401 -0.13328 0.12385 0.03167 -0.40 4.57 135.49*** -35.27*** 0.16Finnish stock returns (OMXH) 0.00183 0.00455 -0.18631 0.22965 0.04108 -0.48 6.30 512.99*** -34.41*** 0.06

Portuguese stock returns (PSI 20) 0.00065 0.00233 -0.18923 0.12115 0.02774 -0.64 6.86 717.90*** -32.41*** 0.33Irish stock returns (ISEQ) 0.00094 0.00299 -0.17576 0.14881 0.03018 -0.72 7.84 1106.49*** -33.90*** 0.07

Greek stock returns (ATHEX 20) 0.00029 0.00220 -0.16173 0.18826 0.04094 -0.16 4.71 131.65*** -31.41*** 0.06German 10-year bond yield changes -0.00006 -0.00014 -0.00323 0.00469 0.00102 0.42 3.98 72.45*** -33.09*** 0.05British 10-year bond yield changes -0.00006 -0.00006 -0.00544 0.00510 0.00123 0.18 4.54 108.88*** -34.89*** 0.05French 10-year bond yield changes -0.00006 -0.00013 -0.00430 0.00493 0.00108 0.49 4.57 150.16*** -33.59*** 0.06Italian 10-year bond yield changes -0.00009 -0.00010 -0.01105 0.01048 0.00167 0.03 11.52 3152.67*** -33.39*** 0.06Dutch 10-year bond yield changes -0.00005 -0.00015 -0.00266 0.00423 0.00099 0.57 3.95 94.53*** - -32.84*** 0.05

Spanish 10-year bond yield changes -0.00007 -0.00003 -0.01187 0.00720 0.00159 -0.56 10.15 2277.97*** -34.40*** 0.05Finnish 10-year bond yield changes -0.00009 -0.00016 -0.00581 0.01100 0.00132 1.16 11.14 3114.85*** -32.85*** 0.15

Portuguese10-year bond yield changes -0.00004 -0.00002 -0.02062 0.02249 0.00240 -0.01 24.11 19371.6*** -31.68*** 0.14Irish 10-year bond yield changes -0.00005 -0.00013 -0.01791 0.01638 0.00183 0.04 23.05 17477.3*** -30.03*** 0.15

Greek 10-year bond yield changes 0.00006 0.00000 -0.28446 0.05714 0.01008 -21.09 613.61 162806*** -35.85*** 0.05

Table 1: Descriptive statistics of stock returns and changes in yields on 10-year government bonds. This table reports descriptive statistics forweekly stock returns and changes in 10-year government bond yields of each country. The sample period is between January 1993 and December2012. They include mean, median, standard deviation (Std. Dev.), minimum (Min.) and maximum (Max.) values, as well as Skewness and Kurtosismeasures. JB refers to the statistic of the Jarque-Bera test for normality. The last three columns present the statistics of the Phillips-Perron (PP)unit root test and the KPSS stationarity test, respectively. *, **, *** indicate statistical significance at the 10%, 5% and 1% level, respectively.

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5 Empirical results

Figures 1 and 2 display the graphs of the estimated wavelet coherence and phase-differencebetween changes in yields on 10-year government bonds and stock returns for each country∗.Following standard practice in the literature, the wavelet coherence is presented by usingcontour plots as it involves three dimensions: frequency, time and wavelet coherence power.Frequency and time are represented on the vertical and horizontal axes, respectively. Withthe aim of easing interpretation, the frequency is converted into time units (years) and itranges from the highest frequency of one week (top of the plot) to the lowest frequency offour years (bottom of the plot). The wavelet coherence is depicted by a gray scale and theextent of coherence is interpreted in terms of the intensity of the gray color. Thus, the shadesfor wavelet coherence range from white (low values) to dark gray (high values). The thinblack line represents the cone of influence below which edge effects become important. Hence,values outside the cone of influence should be interpreted very carefully. The thick black lineisolates regions where the wavelet coherence is significant at the 5% level.

Since the theoretical distribution of the wavelet coherence is not known in general, thestatistical significance is usually assessed by using Monte Carlo simulation methods. Forexample, [48] generate 10000 pairs of white noise time series with the same length as theoriginal samples. In our case, we use normally distributed time series (concretely 1000 pairs)because the samples resemble more closely to follow a normal distribution. Moreover, thevariances of the generated time series affect slightly to the computation of the wavelet powerspectra and, consequently, to the wavelet coherency. Although the differences are not rele-vant, we use time series with the same variances as the original samples because it providesa bit more accurate result and the computational cost is identical.

As mentioned earlier, the phase-difference is a complementary tool to the wavelet co-herence. In essence, it enables studying possible lead-lag effects between 10-year bond ratefluctuations and stock returns in the time-frequency domain and also gives us the sign ofthe link between both variables. Phase information is indicated by arrows on the waveletcoherence plots. Horizontal (or close to horizontal) arrows imply that there is no lead-lagrelationships.

The results of the wavelet coherence analysis show that the connection strength betweenmovements in 10-year government bond yields and equity returns varies from country tocountry and also differs widely over time and across investment horizons. In particular, thehighest degree of linkage between 10-year bond rate changes and stock returns is observed inthe UK. Two reasons help to explain this finding. First, the British stock exchange is a highlydeveloped equity market with a significant level of maturity, in which a large proportion ofmarket participants have rational expectations and use all information available to them.This results in that stock prices tend to follow closely, with the only exception of shorterinvestment horizons which are more influenced by noise, fundamentals like long-term interestrates. Second, the fact that the Bank of England has traditionally had a greater real economyorientation, characterized by more aggressive and forceful monetary policy actions aimedat influencing economic growth, than the European Central Bank (ECB) may also play arole in the close linkage between yields on 10-year bonds and equity market. As is wellknown, the monetary policy of the ECB is conducted on the basis of the entire euro area.This means that the ECB is constrained in its decisions because it must strike a balanceamong competing interests from different member states with very diverse macroeconomicconditions. Thus, the more rigid and conservative operating framework of the ECB may bebehind the lower interest rate-stock market nexus in Eurozone countries. The evidence of astrong interdependence between interest rates and equity prices in the UK presented here is

∗All computations in this study have been performed by using a slightly modified ver-sion of the wavelet MatLab program written by C. Torrence and G. P. Compo available athttp://paos.colorado.edu/research/wavelets/. As described below, the difference with the original programof Torrence and Compo lies in the Monte Carlo simulation employed for the estimation of the statisticalsignificance level of the wavelet coherence.

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in line with the results of [51], [31] and [14].As can be seen in Figure 1, the interest rate-stock market link in the UK is consistent

over the entire sample period, although it tends to be stronger in times of market turmoilsuch as the bond market turbulence of 1994 or, especially, the recent global financial crisisthat began in the summer of 2007. It is also found that the significant linkage betweenmovements in 10-year bond yields and equity returns is mainly concentrated at investmenthorizons from one to two years, while for horizons of less than a month and a half theconnection is very weak. This result is in line with the widely accepted notion that investorswith long-term horizons are more likely to follow macroeconomic fundamentals such as long-term interest rates than investors with a shorter time frame. The phase information indicatesthat the relationship between 10-year bond yield fluctuations and stock returns in the UK hasundergone a remarkable change during the period of study. This connection is negative untilthe late-1990s and stock returns lead changes in 10-year bond rates, while the associationbecomes positive since the early 2000s and stock returns also lead interest rate fluctuations.

A set of core Eurozone countries, including Germany, France, the Netherlands and, toa lesser extent, Finland, also display a significant linkage between 10-year bond yields andstock returns during some specific time periods and frequencies. In particular, the interestrate-stock market nexus in these countries is more pronounced from the beginning of theglobal financial crisis in 2007. The sign of this relationship is generally positive, indicatingthat 10-year bond rates and equity prices have moved in the same direction during recentyears. This positive association shows that the historically low levels of interest rates overthe last few years have not been able to boost European stock markets, highlighting theseverity of the global financial crisis. One possible explanation for this finding is that bothlong-term government bond and stock markets have been primarily driven by the economicoutlook. Furthermore, flight-to-quality episodes from stocks into less risky assets such asgovernment bonds of safer European economies, within which these core Eurozone countriesoccupy a central role, during the recent international financial crisis may also have favouredthis positive linkage. Nevertheless, there does not seem to be clear lead-lag effects betweenchanges in yields on 10-year bonds and stock returns in these countries. In addition, mostof the significant association between both variables is also observed in the frequency bandfrom one to two years.

Interestingly, Italy and Spain exhibit a similar pattern of connection between yields on10-year bonds and stock returns, although the level of linkage seems to be a little higherin Spain. The greatest interdependence is detected in both countries in the run-up to thelaunch of the euro. Specifically, in the pre-euro era the correlation is negative and equityreturns seem to lead changes in 10-year bond rates in the two countries. This finding suggeststhat Italian and Spanish stock markets anticipated the beneficial impact on their respectiveeconomies resulting from sharp drops in long-term interest rates during the 1990s withinthe framework of the European convergence process. Furthermore, the significant interestrate-stock market link is concentrated at time horizons from three months to one year.

Another striking result is the low interdependence between 10-year government bondyields and equity returns for Portugal, Ireland and, to a lesser degree, Greece across mostfrequencies and time periods. This lack of connection may be attributed to the fact that stockexchanges of these countries are characterized by low liquidity, thin trading and relativelysmall capitalization, which leads to an idiosyncratic equity market performance in comparisonwith core Eurozone countries and causes the interest rate-stock market link to be very weakin these countries. Furthermore, from the onset of the global financial crisis in 2007 andespecially since the outbreak of the European sovereign debt crisis in late 2009, equity pricesin these peripheral countries have primarily reflected very poor economic prospects. Interestrate risk has played, by contrast, a marginal role in explaining the behavior of stock marketsof these economies in recent years. As a matter of fact, precisely these three countries were

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Figure 1: Wavelet coherence and phase-difference. The thick black contour denotes the 5%significance level of the wavelet coherence estimated from Monte Carlo simulations. Thecone of influence is shown with a light black line. The phase-difference between interest ratechanges and stock returns is indicated by black arrows.

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Figure 2: Wavelet coherence and phase-difference. The thick black contour denotes the 5%significance level of the wavelet coherence estimated from Monte Carlo simulations. Thecone of influence is shown with a light black line. The phase-difference between interest ratechanges and stock returns is indicated by black arrows.

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finally bailed out by the European Union and International Monetary Fund from 2010.Next, the clear evidence of a time-varying interest rate-stock market link provided by

the wavelet coherence is complemented by a sub-period analysis in Tables 2 and 3. Thesetables† report the values of two of the new cross-wavelet tools based on the concept of waveletcoherence introduced in the methodology section. Specifically, Table 2 shows the PTV ofthe wavelet coherence within the cone of influence for the entire sample period as well asthree sub-periods, namely the pre-euro (January 1993-December 1998), post-euro (January1999-July 2007) and global financial crisis (August 2007-December 2012) periods, for eachcountry. In turn, Table 3 presents the PSA at the 5% level of the wavelet coherence withinthe cone of influence for the full period and the three sub-periods for the different countries‡.

Naturally, given that these novel wavelet tools are summary measures of the waveletcoherence, the results in Tables 2 and 3 corroborate the findings of the wavelet coherenceanalysis with regard to the time evolution of the linkage between changes in yields on 10-year government bonds and stock returns. Therefore, for most countries the greater degree ofconnection between long-term government bond and stock markets occurs during the periodof the recent global financial crisis. However, an important interest rate-stock market linkis also identified in the run-up to the euro adoption for some countries, mainly Italy andSpain. In contrast, the lower interdependence between both markets is found, with very fewexceptions, in the period immediately after the launch of the euro, probably as a result ofthe marked reduction in the level and volatility of interest rates caused by the introductionof the euro [4, 14].

Country Whole sample Pre-euro Post-euro Global financialperiod sub-period sub-period crisis sub-period

Germany 44.03% 39.61% 41.84% 53.96%UK 45.68% 47.38% 43.05% 49.47%

France 41.30% 43.93% 38.56% 44.28%Italy 38.79% 46.58% 34.42% 39.52%Spain 38.24% 47.41% 34.79% 35.29%

Netherlands 40.79% 42.57% 37.77% 45.40%Finland 37.61% 34.31% 37.01% 42.71%Ireland 32.22% 35.74% 29.67% 33.79%Greece 35.51% - 36.53% 33.73%

Portugal 32.61% 34.37% 30.50% 35.41%

Table 2: Percentage of total volume (PTV) of the wavelet coherence for each country. Thistable presents the values of the PTV of the wavelet coherence between changes in yields on10-year government bonds and stock returns inside the cone of influence for the full sampleperiod as well as the pre-euro (January 1993-December 1998), post-euro (January 1999-July2007) and global financial crisis (August 2007-December 2012) sub-periods for each country.

Finally, in order to provide a more detailed picture of the changing nature over timeand across frequencies of the association between 10-year bond rate movements and equityreturns, Figure 3 shows the rolling window estimation of the PTV of the wavelet coherencefor different frequency bands for some of the countries with a more significant connection(the UK, Germany, France and Spain). In particular, a window of fixed size equal to one

†The results for Greece in the pre-euro sub-period are not presented in Tables 2 and 3 as data on yieldson Greek 10-year government bonds start from April 1999.‡The values of the PSV at the 5% level of the wavelet coherence inside the cone of influence are virtually

identical to those reported in Table 3. For the sake of brevity, the results of this cross-wavelet tool are notpresented here, but they are available upon request from the authors.

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Country Whole sample Pre-euro Post-euro Global financialperiod sub-period sub-period crisis sub-period

Germany 14.54% 11.59% 9.49% 29.11%UK 18.73% 23.46% 14.55% 22.48%

France 12.15% 14.67% 8.62% 16.98%Italy 11.47% 21.28% 6.36% 11.44%Spain 13.07% 26.70% 7.27% 10.17%

Netherlands 11.39% 13.62% 8.01% 16.26%Finland 9.82% 8.06% 7.24% 17.48%Ireland 5.25% 8.02% 2.97% 7.09%Greece 8.38% - 9.24% 6.87%

Portugal 4.79% 5.22% 2.31% 9.74%

Table 3: Percentage of significant area (PSA) of the wavelet coherence for each country. Thistable reports the values of the PSA at the 5% level of the wavelet coherence between changesin yields on 10-year government bonds and stock returns inside the cone of influence for thefull sample period as well as the pre-euro (January 1993-December 1998), post-euro (January1999-July 2007) and global financial crisis (August 2007-December 2012) sub-periods for eachcountry.

year is used and three frequency bands are considered to determine the specific time horizonsin which the interest rate-stock market linkage is closer: from two weeks to one month and ahalf, from one month and a half to one year and from one to two years§. The plots in Figure 3confirm that the strongest relationship between 10-year bond yields and stock returns is foundin the frequency band from one to two years, with the only exception of Spain. Furthermore,it is again observed that the connection strength between both markets tends to be higherfrom the outset of the global financial crisis in 2007, except in the Spanish case. In fact,Spain exhibits a peculiar behavior in terms of the interest rate-stock market link by whichthe greater connection is confined to the pre-euro era. Instead, the linkage is extremely weakduring the recent financial crisis period, reflecting that the key driver of the Spanish stockmarket over the last few years has been the poor economic outlook and not the behavior ofinterest rates.

Interestingly, the time interval between the end of 2002 and the first quarter of 2003is characterized by a highly significant interest rate-stock market nexus at shorter horizons(from two weeks to one month and a half) for all these countries. This period is marked bythe confluence of the dramatic decline in stock prices after the bursting of the dotcom bubblein 2000, the terrorist attack on the US on September 11, 2001, and geopolitical tensions inthe Middle East, and the sharp fall in interest rates to record lows for the last forty years.Therefore, it is not surprising to find a significant positive association between long-termgovernment bond and stock markets in such a scenario.

6 Conclusions

This paper investigates the linkage between changes in yields on 10-year government bondsand stock returns for the largest European countries in the time-frequency space by applyinga number of continuous cross-wavelet tools, which include the popular wavelet coherence andphase-difference as well as several novel summary measures based on the wavelet coherence.

The empirical results show that the strength of the relationship between movements in

§It is worth highlighting that shorter frequency bands (e.g. from two to four years) have not been consid-ered due to the scarcity of data in these shorter bands not subject to edge effects.

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Figure 3: Rolling estimates of the percentage of total volume (PTV) of the wavelet coherencefor countries and frequency bands. This figure displays the rolling estimates of the PTV ofthe wavelet coherence between changes in 10-year government bond yields and stock returnsinside the cone of influence for different frequency bands. Percentages and time are repre-sented on the vertical and horizontal axes, respectively. A rolling window of fixed lengthequal to one year is employed. Dotted lines represent the rolling estimates corresponding tothe frequency band from two weeks to one month and a half. Dashed lines show the rollingestimates for the frequency band from one month and a half to one year. Finally, solid linesrefer to the frequency band from one to two years.

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long-term sovereign bond yields and stock returns differs considerably among countries andalso varies over time and depending on the investment horizon under consideration. The UKemerges as the country with greater interconnection between 10-year bond yields and equitymarket across time and frequencies. This finding may be explained for the high level ofmaturity of the British stock market and the more aggressive and proactive monetary policystance of the Bank of England in comparison with the European Central Bank. A significantassociation between long-term bond yields and stock markets is also found for a group of coreEurozone countries such as Germany, France, the Netherlands and Finland, and for Spain.Instead, the linkage is very limited in Portugal, Ireland and Greece.

The highest level of connection between 10-year bond rate fluctuations and stock returnsis observed for the UK and core Eurozone countries during the recent global financial cri-sis period. This finding is consistent with the hypothesis that interactions among financialmarkets tend to increase in times of financial turbulence due to the existence of significantcontagion effects. The positive sign of the interest rate-stock market nexus suggests thatyields on 10-year bonds and equity prices have moved together over the past few years, possi-bly driven by economic prospects and the flight-to-quality wave from stocks into governmentbonds of countries with more solid economic fundamentals. In addition, it is worth notingthat the strongest linkage between changes in 10-year sovereign bond rates and stock returnsis generally observed at time horizons from one to two years.

The evidence presented has important practical implications for market participants.For instance, risk-conscious investors and portfolio managers should take into account thetime- and investment horizon-dependence of the interest rate-stock market nexus in theirinvestment decisions and diversification and risk management strategies. These results canalso help policy makers to gain insight into the transmission mechanism of monetary policy.Finally, this analysis shows the serious shortcomings of prior research focused on standardtime series methods which do not consider the time and frequency-varying features of theinterest rate-stock market link.

Acknowledgments

The authors gratefully acknowledge Jose Miguel Ferrer and Javier Rivaya for their valuablecomments and suggestions.

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