Exchange Rates, Interest Rates, Prices and Expectations: A Study of Determination Models, Lecture notes of Japanese

Download Exchange Rates, Interest Rates, Prices and Expectations: A Study of Determination Models and more Lecture notes Japanese in PDF only on Docsity! III.1 CHAPTER III EXCHANGE RATES, INTEREST RATES, PRICES AND EXPECTATIONS This chapter presents simple models of exchange rate determination. These models apply arbitrage arguments in different contexts to obtain equilibrium relations that determine exchange rates. In this chapter, we define arbitrage as the activity that takes advantages of pricing mistakes in financial instruments in one or more markets, facing no risk and using no own capital. The no own capital requirement is usually met by buying and selling (or borrowing and lending) the same or equivalent assets or commodities. The no risk requirement is usually met by doing the buying and selling (or borrowing and lending) simultaneously. Obviously, arbitrageurs will engage in this activity only if it is profitable, which means there should be a pricing mistake. Financial markets are said to be in equilibrium if no arbitrage opportunities exist. The equilibrium relations derived in this chapter are called parity relations. Because of the underlying arbitrage argument, parity relations establish situations where economic agents are indifferent between two financial alternatives. Thus, parity relations provide an “equilibrium” value or a “benchmark.” These benchmarks are very useful. For example, based on a parity benchmark, investors or policy makers can analyze if a foreign currency is “overvalued” or “undervalued.” I. Interest Rate Parity Theorem (IRPT) The IRPT is a fundamental law of international finance. Open the pages of the Wall Street Journal and you will see that Argentine bonds yield 10% and Japanese bonds yield 1%. Why wouldn't capital flow to Argentina from Japan until this differential disappeared? Assuming that there are no government restrictions to the international flow of capital or transaction costs, the barrier that prevents Japanese capital to fly to Argentina is currency risk. Once yens are exchanged for pesos, there is no guarantee that the peso will not depreciate against the yen. There is, however, one way to guarantee a conversion rate between the peso and the yen: a trader can use a forward foreign currency contract. Forward foreign currency contracts eliminate currency risk. A forward foreign currency contract allows a trader to compare domestic returns with foreign returns translated into the domestic currency, without facing currency risk. Arbitrage will ensure that both known returns, expressed in the same currency, are equal. That is, world interest rates are linked together through the currency markets. The IRPT embodies this relation: III.2 If the interest rate on a foreign currency is different from that of the domestic currency, the forward exchange rate will have to trade away from the spot exchange rate by a sufficient amount to make profitable arbitrage impossible. 1.A Covered interest arbitrage Covered interest arbitrage is the activity that forces the IRPT to hold. Assume that there are no barriers to the free movement of capital across international borders –i.e., there is perfect capital mobility. Consider the following notation: id = domestic nominal risk-free interest rate for T days. if = foreign nominal risk-free interest rate for T days. St = time t spot rate (direct quote: units of domestic currency per unit of foreign currency). Ft,T = forward rate for delivery at date T, at time t. Now, consider the following strategy: (1) At time 0, we borrow from a foreign bank one unit of a foreign currency for T days. At time T, we should pay the foreign bank (1+if x T/360) units of the foreign currency. (2) At time 0, we exchange the unit of foreign currency for domestic currency, that is, we get S units of domestic currency. (3) At time 0, we deposit St units of domestic currency in a domestic bank for T days. At time T, we should receive from the domestic bank St(1+id x T/360) units of domestic currency. (4) At time 0, we also enter into a T-day forward contract to buy foreign (sell domestic currency) at a pre-specified exchange rate (Ft,T). At time T, we exchange the St(1+id) units of domestic currency for foreign currency, using the pre-specified exchange rate in the forward contract. That is, we get St(1+id x T/360)/Ft,T units of foreign currency. This strategy will not be profitable if at time T, what we receive in units of foreign currency is equal to what we have to pay in units of foreign currency. Since arbitrageurs will be searching for an opportunity to make a risk-free profit, arbitrage will ensure that St (1 + id * T/360)/Ft,T = (1 + if * T/360). Solving for Ft,T, we obtain the following expression for the IRPT: 𝐹 , 𝑆 ∗ ∗ ∗ (III.1) III.5 The approximation in (III.2) is quite accurate when id and if are small, usually lower than 10%. The above equation gives us a linear approximation to formula (III.1): 𝐹 ,  St * 1 𝑖 𝑖 ∗ The above formulae assume discrete compounding. We can also use the following continuous formulation: Ft,T = St exp[(id - if) * T/360].  IRPT: Remark IRPT is a mathematical relation. You can think of the forward rate as an identity linking interest rate differentials and currency rates. The economic intuition of this mathematical relation is simple: the forward rate is the rate that eliminates an arbitrage profit.  1.A.1 IRPT: Assumptions Behind the covered arbitrage strategy -steps (1) to (4)-, we have implicitly assumed: (1) Funding is available. That is, step (1) can be executed. (2) Free capital mobility. No barriers to international capital flow –i.e., step (2) and later (4) can be implemented. (3) No default/country risk. That is, steps (3) and (4) are safe. (4) Absence of significant frictions. Typical examples: transaction costs & taxes. Small transactions costs are OK, as long as they do not impede arbitrage. We are also implicitly assuming that the forward contract for the desired maturity T is available. This may not be true. In general, the forward market is liquid for short maturities (up to 1 year). For many currencies, say from emerging market, the forward market may be liquid for much shorter maturities (up to 30 days). 1.B The Forward Premium and the IRPT Recall the definition of forward premium, p: p = , * Using IRPT derived above, we get that the forward premium is a function of the interest rate differential. p = , * ∗ . III.6 If the domestic interest rate is higher (lower) than the foreign interest rates, the forward premium is positive and the foreign currency is called a premium currency. We have seen that the difference between the forward and the spot exchange rates is called forward points (sometimes this difference is also called the swap rate). Using the IRPT, T- days forward points are calculated as: 𝐹 , 𝑆 = 𝑆 ∗ ∗ 𝑇360 1 𝑖𝑓 ∗ 𝑇360 1 . That is, the forward points are a function of the interest rate differential. If the domestic interest rate is higher (lower) than the foreign interest rates, the forward points will be added (subtracted) to the spot rate. Example III.3: Using the information from Example III.1 we can calculate the one-year forward points as follows: 150 ∗ . . 1 = - 2.7523 JPY/USD. ¶ Consider (III.2). That is, ,  𝑖 𝑖 ∗ Suppose, now, that we consider a 360-day forward contract (i.e., T=360), then we can approximate the above equation as: p  𝑖 𝑖 That is, covered arbitrage forces the forward premium to be approximately equal to the interest rate differential. In equilibrium, the forward premium exactly compensates the interest rate differential. Under this equilibrium condition, there are no arbitrage opportunities and no capital flows moving from one country to another due to covered arbitrage strategies. Exhibit III.1 presents the relation between the forward premium and interest rate differentials in equilibrium. Exhibit III.1 IRPT Line III.7 id-if IRPT Line If p > id -if, then domestic capital will fly to the foreign economy. That is, what an investor loses on the lower interest rate from the foreign investment is more than compensated by the high forward premium. Therefore, a point like A in the above graph represents a situation where there are capital outflows from the domestic economy. Note that covered arbitrage strategy will affect id, if, St, and Ft,T. The (id-if) will clearly increase: domestic interest rates will tend to increase (higher demand for domestic loans); while foreign interest rates will tend to decrease (higher bank deposits in the foreign country). On the other side, the forward premium will clearly decrease: the exchange rate will tend to increase (higher demand for the foreign currency); while the forward rate will decrease (higher foreign currency forward sales). Thus, arbitrageurs will force the equilibrium back to the IRPT line. On the other hand, if p < id -if, then foreign capital will fly to the domestic economy. That is, what an investor makes on the high interest rate from the domestic investment is more than what the investor gets by investing in the covered foreign investment. That is, a point like B in the above graph represents a situation where the domestic economy experiences capital inflows. Similar to the previous case, covered interest strategies will move the economy from B to the IRPT line. Example III.4: Suppose you are given the following data -taken from Example III.1: St = 150 JPY/USD iJPY,1-yr = 7% iUSD,1-yr = 9% Ft,1-yr = 140 JPY/USD. With this information, we calculate p and the interest rate differential: p = (140 - 150)/150 = -.06667 (p < 0, a discount) iJPY - iUSD = .07 - .09 = -.02. Since p < iJPY - iUSD, we expect foreign capital to fly to Japan (the domestic country) to buy Japanese assets (we are in a point like B, in Exhibit III.1). For instance, U.S. investors will buy Japanese government bonds or bank deposits, which is consistent with the second part of Example III.1. ¶ 1.C IRPT with Bid-Ask Spreads A  B p (forward premium) III.10 Exhibit III.2 Trading bounds for the Forward bid and the Forward ask Fask,t,T Fbid,t,T Lask Ubid Ft,T Example III.5: Suppose we have the following information: St=1.6540-.0080 USD/GBP, iUSD=7¼- ½, iGBP=8 1/8–3/8, and Ft,one-year=1.6400-.0050 USD/GBP. Given these prices, we should check if there is an arbitrage possibility. If a trader borrows one USD, she will repay USD 1.07500. If she buys GBP, deposit them at the GBP rate, and sells GBP forward, she will obtain (1/1.6620) * (1 + .08125) * 1.64 = USD 1.06694. Therefore, there is no arbitrage opportunity. For each USD the trader borrows, she would lose USD .00806. On the other hand, if the trader borrows one GBP, she will repay GBP 1.08375. If she buys USD, deposit them at the USD rate, and buy GBP forward, she will obtain 1.6540 * (1 + .07250) * (1/1.6450) = GBP 1.07837. Again, there is no arbitrage opportunity. That is, the bid-ask forward quote is consistent with no arbitrage. This is due to the fact that the forward quote is within the IRPT bounds. To check this point, we calculate the bounds for the forward rate, Ubid and L ask. Ubid = Sask,t * , , = 1.6620 USD/GBP * . . = 1.6524 USD/GBP  Ubid  Fbid,t,T = 1.6400 USD/GBP. Lask = Sbid,t * , , = 1.6540 USD/GBP * . . = 1.6368 USD/GBP  Lask  Fask,t,T = 1.6450 USD/GBP. ¶ To check your understanding of the IRPT with bid-ask spreads, do exercises III.4 and III.5. 1.D Synthetic Forward Rates A synthetic asset is a combination of different assets that exactly replicates the cash flows of the original asset. We have already used this concept to construct a covered arbitrage opportunity. We have already constructed synthetic forward rates by combining the spot rate III.11 and the domestic and foreign interest rate. If the synthetic forward rate is cheaper than the market forward rate, then there is an arbitrage opportunity. Note, however, that sometimes it is possible to observe a synthetic forward rate less or more expensive than the forward rate, but there are no arbitrage opportunities, because of transaction costs. In this case, a trader would use the less expensive forward rate. Now, it is possible that for some currencies there is no active market for forward exchange rates. For many currencies, this is the usual case, especially for long-term forward contracts. In general, the majority of the governments around the world issue long terms bonds. A trader can use the yields on long term bonds to obtain a forward rate quote. This trader can replicate the forward contract using a spot currency contract combined with borrowing and lending government bonds. This replication is done using equation (III.1). Example III.6: Replicating a 10-year forward bid quote. A trader at Bertoni Bank is unable to obtain a USD/JOD 10-year forward bid quote (JOD = Jordanian Dinar). She decides to replicate a USD/JOD forward contract using 10-year government bond yields and the spot exchange rate. The yield for 10-year government bonds at the bid is 6% in the U.S. and at the ask 8% in Jordan. The ask USD/JOD spot quote is 1.60 USD/JOD. She shorts the domestic (USD) bond, converts the USD into JOD and buys the Jordanian (JOD) bond. Ignoring transaction costs, she creates a 10-year forward bid quote: Fbid,t,10-year = Sbid,t * [ , , , , ]10 = 1.60 USD/JOD * [1.06/1.08]10 = 1.3272 USD/JOD. ¶ Synthetic forward contracts are very useful for exotic currencies. When countries impose borrowing or lending restrictions, it will be difficult for traders to construct synthetic forward contracts. 1.E IRPT: Evidence Testing IRPT is very simple. Recall the relation between the forward premium and the interest rate differential, p  id -if. Then, we can plot the forward premium and the interest rate differential for several currencies in a graph similar to Exhibit III.1. The visual test would accept the IRPT if we observe a 45% degree line in the plot. A more formal test of the IRPT can be designed by using the following regression: p =  +  𝑖 𝑖 ∗ + , where  represents a regression error term. Under the IRPT, the null joint hypothesis is =0 and =1. An F-test can be used to test this joint hypothesis. Starting from Frenkel and Levich (1975), there is a lot of evidence that supports IRPT. As an example, in Figures III.1 we plot the daily interest rate differential against the annualized forward premium. They plot very much along the 45° line. Moreover, the correlation III.12 coefficient between these two series is 0.995, highly correlated series! FIGURE III.1 USD/GBP premia and interest rates differentials (1990-2015) Using intra-daily data (10’ intervals), Taylor (1989) also find strong support for IRPT. At the tick-by-tick data, Akram, Rice and Sarno (2008, 2009) show that there are short-lived (from 30 seconds up to 4 minutes) departures from IRP, with a potential profit range of 0.0002- 0.0006 per unit. There are, however, small deviations from IRP. What is the meaning of these small deviations? Are arbitrageurs not taking advantages of these departures from IRP? The answer to the last question is no. There are several variables that explain departures from IRP. The first reason behind departures from IRP is the time lag that exists between the observation of an arbitrage opportunity and the actual execution of the covered arbitrage strategy. Once an arbitrageur decides to take advantage of the IRPT not holding, the deviation from IRP has disappeared. That is, the prices we use to test the IRPT --p and (id -if)—are, many times, misleading. Arbitrageurs were not able to use those quoted prices. The second reason, and the most obvious, for observing deviations from the IRPT is transaction costs. Arbitrageurs cannot take advantage of violations of the IRPT that are smaller than the transaction costs they need to pay to carry out a covered arbitrage strategy. That is, the existence of transaction costs would allow deviations from IRP equal or smaller than these transaction costs. There are situations, however, where we observe significant and more persistent deviations from the IRPT line. These situations are usually attributed to monetary policy, credit risk, funding conditions, risk aversion of investors, lack of capital mobility, default risk, country risk, and market microstructure effects. Let’s focus on country risk. The forward contract ‐0.10 ‐0.08 ‐0.06 ‐0.04 ‐0.02 0.00 0.02 ‐0.10 ‐0.08 ‐0.06 ‐0.04 ‐0.02 0.00 0.02 i_ U SD ‐i _G B P premium_1Y III.15 baskets of goods to estimate PPP implied exchange rates. One popular price index is based on the PWT (Penn World Tables) data set. The PWT presents price measures for different countries that are based on a common market basket of approximately 150 detailed categories of goods. Another common market basket was popularized by The British magazine The Economist. The Economist uses the Big Mac as a basket of common goods: beef, cheese, onion, lettuce, bread, pickles and special sauce. The Big Mac is sold in 120 countries around the world and it represents a standardized basket of goods. Most of the ingredients that are used in the Big Mac are traded in international markets. The Economist uses the prices of Big Macs around the world to derive PPP implied exchange rates (relative to the USD) and, then, to derive an indicator of undervaluation or overvaluation of a currency. This indicator is usually called the Big Mac Index. Exhibit III.2 shows the Big Mac Index calculated on January 2012 for different countries. Using 2000 data, Pakko and Pollard (2003), two economists from the Federal Reserve Bank of St. Louis, found a .73 correlation between the PPP measured derives from the PWT and the Big Mac Index.. Example III.9: The Economist reports the price of a Big Mac in the Euro-area and in the U.S. as EUR 3.721 and USD 4.93, respectively. That is, Pf = EUR 3.72 (USD 4.00) Pd = USD 4.93 St = 1.0753 USD/EUR. St PPP = PUSA/ PEUR = USD 4.93/ EUR 3.72 = 1.3253 USD/EUR > St = 1.0753 USD/EUR. Taking the Big Mac as our basket, the EUR is undervalued by 18.9% (=1.0753/1.3253). ¶ In theory, traders can exploit the price differentials in Big Macs. In Example III.9, Euro-area traders can export Big Macs to the U.S. But, this scenario is not realistic; a Big Mac sandwich shipped from the Europe to the U.S. would probably not be very appealing. But, since the components of a Big Mac are traded on world markets, the LOOP suggests that prices of the components should be similar in all markets. 2.A.1 Real v. Nominal Exchange Rates The absolute version of the PPP theory is expressed in terms of St, the nominal exchange rate. This is "nominal" because it is expressed in terms of money rather than in units of a real good or consumption basket. We can modify the absolute version of the PPP relationship in terms of another exchange rate, the real exchange rate, Rt. That is, Rt = St Pf / Pd. The real exchange rate allows us to compare foreign prices, translated into domestic terms with domestic prices. If absolute PPP holds, then Rt should be equal to one. If Rt is different III.16 than one, one country is more competitive than the other is. This is not an equilibrium situation --or at least, a long-run equilibrium situation. If prices and exchange rates are flexible, absolute PPP will force an adjustment via inflation or/and the nominal exchange rate, until Rt is equal to one. Example III.10: Suppose that the cost of the consumer basket represented by the Consumer Price Index (CPI) in Switzerland and in the U.S. is CHF 1241.2 and USD 755.3, respectively. Also, suppose that St = 0.8420 USD/CHF. Then, Rt = St PSWIT / PUS = 0.8420 USD/CHF * CHF 1241.2 / USD 755.3 = 1.3837. We can conclude that Switzerland is less competitive than the U.S. since its prices are 38.37% higher than U.S. prices, after taking into account the nominal exchange rate. Swiss residents will buy more U.S. goods, than U.S. residents buy Swiss goods. This is not an equilibrium situation (under absolute PPP, Rt=1). One way to get back to the equilibrium level is to have the CHF depreciate against the USD, over time. ¶ A currency can experience a real exchange rate appreciation, when a country’s inflation is much higher than that of a foreign trading partner and the exchange rate, St, does not move exactly to compensate for the difference in inflation rates. That is, the real exchange rate can appreciate or depreciate without movements of the nominal exchange rate. For instance, in 1999, the Argentine peso (ARS) experienced a real depreciation against the USD, since the inflation rate in Argentina was -1.8%, the inflation rate in the U.S. was 2.5%, while the ARS/USD exchange rate remained fixed at 1 ARS/USD. The ARS had a real depreciation against the USD. Therefore, U.S. goods for Argentine residents became relatively more expensive, while Argentine goods for U.S. residents became relatively more attractive.  The Real Exchange Rate as an Indicator of a Currency Crisis Recent studies have identified certain variables that signal that a country is vulnerable to a currency crisis. The signal that appears to be the most important is the real exchange rate. A study by the IMF has estimated that when the inflation of a country is much higher than that of its trading partners, and the exchange rate remains fixed, the probability of a currency crisis increases to 67%. That is, when a currency is significantly overvalued, in real terms, it indicates a high chance of a crisis.  2.A.2 Absolute PPP: Does It Hold? The Economist’s Big Mac Index reports the real exchange rate for many countries: Rt = St PBigMac,d / PBigMac,d,f A test of Absolute PPP is very simple: Check if Rt = 1. If big deviations are observed, a shadow is cast over the validity of this simple theory, Exhibit III.2 shows the real exchange rate, as reported by The Economist, for the major currencies in January 2019. III.17 Exhibit III.2 Big Mac Index We observe big departures from Absolute PPP. With some exceptions, the Big-Mac tends to be more expensive in developed countries (U.S.A., Euro area, Australia) than in less developed countries (Egypt, India, South Africa, Venezuela). 2.A.3 Trade Frictions and Other Factors affecting PPP III.20 In Brazil, the expected USD price, given its GDP per capita, is 3.05; while the actual USD price is 5.12, for a 67% overvaluation. On the other hand, we have China; which according to the unadjusted index, its currency is undervalued by 44.1%, but once prices are adjusted by GDP, the undervaluation is only 6.5%. That is, these adjustments to PPP implied exchange rates can be significant. In Exhibit III.3, we report a comparison of both PPP exchange rates for January 2012. Exhibit III.3 Adjusted Big Mac Index Switzerland Brazil Hong Kong 𝛼 + β GDP_p t III.21 PPP: Borders matter You may look at the Big Mac Index and think: “No big deal: there is also a big dispersion in prices within the U.S., within Texas, and, even, within Houston!” It is true that prices vary within the U.S. For example, in 2015, the price of a Big Mac (and Big Mac Meal) in New York was USD 5.23 (USD 7.45), in Texas as USD 4.39 (USD 6.26) and in Mississippi was USD 3.91 (USD 5.69). Engel and Rogers (1996) computed the variance of LOOP deviations for city pairs within the U.S., within Canada, and across the border. They found that distance between cities within a country matter, but the border effect is very significant. To explain the difference between prices across the border using the estimate distance effects within a country, they estimate the U.S.-Canada border should have a width of 75,000 miles! This huge estimate of the implied border width between the U.S. and Canada has been revised downward in subsequent studies, but a large positive border effect remains. 2.B Relative Purchasing Power Parity As noted above, one important criticism of absolute PPP is the assumption of absence of transportation costs, tariffs, or other obstruction to the free flow of trade. Because of these trade frictions, prices can differ from country to country. The relative version of the PPP theory takes into account trade frictions, which will be assumed constant. Thus, relative PPP is a weaker version of PPP. Under the assumption that trade frictions are constant, the difference between the two country’s price indices is constant. Therefore, the rate of change in the prices of products should be similar when measured in a common currency --as long as trade frictions are unchanged. The following formula reflects the relationship between relative inflation rates and changes in exchange rate according to the relative version of PPP: 𝑠 , = (St+T/St) - 1 = 1 (Relative PPP), where 𝑠 , = change in St from t and t+T. If = (Pf,,t+T/Pf,t) -1 = foreign inflation rate from t to t+T; Id = (Pd,,t+T/Pd,t) -1 = domestic inflation rate from t to t+T. We can use a linear approximation to the above formula; similar to the approximation we use for the IRPT formula. This linear approximation works very well for small inflation rates. III.22 Under this approximate formula, the percent change in exchange rates is equal to the inflation rate differential between the two countries. That is, 𝑠 ,  Id - If. Since this relationship is not expected to hold at every time interval, it is usually rewritten in terms of conditional expectations (averages): Et[𝑠 , ]  Et[Id] - Et[If]. We expect to observe a one-to-one relation between (Id-If) and st. For example, if prices are expected to double in the U.S. relative to those in Switzerland, the exchange rate of the CHF with respect to the USD should be expected to double (say, from .75 to 1.50). Example III.11: Forecasting with PPP the USD/South African rand exchange rate (USD/ZAR). You have the following information: CPIUS,2021=104.5 CPISA,2021 = 100.0 S2021 = 0.2035 USD/ZAR. You are given the 2022 forecast for the CPI in the U.S. and South Africa: CPIUS,2022 = 110.8 CPISA,2022 = 102.5. You want to forecast S2000 using the relative (linearized) version of PPP. SF 2022 = S2021 * [1 + IF US, 2022 – IF SA, 2022] = .2035 USD/ZAR * [1 + .06029 -.025] = .2107 USD/ZAR. You forecast an appreciation of the ZAR against the USD. ¶ As long as there are no changes in transportation costs, obstructions to trade, or the ratio of traded goods to non-traded goods, the change in the exchange rate should be roughly proportional to the change in the ratio of the two countries' general price levels. That is, under the relative version of PPP, the real exchange rate, Rt, remains constant. Relative PPP is also used to classify a currency as overvalued or undervalued. The term overvalued or undervalued insinuates that exchange rates are not supposed to be what the free-market rates are. For example, suppose that, over time, domestic inflation is higher than foreign inflation. According to PPP, we should expect a depreciation of the domestic currency. If the domestic currency depreciates less than what PPP suggests, it is said that the domestic currency is overvalued. Similarly, if the domestic currency depreciates by more than what PPP suggests, the domestic currency is undervalued. 2.C PPP: Implications III.25 Economists usually report the number of years that a PPP deviation is expected to decay by 50% (the half-life) is in the range of 3 to 5 years for developed currencies. Very slow! Long-term contracts and implicit price agreements make many prices in the economy sticky, in the short- and medium-run. Thus, since prices, trade, and commodity arbitrage respond sluggishly, PPP is not expected to be a good model. In the long-run, however, there is evidence supporting a role for inflation rate differentials. Over time, countries with persistent positive inflation rate differentials tend to see a depreciation of their domestic currencies. Similarly, over time, countries with persistent negative inflation rate differentials tend to see an appreciation of their domestic currencies. Over the years, relative price levels matter. The experience with other currencies is similar to the experience with the JPY/USD exchange rate displayed above. 2.D.1 PPP: Formal Statistical Evidence Let’s look at the usual descriptive statistics for (Id - If)t and ef,t. For the JPY/USD, they have similar means, but quite different standard deviations (look at the very different minimum and maximum stats). A simple t-test for equality of means (t-test=0.175) cannot reject the null hypothesis of equal means, which is expected given the large SDs, especially for ef,t. I JPY I USD I JPY -I USD e f,T (JPY/USD) Mean 0.0021 0.0033 -0.0012 -0.0015 SD 0.0063 0.0038 0.0061 0.0316 Min -0.0107 -0.0191 -0.0192 -0.1474 Median 0.0010 0.0030 -0.0019 -0.0001 Max 0.0431 0.0177 0.0346 0.1092 But, the average relation over the whole sample is not that informative, especially with such a big SD. We are more interested in the contemporaneous relation between ef;T and (Id - If)t. That is, what happens to ef;T when (Id - If)t jumps? To test the contemporaneous relation we have a more formal test, a regression: st = (St+T - St)/St = α + β (Id - If ) t + εt, where εt is the regression error, with mean 0 –i.e., E[εt]=0. To do this regression we need to collect data on exchange rates and inflation rates for the two countries involved. We will estimate two parameters, α and β. We will use the following notation: K refers as the number of parameters in a regression and N refers to the number of observations used in a regression. III.26 Under relative PPP, we have the following null hypothesis: H0 (Relative PPP holds): α=0 and β=1 H1 (Relative PPP does not hold): α≠0 and/or β≠1 The statistical tests are t-tests, for the individual estimated coefficients α and β, and F-tests, for a joint test on the estimated coefficients α and β: 1) t-test = [Estimated coeff. – Value of coeff. under H0]/S.E.(coeff.) The t-test follows a tv distribution, where v=N-K refers to the degrees of freedom. The decision rule is simple: if |t-test| > tv,α/2, reject H0 at the α level. Usually, α = .05 (5 %). (2) F-test = {[SSR(H0)-SSR(H1)]/J}/{SSR(H1)/(N-K)} The F-test follows an FJ,N-K distribution, where J is equal to the number of restrictions imposed by H0, and SSR refers to the sum of squared residuals of the regression. The decision rule is simple: if F-test > FJ,N-K,α , reject H0 at the α level. Example III.12: Using monthly Japanese and U.S. data from the graph (1/1971-12/2015), we fit the following regression: st (JPY/USD) = (St - St-1)/St-1 = α + β (IJAP – IUS) t + εt. R2 = 0.000123 Standard Error (σ) = 0.0316 F-stat (slopes=0 –i.e., β=0) = 0.0657 (p-value = 0.7978) F-test (H0: α=0 and β=1) = 14.583 (p-value: lower than 0.0001) => reject at 5% level (F2,477,.05= 3.015) Observations = 537 Coefficients Stand Error t Stat P-value Intercept (α) -0.00155 0.001391 -1.11501 0.265345 (IJAP – IUS) (β) -0.05749 0.224279 -0.25635 0.79778 Let’s test H0, using t-tests (t477.05=1.96 –when N-K>30, t.05 = 1.96): tα=0 (t-test for α = 0): (-0.00155–0)/ 0.001391= -1.11501 (p-value = .265) => cannot reject at the 5% level tβ=1 (t-test for β = 1): (-0.05749-1)/ 0.224279 = -4.7150 (p-value = .00001) => reject at the 5% level Regression Notes: ⋄ If we look at the R2, the variability of monthly (IJAP – IUS) explain very little, 0.01%, of the variability of monthly st. ⋄ We can modify the regression to incorporate the Balassa-Samuelson effect, by incorporating GDP differentials. Say, st (JPY/USD) = α + β (IJAP – IUS) t + δ (GDP_capJAP – GDP_capUS) t + εt. ¶ Example III.12 formally rejects relative PPP. Formal tests of PPP arrive to similar conclusions for other currencies: Relative PPP tends to be rejected in the short-run.. In the long-run, there is a debate about its validity. As mentioned above there is some evidence of (slow) mean reversion. In the long-run, inflation differential matter: Currencies with high inflation rate differentials tend to depreciate. III.27 Taylor (2002), using real exchange rates for 20 countries for over 100 years, finds strong evidence for PPP. However, deviation from PPP parity can be substantial in the short-run. In a survey of the PPP literature, Rogoff (1996) describes a consensus among PPP researchers that half the deviation from the PPP parity disappears between 3 to 5 years. It can take 5 to 10 years for the real exchange rate to revert back to its equilibrium level. Officer, in a paper published in the IMF Staff Papers, in 1976, points out that PPP emphasizes monetary demand and supply disturbances. For instance, other factors being constant or negligible, a tight domestic money supply policy decreases the rate of inflation and, therefore, leads to a higher value for the domestic currency. In the short-run, other factors are not constant and changes in price levels are not solely determined by monetary factors. In the short-run, the existence of contracts makes prices sticky. In the long run, however, monetary factors are the main determinant of the inflation rate, therefore, PPP tends to hold in the long run.  PPP and High Inflation Officer's considerations help PPP to provide a good description of exchange rates movements in high inflation countries, even in the short-run. Under high inflation, all other factors that influence prices become relatively negligible. In high inflation countries, contracts are written to adapt to the high inflation conditions. Economic agents are very sensitive to price changes and, thus, prices adjust very rapidly in response to monetary disturbances.  2.D.2 PPP: Rt, St and Sticky Prices Research shows that Rt is much more variable when St is allowed to float. Rt‘s variability tends to be highly correlated with St’s variability. This finding comes from Mussa (1986). Figure III.4 shows the finding of Mussa (1986) for the USD/GBP exchange rate: After 1973, when floating exchange rates were adopted, Rt moves like St. As a check to the visual evidence: the monthly volatility of changes in Rt is 2.96 and the monthly volatility of changes in St is 2.93, with a correlation coefficient of .979. Almost the same! III.30 FIGURE III.5 MXN/USD exchange rate: Does PPP hold in the long run? In the short-run, Relative PPP is missing the target, St. But, in the long-run, PPP gets the trend right. That is, inflation rate differentials matter: as predicted by PPP, the high Mexican inflation rates differentials against the U.S depreciate the MXN against the USD. Similar behavior is observed for the JPY/USD, see Figure III.6, using data from 1971-2018. The inflation rates in the U.S. have been consistently higher than in Japan, then, according to Relative PPP, the USD should depreciate against the JPY. PPP gets the long term trend right, but misses St in the short-run. 0 5 10 15 20 25 30 1 2 /1 /1 9 8 7 1 /1 /1 9 8 9 2 /1 /1 9 9 0 3 /1 /1 9 9 1 4 /1 /1 9 9 2 5 /1 /1 9 9 3 6 /1 /1 9 9 4 7 /1 /1 9 9 5 8 /1 /1 9 9 6 9 /1 /1 9 9 7 1 0 /1 /1 9 9 8 1 1 /1 /1 9 9 9 1 2 /1 /2 0 0 0 1 /1 /2 0 0 2 2 /1 /2 0 0 3 3 /1 /2 0 0 4 4 /1 /2 0 0 5 5 /1 /2 0 0 6 6 /1 /2 0 0 7 7 /1 /2 0 0 8 8 /1 /2 0 0 9 9 /1 /2 0 1 0 1 0 /1 /2 0 1 1 1 1 /1 /2 0 1 2 1 2 /1 /2 0 1 3 1 /1 /2 0 1 5 2 /1 /2 0 1 6 3 /1 /2 0 1 7 4 /1 /2 0 1 8 Actual vs Long Run PPP: MXN/USD PPP MX/US MXN/USD III.31 FIGURE III.6 JPY/USD exchange rate: Does PPP hold in the long run? Note that in both graphs, St PPP is smoother than St, which is why a very poor model to explain the short-term MXN/USD movements. Both exchange rates, however, share the same long- run trend. 2.F PPP: Two practical applications Given the basic economic intuition behind PPP and the empirical evidence that gives PPP some long-run support, many analysts use PPP exchange rates to compare economic fundamentals across countries. In addition, PPP exchange rates are more stable than actual exchange rates and, thus, big swings in actual exchange rates do not affect PPP valuations of economic fundamentals very much. For example, GDP is usually reported in both actual and PPP figures. Example III.13: In 2011, using market prices –actual exchange rates- the U.S. GDP was USD 15.06 trillion, which amounted to a 23.1% share of the world’s GDP (27.5% share in 1996), while China had a GDP equal to USD 6.99 trillion, for a 9.3% share of the world’s GDP (3.1% share in 1996). If PPP exchange rates were used, the U.S. GDP was the same, USD 15.06 trillion for a 20% share of the world’s GDP and the Chinese GDP was USD 11.3 trillion, for a 14.4% share. ¶ Many central banks follow a very simple rule to establish a crawling peg. They adjust the domestic currency, with the goal of maintaining a stable real exchange rate. In this way, the exchange rate becomes the inflationary anchor as the nominal depreciation (appreciation) rate matches the growth in domestic prices, thus reducing expectations and loss of competitiveness. 0 50 100 150 200 250 300 350 400 450 500 1 /1 /1 9 7 1 9 /1 /1 9 7 2 5 /1 /1 9 7 4 1 /1 /1 9 7 6 9 /1 /1 9 7 7 5 /1 /1 9 7 9 1 /1 /1 9 8 1 9 /1 /1 9 8 2 5 /1 /1 9 8 4 1 /1 /1 9 8 6 9 /1 /1 9 8 7 5 /1 /1 9 8 9 1 /1 /1 9 9 1 9 /1 /1 9 9 2 5 /1 /1 9 9 4 1 /1 /1 9 9 6 9 /1 /1 9 9 7 5 /1 /1 9 9 9 1 /1 /2 0 0 1 9 /1 /2 0 0 2 5 /1 /2 0 0 4 1 /1 /2 0 0 6 9 /1 /2 0 0 7 5 /1 /2 0 0 9 1 /1 /2 0 1 1 9 /1 /2 0 1 2 5 /1 /2 0 1 4 1 /1 /2 0 1 6 9 /1 /2 0 1 7 Actual vs Long Run PPP: JPY/USD PPP JPY/USD JPY/USD III.32 Example III.14: The Bolivian Central Bank followed a crawling peg from 1985 to 1994, through a system of mini-devaluations of the peso boliviano (BOB) against the USD to achieve a stable real exchange rate. The following table shows the changes from 1992 to 1994 in exchange rates (BOB/USD), Bolivian inflation and U.S. inflation: 1992 1993 1994 St (BOB/USD) 4.10 4.48 4.70 st (%) 9.33 9.27 4.91 IBOL 10.46 9.31 8.52 IUS (%) 1.73 0.85 2.44 The depreciation of the BOB closely followed the inflation rate differential. From June 1994 on, the Bolivian Central Bank has devalued its domestic currency to maintain a stable exchange rate against a basket of currencies. The basket of currencies represents a weighted average of the currencies in Bolivia’s six largest trading partners. ¶ III. International Fisher Effect (IFE) Along with the PPP theory, another major theory is the International Fisher Effect (IFE) theory. It uses nominal interest rate differentials rather than inflation rate differentials to explain why exchange rates change over time, but it is closely related to the PPP theory because nominal interest rates are highly correlated with inflation rates. Recall that PPP emphasizes trade as the determinant of supply and demand for foreign exchange. IFE, on the other hand, emphasizes financial transactions. 3.A Arbitrage in Perfect Financial Markets Assume that there are perfect international capital markets. That is, there are no restrictions to the free flow of capital across national borders. Also, assume that investors consider a foreign asset a perfect substitute of a similar domestic asset. Then, under the IFE, the expected return to investors who invest in money markets in their home country should be equal to the return to investors who invest in foreign money markets once adjusted for currency fluctuations. For example, using equation (I.1) and ignoring transactions costs, taxes and uncertainty, the "effective" T-day return on a foreign bank deposit is given by rd = (1 + if * T/360) (1 + E[st,T]) -1. where, if = foreign interest rate for T days; id = domestic interest rate for T days. On the other hand, the effective T-day return on a home bank deposit is: rd = id * T/360. III.35 Note: This strategy worked well for a couple of years, when Et[st,T] = 5%. But, it failed big in December 1994, when the MXN lost 40% of its value and the accumulated gains were wiped out in a matter of days. ¶ The IFE pseudo-arbitrage strategy differs from covered arbitrage in the final step. Step (4) involves no coverage. It’s an uncovered strategy. IFE is also called Uncovered Interest Rate Parity (UIRP). 3.D IFE: Evidence Testing IFE is more complicated than PPP, since IFE involves an expectation (an unobservable). In general, we test IFE assuming that on average what we expect occurs. That is, the observed average st,T equals the expected change at time t, or Et[st,T]. IFE has been extensively tested, in general, assuming that on average what we expect occurs (“rational expectations” assumptions). Similarly to the PPP formal tests, a formal test of IFE can be done with a regression based on the linearized version of equation (III.3). That is: st,T =  + ß [(id,t-1 - if,t-1) * T/360] + t. The test is based on the following null joint hypothesis H0: =0 and ß=1. An F-test can be used to test this null hypothesis. The null hypothesis has been soundly rejected by the data. In general the rejection arises because ß is not statistically different from zero. Example III.17: Short-run tests of IFE for the USD/EUR We collected monthly interest rates differentials (iUSD – iEUR) and ef (USD/EUR) from 01/99 to 12/17. III.36 We immediately see that there is no clear 45 degree line. The visual evidence rejects IFE. For a formal test of IFE, we estimate the following regression: st = (St+T - St)/St = α + β (iUSD – iEUR)t + εt R2 = 0.00641 Standard Error = 0.02907 F-statistic (slopes=0) = 0.1459 (p-value=0.70305) F-test (α=0 and β=1) = 68.63369 (p-value= lower than 0.0001) => rejects H0 at the 5% level (F2,193,.05=3.05) Observations = 228 Coefficients Stand Error t-Stat P-value Intercept (α) 0.000588 0.001935 0.30400 0.76141 (iUSD – iEUR ) (β) -0.05477 0.14350 -0.38169 0.70305 Let’s test H0, using t-tests (t104,.05 = 1.96) : tα=0 (t-test for α = 0): (0.000588 – 0)/0.001935 = 0.304  cannot reject at the 5% level. tβ=1 (t-test for β = 1): (-0.05477 – 1)/0.14350 = -8.045  reject at the 5% level. Formally, IFE is rejected in the short-run (both the joint test and the t-test reject H0). Also, note that β is negative, not positive as IFE expects. Note: During the 1999-2017 period, the average monthly (iUSD – iEUR) was -0.00164/12 = -.00015. That is, st IFE = -0.015% per month (IFE expects a 0.015% monthly depreciation of the EUR). But, the actual average monthly st was .0007 (st =0.07% per month; statistically speaking not different from zero), which is different from st IFE. ‐0.05 ‐0.04 ‐0.03 ‐0.02 ‐0.01 0 0.01 0.02 0.03 0.04 0.05 ‐0.12 ‐0.07 ‐0.02 0.03 0.08 IFE: USD/EUR iUSD – iEUR st,T III.37 If we use the regression to derive an expectation, the regression expects Et[st] = .000588-.005477*(- .00164) = 0.0006, which is statistically speaking not different from zero. That is, we expect a very close to zero monthly change in the EUR against the USD. This zero change is still different from st IFE = - 0.15%, which is statistically significantly different from 0. Recall that consistent deviations from IFE point out that carry trades are profitable: During the 1999- 2017 period, USD-EUR carry trades should have been profitable. ¶ Similar to PPP, there is no short-run evidence. As pointed out above, consistent IFE departures make carry trades profitable: Burnside (2008) show that the average excess return of an equally weighted carry trade strategy, based on up to 20 currencies and executed monthly over the period 1976–2007, was about 5% per year. Lower than excess returns for equity markets, but with a Sharpe ratio twice as big as the S&P500! (Annualized volatility of the carry trade returns was much less than that for stocks). IFE, however, has some empirical support in the long run: interest rate differentials have some power to predict exchange rates movements. As predicted by the IFE, we find over extended periods of time (5, 10 years) that currencies with relatively high interest rates tend to depreciate and currencies with relatively low interest rates tend to appreciate. Chinn and Meredith (2004) find that estimates of the β are usually not significantly different from 1, at 5 and 10 year horizons. A different test of IFE is provided by dropping the rational expectations assumption. Froot and Frankel (1989) rely upon survey-based measures of exchange rates to calculate expected depreciation. They find that for reserve currencies (against the U.S. dollar) it is much more difficult to reject the null hypothesis that β=1. But, for other currencies and using a more recent extended sample period, the evidence for IFE is not very strong. Frankel and Poonawala (2006) find that support on IFE depends to some extent on the exchange rate system: highly managed exchange rate regimes are associated with currencies that show greater deviations from IFE. Some practitioners use rule of thumbs based on long-run IFE. For example, a 1% change in the nominal 10-year bond yield differential -between USD bonds and EUR bonds- is used to forecast a change in the USD/EUR exchange rate of 10%. We should note that in Section 1.B we mentioned that PPP is not supported by the data, especially in the short-run. Since IFE is based on some form of purchasing power parity, it should not be surprising that IFE is also rejected by the data. IV. Expectations Hypothesis of Exchange Rates The expectations hypothesis of exchange rates states that the expected spot rate T periods from now (St+T) is equal to today's forward rate for delivery T periods from now (Ft,T): III.40 Given that the forward rate is not a good predictor of futures spot rates, many economists have attempted to provide rational explanations for this counterintuitive result. 4.C.1 Risk Premium A possible explanation for the failure of the expectations hypothesis is the existence of a risk premium. Recall that the risk premium of a given security is defined as the return on this security, over and above the risk-free return. A foreign exchange risk premium induces risk- averse agents to take a risk in the foreign exchange market. Thus, the existence of a divergence between Et[St+T] and Ft,T can be justified by risk-aversion. Now, let us formalize the idea of a risk premium in the foreign exchange market. After some simple algebra, we find that the expected excess return on the foreign exchange market is given by: (Et[St+T] - Ft,T)/St = RPt,t+T, where RPt,t+T represents the foreign exchange risk premium. Example III.19: Understanding the meaning of the foreign exchange risk premium. Suppose you have the following data: St=1.58 USD/GBP, Et[St+6-mo]=1.65 USD/GBP and Ft,6-mo=1.62 USD/GBP. The expected change in the exchange rate is equal to (Et[St+6-mo] - St)/St = (1.65-1.58)/1.58 = 0.0443. The 6-mo foreign exchange forward premium on the GBP is p6-mo = (Ft,6-mo - St)/St = (1.62-1.58)/1.58 = 0.0253. According to this example, in the next 6-month period, the GBP is expected to appreciate against the USD by 4.43%, while the forward premium suggests a GBP appreciation of 2.53%. The discrepancy arises from the presence of a foreign exchange risk premium, RPt,t+6-mo, which makes the forward rate a biased predictor of the exchange rate six months from now. Given the positive risk premium on the GBP, the expected (USD) return from holding a GBP deposit will be more than the USD return from holding a USD deposit. This non-zero return differential might be an equilibrium result consistent with rational investor behavior. The higher return from holding a GBP deposit is necessary to induce investors to hold the riskier GBP denominated investments. ¶ A risk premium in foreign exchange markets implies that the expectation hypothesis should be written differently: Et[St+T] = Ft,T + St RPt,t+T. As long as the risk premium, RPt,t+T is consistently different from zero, foreign exchange markets will display a forward bias. III.41 The empirical evidence for a risk premium in foreign exchange markets is weak. Several researchers have assumed that the forward rate is an unbiased predictor of future spot rates. Then, they have tried to explain the risk premium using the fundamental variables used in the finance literature to explain risk premia in financial assets, such as volatility. No significant relation has been found between the foreign exchange risk premium and fundamental variables.  Risk Premium and Diversifiable Risk Note that the existence of a divergence between Et[St+T] and Ft,T can be justified by the existence of a risk premium. Many economists claim, however, that a risk premium is justified if exchange rate risk is not diversifiable. If a risk is diversifiable, then there is no need to expect a compensation for holding it.  4.C.2 Errors in Forming Expectations In an uncertain environment, economic agents are expected to make forecasting mistakes. Rational agents, however, will eventually learn and, thus, errors will not consistently persist. Nevertheless, some economists have argued that investors make consistent errors in forecasting exchange rates. One explanation for these consistent mistakes relies on the assumption that it takes time for investors to learn about new market conditions. For example, suppose there is a new chairman on the Bank of Japan. It might take years for economic agents to learn the Bank of Japan's new monetary policy. That is, there is "slow learning." Karen Lewis, in a paper published in the Journal of Monetary Economics in 1989, showed that even when slow learning of money supply rules is taken into account the forward bias observed in the early 1980s did not disappear. 4.C.3 The "Peso Problem" A peso problem is a very specific form of a small sample problem that affects statistical inference. According to this view, for long periods of time investors assign a small but positive probability to an extreme change in the asset price (such as a devaluation or a stock market crash), which may never materialize in a limited sample period. The frequency of the extreme events in the sample studied does not equal the ex ante anticipated probability. The forward rate, however, will reflect the ex-ante probability distribution. Since the event may never materialize, markets will observe a persistent forward bias. The small sample problem is called peso problem, in reference to the discrete changes in the Mexican peso in 1976. Before 1976, the Mexican peso had been successfully pegged to the USD for 23 years. Mexican interest rates were substantially higher than U.S. interest rates, creating a MXP/USD forward rate higher from the MXP/USD spot rate. Therefore, the MXP/USD showed a persistent premium. The peso problem, however, is not a new problem, nor it is constrained to developing economies. It applies to any situation in which there can be a discrete jump in prices or shift in policy regimes. III.42 Example III.20: Peso problem: Now and then. The Mexican peso used to show a real and continuous appreciation until the Mexican government finally devalued the peso (generally after an election). Before the devaluation, since markets were expecting a devaluation, the peso used to have a strong forward bias. During the period 1890-1908 the USD/GBP showed a peso problem. That is, during that period financial markets expected the USD to depreciate against the GBP, but this never happened –i.e., expectations were persistently biased. Different events created this bias. One of them was the 1896 Presidential Election, in which the U.S. adherence to the gold standard was in question. ¶ V. Looking Ahead The exchange rate models based on arbitrage that we have studied do not enjoy a strong support from the data, especially in the short-run. Note that we have not explicitly mentioned supply and demand factors when the parity relations were developed. In Chapter I, however, we emphasized that exchange rates are just prices. In the next chapter, we are going to explicitly model supply and demand for the foreign currency to gain more insight into exchange rates. Interesting readings: International Financial Markets, by J. Orlin Grabbe, published by McGraw-Hill. International Financial Markets and The Firm, by Piet Sercu and Raman Uppal, published by South Western. International Investments, by Bruno Solnik, published by Addison Wesley. Cassel, Gustav (1918), “Abnormal deviations in international exchanges,” The Economic Journal. Hummels, David (2001). “Toward a Geography of Trade Costs.” Unpublished manuscript, Purdue University, September. Parsley, D. and Wei, S., 2007. A Prism into the PPP Puzzles: The Micro-foundations of Big Mac Real Exchange Rates, The Economic Journal, October, 117, 1336-1356. Rogoff, Kenneth (1996), “The purchasing power parity puzzle”, Journal of Economic Literature, June 1996, 647-68. Taylor, Alan M. (2002), “A century of purchasing power parity”, Review of Economics and Statistics, 84: 139-150