multi-period portfolio choice and the intertemporal hedging demands for stocks and bonds:...

Journal of International Money and Finance 28 (2009) 427–453

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Journal of International Moneyand Finance

journal homepage: www.elsevier .com/locate/ j imf

Multi-period portfolio choice and the intertemporalhedging demands for stocks and bonds:International evidenceq

David E. Rapach a,*, Mark E. Wohar b,1

a Department of Economics Saint Louis University, 3674 Lindell Boulevard, Saint Louis, MO 63108-3397, USAb Department of Economics, University of Nebraska at Omaha, RH-512K, Omaha, NE 68182-0286, USA

JEL classification:C32G11

Keywords:Intertemporal hedging demandMulti-period portfolio choice problemParametric bootstrapReturn predictability

q The results reported in this paper were generatprograms are available at http://pages.slu.edu/faculavailable on John Campbell’s web page at http://ku

* Corresponding author. Tel.: þ1 314 977 3601;E-mail addresses: [email protected] (D.E. Rapa

1 Tel.: þ1 402 554 3712; fax: þ1 402 554 2853.

0261-5606/$ – see front matter � 2008 Elsevier Ldoi:10.1016/j.jimonfin.2008.12.004

a b s t r a c t

We investigate the intertemporal hedging demands for stocks andbonds for investors in the U.S., Australia, Canada, France, Germany,Italy, and U.K. Using the methodology of Campbell et al. [Campbell,J.Y., Chan, Y.L., Viceira, L.M., 2003a. A multivariate model of stra-tegic asset allocation. Journal of Financial Economics 67(1), 41–81],we solve a multi-period portfolio choice problem for an investor ineach country with an infinite horizon and Epstein–Zin–Weil utility,where the dynamics governing asset returns are described bya vector autoregressive process. We find sizable mean inter-temporal hedging demands for domestic stocks in the U.S. and U.K.and considerably smaller mean hedging demands for domesticstocks in the other countries. An investor in the U.S. who hasaccess to foreign stocks and bonds displays small mean inter-temporal hedging demands for foreign stocks and bonds, whileinvestors in Australia, Canada, France, Germany, Italy, and the U.K.who have access to U.S. stocks and bonds all exhibit sizable meanhedging demands for U.S. stocks.

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ed using GAUSS 6.0. A data appendix, all unreported results, and the GAUSSty/rapachde/. Some of the GAUSS programs are based on MATLAB programsznets.fas.harvard.edu/~campbell/data.html.

fax: þ1 314 977 1478.ch), [email protected] (M.E. Wohar).

td. All rights reserved.

http://pages.slu.edu/faculty/rapachde/

http://kuznets.fas.harvard.edu/~campbell/data.html

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D.E. Rapach, M.E. Wohar / Journal of International Money and Finance 28 (2009) 427–453428

1. Introduction

There has been a recent resurgence of interest in portfolio choice problems. In particular, the body ofempirical evidence accumulated over the last two decades indicating that stock and bond returns haveimportant predictable components has reinvigorated interest in multi-period portfolio choiceproblems. As initially recognized by Samuelson (1969) and Merton (1969, 1971), return predictabilityhas potentially important implications for multi-period portfolio choice problems.2 More specifically,return predictability can give rise to intertemporal hedging demands for assets, so that – in contrast tothe canonical static portfolio choice problem of Markowitz (1952) – investors look beyond one-period-ahead when optimally allocating across assets. Intuitively, investors may want to hedge against adversefuture return shocks, and return predictability provides a temporal mechanism to accomplish this.

While return predictability can have important implications for multi-period portfolio choiceproblems, a difficulty in studying these problems is that exact analytical solutions are generally notavailable.3 This has led researchers to use different approaches in order to solve multi-period portfoliochoice problems in empirical applications. A number of researchers take advantage of gains incomputing power and employ computationally intensive numerical procedures to approximate thesolutions to multi-period portfolio choice problems in the presence of return predictability. Forexample, Brennan et al. (1997), Balduzzi and Lynch (1999), Barberis (2000), Lynch and Balduzzi (2000),Lynch (2001), and Lynch and Tan (2006) use discrete-state approximations to numerically solveportfolio choice problems for investors with long horizons when returns are predictable. Anotherapproach uses approximate analytical methods to solve portfolio choice problems for investors withinfinite horizons when returns are predictable in neighborhoods of known exact solutions; seeCampbell and Viceira (1999, 2001, 2002).4

In a recent extension of Campbell and Viceira (1999), Campbell et al. (2003a; henceforth, CCV)develop an approach that combines an approximate analytical method with a relatively simplenumerical procedure. This approach has the advantage of being able to accommodate multi-periodportfolio choice problems with a relatively large number of assets and potential return predictors,whereas such problems can quickly become intractable using approaches based on more computa-tionally intensive numerical procedures. CCV use their approach to analyze optimal dynamic assetallocation across U.S. bills, stocks, and bonds when return predictability is described by a first-ordervector autoregressive (VAR(1)) process in real bill returns, excess stock returns, excess bond returns, aswell as the nominal bill yield, dividend yield, and term spread.5 CCV consider an investor whomaximizes the expected utility of lifetime consumption over an infinite horizon, where the utilityfunction is of the Epstein–Zin–Weil (Epstein and Zin, 1989, 1991; Weil, 1989) form. Interestingly, CCVfind that return predictability should lead an investor in the U.S. to have a sizable positive meanintertemporal hedging demand for domestic stocks for a range of values of the coefficient of relativerisk aversion (CRRA) and a unitary elasticity of intertemporal substitution (EIS). Overall, the empiricalresults in CCV, as well as the other studies cited above, indicate that return predictability can generateimportant intertemporal hedging demands for U.S. assets, especially U.S. stocks.

While the existing empirical literature on multi-period choice problems contains important find-ings relating to the implications of return predictability, the literature focuses almost exclusively on

2 Note that for our purposes, it is the existence of return predictability itself – and not the reason for its existence – that haspotentially important implications for multi-period portfolio choice problems, so we can sidestep the thorny issue of whetherreturn predictability is due to time-varying equilibrium returns or market inefficiencies (Fama, 1991). Campbell (2000) providesa survey of the predictability literature.

3 Kim and Omberg (1996), Wachter (2002), Bhamra and Uppal (2006), and Liu (2007) obtain exact analytical solutions incertain special classes of multi-period portfolio choice problems.

4 Employing another computationally intensive approach, Brandt (1999) and Aıt-Sahalia and Brandt (2001) use non- andsemi-parametric procedures to analyze Euler equations and approximate the solutions to portfolio choice problems forinvestors with long horizons in the presence of return predictability. See Brandt (forthcoming) for an extensive survey of theliterature on both static and multi-period portfolio choice problems.

5 A number of studies find that the latter three variables have predictive ability with respect to stock and bond returns; see,for example, Rozeff (1984), Campbell (1987), Campbell and Shiller (1988), Fama and French (1988, 1989), Hodrick (1992), Solnik(1993), Lewellen (2004), and Campbell and Yogo (2006).

D.E. Rapach, M.E. Wohar / Journal of International Money and Finance 28 (2009) 427–453 429

domestic investments in U.S. assets. In the present paper, we extend the extant empirical literature anduse the CCV approach to analyze dynamic asset allocation across domestic bills, stocks, and bonds forinvestors in Australia, Canada, France, Germany, Italy, and the U.K., as well as the U.S., where the returnsdynamics are characterized by a VAR(1) process that includes three instruments (nominal bill yield,dividend yield, and term spread). For a set of plausible values for the parameters relating tointertemporal preferences, we use the CCV approach to estimate the mean total, myopic, and inter-temporal hedging demands for domestic bills, stocks, and bonds in each country. We also presentestimates of the intertemporal hedging demands for domestic stocks and bonds for each month overthe sample in each country.

In addition to examining the implied intertemporal hedging demands for domestic stocks andbonds for investors in a number of different countries, we consider a multi-period portfolio choiceproblem for an investor in the U.S. who can invest in stocks and bonds from a foreign country. It is quitefeasible to use the CCV approach to solve a multi-period portfolio choice problem with five risky assetsand six instruments. This allows us to extend the empirical application in CCV and analyze a multi-period portfolio choice problem for an investor in the U.S. who, in addition to domestic bills, stocks, andbonds, has access to stocks and bonds from a foreign country (Australia, Canada, France, Germany, Italy,or the U.K.), and where the investor considers six instruments (the domestic and foreign nominal billyields, dividend yields, and term spreads) that potentially contribute to return predictability.6 We usethe CCV approach to estimate the total, myopic, and intertemporal hedging demands for domestic bills,stocks, and bonds and foreign stocks and bonds for an investor in the U.S. when the return dynamicsare characterized by a VAR(1) process that includes the five returns and six instruments. In anotherextension, we use the CCV approach to analyze a multi-period portfolio choice problem for an investorin Australia, Canada, France, Germany, Italy, or the U.K. who, in addition to domestic bills, stocks, andbonds, can invest in U.S. stocks and bonds.

An important econometric contribution of the present paper is the development of a parametricbootstrap procedure that allows us to account for sampling uncertainty when estimating mean assetdemands. More specifically, we augment the CCV approach with a parametric bootstrap procedure thatenables us to compute 90% confidence intervals for the mean total, myopic, and intertemporal hedgingdemands in each country. To our knowledge, this is the first procedure that allows confidence intervalsto be computed for asset demands in the context of multi-period portfolio choice problems.

Previewing our empirical results, we find sizable mean intertemporal hedging demands fordomestic stocks for investors in the U.S. and U.K., while the mean hedging demands for domesticstocks are decidedly smaller for investors in Australia, Canada, France, Germany, and Italy. When aninvestor in the U.S. has access to foreign stocks and bonds in addition to domestic bills, stocks, andbonds, the mean intertemporal hedging demands for domestic stocks remain sizable; in contrast,with the exception of U.K. stocks, the mean hedging demands for foreign stocks and bonds are smallfor all countries. When investors in Australia, Canada, France, Germany, Italy, and the U.K. haveaccess to U.S. stocks and bonds, we find substantial mean intertemporal hedging demands for U.S.stocks. The 90% confidence intervals show that sampling uncertainty can make it difficult to estimateasset demands with precision. Nevertheless, the estimated mean intertemporal hedging demands forU.S. stocks are consistently significant according to the confidence intervals. Overall, we discoverimportant similarities and differences in the optimal intertemporal hedging demands for stocks andbonds across countries, and our results point to especially important intertemporal hedgingdemands for U.S. stocks.

It is important to emphasize that the asset demands derived from multi-period portfolio choiceproblems in the recent empirical literature (including the present paper) are partial equilibrium in

6 Ang and Bekaert (2002) consider a multi-period portfolio choice problem where an investor in the U.S. can invest indomestic stocks as well as stocks from one or two foreign countries. Unlike most of the literature, Ang and Bekaert (2002) donot characterize return predictability using a VAR(1) process that includes instruments such as the dividend yield, but insteadthey use a Markov-switching process for the moments of the returns. Also see Campbell et al. (2003b), who use the CCVapproach to study a multi-period portfolio choice problem where an investor in the U.S. has access to domestic bills and billsfrom a foreign country (the U.K., Germany, or Japan). The present paper extends these studies by considering a broader range ofdomestic and foreign assets and a larger number of countries.


nature, as the return processes are treated as exogenous. That is, given an exogenous return process(usually calibrated to U.S. data), researchers calculate the implied optimal asset demands for anindividual investor with a long horizon and an assumed set of preferences; no attempt is made to usethe implied model of investor behavior to explain observed asset returns.7 Two ways of interpretingthe estimated asset demands in the extant empirical literature have been offered. First, Campbell andViceira (2002) suggest viewing the estimated asset demands as normative descriptions of investorbehavior, so the estimated asset demands are those that an investor with an assumed set of preferencesshould have for a given return process. Alternatively, we can follow the suggestion of Lynch (2001) andview the estimated asset demands as a positive description of the behavior of a unique individual orsmall group (rather than a representative agent) in the economy who exploits the return predictabilitycreated by a large number of other investors with different preferences. These different preferencesmay be created by habit persistence, as in Campbell and Cochrane (1999), or they may be of the typeassumed in models of behavioral finance, such as Barberis et al. (2000).

The rest of the paper is organized as follows: Section 2 describes our empirical approach, includingthe CCV framework and our parametric bootstrap procedure; Section 3 presents our empirical results;and Section 4 concludes.

2. Empirical approach

2.1. Multi-period portfolio choice problem

In this subsection, we outline the multi-period portfolio choice problem faced by the investor;additional details are provided in Section 2 of CCV. The investor has access to n risky assets. Let R1,tþ1 bethe real return on a benchmark asset (usually a Treasury bill) from time t to time tþ 1, and let Ri,tþ1,i¼ 2,.n, be the real returns on the n� 1 additional assets.8 The real return on the investor’s portfolio(Rp,tþ1) can be expressed as

Rp;tþ1 ¼Xn

i¼2

ai;t�Ri;tþ1 � R1;tþ1

�þ R1;tþ1; (1)

where ai;t is the portfolio weight on asset i at time t.9 Letting ri;tþ1 ¼ logðRi;tþ1Þ, define the vector of logexcess returns as xtþ1 ¼ ½r2;tþ1 � r1;tþ1;.; rn;tþ1 � r1;tþ1�0. In addition to the n returns, a vector ofinstruments stþ1 comprises the state variables. The m-vector of state variables is given by

ztþ1 ¼�r1;tþ1; xtþ1; stþ1

�0: (2)

CCV assume that the dynamics of the system of state variables are well characterized by a VAR(1)process,10 so that the data-generating process for the state vector ztþ1 is given by

ztþ1 ¼ F0 þ F1zt þ vtþ1; (3)

where vtþ1 is an m-vector of VAR innovations that are independently and identically distributed asNð0m�1;SvÞ.11 The variance–covariance matrix Sv can be partitioned such that

7 Explaining observed asset returns requires embedding a representative investor of this type in a general equilibriumframework where all markets clear. Lynch (2003) makes important progress in this direction by embedding a representativeinvestor with a long horizon and access to three stock portfolios sorted on book-to-market ratios in a general equilibrium modeland comparing the quantitative properties of asset returns implied by the model to actual U.S. asset returns.

8 While we follow CCV and use a 3-month Treasury bill as the benchmark asset, the designation of the benchmark asset isarbitrary.

9 The portfolio weight on the benchmark asset at time t is 1�Pn

i¼2 ai;t .10 This follows, for example, Campbell (1991), Balduzzi and Lynch (1999), Kandel and Stambaugh (1996), Campbell and Viceira

(1999), Barberis (2000), Lynch and Balduzzi (2000), and Lynch (2001).11 Note that the VAR innovations are assumed to be homoskedastic, following the studies cited in footnote 10 above. Chacko

and Viceira (2005) solve a multi-period portfolio choice problem with stochastic volatility.


Sv ¼

24 s2

1 s01x s01ss1x Sxx S0xss1s Sxs Sss

35; (4)

where s21 is the variance of the innovation to the benchmark asset return; s1x is an (n� 1)-vector of

covariances between innovations to the benchmark asset return and innovations to the excess returnson the remaining assets; s1s is an (m� n)-vector of covariances between innovations to the benchmarkasset return and innovations to the instruments; Sxx is the ðn� 1Þ � ðn� 1Þ variance–covariancematrix for the innovations to the excess returns; Sxs is the ðm� nÞ � ðn� 1Þ matrix of covariancesbetween innovations to the excess returns and innovations to the instruments; Sss is theðm� nÞ � ðm� nÞ variance–covariance matrix for the innovations to the instruments.

The investor is assumed to have Epstein–Zin–Weil utility, which she maximizes over an infinitehorizon. The recursive preferences that characterize Epstein–Zin–Weil utility are given by

U½Ct ; EtðUtþ1Þ� ¼�ð1� dÞCð1�gÞ=q

t þ dhEt

�U1�g

tþ1

�i1=qq=ð1�gÞ

; (5)

where Ct is consumption at time t; Etð Þ is the expectation operator conditional on information availableat time t; g > 0 is the CRRA; j > 0 is the EIS; 0 < d < 1 is the time discount factor; q ¼ ð1� gÞ=ð1� j�1Þ.As emphasized by CCV, Epstein–Zin–Weil utility severs the tight link between the CRRA and EIS thatcharacterizes the popular time-separable power utility function.12 This is a nice feature of Eq. (5), as theCRRA and EIS are conceptually distinct notions relating to risk and intertemporal preferences, respec-tively. At each time t, the investor selects Ct and at ¼ ½a2;t ;.;an;t �0 in order to maximize Eq. (5), using allavailable information at time t, subject to the intertemporal budget constraint,

Wtþ1 ¼ ðWt � CtÞRp;tþ1; (6)

where Wt is wealth at time t.With time-varying investment opportunities, exact analytical solutions for this problem are

generally not available. CCV combine an extension of the Campbell and Viceira (1999) approximateanalytical solution with a relatively simple numerical procedure to compute the investor’s optimalasset allocation and consumption policies. A key approximation used by CCV involves the log realreturn on the portfolio. This approximation is exact in continuous time, and CCV observe that it ishighly accurate for short time intervals.13 CCV also employ first- and second-order log-linearapproximations of the budget constraint and Euler equation, respectively, that are exact when j ¼ 1.CCV show that the solution to the approximate model can be expressed in terms of policy functions forat and ct�wt, where ct and wt are the log-levels of Ct and Wt, respectively:

at ¼ A0 þ A1zt ; (7)

ct �wt ¼ B0 þ B01zt þ z0tB2zt ; (8)

where A0 ½ðn� 1Þ � 1�, A1 ½ðn� 1Þ �m�, B0 (1�1), B1 (m�1), and B2 (m�m) are coefficient matrices thatare constant through time and functions of g, j, d, r, F0, F1, and Sv, where r ¼ 1� exp½Eðct �wtÞ�.14

CCV use an iterative numerical procedure to compute estimates of A0, A1, B0, B1, and B2. We are

12 When g ¼ j�1, Eq. (5) reduces to the familiar case of time-separable power utility; when g ¼ j�1 ¼ 1, Eq. (5) reduces tolog utility.

13 Our use of monthly data in Section 3 below should help to ensure the accuracy of the approximation in our empiricalapplications. While the CCV framework does not impose borrowing or short-sales constraints, the approximation to the log realreturn on the portfolio used by CCV has the effect of ruling out the possibility of bankruptcy; see Campbell and Viceira (2002,pp. 28–29).

14 The coefficient matrices are constant through time due to the infinite-horizon assumption. This assumption means that wedo not have to solve the problem backward recursively starting from the terminal date.


primarily interested in the estimates of the parameters in Eq. (7), which govern the investor’s optimalasset allocations.

In our applications in Section 3 below, we use the CCV approach to estimate Eqs. (7) and (8) fora infinitely lived investor in the U.S., Australia, Canada, France, Germany, Italy, and U.K. (in turn) who caninvest in domestic 3-month Treasury bills, a broad domestic stock market index, and domestic 10-yeargovernment bonds. We treat the log real return on a 3-month Treasury bill (rtbrt) as the return on thebenchmark asset, so that the two log excess real returns are those on the stock market index anda 10-year government bond (xsrt and xbrt, respectively). In addition to lagged returns, three domesticinstruments serve as potential return predictors: the nominal yield on a 3-month Treasury bill (billt),the log of the dividend yield on the stock market index (divt), and the term spread (spreadt). Given thesereturns and instruments, the state vector is ztþ1 ¼ ½rtbrtþ1; xsrtþ1; xbrtþ1;billtþ1;divtþ1; spreadtþ1�0.We assume d ¼ 0:92 on an annual basis (so that the discount factor equals 0.921/12 on a monthly basis)and j ¼ 1,15 and we estimate the VAR parameters in Eq. (3) using maximum likelihood.16 We considerthree values for g: 4, 7, and 10. These g values are similar to those considered in other studies,17 and theyrepresent plausible values for the CRRA. We report estimates of the mean asset demands for domestic3-month Treasury bills, stocks, and 10-year government bonds over the sample for each g value usinga ¼ bA0 þ bA1z, where bA0 and bA1 are the estimates of A0 and A1, respectively, in Eq. (7) obtained using theCCV numerical procedure and z ¼

PTt¼1 zt , where T is the number of available sample observations for

the state vector.We are especially interested in the intertemporal hedging demands for the various assets. Following

Merton (1969, 1971), CCV derive the following result that is useful for dividing the total demand into itsmyopic and intertemporal hedging components:

A0 ¼ ð1=gÞS�1xx

hHxF0 þ 0:5s2

x þ ð1� gÞs1x

iþ ½1� ð1=gÞ�S�1

xx ½ � L0=ð1� jÞ�; (9)

A1 ¼ ð1=gÞS�1xx HxF1 þ ½1� ð1=gÞ�S�1

xx ½ � L1=ð1� jÞ�; (10)

where Hx ¼ ½0ðn�1Þ�1; In�1;0ðn�1Þ�ðm�nÞ� is a matrix that selects xt from the state vector zt; s2x is the

vector of diagonal elements in Sxx; L0 and L1 are matrices that depend on B0, B1 and B2, as well as g, j,d, r, F0, F1, and Sv. The first term on the right-hand-side (RHS) of Eqs. (9) and (10) represents themyopic part of asset demand. The myopic component focuses solely on a single-period-ahead andessentially corresponds to the asset demand generated under the static Markowitz problem. Thesecond term on the RHS of Eqs. (9) and (10) represents the intertemporal hedging component of assetdemand. In contrast to the static Markowitz problem, an intertemporal hedging demand can arise ina multi-period portfolio choice problem, as a risk-averse investor in a multi-period setting may lookbeyond a single-period-ahead and be interested in hedging her exposure to adverse future returnshocks.18

The CCV approximations are exact when j ¼ 1, and an EIS value of unity is reasonably consistentwith recent EIS estimates for stockholders reported in Vissing-Jorgensen (2002) and Vissing-Jorgensen and Attanasio (2003). However, EIS estimates for non-stockholders are typically well

15 We discuss alternative values of j below.16 OLS estimation of F0 and F1 in Eq. (3) is equivalent to maximum-likelihood estimation.17 For example, CCV include tabulated results for g ¼ 5; Balduzzi and Lynch (1999) consider g ¼ 6; Barberis (2000) considers

g ¼ 5; 10; Lynch (2001) considers g ¼ 4.18 Note that a multi-period choice problem is a necessary, but not sufficient, condition for the existence of an intertemporal

hedging demand. When g ¼ 1, the second term on the RHS of Eqs. (9) and (10) vanishes, so that there is no intertemporalhedging demand, as the investor is not sufficiently risk-averse to generate an intertemporal hedging demand. If the matrix ofVAR slope coefficients ðF1Þ is a zero matrix – so that there is no return predictability – the second term on the RHS of eachequation will also vanish. Thus, in order for an intertemporal hedging demand to exist in a multi-period portfolio choiceproblem, the investor must be sufficiently risk-averse and returns must be predictable. Actually, an additional condition needsto be satisfied: the variance–covariance matrix for the VAR innovations, Sv , cannot be diagonal; see, for example, Brandt(forthcoming, Section 2.3). The importance of this last condition will become evident in the discussion of the empirical resultsin Section 3 below.


below unity, as are EIS estimates based on aggregate data (Hall, 1988; Yogo, 2004). Given the rangeof estimates for the EIS in the extant literature, we also calculate mean asset demands for j values of0.3 and 1.5 (assuming g¼ 7).19

2.2. Parametric bootstrap procedure

Apparently due to computational costs, extant studies of multi-period portfolio choice problems(including CCV) only report point estimates of asset demands. To get a sense of the samplinguncertainty associated with the point estimates of the mean total, myopic, and hedging demands foreach asset in each country, we construct 90% confidence intervals for the mean demands usinga parametric bootstrap procedure. We assume that the state vector ztþ1 is generated by Eq. (3), wherethe parameters of the VAR are set to their maximum-likelihood estimates. In order to generate a seriesof innovations to use in constructing a pseudo-sample of observations for zt, we make T þ 100independent draws from a Nð0m�1;

bSvÞ distribution. Using the randomly drawn innovations, Eq. (3)with F0 ¼ bF0 and F1 ¼ bF1, and setting the initial zt observations to zero, we can build up a pseudo-sample of T þ 100 observations for zt. We drop the first 100 transient start-up observations in order torandomize the initial zt observations, leaving us with a pseudo-sample of T observations for zt, matchingthe original sample. For the pseudo-sample, we use the CCV approach to estimate Eqs. (7) and (8) andthe mean total, myopic, and hedging demands for each asset. We repeat this process 500 times, givingus an empirical distribution for each of the mean asset demands. We construct 90% confidence intervalsfor each mean asset demand from the empirical distributions using the percentile method described inDavidson and MacKinnon (1993, p. 766).

3. Empirical results

3.1. Data

The data for the U.S., Australia, Canada, France, Germany, Italy, and U.K. are from Global FinancialData. Following CCV, our sample begins in 1952:04 for each country, with the exceptions of France andGermany, where, due to data availability, the sample begins in 1961:01 and 1967:02, respectively.20 Thesample ends in 2004:05 for each country. We measure the log real return on a 3-month Treasury bill fora given month as the difference in the logs of the total return index for bills for the given and previousmonths minus the difference in the logs of the consumer price index for the given and previousmonths.21 The log excess stock (bond) return for a given month is the difference in the logs of the totalreturn index for stocks (10-year government bonds) for the given and previous months minus thedifference in the logs of the total return index for bills for the given and previous months. The nominalbill yield is the yield on a 3-month Treasury bill, and the term spread is the difference between theyields on a 10-year government bond and 3-month Treasury bill.22

Table 1 reports summary statistics (mean, standard deviation, first-order autocorrelation coeffi-cient) for the three risky asset returns and three instruments for each of the seven countries. The meanand standard deviation for the returns are expressed in annualized percentage units, and we includethe Sharpe ratio (the ratio of the annualized mean to the annualized standard deviation) for the excessstock and bond returns. The mean excess stock return for the U.S., Australia, and U.K. is between 5% and6%. Canada, Germany, and France exhibit lower mean excess stock returns, while Italy has the lowest

19 When j ¼ 1, the optimal consumption-wealth ratio is constant (Giovannini and Weil, 1989), and this simplifies thenumerical procedure used to solve the optimal portfolio choice problem. When js1, the numerical procedure needs to includean additional recursion; see Campbell et al. (2002, p. 12).

20 We had originally planned to include Japan in order to include all of the G-7 countries, but data for all of the necessaryseries for Japan are not available for a sufficiently long period from Global Financial Data.

21 Due to data availability, we use the wholesale price index for Australia.22 Following a number of other studies, we use deviations in the nominal 3-month Treasury bill yield from a 1-year back-

ward-looking moving average. Names and descriptions of the Global Financial Data files used to construct all of the variablesare provided in an on-line data appendix available at http://pages.slu.edu/faculty/rapachde/.


Table 1Summary statistics, 1952:04–2004:05.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)Variable Mean Standard

deviationSharperatio

r1 Variable Mean Standarddeviation

Sharperatio

r1

U.S., 1952:04–2004:05 Germany, 1967:02–2004:05rtbrt 1.34 0.99 0.37 rtbrt 2.09 1.15 0.29xsrtt 5.69 14.62 0.39 0.03 xsrtt 2.98 18.28 0.16 0.08xbrt 1.01 5.65 0.18 0.14 xbrt 2.63 5.97 0.44 0.21billt �0.01 1.03 0.88 billt �0.04 1.02 0.93divt 1.16 0.39 0.99 divt 1.19 0.30 0.98spreadt 1.39 1.16 0.94 spreadt 1.98 1.27 0.95

Australia, 1952:04–2004:05 Italy, 1952:04–2004:05rtbrt 2.30 2.86 0.37 rtbrt 1.91 1.62 0.49xsrtt 5.54 17.43 0.32 0.07 xsrtt 1.89 21.90 0.09 0.09xbrt 1.14 7.15 0.16 0.05 xbrt 1.18 7.08 0.17 0.31billt 0.05 1.26 0.89 billt �0.01 1.42 0.90divt 1.66 0.29 0.99 divt 1.14 0.40 0.98spreadt 1.35 1.67 0.95 spreadt 1.22 1.71 0.94

Canada, 1952:04–2004:05 U.K., 1952:04–2004:05rtbrt 2.36 1.41 0.21 rtbrt 1.59 2.07 0.28xsrtt 3.17 15.60 0.20 0.09 xsrtt 5.30 18.53 0.29 0.11xbrt 1.10 7.38 0.15 0.10 xbrt 0.92 4.83 0.19 0.32billt 0.01 1.33 0.92 billt 0.03 1.36 0.90divt 1.15 0.36 0.99 divt 1.51 0.28 0.98spreadt 1.30 1.48 0.95 spreadt 0.84 2.07 0.97

France, 1961:01–2004:05rtbrt 2.23 1.15 0.51xsrtt 2.12 19.44 0.11 0.12xbrt 1.21 6.36 0.19 0.17billt �0.02 1.33 0.93divt 1.28 0.47 0.99spreadt 0.94 1.45 0.95

Notes: rtbrt¼ log real 3-month Treasury bill return; xsrtt¼ log excess stock return; xbrtt¼ log excess bond return; billt¼ 3-monthTreasury bill yield (deviations from 1-year backward-looking moving average); divt¼ log dividend yield; spreadt¼ 10-yeargovernment bond yield� 3-month Treasury bill yield. Sharpe ratio is the mean divided by the standard deviation. r1 is the first-order autocorrelation coefficient.


mean excess stock return of 1.89%. The mean excess bond returns range from 0.92% to 2.63%. The meanexcess bond return is lower than the mean excess stock return for each country, although the meanexcess stock return is less than a single percentage point higher than the mean excess bond return forFrance, Germany, and Italy. For all seven countries, the standard deviation of excess stock returns isapproximately 2–4 times larger than the standard deviation of excess bond returns, and the standarddeviation of the real bill return is always considerably below that of excess bond return for eachcountry.

The U.S., Australia, and U.K. have the highest Sharpe ratios for excess stock returns (0.39, 0.32, and0.29, respectively), while Italy has the smallest (0.09). The Sharpe ratios for excess bond returns arevery similar across all of the countries (with the exception of Germany), ranging from 0.15 to 0.19. ForGermany, the Sharpe ratio for excess bond returns is considerably higher at 0.44. Note that the Sharperatio for excess stock returns is approximately 1.5–2.5 times higher than the Sharpe ratio for excessbond returns for the U.S., Australia, Canada, and U.K., while for France, Germany, and Italy, the Sharperatio for excess stock returns is actually less than the Sharpe ratio for excess bond returns. All else equal,the Sharpe ratios lead us to expect a higher demand on average for stocks in the U.S., Australia, Canada,and U.K. This is borne out in the empirical results reported in Section 3.3 below.

Excess stock returns exhibit fairly limited persistence in all seven countries (first-order autocor-relation coefficients between 0.03 and 0.12). Excess bond returns typically appear more persistent thanexcess stock returns, with Italy and the U.K. displaying the most persistent excess bond returns. The


real bill return is moderately persistent for all countries. In contrast to the returns, the instrumentsappear quite persistent for all countries, with the first-order correlation coefficients ranging from 0.88to 0.93 for the nominal bill yield, 0.98–0.99 for the dividend yield, and 0.94–0.97 for the term spread.

3.2. Domestic asset demands for investors in different countries

Table 2 reports the mean total, myopic, and intertemporal hedging demands (in percentages) fordomestic bills, stocks, and bonds in each country generated using the CCV approach for g values of 4, 7,and 10 and j ¼ 1. Table 2 also reports 90% confidence intervals for the mean asset demands generatedusing the parametric bootstrap procedure described in Section 2.2 above. Of course, the total meandemands across the three assets sum to 100; the mean myopic demands across assets also sum to 100,while the mean hedging demands sum to 0.

For the U.S., there are large mean total and intertemporal hedging demands for stocks for eachreported g value. As we would expect, the mean total demand for stocks – the most risky asset –decreases as g increases. In addition, the mean hedging demand for stocks also decreases as g increases.The mean hedging demands for bonds are negative and fairly large in magnitude, contributing to thesmaller total demands for bonds vis-a-vis stocks. The mean total demand for bills is negative for eachreported g value, so that the investor typically shorts bills. The results in Table 2 for the mean hedgingdemands for stocks in the U.S. are similar to the mean hedging demand for stocks (100.84) reported inCCV for the U.S. using quarterly data for 1952:2–1999:4, g ¼ 5, and j ¼ 1; the mean hedging demandsfor bonds in the U.S. in Table 2 are smaller in magnitude than the mean hedging demand for bonds(�122.57) reported in CCV. The most striking result for the U.S. in Table 2 (and in CCV) is the substantialmean total and intertemporal hedging demands for domestic stocks for an investor in the U.S.

What explains the sizable intertemporal hedging demand for domestic stocks in the U.S.? Whilea number of factors are at work in this multivariate analysis, as emphasized by CCV and others,23 twofactors appear to play especially important roles: (i) the positive coefficient on the lagged dividendyield in the excess stock return equation of the VAR and (ii) the strong negative correlation betweeninnovations to excess stock returns and the dividend yield.24 To see how these factors generate a strongintertemporal hedging demand for stocks, consider a negative innovation to excess stock returns nextperiod. Due to the large Sharpe ratio for stocks in the U.S., investors are usually long in stocks, so thatthe negative innovation to excess stock returns represents a worsening of the investor’s investmentopportunities next period. However, a negative innovation to excess stock returns next period tends tobe accompanied by a positive innovation to the dividend yield next period, and according to thepositive coefficient on the lagged dividend yield in the excess stock return equation of the VAR, thehigher dividend yield next period leads to higher expected stock returns two periods from now.25 Thus,by looking beyond one-period-ahead – as an investor with g > 1 will do – and taking into accountreturn predictability, as well as the negative correlation between innovations to stock returns and thedividend yield, stocks become a good hedge against themselves, in that they hedge exposure to futureadverse return shocks.

As a cautionary note, observe that the 90% confidence intervals for the mean asset demands tend tobe quite wide for the U.S. in Table 2, especially with regard to the mean total demands for each assetand the mean myopic demands for bonds and bills. This suggests that the reporting of point estimatesalone can mask considerable sampling uncertainty in empirical multi-period portfolio choice prob-lems. Of course, substantial sampling uncertainty can be a problem in general with regard to assetallocation problems; see, for example, Brandt (forthcoming, Section 3.1.2). Nevertheless, it is importantto observe that while many of the confidence intervals for the U.S. are quite wide in Table 2, the 90%

23 See, for example, the discussion in Brandt (forthcoming, Section 2.2.1).24 To conserve space, we do not report the complete VAR estimation results for each country. They can be found in Tables

A1–A7 in an on-line appendix of VAR results available at http://pages.slu.edu/faculty/rapachde/. The results for the U.S. arecontained in Table A1.

25 Furthermore, divt is a persistent process, so that there will be a persistent increase in the expected dividend yield, leading toa persistent increase in expected excess stock returns.


Table 2Mean demands for domestic assets for investors in different countries.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Stocks Bonds Bills

CRRA Total demand Myopic demand Hedging demand Total demand Myopic demand Hedging demand Total demand Myopic demand Hedging demand

U.S., 1952:04–2004:04g¼ 4 173.79 78.28 95.51 35.59 70.15 �34.56 �109.37 �48.42 �60.95

[68, 230] [39, 95] [31, 149] [�102, 198] [�86, 222] [�55, 11] [�295, 28] [�194, 104] [�155, �6]g¼ 7 124.65 44.58 80.07 15.49 40.21 �24.72 �40.14 15.21 �55.35

[50, 183] [22, 54] [25, 134] [�65, 107] [�49, 127] [�40, 8] [�182, 36] [�68, 103] [�135, �8]g¼ 10 98.08 31.10 66.97 8.81 28.23 �19.42 �6.89 40.66 �47.55

[40, 151] [15, 38] [23, 120] [�46, 75] [�34, 89] [�31, 6] [�96, 73] [�17, 102] [�118, �5]

Australia, 1954:02–2004:04g¼ 4 73.60 55.35 18.25 36.73 28.88 7.86 �10.33 15.78 �26.11

[14, 114] [16, 83] [�5, 39] [�54, 131] [�69, 139] [�26, 35] [�102, 92] [�69, 105] [�71, 22]g¼ 7 40.34 31.52 8.81 13.63 15.57 �1.93 46.03 52.91 �6.88

[3, 68] [10, 48] [�8, 28] [�37, 72] [�41, 78] [�26, 19] [�12, 111] [4.5, 105] [�45, 28]g¼ 10 25.64 22.00 3.64 3.06 10.24 �7.18 71.30 67.76 3.54

[�1, 50] [7, 33] [�11, 20] [�36, 44] [�30, 54] [�23, 15] [24, 115] [34, 105] [�29, 34]

Canada, 1954:02–2004:04g¼ 4 59.46 40.46 19.00 29.99 43.52 �13.53 10.55 16.02 �5.47

[13,109] [9, 66] [0, 61] [�60, 158] [�79, 170] [�30, 16] [�105, 128] [�86, 134] [�60, 19]g¼ 7 34.96 23.07 11.89 16.22 24.73 �8.51 45.82 52.21 �3.38

[10, 76] [5, 37] [�3, 44] [�35, 89] [�46, 97] [�18, 13] [�26, 117] [�6, 120] [�48, 15]g¼ 10 24.16 16.11 8.05 11.11 17.21 �6.10 64.73 66.68 �1.95

[3, 55] [3, 26] [�2, 37] [�24, 63] [�31, 69] [�12, 11] [5, 111] [26, 114] [�39, 14]

France, 1961:01–2004:04g¼ 4 23.78 20.48 3.30 56.38 80.05 �23.67 19.84 �0.53 20.37

[�10, 74] [�11, 49] [�10, 26] [�84, 209] [�112, 244] [�55, 16] [�126, 163] [�170, 172] [�26, 48]g¼ 7 13.11 11.67 1.44 30.82 45.98 �15.16 56.07 42.35 13.72

[�9, 43] [�6, 28] [�9, 18] [�46, 118] [�63, 140] [�35, 10] [�27, 139] [�55, 141] [�16, 32]g¼ 10 8.70 8.14 0.56 21.11 32.36 �11.24 70.19 59.50 10.68

[�6, 32] [�5, 20] [�5, 17] [�34, 79] [�44, 98] [�25, 7] [7, 123] [�3, 134] [12, 24]

Germany, 1967:02–2004:04g¼ 4 49.85 25.77 24.08 151.76 201.22 �49.46 �101.61 �126.98 25.38

[0, 105] [�7, 53] [�2, 61] [29, 330] [41, 384] [�84, 4] [�283, 36] [�306, 21] [�44, 68]g¼ 7 29.86 14.80 15.06 83.52 114.03 �30.52 �13.38 �28.84 15.46

[0, 105] [�4, 30] [�5, 41] [0, 170] [23, 218] [�55, 3] [�123, 56] [�132, 55] [�35, 40]g¼ 10 20.92 10.42 10.50 57.66 79.16 �21.50 21.41 10.42 10.99

[1, 52] [�3, 21] [�5, 31] [0, 118] [13, 150] [39, 3] [�59, 68] [�62, 69] [27, 29]

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Italy, 1952:04–2004:04g¼ 4 15.68 19.21 �3.53 46.89 72.15 �25.27 37.43 8.63 28.80

[�10, 50] [�4, 44] [�11, 13] [�79, 187] [�114, 241] [�68, 29] [�99, 162] [�185, 173] [�28, 77]g¼ 7 6.48 11.03 �4.55 23.47 41.12 �17.64 70.04 47.85 22.19

[�10, 26] [�2, 25] [�10, 7] [�47, 101] [�65, 138] [�44, 17] [�9, 136] [�62, 143] [�17, 49]g¼ 10 2.75 7.76 �5.01 14.44 28.70 �14.26 82.81 63.54 19.27

[�9, 18] [�1, 18] [�9, 5] [�37, 66] [�46, 96] [�35, 10] [27, 127] [�13, 129] [�7, 42]

U.K., 1952:04–2004:04g¼ 4 125.59 48.63 76.97 75.50 73.72 1.78 �101.09 �22.34 �78.75

[62, 177] [25, 69] [40, 122] [�114, 218] [�106, 258] [�28, 50] [�294, 83] [�196, 169] [�140, �20]g¼ 7 88.00 28.27 59.73 44.30 42.36 1.94 �32.30 29.37 �61.67

[43, 132] [16, 41] [25, 96] [�55, 136] [�61, 153] [�21, 34] [�143, 90] [�70, 138] [�114, �13]g¼ 10 66.61 20.13 46.48 30.79 29.82 0.97 2.60 50.05 �47.45

[33, 105] [11, 29] [15, 76] [�37, 99] [�43, 107] [�18, 26] [�82, 88] [�20, 126] [�92, �8]

Notes: The table reports mean monthly total, myopic, and hedging asset demands in percentages for stocks, 10-year government bonds, and 3-month Treasury bills for an investor witha unitary elasticity of intertemporal substitution (j¼ 1); a discount factor (d) equal to 0.921/12; and a coefficient of relative risk aversion (g) equal to 4, 7, or 10. Bootstrapped 90% confidenceintervals for the mean asset demands are given in brackets. A bold entry indicates significance according to the 90% confidence interval.

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confidence intervals for the mean hedging demands for stocks are tight enough to conclude that themean hedging demands for stocks are significant and large in the U.S. for the reported g values.However, we cannot reject the null hypothesis that the mean hedging demands for bonds are zeroaccording to the 90% confidence intervals.

In order to glean additional insight into the intertemporal hedging demands for domestic stocks andbonds in the U.S., Panel A of Fig. 1 portrays the estimated hedging demands for stocks and bonds for eachmonth over the sample in the U.S. when g ¼ 7. Overall, the hedging demand for stocks appearsconsiderably less volatile than the hedging demand for bonds. The hedging demand for stocks is typicallywell above the hedging demand for bonds over the sample, with the exception that the hedging demandfor bonds does move above the hedging demand for stocks during the late 1990s and 2000.

A B

C D

E F

G

Fig. 1. Historical intertemporal hedging demands for domestic stocks (solid lines) and bonds (dashed lines) for investors in differentcountries when g = 7 and j = 1.


Turning to the results for Australia in Table 2, while the mean total demands for domestic stocks aremoderately large, they are considerably smaller than the mean total demands for domestic stocks inthe U.S. The mean hedging demands for stocks in Australia are also much smaller than thecorresponding demands in the U.S., and the 90% confidence intervals for the mean hedging demandsfor stocks in Australia do not lead to rejection of the null hypothesis of a zero mean hedging demand.The mean total demands for bonds in Australia are very similar to those in the U.S., while the meanhedging demands for bonds in Australia are closer to zero than in the U.S. As in the U.S., the 90%confidence intervals for the mean hedging demands for bonds do not indicate rejection of the nullhypothesis of zero mean hedging demand for bonds. From Panel B of Fig. 1, we see that the hedgingdemand for stocks is always above the hedging demand for bonds when g ¼ 7, although a drop in theaverage hedging demands for both stocks and bonds in Australia is evident in the early 1980s.

The results for Canada in Table 2 are similar to those for Australia, in that the mean total and hedgingdemands for stocks are positive but considerably smaller than the corresponding demands for the U.S.,while the mean total demands for bonds are similar to, and the mean hedging demands for bonds areconsiderably smaller in magnitude than, those for the U.S. We cannot reject the null hypothesis of zeromean hedging demands for stocks and bonds in Canada according to the 90% confidence intervals.Panel C of Fig. 1 shows that the hedging demand for bonds is much more volatile than the hedgingdemand for stocks in Canada when g ¼ 7.

With respect to the results for France in Table 2, the mean total and hedging demands for stocks aremuch smaller in magnitude than those in any country, with the exception of Italy, and we clearlycannot reject the null hypothesis that the mean total and hedging demands for stocks are zero in Franceaccording to the 90% confidence intervals. The mean hedging demands for bonds in France are similarto, but somewhat smaller in magnitude than, those in the U.S., and we cannot reject the null hypothesisof zero mean total and hedging demands for bonds in France. Panel D of Fig. 1 shows that the hedgingdemand for stocks is always very close to zero in France when g ¼ 7, and that the hedging demand forbonds is much more volatile than the hedging demand for stocks.

The mean total and hedging demands for stocks in Germany in Table 2 are similar to thecorresponding demands in Australia and Canada, while the mean hedging demands for bonds inGermany are similar to those in the U.S. We cannot reject the null hypothesis of zero mean hedgingdemands for stocks and bonds in Germany. From Panel E of Fig. 1, we also see that the hedging demandfor bonds is more volatile than the hedging demand for stocks in Germany when g ¼ 7.

The mean total demands for stocks in Italy in Table 2 are even lower than those in France, and themean hedging demands for stocks are also very small in magnitude in Italy. Not surprisingly, we cannotreject the null hypothesis that the mean total and hedging demands for stocks are zero in Italyaccording to the 90% confidence intervals. The mean hedging demands for bonds in Italy are similar to,although slightly smaller in magnitude than, those in the U.S., and as in the U.S., we cannot reject thenull hypothesis of zero mean hedging demands for bonds. We see from Panel F of Fig. 1 that thehedging demand for stocks in Italy is always very near zero over the sample period when g ¼ 7, whilethe hedging demand for bonds is quite volatile.

The final country for which we report results in Table 2 is the U.K. With respect to the mean totaland hedging demands for stocks, the results for the U.K. are very similar to those for the U.S., in that thedemands for stocks are substantial. According to the 90% confidence intervals, it is reasonable toconclude that, as in the U.S., the mean hedging demands for stocks in the U.K. are significant andsizable. The mean hedging demands for bonds are very small in magnitude in the U.K., and we cannotreject the null hypothesis that they are zero according to the 90% confidence intervals. Similar to thecase for the U.S., we see from Panel G of Fig. 1 that the hedging demand for stocks is typically above thehedging demand for bonds over the sample period in the U.K. when g ¼ 7. Again similar to the U.S.,there is a period in the late 1990s and 2000 where the hedging demand for bonds moves above thehedging demand for stocks.

We next discuss factors contributing to the smaller intertemporal hedging demands for domesticstocks in some countries vis-a-vis the U.S. As we mentioned above, a large Sharpe ratio for stocks is likelyto lead to a long position in stocks, while a positive coefficient on the lagged dividend yield in the excessstock return equation of the VAR, combined with a negative correlation between innovations to excessstocks returns and the dividend yield, help to make stocks a good hedge against future adverse return


shocks for an investor long in stocks. With respect to Canada, France, Germany, and Italy, recall fromTable 1 that these countries have Sharpe ratios for domestic stocks that are considerably smaller than theSharpe ratio for stocks in the U.S., so that investors in these countries are likely to hold fewer stocks thanan investor in the U.S. This will make investors in these countries less concerned with hedging againstadverse future stock return shocks using the dividend yield-excess stock return relationship, therebycontributing to a lower hedging demand for stocks in these countries. This is especially likely to be thecase for France and Italy, where the Sharpe ratios for stocks are the lowest. In addition, the coefficients onthe lagged dividend yield in the excess stock return equations are the smallest for France and Italy,limiting the hedging ability of the dividend yield in France and Italy and thereby contributing further tothe weak hedging demands for stocks in these two countries relative to the U.S.

Recall from Table 1 that the Sharpe ratio for stocks in the U.K. is relatively large. The coefficient onthe lagged dividend yield in the excess stock return equation is also large relative to the other countries,and innovations to excess stock returns and the dividend yield are strongly negatively correlated(�0.79).26 These factors help to explain why, like the U.S., there are sizable hedging demands fordomestic stocks in the U.K.

Also note that the hedging demands for stocks decrease as g increases in Table 2, and the declinesare especially noticeable for the U.S. and U.K. This is related to the idea that stocks are the riskiest assetand can be a good hedge against themselves: as risk aversion increases, the investor’s demand for theriskiest asset (stocks) decreases, and the investor can hold fewer stocks as a hedge against adversestock returns.

As we discussed in Section 2.1 above, recent empirical evidence suggests that the EIS varies acrosseconomic agents. We thus compute mean asset demands for the same seven countries in Table 2 foralternative j values of 0.3 and 1.5 (and g¼ 7) in Table 3. To facilitate comparisons with the results inTable 2, we again report the mean asset demands for j ¼ 1 in Table 3. The mean total and hedgingdemands for stocks change little as we vary j for Australia, Canada, France, Germany, and Italy.However, there are sizable increases in the mean total and hedging demands for stocks as j increasesfor the U.K. and especially the U.S. Intuitively, as the EIS increases, agents become more willing to tradefuture for current consumption, and they hold more stocks, which have a relatively high expectedreturn. As they hold more stocks (for a given degree of relative risk aversion), the hedging demand forstocks increases, as stocks are a good hedge against themselves in the U.S. and U.K. It is also interestingto note that the patterns in Table 3 generally agree with theoretical results derived by Bhamra andUppal (2006). They consider a multi-period portfolio choice problem for an agent with recursive utilityin a highly stylized setting with three periods, a risk-free asset, and a one-period risky asset. In contrastto the more general setting used in the present paper, they assume a highly stylized setting in order toobtain analytical solutions for the policy functions. They show that the EIS affects the magnitude, butnot the sign, of the intertemporal hedging demand for the risky asset. In line with their theoreticalresults, we see from Table 3 that changes in the value of j only affect the magnitude, and not the sign, ofthe mean hedging demands for stocks.

3.3. Asset demands for an investor in the U.S. who can also invest in foreign stocks and bonds

We next use the CCV approach to analyze a multi-period portfolio choice problem for an investor inthe U.S. who, in addition to domestic bills, stocks, and bonds, has access to stocks and bonds froma foreign country. We take, in turn, Australia, Canada, France, Germany, Italy, and the U.K. to be theforeign country. We take the countries in turn to keep the VAR parameter space to a reasonable size.The log real return on a 3-month U.S. Treasury bill again serves as the return on the benchmark asset,and the log excess returns on U.S. stocks and bonds and foreign stocks and bonds constitute the excessreturns on the other four assets.27 The instrument set includes the U.S. nominal bill yield, dividend

26 See Table A7 in the on-line appendix of VAR results.27 As in, for example, Harvey (1991), we measure the log excess return on foreign stocks or bonds by first converting the

foreign stock or bond return to U.S. dollars using exchange rates and then computing the U.S. dollar return in excess of the U.S.dollar return on a U.S. 3-month Treasury bill.

Table 3Mean demands for domestic assets for investors in different countries assuming different values for the elasticity of intertemporal substitution (j) and a coefficient of relative risk aversion(g) equal to 7.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Stocks Bonds Bills

CRRA Total demand Myopic demand Hedging demand Total demand Myopic demand Hedging demand Total demand Myopic demand Hedging demand

U.S., 1952:04–2004:04j¼ 0.3 96.26 44.58 51.68 8.36 40.21 �31.85 �4.62 15.21 �19.83j¼ 1.0 124.65 44.58 80.07 15.49 40.21 �24.72 �40.14 15.21 �55.35j¼ 1.5 197.52 44.58 152.93 34.79 40.21 �5.42 �132.31 15.21 �147.52

Australia, 1954:02–2004:04j¼ 0.3 40.61 31.52 9.09 13.83 15.57 �1.74 45.56 52.91 �7.35j¼ 1.0 40.34 31.52 8.81 13.63 15.57 �1.93 46.03 52.91 �6.88j¼ 1.5 40.13 31.52 8.61 13.49 15.57 �2.08 46.38 52.91 �6.53

Canada, 1954:02–2004:04j¼ 0.3 32.70 23.07 9.63 15.26 24.73 �9.46 52.04 52.21 �0.17j¼ 1.0 34.96 23.07 11.89 16.22 24.73 �8.51 45.82 52.21 �3.38j¼ 1.5 37.41 23.07 14.34 17.28 24.73 �7.44 45.31 52.21 �6.90

France, 1961:01–2004:04j¼ 0.3 13.00 11.67 1.33 30.75 45.98 �15.23 56.25 42.35 13.90j¼ 1.0 13.11 11.67 1.44 30.82 45.98 �15.16 56.07 42.35 13.72j¼ 1.5 13.20 11.67 1.53 30.87 45.98 �15.11 55.93 42.35 13.57

Germany, 1967:02–2004:04j¼ 0.3 28.84 14.80 14.03 83.04 114.03 �30.99 �11.88 �28.84 16.96j¼ 1.0 29.86 14.80 15.06 83.52 114.03 �30.52 �13.38 �28.84 15.46j¼ 1.5 30.74 14.80 15.93 83.95 114.03 �30.08 �14.69 �28.84 14.15

Italy, 1952:04–2004:04j¼ 0.3 6.84 11.03 �4.19 23.78 41.12 �17.34 69.38 47.85 21.53j¼ 1.0 6.48 11.03 �4.55 23.47 41.12 �17.64 70.04 47.85 22.19j¼ 1.5 6.19 11.03 �4.84 23.25 41.12 �17.87 70.56 47.85 22.71

U.K., 1952:04–2004:04j¼ 0.3 71.14 28.27 42.87 37.87 42.36 �4.49 �9.01 29.37 �38.38j¼ 1.0 88.00 28.27 59.73 44.30 42.36 1.94 �32.30 29.37 �61.67j¼ 1.5 105.98 28.27 77.71 51.06 42.36 8.70 �57.04 29.37 �86.40

Notes: The table reports mean monthly total, myopic, and heding asset demands in percentages for stocks, 10-year government bonds, and 3-month Treasury bills for an investor with anelasticity of intertemporal substitution (j) equal to 0.3, 1.0, or 1.5; a discount factor (d) equal to 0.921/12; and a coefficient of relative risk aversion (g) equal to 7.

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yield, and term spread, as well as their foreign counterparts. The state vector for the multi-periodportfolio choice problem is now given by

ztþ1¼�rtbrtþ1;xsrtþ1;xbrtþ1;xsr�tþ1;xbr�tþ1;billtþ1;divtþ1;spreadtþ1;bill�tþ1;div�tþ1;spread�tþ1

�0;

(11)

where xsr�tþ1ðxbr�tþ1Þ is the log excess return in U.S. dollars on foreign stocks (bonds) relative to the U.S.3-month Treasury bill return, and bill�tþ1, div�tþ1, and spread�tþ1 are the instruments in the foreigncountry. We again assume that the state vector is generated by a VAR(1) process, d¼0:92 on an annualbasis, and j¼1. We again consider g values of 4, 7, and 10 and use the CCV numerical procedure toestimate the asset demands. The mean asset demands, along with 90% confidence intervals (generatedusing a suitably modified version of the parametric bootstrap procedure described in Section 2.2above), are reported in Table 4.28

A number of results stand out in Table 4. First, an investor in the U.S. continues to have substantialmean total and intertemporal hedging demands for domestic stocks, regardless of the foreign country.That is, it is still optimal for an investor in the U.S. with access to foreign stocks and bonds fromAustralia, Canada, France, Germany, Italy, or the U.K. to have sizable mean total and hedging demandsfor domestic stocks. While the confidence intervals are again generally quite wide in Table 4, thehedging demands for domestic stocks are significant according to the 90% confidence intervals whenAustralia, Canada, France, Italy, or the U.K. serves as the foreign country. We also see fairly largenegative intertemporal hedging demands for domestic bonds when France, Germany, or Italy is theforeign country, although these demands are not significant according to the 90% confidence intervals.It is interesting to note that, with the possible exceptions of Canadian stocks and U.K. stocks and bonds,the intertemporal hedging demands for the foreign assets are quite small in magnitude. In no case canwe reject the null hypothesis of zero mean hedging demands for foreign stocks and bonds according tothe 90% confidence intervals, although the mean intertemporal hedging demands for foreign stocks arenearly significantly positive when the U.K. serves as the foreign country.

Fig. 2 presents the intertemporal hedging demands for domestic and foreign stocks and bonds foreach month over the sample when g ¼ 7. The profiles of the hedging demands for domestic stocks andbonds in each panel of Fig. 2 are similar to those for the U.S. in Panel A of Fig. 1. We see that the hedgingdemands for foreign stocks and foreign bonds are typically small in magnitude throughout the samplefor all foreign countries with the exception of the U.K.

The sizable intertemporal hedging demandsfor domestic stocks for U.S. investors who alsohave access toforeign stocks and bonds can again be largely explained by the relatively high Sharpe ratio for U.S. stocks inconjunction with the relationship between excess stock returns and the dividend yield. The high Sharperatio makes domestic stocks attractive to U.S. investors, while the positive effect of the lagged dividend yieldon excess stock returns and the strong negative correlation between innovations to excess stock returns andthe dividend yield make domestic stocks a good hedge against themselves. This creates a strongintertemporal hedging demand for domestic stocks for investors in the U.S., and the strong hedging demandfor domestic stocks is evident whether or not an investor in the U.S. has access to foreign stocks and bonds.29

One way to quantify the gains from international diversification is to compute the expected value ofthe value function per unit of wealth, which is readily available using the CCV approach; see Campbellet al. (2002). For a CRRA equal to 7, an investor in the U.S. who invests in domestic assets alone wouldrequire the following factor increases in wealth in order to enjoy the same utility as she realizes whenshe can invest in both domestic assets and assets from a foreign country: Australia, 2.47; Canada, 7.63;France, 3.00; Germany, 2.53; Italy, 3.26; U.K., 4.89.

28 In order to conserve space, we do not report summary statistics or the complete VAR estimation results for these models.They are available at http://pages.slu.edu/faculty/rapachde/.

29 As observed in Section 3.2 above, there are numerous factors at work in determining hedging demands in this multivariateanalysis. We focus on factors that are likely to be of prime importance in explaining the strong hedging demands for U.S. stocks. It ismore difficult to provide intuition for certain results, such as why the mean hedging demand for domestic bills is non-monotonic ing when France is the foreign country in Table 4. This is especially true when the changes in mean asset demands are small inmagnitude, as they are in this case.


Table 4Mean asset demands for an investor in the U.S. who can also invest in foreign stocks and bonds.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 12) (13) (14) (15) (16)

Domestic stocks Domestic bonds Foreign stocks Foreign bon s Domestic bills

CRRA Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

yopicemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Foreign country: Australia, 1952:04–2004:04g¼ 4 148.94 59.39 89.55 75.97 92.51 �16.54 50.36 37.52 12.85 �36.54 33.26 �3.28 �138.73 �56.15 �82.58

[58, 218] [10, 73] [30, 153] [�35, 271] [�50, 270] [�56, 40] [�2, 93] [�10, 66] [�5, 33] [�102, 71] �106, 59] [�22, 10] [�349, 54] [�274, 102] [�175, �5]g¼ 7 110.73 33.68 77.05 40.56 53.14 �12.58 30.18 21.61 8.57 �21.51 19.21 �2.30 �59.95 10.78 �70.74

[45, 170] [5, 41] [25, 132] [�24, 157] [�28, 154] [�35, 30] [�2, 56] [�6, 38] [�4, 24] [�63, 37] �61, 33] [�16, 7] [�210, 40] [�114, 100] [�146, �8]g¼ 10 89.40 23.39 66.01 26.76 37.39 �10.64 21.14 15.24 5.89 �15.15 13.59 �1.56 �22.15 37.56 �59.70

[35, 141] [4, 28] [20, 113] [�36, 91] [�19, 108] [�28, 23] [�2, 41] [�4, 27] [�3, 19] [�45, 26] �43, 23] [�13, 6] [�141, 48] [�50, 100] [�128, �12]

Foreign country: Canada, 1952:04–2004:04g¼ 4 172.01 95.82 76.19 46.28 56.64 �10.36 5.50 �22.32 27.81 13.11 7.27 �4.16 �136.90 �47.42 L89.48

[31, 268] [12, 135] [�3, 140] [�145, 238] [�67, 197] [�84, 74] [�60, 93] [�81, 43] [�12, 79] [�106, 133] �84, 125] [�32, 36] [�349, 52] [�192, 96] [�205, �12]g¼ 7 119.81 54.25 65.57 29.22 32.97 �3.75 9.53 �12.31 21.84 6.24 .50 �3.26 �64.80 15.59 �80.40

[26, 195] [6, 77] [7, 124] [�98, 144] [�38, 113] [�54, 66] [�24, 73] [�44, 27] [�10, 56] [�68, 77] �48, 71] [�24, 29] [�206, 59] [�67, 97] [�175, �14]g¼ 10 92.87 37.62 55.25 24.18 23.50 0.67 9.65 �8.31 17.95 2.81 .39 �3.58 �29.50 40.79 �70.30

[20, 155] [4, 53] [8, 108] [�74, 107] [�26, 80] [�40, 55] [�15, 57] [�30, 19] [�7, 46] [�51, 55] �34, 50] [�19, 23] [�138, 67] [�17, 97] [�156, �18]

Foreign country: France, 1961:01–2004:04g¼ 4 145.76 62.97 82.78 �8.42 45.14 �53.56 �15.99 �3.33 �12.66 63.53 7.29 6.24 �84.87 �62.07 �22.79

[21, 216] [9, 86] [4, 143] [�122, 288] [�78, 228] [�94, 32] [�61, 49] [�45, 33] [�28, 21] [�36, 173] �32, 161] [�15, 29] [�387, 64] [�276, 123] [�133, 48]g¼ 7 105.92 35.89 70.03 �13.45 25.93 �39.37 �13.08 �2.06 �11.02 38.06 2.86 5.20 �17.45 7.38 �24.84

[17, 163] [5, 49] [15, 134] [�74, 136] [�44, 131] [�62, 24] [�40, 29] [�26, 19] [�21, 13] [�20, 101] �18, 92] [�12, 19] [�178, 90] [�115, 113] [�116, 17]g¼ 10 84.68 25.06 59.62 �13.39 18.24 �31.64 �11.29 �1.56 �9.74 27.66 3.09 4.57 12.34 35.16 �22.82

[13, 135] [1, 32] [8, 106] [�55, 95] [�31, 92] [�47, 17] [�33, 17] [�18, 13] [�19, 9] [�14, 71] �13, 65] [�9, 14] [�120, 77] [�53, 107] [�98, 14]

Foreign country: Germany, 1967:02–2004:04g¼ 4 122.46 50.90 71.56 �18.10 41.65 �59.75 �5.62 4.64 �10.26 66.91 2.96 3.95 �65.66 �60.16 �5.50

[11, 203] [�4, 73] [�5, 142] [�161, 200] [�95, 193] [�93, 46] [�71, 44] [�64, 44] [�31, 13] [�16, 163] �8, 155] [�15, 30] [�301, 102] [�230, 84] [�129, 67]g¼ 7 85.28 28.86 56.41 �16.92 24.28 �41.20 �5.13 2.62 �7.75 38.58 5.80 2.78 �1.80 8.44 �10.24

[�1, 149] [�2, 41] [�12, 112] [�103, 110] [�54, 111] [�65, 29] [�45, 23] [�37, 25] [�21, 8] [�12, 91] �5, 89] [�14, 17] [�149, 109] [�88, 91] [�98, 48]g¼ 10 66.21 20.05 46.16 �14.16 17.33 �31.49 �4.61 1.82 �6.43 27.27 4.93 2.34 25.29 35.87 �10.58

[�1, 121] [�2, 29] [�11, 94] [�76, 77] [�37, 77] [�48, 24] [�33, 15] [�26, 17] [�17, 5] [�7, 65] �3, 62] [�10, 13] [�93, 101] [�31, 94] [�81, 37]

Foreign country: Italy, 1952:04–2004:04g¼ 4 169.43 77.64 91.79 �6.46 45.59 �52.04 8.26 4.71 3.55 41.14 4.04 �12.90 �112.37 �81.98 �30.39

[83, 241] [34, 95] [46, 160] [�117, 171] [�105, 171] [�98, 24] [�28, 47] [�23, 36] [�11, 19] [�31, 140] �33, 158] [�31, 7] [�312, 61] [�277, 94] [�154, 34]g¼ 7 122.73 44.28 78.45 �8.42 26.07 �34.49 4.01 2.58 1.43 23.05 0.94 �7.90 �41.36 �3.88 �37.48

[71, 194] [20, 54] [40, 139] [�85, 85] [�60, 97] [�66, 17] [�16, 30] [�13, 20] [�7, 16] [�19, 78] �19, 91] [�21, 6] [�181, 62] [�115, 97] [�146, 2]

(continued on next page)

D.E.R

apach,M.E.W

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(

d

Md

�[�[�[

1[9[6[

5[3[2[

6[3[2[

5[3[

Table 4 (continued)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

Domestic stocks Domestic bonds Foreign stocks Foreign b ds Domestic bills

CRRA Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

g¼ 10 98.25 30.94 67.31 �8.07 18.27 �26.34 1.69] 1.73 �0.04 16.55 21.70 �5.15 �8.41 27.36 �35.78[56, 159] [13, 38] [35, 123] [�59, 62] [�42, 68] [�45, 19] [�13, 21 [�9, 14] [�7, 11] [�15, 54] [�13, 64] [�17, 4] [�111, 69] [�51, 98] [�122, �1]

Foreign country: U.K., 1952:04–2004:04g¼ 4 163.61 64.96 98.65 38.09 53.60 �15.51 57.63 21.38 36.25 17.76 39.82 �22.06 �177.10 �79.77 �97.33

[82, 231] [28, 87] [41, 154] [�114, 194] [�90, 205] [�49, 45] [�25, 108] [�23, 49] [�1, 70] [�72, 130 [�44, 135] [�46, 5] [�367, 12] [�258, 81] [�188, �17]g¼ 7 125.07 37.13 87.94 22.13 30.72 �8.59 40.86 11.97 28.90 5.13 22.96 �17.83 �93.19 �2.77 �90.42

[64, 183] [16, 50] [40, 141] [�68, 114] [�51, 117] [�36, 33] [�11, 78] [�13, 28] [�2, 54] [�49, 73] [�25, 77] [�38, 3] [�221, 25] [�105, 89] [�167, �20]g¼ 10 102.70 25.99 76.70 15.91 21.57 �5.66 31.00 8.20 22.81 2.02 16.22 �14.19 �51.64 28.02 L79.66

[56, 158] [11, 35] [36, 125] [�49, 80] [�36, 82] [�31, 25] [�5, 65] [�9, 19] [0, 46] [�39, 50] [�17, 54] [�30, 4] [�151, 35] [�43, 93] [�148, �23]

Notes: The table reports mean monthly total, myopic, and hedging asset demands in percentages for domestic stocks, dom stic 10-year government bonds, foreign stocks, foreign 10-yeargovernment bonds, and domestic 3-month Treasury bills for an investor with a unitary elasticity of intertemporal substitut n (j¼ 1); a discount factor (d) equal to 0.921/12; and a coefficientof relative risk aversion (g) equal to 4, 7, or 10. Bootstrapped 90% confidence intervals for the mean asset demands are give in brackets. A bold entry indicates significance according to the90% confidence interval.

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on

]

eion

A B

C D

E F

Fig. 2. Historical intertemporal hedging demands for domestic stocks (solid lines), domestic bonds (dashed lines), foreign stocks(dotted lines), and foreign bonds (closely spaced dotted lines) for an investor in the U.S. who can also invest in foreign stocks andbonds when g = 7 and j = 1.


Table 5 reports mean asset demands for j values of 0.3 and 1.5 and g ¼ 7 for an investor in the U.S.who can also invest in foreign stocks and bonds. (The mean asset demands for j ¼ 1 from Table 4 areincluded in Table 5 to facilitate comparison of results.) The pattern of results is similar to that in Table 3for an investor in the U.S.: the mean hedging demand for domestic stocks increases as j increases, andthe sign of the hedging demand is unchanged for different values of j. The same intuition for theformer results also applies: investors become more willing to trade future for current consumption as

Table 5Mean asset demands for an investor in the U.S. who can also invest in foreign stocks and bonds assuming different values for the elasticity of intertemporal substitution (j) and a coefficientof relative risk aversion (g) equal to 7.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

Domestic stocks Domestic bonds Foreign stocks Foreign bonds Domestic bills

CRRA Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

TotalDemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Foreign country:Australia, 1952:04–2004:04

j¼ 0.3 67.81 33.68 34.13 25.43 53.14 �27.72 26.08 21.61 4.48 �18.87 �19.21 0.34 �0.45 10.78 �11.23j¼ 1.0 110.73 33.68 77.05 40.56 53.14 �12.58 30.18 21.61 8.57 �21.51 �19.21 �2.30 �59.95 10.78 �70.74j¼ 1.5 205.67 33.68 171.99 75.72 53.14 22.58 39.29 21.61 17.68 �27.43 �19.21 �8.22 �193.24 10.78 �204.02

Foreign country:Canada, 1952:04–2004:04

j¼ 0.3 79.89 54.25 25.64 5.52 32.97 �27.45 �7.48 �12.31 4.83 4.30 9.50 �5.20 17.77 15.59 2.18j¼ 1.0 119.81 54.25 65.57 29.22 32.97 �3.75 9.53 �12.31 21.84 6.24 9.50 �3.26 �64.80 15.59 �80.40j¼ 1.5 163.64 54.25 109.39 48.54 32.97 15.57 28.80 �12.31 41.11 11.62 9.50 2.12 �152.61 15.59 �168.20

Foreign country:France, 1961:01–2004:04

j¼ 0.3 71.61 35.39 35.71 �22.95 25.93 �48.87 �12.27 �2.06 �10.20 37.82 32.86 4.95 25.79 7.38 18.41j¼ 1.0 105.92 35.89 70.03 �13.45 25.93 �39.37 �13.08 �2.06 �11.02 38.06 32.86 5.20 �17.45 7.38 �24.84j¼ 1.5 186.27 35.89 150.38 12.00 25.93 �13.93 �10.06 �2.06 �7.99 36.02 32.86 3.16 �124.23 7.38 �131.62

Foreign country:Germany, 1967:02–2004:04

j¼ 0.3 56.98 28.86 28.11 �24.55 24.28 �48.83 �4.35 2.62 �6.97 39.63 35.80 3.83 32.29 8.44 23.86j¼ 1.0 85.28 28.86 56.41 �16.92 24.28 �41.20 �5.13 2.62 �7.75 38.58 35.80 2.78 �1.80 8.44 �10.24j¼ 1.5 163.21 28.86 134.35 7.09 24.28 �17.18 �5.91 2.62 �8.54 35.28 35.80 �0.52 �99.67 8.44 �108.10

Foreign country:Italy, 1952:04–2004:04

j¼ 0.3 79.94 44.28 35.65 �21.69 26.07 �47.76 �0.17 2.58 �2.75 27.32 30.94 �3.62 14.60 �3.88 18.48j¼ 1.0 122.73 44.28 78.45 �8.42 26.07 �34.49 4.01 2.58 1.43 23.05 30.94 �7.90 �41.36 �3.88 �37.48j¼ 1.5 227.47 44.28 183.19 29.09 26.07 3.02 16.32 2.58 13.74 11.33 30.94 �19.61 �184.22 �3.88 �180.34

Foreign country: U.K., 1952:04–2004:04j¼ 0.3 70.73 37.13 33.61 3.91 30.72 �26.81 27.51 11.97 15.55 14.64 22.96 �8.32 �16.80 �2.77 �14.02j¼ 1.0 125.07 37.13 87.94 22.13 30.72 �8.59 40.86 11.97 28.90 5.13 22.96 �17.83 �93.19 �2.77 �90.42j¼ 1.5 168.79 37.13 131.67 36.59 30.72 5.86 50.78 11.97 38.82 �2.22 22.96 �25.18 �153.94 �2.77 �151.17

Notes: The table reports mean monthly total, myopic, and hedging asset demands in percentages for domestic stocks, domestic 10-year government bonds, foreign stocks, foreign 10-yeargovernment bonds, and domestic 3-month Treasury bills for an investor with an elasticity of intertemporal substitution (j) equal to 0.3, 1.0, or 1.5; a discount factor (d) equal to 0.921/12; anda coefficient of relative risk aversion (g) equal to 7.

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the EIS increases and thus hold more stocks, which have a relatively high expected return; as they holdmore stocks, the hedging demand for stocks increases.

3.4. Asset demands for investors in Australia, Canada, France, Germany, Italy, and the U.K.who can also invest in U.S. stocks and bonds

Our final empirical exercise is another extension that analyzes asset demands for an investor inAustralia, Canada, France, Germany, Italy, or the U.K. who has access to domestic bills, stocks, andbonds, as well as stocks and bonds from the U.S. The log real return on a domestic 3-month Treasury billacts as the return on the benchmark asset, and the log excess returns on domestic stocks and bonds andU.S. stocks and bonds comprise the excess returns on the other four assets.30 The investor considers sixinstruments: the domestic and U.S. nominal bill yields, dividend yields, and term spreads. We continueto assume a VAR(1) structure for the state vector, j ¼ 1, and d ¼ 0:92 on an annual basis. Table 6reports the mean asset demands and corresponding 90% confidence intervals (again generated usinga suitably modified parametric bootstrap procedure) for g values of 4, 7, and 10.31

A number of results stand out in Table 6. With the exception of the U.K., the mean intertemporalhedging demands for domestic stocks are fairly small in magnitude in all of the countries. According tothe 90% confidence intervals, we cannot reject the null hypothesis that the mean hedging demands fordomestic stocks are zero in Australia, Canada, France, Germany, and Italy. A similar situation holds withrespect to the mean intertemporal hedging demands for domestic bonds, and although these demandsare typically larger in magnitude than the mean hedging demands for domestic stocks, we cannotreject the null hypothesis that the mean hedging demands for domestic bonds are zero for any country.The only case where we see a sizable and significant mean hedging demand for a domestic asset isdomestic stocks in the U.K.

While the mean intertemporal hedging demands for domestic assets are typically small in Table 6,the mean hedging demands for U.S. stocks are substantial. The mean hedging demands for U.S. stocksare significant according to the 90% confidence intervals in Australia, Canada, Italy, and the U.K. Thereare also sizable and sometimes significant mean hedging demands for U.S. bonds. Overall, the results inTable 6 indicate that access to U.S. stocks and bonds for investors in Australia, Canada, France, Germany,Italy, and the U.K. generates sizable intertemporal hedging demands for U.S. assets. Fig. 3 presents theintertemporal hedging demands for domestic and foreign stocks and bonds for each month over thesample in each country when g ¼ 7, and the figure reinforces the conclusions from Table 6. The largestintertemporal hedging demand among domestic and U.S. stocks and bonds over most of the sample foreach country is that for U.S. stocks.

What accounts for the sizable intertemporal hedging demand for U.S. stock that emerges wheninvestors in Australia, Canada, France, Germany, Italy, and the U.K. have access to U.S. stocks and bonds?In the previous two subsections, we have discussed how the high Sharpe ratio for U.S. stocks, togetherwith the relationship between U.S. excess stock returns and the dividend yield, create sizableintertemporal hedging demands for domestic stocks for investors in the U.S. When investors outsidethe U.S. have access to U.S. stocks, the relatively high (local currency) Sharpe ratio for U.S. stocks makesU.S. stocks relatively attractive to these investors. From the standpoint of investors outside the U.S.,there is again a positive relationship between the lagged U.S. dividend yield and excess U.S. stockreturns and a strong negative correlation between innovations to U.S. excess stock returns and the U.S.dividend yield, making U.S. stocks an attractive hedge against themselves. All of this creates a strongintertemporal hedging demand for U.S. stocks for investors outside the U.S.

We can again quantify the gains from international diversification using the expected value of thevalue function per unit of wealth. For a CRRA equal to 7, an investor in a given country who invests in

30 We measure the log excess return on U.S. stocks or bonds by first converting the U.S. dollar stock or bond return to localcurrency using exchange rates and then computing the local currency return in excess of the local currency return ona domestic 3-month Treasury bill.

31 Again in order to conserve space, we do not report summary statistics or the complete VAR estimation results for thesemodels. They are available at http://pages.slu.edu/faculty/rapachde/.


Table 6Mean asset demands for investors in Australia, Canada, France, Germany, Italy, and the U.K. who can also invest in U.S. stocks and bonds.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)


CRRA Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Domestic country: Australia, 1952:04–2004:04g¼ 4 75.42 49.22 26.20 50.28 31.83 18.45 142.81 60.94 81.87 �9.04 67.51 �76.54 �159.48 �109.50 �49.98

[11, 135] [1, 79] [�8, 57] [�76, 201] [�88, 188] [�25, 62] [39, 204] [12.81] [7, 135] [�68, 75] [23, 132] [�120, �20] [�319, �21] [�253, 25] [�116, 23]g¼ 7 44.29 28.16 16.14 26.79 17.13 9.66 100.97 34.60 66.36 �21.61 39.23 �60.85 �50.44 �19.13 �31.31

[2, 82] [1, 46] [�7, 40] [�52, 110] [�51, 106] [�19, 43] [23, 147] [6, 46] [9, 114] [�68, 34] [14, 77] [�98, �116] [�150, 30] [�104, 54] [�80, 25]g¼ 10 29.72 19.73 9.99 14.97 11.26 3.71 79.38 24.07 55.31 �22.21 27.92 �50.13 �1.86 17.02 �18.88

[�2, 59] [0, 33] [�7, 31] [�36, 80] [�36, 74] [�21, 29] [16, 118] [4, 32] [7, 96] [�54, 26] [11, 55] [�75, �4] [�76, 56] [�43, 67] [�63, 23]

Domestic country: Canada, 1952:04–2004:04g¼ 4 �12.69 �10.54 �2.15 168.83 146.24 22.59 161.16 74.58 86.58 �270.70 �179.30 L91.41 53.40 69.01 �15.61

[�86, 91] [�64, 52] [�38, 73] [26, 330] [�13, 294] [�12, 64] [23, 244] [�7, 97] [7, 144] [�428, �34] [�285, 8] [�156, �14] [�213, 176] [�100, 210] [�87, 51]g¼ 7 �6.72 �5.79 �0.93 102.19 83.63 18.55 110.80 42.26 68.54 �164.37 �102.64 �61.73 58.10 82.53 �24.43

[�52, 63] [�36, 30] [�31, 51] [23, 199] [�7, 168] [�8, 51] [10, 168] [�5, 55] [9, 121] [�267, �18] [�168, 0] [�108, 5] [�94, 147] [�13, 164] [�82, 25]g¼ 10 �3.72 �3.89 0.17 72.70 58.59 14.11 83.51 29.33 54.18 �113.95 �71.97 �41.97 61.46 87.94 �26.49

[�37, 49] [�26, 21] [�26, 37] [19, 147] [�5, 118] [�5, 44] [12, 136] [�3, 38] [3, 96] [�197, �12] [�118, 0] [�93, 0] [�44, 130] [21, 145] [�75, 13]

Domestic country: France, 1961:01–2004:04g¼ 4 �10.59 �1.14 �9.44 85.10 99.83 �14.73 109.18 50.08 59.10 �98.47 �45.60 �52.87 14.78 �3.16 17.94

[�53, 52] [�37, 37] [�28, 21] [�45, 311] [�85, 301] [�35, 29] [�28, 156] [�20, 68] [�10, 98] [�194, 49] [�112, 57] [�79, 13] [�210, 160] [�227, 177] [�37, 54]g¼ 7 �8.32 �0.71 �7.61 48.51 57.40 �8.89 75.63 28.64 46.99 �66.74 �26.20 �40.53 50.92 40.88 10.04

[�34, 32] [�21, 21] [�19, 16] [�28, 174] [�48, 172] [�30, 13] [�15, 114] [�12, 38] [�7, 81] [�130, 28] [�64, 32] [�64, 11] [�76, 139] [�87, 144] [�34, 29]g¼ 10 �6.77 �0.54 �6.23 34.11 40.43 �6.32 58.38 20.06 38.32 �50.85 �18.44 �32.41 65.12 58.49 6.63

[�24, 24] [�15, 15] [�16, 11] [�19, 123] [�33, 121] [�17, 16] [�10, 90] [�9, 26] [�4, 69] [�96, 20] [�45, 22] [�54, 8] [�31, 122] [�34, 128] [�25, 26]

Domestic country: Germany, 1967:02–2004:04g¼ 4 14.65 6.86 7.79 201.53 223.50 �21.97 110.78 42.63 68.15 �104.53 �37.28 �67.26 �122.43 �135.71 13.28

[�64, 59] [�71, 37] [�13, 37] [0, 403] [�9, 384] [�70, 38] [�24, 162] [�7, 70] [�23, 11] [�164, 87] [�85, 78] [�111, 7] [�330, 89] [�304, 96] [�79, 55]g¼ 7 7.99 3.99 4.00 116.64 126.80 �10.16 78.31 24.33 53.99 �72.36 �21.31 �51.05 �30.58 �33.81 3.22

[�38, 34] [�39, 22] [�11, 23] [5, 236] [�6, 220] [�38, 33] [�18, 116] [�4, 40] [�10, 94] [�117, 47] [�49, 44] [�86, 7] [�160, 86] [�130, 98] [�59, 35]g¼ 10 5.03 2.84 2.18 82.67 88.11 �5.44 61.17 17.00 44.17 �55.76 �14.92 �40.84 6.89 6.96 �0.07

[�28, 23] [�28, 16] [�6, 19] [3, 166] [�5, 153] [�25, 28] [�11, 97] [�3, 28] [�9, 77] [�92, 31] [�34, 30] [�68, 7] [�83, 92] [�59, 102] [�46, 27]

Domestic country: Italy, 1952:04–2004:04g¼ 4 3.26 5.00 �1.74 59.66 78.57 �18.91 126.25 64.07 62.18 �100.43 �53.56 �46.87 11.27 5.92 5.35

[�34, 39] [�20, 39] [�17, 12] [�107, 234] [�133, 283] [�66, 26] [53, 188] [22, 87] [18, 105] [�183, 16] [�128, 30] [�77, �11] [�196, 165] [�242, 191] [�56, 50]g¼ 7 0.24 2.92 �2.68 32.87 44.93 �12.06 88.90 36.35 52.55 �68.70 �30.28 �38.42 46.69 46.08 0.61

[�21, 23] [�12, 22] [�12, 9] [�67, 127] [�76, 162] [�45, 16] [35, 132] [12, 49] [13, 84] [�126, 2] [�72, 18] [�63, �11] [�71, 133] [�96, 152] [�39, 32]

D.E.Rapach,M

.E.Wohar

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ofInternational

Money

andFinance

28(2009)

427–4534

48

g¼ 10 �1.17 2.09 �3.26 22.50 31.47 �8.98 70.56 25.26 45.30 �53.59 �20.96 L32.63 61.71 62.14 �0.43[�16, 16] [�8, 15] [�12, 5] [�47, 88] [�53, 113] [�32, 15] [26, 104] [8, 34] [13, 72] [�95,1] [�50, 13] [�53, �8] [�20, 123] [�37, 137] [�32, 23]

Domestic country: U.K., 1952:04–2004:04g¼ 4 80.43 24.44 56.00 131.34 113.69 17.65 134.35 52.63 81.73 �104.36 �44.06 �60.29 �141.77 �46.69 �95.08

[�4, 128] [�14, 54] [1, 76] [�54, 346] [�113, 313] [�22, 59] [48, 183] [12, 77] [16, 111] [�172, �19] [�109, 37] [�91 �16] [�342, 146] [�279, 211] [�152, 0]g¼ 7 60.58 14.42 46.17 80.50 65.40 15.10 102.43 30.13 72.30 �77.66 �25.96 �51.71 �65.85 16.01 �81.86

[�8, 86] [�8, 32] [2, 65] [�38, 192] [�64, 179] [�22, 42] [36, 138] [7, 44] [18, 100] [�119, �22] [�64, 20] [�77, �14] [�176, 123] [�113, 168] [�122, 5]g¼ 10 47.84 10.41 37.43 57.47 46.08 11.39 83.91 21.13 62.78 �63.00 �18.72 �44.28 �26.22 41.09 �67.32

[�2, 71] [�5, 23] [�1, 52] [�23, 143] [�44, 126] [�18, 33] [28, 113] [5, 31] [18, 88] [�94, �18] [�45, 14] [�66, �12] [�113, 112] [�49, 147] [�107, 2]

Notes: The table reports mean monthly total, myopic, and hedging asset demands in percentages for domestic stocks, domestic 10-year government bonds, foreign stocks, foreign 10-yeargovernment bonds, and domestic 3-month Treasury bills for an investor with a unitary elasticity of intertemporal substitution (j¼ 1); a discount factor (d) equal to 0.921/12; and a coefficientof relative risk aversion (g) equal to 4, 7, or 10. Bootstrapped 90% confidence intervals for the mean asset demands are given in brackets. A bold entry indicates significance according to the90% confidence interval.

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A

C

E

B

D

F

Fig. 3. Historical intertemporal hedging demands for domestic stocks (solid lines), domestic bonds (dashed lines), foreign stocks(dotted lines), and foreign bonds (closely spaced dotted lines) for investors in Australia, Canada, France, Germany, Italy, and the U.K.who can also invest in U.S. stocks and bonds when g=7 and j=1.


domestic assets alone would require the following factor increases in wealth in order to enjoy the sameutility as she realizes when she can invest in both domestic and U.S. assets: Australia, 7.50; Canada,9.08; France, 4.00; Germany, 4.09; Italy, 4.00; U.K., 7.52. Note that these factors are larger than thecorresponding factors reported in Section 3.3 above, implying that investors outside the U.S. have moreto gain by having access to U.S. stocks and bonds than U.S. investors do by having access to foreignstocks and bonds. This reiterates the idea that U.S. stocks are attractive to investors outside of theU.S. because U.S. stocks offer them access to an asset with a high expected return (and Sharpe ratio)that also serves as a good hedge against itself, and investors outside the U.S. typically do not havedomestic assets available to them with these attributes.

Table 7Mean asset demands for investors in Australia, Canada, France, Germany, Italy, and the U.K. who can also invest in U.S. stocks and bonds assuming different values for the elasticity ofintertemporal substitution (j) and a coefficient of relative risk aversion (g) equal to 7.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)


CRRA Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Totaldemand

Myopicdemand

Hedgingdemand

Domestic country: Australia, 1952:04–2004:04j¼ 0.3 34.01 28.16 5.85 13.54 17.13 �3.60 71.20 34.60 36.59 �0.40 39.23 �39.63 �18.34 �19.13 0.79j¼ 1.0 44.29 28.16 16.14 26.79 17.13 9.66 100.97 34.60 66.36 �21.61 39.23 �60.85 �50.44 �19.13 �31.31j¼ 1.5 72.52 28.16 44.37 63.93 17.13 46.79 177.25 34.60 142.64 �75.00 39.23 �114.23 �138.70 �19.13 �119.57

Domestic country: Canada, 1952:04–2004:04j¼ 0.3 �15.65 �5.79 �9.86 83.41 83.63 �0.23 75.50 42.26 33.24 �146.18 �102.64 �43.54 102.92 82.53 20.39j¼ 1.0 �6.72 �5.79 �0.93 102.19 83.63 18.55 110.80 42.26 68.54 �164.37 �102.64 �61.73 58.10 82.53 �24.43j¼ 1.5 �3.06 �5.79 8.85 123.29 83.63 39.66 145.10 42.26 102.84 �186.58 �102.64 �83.94 15.13 82.53 �67.40

Domestic country: France, 1961:01–2004:04j¼ 0.3 �8.90 �0.71 �8.19 42.95 57.40 �14.44 56.63 28.64 28.00 �55.17 �26.20 �28.97 64.48 40.88 23.61j¼ 1.0 �8.32 �0.71 �7.61 48.51 57.40 �8.89 75.63 28.64 46.99 �66.74 �26.20 �40.53 50.92 40.88 10.04j¼ 1.5 �0.95 �0.71 �0.24 66.51 57.40 9.11 125.50 28.64 96.86 �96.27 �26.20 �70.07 5.22 40.88 �35.66

Domestic country: Germany, 1967:02–2004:04j¼ 0.3 5.60 3.99 1.61 99.69 126.80 �27.11 50.36 24.33 26.04 �53.38 �21.31 �32.08 �2.26 �33.81 31.54j¼ 1.0 7.99 3.99 4.00 116.64 126.80 �10.16 78.31 24.33 53.99 �72.36 �21.31 �51.05 �30.58 �33.81 3.22j¼ 1.5 17.01 3.99 13.02 172.37 126.80 45.57 166.77 24.33 142.44 �131.52 �21.31 �110.22 �124.63 �33.81 �90.82

Domestic country: Italy, 1952:04–2004:04j¼ 0.3 �0.98 2.92 �3.90 32.92 44.93 �12.01 66.11 36.35 29.76 �56.94 �30.28 �26.67 58.90 46.08 12.82j¼ 1.0 0.24 2.92 �2.68 32.87 44.93 �12.06 88.90 36.35 52.55 �68.70 �30.28 �38.42 46.69 46.08 0.61j¼ 1.5 5.00 2.92 2.08 32.46 44.93 �12.47 144.70 36.35 108.35 �95.90 �30.28 �65.62 13.74 46.08 �32.34

Domestic country: United Kingdom, 1952:04–2004:04j¼ 0.3 33.66 14.42 19.24 55.16 65.40 �10.24 54.55 30.13 24.43 �53.96 �25.96 �28.01 10.59 16.01 �5.42j¼ 1.0 60.58 14.42 46.17 80.50 65.40 15.10 102.43 30.13 72.30 �77.66 �25.96 �51.71 �65.85 16.01 �81.86j¼ 1.5 80.26 14.42 65.84 100.91 65.40 35.51 141.41 30.13 111.28 �96.55 �25.96 �70.59 �126.03 16.01 �142.04

Notes: The table reports mean monthly total, myopic, and hedging asset demands in percentages for domestic stocks, domestic 10-year government bonds, foreign stocks, foreign 10-yeargovernment bonds, and domestic 3-month Treasury bills for an investor with an elasticity of intertemporal substitution (j) equal to 0.3, 1.0, or 1.5; a discount factor (d) equal to 0.921/12; anda coefficient of relative risk aversion (g) equal to 7.

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Parallel to Tables 3 and 5, Table 7 reports mean asset demands for j ¼ 0:3;1;1:5 and g ¼ 7.(The results for j ¼ 1 are reported to facilitate comparison with the results in Table 6.) Focusing on themean hedging demands for U.S. stocks (foreign stocks in Table 7), we again see that the hedgingdemand increases as the EIS increases.

4. Conclusion

We investigate the intertemporal hedging demands for stocks and bonds for investors withEpstein–Zin–Weil preferences and infinite horizons in the U.S., Australia, Canada, France, Germany,Italy, and U.K. When we analyze allocations across domestic bills, stocks, and bonds for investors ineach country, the sizable mean intertemporal hedging demands for domestic stocks in the U.S. and U.K.stand out. When an investor in the U.S. also has access to foreign stocks and bonds, she continues tohave sizable mean hedging demands for U.S. stocks, and the only foreign asset for which she exhibitsrelatively large intertemporal hedging demands is U.K. stocks. When investors outside of the U.S. haveaccess to U.S. stocks and bonds, investors in all of these countries display substantial intertemporalhedging demands for U.S. stocks. The excess stock return–dividend yield relationship in the U.S. helpsto account for the strong intertemporal hedging demands for U.S. stocks. Overall, our results indicatethat U.S. stocks provide especially attractive intertemporal hedging instruments for internationalinvestors with Epstein–Zin–Weil preferences and infinite horizons.

Acknowledgments

Rapach acknowledges financial support from a John Cook School of Business Summer ResearchGrant. We thank an anonymous referee for very helpful comments, as well as seminar participants atSaint Louis University, the 2005 Missouri Economics Conference, and 11th International Conference onComputing in Economics and Finance in Washington, D.C. The usual disclaimer applies.

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