Determinants of Foreign Portfolio Investment: Another Approach

John Doukas Concordia University*

This paper develops a relatively simple model of net foreign investment in the context of a two-country world where markets are not necessarily integrated. The model builds on the assumption of the stochastic nature of exchange rates and inflation rates, in the absence of symmetric initial country-endowment restrictions. The analysis is conducted on the basis of a multi-period continuous time framework allowing for changes in the investment opportunity set over time. It then proceeds to analyze the factors determining the demand for foreign and domestic risky assets. Finally, conditions for positive net foreign risky investment are derived and dis- cussed.

Introduction Since the pioneering work of Grubel (1968) on international investment, numerous researchers have tried to explain the determinants of inter- national investment. Lee (1969) used a mean-variance framework to ex- plain the well-known “perverse” movement of capital from a high-interest country to a low-interest country. T h e central implication of Lee’s study as well as of Miller and Whitman’s (1970) study among others, was that diversification desired by investors causes such international capital flows. Ragazzi (1973) and Black (1974) addressed the issue qualitatively and demonstrated that distribution of investment opportunities, distribution of wealth across countries, taxes and threat of expropriation also affect foreign investment decisions. Kecently, Stulz (1 983b), approached the topic of international investment by using the continuous time inter-tem- poral framework. He develops a simple model that links net foreign in- vestment to expected returns and variances of the returns in foreign and domestic technology. Assuming a non-stochastic exchange rate environ- ment in a two-country world, he shows that: (1) an increase in expected return on the foreign technology decreases net domestic investment by foreign investors; (2) an increase in foreign wealth along with an equiv- alent reduction in domestic Ivealth has the opposite effect; ( 3 ) an increase

in the variance of returns on foreign technology decreases net investment domestically by foreigners; and (4) a change in the risk tolerance coeffi- cient of the domestic investor has no effect on the net investment by foreign investors.

Stulz’s (1983b) paper makes a number of unrealistic assumptions as Macedo (1983) rightly pointed out. First, the role of exchange rates is eliminated and his model allows for the existence of an international riskless asset. Second, the model builds on the assumption that both coun- tries are identical in terms of initial endowment. This assumption is sim- plistic and excessive as Macedo (1983) shows that even in the absence of initial symmetry the results obtained by Stulz hold.

The purpose of this paper is to analyse the determinants ofthe portfolio demand for domestic and foreign risky assets. Unlike the Stulz model this paper allows exchange rates to vary stochastically. Further, it allows for differing inflation rates in each country. The presence of stochastic ex- change rates rules out the possibility of a riskless international asset. Therefore, this paper is concerned with the net rzsky investment in each country. More specifically, the following questions are addressed:

1 What are the factors that determine the proportion an investor will invest in risky assets in his own country and the proportion he will invest in a foreign country?

2 What determines the net foreign risky investment in a given country and how does this change when some of the underlying factors change?

The term net foreign risky investment will be used in the context of portfolio investments by individual investors and not in the context of direct investment. As in the previous studies, we assume a two-country world where markets in these countries are not necessarily integrated. Studies reported in Aliber (1977) indicate that world markets are not necessarily integrated. The model developed in this paper allows for mar- ket segmentation. Exchange rates and inflation rates are assumed to be uncorrelated. This is because studies by Kravis and Lipsey (1978), and Frenkel (1981) among others on the purchasing power parity condition show that although it may hold in the long run it does not hold in the short run. Furthermore, this study allows changeover time in investors’ opportunity set, defined in terms of state variables which determine the properties of the technologies, the stock of goods, and domestic or foreign wealth.

The following section develops the basic model giving the risk-return trade-off for domestic and foreign assets. Then the factors determining the demand for foreign and domestic risky assets are investigated. Con- ditions for positive net foreign risky investment in a given country are also derived. Then the factors affecting the net foreign portfolio risky investments are discussed. Finally results are summarized and the paper concluded.

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The Risk-Return Trade-off: A Continuous Time Model The analysis of this study is based on a multi-period continuous time framework where every investor is assumed to optimize his consumption and investment decisions in an infinitesimal planning horizon. The anal- ysis permits continuous changes in the investment opportunity set over time. In addition, all assets are assumed to be infinitely divisible and continuously traded in time. Typically, portfolio investments are more sensitive to short-run economic changes because they are of shorter du- ration compared to direct investments which tend to be of large duration. Hence the continuous time methodology seems to be the appropriate approach in analyzing portfolio investments. In addition, the advantage ofthe continuous time framework is that it makes fewer and less restrictive assumptions about investors’ risk preferences. The continuous time inter- temporal framework assumes only “strict concavity” of the utility function and not the restrictive quadratic utility function underlying the conven- tional mean-variance analysis. There are no transaction costs or taxes. There exists a risk-free asset in each country whose returns are uncor- related with those of all their assets. Investors are risk-averse and have a strictly concave utility function of wealth and lifetime consumption. Fur- ther, we deal with the more complex but the realistic situation in which investors in a given country are price takers and are allowed to have homogeneous consumption preferences, but possess differing consump- tion preferences when considered across countries. Finally, the paper uses techniques developed in the finance literature which build on Merton’s (197 I ; 1973) work and the equilibrium concept developed in Lucas (1978), Cox, Ingersoll, and Ross (1978), and Breeden (1979).

The model developed below includes exchange rates and inflation rates which change over time. Exchange rate and inflation dynamic affect the demands for risky assets, and, therefore, affect net foreign investment. To simplify the analysis but without loss of generality, it is assumed that there are only two countries denoted by 1 and 2. Therefore, the logarithm of asset prices, the percentage change in exchange rates, and the inflation rate in each country follow stochastic differential equations of the It8 type written asy

(1) R,,n = pJ,dt + UjidZ,,

x. = ~ m d t + UxidZx,

n , = Pnidt + uTidZmi

fo r i = 1, 2

(2)

(3) where Rii is the nominal rate of return on the risky asset j in country i, kji is the drift of the process or the instantaneous expected nominal rate of return for risky asset j in country i and uii is the diffusion of the process per unit of time or the instantaneous standard deviation of the nominal returns on asset j in country i. Xi stands for the percentage change in the

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instantaneous exchange rate between countries 1 and 2 from country i's standpoint; IJ.,, is the drift of the exchange rate process; and uy, is the diffusion of the exchange rate process per unit of time. n, represents the instantaneous rate of uncertain inflation in country i; p,, is the drift of the uncertain inflation process in country i; and uTl is the diffusion of the uncertain inflation rate in country i. dZ,,, dZ,, and dZ,, are the white noise terms in a standard Wiener process so that

E(dZ,,, dZ,,,) = 0

E(dZ,,, dZ,,) = 0

E(dZ,, dZ,,) = 0

The expression E(dZ,, dZ,,) = 0 for all i reflects the assumption that the changes in exchange rates and uncertain inflation are uncorrelated con- temporaneously. Although this may seem a strong assumption, empirical evidence indicates that the correlation between the two is at best very weak in the short run (e.g., see Frenkel (1981)).

Let R, denote the nominal instantaneous rate of return on the riskless asset in country i. An investor is assumed to borrow or lend at this risk- free rate only in his country. An investment in the risk-free asset of a foreign country would still be considered risky due ta the exchange rate risk. Let D,, denote the proportion of wealth of the investor k in country i that is invested in the domestic risky assets and let F,, denote the pro- portion of wealth of the investor k in country i that is invested in the foreign risky assets. Also let $',1(q21r) denote the proportion of Dkl(DkP) invested in asset j of country 1(2), and $'J2(q21,) represent the proportion of Fkl(FkP) invested in asset j of country 2(1). Finally, assume that there are n, assets in country 1 and n2 assets in country 2.

The wealth dynamics of investor k in country 1 [wk](t + dt)] can be expressed as

for X, # X,

for T, # TT~

for all i

(RIP - nl - X,)dt (1 + II,dt)(l + X,dt) + FkI 5 $'j2 (

j = 1

where Wk1.t is the wealth of investor k in country 1 in period t, and ckl is the investor k's instantaneous consumption rate in country 1 in the period t+dt , and Wkl,t(t + dt) represents the terminal wealth of investor k. The second term within the parentheses represents the real return for in- vesting in the domestic risky assets. The third term is the real return on the foreign investment adjusted for the changes in the exchange rate.

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l’he final term gives the real return on the domestic risk-free asset. As mentioned previously, the real returns on the risk-free asset are not free of risk due to the stochastic nature of inflation. To simplify the notation Jve drop the subscripts k and 1 that describe the investor k in country 1. Hence, the investor’s objective is to maximize the objective function

where U(C,) is a strictly concave Von Newmann-Morgenstern utility func- tion of consumption C,; J ( ) is a strictly concave utility function of terminal Jvealth, .r is the investor’s time horizon and E is the conditional expectation operator. ’The optimization problem in (5) is solved by using the Bellniari principle of dynamic programming applied to the continuous time framework. Investor’s optimization problem then can be restated in ternis of finding the optimum value function J,(W,, t, SJ, where S suffi- ciently describes the investor’s opportunity set, such that

-1’he opportunity set may change over time and the changes in the op- portunity set are assumed to follow an It6 process as shown in (7).

d S S - = E,dt + u,dZ, ( 7 ) Jvhere E, and u, are the instantaneous expected values and standard de- viation of the changes in the opportunity set and dZ, is the random term as in a standard Wiener process. The investor’s objective is to maximize the expected utility of consumption in the period t + dt and also maximize util i ty of next period Jvealth for future consumption, taking into consid- eration the various changes including those in the opportunity set.

Using a result by Merton (1971), the necessary optimality conditions for investor’s consumption-investment decision at time t can be written a S

here the subscript on J denotes the partial derivatives. Using the first-order conditions of the investor’s optimization problem,

t.\e get’ ” I

F ~ I - R ~ I - Cov ( R ~ I , n ~ ) = X I {DI C $ ’ k l COv ( R ~ I , R k l ) k = I

P I X I Wl,

+ - COV (R,,, ds) (9)

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Define

J w w W i t - _ - U"(dC/dW)Wl, Al = -

where A, is the Arrow-Pratt measure of relative risk aversion

J w u c

JIW - u c c . ac - h P 1 - Jw Uc Wit

where P I represents the ratio ac/as / dC/aW in which the numerator (dC/dS) represents the sensitivity of consumption to changes in the op- portunity set, and the denominator (aC/aW) the marginal propensity to consume out of wealth. Alternatively, PI can be viewed as a measure of sensitivity of wealth to changes in the opportunity set. Equations (9) and (10) yield the first significant result of this paper. They represent the risk- return trade-offs for the domestic and foreign risky assets respectively, as perceived by an individual investor in country 1. While these equations do not represent equilibrium valuation relationships for those assets, they can be used to derive the individual demand functions for domestic and foreign risky investments.

Demand for Domestic and Foreign Investment This section is concerned with the derivation of the determinants of the demand for domestic and foreign risky assets. Multiplying (9) by JrIl and aggregating over all the assets (i = 1, n) and then multiplying the same by each investor's wealth relative 8,, and aggregating over all investors in that country (l), we obtain

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where p,l,I is the expected return on the market portfolio of country 1, p,,). is the expected return in the market portfolio of country 2, Cov(R,,, HI) is the covariance between the domestic market portfolio and the do- mestic inflation rate; and Var(R,,) is the variance of the market portfolio of country 1. The remaining covariance and variance terms have similar interpretations.

is a weighted average relative risk aversion measure for country 1's inves- tors, where K, is the number of investors in that country;

K ' O k l A k l D k l DT = 2 k = l AT

is a weighted average of the demand for domestic risk assets by domestic investors;

" 0 A F FT = c k l k l k 1

k = l A:

is a weighted average of the demand for foreign risky assets by domestic investors; and

is a weighted average of the compensation for bearing a unit of risk attributable to the change in the opportunity set. Note that the subscript 1 in the above variables indicates that these are specific to investors of country 1 only.

Solving for DT and FS f-rom the system (1 1) and (12), we have

PT Wl

- Cov (Rm2, XI) + Var(X,)] - ~ C o v (Rm2 - X I , ds)

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DT and FT represent the aggregate demand for domestic and foreign risky assets as a proportion of country 1’s wealth. Apparently, the sign of DT and F: depends on the sign of the numerator since the Z, is always n~n-negative.~ The demand for domestic and foreign risky assets derived for the domestic investor yields the second important result of this paper.

Proposition 1. An increase in the expected return on the domestic (for- eign) market wml ( p,,) increases the proportion invested domestically (abroad).

The proof is straightforward. This result obtains since the aggregate demand for domestic (foreign) risky assets is a positive (negative) function of T~ and negative (positive) function of El which, in turn, are positive functions of the expected domestic and foreign market rate of return, respectively.

Proposition 2. An increase in the domestic risk free rate, ceteris paribus, will cause both foreign and domestic investments (FT, DT) to decrease.

To illustrate this, assume that Var(R,, - XI) Var(R,,).5 Hence, the [Var(R,, - X,)] Var(Rm,)] will exceed Cov(R,,, R,, - X,), since the correlation between (R,,,, - XI) and Rml is less than perfect. If an increase in Rfl does not affect the market returns then T , and 5, will decrease by identical amounts. Since the variances are greater than the covariances the net effect would be a decline in both DT and FT.

To understand equations (13) and (14), T~ and 5, can be interpreted as the investment factors induced by markets 1 and 2, respectively. The domestic investment factor (7,) is a function of (1) the expected domestic market return, (2) the domestic risk free rate, (3) the weighted average relative risk aversion AT, (4) the covariance between the domestic market return (R,J and the domestic inflation rate [COV(R,,~, T,)], (5) the co- variance between RmI and the dynamic change in the opportunity set [Cov(R,,, ds)], and (6) the consumption related variable PT. Any change in these variables that increases (decreases) T , will result in an in- crease(decrease) in domestic investment (DT). For instance, if the risk due to changes in the opportunity set in the domestic country [Cov(R,,, ds)] increases, the T~ decreases. This implies a decrease in domestic investment and an increase in foreign investment. However, if the Cov(R,,,,, ds) de- clines, there will be a rise in domestic investment and a fall in foreign investment. Similarly, the foreign market investment factor ( E l ) is affected by factors such as p,,, Rrl, AT, Cov(R,,, IIJ, Cov(R,,, XI), Var(X,), PT, and Cov(R,, - XI, ds). Alterations in these factors that affect the foreign market investment factor will cause changes in domestic and foreign in- vestment. For instance, if the variance of the percentage change in the exchange rate [Var(X,)] increases, these will cause an increase in E l . Con- sequently, foreign investment (FT) increases while domestic investment

27 1

(Uy) decreases. This result may seem counter-intuitive at first glance al- though it is really not. Consider the U.S. and France for instance. If the variance of the U.S. dollar decreased with respect to the French franc then we nmild say that the dollar is more stable than before. A stable dollar would induce French citizens to invest more in dollars than in French francs. This would increase the investment in US. and decrease the in- vestment in France.

Corollary 1. I f the risks induced by the domestic and foreign market are the same (7, - t ,), then the proportion invested in each market would depend on the exchange-rate-adjusted volatility of each market. In gen- eral, if the domestic market is less volatile than the exchange-rate-adjusted foreign market [Var(K,,,,) < Var(R,,,, - X,)], the greater will be the pro- portion invested domestically. Also, dDT/dkn,, > 0 and dFT/al~’~,,~ > 0. That is, an increase in expected return in a given market will cause a greater proportion to be invested in that market. However, the signs of dDT/dk,, and d F ~ / d l ~ ’ , , , ~ are not clear since they depend on the covariance and the foreign market return adjusted for the percentage change in the exchange rate, Cov(K,,,,, K,,,, - X,).”

Corollary 2. Assume that there is no percentage change in the exchange rate (XI = 0). Then, if the covariance between the domestic and foreign market returns, Cov(K,,,, RRII)), is positive, domestic investment (DT) is a decreasing function of the expected foreign market return [(dDTidk,,,) = - Cov(R,,,,, R,,,,)/ATZ,) < 0, X = 01. As the expected foreign market return (p.,,,.) rises (falls), domestic investment declines(increase). Similarily, an increase in the expected domestic market return leads to a decrease in foreign investment [(dF?/dl~’,,,~) < 01. These results explain investment ac- tivity among free industrial nations. When the U.S. market becomes at- tractive ( l ~ ’ , , ~ , increases), the flow of capital from the rest of the industrial community to the C.S. rises.

Finally, it follows from equations (13) and (14), given DT > 0 and FT < 0, that dDT/dAT < 0 and dFT/dAT < 0. That is, an increase in the relative risk aversion of the domestic investors decreases the proportion invested in risky assets both at home and abroad.

Net Foreign Risky Investment Follorving the same procedure for country 2, the first-order conditions can be used to obtain the foreign investor’s investments in the domestic country and abroad:

Var(R,,,, - X,) T? - Cov (Rm2, R,,, - X,)i& z2

D$ =

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B; W;

+ Var(X,)] - - Cov (R,,,, - X,, ds)

The determinants of D; and F; can be interpreted similarly to those of DT and FT. Now, let Wi(i = 1, 2) be the total wealth of country i. The net foreign risky investment in country 1, NF, is

Proposition 3. The necessary condition for the net foreign risky investment in country 1 to be positive is W2F; > W,FT or equivalently, W,/W, > FTIF;.

In words, this condition means that for the net risky investment in a country to be positive, the ratio of the initial wealth of the two countries must exceed the ratio of the proportional foreign investment in those two countries. Assuming that FT > 0 and F; > 0, the ratio FTIF; can be viewed as a measure of the “relative attractiveness” of foreign investment in country 2 relative to that in country 1 . If this ratio is greater than 1 then this would imply that country 2 is more attractive to foreign investments than country 1, and vice versa. If the relative attractiveness ratio is equal to unity, then the net investment in a given country is determined solely by the initial endowments of the two countries. In general, when FT/F; = 1, the country with the lower initial endowment will experience a positive net foreign risky investment.

Corollary 3. Assume that both countries are equally endowed. If the return on the market of country 1 increases then F; will increase. However, FT may increase or decrease, depending on the correlation between the mar- ket returns of the two countries, assuming T,, T ~ , t, and t2 are all positive. If the correlation between the two is negative then FT will increase and hence NFT will decrease. If the correlation is positive then NF* will increase.

Corollary 4. If the two markets are uncorrelated, then NF, will increase because FZ increases. Consequently, the net foreign investment in the

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home country (NFT) depends on the correlation (covariance) between the domestic and foreign market return, as well as factors influencing FT and F$ as indicated above.

Conclusions In this paper, a relatively simple model of net foreign investment was derived. It was shown that the greater the expected return in a given country the greater will be the proportion invested there. Net foreign investment in a country is positive when the initial endowment ratio ex- ceeds the “relative attractiveness ratio” (FT/F;). An increase in foreign wealth accompanied by an equivalent decrease in domestic wealth is shown to increase net foreign investment in the domestic country if and only if the “relative attractiveness ratio” is unity. Factors affecting FT and F: also influence the net foreign risky investment in the domestic country. In addition, it was shown that foreign investment varies inversely with the investors’ risk aversion and the country’s exchange rate sensitivity. Finally, it was demonstrated that a country that offers a better hedge against inflation is expected to be preferred by investors. The magnitude of the variances of RmI, Rm2, X I and ds play an important role in determining the domestic and foreign investments and so do the interaction of indi- vidual effects with one another.

Methodologically, the paper uses the Bellman equation in two different ways. First, the first-order conditions are explored. The advantage of this is that it is not necessary to solve a non-linear partial differential equation, which often has no known solution. Using the Bellman equation for a representative investor makes it possible to solve for the determinants of the demand for domestic and foreign investment. One direction in which the present model would be extended consists in taking into account barriers to international investment such as taxes, transactions costs, and national or political risks.

Appendix

Applying Taylor series expansion to (1 + x, dt)-’ and (1 + II, dt)-l we obtain

( 1 + X I d t ) I 1 - k,, dt - uXl dZ,, + 6, ( 1 ’ )

(1 + n2 dt) ’ = 1 - pnl dt - u,, dZ,, + 6, ( 2 ’ )

where 6, arid 6, are of‘ order (dt)’i’2 and higher which will be ignored in the analysis below.

Using (4), (1;) and (2’) equation (8) can be rewritten as follows: n .

0 = Max {U,(C) + J, + J x [W,{D 2 QJI (kJl dt - R,, dt - Cov (R,,, II,)dt (C DC P W J = I

” 2

+ F (w12 dt - R,,dt - Cov(R,,, II,)dt) - Cov (R,,, XJdt + Var (X,)dt

+ (R,,dt - p,,dt + Var (n,)dt)} - C,dt] + J,dt - J7X, I= I

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the (n, + n2 + 1) necessary first-order conditions can then be written as

U, - J w = 0 (4')

(R,,, Rk2 - XI) - Cov (R,,, II,)} + JSx Cov (R,,, ds) = 0 for j = 1, n, ( 5 ' )

Equation (4') demonstrates the familiar equality between the marginal utility of present and future consumption. The equations ( 5 ' ) and (6') define the risk- return trade-off for the domestic and foreign risky assets from the domestic country's point of view.

Notes I am grateful to an anonymous referee for useful comments.

1 Since investors differ across countries, the distribution of wealth across countries, for instance, can be viewed as a state variable.

2 For an exposition of the methodology used here, and the properties of sto- chastic differential equations, see Merton (1971).

3 See Appendix for derivation of the first-order conditions. 4 Dividing Z, by Var(R,,) Var(R,, - XI), we have 1 - {(Corr(R,,, R,,, -

XI)}* 2 0. Since Corr (.) = 1 would imply DT = * and FT = ? 00 this case is ruled out. Hence Z, > 0.

5 This assumption is being made to facilitate easier understanding. If Var(R,, - X,) f Var (R,J but Var(R,, - X,) > Cov(R,, - X I , R,,,) and Var(R,,,,) Cov(R,, - XI, R,,J the results still hold. However, when Var(R,,,) < Cov(R,, - XI, Rml) then DT will increase with increases in Rf,. Similarly when Var(R,, - XI) < Cov(R,, - XI, Rml) an increase in R,, will increase FT. But these two cases are ruled out, since they are not justifiable by logic.

6 (aDT/a~,z = - [COV (R,i, R,, - XiY(AZi)I = ( d F i / P ~ , i ) 7 If cmrelation or Cov (R,,, R,, - XI) in (14) is positive, an increase in kml

will raise T, and therefore FT will drop.

275

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Resume Ce memoire expose un modPle relativement simple d’investissement etranger net dans le contexte d’un monde oh il n’existe que deux pays et oh les marches ne sont pas necessairement integres. Le modele est construit sur la presomption de la nature stochastique des taux de change et d’inflation, sans qu’il ne soit toutefois impose de restrictions symetriques initiales quant aux ressources des deux pays. L’analyse est menee dans un cadre temporel continu a periodes multiples qui permet au cours d’un certain temps des changements dans les possibilites d’in- vestissement. On analyse ensuite les facteurs determinant la demande d’actif domestique et etranger a risque eleve.

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