Research in International Business and Finance 25 (2011) 53–63

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Research in International Businessand Finance

journal homepage: www.elsevier.com/locate/r ibaf

Global trends in real risk free rates

Hui Hea,∗, Peter Lockeb

a School of Business, Finance and Business Law, James Madison University, MSC0203, Harrisonburg, VA 22807, United Statesb M. J. Neeley School of Business, TCU Box 298530, Fort Worth, TX 76129, United States

a r t i c l e i n f o

Article history:Received 13 February 2009Received in revised form 21 June 2010Accepted 10 July 2010Available online 30 July 2010

JEL classification:F20F21F33

Keywords:Real interest ratesCommon trendsCointegrationGlobal finance

a b s t r a c t

We examine real returns to government debt of the G7 countries,for both short and long maturities. Our focus is on returns to fixedincome investing rather than contemporaneous yields. We find evi-dence that investments in the same maturity across countries maybe modeled as a cointegrated process, in a vector error correctionframework, with common trends separated into their permanentand transitory components for the system. Our findings are basedon analysis of both short-term maturities and long-term maturi-ties. However, the structure varies excessively across maturitiesand time frames, with recent data showing less integration.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Recent events in the Euro zone (Greece debt crisis) show us that government bond markets arefar from fully convergence and we are still in search of a single global risk free rate. Theoreticallyreal risk free returns among comparable countries ought to be highly related. In particular, investorsseeking a risk free return over a particular maturity can easily transfer funds among the G7 (pre-Russia) countries. A higher expected return in any of these countries ought to draw funds away fromthe other six countries, if these are all considered risk free. Such global arbitrage or quasi-arbitragewill result in similar expected real returns across these countries. Past studies have found unusuallysmall cross-country correlations among risk free yields (e.g., Hardouvelis, 1994). Yields on debt areonly ex ante expectations of a nominal return to maturity. Other studies have assumed away the issue

∗ Corresponding author.E-mail address: hehx@jmu.edu (H. He).

0275-5319/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.ribaf.2010.07.001

54 H. He, P. Locke / Research in International Business and Finance 25 (2011) 53–63

of dynamic global integration and constructed a “world interest rate” by forming a weighted averageof individual country rates (e.g., Plosser and Rouwenhorst, 1994). Thus, there is plenty of room formore knowledge regarding the degree of integration of global risk free rates.

We present a theoretical argument regarding global risk free returns, and develop tests based onthat argument. Essentially, total real returns from investing in global sovereign debt should be drivenby term premia plus a nonstationary term (random walk component). We apply the Gonzalo andGranger (1995) permanent and transitory components separation method and decompose the returnsto G7 sovereign debt (one particular maturity group) into stationary (transitory) and nonstationaryrandom walk (permanent) components. Our permanent components will be driven by common globalstochastic trends. There may be only one single global stochastic trend, and if this is the case we wouldconsider this a finding of a strong form of integrated global sovereign debt. If there is more than onecommon trend, but there still exist some long-term equilibrium relationships across countries, wewould consider this a weak form of integrated global real risk free real rates as there is still certainpredictability in the system. Finally, if we fail to find any equilibrium relationship among the returnsto sovereign debt, there is no common global trend.

The purpose of the paper is to uncover evidence of convergence of global real risk free rates towarda common trend or set of trends. This is not the same as finding a unique real risk free rate for allcountries. A unique risk free rate can be viewed as a special case of market integration, but in this casethere will be no arbitrage opportunities, such as in the “ideal” Euro zone. Although we are not likelyto find evidence of a global risk free rate, we may find that there is a common global trend or trendsin real risk free rates. With a finding of this commonality, an investor seeking a real risk free returnof a particular maturity should be indifferent between investing in any of these countries’ sovereigndebt, with appropriate currency risk management.

Most of recent interests in market integration have been focused on international equity markets,and only a few of them on international government bond markets (Clare et al., 1995; Smith, 2002; Barrand Priestley, 2004; Yang, 2005; Davies, 2007). But even these studies do not agree with each other.Smith (2002) find cointegrating relationships across market, while Clare et al. (1995) and Yang (2005)fail to find common trends. Davies (2007) uses daily frequency data and finds a common long-runrelationship which subjects to structural change. Barr and Priestley (2004) suggest that world bondmarkets might not be fully integrated due to home bias or institutional factors. Our research adds tothis unsettled field by looking at total real risk free rates and separating long-term and short-termmaturities.

There have been several cross-country interest rate studies in the literature, typically compar-isons of international term structure models and pursuits of economic factors that help to predictrates (e.g., Hardouvelis, 1994; Jorion and Miskin, 1991; Plosser and Rouwenhorst, 1994). Economet-ric techniques such as principal component analysis (PCA) or inter-battery factor analysis have beenapplied to international bond returns with different maturities and find various factors that explaincross-country variations (e.g., Driessen et al., 2003; Perignon et al., 2007). Cointegration tests in inter-national equity markets have shown common trends driving these markets (Kasa, 1992; Richards,1995; Fraser and Oyefeso, 2005). Studies have also been done using cointegration technique to modelgovernment bonds and find endogenous relationships that help to forecast (Hall et al., 1992; Bradleyand Lumpkin, 1992; Gonzalo and Granger, 1995; Smith, 2002). The advantage of cointegration tests isto explore a long-term equilibrium in which endogenous variables are predictable, hence the benefitsof international diversification are reduced (Smith, 2002).

However, most interest rate analysis uses nominal, ex ante, yields as the subject of analysis, evenwhen applying cointegration. This raises a specification issue. The proper series to begin with in acointegration analysis of returns to holding debt are the values of fixed income investments. In otherwords, an investor will be concerned with the total real return on investments in any country, irre-spective of contemporaneous ex ante yields. This is particularly true when there is a common realrisk free return process. In this case, total real cross-country accumulations will be cointegrated, anddriven by a common global stochastic trend or trends. Therefore, we use ex-post total real returnson sovereign debt, with adjustments for maintaining a constant maturity, rather than ex ante yields.We calculate total real returns by adjusting the total return on a constant maturity bond index by thecorresponding exchange rates and consumer price changes. If the total real return are nonstationary,

H. He, P. Locke / Research in International Business and Finance 25 (2011) 53–63 55

as we expect, then the real returns, the first differences of the total returns, may be stationary. In otherwords the real returns may be modeled as a vector error correction process, with the error(s) a linearcombination of the total returns.

In searching for the common global trend(s) among G7 sovereign debts, we first determine whetherthere is a cointegrating relationship among the total real returns, and then use the cointegrating vec-tors to construct the global trends during the periods December 1993 to December 2009 for both longand short maturities. Since the Euro area adopted a unified currency and macroeconomic policies in1999, we divide the period into 1993–1998 and 1999–2009. We find evidence of strong form integra-tion of returns to global debt long maturities rates for the period 1993:12–1998:12, but not for thelatter period. For short-term maturities, only the weak form integration for the global risk free rate isconfirmed in both periods.

2. The estimation and testing procedures

The hypothesis of a global real risk free process implies that if the real values of governmentsecurities (with reinvested coupons) are nonstationary, changes in these values will be stationary,and may be modeled jointly in a cointegrated framework. We use the Johansen (1988, 1991) rank testand maximum eigenvalue test to find the cointegration rank, i.e., the cointegrating relationships in thetotal real treasury indices. This section will discuss briefly these tests and also show how to constructthe Gonzalo and Granger (1995) common long-memory components using these test results.

We define for n countries a vector of real returns �Yt as the real percentage changes (logarithmicdifferences) in constant maturity sovereign debt with reinvested coupons. For our analysis, the “real”return will be from the point of view of a U.S. investor. Other home countries could be used withoutchanging the analysis. If the logarithms of total real returns to sovereign debt, Yt, are integrated oforder one and cointegrated then we may describe their evolution by a vector error correction model(VECM):

�Yt = ˘Yt−1 + �1�Yt−1 + · · · + �p�Yt−p + zt (1)

with the associated vectors of real innovations, z1, . . ., zT, assumed to be IINn(0, �). If real yieldsare constant, then the real returns given by �Yt would be identical to a vector of the real yields onsovereign debt at time t − 1. In other words, $100 invested in a government security with a 5% annualyield would grow to $105 in a year. Instead, since real yields are likely to change over time, we focuson a vector of the ex-post changes in total real returns to investing in sovereign debt, given by �Yt,to obtain a comparable measure across countries for the rationally expected return to a U.S. investorfrom investing in those countries’ risk free debt over a time interval.

The estimated rank of ˘ in Eq. (1), which we denote r, provides the number of cointegratingrelationships. ˘ may be considered the “long-run impact matrix”, to distinguish these effects fromthe short run autocorrelation terms � s, s = 1, 2, . . ., p. If none of the eigenvalues of ˘ are significantlydifferent from zero, then r = 0. We would be faced with no cointegration, and no global real risk freerate integration. At the other extreme, if all of the eigenvalues of ˘ are significantly different fromzero (and less than unity), then the rank of ˘ is equal to the number of series (r = n) and we have astationary process with no long-term expected changes in real returns. Of most interest is the casewhere 0 < r < n, so that the vector of total returns on sovereign debt, Yt is nonstationary, and thereare r cointegrating relationships among the elements of Yt, or, equivalently, there are (n − r) global orcommon stochastic risk free trends.

In the presence of cointegrated real total returns, the long-term impact matrix ˘ may be factoredinto the product of two n × r matrices ˛ and ˇ so that ˘ = ˛ˇ′. The matrix ˛ can be considered theadjustment matrix which acts on the vector series ˇ′Yt, a set of linearly independent combinations ofthe real country debt indices that are stationary. Since the decomposition into ˛ and ˇ is not unique,the specific elements of ˛ and ˇ are generally not of particular interest. Instead, the dimensionalityand structure of the space spanned by their respective columns is of interest (e.g., Kasa, 1992).

56 H. He, P. Locke / Research in International Business and Finance 25 (2011) 53–63

Substituting arbitrary ˛ and ˇ for ˘ , we may rewrite the VECM as:

�Yt = c + ˛ˇ′Yt−1 +p∑

i=1

�i�Yt−i + zt (2)

We use Johansen (1988, 1991) and common interpretations of the statistics on ˘ to select a rank rand lag structure for the system of total real risk free returns.

After we identify the rank of the cointegration relationship, we obtain the estimates of the r coin-tegrating relationships among the total returns. But this tells us little about what the global stochastictrends will look like and how each country’s real debt index is related to the global stochastic trends. Wealso want to know how the global stochastic trends affect the permanent components each country’sreal debt index. For example, if real returns are mainly determined by innovations to the long-memorycomponents, then the short run components will be quite small. Here we follow Gonzalo and Granger(1995) and separate the permanent and transitory components in each subsystem of real debt indices:

Yt = A1˛′⊥Yt + A2ˇ′Yt (3)

where A1 = ˇ⊥(˛′⊥ˇ⊥)−1 and A2 = ˛(ˇ′˛)−1. The matrices ˛⊥(n × (n − r)) and ˇ⊥(n × (n − r)) denote the

orthogonal complements to ˛ and ˇ respectively. The row or rows of ˛′⊥ define the space of the global

stochastic trend ft = ˛′⊥Yt and indicate the relative contribution of each country’s real index to the

particular global trend associated with the row.According to the rational expectations theory of the term structure (Shiller, 1990), term premia

are assumed to be constant through time, but vary with maturities of one country’s term structure.Similarly in our cross-country case, we argue that if there is indeed a global real risk free process,we may separate real total return indices into a global factor plus risk premiums for that particularconstant maturity group. The global real risk free component of each country’s rate is free to evolve,but the risk premia depend only on the country and maturity and not time. We will refer to this as therational expectation hypothesis of global real risk free rates. Consider the following representation:

Yi,t = RGi,t + �i,t (4)

where Yi,t represents the vector of real total returns on sovereign debt in the G7 countries, the subscripti indicates the maturity, and t indicates time. Recall that, for convenience, the real return is from the U.S.investor’s perspective. The term RG

i,trepresents the common or global component of the ith maturity

group at time t, while �i,t represents a vector of premia for each country of the ith maturity group attime t. With a global risk free term structure, Yi,t and RG

i,tare nonstationary processes but the �i,t may

be stationary processes as they are often discussed in the term premia literature.In our strong rational expectations hypothesis of the integration of global risk free rates, the global

component RGi,t

will be a univariate nonstationary process for each maturity, or a single global factor.

More generally, RGi,t

can be expressed as a vector parameter times a nonstationary process. The commoncomponent of the real total return process is driven by the common trend, i.e.,

RGi,t =

⎡⎢⎣

aCanada

aGermany

· · ·aU.S.

⎤⎥⎦ × fi,t =

⎡⎢⎣

a1,Canada· · ·ak,Canada

a1,Germany· · ·ak,Germany

· · ·a1,U.S.· · ·ak,U.S.

⎤⎥⎦ ×

⎡⎢⎣

f1,t

f2,t

· · ·fk,t

⎤⎥⎦ (5)

where fi,t represents the common trend that drives the nonstationary processes in the strong form ofrational expectation hypothesis, and aj represents the country loadings of the global trend for eachcountry j. If aj = a ∀ j, then we have the RG

i,tas a single process. fi,t can be expressed as fi,t = (f1,i,t, f2,i,t, . . .,

fk,i,t) with k (=n − r) representing the number of common trends found in the cointegrating system.In the weak form of the rational expectation hypothesis of the global interest rate integration, thereremain common long-term equilibriums in RG

i,t, but there is more than one global factor. These other

factors may be due to regional or country institutional issues which lead to longer term or persistentcapital market imperfections. The representation of RG

i,tgiven by (5) is estimated by Gonzalo and

Granger (1995) P–T separation.

H. He, P. Locke / Research in International Business and Finance 25 (2011) 53–63 57

Given the general level of global financial integration, we expect at least one common global factor.If other country-specific risk factors persist, such as any regional institutional constraints or liquidityconcerns, we could find more global factors but not less. More cointegrating relationships indicatethere are more linear constraints on the variables in the system so that they may not “wander” exces-sively from each other. For the case of no cointegration, each individual series will appear to be aunique stochastic nonstationary process, i.e., there is no predictability in the system and the marketis efficient.

3. Data

Monthly data of constant maturity total return indices are obtained from DataStream All Tradedindices for G7 countries (Canada, France, Germany, Italy, Japan, U.K. and U.S.) and for each country:maturities of 1–3 years (short-term) and over 10-year (long-term) indices from December 1993 toDecember 2009.1 These indices include cash flows reinvested into the index on the day they takeplace and adjustments for serial redemption bonds to account for price movements resulting fromdrawings and for bonds with ex-coupon periods. The U.S. consumer price index is collected from USDepartment of Labor and foreign exchange rate data is retrieved from the U.S. Federal Reserve BankReports.

To calculate constant maturity total return indices, DataStream takes into account the effect ofre-investing all the gross coupons received back into the bonds of the index and then adjusts the mixof bonds in the index in order to hold the maturity constant. All indices are valued in local currencyterms.2 To construct our real total return indices, we first adjust the total return index RIj,i,t for eachcountry j, maturity range i, and time t, by the foreign exchange rate to obtain a dollar equivalent totalreturn index RI$. For any investors in one G7 investing in another G7, a currency risk is embedded inthe total returns after adjusting for exchange rate changes. Thus, for example, U.K. default and U.S.default risk may be comparably near 0, but there is the issue of translating the risk free pound returninto dollars. Our argument is if there is global financial convergence, that rational expectations willforce the return on all sovereign debt with similar default risks, such as those in the G7, to be thesame for any investor in any of the G7 countries. We choose, without loss of generality, a U.S. investoras our representative investor. Next we turn the series to real returns, or constant U.S. dollars, usingthe U.S. consumer price index. Finally we apply logarithms, so that the error correction model usingchanges in these log indices has a vector of ex-post real (constant dollar) returns on the left handside. There is a structural change in January 1999 for Italy, Germany, France, due to the introduction ofEuro. Therefore we break our sample into two subperiods: 1993:12–1998:12 and 1999:1–2009:12. Weexamine these two subsamples independently for both maturity ranges. Table 1 gives the correlationsamong countries for the two maturity groups in these two periods for both total real risk free returnindices and their first differences (real returns). The high correlation coefficients among total returnindices also justify the application of cointegration below.

1 Data for Italy in the long-term index is available from December 1, 1993.2 Datastream provides the calculation of the total return index (RI) as follows. A simple form is given by:

RI0 = 100

RIt = RIt−1 ×∑

i(P∗

i,t+ Ai,t + CPi,t + Gi,t ) × Ni,t−1∑

i(Pi,t + Ai,t−1 + CPi,t ) × Ni,t−1

where the summations are over the bonds currently in the index. Pi,t is the clean price of the ith bond at time t; Pi,t* is theclean price of the ith bond at time t, adjusted for a partial serial redemptions. At all other times it is the same as the unadjustedprice; Ai,t is the accrued interest to the ‘normal’ settlement date; Gi,t is the value of any coupon payment received from the ithbond at time t or since time (t − 1). If none, the value =0; N is the nominal value of amount outstanding if known, otherwisethe issued amount. CP is an adjustment made for bonds which have ex-dividends periods: when a bond goes ex-dividend, CPhas a value equal to the next coupon payment; outside the ex-dividends period, CP = 0. This compensates for the sharp drop inaccrued interest when a bond goes ex-dividend. For any bonds currently in the index that have serial redemption features, anadjustment is made when t falls within the period between the drawing date and the next serial redemption date. DataStreamAll-traded Indices cover all traded bonds, irrespective of issue size. The start date for the entire set is December 30, 1988, whichis the benchmark date for the indices (index = 100).

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25 (2011) 53–63Table 1Correlation matrix for monthly total real returns indices.

Total Real Return Indices Differences of total real returns

CAN FRA GER ITA JAP UK US CAN FRA GER ITA JAP UK US

1993:12–1998:12 1993:12–1998:121–3 year

CANADA 1.00 1.00FRANCE 0.38 1.00 0.16 1.00GERMANY 0.14 0.95 1.00 0.13 0.96 1.00ITALY 0.80 0.56 0.28 1.00 0.13 0.48 0.38 1.00JAPAN −0.60 0.18 0.45 −0.66 1.00 0.15 0.58 0.64 0.12 1.00UK 0.75 0.33 0.08 0.87 −0.74 1.00 0.35 0.57 0.56 0.38 0.36 1.00US 0.74 0.29 0.03 0.86 −0.78 0.97 1.00 0.31 0.27 0.26 0.17 0.31 0.33 1.00

Over 10 yearCANADA 1.00 1.00FRANCE 0.90 1.00 −0.02 1.00GERMANY 0.81 0.97 1.00 0.10 0.97 1.00ITALY 0.93 0.87 0.77 1.00 −0.07 0.99 0.94 1.00JAPAN 0.37 0.58 0.68 0.22 1.00 0.10 0.18 0.26 0.14 1.00UK 0.94 0.89 0.84 0.94 0.38 1.00 0.46 0.00 0.17 −0.08 0.23 1.00US 0.93 0.94 0.92 0.89 0.48 0.94 1.00 0.06 −0.09 −0.04 −0.11 0.35 0.01 1.00

1999:1–2009:12 1999:1–2009:121–3 year

CANADA 1.00 1.00FRANCE 0.98 1.00 −0.02 1.00GERMANY 0.98 1.00 1.00 0.10 0.97 1.00ITALY 0.98 1.00 1.00 1.00 −0.07 0.99 0.94 1.00JAPAN 0.59 0.67 0.67 0.67 1.00 0.10 0.18 0.26 0.14 1.00UK 0.97 0.95 0.95 0.95 0.47 1.00 0.46 0.00 0.17 −0.08 0.23 1.00US 0.93 0.93 0.93 0.93 0.62 0.88 1.00 0.06 −0.09 −0.04 −0.11 0.35 0.01 1.00

Over 10 yearCANADA 1.00 1.00FRANCE 0.98 1.00 0.06 1.00GERMANY 0.98 1.00 1.00 0.27 0.94 1.00ITALY 0.98 1.00 1.00 1.00 −0.03 0.99 0.89 1.00JAPAN 0.78 0.82 0.82 0.79 1.00 0.27 0.17 0.30 0.10 1.00UK 0.95 0.95 0.95 0.97 0.66 1.00 0.61 0.11 0.34 0.00 0.39 1.00US 0.94 0.95 0.96 0.94 0.88 0.87 1.00 0.56 0.06 0.25 −0.04 0.42 0.52 1.00

Note: The Pearson Correlation coefficients are presented above. The total real return index is calculated by ri$j,i,t

= LN(RI$j,i,t

/CPIUS,t ) for each country j maturity i and time t, where

RI$j,i,t

= RIj,i,t ∗ S$j,t

/S$j,0

. RIj,i,t represents the DataStream total return index, in local currency. S$j,t

and S$j,0

represent the exchange rates in terms of local currency per U.S. dollar at time t andtime 0 (index base period). The differences of total real return indices represent the monthly real returns for each bond indices.

H. He, P. Locke / Research in International Business and Finance 25 (2011) 53–63 59

4. Results

To reach our objective of finding any integration of global real risk free rate process, we first testthe earlier period for the two groups, then use the second period as a robustness check. We also usethe sample for the G4 (Germany, Japan, UK and US) for the whole sample period as another robustnesscheck.

We perform augmented Dickey-Fuller (ADF) unit root tests for the real total return indices and theirfirst differences, and we conclude that that the real total return series are integrated processes of orderone, i.e., I(1) processes. We assume no separate drift in the VECM (p) form, while a constant entersvia the error correction term. We determine a lag length based on the Akaike Information Criterion(AIC).3 Table 2 reports both the trace statistics and the maximum eigenvalue statistics for the sampleperiod of December 1993 to December 1998. The eigenvalues range from 0.0523 to 0.6187 for theshort-term indices, The trace test rejects the null hypothesis at r = 4 but not at r = 5, which indicatesa rank of 5. On the other hand, the maximum eigenvalue test rejects the null hypothesis at r = 1 butnot at r = 2, which indicates a rank of 2. Since the eigenvalues do not show a clear clustering, we relyon the trace test.4 For the long-term group, both the trace statistics and eigenvalues indicate r = 6. Interms of common factors, we find two global I(1) factors for short-term returns, and one global I(1)factor for the long-term group. The long-term indices conform to our strong form rational expectationhypothesis of a global (G7) real risk free rate. That is, there is one long-run global stochastic trendaffecting all the G7 countries’ real total return indices.

4.1. Common trends (1993:12–1998:12)

Looking at the structure of the common stochastic trend f for the long-term G7 government bonds,we note that the common trend ft = ˛′

⊥Yt can be written as5

flong = −0.38071yCanada,t + 0.21774yFrance,t − 0.35362yGermany,t − 0.47358yItaly,t

− 0.22227yJapan,t + 0.49555yUK,t + 0.40416yUS,t (6)

U.K. and U.S. both have a large positive component in this common trend, while Italy has a largenegative component. Other countries, Canada, France, Germany and Japan have absolute componentsabove 0.21. As shown in Fig. 1a, the trend moves downward through this 5-year period, with a smallboom in the end of 1994 and a lower bottom around the latter part of 1996, and then it has been quitelow throughout 1997–1998. One interpretation of this relationship is, in the case where r = n − 1, thatthere is one common stochastic trend, and investors seeking the (temporary) highest “risk free” realreturn across countries force the returns on all countries debt to be driven by this common rate. Atthe very least this quasi-arbitrage, or value seeking, will cause total real risk free returns to converge.More sophisticated short/long arbitrage strategies may also be in play, hastening the convergence.

Compared with the “world interest rate” computed in Plosser and Rouwenhorst (1994), our com-mon total stochastic trend carries more moments and more closely captures the G7 real total returns,from which we can easily extract the permanent components for these G7 long-term governmentbonds.

As we find two global trends for the short-term group, we focus on the components of these twoglobal stochastic trends and how they relate to the permanent components of the series. As shown

3 Fraser and Oyefeso (2005), show that the rank test is not particularly sensitive to over parameterization arising from amisspecified short-run structure. As a result, the most efficient strategy is to select the lag length based on AIC. Simulationsrun by Schwert (1989) show that in small and moderately large samples (from t = 25 to 1000), including enough lags in theautoregression is important to minimize size distortions.

4 As noted in Kasa (1992), the trace statistics takes account of all the n − r of the smallest eigenvalues; it will tend to begreater than the maximum eigenvalue statistics when the eigenvalues are evenly distributed. However, when eigenvalues areclustered close to one or zero, maximum eigenvalue statistics will tend to give better test results. As a result, the decision onthe system rank is a judgment considering the two statistics along with an inspection of eigenvalues.

5 Due to page limit, the results on the alpha and beta are not reported but available upon request.

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Table 2Cointegration test statistics.

H0: rank = r HI: rank > r (trace test) orrank = r + 1 (max eigen)

Eigenvalue Trace test Maximum eigenvalue

Statistics 5% Criticalvalue

Statistics 5% Criticalvalue

Panel A: 1993.12–1998.12

Short-term indices

0 0 0.62 166.30 109.93 56.89 41.511 1 0.47 109.41 82.61 37.53 36.362 2 0.39 71.88 59.24 29.19 30.043 3 0.27 42.68 39.71 18.42 23.804 4 0.19 24.27 24.08 12.75 17.895 5 0.13 11.51 12.21 8.34 11.446 6 0.05 3.17 4.14 3.17 3.84

Long-term indices

0 0 0.81 324.38 109.93 91.01 41.511 1 0.77 233.37 82.61 79.98 36.362 2 0.72 153.40 59.24 70.32 30.043 3 0.55 83.08 39.71 43.33 23.804 4 0.35 39.75 24.08 23.30 17.895 5 0.25 16.45 12.21 16.03 11.446 6 0.01 0.42 4.14 0.42 3.84

Panel B: 1999.1–2009.12

Short-term indices

0 0 0.20 61.20 68.68 29.31 33.461 1 0.12 31.89 47.21 16.99 27.072 2 0.07 14.91 29.38 8.96 20.973 3 0.04 5.95 15.34 5.56 14.074 4 0.00 0.39 3.84 0.39 3.76

Long-term indices

0 0 0.24 61.94 68.68 34.59 33.461 1 0.09 27.35 47.21 12.12 27.072 2 0.07 15.23 29.38 9.54 20.973 3 0.04 5.70 15.34 4.82 14.074 4 0.01 0.88 3.84 0.88 3.76

Note: The critical values are at the 5% level. Short-term indices are total real return indices of government bonds with maturity 1–3 years; while long-term indices are total real returnindices of government bonds with maturity over 2 years.

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Fig. 1. Common trends and real total return indices for G7 government bonds, 1993:12–1998:12. (a) Long-term maturity.(b) Short-term maturity. Flong , Fshort1 and Fshort 2 refer to the common I(1) factor. The real total return index is calculated byri$

j,i,t= LN(RI$

j,i,t/CPIUS,t ) for each country j maturity i and time t, where RI$ = RIj,i,t × SLC/USD,0/SLC/USD,t − RIj,i,t represents the total

return index, LC represents the local currency, 0 is the starting time point, so SLC/USD,0 represents the exchange rate of localcurrency per US dollar at time 0. Sources: total return indices, exchange rate data and CPI data are obtained from Datastream,Federal Reserves Reports and IFS statistics respectively.

in Fig. 1b, fshort1 has an upward trend above all the original series while fshort2 appears somewhat flat.These two factors are generated by the following equations:

fshort1 = 0.703yCanada,t + 0.043yFrance,t − 0.052yGermany,t + 0.387 yItaly,t − 0.077yJapan,t

+ 0.577yUK,t + 0.112 yUS,t (7)

fshort2 = −0.042yCanada,t + 0.368 yFrance,t − 0.294yGermany,t − 0.224 yItaly,t + 0.138yJapan,t

+ 0.002yUK,t + 0.841 yUS,t (8)

Among G7, Canada and U.K. have the largest weights toward fshort1, while the others are relatively low.For the second common trend (fshort2), France, Germany, Italy and U.S. have larger weights. The U.S.,in particular, has a component of 0.84 in fshort2.

4.2. Permanent and transitory decomposition (1993:12–1998:12)

We next examine the residual components in the system – these are what Gonzalo and Grangerrefer to as transitory components of these series.6 From the global real return factor ft, we calculateRG

i,t= A1,i × ft which are the permanent components for each of the total return series for these G7

long-term government bonds. The common trend of the long-term group has the largest impact on thepermanent component of the U.S., which is 4.456ft and the lowest impact on that of Italy, 0.434ft. Thetransitory components of the long-term group appear to be trend-stationary, while those of the short-term group have a more “messy” graph. The transitory components of the long-term group have a lowermean (except for Italy) but a higher standard deviation compared with the permanent components,which indicates that the short run fluctuations of the real total returns of these long-term governmentbonds are relatively small and mainly transitory which disappear very quickly. These results confirmthe findings in Hardouvelis (1994). Short run movements of real prices are mainly caused by transitorycomponents, and are not caused by the permanent components.

6 Due to the page limit, we did not report statistics for the permanent-transitory components after the Gonzalo and Grangerseparation but it is available upon request.

62 H. He, P. Locke / Research in International Business and Finance 25 (2011) 53–63

These results are interesting when compared to the results of Gonzalo and Granger (1995). Theyfound that for monthly long-term U.S. yields, the permanent components are very close to thelong-term rate series, while the transitory components are almost close to zero during the period1969–1988. We find that the transitory component is quite volatile and has a trend pattern. It is plau-sible that instead of convergence, volatility in the long-term government bonds has increased sincethe 1990s. As for the short-term group, our ex-post short-term real risk free returns in the early perioddo not appear to follow a global trend, with patterns very similar to the results of Gonzalo and Granger(1995). The transitory components fluctuate around zero, while the permanent components appearsimilar to the real total return series.

4.3. Robustness checks using the latter period and G4

For the period of January 1999 to December 2009, we use Germany as a representative for theEuropean group, and we have a “G5” instead of G7. With the same procedure, we find no cointegrationusing both trace and maximum eigenvalue statistics for the short-term returns (Panel B of Table 2).For the long-term group, we find one cointegrating relationship for the five countries. Our results forthe long-term maturity confirm Smith (2002) and Davies (2007) with most recent data. We apply thisprocedure to a subsample of countries, the four largest global economies – Germany, U.S, U.K. andJapan with whole period and we find no cointegration for both maturities.7

5. Summary

The contribution of this study is two folded. Firstly, we examine a rational expectation hypothesisfor global trends in real risk free interest rates in seeking convergence and predictability in the globalrisk free asset markets. Our focus is on returns to fixed income investing rather than contemporaneousyields. Secondly, we add new evidence to the paradox of diversification in international risk free invest-ments. We use monthly G7 sovereign debt real total return series for the periods 1993:12–1998:12and 1999:1–2009:12. Our results are mixed, and somewhat surprising. We expect more integratedrisk free rates, especially over time but find less.

For our early period (1993:12–1998:12), we find one common trend for long-term G7 governmentbonds and two for the short-term group, indicating that these investments are highly linked during1990s. The cointegrating constraints found in these groups indicate that the individual series do notwander far from the system and that the integration of global risk free rates is high. The short runfluctuations of real total returns of these long-term government bonds are relatively small and mainlytransitory which disappear very quickly. As for the short-term group, the transitory components showa much lower mean compared with the permanent components.

However, for the recent period from 1999 to 2009, only the long-term group reveals only very weakintegration and we find no cointegration with subsample of four countries over the entire period. Onepossibility is that as we adjust the total return indices into constant dollar terms, the exchange raterisk and country-specific inflation risk are embedded into these return series. These embedded factorsbring idiosyncratic risks to each country government bond indices and we do not have “pure” real riskfree rates. The global factor, such as it is, varies with each maturity group and each period. Thus,there appears to be clear diversification benefits from investing across international risk free treasurysecurities, even among these most integrated developed markets.

Further studies could expand the country selections to some of the emerging markets. Of course,failure to find integration in real risk free rates among the G7 gives little hope for finding integratedreal rates among emerging markets, whose sovereign debt is less likely to be considered risk free.We also want to point out that this second period corresponds to the general market decline of 2000,as well as the September 2001 turmoil and 2008 financial crisis. These and other effects may haveallowed real rates to differ substantially during this sample period. Our expectation is that this findingof a lack of convergence will be a temporary phenomenon.

7 Results for Johansen tests are not reported but available upon request.

H. He, P. Locke / Research in International Business and Finance 25 (2011) 53–63 63

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