# Chap. 7 Parity Conditions and Currency Forecasting. The Law of One Price. The Law of One Price and Purchasing Power Parity. The Law of One Price

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3 Forecasting with (Relative) PPP PPP says the currency with the higher inflation rate is expected to depreciate relative to the currency with the lower rate of inflation. A simple (rule-of-thumb) version of the PPP is (e 1 - e 0 )/ e 0 = i h - i f where e 1 = expected future spot rate, e 0 = spot rate, i h = expected home inflation, and i f = expected foreign inflation. Uncertainties are represented by italics. Forecasting with (Relative) PPP (e 1 - e 0 )/ e 0 = i h - i f E.g., if you expect Mexico to have 12% inflation this year and the U.S. only 2%, the difference is what you expect the Peso to depreciate by: 10%. Note that reversing the home currency means that Mexicans would expect the Dollar to appreciate by 10%! Exhibit 7.3 shows a relationship between changes in exchange rates and relative inflation rates. Yen: lowest inflation, held value best. Relative Purchasing Power Parity To understand relative PPP, consider Brazil with 10% inflation and Argentina with no inflation (Argentina uses a currency board). A market basket of goods worth 1,000 Reals today will cost 1,100 Reals next year. In Argentina, a 100 Peso Argentine basket will cost still cost 100 Pesos one year later. Argentines will now find Brazil 10% more expensive and Brazilians will find Argentina 10% less expensive Relative Purchasing Power Parity Assuming that there is free trade between the countries, Argentines will buy less in Brazil, and Brazilians will buy more in Argentina. This means Brazilians will demand more Pesos and Argentines will demand fewer Reals, the exchange equilibrium between the currencies should results in a cheaper Real (or more expensive Peso). Potentially the peso could rise by ten percent (in Brazilian terms) which would leave the relative cost of Brazilian/Argentine goods unchanged. Forecasting with Relative PPP The more accurate version of the PPP for the one-period spot rate is e 1 = e 0 (1 + i h )/(1 + i f ) or the percent change is (e 1 /e 0 ) - 1 = (1 + i h )/(1 + i f ) -1 In the previous example, we would expect the peso to depreciate by (1.02)/(1.12) -1 = -8.93%. The Mexican view: the dollar appreciates by (1.12)/(1.02) -1 = = 9.80%. Forecasting with Relative PPP The general version for t periods is e t = e 0 (1 + i h ) t /(1 + i f ) t or e t /e 0-1 = (1 + i h ) t /(1 + i f ) t -1 In the previous example, if the Peso is worth ten cents today, we would estimate that the peso would be worth \$0.10(1.02)/(1.12) = \$ in one year \$0.10(1.02) 2 /(1.12) 2 = \$ in two years \$0.10(1.02) 3 /(1.12) 3 = \$ in three years 3

4 PPP: Real Exchange Rates PPP predicts that real exchange rates ( e t ) stay the same. (that is, exchange rates adjust to differences in inflation) e t = e t (1 + i h ) t /(1 + i f ) t is constant. This property is usually violated in the short run but appears to hold up in the long run. Look at the 9 graphs in Exh. 7.4: the dark steady lines represent a constant real (inflation-adjusted) exchange rate. The blue, wiggly lines are the actual rates. PPP: Real Exchange Rates Suppose Mexico has 12% inflation, we have 2%. From the previous example, we found that we could expect the Peso to fall from \$0.10 to \$ Plugging in the PPP prediction e t = e t (1 + i f ) t /(1 + i h) t = (1 +.12)/(1 +.02) =.10 If the peso falls more than , we say it had a fall in the real exchange rate, if it stays at 10 cents we say it has risen in real terms. PPP: Real Exchange Rates Ex: Suppose Mexico has 12% inflation, we have 2%. From the previous example, we found that we could expect the Peso to fall from \$0.10 to \$ If the peso stays unchanged at 10 cents in nominal, find the change in real terms. \$( )/\$ =.098..or 9.8% real appreciation Inflation and Real Interest Rates We often make a distinction between the normal everyday contract rate of interest, and the interest rate that is adjusted for any loss of purchasing power caused by inflation. The everyday contract rate is called the nominal interest rate, denoted in the text as r. The rate reduced to reflect loss of purchasing power is the real rate, denoted as a. Irving Fisher popularized the use of the real rate and thus the relationship was named after him. Inflation and Real Rates A simple version of this Fisher Effect is a = r- i, where a is the expected real rate, r is the nominal rate, and i is expected inflation. Under current conditions this would be: the T- bond YTM is 6.5%, inflation is expected to 2%, therefore the expected real rate on the T-bond is 4.5% (awfully high). Note: the real rate is usually the residual of the nominal rate and inflation. Inflation and Real Rates The T-bill rate is more like 5%, so the real T-bill rate is expected to be 3%. Note that each nominal rate has its own corresponding real rate. There is, however, a compounding effect that is reflected in the full, more correct formula: (1+a) = (1 + r)/(1 + i) or (1+r) = (1 + a)(1 + i) = 1 + a + i + a*i r = a + r + a*r where a*i is analogous to compounding between i and a. 4

5 Inflation and Real Rates Example: Suppose we expect Russia to have 100% inflation next year and we want to increase our purchasing power by 5% (real return). What (nominal) rate does a Russian bank need to offer you? Inflation and Real Rates Not just 105% (1+r) = (1 + a)(1 + i) r = (1 + a)(1 + i) -1 r= (1 +.05)( ) - 1 = = 1.10 = 110% Note you need 100% to buy what you bought last year, 5% to buy 5% more goods, and another 5% to pay for the inflation on the extra goods. PPP: Real Interest Rates Suppose Mexico has 12% inflation, we have 2%. From the previous example, we found that we could expect the Peso to fall from \$0.10 to \$ Plugging in the PPP prediction. Suppose we borrow pesos at 16% when the peso was at 10 cents and falls to 8 cents. What is the USD cost of borrowing? In real terms? In nominal terms r usd = (1 + r mex ) (e 1 /(e 0 ) - 1 = 1.16(.08/.10) = = = -7.2% In real terms 0.928/(1.02) - 1 = = -9.02% Inflation and Real Rates The Fisher Effect says that the real rate is somewhat steady over time: at least steadier than the nominal rate. We find that most of the changes in interest rates can be explained by changes in inflation. Exhibit 7.6. Shows the close relation between the level of inflation and the level of nominal rates In the U.S. the Real T-bill rate is usually about 1 or 2 %. Lower values encourage people to borrow, leading to inflation. Higher real rates should encourage saving, and thus reduce inflation. The International Fisher Effect (IFE) A syllogism: 1) FE: differences in nominal interest rates reflect different expectations of inflation, 2) PPP: the inflation rate differential between two countries predicts the future exchange rate 3) IFE: therefore differences in nominal interest rates predict future exchange rate. The International Fisher Effect (IFE) IFE requires both PPP and FE Another way of stating IFE is we assume that PPP holds and that the real rate a is the same in all countries. Thus we have two questionable assumptions: PPP is true and real rates are the same everywhere. 5

6 Forecasting With the IFE The IFE says the currency with the higher interest rate is expected to depreciate relative to the currency with the lower rate of interest. A simple (rule-of-thumb) version of the IFE is (e 1 - e 0 )/ e 0 = r h - r f where e 1 = expected future spot rate, e 0 = spot rate, r h = home interest rate, and r f = foreign interest rate. Uncertainties are represented by italics. Forecasting With the IFE (e 1 - e 0 )/ e 0 = r h - r f E.g., if you see that Mexico has19% interest and the U.S. only 5%, the difference is what you expect the Peso to depreciate by: 14%. Note that reversing the home currency means that Mexicans would expect the Dollar to appreciate by 14%! Forecasting With the IFE The more accurate version of the IFE for the one-period spot rate is e 1 = e 0 (1 + r h )/(1 + r f ) or the percent change is (e 1 /e 0 ) - 1 = (1 + r h )/(1 + r f ) -1 In the previous example, we would expect the peso to depreciate by (1.05)/(1.19) -1 = %. The Mexican view: the dollar appreciates by (1.19)/(1.05) -1 = = 13.33%. PPP and IFE To see why IFE is a combination of Fe and PPP replace (1 + r) below with (1 + a)(1 + i) e 1 = e 0 (1 + r h )/(1 + r f ) e 1 = e 0 (1 + a)(1 + i h )/(1 + a)(1 + i f ) canceling the (1 + a) we get e 1 = e 0 (1 + i h )/(1 + i f ) Forecasting with the IFE The general version for t periods is e t = e 0 (1 + r h ) t /(1 + r f ) t or e t /e 0-1 = (1 + r h ) t /(1 + r f ) t -1 In the previous example, if the Peso was worth ten cents today, we would estimate that the peso would be worth \$0.10(1.05)/(1.19) = \$ in one year \$0.10(1.05) 2 /(1.19) 2 = \$ in two years \$0.10(1.05) 3 /(1.19) 3 = \$ in three years Forecasting with the IFE Note: the future spot rate predicted by IFE is lower than the rate predicted by PPP. This was of course a function of the examples choices of values for r and i. However, we normally find that real rates tend to be higher in LDCs like Mexico and Brazil, where the political risk and the chances of currency controls, poor financial policies is thought to be higher than in developed countries. This means that IFE-based predictions for the Peso and Real will be too low. 6

7 Forecasting with the IFE Another source of violation of the same real rate assumption is monetary policy. If the local central bank is fighting an overheating economy (the U.S), real rates may tend to be too higher than the usual. Rates would be lower if the central bank was dealing with a sluggish economy (Japan and Switzerland). Exhibit 7.8 is demonstrates how the constant a assumption is violated for less stable nations. Forecasting with the Forward Rate The unbiased forward rate: In this theory, the forward rate may not be an accurate predictor of the future spot rate but it is at least unbiased. That is, the average of its mistakes tends toward zero. Stated as f t = E(e t ) The trouble is that we expect the forward rate to be set by IRP, so this relationship is almost impossible to distinguish from IFE. Forecasts based on the Forward or the Future Rates If a currency sells at a forward discount (like Mexico s), then the forward rate is predicting that the Peso will fall. There is a reasonable likelihood that the Peso will fall because (1) the discount means interest rates are higher in Mexico; (2) higher interest rates could mean higher expected inflation, and (3) higher inflation could mean a fall in the Peso. Forecasting with the Forward Rate IFE has a problem with forecasting as real rates tend to be higher for LDCs, and, when a central bank in a developed country is fighting inflation/hot economy. The forward rate is set by these same interest rates so the forward discount for the Peso will tend to overstate the future depreciation. Potential arbitrage? Buying cheap forward Peso and shorting expensive spot peso? Forecasts based on the Forward or the Future Rates Remember that for all financial assets, the forward/futures rate is determined by the spot rate and the differences in interest rates (IRP). The spot rate (like a price on the stock market) already reflects all relevant public information that may affect the future spot price. So why should the forward rate on any financial futures contain any additional information? Forecasts based on the Forward or the Future Rates Commodity futures are a different situation. For some commodities such as gold, that is held for investment purposes, storage costs are very low, and it is not perishable. Perhaps other metals as well: platinum, silver? For these commodities, IRP usually holds. Most other commodities are perishable or expensive to store: IRP still applies (the opportunity cost of capital), but the other factors can dominate. 7

8 Forecasts based on the Forward or the Future Rates If the forward moves up to predict a future gain in the exchange rate, does the spot market eventually follow? That would be a case of the tail wagging the dog. The two markets should move together through IRP. Comparing the 3 parity equations IRP: interest rates, (with the spot rate) set today s forward rate. PPP: inflation rates predict the future spot rate. IFE: interest rates (as proxy for expected inflation) may also be used to predict the future spot rate. You might argue that IFE is a lazy man s PPP, or you might argue that the market, through setting the nominal rate, does a better job of forecasting inflation than any economic model Five Key Relationships Other Currency Forecasting Methods forward discount or premium UBE PPP IRP %e IFE interest rate differential MODEL-BASED FORECASTS: A. Fundamental analysis relies on key macroeconomic variables and policies which most like affect exchange rates. (ex: trade deficits, savings rates, interest rates, government deficits) expected inflation differential FE B.Technical analysis relies on use of 1) historical volume and price data 2) Charting and trend analysis The trend is your friend Problem Sets: Questions: 4, 5 Problems: 3-6, 8-12, 14 (either text) 8

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