An expanded graphical representation of the portfolio balance model of exchange rate determination.

Zietz, Joachim

1. Introduction

The portfolio balance model of exchange rate determination was developed, inter alia, by Allen (1973), Branson (1975), and Allen and Kenen (1976). The model evolved from a general dissatisfaction with the implications of the flow model of exchange rate determination that is at the heart of the well-known Mundell-Fleming model.(1) If one assumes perfect substitutability between domestic and foreign bonds as well as rational expectations, the portfolio balance model reduces to the monetary model of exchange rate determination based on uncovered interest parity (UIP). Although the latter two assumptions simplify the exposition, there is growing empirical evidence that the monetary model based on UIP is not consistent with the data.(2) This leaves the portfolio balance model (PBM) as the major alternative asset market model of short-run exchange rate determination.

The PBM's one-diagram graphical representation (e.g. Cuthbertson and Taylor 1987, MacDonald 1988), whose style resembles that of the IS/LM model, gets high marks for conciseness and efficiency but falls short in providing an intuitive understanding of the forces that drive the model. The purpose of this paper is to offer a somewhat expanded graphical representation of the PBM. It features a diagram for each of the three markets considered by the PBM, domestic bond market, foreign bond market, and money market. Compared to the traditional one-diagram version, the expanded graph adheres more closely to the basic purpose of graphical representations of complex models: to provide intuition rather than add another layer of conciseness that may not be easily understood by the non-specialist.

2. The Model in Equation Form

To appreciate the logical connection between graph and mathematical model, it is useful to write down the PBM's underlying equations. The model consists in its core of three equilibrium conditions for three asset markets, domestic money (M), domestic bonds (B), and foreign bonds held domestically (F),(3)

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where the left-hand sides of equations (1) through (3) represent asset supplies, the right-hand side asset demands, and where signs indicate the signs of partial derivatives. A rise in its own rate raises the stock demand for an asset, an increase in its cross rate lowers it. Money is assumed to have no own rate of return.(4)

Money, domestic bonds, and foreign bonds are subject to an adding up constraint: the three assets sum to private domestic financial wealth (W),

W [is equivalent to] M + B + F/e (4)

By setting demand equal to supply for each asset we assume that all three markets are in stock equilibrium at all times.(5)

The three asset demands have the same form. Demand is a function of the rate of return on domestic bonds (i) and the expected rate of return on foreign bonds (j),(6) which is given as

j = i* - [e.sup.e] - e/e (5)

where i* is the rate of return on foreign bonds in foreign currency and where the term ([e.sup.e] - e)/e represents the expected rate of appreciation of the home currency (US$). The variable [e.sup.e] denotes the expected exchange rate (foreign currency per $US). For given e, a rise in [e.sup.e] signifies an increase in the expected rate of appreciation of the dollar.(7) According to equation (5), such an increase reduces the expected rate of return on foreign bonds, and, hence, makes them less desirable to domestic investors. Wealth is another determinant of asset demand. The variable enters as a scaler determining the size of desired asset holdings without affecting their composition.

The model consists of three endogenous variables, the domestic bond return (i), the exchange rate (e), and wealth (W). All other variables are treated as exogenous, including exchange rate expectations.(8) F changes only as a result of current account surpluses or deficits. Domestic bonds and money are assumed non-tradable. They are held only by domestic residents. The assumed small size of the country, which allows us to treat i* as exogenous, makes them unattractive to foreigners. Since the model's purpose is to identify the short-run determinants of the exchange rate behavior, changes in income and price are ignored for simplicity.

3. Graphical Representation

Figure 1 provides a graphical representation of the model set out in equations (1) to (5). Panel (a) represents the equilibrium condition for the foreign bond market (equation 3) in terms of the variables i and e. This curve also forms part of the received graph of the PBM. Figure 1 deviates from the traditional BPM graph in that the markets for domestic money and bonds enter in their simple demand/supply representations (panels b and c, respectively) rather than as additional market equilibrium curves in panel (a).

To see the economic logic of the positive slope of the foreign bond equilibrium curve in panel (a) it is useful to convert equation (3) from level to share form and to make use of equations (4) and (5)

[Mathematical Expression Omitted]

From equation (6) it is easy to see that a rise in e lowers the left-hand side of (6), that is, the supply of foreign bonds drops relative to domestic financial wealth. The rise in e increases the right-hand side of (6); the desired share of foreign bonds in total financial wealth goes up. Foreign bonds become more attractive because an increase in e lowers, for given [e.sup.e], the expected appreciation of the dollar. To eliminate the excess demand for foreign bonds and restore stock equilibrium, domestic residents have to be induced to lower their desired share of foreign bonds in favor of dollar-denominated bonds. This requires a rise in the domestic bond rate (i). Hence, a rise in e has to be accompanied by an increase in i to maintain equilibrium in the market for foreign bonds. Therefore, the curve in panel (a) of Figure 1 is upward sloping.(9)

The curve representing equilibrium in the foreign bond market shifts as variables other than i and e change. The direction of these shifts can be derived from equation (6) similar to its slope. Consider, for example, an increase in the money supply (M). A rise in M reduces the actual share of foreign bonds in total wealth (left side of equation 6). Without a corresponding decrease in the desired share of foreign bonds in wealth (right side of equation 6), excess demand will develop in the market for foreign bonds. Investors can be induced to lower their desired share of foreign bonds in favor of domestic bonds with an increase in the rate of return on domestic bonds (i). In sum, a rise in M requires i to go up for given e or, as depicted in panel (a) of Figure 1, e to go down for given i. The equilibrium curve for foreign bonds shifts up or to the left as the money supply rises (M [arrow up]). The shifts caused by changes in exogenous variables other than M can be derived in a completely analogous manner. The arrow in panel (a) of Figure 1 illustrates what will happen to the equilibrium curve in each case.

Panels (b) and (c) of Figure 1 assume for simplicity that the supplies of domestic bonds and money are unresponsive to domestic interest rate changes. This allows a clearer focus on the absolute and relative sizes of the slopes of the asset demand curves. They are constrained by the adding-up condition implicit in equation (4). By the adding-up condition, the shares of total wealth held in money (m), domestic bonds (b), and foreign bonds (f) sum to unity for all rates of return i and j,

m(i,j) + b(i,j) + f(i,j) = 1 (7)

For given W, a change in i or j will not increase total asset demand, or more formally,

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where the partial derivative of the desired share of money in the portfolio with respect to i ([m.sub.i]) represents the slope of the money demand curve in panel (b) of Figure 1. The other terms in equations (8) and (9) have to be interpreted equivalently. To see more clearly how the slopes are related, solve equation (8) for [b.sub.i],

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The demand for domestic bonds reacts to a change in its own rate of return as much as the demands for money and foreign bonds taken together. A similar condition holds for the desired share of foreign bonds. A short form of expressing the economic content of equation (10) is to say that "own rate effects" dominate "cross rate effects" for desired changes in asset shares. For panels (b) and (c) of Figure 1, the adding-up condition implies that the slope of the curve for bond demand in panel (c) has to be more responsive to i than the demand curve for money.

4. Some Applications of the Graphical Model

Figure 2 illustrates the impact of an open-market purchase of domestic bonds by the Federal Reserve.(10) To induce investors to swap domestic bonds for money willingly, the Federal Reserve has to offer investors a sufficiently high bond price. An increase in the bond price is equivalent to a drop in the rate of return on domestic bonds. The drop in i increases the relative attractiveness to investors of both non-interest bearing money (panel b) and foreign bonds. Since investors desire to hold more foreign bonds in their portfolio, excess demand develops in the market for foreign bonds. For given levels of [e.sup.e] and i*, stock equilibrium can be restored in the market for foreign bonds only if the dollar depreciates (e [down arrow] in panel a). The dollar depreciation has both a substitution and a wealth effect. The substitution effect induces investors to reduce their stock demand for foreign bonds in favor of domestic bonds and money because the dollar depreciation increases, for given [e.sup.e], the expected rate of appreciation of the dollar. In panels (b) and (c) of Figure 2, this is depicted as a rightward shift of the demand curves for both money and domestic bonds. The wealth effect comes about because the dollar depreciation raises the dollar value of foreign bond holdings (F/e [arrow up]). An increase in wealth, however, raises investors' desired holdings of all three assets. The end result of the expansionary open market operations is a depreciation of the dollar, a decrease in the rate of return on domestic bonds, and an increase in wealth.

To avoid a multitude of shifting curves in Figure 2, each demand curve is moved only once. Corresponding arrows indicate the economic reasoning. For example, the rightward shift in the demand curve for bonds in panel (c) is the result of a wealth effect and an exchange rate effect operating in the same direction.

Next, consider briefly the financing of a government budget deficit by bonds. To induce investors to hold a larger supply of domestic bonds, the own rate of return on domestic bonds has to increase (panel c). This lowers the willingness of investors to hold the two alternative assets, money and foreign bonds. Taken on its own, the substitution effect would create excess supplies in the markets for both money and foreign bonds. However, since wealth has gone up at the same time, excess supplies need not develop. The increase in wealth has made investors willing to hold more of all three assets at the given rate of return on domestic bonds.(11) Consequently, the demand curves for the two assets shift to the right in Figure 3 and the effect of a bond-financed budget deficit on the exchange rate is ambiguous. To simplify the graph, panel (a) of Figure 3 is drawn such that the exchange rate does not change. This assumes that the reduction in the actual share of foreign bonds in wealth (F/e W) is just matched by a reduction in their desired share (f), following the increase in the rate of return on domestic bonds.

Numerous other changes in policy or exogenous variables can be analyzed with the apparatus of Figure 1. Some cases together with their impact on the three endogenous variables i, e, and W are contained in Table 1. The signs have to be interpreted as in the previous equations. A question mark signifies an ambiguous effect. The first two columns, identified as (1) and (2), simply repeat the results obtained from Figures 2 and 3, with [Delta]M = -[Delta]B indicating an expansionary open market purchase of domestic bonds by the Federal reserve, and G - T = [Delta]B the financing of a government budget deficit (G-T) by bonds. Column (3) provides the results for a budget deficit financed by money. Column (4) assumes the Federal Reserve buys foreign bonds to intervene in the foreign currency market and sterilizes the effect of those purchases on the domestic money supply by selling domestic bonds. Column (5) again assumes the Federal Reserve buys foreign bonds. This time, however, there is no sterilization of the resulting increase in the domestic money supply. Column (6) refers to an increase in foreign bonds as resulting from a current account surplus and the last column looks at the effect of an increase in the foreign interest rate.(12)

Notes

1. The problems of the flow model of exchange rate determination were identified early on by McKinnon and Oates (1966).

2. A summary of recent empirical evidence can be found in Boothe and Longworth (1986) and Froot and Frankel (1989).

3. F is denominated in foreign currency. Dividing F by the exchange rate (e [is equivalent to] foreign currency units per US$) converts it into domestic currency.

4. We abstract here from the fact that some monetary assets, such as NOW accounts, do pay interest in the U.S.

5. It is assumed that information and transaction costs are minimal so that one need not worry about the dynamics of stock adjustment. All adjustments take place instantaneously. For empirical applications of the PBM, such an assumption oversimplifies matters; see, for example, Zietz and Weichert (1988).

6. Asset demands are specified here in the simplest possible form. At the expense of more complexity, one could add other determinants of asset demands, such as perceived risk.

7. Depending on the numerical values of e and [e.sup.e], an increase in the term ([e.sup.e] - e)/e could also be interpreted as (i) a reduction in the expected depreciation of the dollar, (ii) an increase in the expected depreciation of the foreign currency, or (iii) a reduction in the expected appreciation of the foreign currency.

8. Expectations play a trivial role in the PBM compared to rational expectations models, where they are generated from within the model.

9. See the appendix for a mathematical derivation of this result.

10. This example also illustrates the point made earlier on the restrictions placed by the adding-up condition on the slopes of asset demand curves. If one makes the slope of the bond demand curve steeper than the one for money, the model will be unstable.

11. One may want to point out in this context that, while the supply of bonds has increased, no other component of wealth has been reduced in size. In particular, the money received by the government in return for its new bonds has not left the private sector but has been spent immediately. Hence, wealth has unambiguously gone up. We ignore here the neo-Ricarian debate over whether government bonds are indeed considered a component of wealth by the private sector.

12. Corresponding graphical illustrations along the lines of Figures 2 and 3 will be provided by the author upon request.

Appendix

To verify the slope and shift parameters of the equilibrium curve in panel (a) of Figure 1, we differentiate the equilibrium condition for the foreign asset market.

[Mathematical Expression Omitted]

totally to find

[f.sub.i]W di + ((1 - f)F/[e.sup.2] + [f.sub.e]W)de = -[f.sub.i*]W di*

-[f.sub.e]e W d[e.sup.e] - fdM - f dB + (1-f)/e dF. (A2)

Setting di* = d[e.sup.e] = dM = dB = dF = 0, one can derive the curve's slope as

di/de = 1/-[f.sub.i]W((1-f)F/[e.sup.2] + [f.sub.e]W) [is greater than] 0 (A3)

Its sign is uniquely determined given the signs of the partial derivatives of equation (A1). Setting, alternatively, di = d[e.sup.e] = dM = dB = dF = 0 in equation (A2), one can identify how the equilibrium curve in panel (a) responds to a change in foreign interest rates (i*).

de/di* = -[f.sub.i]W/((1-f)F/[e.sup.2] + [f.sub.e]W) [is less than] 0 (A4)

that is, the equilibrium curve in panel (a) shifts to the left for an increase in foreign interest rates. Manipulating equation (A2) in a similar fashion for the other shift variables one can derive the comparative statistics indicated by the arrow in panel (a) of Figure 1.

References

Allen, P. R. "A Portfolio Approach to International Capital Flows." Journal of International Economics 3 (May 1973): 135-60.

Allen, P. R., and P. B. Kenen. "Portfolio Adjustment in Open Economies: A Comparison of Alternative Specifications." Weltwirtschaftliches Archiv 112 (1976): 34-71.

Boothe, P., and D. Longworth. "Foreign Exchange Market Efficiency Tests: Implications of Recent Empirical Findings." Journal of International Money and Finance 5 (June 1986): 136-52.

Branson, W. "Stocks and Flows in International Monetary Analysis." In International Aspects of Stabilization Policies, edited by A. Ando et al. Boston: Federal Reserve Bank of Boston, 1975.

Cuthbertson, K., and M. P. Taylor. Macroeconomic Systems. New York: Basil Blackwell, 1987.

Froot, K. A., and J. A. Frankel. "Forward Discount Bias: Is It an Exchange Risk Premium?" Quarterly Journal of Economics 104 (February 1989): 139-161.

MacDonald, R. Floating Exchange Rates: Theories and Evidence. London: Unwin and Hymann, 1988.

McKinnon, R. I., and W. Oates. The Implications of International Economic Integration of Monetary, Fiscal, and Exchange Rate Policy. Princeton Studies in International Finance No. 16, 1966.

Zietz, J. and R. Weichert. "A Dynamic Singular Equation System of Asset Demand." European Economic Review 32 (July 1988): 1349-57.