A Decomposition of International Capital Flows

Abstract

We propose a method to break down capital flows into portfolio growth and portfolio reallocation components and apply it to data on US equity and bond outflows. The decomposition is part of an integrated approach that decomposes purchases of any asset into portfolio growth and reallocation components. US equity and bond outflows depend not just on portfolio growth and the reallocation between US and foreign equity and bonds, but also on reallocation decisions higher up on the decision tree. This includes reallocation between portfolio and non-portfolio assets and between equity and bonds. We also consider the decomposition of US equity and bond outflows to individual foreign countries. The data shed light on the importance of the various components in accounting for capital flows over both the short and long run.

Appendices

Appendix A: Equity Outflow Decomposition

In this Appendix, we derive the decompositions (15) and (18) of equity outflows. First consider the decomposition in Sect. 2.2. Differentiating (14), we have

$$\begin{aligned} Q_{t}^{e,F} \Delta X_{t+1}^{e,F}+X_{t}^{e,F} \Delta Q_{t+1}^{e,F}= k_{t}^{e,F} \Delta A_{t+1}+\left( \frac{\Delta k_{t+1}^{p}}{k_{t}^{p}}+\frac{\Delta k_{t+1}^{e|p}}{k_{t}^{e|p}}+\frac{\Delta k_{t+1}^{F|e}}{k_{t}^{F|e}}\right) k_{t}^{e,F} A_{t} \end{aligned}$$

(A.1)

Using (3) we have

$$\begin{aligned} \Delta A_{t+1}=(1-k_{t}^{p}) \left( \frac{\Delta Q_{t+1}^{np}}{Q_{t}^{np}}-\frac{\Delta Q_{t+1}^{p}}{Q_{t}^{p}} \right) A_{t}+ \frac{\Delta Q_{t+1}^{p}}{Q_{t}^{p}} A_{t} +S_{t+1} \end{aligned}$$

(A.2)

Using (5)–(7), this can be written as

$$\begin{aligned} \Delta A_{t+1}= & {} (1-k_{t}^{p}) \left( \frac{\Delta Q_{t+1}^{np}}{Q_{t}^{np}}-\frac{\Delta Q_{t+1}^{p}}{Q_{t}^{p}} \right) A_{t}+ (1-k_{t}^{e|p}) \left( \frac{\Delta Q_{t+1}^{b}}{Q_{t}^{b}} -\frac{\Delta Q_{t+1}^{e}}{Q_{t}^{e}} \right) A_{t} \nonumber \\&+\, (1- k_{t}^{F|e}) \left( \frac{\Delta Q_{t+1}^{e,H}}{Q_{t}^{e,H}}- \frac{\Delta Q_{t+1}^{e,F}}{Q_{t}^{e,F}} \right) A_{t}+ \frac{\Delta Q_{t+1}^{e,F}}{Q_{t}^{e,F}}+S_{t+1} \end{aligned}$$

(A.3)

We can use the definition of changes in passive portfolio shares to write this as

$$\begin{aligned} \Delta A_{t+1}=-\frac{\Delta {\tilde{k}}_{t+1}^{p}}{k_{t}^{p}} A_{t}-\frac{\Delta {\tilde{k}}_{t+1}^{e|p}}{k_{t}^{e|p}} A_{t}- \frac{\Delta {\tilde{k}}_{t+1}^{F|e}}{k_{t}^{F|e}}A_{t}+ \frac{\Delta Q_{t+1}^{e,F}}{Q_{t}^{e,F}}A_{t}+S_{t+1} \end{aligned}$$

(A.4)

Substituting this into (A.1), using that $X_{t}^{e,F} \Delta Q_{t+1}^{e,F}=k_{t}^{e,F} A_{t} \Delta Q_{t+1}^{e,F}/Q_{t}^{e,F}$, we obtain the decomposition of equity outflows $Q_{t}^{e,F} \Delta X_{t+1}^{e,F}$ in (15).

We can similarly derive the decomposition of equity outflows to individual countries. Differentiating (17), we have

$$\begin{aligned}&Q_{t}^{e,F,n} \Delta X_{t+1}^{e,F,n}+X_{t}^{e,F,n} \Delta Q_{t+1}^{e,F,n}= k_{t}^{e,F,n} \Delta A_{t+1}\nonumber \\&\quad + \left( \frac{\Delta k_{t+1}^{p}}{k_{t}^{p}}+\frac{\Delta k_{t+1}^{e|p}}{k_{t}^{e|p}}+\frac{\Delta k_{t+1}^{F|e}}{k_{t}^{F|e}}+\frac{\Delta k_{t+1}^{n|e,F}}{k_{t}^{n|e,F}} \right) k_{t}^{e,F,n} A_{t} \end{aligned}$$

(A.5)

We can write $X_{t}^{e,F,n} \Delta Q_{t+1}^{e,F,n}=k_{t}^{e,F,n} A_{t} \Delta Q_{t+1}^{e,F,n}/Q_{t}^{e,F,n}$. From (A.4) we have

$$\begin{aligned} k_{t}^{e,F,n} \Delta A_{t+1}-k_{t}^{e,F,n} A_{t} \frac{\Delta Q_{t+1}^{e,F,n}}{Q_{t}^{e,F,n}}= & {} -\frac{\Delta {\tilde{k}}_{t+1}^{p}}{k_{t}^{p}}k_{t}^{e,F,n} A_{t}-\frac{\Delta {\tilde{k}}_{t+1}^{e|p}}{k_{t}^{e|p}} k_{t}^{e,F,n}A_{t}\nonumber \\&-\, \frac{\Delta {\tilde{k}}_{t+1}^{F|e}}{k_{t}^{F|e}}k_{t}^{e,F,n}A_{t}- \frac{\Delta {\tilde{k}}_{t+1}^{n|e,F}}{k_{t}^{n|e,F}}k_{t}^{e,F,n}A_{t}+k_{t}^{e,F,n} S_{t+1} \end{aligned}$$

(A.6)

Substituting this into (A.5) gives the decomposition of equity outflows $Q_{t}^{e,F,n} \Delta X_{t+1}^{e,F,n}$ to country n shown in (18).

Appendix B: Portfolio Reallocation and Portfolio Rebalancing

As pointed out in Sect. 2.4, the relationship between portfolio rebalancing and portfolio reallocation depends on the nature of the shocks and parameters. A couple of examples will help illustrate the point. First consider an increase in the relative supply of country n equity. This will lower its relative price. There will then be a demand shift toward country n for two reasons: The lower price raises its expected return and the lower price creates a desire to rebalance the portfolio toward country n. Which of these dominate depends on how sensitive portfolios are to expected returns, which depends for example on risk aversion. If portfolios are not very sensitive to expected returns, almost all of the reallocation is associated with portfolio rebalancing. If portfolios are very sensitive to expected returns, equilibrium prices change very little and very little of the portfolio reallocation is associated with rebalancing.

A different type of shock is one where asset supplies do not change, but instead there is a portfolio demand shift toward country n by all investors, for example as a result of reduced risk of country n equity. Since asset supplies do not change, in equilibrium portfolio demand will not change either. There will be no portfolio reallocation toward or away from country n. But this is because rebalancing away from country n (due to a higher equity price) is offset by a higher equilibrium portfolio share invested in country n (supply has not changed and the price has increased).

As a final example, consider a portfolio shift toward foreign equity and away from domestic equity by investors in all countries. For each country, foreigners want to hold more of its equity and domestic investors less. If total equity demand (by domestic and foreign investors) does not change in any country, equity prices do not change either. In this case there is positive reallocation toward foreign equity, but no portfolio rebalancing as equity prices do not change.

The reported results on the relationship between portfolio reallocation and rebalancing vary by type of reallocation as well. Portfolio–non-portfolio reallocation is more closely associated with rebalancing than is the case for the other reallocation components. When two assets are not close substitutes, investors are less willing to change their allocation between the assets. A change in relative asset supplies will then lead to greater rebalancing. Portfolio rebalancing plays a less important role when agents for various reasons wish to change their allocation across the assets a lot. Since portfolio and non-portfolio assets are very different in nature (non-portfolio assets include for example real estate and bank deposits), they are not close substitutes, explaining the larger importance of rebalancing.