The three models of exchange rate determination
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Published: Mon, 5 Dec 2016
This paper presents three models of exchange rate determination. Each models are based on the equilibrium of markets in the international economy. The equilibrium of goods market determine exchange rate according to purchasing power parity; the equilibrium of money market determine exchange rate according to monetary model; the equilibrium of asset markets determine exchange rate according to portfolio model.
It is in the interest of a variety of parties to understand the determinants of exchange rates. For economists, it is for their intellectual and academic pursuit to uncover the economic mechanism determining exchange rates. Policymakers would like to understand the impacts and consequences of exchange rates to the policies and vice versa. Finance managers would like analyze the fundamental factors determining exchange rates and incorporate these factors in their financial or investment decision making. Speculators in foreign exchange market would like to know the direction of exchange rate movement aforehand to make profit. In the following, we explain three models of exchange rate determination, namely, the purchasing power parity(PPP), the monetary model and the portfolio balance theory.
Purchasing Power Parity
The theoretical assumption of Purchasing Power Parity starts from the Law of One Price. The Law of One Price in open economy states that, if the market is competitive, no transaction cost and no barriers of trade, then identical products in different countries should be sold at the same prices, adjusted by exchange rate, i.e. under the same currency denomination. Otherwise, there is arbitrage opportunity. In notation,
pi =spi* (1)
for pi = price of good i at home country, pi*= price of good i at foreign country, s = exchange rate
For example, the price an ounce of gold quoted at London in GBP should be the same as an ounce of gold quoted at New York in USD times exchange rate of GBP/USD.
Next, we consider a model with two countries. Both of them have the floating exchange rate-regimes and Law of One Price holds for all goods in the two counties. Then, the general price level of home country is should be the same as the general price level of foreign country, adjusted by exchange rate. In notation,
for P= general price level at home country, P*= general price level at foreign country
P and P*, the general price level is the weighted average of all prices of goods. So if (1) holds for all goods, (2) will holds. (2) is what we called the absolute Purchasing Power Parity (absolute PPP): the general price level of every country should be the same if adjusted to the same currency. In other words, the exchange rate should be determined by the relative price level of two countries. If you can use $1 of home currency to buy a basket of goods at home country, then the $1 converted to foreign currency should be able to buy the same basket of products in foreign country, i.e. they have the same purchasing power.
We can interpret that PPP is a long-run equilibrium level of exchange rate that there is fundemental force of demand and supply in goods market to retain it. For example, assume that the domestic price level is higher than the foreign price level under the same currency measure, i.e. P > sP*. If goods are identical and there is transaction cost and barriers of trade, then consumers from domestic country will not buy local products. They will use their domestic currency to exchange to foreign currency to buy foreign products, which is cheaper. The force of supply and demand of currency will drives down exchange rate to depreciate. In turn, depreciation of exchange rate will lower the price of domestic products(under the same currency measure) and then the PPP equilibrium, P = sP* is retained.
Yet the absolute PPP to be too strict, economists considers a weaker form, called the relative PPP. It states that percentage changes in price levels of two countries determine the percentage change in exchange rate. In notation,
Î”P/P = Î”s/s +Î”P*/P* (3)
The relative PPP is a weaker form of absolute PPP because if absolute PPP holds true, the relative PPP holds true also but not vice versa. Moreover, change in price level is indeed the inflation rate. The relative PPP implies that exchange rate should be adjustedÎ”e/e to the difference between two countries’ inflation rates. For example, a country with hyperinflation should encounter substantial depreciation in its currency.
The Purchasing Power Parity states that relative price level is a fundamental determinant of exchange rate. An empirical test would like to see whether there is such a relationship in historical data. The PPP hypothesis has be enormously and extensively tested empirically by economists. The extensive tests by economists found very little empirical support to PPP. Exchange rate and the relative price level are unrelated in short run and medium run. In the long run, results found that exchange rate would converge to the theoretical equilibrium value from PPP, but at a very slow rate.
At the first glace, PPP seems to be a too strict hypothesis that it’s assumption is unlikely to hold. In reality, there is transaction cost and barriers of trade. The general price levels indeed include non-tradable goods and different countries have different components in their general price level. These deviations of the theoretical PPP will cause the domestic price level and foreign price level not converges, but retain at some deviated level.
Officer (1982) contains a detailed summary on the theoretical and empirical works on PPP at early stage. Rogoff (1996) provides a more update survey on PPP and their empirical tests. Taylor & Taylor (2004) uses more complete data and more powerful econometric tests, as they describe, retain similarly result as previous scholars.
As exchange rate is the relative price of two currencies, it is reasonable to consider the supply and demand of money be an important determinant of exchange rates. Introduction of money supply and money demand, two very fundamental macroeconomic variables, into our models
The monetary approach rests on the quantity theory of money in macroeconomics. Firstly, Money supply (Ms) is a quantity determined by the central bank. In the quantity theory, money is for the purpose of medium of exchange. Money demand of an economy is directly proportional to the general price level and also the quantity of real output. For example, if the general price level is doubled, then the economy would need double amount of money for their transactions. The same idea holds for quantity of real output. Then,
Md = kPy (4)
Where Md is money demand, P is the price level, y is the real output and k is the velocity of money. In equilibrium, Money supply must be equal money demand, and so:
Ms = kPy (5)
By rearranging, we have
P= Ms/ky (6)
By this form, we can interpret that given a level of real output of the economy and a given level of money supply determined by the central bank, the price level of the economy will be adjusted to Ms/ky.
Let * denotes the foreign currency variables. We assume the quantity theory of money holds true to foreign country also. We have
Ms*= k*P*y* (7)
The second important assumption of the monetary approach is that PPP holds true. The exchange rate always attains its PPP equilibrium level, as in (2).
In the monetary approach, we have three relationships of variables now: the quantity money of home country, quantity money of foreign country, and PPP. Combining there three relationships and rearranging the three equations, we have:
Ms/ ky = S Ms*/ k*y* (8)
The quantity theory of money and PPP are two building blocks of the monetary approach. The PPP tells us that at the long run equilibrium, the exchange rate should be equal to the ratio of home and foreign price level. The quantity theory of money marcoeconomics describes that price level of a country is related to money supply of central bank and real output of the economy. Combining them, the monetary approach concluded that exchange is determined by domestic and foreign money supply (Ms & Ms*), domestic and foreign real output (y & y*), and domestic and foreign velocity of money(k & k*).
An important implication of the monetary approach is that central bank’s money supply policy would have primary impact to exchange rate.
Start with the domestic central bank suddenly increase the money supply by a substantial amount, with all other domestic and foreign variables keep unchanged. The quantity theory of money implies that the rise of money supply without increase in real output will drives up the domestic price level, which means inflation also. The increase in domestic price level will induce domestic people to buy more foreign products and cause the exchange rate to depreciate. This is the same equilibrating mechanism described in PPP.
We may consider the magnitude of depreciation of currency by increase of domestic money supply. According to equation (x), exchange rate, s, is directly proportional to Ms. So in the monetary approach, a given percentage increase in money supply will leads to the same percentage of depreciation of currency.
A natural consequence of the above analysis is to see if foreign money supply would leads to what kind change of exchange rate. From equation (x), we can see that foreign money supply Ms* comes into determining the exchange rate. If the foreign central bank increase money supply, the foreign currency would depreciate as by our previous analysis. Then, in turn, the domestic currency would appreciate relatively.
On the other hand, we may consider the effect of an increase in real output on exchange rate in the monetary approach. Given a fixed level of money supply, real output increase will leads to lowering price level, as described in the quantity theory of money. Then, on the open economy side, the exchange rate must appreciate, making the local products more expensive, to preserve the PPP equilibrium. So we can conclude that a rise in real output(GDP) will leads to appreciation of the domestic currency, given other thing else constant.
The monetary approach is largely based on PPP. Given the failure of PPP on empirical testing, it is not difficult to imagine that empirical test on the monetary model of exchange rates should found little support. Extensive tests have been carried out to examine the relationship between exchange rate vs. money supply and exchange rate vs. real output. As representative, Frenkel (1976) and Meese & Rogoff (1983) shows little empirical support on the Monetary approach.
Johnson (1977) portrays a model treatment of the monetary model of exchange rates. Frenkel (1976) and Meese & Rogoff (1983) are representative empirical works on the monetary approach.
Portfolio Balance Model
In the monetary model, the global economy is simplified as having goods and money only, and money is the medium of exchange to buy domestic and foreign goods. Exchange rates are determined by the relative demand and supply of money, domestic and foreign.
The portfolio balance model takes a further step from the monetary model that there are investment assets in the global economy for people to hold. People would consider holding money, domestic assets and foreign assets alternatively on their portfolio balance. Then the relative demand and supply of these investment assets would determine the exchange rate.
The portfolio balance model assumes there are three kinds of assets for people to allocate their total wealth: Domestic money (M), domestic bond (B), and foreign bond (FB). Domestic money (M), pays no interest, is a riskless asset. In term of finance, the risk-free rate is zero in this simplified model. Domestic bond and foreign bond are risky assets that payout with, with interest rate rand r* respectively. Then the actual interest rate individual receive from foreign bond is sr*.
The portfolio balance model of exchange rate makes further assumption in line with modern portfolio theory. Domestic bond and foreign bond are not perfect substitutes. Holding domestic and foreign bond together in the portfolio would reduce the unsystematic risk. So people would not simply hold the bond with higher yield only, but hold a portfolio of domestic and foreign bonds. Moreover, the individuals, being are risk-averse and so they would hold some portion of riskless asset, the money.
The individuals have a total wealth of W would decide how to allocate them into money, domestic bond and foreign bond respectively based on his risk preference and the returns of different assets, as in modern portfolio theory. He would purchase more of one asset if the return of the asset increase, or if the return of the alternative assets decrease. In summary,
Demand of money = M(r, sr*) is decreasing in r and sr*
Demand of domestic bond = B(r, sr*) is increasing in r and decreasing in sr*.
Demand of foreign bond = FB(r, sr*) is increasing in sr* and decreasing in r.
Total wealth, the supply of various assets, would equal to the demand of various assets., such that
W = M(r, sr*) + B(r, sr*) + BF(r, sr*) (9)
It means that, in equilibrium, there would be some equilibrium value of r, r* and s to balance demand and supply.
To focus on the role of exchange rate in this model, we may consider r and r* as given to be stable by the bond markets and only the exchange rate varies. The equation above can be simplified as:
W = M(s) + B(s) + BF(s) (10)
Then, there will be a value of s to equalize the demand of various assets to total wealth. In other words, the exchange rate is determined by the equilibrium across the money, domestic bond and foreign bond markets in this portfolio balance model.
Implications and evidence of portfolio balance model
One of the most important implications from the portfolio balance model is that current account surplus will be associated with depreciation of currency. Current account surplus must be associated with capital account deficit, which means that the country is a net purchaser of foreign assets. The demand of foreign bond increase and so exchange rate would depreciate for the equilibrium in asset markets to restore.
However, as noted by Copeland (2008), the tests of portfolio balance model, is far from satisfactory.
Several articles by Branson propelled the portfolio balance model, and include empirical evidence also. Branson (1983) provides a good account of summary.
We have reviewed three different models on exchange rates. The PPP, the most fundamental one, claims that price level is the fundamental determinant of exchange rates in the long run. The market force of goods arbitrage would push the exchange rate to the equilibrium level that balance the purchasing power of the different currency to the same level. The monetary model incorporates the classical quantity theory of money in marcoeconomics with purchasing power parity. It predicts that money supply, determined by the central bank, and real output are the determinants of exchange rate. The third theory, the portfolio balance model extends the monetary model from considering the money market to the markets of a number of assets. Individuals demand each type of assets and exchange rate is determined as the equilibrium price of various asset markets.
All of the models we discussed are laid on fundamental economic theory and are conceptually sound. Unfortunately, economists found little direct empirical support to these models.
We should not consider rejecting these three models because of the lack of empirical support. Firstly, these three models are conceptually fundamental and shape our thinking in exchange rates. They will be extremely useful when we extend our analysis with specifications in further detail and seek more specific implications in exchange rate. Secondly, these models portray the long-run equilibrium behavior of the exchange market. It is difficult to consider the volatile, second-to-second changing exchange rate market behavior would be consistent with these models. There may exists random shocks to the exchange rate market that consistently propel the exchange rate to move in a random style and so the long-run equilibrium of the models cannot be attained.
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