International Portfolio Choice with Frictions: Evidence from Mutual Funds

Abstract

An extensive literature has introduced frictions into models of portfolio choice that lead to deviations from the standard Markowitz mean-variance portfolio.¹ This is supported by micro evidence of sluggish portfolio decisions by households and helps explain various asset pricing facts. In this paper we focus on international portfolio decisions. The objective is to provide evidence on how U.S. mutual funds allocate their equity portfolios across countries, and specifically to what extent this is affected by portfolio frictions that lead to a weaker and more gradual response to changes in expected returns. It has frequently been suggested that global investors are slow to adjust their portfolios in response to new information. In the context of U.S. external equity investments, Bohn and Tesar (1996, p. 80) comment that “we suspect that investors may adjust their portfolios to new information gradually over time, resulting in both autocorrelated net purchases and a positive linkage with lagged returns.” Froot, and O’Connell and S Seasholes (2001) provide similar evidence. Froot and Thaler (1990), in attempting to explain the forward discount puzzle of excess return predictability in the foreign exchange market, hypothesize that “...at least some investors are slow in responding to changes in the interest differential.” More formally, Bacchetta and van Wincoop (2010, 2021), Bacchetta, Davenport, and van Wincoop (2022), and Bacchetta, van Wincoop, and Young (2023) show that open economy models with portfolio frictions can explain a variety of evidence related to excess return predictability in foreign exchange and equity markets as well as various data moments involving capital flows, saving, investment and aggregate U.S. equity portfolios.² Nonetheless, none of the existing literature has provided direct evidence of portfolio frictions in international portfolio allocation data. This paper aims to fill that gap.

Our evidence is based on 15 years of monthly equity portfolio allocation data across 36 countries for 316 U.S. mutual funds that report to EPFR (Emerging Portfolio Fund Research). Mutual funds are the most important players in external U.S. equity holdings, accounting for 50|$\%$| of all U.S. foreign equity holdings at the end of 2021.³ To structure the analysis, we present a portfolio choice model that enables us to derive a simple and testable portfolio equation. While the standard Markowitz mean-variance portfolio is embedded as a special case, the model allows for deviations from the Markowitz portfolio as a result of portfolio frictions that involve costs of deviating from two benchmark portfolios. The optimal portfolio share then depends on both of these benchmark portfolios and a present discounted value of expected future excess returns. We first document that international differences in stock returns are predictable and that predictability improves over longer horizons. We then use the estimates for these expected excess returns in our portfolio regressions. We find that portfolios respond to expected return differentials, but deviate gradually from benchmark portfolios. The results from the portfolio regressions are used to obtain estimates of the structural parameters of the model, such as the two portfolio frictions and risk aversion.

The simple theoretical portfolio choice model that structures the empirical analysis is analogous to Gârleanu and Pedersen (2013). It assumes that funds (investors) maximize the present discounted value of risk-adjusted portfolio returns minus quadratic costs of deviating from two benchmark portfolios. The first portfolio friction is a cost of deviating from the portfolio share during the previous month, which is the portfolio under complete rebalancing. The second is a cost of deviating from a buy-and-hold portfolio. The more important these portfolio frictions are, the more the optimal portfolio share depends on the two benchmark portfolios and the less it depends on expected excess returns. In addition, the portfolio frictions imply that the optimal portfolio depends not just on expected excess returns over the next period (as in the Markowitz portfolio), but on a present discounted value of future excess returns. The frictions lead to a more gradual response of portfolio shares to changes in expected returns.⁴

One difficulty with estimating portfolio expressions is endogeneity. The error term of the portfolio expression, which, for example, captures latent time-varying risk, can lead to portfolio shifts that affect equity prices. While individual mutual funds are too small to affect equity prices, there may be common components across investors of such portfolio shifts. This leads to endogeneity of our explanatory variables (the lagged portfolio share, buy-and-hold portfolio share, and the measure of expected excess returns). We address this issue by using two-stage least squares (2SLS), using instruments that have explanatory power for the endogenous regressors but are unrelated to time-varying risk that enters the error term. The portfolio theory provides guidance in computing a fund-specific measure of portfolio risk that affects the portfolio of a fund.

We find that the funds respond to the discounted expected excess return with strong statistical significance. We also find that both benchmark portfolios are very important, so that portfolio frictions matter. Our estimates imply a humped shaped portfolio response to an expected excess return innovation. The initial portfolio response is weaker than in the absence of portfolio frictions, while the portfolio response builds gradually as a result of the frictions. The regression estimates imply a plausible rate of risk aversion of 3.2. We also find that the lagged portfolio share is at least as important as the buy-and-hold portfolio, which is consistent with extensive portfolio rebalancing by the mutual funds.

The paper is related to various strands of literature. The first is the literature on excess return predictability. While the evidence we report on the predictability of international stock return differentials is new, the evidence on the predictability of international short term bond return differentials (UIP deviations) has been known since Fama (1984). Predictability also has been widely documented in the context of country or individual stock returns or the excess of stock returns over bond returns (for a textbook discussion, see Campbell, Lo, and MacKinlay 1997). While the latter literature mainly focuses on the United States, some papers document stock return predictability in other countries or show, by pooling the data, the global predictability (Hjalmarsson 2010). Cenedese et al. (2016) consider the profitability of trading strategies that exploit international equity return differentials. They sort countries into various “bins” based on the realization of variables like the dividend yield that are likely to predict future equity returns. They do not estimate portfolio expressions or excess returns, but find substantial Sharpe ratios from trading strategies that exploit in which bin countries are located.

In terms of estimation of portfolio regressions, the open economy literature is very limited, which is the motivation behind this project. Frankel and Engel (1984) invert the portfolio expression obtained from a simple frictionless mean variance portfolio model, relating expected returns on various currencies to asset supplies. They strongly reject the model. Also relevant is recent work by Koijen and Yogo (2020). As we do, they adopt a two-step approach, first estimating expected excess returns and then estimating portfolio expressions. They differ in that they do not allow for portfolio frictions and use aggregate bilateral portfolio shares in three asset classes. They also handle the endogeneity issue differently as they can use their global demand system to instrument asset prices with macroeconomic variables. Some papers have investigated the link between international capital flows (as opposed to portfolio shares) and past returns as well as expected future returns (see, e.g., Bohn and Tesar 1996; Froot, and O’Connell and S Seasholes 2001; Didier and Lowenkron 2012).

One vein of literature also has investigated international portfolio allocation using fund-level data. Raddatz and Schmukler (2012) use EPFR data to regress portfolio allocation of mutual funds across countries on either the lagged portfolio share or the buy-and-hold portfolio share, as well as the most recent return differential (of a destination country relative to the overall fund return). But the optimal portfolio does not depend on expected excess returns.⁵ Portfolios therefore only have backward-looking elements (past portfolios and past returns) and no forward-looking element. Raddatz, Schmukler, and Williams (2017) also estimate portfolio regressions for EPFR funds. Portfolio shares are related to benchmark portfolios in the form of indexes, such as global or regional stock or bond indexes, as well as relative returns. Again, there are no forward-looking elements. Disyatat and Gelos (2001) compare EPFR portfolio weights to the predictions of a simple mean-variance portfolio model, where the variance of the portfolio return is replaced by the variance of a tracking error relative to a benchmark index and expected returns are based on historical returns.

Camanho et al. (2022) use data for equity funds in various countries to consider the extent of portfolio rebalancing by different funds. They use the FactSet/LionShares database, also used by Ferreira and Matos (2008). These data cover more funds than EPFR data, as they include a broad set of institutional investors rather than just mutual funds (including pension funds, bank trusts, and insurance companies). In addition, only a limited set of mutual funds that report to EPFR indicate their country allocation. Nonetheless, EPFR data have several advantages over FactSet data that make them attractive for our purpose. Since we are interested in the speed of portfolio adjustment, it is attractive that the EPFR data are available at the monthly frequency. The reporting frequency of the FactSet data is quarterly, semiannual, or annual, dependent on the country. In addition, funds reporting to EPFR report their portfolio allocation at the end of each month. Reporting dates vary significantly across funds in the FactSet data. It is important that the time of reporting matches up to the date at which we compute expected excess returns. Finally, just using data for mutual funds has the advantage that they are more homogenous than the entire set of institutional investors. They are also the most internationally oriented.

Outside the open economy literature, one line of literature tackles individual portfolio choice and has documented significant portfolio inertia. This literature (see, e.g., Ameriks and Zeldes 2004; Bilias et al. 2010; Brunnermeier and Nagel 2008; Mitchell et al. 2006) uses data on portfolio allocation by individual households. It is consistent with gradual portfolio adjustment, although (like the other literatures discussed above) it does not relate portfolio allocation to expected excess returns as in standard portfolio theory. An exception is the recent paper by Giglio et al. (2021), which relates equity portfolio shares to expected returns based on survey data of U.S.-based Vanguard investors. They find that portfolio shares depend positively on reported equity return expectations, but that responsiveness to expected equity returns is too weak to make sense in the context of the frictionless mean-variance portfolio choice model (implied risk aversion is excessive). They further provide evidence that changes in expected returns have limited explanatory power for when investors trade, but help predict the direction and the magnitude of trading conditional on its occurrence. They argue that the evidence is consistent with infrequent trading.

1. A Model of Portfolio Allocation with Financial Frictions

1.1 Portfolio objective

Consider a fund |$i$| that allocates its equity portfolio to |$N$| countries. We will focus the analysis here on a specific fund |$i$|⁠, although one should keep in mind that the investment universe varies across funds. We take the set of countries that a fund invests in as given and consider the portfolio allocation across these countries. One should therefore keep in mind that the number of countries |$N$| in the investment universe will vary across funds.

The vectors of portfolio shares and country equity returns are

where |$z_{i,n,t}$| is the share that fund |$i$| allocates to country |$n$| at time |$t$| and |$R_{n,t+1}$| is the country |$n$| equity return from |$t$| to |$t+1$|⁠.

Define the buy-and-hold portfolio as

This is the portfolio held at time |$t$| in the absence of asset trade at time |$t$|⁠. The buy-and-hold portfolio share differs from the lagged portfolio share |$z_{i,n,t-1}$| only because of the valuation effects associated with equity returns.

We consider a structure similar to that of Gârleanu and Pedersen (2013), where funds maximize the present discounted value of risk-adjusted portfolio returns, but face costs of deviating from benchmark portfolios. The objective of the fund is to maximize

Here, |$E_{i,t}$| is the expectation of fund |$i$| at time |$t$| and |${\bf \Omega}_{i,t}=var_{i,t}({\bf R}_{t+s+1})$| is the variance at time |$t$| of the time |$t+s+1$| portfolio return. While we allow for time-varying risk, we assume that the variance of future returns does not vary by horizon. Gârleanu and Pedersen (2013) assume that |${\bf \Omega}_{i,t}$| is a constant, so that there is no time-varying risk.

The first line of (3) is a present-value version of a standard mean-variance objective. The discount rate is |$\beta$| and the rate of risk aversion is |$\gamma_{i}$|⁠. The last two lines capture the cost of deviating from the benchmark portfolios, respectively, the lagged portfolios and the buy-and-hold portfolios. The parameters |$\mu_{1,i}$| and |$\mu_{2,i}$| determine the cost of deviating from, respectively, the lagged portfolios and the buy-and-hold portfolios. We follow Gârleanu and Pedersen (2013) by assuming |$A_{i,t}={\bf \Omega}_{i,t}$|⁠, for which they provide micro foundations.

Multiple underlying frictions can generate the gradual portfolio adjustment implied by (3). Gârleanu and Pedersen (2013) think of it as transaction costs. This is particularly relevant when deviating from the buy-and-hold portfolio, which involves asset trade. There also may be costs to acquiring information or costs to portfolio reoptimization that make fund managers more conservative and stick closer to prior benchmarks. In addition, it is possible that fund managers are penalized more for bad performance if this happens after they make significant portfolio changes relative to benchmark portfolios. This can take the form of fund outflows that affect manager compensation or lead to the replacement of managers.

1.2 Optimal portfolio

For a given optimal portfolio share |$z_{i,n,t}$|⁠, consider the allocation of the remaining portfolio share |$1-z_{i,n,t}$| among countries other than |$n$|⁠. Specifically, define

This is the share allocated to country |$m$| of the equity portfolio outside country |$n$|⁠. We can define a vector |${\bf z}_{i,-n,t}$|⁠, where element |$m$| is equal to |$z_{i,m,-n,t}$| if |$m \ne n$| and element |$n$| is zero. Choosing the optimal |${\bf z}_{i,t}$| is equivalent to choosing |$z_{i,n,t}$| and |${\bf z}_{i,-n,t}$|⁠. The first-order condition with respect to |$z_{i,n,t}$| therefore takes the optimal |${\bf z}_{i,-n,t}$| as given.

We define a reference portfolio return for fund |$i$| and country |$n$| as the return on the portfolio of countries other than country |$n$|⁠, with portfolio weights |$z_{i,m,-n,t}$|⁠:

The excess return of country |$n$| relative to the reference portfolio is

where |${\bf e}_{n}$| is a vector of size |$N$| with element |$n$| equal to one and zero otherwise.

We maximize (3) with respect to |$z_{i,n,t}$| after substituting the identity

Appendix A derives the first-order condition with respect to |$z_{i,n,t}$|⁠. After some rewriting, it becomes⁶

Here, |$\hat{z}_{i,n,t}=z_{i,n,t}-\bar{z}_{i,n}$|⁠, where

is the mean (over time) of the portfolio share allocated to country |$n$| by fund |$i$|⁠, where |$\sigma_{n,ref(i,n)}$| is the mean value of |$cov_{i,t}(er_{i,n,t+1},R_{ref(i,n),t+1})$| and |$\sigma_{i,n}^{2}$| is the mean value of |$var_{i,t}(er_{i,n,t+1})$|⁠. While |$\bar{z}_{i,n}$| depends on the mean level of risk, the term |$u_{i,n,t}$| in (8) captures time-varying risk in deviation from its mean:

Appendix B derives the following solution to the second-order difference Equation (8) in |$\hat{z}_{i,n,t}$|⁠:

where |$\theta_{i}=\mu_{1,i}+\mu_{2,i}$|⁠,

and

In (12) the term |$E_{i,t} er_{i,n,t+s}-E_{t} er_{i,n,t+s}$| captures the expected excess return by fund |$i$| minus the expected excess return by the econometrician. The expectation operator for the latter is denoted as |$E_{t}$|⁠.

In general there is heterogeneity across funds in |$\gamma_{i}$|⁠, |$\mu_{1,i}$| and |$\mu_{2,i}$|⁠, as well as heterogeneity across |$(i,n)$| pairs with respect to |$\sigma_{i,n}^{2}$| and risk that determines |$\bar{z}_{i,n}$|⁠. The same parameters without the |$i$| and |$n$| subscripts will refer to their mean across funds and countries. Since we will not be able to precisely characterize the heterogeneity across funds in the data, we focus on the mean of these parameters. To do so, we linearize the optimal portfolio expression (11) with respect to these parameters equal to their mean and all |$\hat{z}$| variables and excess returns equal to zero.⁷

Defining |$\delta=\beta \omega$|⁠, this gives

where

Here, |$\omega$|⁠, |$\theta$|⁠, |$\gamma$|⁠, |$\mu_{1}$| and |$\mu_{2}$| refer to the mean across funds of |$\omega_{i}$|⁠, |$\theta_{i}$|⁠, |$\gamma_{i}$|⁠, |$\mu_{1,i}$| and |$\mu_{2,i}$|⁠, while |$\sigma^{2}$| is the mean across all |$(i,n)$| of |$\sigma_{i,n}^{2}$|⁠. The present discounted value of future expected excess returns is defined as

Note that it is defined such that the weights on all future expected excess returns sum to one.

1.3 Intuition

Equation (14) writes the optimal portfolio as a linear function of the two benchmark portfolios, |$z_{i,n,t-1}$| and |$z^{bh}_{i,n,t}$|⁠, the expected present discounted value of the excess return on country |$n$| equity relative to the reference portfolio, and a time-varying error term.

First, consider the role of risk aversion. In general, investors face a trade-off between risk, expected returns and the cost of deviating from the benchmark portfolios. A rise in risk aversion implies that investors are more concerned with risk, and therefore relatively less concerned with deviating from the benchmark portfolios and expected excess returns. This therefore reduces |$b_{1}$|⁠, |$b_{2}$| and |$b_{3}$| (through |$\omega$|⁠).⁸

Next, consider the role of the portfolio frictions. A higher relative cost of deviating from the lagged portfolio compared to the buy-and-hold portfolio leads to a higher relative weight on the lagged portfolio in the portfolio expression: |$b_{1}/b_{2}=\mu_{1}/\mu_{2}$|⁠. An increase in the aggregate portfolio friction |$\theta$| implies that investors are relatively more concerned with deviating from the benchmark portfolios and therefore relatively less concerned with risk and expected returns. It therefore raises the weight on both benchmark portfolios (⁠|$b_{1}$| and |$b_{2}$|⁠), while it lowers the weight on the present discounted value of expected future excess returns (⁠|$b_{3}$|⁠).

Portfolio frictions also have the implication that the optimal portfolio gives relatively more weight to expected excess returns further into the future. The weight on the expected excess return |$s$| periods into the future is |$(1-\delta) \delta^{s-1}$| with |$\delta=\beta \omega$|⁠. A higher average portfolio friction |$\theta$| raises |$\delta$|⁠, which implies more weight on expected returns further into the future.⁹ Without portfolio frictions, all the weight is on the expected excess return in the immediate future:

Gârleanu and Pedersen (2013) explain the impact of the portfolio friction in terms of an “aim portfolio.” Their setup has only a cost of deviating from the lagged portfolio. They show that the optimal portfolio is a weighted average of the lagged portfolio share and that of the aim portfolio. The aim portfolio is forward-looking. It is a portfolio toward which investors trade each period. The aim portfolio is a weighted average of the Markowitz portfolio and the expected aim portfolio during the next period. One can write it as the present discounted value of the current and all future Markowitz portfolios. Since the Markowitz portfolio depends on the one-period expected excess return, the aim portfolio depends on the expected present discounted value of all future expected excess returns. As a result, the optimal portfolio has both a backward-looking part (the lagged portfolio) and a forward-looking part that depends on all future Markowitz portfolios and therefore all future expected excess returns. As a result of the portfolio friction, investors wish to smooth the transition between the past portfolio and future Markowitz portfolios.¹⁰

Some comments are in order regarding the error term |$\epsilon_{i,n,t}$| defined in (12). The first term is the expected present discounted value of risk in deviation from its mean. Risk is defined by |$u_{i,n,t}$| in (10) and depends on both the variance of the excess return and the covariance of the excess return with the reference portfolio. The second term captures differences between expected excess returns by fund |$i$| and that by the econometrician (denoted as the expectation operator |$E_{t}$|⁠). This may include deviations from rational expectations of excess returns that lead to noise trade. It may also capture changes in expected excess returns associated with information that is not easily available to an econometrician. Waves of optimism or pessimism about a country that we cannot easily measure therefore go into the error term as well.

Finally, we should point out the alternative way of writing the optimal portfolio expression. Define a valuation effect variable as the difference between the buy-and-hold portfolio and the lagged portfolio:

Linearizing, we have |${\it val}_{i,n,t}=\bar{z}_{i,n}(1-\bar{z}_{i,n})er_{i,n,t}$|⁠. It tells us how much the portfolio share increases due to an increase in the excess return in the absence of any asset trade.

The portfolio then can be written as

where |$a_{1}=b_{1}+b_{2}$|⁠, |$a_{2}=b_{2}$|⁠, |$a_{3}=b_{3}$|⁠. The coefficient for the valuation effect therefore corresponds to the coefficient for the buy-and-hold portfolio in (14), while the coefficient for the lagged portfolio is now the sum of the coefficients for the lagged and buy-and-hold portfolios in (11). Raddatz and Schmukler (2012) estimate a portfolio expression for mutual funds that includes the lagged portfolio and valuation effect, but not the expected excess returns.

Given estimates of the reduced-form parameters |$a_{1}$|⁠, |$a_{2}$|⁠, and |$a_{3}$| from the portfolio regression, we can then extract the structural parameters. We will use scaled structural parameters for the portfolio frictions, defined as |$\lambda_{1}=\mu_{1} \sigma^{2}$| and |$\lambda_{2}=\mu_{2} \sigma^{2}$|⁠. Then

We also have |$\delta=\beta a_{1}$|⁠.

2. Predicting Cross-Country Equity Return Differentials

A key variable in the portfolio expression derived above is the expected excess return variable |${\it ER}_{i,n,t}$|⁠. This section describes how we construct estimates for expected excess return differentials. After reviewing the empirical strategy, we show that return differentials can be predicted by standard variables: dividend-price, earnings-price and momentum. We present results for pooled linear regressions.

2.1 Outline

The excess return in a specific country is fund specific, meaning it depends on equity returns in the reference countries and the portfolio weights of the fund in these countries. In this section, rather than considering expected excess returns for individual funds, we consider return differentials relative to the United States. Specifically, the excess return for country |$n$| at |$t+s$| is |$er_{n,t+s} =R_{n,t+s}-R_{US,t+s}$|⁠, where |$R_{n,t+s}$| and |$R_{US,t+s}$| are the equity returns of country |$n$| and the United States at |$t+s$|⁠. As will be further discussed in the next section, the expected excess return for a specific fund can easily be computed once we know the expected excess returns relative to the United States for individual countries. For a fund |$i$|⁠, it is simply the expected excess return |$E_{t} er_{n,t+s}$| for country |$n$| minus the weighted average of expected excess returns |$E_{t} er_{m,t+s}$| of the reference countries, using the portfolio shares for the reference portfolio of fund |$i$| as weights.

In the theory, portfolio shares depend on a present discounted value of expected excess returns at all future dates. We will indeed use such present values when applying the theory to U.S. mutual fund portfolio shares in the next section. But in this section we consider either the predictability of excess returns |$er_{n,t+1}$| over the next month or cumulative excess returns |$er_{n,t,t+k}=er_{n,t+1}+...+er_{n,t+k}$| over the next |$k$| months. We use panel regressions to report predictability at different horizons.

2.2 Panel regressions

We use pooled regressions over 73 countries with monthly data from January 1970 to March 2019. All data in the baseline regressions come from MSCI, using the last trading day of the month. Since data availability starts later for many countries, this gives us an unbalanced panel with more than 22,000 observations.¹¹ Returns are computed from the MSCI total return index. We consider the following benchmark regression:

where |$X_{n,t}$| is a set of explanatory variables known at time |$t$|⁠. Following Petersen (2009), we include a country fixed effect and cluster standard errors by month.¹² Using pooled data and assuming a common coefficient |$\beta$| allows us to get more precise estimates.¹³

The explanatory variables in the benchmark specification are standard in the literature on stock return predictability,¹⁴ but here we consider the differential with the United States. These variables are the differential in the logarithm of the earning-price ratio |$dep_{n,t}= ln(E/P)_{n,t}-ln(E/P)_{US,t}$|⁠; the differential in the logarithm of the dividend-price ratio |$ddp_{n,t} = ln(D/P)_{n,t}-ln(D/P)_{US,t}$|⁠; and momentum, measured by the current return differential |$er_{n,t-1,t}$|⁠. Since we take the logarithm of the earning-price ratio, we omit the periods with a negative value.¹⁵

Table 1 shows the results of regression (22) for one-period ahead returns |${er}_{n,t,t+1}$|⁠. We see that the three variables are strongly significant and have the expected sign. From the first column, it is interesting to notice that the small coefficient of 4.26 on momentum implies that excess returns are not very persistent. In line with the literature on return predictability, the |$R^2$| is extremely low for short-horizon predictions.

The fit of Equation (22) significantly improves when the horizon increases. Table 2 shows the results for 1- (as in Table 1), 12-, 24-, and 36-month excess returns, using the three variables in the regression. We see that coefficient values increase with the horizon. Moreover, the |$R^{2}$| increases significantly, reaching |$13.7\%$| at the 36-month horizon.

The results in Tables 1 and 2 indeed show the predictability of stock market return differentials, which are particularly strong at longer horizons.¹⁶ In Appendix C we show that the predictability is also economically significant, following an approach similar to that of Cenedese et al. (2016). When sorting countries each month into quintiles based on their values of momentum, dividend-price differential, or earning-price differential, we show that returns are substantially higher for higher quintiles; that is, higher values of momentum, dividend-price, or earning-price are associated with higher returns.

3. Results for Mutual Fund Portfolios

In this section we use panel data of U.S.-based equity funds that report country portfolio allocations to EPFR. These data are used to estimate portfolio equations implied by the model developed in Section 1. The sample runs from January 2002 to July 2016.

3.1 Sample selection and portfolio shares

The U.S. mutual funds that report their country allocation to EPFR are mostly globally or regionally oriented, with a relative small average share allocated to U.S. equity. The funds report their equity holdings in 135 countries (including the United States) and cash holdings. Cash holdings are relatively small, on average 2.8|$\%$| of total assets under management (AUM). In what follows we focus on the noncash component of AUM, the equity holdings in the 135 countries. Aggregating across all funds, during an average month 7.5|$\%$| of equity holdings are allocated to the United States. This shows the strong global bias of our funds. As discussed further below, the far majority of the funds have no U.S. equity holdings at all.

It is useful to place the foreign equity holdings of these funds in a broader perspective. At the end of the sample, July 2016, total U.S. foreign equity holdings was $7,045 billion.¹⁷ Of that, $3,394 billion (47|$\%$|⁠) was held by U.S. mutual funds.¹⁸ U.S. equity mutual funds that report their country allocation to EPFR report a $436 billion allocation to foreign equity, which is 13|$\%$| of all foreign equity held by U.S. mutual funds. The remaining 87|$\%$| is held by U.S. funds that do not report to EPFR and funds that do report to EPFR, but do not report their country allocation.

We clean the original sample. We focus on a subset of 316 funds investing in 36 countries and remove very small portfolio shares.¹⁹ The sample selection is described below. In July 2016, this cleaned sample has a $395 billion foreign equity allocation. This is 91|$\%$| of the full sample AUM, so not much AUM is lost by cleaning the sample.

In the Internet Appendix, we report some evidence of how representative this sample is in terms of the allocation across foreign countries. For July 2016, we report the portfolio shares allocated to the 35 foreign countries. We do this both for our sample of 316 mutual funds and for total U.S. foreign equity holdings. The correlation is 0.88. Our mutual fund sample invests a higher share in emerging markets, particularly in Asian and Latin American countries. We also report time series of portfolio shares allocated to three regions (Europe, Asia, and Latin America) from January 2002 to July 2016. These portfolio shares look quite similar to those based on total foreign equity holdings, with correlations of 0.67, 0.56, and 0.79, respectively, for the three regions.²⁰

Regarding the selection of funds, we only include U.S. equity funds with more than $5 million in AUM at the end of the sample. In addition, we impose that the fund must report its global equity allocations for at least 12 consecutive months during the sample. This leaves us with a total of 316 funds. We then drop very small portfolio shares and countries in which very few funds invest or for which we have insufficient MSCI data. There are two problems with small portfolio shares. First, valuation effects are very close to zero. As discussed, after linearizing we can write the valuation effect as |${\it val}_{i,n,t}=\bar{z}_{i,n}(1-\bar{z}_{i,n}) er_{i,n,t}$|⁠. When |$\bar{z}_{i,n}$| is very close to zero, the valuation effect is essentially zero. This makes it difficult to determine the coefficient for the valuation effect, which is needed to determine the relative importance of the two portfolio costs (deviation from the lagged portfolio and the buy-and-hold portfolio). Second, as will be discussed below, we assume that the return of a fund in country |$n$| is the MSCI return of country |$n$|⁠. If a fund invests very little in country |$n$|⁠, its country |$n$| portfolio is much less likely to be well represented by the MSCI return in that country.²¹

To be more precise with our selection, consider the allocation of each fund to the aggregate of the 135 countries. Write the corresponding portfolio share as |$k_{i,n,t}$|⁠, which is the investment by fund |$i$| in country |$n$| in month |$t$| as a share of the total allocation by fund |$i$| to the 135 countries in month |$t$|⁠. Let |$\bar{k}_{i,n}$| be the sample average, with the average taken over the months for which the fund reports the country allocation. Here, we use the letter |$k$| instead of |$z$| to be clear that these are not the shares that will be used in the regression analysis.

For portfolio shares, we only consider |$(i,n)$| pairs for which |$\bar{k}_{i,n}$| is at least 2|$\%$|⁠. Regarding countries, only 74 of the 135 have complete MSCI equity return data that are needed to compute excess returns and valuation effects. Many of these countries have very few funds that invest in them. We only include countries |$n$| in which at least 10 of the 316 funds invest during some month of the sample, not including the |$(i,n)$| pairs we removed. This reduces the sample to 36 countries. The Internet Appendix extends our main analysis by including smaller portfolio shares.

Funds enter and exit the sample. The 316 funds are never all reporting simultaneously. The number of funds reporting rises over time. During the first 12 months of the sample an average of 46 funds report each month, while during the last 12 months of the sample an average of 222 funds report each month. For an average country and an average month, 29|$\%$| of reporting funds report a portfolio allocation to that country. For an average month, only 21|$\%$| of reporting funds report a portfolio allocation to the United States. Therefore, 79|$\%$| of the funds only invest in foreign equity.

For our sample of 316 funds and 36 countries we compute |$z_{i,n,t}$| as the equity that fund |$i$| holds in country |$n$| during month |$t$|⁠, divided by the total equity allocation of fund |$i$| to the 36 countries in the sample during month |$t$|⁠. In addition, as necessary for the regressions, we only use observations |$z_{i,n,t}$| when data are available for both |$z_{i,n,t}$| and |$z_{i,n,t-1}$|⁠. This results in a total of 154,407 observations.

3.2 Equity returns

We need data on equity returns both to compute an estimate of expected excess returns and to compute the buy-and-hold portfolio. We would preferably use equity returns of individual funds in each country. Unfortunately, EPFR does not provide the return of funds in individual countries. We therefore follow Raddatz and Schmukler (2012) and use MSCI returns in each country. They argue that this provides a reasonable approximation. In what follows it is therefore assumed that |$R_{n,t}$| is the equity return of country |$n$| for all funds.

We construct the expected excess return variable |${\it ER}_{i,n,t}$| as follows. We use the average sample weights |$\bar{z}_{i,m,-n}$| to compute the reference portfolio. In the Internet Appendix, we show that results are similar when we use the contemporaneous weights |$z_{i,m,-n,t}$|⁠. The excess return relative to the reference portfolio then can be written as

where |$er_{n,t+s}$| is the excess return at |$t+s$| of country |$n$| relative to the United States. It then follows that

First, some comments are in order regarding |$k$| and |$\delta$|⁠. While in the theory |$k=\infty$|⁠, in the empirical applications |$k$| is necessarily finite. We assume |$k=24$|⁠. We consider lower and higher values in the Internet Appendix. The estimation of |$\delta$| is discussed in Section 4.5. It is consistent with its theoretical value of |$\beta a_{1}$|⁠, where |$a_{1}$| is the coefficient for the lagged portfolio share in (18).

We compute the expectation in (23) by estimating a panel regression of

on the same variables that we regressed |$er_{n,t,t+k}$| on in Section 3. However, in contrast to the previous section, here we create true forecasts using recursive regressions up to the time of the forecast rather than using the entire sample. The sample starts in January 1970. As shown in the Internet Appendix, we still find predictability when restricting the sample to the 35 foreign countries and using discounted returns. The three variables, momentum, dividend-price and earning-price are all significant.

The buy-and-hold portfolio is computed as follows. We have |$z^{bh}_{i,n,t}=z_{i,n,t-1}\frac{1+R_{n,t}}{1+R_{i,p,t}}$|⁠. For |$R_{i,p,t}$| we use the portfolio return of fund |$i$| obtained from EPFR. Since EPFR does not provide the return of funds in individual countries, we again use the country |$n$| equity return from MSCI for |$R_{n,t}$|⁠. We compute the valuation effect as |${\it val}_{i,n,t}=z^{bh}_{i,n,t}-z_{i,n,t-1}$|⁠.

3.3 Endogeneity

An endogeneity problem arises when the error term of the portfolio regression (18) is correlated with |$z_{i,n,t-1}$|⁠, |${\it val}_{i,n,t}$| or |${\it ER}_{i,n,t}$|⁠. It is not difficult to see how this could happen. Portfolio shifts that enter through the error term affect asset demand, which leads to changes in equilibrium equity prices.²² This is particularly the case when such portfolio shifts are common not just across the U.S. mutual funds in our sample, but the broader class of investors in these equity markets.²³ The three explanatory variables |$z_{i,n,t-1}$|⁠, |${\it val}_{i,n,t}$|⁠, or |${\it ER}_{i,n,t}$| depend on the level of equity prices, changes in equity prices, or both. All three therefore may be correlated with the error term.

We therefore seek instruments that are plausibly correlated with either the level or changes of equity prices, and therefore with our three explanatory variables, while at the same time being uncorrelated with the error term. The theory in Section 2 provides guidance about the nature of the error term. Equation (12) provides an expression of the error term, which is related to both risk and expectations of the excess returns that are not captured by the econometrician.

Risk depends on the present discounted value of |$u_{i,n,t}$|⁠, which from (10) is a linear function of

Using that |$er_{i,n,t+1}=R_{n,t+1}-R_{ref(i,n),t+1}$|⁠, we can also write this as

Country |$n$| equity is therefore riskier from the perspective of fund |$i$| when its excess return over the reference portfolio of fund |$i$| is more correlated with the overall portfolio return of the fund. The fund’s overall portfolio return is |$\bar{z}_{i,n}R_{n,t+1}+(1-\bar{z}_{i,n}) R_{ref(i,n),t+1}$|⁠.

We obtain a proxy for this risk by computing for each country |$n$|⁠, fund |$i$| and month |$t$| the following risk measure:

Here, |$d \in t+1$| refers to days |$d$| during month |$t+1$|⁠. We therefore compute the covariance using daily values of the excess return and overall portfolio return of the fund for the days during month |$t+1$|⁠. This uses MSCI data on daily equity returns for all countries in the sample.

The other component of the portfolio error term in (12) relates to the difference between expectations of the funds and our expectations as econometricians. We compute |${\it ER}_{i,n,t}$| by regressing future excess returns on momentum, the dividend-price ratio and the earnings-price ratio. Expectations of future excess returns by funds that are orthogonal to these variables end up in the error term. But this implies that this part of the error term is by construction uncorrelated with |${\it ER}_{i,n,t}$|⁠. In what follows we will therefore not be concerned with this component of the error term.

The endogeneity problem is not easy to tackle. We are looking for instruments that have independent explanatory power for all three of our endogenous explanatory variables, while at the same time none of them can be correlated with risk. We identify a set of instruments in several steps. We start with a set of 13 variables that are plausibly correlated with equity prices or changes in equity prices, and therefore with our endogenous regressors. Next, we reduce this to a set of eight instruments that satisfy two criteria: (a) each instrument must have statistically significant explanatory power for at least one of the three endogenous regressors, and (b) none of the instruments is significant when regressing |${\it risk}_{i,n,t+1}$| on the set of instruments. We also check that the F-tests of the first-stage regressions of the regressors on the instruments are acceptable. Only when all of this is satisfied do we estimate the portfolio regression with two-stage least squares (2SLS).

We first discuss the set of 13 variables that we start with. Details regarding their computation can be found in the Appendix. Since the endogenous regressors depend on both the level and first difference of equity prices, we consider both the level and first difference for most of the variables. The first difference for a variable |$x_{i,n,t}$| is defined as |$\Delta x_{i,n,t}=x_{i,n,t}-x_{i,n,t-1}$|⁠. We focus the description here on the levels of the variables.

First, |$e_{i,n,t}$| is the logarithm of earnings for country |$n$| equity, minus the portfolio-weighted average of the logarithm of earnings for the reference countries with portfolio weights |$\bar{z}_{i,m,-n}$|⁠. Earnings is obtained by multiplying the earnings-price ratio with the price index. |$d_{i,n,t}$| is defined analogously for dividends. |$i_{i,n,t}$| is the 3-month Euro Libor interest rate for country |$n$| minus the portfolio weighted average interest rates of the reference countries. |$y_{i,n,t}$| is the monthly logarithm of the industrial production index for country |$n$|⁠, minus the portfolio weighted average for reference countries. |$b_{i,n,t}$| is the logarithm of the book value for country |$n$| equity (market value divided by the price) minus the portfolio-weighted average for the reference countries. |$h_{i,n,t}$| is the logarithm of the bond price index (mostly for 10-year government bonds) minus the weighted average for reference countries.

Together with their first differences, this gives 12 variables, all of which are naturally correlated with equity prices and equity returns. While our mutual funds invest in equity of different countries, other investors arbitrage equity returns relative to short- and long-term bond returns. Such arbitrage leads to a relationship between equity prices, interest rates and long-term bond prices. The industrial production variable may be correlated with equity prices in various ways. It may be related to the wealth of domestic investors, which affects demand for domestic equity and therefore the price. Industrial production growth may also affect equity prices to the extent that it helps predict future dividends. A change in the book value affects equity prices through a change in the equity supply. Finally, earnings and dividends naturally affect equity prices through the income component of equity returns.

The last variable we consider is the 1-month-lagged valuation effect, |${\it val}_{i,n,t-1}$|⁠. It is associated with relative equity price changes from |$t-2$| to |$t-1$|⁠. It will be correlated with |${\it val}_{i,n,t}$|⁠, one of our endogenous regressors, to the extent that there is any autocorrelation of equity price changes. Gabaix and Koijen (2022) find that financial shocks, which capture a common component of portfolio shifts across investors, have a very persistent effect on equity prices. While the error term of the portfolio expression can then be expected to be correlated with the level of the equity price, as well as the contemporaneous equity price change, it will not be much correlated with lagged price changes.

All of these variables could be potentially correlated with portfolio risk that enters the error term. For example, monetary policy may respond to a change in risk, which could lead interest rates and bond prices to be correlated with risk. Similarly, periods of low corporate earnings may coincide with periods of increased risk. The relationship between the instruments and portfolio risk is ultimately an empirical matter that we need to check with our measure of risk.

While we will check that the instruments are uncorrelated with risk, we cannot rule out that they are correlated with the remainder of the error term, which is unrelated to both risk and expected returns. But it is not obvious why the instruments would be correlated with the error term of the portfolio share of the mutual funds. For example, a variable like book value is related to asset supply rather than demand. Interest rates and bond prices, among other variables, naturally affect equity portfolio demand by investors that arbitrage equity and bond returns, which our equity mutual funds do not do. Equity demand depends on the product of financial wealth and portfolio shares. We think of industrial production as a proxy for wealth. Nonetheless, we have no way to rule out for certain that components of the error term, other than risk and expected returns, may be correlated with some of the instruments.

We first conduct first-stage regressions of the three endogenous regressors on each of the 13 variables. We run regressions of |$z_{i,n,t-1}$|⁠, |${\it val}_{i,n,t}$| and |${\it ER}_{i,n,t}$| on one variable at a time and remove variables that are not statistically significant at the 5|$\%$| level for all of the endogenous regressors. This is the case for |$\Delta i_{i,n,t}$|⁠, |$\Delta b_{i,n,t}$| and |$\Delta y_{i,n,t}$|⁠. This leaves us with 10 variables. Since we prefer to have as many instruments as possible, within the remaining set of 10 variables we identify the largest subset such that none of the variables is significant in a regression of the |${\it risk}_{i,n,t+1}$| on these variables. For this we conservatively use 10|$\%$| significance as a cutoff, which leads us to more easily reject variables as valid instruments than a 5|$\%$| significance cutoff. The largest set includes eight instruments:

In going from 10 to 8 variables, we remove |$i_{i,n,t}$| and |$d_{i,n,t}$|⁠.

Table 3 reports regressions showing that the instruments have significant explanatory power for the endogenous variables, while they are unrelated to risk. The first three columns report first stage regressions of the three endogenous regressors on the eight instruments. Each of the instruments in the set |$I_{i,n,t}$| is statistically significant for at least one of the endogenous regressors, and in most cases for two or all three of the endogenous regressors. For the regressors |$z_{i,n,t-1}$|⁠, |${\it val}_{i,n,t}$| and |${\it ER}_{i,n,t}$|⁠, 4, 4, and 8 instruments, respectively, are statistically significant at the 1|$\%$| level. The first stage Sanderson-Windmeijer F-statistics are all well above 10, again suggesting strong instruments. The last column of Table 3 reports the regression of |${\it risk}_{i,n,t+1}$| on this set of instruments |$I_{i,n,t}$| as well as the lagged risk |${\it risk}_{i,n,t}$|⁠. While lagged risk is significant, none of the instruments is significant.

3.4 Benchmark results

Columns 1 and 2 of Table 4 report the benchmark estimation of Equation (18), both with ordinary least squares (OLS) and instrumental variables (IV). The regression includes a country-fund dummy |$b_{i,n}$|⁠. As we have seen in the theory, this is related to differences in mean portfolio shares |$\bar{z}_{i,n}$|⁠. We get the same results when we simply subtract the mean of all variables for each country-fund pair.²⁴Column 3 reports the regression (14) with IV, which replaces |${\it val}_{i,n,t}$| with |$z^{bh}_{i,n,t}$|⁠. As previously discussed, these are identical regressions, just written slightly differently.

The role of endogeneity can be seen by comparing columns 1 and 2. While the differences are not large, we see that the coefficient for the valuation effect is a bit higher under OLS, while the coefficient for the expected excess return variable is a bit lower. This is intuitive. The error term is positively correlated with the valuation effect, as an exogenous financial flow toward country |$n$| raises the country |$n$| equity price. This leads to upward bias of the coefficient for |${\it val}_{i,n,t}$|⁠. At the same time, the higher price lowers the dividend-price and earnings-price ratios, which lowers the expected excess return. The negative correlation between the expected excess return and the error term therefore leads to a downward bias of the coefficient for the expected excess return variable under OLS.

Nonetheless, both the OLS and IV results yield a similar message. The weight on the lagged portfolio in (18) is highly significant and large, respectively, 0.93 under OLS and 0.948 under IV. The coefficient for the valuation effect is positive and also highly significant. For the regression (14) in column 3 this means a substantial weight on both the lagged portfolio and the buy-and-hold portfolio. It appears therefore that both portfolio frictions are important, though the weight on the lagged portfolio is more than twice as big as on the buy-and-hold portfolio.²⁵

The coefficient for the expected excess return is also highly significant, 8.2 under OLS and 9.4 under IV. The standard error is substantially larger under IV, but the t-value is still a respectable 3.4 (vs. 10 under OLS). The IV coefficient for the expected excess return variable is similar to findings by Giglio et al. (2021), even though their data are completely different. They regress the equity share of Vanguard investors on 1-year expected excess returns on equity. The cross-sectional variation of expected excess returns, obtained from survey data, is key to their results. They implicitly assume that the portfolio depends on expected excess returns over the next 12 months, with equal weight on each month. In our expected excess return variable, most of the weight is also on the first 12 months (71|$\%$| under OLS and 63|$\%$| under IV).

For comparability, we multiply the coefficient for the expected excess return in Giglio et al. (2021) by 12 to translate to monthly expected excess returns. Their coefficient is then 8.3. It is 13.9 when they remove some outliers from their data. These numbers are broadly consistent with our estimates. They emphasize that this weight on the expected excess return is substantially lower than what one might expect in a frictionless model.

In the last two columns of Table 4, we control for risk. The covariance (24) is conditional on information by the fund at time |$t$|⁠. We do not know this information set. We will use two measures of |${\it risk}_{i,n,t+1}$|⁠. The first is the unconditional covariance based on daily returns during month |$t+1$|⁠. This is appropriate when funds have so much information that at time |$t$| they know the ex-post covariance measured at |$t+1$|⁠. The second measure regresses |${\it risk}_{i,n,t+1}$| on |${\it risk}_{i,n,t}$| as well as the two variables we removed because of their significance in the risk regression, |$i_{i,n,t}$| and |$d_{i,n,t}$|⁠. These time |$t$| variables are in the information set of funds. We write the risk conditional on these three variables as |$\widehat{\it risk}_{i,n,t+1}$|⁠. Table 4 shows that both |$\widehat{\it risk}_{i,n,t+1}$| and |${\it risk}_{i,n,t+1}$| are significant, with the expected negative sign. Controlling for risk has virtually no effect on the other estimated coefficients. This is not surprising as the instruments that we use have no predictive power for risk.

3.5 Retrieving the structural parameters

In connecting the estimates in Table 4 to the structural parameters from the theory, we will assume a time discount rate of |$\beta=0.97$|⁠. The time discount rate is not identified by the reduced-form parameter estimates. While |$\beta=0.97$| may seem low with monthly data, it is important to keep in mind that the average turnover of portfolio managers is 2|$\%$| per month (see Kostovetsky and Warner 2015). An even lower |$\beta$| may need to be assumed if we take into account that many funds have short lives. In the Internet Appendix, we consider alternative values for |$\beta$|⁠.

In the theory, |$\delta$| is equal to |$\beta a_{1}$|⁠, where |$a_{1}$| is the coefficient for |$z_{i,n,t-1}$| in (18). The estimate of |$a_{1}$| is virtually unaffected by the assumed |$\delta$|⁠. In the regression, we first set |$\delta=0.9$|⁠, then estimate (18) to obtain an estimate of |$a_{1}$| and therefore |$\delta$|⁠. We then estimate (18) again when |${\it ER}_{i,n,t}$| is computed with this estimate of |$\delta$|⁠.

Next, we use Equations (19), (20), and (21) to obtain point estimates and confidence intervals for the structural parameters |$\lambda_{1}$|⁠, |$\lambda_{2}$| and |$\gamma$| from the point estimates and variance matrix of |$a_{1}$|⁠, |$a_{2}$| and |$a_{3}$|⁠. We set |$\sigma^{2}=0.00172$|⁠, which is the mean variance of the excess return across |$(i,n)$|⁠. Table 5 reports results based on the OLS and IV estimates of Table 4 (columns 1 and 2). It reports both the point estimates for the structural parameters and 95|$\%$| confidence intervals.

A first point to note is that the confidence intervals for |$\lambda_{1}$|⁠, |$\lambda_{2}$|⁠, as well as |$\lambda_{1}-\lambda_{2}$|⁠, are much tighter based on the OLS than IV estimates. This is because in (19) and (20) the parameters |$\lambda_{1}$| and |$\lambda_{2}$| depend inversely on |$a_{3}$|⁠, the coefficient for the expected excess return. We can see from Table 4 that while the magnitudes of the expected excess return coefficients for OLS and IV do not differ a lot, the standard error is much smaller for OLS, leading to substantially tighter estimates of the structural parameters.

Several points are immediate from Table 5. First, for both OLS and IV we see that |$\lambda_{1}$| and |$\lambda_{2}$| are positive and significant. Therefore, strong evidence indicates that both portfolio frictions are important. Second, |$\lambda_{1}-\lambda_{2}$| is positive. It is significant under IV, which means that there is a larger cost of deviating from the lagged portfolio than the buy-and-hold portfolio. This relates to the substantially higher coefficient for the lagged portfolio than the buy-and-hold portfolio in column 3 of Table 4.

Finally, the point estimate of |$\gamma$| is 3.2 under IV, with a 95|$\%$| confidence interval of [2.4, 5.0]. This is quite reasonable. By contrast, if we just regress on the 1-month expected excess return (plus the fund-country fixed effect), as would be appropriate in the absence of portfolio frictions, we obtain a coefficient of 4.2 (SE = 0.25) with IV. Since the coefficient for the 1-month expected excess return in the frictionless model is |$1/(\gamma \sigma^{2})$|⁠, it would imply |$\gamma=138$|⁠, which is clearly excessive.²⁶ For a more reasonable level of risk aversion, the coefficient for the expected excess return would be far higher in the frictionless model. Therefore, the estimates of |$\lambda_{1}$|⁠, |$\lambda_{2}$| and |$\gamma$| all provide evidence of the importance of portfolio frictions.

3.6 Portfolio dynamics

It is useful to consider the implication of the results above for the dynamic response of portfolios to an expected excess return innovation and compare the case with the estimated portfolio frictions to the frictionless case. For the case with frictions, the expected excess return variable is |${\it ER}_{i,n,t}$|⁠. Using the pooled data, we estimate an AR(1) process |${\it ER}_{i,n,t}=a_{i,n}+\rho {\it ER}_{i,n,t-1}+\nu_{i,n,t}$|⁠, where |$a_{i,n}$| is a fund-country dummy. We obtain an AR coefficient |$\rho=0.87$| and an average standard deviation of |$\nu_{i,n,t}$| across |$(i,n)$| pairs of 0.00076. In the frictionless case the expected excess return is |$E_{t} er_{i,n,t+1}$|⁠, for which we analogously obtain an AR coefficient of 0.51 and average standard deviation 0.0017 of the expected excess return innovation.

We make two additional assumptions. First, for the purpose of this exercise we only include the lagged portfolio share in the regression in order to abstract from valuation effects in the buy-and-hold portfolio. The coefficient for the lagged portfolio share is then 0.88, and the coefficient for the expected excess return variable is 19.6. Second, we need to make an assumption about the portfolio response in the frictionless case. We cannot use the estimated response when regressing |$z_{i,n,t}$| on |$E_{t} er_{i,n,t+1}$| as that is based on data that provide strong evidence of portfolio frictions.

As shown in (16), in the frictionless case the coefficient for the expected excess return is equal to |$1/(\gamma \sigma^{2})$|⁠. We again set |$\sigma^{2}=0.00172$| and assume a rate of risk aversion of |$\gamma=10$|⁠. We can scale the portfolio response in the frictionless case up or down by lowering or raising, respectively the rate of relative-risk aversion.

Figure 1 shows the results. The initial portfolio response to a one-standard-deviation expected excess return innovation is much larger in the frictionless case. If we set the risk aversion equal to the |$\gamma=3.2$| implied by estimates for the model with frictions, the initial response in the frictionless case would be even much higher by a factor of three. Apart from the initial portfolio inertia with the estimated frictions, we also see significant portfolio persistence. The portfolio response peaks after 9 months, while in the frictionless case it peaks at the time of the shock and dies out quickly.

Figure 1

Impulse response portfolio share to expected return shock

Figure 1 shows the impulse response of portfolio shares to a one-standard-deviation shock in expected excess return innovation. The solid black line corresponds to the frictionless case. The dashed gray line corresponds to the case with estimated portfolio frictions.

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Bacchetta and van Wincoop (2021) refer to the initial portfolio response as return sensitivity and the gradual portfolio response as portfolio persistence. They show in a model for the foreign exchange market with portfolio frictions that both diminished return sensitivity and increased portfolio persistence are key to accounting for a variety of currency excess return predictability puzzles.

3.7 Robustness analysis

We now discuss extensions and robustness analysis. We first consider results when using subsets of the eight instruments. After that we consider heterogeneity of the portfolio response to the expected excess return variable |${\it ER}_{i,n,t}$| associated with the size of the average portfolio share |$\bar{z}_{i,n}$|⁠. Then we discuss alternative regression specifications as well as alternative subsamples.

With three endogenous regressors, we need at least three instruments for the portfolio regressions. Within our set of eight instruments, we identify four instruments that account for most of the explanatory power of the endogenous regressors. We will refer to these as the four strong instruments. We refer to the other four instruments as weak instruments:

While the weak instruments have statistically significant explanatory power for the endogenous regressors, their joint explanatory power is significantly weaker than for the four strong instruments. This can be seen in Table 6. When regressing |${\it val}_{i,n,t}$| and |${\it ER}_{i,n,t}$| on the four strong instruments, the adjusted |$R^{2}$| is .07 and .27, respectively. By contrast, when regressing on the four weak instruments the adjusted |$R^{2}$| is .01 and .06, respectively.

This significantly weaker explanatory power affects the portfolio regression. Table 7 reports the portfolio regression with all eight instruments (benchmark), the four strong instruments and the four weak instruments. The results with the four strong instruments are not significantly different from those with all eight instruments. But when we use the four weak instruments, the coefficient for |${\it ER}_{i,n,t}$| is close to zero and insignificant. The coefficient for |${\it val}_{i,n,t}$| more than triples. We find that the four strong instruments are the minimum set of instruments that we need. Adding further instruments from the set of four weak instruments has little effect on the portfolio regressions. Reducing the set of instruments to three, the minimum needed to conduct the portfolio regressions, gives insufficient independent predictive power for the three endogenous regressors, making the portfolio regressions unreliable.

We further investigate the portfolio response to the expected excess return variable by considering heterogeneity. So far we have focused on the average coefficients |$a_{1}$|⁠, |$a_{2}$| and |$a_{3}$| that describe the relationship between |$z_{i,n,t}$| and |$z_{i,n,t-1}$|⁠, |${\it val}_{i,n,t}$| and |${\it ER}_{i,n,t}$|⁠, respectively. But these coefficients will in general vary across funds and countries. Here, we consider one dimension of this heterogeneity, associated with the average portfolio share |$\bar{z}_{i,n}$| of a fund in a country. These mean portfolio shares vary substantially, as shown in the Internet Appendix. The 10th, 50th, and 90th percentiles are 2.7|$\%$|⁠, 6|$\%$|⁠, and 20.4|$\%$|⁠.

The theory in Section 2 implies that both |$\bar{z}_{i,n}$| and the coefficient for |${\it ER}_{i,n,t}$| will be higher when |$\sigma_{i,n}^{2}$| is lower. This is the perceived risk of the excess return of country |$n$| by fund |$i$|⁠. The coefficients for the lagged portfolio and the buy-and-hold portfolio (or the lagged portfolio and the valuation effect) are unaffected by |$\sigma_{i,n}^{2}$|⁠. This suggests that when the portfolio share |$\bar{z}_{i,n}$| is larger, due to lower perceived risk, the portfolio response to the expected excess return |${\it ER}_{i,n,t}$| will be larger as well.

This is tested in column 4 of Table 7. Apart from the three standard regressors |$z_{i,n,t-1}$|⁠, |${\it val}_{i,n,t}$|⁠, and |${\it ER}_{i,n,t}$|⁠, we add an interaction |$\bar{z}_{i,n} {\it ER}_{i,n,t}$|⁠. We then need to double the number of instruments as we need to add interactions of the instruments with |$\bar{z}_{i,n}$|⁠. To avoid using an excessively large number of instruments (16 when we use the benchmark instruments), we only use the four strong instruments discussed above. This gives us eight instruments after including the interactions with |$\bar{z}_{i,n}$|⁠. Table 7 shows that the coefficient of the interaction term is 95.6 and is significant. This confirms that the portfolio response to |${\it ER}_{i,n,t}$| is larger when the portfolio share |$\bar{z}_{i,n}$| is larger. Specifically, the coefficient for |${\it ER}_{i,n,t}$| varies from 6.4 for small |$\bar{z}_{i,n}$| (10th percentile) to 9.5 for the median |$\bar{z}_{i,n}$| and 23.3 for large |$\bar{z}_{i,n}$| (90th percentile). The benchmark regression result in column 1 of Table 7 is consistent with the median fund.

The Internet Appendix describes additional robustness analysis for regression (18), comparable to column 2 of Table 4, both for alternative regression specifications and alternative samples. We consider seven alternative regression specifications. The first uses country-time fixed effects instead of the instruments to deal with endogeneity. This is intended to sweep up common portfolio shifts in and out of each country in the error term. The next two consider |$k=12$| and |$k=36$|⁠, respectively, to compute |${\it ER}_{i,n,t}$| in (23). Then we assume alternative values of |$\beta$| of 0.96 and 0.98, respectively. Next, we consider weights |$z_{i,m,-n,t-1}$| to compute the reference portfolio instead of sample average weights |$\bar{z}_{i,m,-n}$|⁠. We finally report a nonrecursive regression, which uses data over the entire sample to compute expected excess returns as opposed to true forecasts using recursive regressions up to the time of the forecast.

We also consider three alternative samples. The first has a start date of January 2012 as fewer funds report country allocations at the beginning of our sample. Next, we restrict the sample to funds that report their global equity allocation for at least 24 consecutive months (as opposed to 12). Finally, we consider a sample where |$\bar{k}_{i,n}$| is at least 1|$\%$|⁠, as opposed to 2|$\%$|⁠, which leads to the inclusion of smaller portfolio shares.

In almost all of these regressions the results remain quite similar to the benchmark results. The largest changes occur when we use country-time fixed effects instead of the instruments and when we start the sample in January 2012. In both cases the coefficient for the expected excess return |${\it ER}_{i,n,t}$| doubles. When starting the sample in 2012, the coefficient for |${\it val}_{i,n,t}$| drops to 0.18 and is only significant at the 10|$\%$| level. In all other regressions, the coefficients remain significant at the 1|$\%$| level.

The Internet Appendix describes two further types of robustness analysis. One type includes additional instruments: the consumer confidence index, the business confidence index, or leading indicators. These data are not available for all countries and reduce our sample by about 20|$\%$|⁠. When adding these to the benchmark eight instruments, one at a time, the results remain virtually unchanged. We finally report the results for a log-portfolio regression, where we regress |$\ln(z_{i,n,t})$| on |$\ln(z_{i,n,t-1})$|⁠, |${\it val}_{i,n,t}$| and |${\it ER}_{i,n,t}$|⁠. While the theory implies that we should estimate the portfolio expression in levels, in practice portfolio shares are closer to being lognormal. The coefficients for all three regressors remain highly significant.²⁷

4. Conclusion

The objective of the paper was to provide evidence on international portfolio choice and specifically the role of portfolio frictions. We developed a simple optimal portfolio expression that relates portfolio choice to the present discounted value of expected excess returns and two benchmark portfolios, the lagged portfolio share, and the buy-and-hold portfolio. We estimated the reduced-form parameters of the portfolio expression with international equity portfolio data from U.S. mutual funds, using instrumental variables to address endogeneity. We find that portfolio shares of U.S. mutual funds significantly depend on both benchmark portfolios, with coefficients that are quite precise.

We also find that international equity return differentials are predictable and that mutual fund portfolios respond to expected excess returns. The results are consistent with a reasonable rate of risk aversion of 3.2. While the responsiveness to the present value of expected excess returns is strongly statistically significant, we also find that quantitatively the portfolio response to expected returns is much smaller than it would be in a frictionless portfolio model. Portfolio frictions make the response to changes in expected returns smaller initially and more gradual.

There is a clear need to introduce these portfolio frictions into open economy models, as recently done by Bacchetta and van Wincoop (2021), Bacchetta, Davenport, and van Wincoop (2022), and Bacchetta, van Wincoop, and Young (2023) for the foreign currency market, global financial markets broadly, and the global equity market, respectively. It has significant implications for the response of asset prices, capital flows, saving and investment to shocks. A weaker and more gradual portfolio response to expected returns implies more excess return predictability in both the foreign exchange market and global equity markets. It also implies a much larger impact of exogenous portfolio shocks, including also central bank asset purchases, on asset prices and capital flows. Gabaix and Maggiori (2015) and Itskhoki and Mukhin (2021) have recently emphasized the importance of such financial shocks for exchange rates and capital flows, which they point out is consistent with a variety of evidence.

Appendix A: First-Order Condition Optimal Portfolio

After working (7) into (3), we maximize with respect to |$z_{i,n,t}$|⁠. To do so, use that for a function |$f({\bf z}_{i,t})$| we have

The first-order condition with respect to |$z_{i,n,t}$| is then

The last line uses that |$z_{i,n,t+1}^{bh}$| can be written as |$z_{i,n,t}$| plus a time |$t+1$| valuation effect (see (17)).

Using that |${\bf z}_{i,t}={\bf z}_{i,-n,t}+\left({\bf e}_{n}-{\bf z}_{i,-n,t}\right) z_{i,n,t}$|⁠, the first line can be written as

In what follows we will think of moments involving the reference portfolio as evaluated at mean portfolios |$\bar{z}_{i,m,-n}$| as in the data such covariances are virtually identical whether evaluated at portfolios |$z_{i,m,-n,s}$| for |$s=t-1,t,t+1$| or |$\bar{z}_{i,m,-n}$|⁠. The same applies to moments with the excess return, which depend on the reference portfolio.

Next, take the first term of the second line of (A2), substituting |${\bf z}_{i,t}={\bf z}_{i,-n,t}+\left({\bf e}_{n}-{\bf z}_{i,-n,t}\right) z_{i,n,t}$| and |${\bf z}_{i,t-1}={\bf z}_{i,-n,t-1}+\left({\bf e}_{n}-{\bf z}_{i,-n,t-1}\right) z_{i,n,t-1}$|⁠. We can then write it as

In analogy, the second term of the second line of (A2) can be written as

The second line of (A2) then becomes

Approximating the buy-and-hold portfolio as |$z^{bh}_{i,n,t}-z_{i,n,t-1}=z_{i,n,t-1} \left(R_{n,t}-R_{t}^{i,p}\right)$|⁠, with |$R_{t}^{i,p}=\sum_{m=1}^{N} z_{i,m,t-1} R_{m,t}$|⁠, we can write the last line of (A2) as

The first line is the same as the second line of (A2), with |$\mu_{1,i}$| replaced by |$\mu_{2,i}$|⁠. It can therefore be written as

Take the second line of (A7). This can be written as

One can think of the summation in the last line as a cross-sectional covariance |$E(xy)$|⁠, with |$x=cov_{i,t}(er_{i,n,t+1},R_{m,t+1})$| and |$y=(R_{m,t}-R^{i,p}_{t})$| and |$z_{i,m,-n,t-1}$| the probability. Since there is no reason the |$x$| and |$y$| would be correlated, when the number of countries is large enough, we can write this as |$E(x)E(y)$|⁠. (A9) then becomes

Using that |$(1-z_{i,n,t-1})(R_{ref(i,n),t}-R^{i,p}_{t})=z_{i,n,t-1}(R^{i,p}_{t}-R_{n,t})$|⁠, and that |$cov_{i,t}(er_{i,n,t+1},R_{n,t+1})=var_{i,t}(er_{i,n,t+1})+cov_{i,t}(er_{i,n,t+1},R_{ref(i,n),t+1})$|⁠, this becomes

Analogously, the last line of (A7) is

We can write the difference between the buy-and-hold portfolio and lagged portfolio as

Combining (A11) and (A12), we then have

We can now combine all the terms from (A2), which gives us

Define |$\sigma_{i,n}^{2}$| as the mean of |$var_{i,t}(er_{i,n,t+1})$| and |$\sigma_{n,ref(i,n)}$| as the mean value of |$cov_{i,t}(er_{i,n,t+1},R_{ref(i,n),t+1})$|⁠. The mean of the excess return is zero. From (A15) the steady state portfolio is then

Linearizing (A15) around the second moments equal to their mean, the portfolio shares equal to |$\bar{z}_{i,n}$| and the excess returns equal to zero, we have

where |$\hat{z}_{i,n,t}=z_{i,n,t}-\bar{z}_{i,n}$| and

Appendix B: Solution Optimal Portfolio

We now solve the second-order difference Equation (A17) in the portfolio share |$\hat{z}_{i,n,t}$|⁠. Collecting terms, we can write (A17) as

where |$D_{i}=\gamma_{i} +\theta_{i} (1+\beta)$|⁠.

This can be written as

where |$L^{-2} z_{i,n,t-1}=E_{i,t} z_{i,n,t+1}$| and |$L^{-1} z_{i,n,t-1}=z_{i,n,t}$|⁠. Factoring gives

where |$\omega_{1,i}$| and |$\omega_{2,i}$| are the roots of the characteristic equation

These roots are

For convenience, we will refer to the stable root (with the minus sign) simply as |$\omega_{i}$| and the unstable root (with the positive sign) as |$\omega_{2,i}$|⁠.

Now write the solution as

This implies

To summarize, we have

where |$E_{t}$| is the expectation operator of the econometrician based on public information and

and

The second term in |$a_{3,i,n}$| is very close to zero. We therefore abstract from it in what follows.

We can also write the solution for |$z_{i,n,t}$| as a function of the lagged portfolio and the buy-and-hold portfolio. For this, use that from linearizing (A13) |$z_{i,n,t}^{bh}=z_{i,n,t-1}+\bar{z}_{i,n}(1-\bar{z}_{i,n}) er_{i,n,t}$|⁠, so that

We then have

where |$\hat{z}_{i,n,t}^{bh}=z_{i,n,t}^{bh}-\bar{z}_{i,n}$|⁠. Use that |$\omega_{2,i}=1/(\beta \omega_{i})=1/(\beta a_{2,i})$|⁠. This gives us

with

Using the expression for |$D_{i}$|⁠, we can also write the stable root (B3) as

Appendix C: Trading Strategies

To evaluate the prediction performance and estimate the economic significance of predictability reported in Section 3, we follow the literature in building trading strategies based on the three predictors used in the regressions. The analysis is close to that of Cenedese et al. (2016). For each month, we sort countries into quintiles based on their values of momentum, dividend-price differential, or earning-price differential. The one fifth of countries whose predictors have the lowest value are allocated to the first quintile Q1, the next fifth to the second quintile Q2, and so on. Thus, Q1 should contain low excess returns and Q5 high excess returns. For each pair month-quintile, we take the equally weighted average equity return differential with the United States. Then, for each predictor variable we form a long-short HML portfolio, obtained by going long on Q5 and short on Q1. The sample is January 1970 to February 2019.

Table C1 reports the average annualized equity return by quintile and the portfolio return when the predictor is momentum, the dividend-price ratio, or the earning-price ratio (it is also possible to build strategies based on a combination of the three variables). The table shows that returns tend to be higher for higher quintiles; that is, higher values of momentum, dividend-price, or earning-price are associated with higher returns. This is confirmed by the results in the last column that show large returns from HML portfolios. These results are in line with those of Cenedese et al. (2016), who use a more restricted sample. These results therefore demonstrate the economic significance of equity return predictability, which justifies that time-varying expected excess returns are taken into account in actual portfolio allocations.

Appendix D: Data Appendix

Here, we describe the data used (other than the portfolio data from EPFR that are described in detail in Section 4.1).

We obtain the following monthly MSCI data: monthly total return index, price index, earning-price ratio, dividend-price ratio, and market value (market capitalization). The total return index includes both the capital gains and dividend component of the return. All data are denominated in dollars. From these MSCI data we also compute

• Equity return: relative change of the total return index from the prior month.

• Earnings: earning-price ratio multiplied by the price index.

• Dividend: dividend-price ratio multiplied by the price index.

• Book value: market value divided by the price index.

• Volatility: for each country and each month, we compute the standard deviation of the daily returns, using the daily total return index from MSCI.

In addition to these MSCI data, we obtain the following variables from other sources:

• Industrial production. The main source is the industrial production index from IFS. If not available, we use the manufacturing or the retail index from the IFS. When countries do not report the industrial production, the manufacturing nor the retail index, we use the monthly gross domestic product index obtained from the Leading Indicators of the OECD. Finally, for Hong-Kong and Thailand, we use quarterly real GDP data from OECD, interpolated to a monthly series. We transform the final series for each country into an index equal to 100 in July 2016.

• Nominal interest rate. We use the 3-month Eurorates obtained from Datastream. The data are midpoint of the offer and bid rates. Original data are expressed at annual rates as percentages. We transform the data into a monthly rate by dividing by 1,200.

• Bond price index. We obtain the data on bond price index from J.P. Morgan and Merrill Lynch obtained through Datastream. We use the price index of a 10-year government bond provided by JPM. For emerging economies, when the price index of the 10-year government bond is not available, we use the Emerging Market Bond Index provided by JPM. For Taiwan and Thailand, we use the Government Bond Index provided by Merrill Lynch. When the bond price index is in local currency, we convert to dollars.

Acknowledgement

We would like to thank Margaret Davenport, Amit Goyal, and Julien Pénasse, seminar participants at the universities of Bocconi, Wisconsin, and Zurich and at the European Finance Association in Barcelona, and two referees for useful comments. We gratefully acknowledge financial support from the Swiss National Science Foundation and the Bankard Fund for Political Economy. Supplementary data can be found on The Review of Financial Studies web site.

Footnotes

References