Expected Returns on Currencies FX Carry Trade

Transcription

1 Expected Returns on Currencies FX Chris Telmer May / 57

2 Basic Question Popular Wisdom Evidence Economic Reasoning? How are interest rates and exchange rates related? CIP gives an answer, but not a very interesting one F t S t = 1+i t 1+i t The issue: interest rates today vs. future exchange rates? i t i t versus S t+1 S t 2 / 57

3 Popular Wisdom Popular Wisdom Evidence Economic Reasoning? How are interest rates and exchange rates related? What do we read in the newspapers? High rates imply strong currencies. Indeed, the main cause of the dollar s recent slide has been the ECB s refusal to cut interest rates (because of its inflation concerns) while the Fed is slashing rates to support growth.... The Economist, March 27, Low Swiss interest rates and the carry trade weaken the Swiss currency because essentially, speculators are selling it in order to buy, say, Icelandic kronur; all that selling pushes down the franc s value.... Wall Street Journal, March 16, 2007 Another reason (for the devaluation of GBP) is UK interest rates. In January, these were very high. Now they are relatively low. So some of the pound s continued fall is due to momentum selling from foreign investors.... Financial Times, December 12, / 57

4 Evidence Popular Wisdom Evidence Economic Reasoning? First... what are the facts? High rates are associated with strong currencies. But there is tons of variability around this association. This is made precise below. It is the main point. Also, the reasoning we often find in newspapers is, at best, incomplete So, what is the economic reasoning that we often read? 4 / 57

5 Economic Reasoning? Popular Wisdom Evidence Economic Reasoning? Supply and demand story: Bank of England increases interest rates U.S. investors want to buy British, high-yielding bonds Creates demand for GBP... drives up value of GBP What s wrong with this story? It s right! Money is not tomatoes Unclear about returns. Suppose that the story is right. Isn t this too-good-to-be-true for these U.S. investors? Perhaps there s something else going on? Risk! We need to think more seriously about currency returns 5 / 57

6 Currency Explanation UIP Forward Rate Approximations Returns on Foreign Exchange 6 / 57

7 Currency Returns Currency Explanation UIP Forward Rate Approximations i t = one-year riskless USD interest rate (from date t to t+1) i t = one-year riskless GBP interest rate (from date t to t+1) S t = USD/GBP spot exchange rate, date t. Suppose you are a U.S. investor. What s the return on 1-year USD? What s the USD-denominated return on 1-year GBP? We call this the uncovered USD return on a foreign bond. 7 / 57

8 Explanation: Currency Returns Currency Explanation UIP Forward Rate Approximations The uncovered, USD-denominated return to investing one dollar in a GBP-denominated eurodeposit (a eurosterling deposit ) between time t and one-year hence, time t + 1, is: (1 + i t )S t+1 S t, where S t is USD/GBP. It is crucial to note that this is a USD-denominated return. The derivation goes as follows. Start with USD 1.0 at time t and convert it to sterling at today s exchange rate, S t. This gives an amount of sterling equal to GBP 1/S t Deposit the sterling in a riskless eurosterling deposit. After one year you will have GBP (1 + i t )/S t. What does riskless mean here? It means that you know the interest rate at time t, for sure, and there s no credit risk (not quite true in eurocurrencies, but close). Repatriate the proceeds back into USD at the time t + 1 exchange rate. This gives you USD (1 + i t )S t+1 S t. Since we started with USD 1.0, this is also the rate-of-return. The language uncovered should now be clear. In deriving CIP we covered ourselves against currency risk by selling the proceeds of our international investment in the forward market. Here we let-it-ride and take our chances on what will happen to the exchange rate. That is, we have an uncovered investment. 8 / 57

9 Back to the Supply-Demand Story Currency Explanation UIP Forward Rate Approximations What would make it NOT to-good-to-be-true? USD-return on GBP bond equals USD-return on USD bond. Makes no sense. S t+1 is random: not known until (t+1). Instead, how about expected returns equal?* 1+i t = E t (1+i t)s t+1 S t This is called uncovered interest rate parity (UIP). Rewriting slightly: 1+i t 1+i t = E t S t+1 S t High GBP interest rates reflect expected depreciation of GBP. * The notation E t means expected value, conditional, on what you know at date t. Therefore, E t (x t ) = x t for any variable x t, because it is known at date t. Similarly, E t (x t y t+1 ) = x t E t y t+1, since x t is known but y t+1 is not. 9 / 57

10 Expectations Hypothesis Currency Explanation UIP Forward Rate Approximations CIP: UIP: Combined: 1+i t 1+i t 1+i t 1+i t F t = F t S t = E t S t+1 S t S t+1 = E t S t S t = F t = E t S t+1 Two ways to say the same thing: interest rate differential equals expected deprecation is the same as to say forward rate equals expected future spot rate. 10 / 57

11 Approximations Currency Explanation UIP Forward Rate Approximations 1+i t 1+i t = E t S t+1 S t 1. Subtract 1.0 from both sides and use 1+i t 1. i t i t 1+i t = E t S t+1 S t S t = i t i t E t S t+1 S t S t U.S. minus foreign interest rate differential equals expected depreciation rate of USD. 11 / 57

12 (continued) Currency Explanation UIP Forward Rate Approximations 2. Take (natural) logarithms* and use log(1+i) i Log of USD return: log(1+i t) i t Log of USD-denominated return on GBP log ( (1+i t )S ) t+1 S t ) i t +log (S t+1 /S t As before, UIP results from equalizing expected returns ) i t i t = E t log (S t+1 /S t where lower-case letters denote logs = E t s t+1 s t Interest rate differential equals expected, continuously-compounded depreciation rate of USD. *Another alternative is to use the abstraction of continuously-compounded returns and rates of change. In this case no approximations would be required and the above expressions would hold exactly. 12 / 57

13 : Uncovered Interest Rate Parity (UIP) Currency Explanation UIP Forward Rate Approximations Uncovered interest rate parity: expected returns are equal. Different ways to express this are: 1. Exact UIP: 1+i t 1+i t = E t S t+1 S t 2. Approximate UIP: i t i t E t S t+1 S t S t E t log ( S t+1 /S t ) Et s t+1 s t (1) 3. Forward rate expectations hypothesis (EH): Exact: F t = E t S t+1 Approximate:* log(f t ) f t = E t s t+1 *To derive this, take logs of both sides of the CIP formula and then use the approximation log(1 + i) i. This will tell you that f t s t = i t i t. Then equate the LHS of this expression to the RHS of equation (1). Again, if we use continuous compounding everything holds exactly and none of the approximations are required. 13 / 57

14 Organization Uncond evidence Conditional Evidence Regressions Undeniable 14 / 57

15 Questions, Organization Organization Uncond evidence Conditional Evidence Regressions Undeniable Organization of evidence: 1+i t 1+i t = E t S t+1 S t 1. Unconditional ( long-term ) evidence 2. Conditional ( short-term ) evidence 3. Undeniable situations 15 / 57

16 Unconditional Evidence Organization Uncond evidence Conditional Evidence Regressions Undeniable By unconditional evidence I mean the following. Suppose that there is a country that seems to always have low interest rates (e.g., Japan). Then we say that the unconditional mean of their interest rate appears to be relatively low... they have low interest rates on average. The unconditional evidence asks what the unconditional mean the long-term average of this country s currency depreciation rate is? Empirically, we can estimate these things by computing long-term averages of interest rate differentials and FX depreciation rates and doing a scatter plot. See the next page The answer in contrast to the popular wisdom is we should expect the low interest rate currency to appreciate. This is what UIP predicts: i t i t = E t s t+1 s t = E(i t i t ) = E(E ts t+1 s t ) This shouldn t really surprise you. It s just saying that low-inflation countries tend to have low interest rates and strong currencies. Punchline. In the long-term, UIP gets it right. High interest rates are associated with weak currencies. 16 / 57

17 (continued) 3 Unconditional Evidence: UIP Relation Organization Uncond evidence Conditional Evidence Regressions Undeniable Mean Depreciation Rate (Annual Percentage) Italy UK Canada France Germany Japan Mean Interest Rate Differential (Annual Percentage) 17 / 57

18 18 / 57 # "! (continued) Organization Uncond evidence Conditional Evidence Regressions Undeniable % % ' & $ U.S. minus Foreign Interest Rate Differential Depreciation Rate of USD (% p.a.) $

19 Conditional Evidence Organization Uncond evidence Conditional Evidence Regressions Undeniable Now we know what the long-term average looks like: high rate currencies tend to devalue What do we mean by conditional evidence? Are there times when a country s interest rate is higher than average. Of course. There is always variation around the average. When a country s interest rate is higher than average or, synonymously, conditional on a country s interest rate being higher than average what should we expect its exchange rate to do? How do we measure this sort of thing? What should we expect variable y to do, conditional on observing variable x? Regression 19 / 57

20 (continued) Organization Uncond evidence Conditional Evidence Regressions Undeniable Recall that s t logs t. UIP is: E t s t+1 s t = i t i t. (2) Define the forecast error as ε t+1 ε t+1 = s t+1 E t s t+1. (3) Substitute equation (3) into equation (2) s t+1 s t = i t i t +ε t+1. Consider the OLS regression: s t+1 s t = a+b(i t i t)+residuals. What does UIP imply about the coefficients a and b? a = 0 b = 1 20 / 57

21 Regression Results Organization Uncond evidence Conditional Evidence Regressions Undeniable Standard Currency a b Error R 2 British Pound (0.0028) (0.862) Canadian Dollar (0.0009) (0.581) French Franc (0.0032) (0.928) German Mark (0.0043) (1.348) Japanese Yen (0.0032) (0.719) Source: Backus, Foresi and Telmer (2010), Affine Models of Currency Pricing. Standard errors in parentheses. Monthly data on 1-month forward rates. 21 / 57

22 Some Undeniable Situations Organization Uncond evidence Conditional Evidence Regressions Undeniable Finally, there are some situations in which UIP is just undeniably the first-order issue. 22 / 57

23 Hong Kong Organization Uncond evidence Conditional Evidence Regressions Undeniable 23 / 57

24 Thailand Organization Uncond evidence Conditional Evidence Regressions Undeniable 24 / 57

25 25 / 57 ISK USD One Month Interest Rate Diffential Iceland Organization Uncond evidence Conditional Evidence Regressions Undeniable Icelandic Krona per USD (red line) Source: Datastream. Weekly data, June 2006 September 2010, interbank interest rates.

26 Other Organization Uncond evidence Conditional Evidence Regressions Undeniable Mexico 1994 Argentina 2001 Turkey, now Any emerging-market currency crisis you can think of (some emerged countries too) Why were Argentinean interest rates so high in 2002? Because investors anticipated a devaluation of the peso and required high interest on Argentinean bonds as compensation. 26 / 57

27 Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics 27 / 57

28 Recap The regression Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics s t+1 s t = a+b(i t i t)+residuals. b < 0 = interest rate differentials are negatively correlated with depreciation on the dollar! An investor who holds funds in a high yield currency not only benefits from higher yields, but also tends to benefit from an appreciation in the long currency. Implies that there exist excess expected returns in currency prices. 28 / 57

29 Interpretation Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics First, what are currency excess returns? Carry trade (like money-market hedge): Borrow USD 1, convert into AUD, invest, repatriate at maturity. Payoff: ( (1+i USD t )S ) t+1 (1+i t ) S t Forward trade (like forward hedge): Sign contract to receive AUD 1.0, deliver USD F t. Payoff: ( USD 1.0 ( ) ) S t+1 F t Currency risk premiums are the expected values of these things. 29 / 57

30 (continued) Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics Now, we ve seen that, conditionally, UIP is robustly rejected. This means: i t i t = f t s t = E t (s t+1 s t ) }{{} q t f t logf t, s t logs t q t expected depreciation rate p t risk premium +f t E t (s t+1) }{{} p t UIP implies p t = 0. The regression evidence, in contrast, indicates p t 0. Not surprising. Suggests that currency risk matters. 30 / 57

31 Forward Rate Note: the evidence necessarily means that Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics F t E t S t+1 Forward rate is not an unbiased predictor of the future spot rate. Similar evidence holds for interest rates, commodities. Again, shouldn t necessarily surprise you. Expectations hypothesis relies on risk neutrality. 31 / 57

32 Prediction Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics Above we have the carry-trade payoffs. The trick is to know when they are expected to be positive. Simplest approach: go long the high rate currency. Inadequate. Need to go long when interest rate difference is higher than average Example: JPY rates are always low Standard approach: Use regression to predict the above excess returns 32 / 57

33 Predictive Regression Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics The standard UIP regression: s t+1 s t = a+b(i t i t)+residuals Invoke CIP s t+1 s t = a+b(f t s t )+residuals Algebra: s t+1 s t (f t s t ) = a+b(f t s t ) (f t s t )+residuals = s t+1 f t = a+(b 1)(f t s t )+residuals (4) UIP implies b 1 = 0: unpredictable excess returns to currency speculations (risk premiums). We ve found that b 1 < 2. Double whammy! Implication: can use the regression (4) to build dynamic trading strategy. 33 / 57

34 Carry Trading Strategy Run the regression: Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics s t+1 f t = a+(b 1)(f t s t )+residuals Obtain estimates of a and (b 1) Each month, observe f t s t and compute the predicted excess return: x t E t (s t+1 f t ) = a+b(f t s t ) If x t > 0 go long GBP, short USD. If x t < 0 do the opposite. 34 / 57

35 ... Gives Pretty Good-Looking Payoffs Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics From The Returns to Currency Speculation, by Burnside, Eichenbaum, Kleshchelski and Rebelo, August / 57

36 Sharpe Ratios Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics S&P Fama-French Diversified Equity 0.48 U.S. Treasury 0.40 Single-Currency FX 0.39 Portfolio-Based FX / 57

37 During the Crisis 0.1 Mortgage Crisis (July 2007 March 2008, One Month Returns) HML MSCI Recap Interpretation Forward Rate 0.05 Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics corr(hml,msci) = Jul Aug Sep Oct Nov Dec Jan Feb Mar 37 / 57

38 Mechanics Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics Which financial products to use? Interest rates: Borrow in low rate currency and invest in high rate currency. This is where the term carry comes from Forward contracts: deliver low rate FX and receive high-rate FX. Same as saying deliver the currency that has a HIGH forward value relative to its spot value. How to trade? Simplest way: short the low rate FX More sophisticated: run a regression, do what the fitted-value (conditional mean) tells you. Portfolio-based approach: short the 3 low yielders in the G10, go long the 3 high-yielders. Basis of ETFs. Many more sophisticated variations on these themes (e.g., PPP deviations, regression-based portfolio approaches) 38 / 57

39 Recap Interpretation Forward Rate Prediction Predictive Regression Payoffs Sharpe Ratios During the Crisis Mechanics Expected returns on currencies vary across currencies and across time. Long-term they are all the same Short-term, the currencies with interest rates that are higher than average tend to appreciate The short-term variation, by definition, is variation around the long-term averages. Therefore currency risk premiums change sign and average-out to zero. The risks are an open question: Crash risks are very real. Beware. The Fall of 2008 made this clearer than ever. But do currencies look good relative to other risky assets? 39 / 57

40 Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements 40 / 57

41 Interest Rate Parity Relations Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements The interest rate parity relationships Covered: (1+i t )/(1+i t) = F t /S t Not much different that saying interest rates equal at different banks. Uncovered. Three ways to say it: High rates signal depreciation: (1 + i t )/(1 + i t ) = E t S t+1 /S t Forward rate is expected future spot: F t = E t S t+1 Expected returns equal on all currencies... FX risk doesn t matter. The evidence? CIP holds except during crises. UIP? Sometimes UIP is first-order: high rates signal depreciation. Examples: Hong Kong, Argentina, Iceland during the crisis. Other times currency risk is first-order: high rates signal appreciation. Examples: JPY/AUD, JPY/CHF, Iceland, before the crisis. Beware the crash risk. 41 / 57

42 Two Pieces We started by asking how are interest rates and exchange rates related? What s our answer? Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements Interest rate differences contain two pieces: i t i t = f t s t = E t (s t+1 s t ) }{{} q t +f t E t (s t+1) }{{} p t 42 / 57

43 Popular Wisdom? Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements We started by asking how are interest rates and exchange rates related? What s our answer? The popular wisdom is right high rates do (sometimes) signal strong currencies but for the wrong reasons. You must go beyond the supply-demand platitudes and think about risk. Learn to think about exchange rates like you think about stocks and bonds. Thinking like this is really important. For example: Consider a U.S. company that is expanding in Australia. Rates are low in JPY, high in AUD. So they decide to do the corporate carry trade: borrow JPY to finance AUD investments. The evidence suggests that they should expect JPY to devalue relative to AUD, thus reducing their financing costs. But the evidence also suggests that this is risky. Will a crash in AUD (an upward crash in JPY) bankrupt them? 43 / 57

44 Picture: AUD Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements i t i t = E t s t+1 s t +p t = E t s t+1 = s t +i t i t p t = s t+1 = s t +i t i t p t +ε t+1 where ε t+1 s t+1 E t s t+1, the forecast error. Random Walk UIP 44 / 57

45 Brazil Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements Right now, Brazil has high interest rates. How should you think about this? Short-term: BRL is expected to appreciate over the next month, quarter, maybe 1 year. But it s not a sure thing. It s risky. 57 months out of 100 BRL will go up. Crash risk makes this even riskier. Long-term: If BRL rates stay high for a long time, BRL will devalue in the long run. Popular wisdom, supply-demand rhetoric, etc.. Does it make sense? Maybe. But is it helpful? For what? 45 / 57

46 Picture: Brazil PPP and Carry Trading (Bubbles?) in Brazil Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements 46 / 57

47 Random Walk Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements You will often hear FX rates are a random walk. This means that the line in the above picture is FLAT. Is it? No. There s robust evidence that it is not. Nevertheless, the uncertainty is huge. For some questions the uncertainty is so dominant that saying random walk, is fine. Anything else you want to throw in for forecasting (e.g., PPP, current account deficits, non-linear momentum trends, candlesticks, break-out points, etc.), goes into what we ve defined here as the risk premium. None of it will change the fact that the forecast error is HUGE.* *Strictly speaking, I should say anything that forecasts s t+1 s t to be different that i t i t goes into the risk premium. 47 / 57

48 Announcement Effects and Monetary Policy Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements Often, what we read in the newspapers the Fed raises interest rates and USD goes up is a statement about a surprise at 11:00am in i t and how it affects S t at 11:01am. This is a contemporaneous announcement effect: how today s interest rate change (or some other surprise) affects today s spot exchange rate. It is not a statement about the currency returns that we ve been talking about, which involve both S t and S t+1. Announcement effects are explanations of what happens to exchange rates. They are not forecasts. An announcement effect describes how a surprise affects exchange rates. Unless you can forecast surprises which are basically defined as unforecastable then understanding announcement effects can t help you forecast exchange rates. They make good cocktail conversation, but not good trading rules. There is robust evidence that exchange rates react to the usual cast of characters interest rates, announcements about employment, inflation, GDP, etc. in the way that you think (e.g., good news implies an appreciating currency). But there s also robust evidence that the effects don t last very long (e.g., hours, maybe). 48 / 57

49 (continued) Parity Relations Two Pieces Popular Wisdom? Picture Brazil Picture: Brazil Random Walk Announcements Why? Macroeconomics can t really tell you. Some people call it market sentiment or something. I like to think that people have good reasons to bet on macro variables, but there aren t liquid markets in which to place these bets. So the FX market provides a proxy. Where one gets into trouble is by using the announcement-effect evidence to think about longer-term FX phenomenon. A bad trade deficit number may cause an instantaneous devaluation. But if one extrapolates and says that the USD has been weak these last years because of the trade deficit, one is immediately confronted with the fact that (i) there s been lots of times when USD has been going up and the trade deficit has been going down, (ii) for every country with a weak currency and a trade deficit one can usually find a country with a weak currency and a trade surplus. 49 / 57

50 Annual 9 Month Exercises 50 / 57

51 Annual Annual 9 Month Exercises Suppose that 12-month EUR interest rates are high: i eur = 5.0%, i usd = 2.0% USD/EUR = 1.3 What is the UIP-implied, expected spot exchange rate in 12 months? E t S t+1 = S t 1+i t 1+i t = = We call S t+1 /S t the depreciation rate of USD. Equivalently, it is the appreciation rate of EUR. We say that UIP implies a depreciation of USD of ( /1.3 1) = 2.857%. Back-of-the-envelope? UIP says that USD is expected to depreciate by 2 5 = 3%, which would be USD/EUR. Not quite right, but close enough for 9/10 on a test. Finally, if you find the language irritating calling an appreciation of USD a negative depreciation sorry. That s life with FX rates. We fix the units and struggle with the language. 51 / 57

52 9 Month Annual 9 Month Exercises Suppose that 9-month EUR interest rates are high: i eur = 5.0%, i usd = 2.0% USD/EUR = 1.3 What is the UIP-implied, expected spot exchange rate in 9 months? E t S t+1 = S t 1+ i t 12/9 1+ i t 12/9 = / /9 = This example is just to remind you that interest rates are always quoted in annualized terms. Here, the UIP-implied depreciation rate of USD is 2.169% over 9 months. The back-of-the-envelope answer is a depreciation of 2.25% to a level of USD/EUR. 52 / 57

53 Break Even Annual 9 Month Exercises Don t like thinking of UIP as a prediction? You can also think of it as a break-even investment rule. Suppose that one-year interest rates are 3% in the U.S. and 7% in Mexico. The exchange rate is 12.5 MXN/USD. What is the break-even future exchange rate? This is the exchange rate that would make you (after-the-fact) indifferent between investing a dollar in the U.S. or investing it in Mexico. The answer is just the UIP-implied rate of MXN/USD: S t+1 = 1+iMXN = MXN/USD 1+iUSD As an exercise, prove this. Calculate the USD-denominated return on investing in Mexico, given that S t+1 = You should find that it is 3%. The wrinkle here is that, as is the case in the market, the exchange rate is not quoted in USD units. The usefulness of the break-even context is this. We often hear the language the market s view, or the view that s priced into the market. For exchange rates, this is the UIP-implied rate. A trader who s view is the UIP-implied rate has no basis for a trade. 53 / 57

54 Exercises Annual 9 Month Exercises (a) Continuing with the above MXN/USD data, suppose that the interest rates are 3-month rates. What s the break-even FX rate? (b) The JPY/USD spot rate is 95. Japanese 6-month and 1-year interest rates are 1% and 2%. U.S. 6-month and 1-year interest rates are 2% and 1%. What is the UIP-implied path or, equivalently, the break-even path of the FX rate? (c) Continuing with (b), suppose that your view is that JPY will go up by 1/2% and then up by another 1/2%. What trade should you do? (d) Continuing with (c), suppose that you are pretty sure that JPY will go up by 1/2% starting in 6-months, but aren t sure what will happen between now and 6 months from now. What trade should you do? Answers: (a) , (b) JPY appreciates to and then devalues to or JPY appreciates by %, then depreciates by %, for a total of a % depreciation. Or (roughly), JPY goes up by 1/2% then falls by 1.5% for a total of a 1% fall. Note that I m not annualizing rates-of-change, (c) short 1-year USD and go long in 1-year JPY, (d) same as part (c), but ADD to it short 6-month JPY, long 6-month USD. 54 / 57

55 Logs Cont. Compounding 55 / 57

56 Logarithms The rate of change of some variable can be approximated as the change in the natural logarithm. This a useful approximation. It turns multiplicative into additive things. It makes statistics nice and normal (i.e., as in N(µ,σ 2 )). The idea is simple. The definition of a rate of change, d t,t+1, for a variable x t is simply 1 + d t,t+1 x t+1 x t. Logs Cont. Compounding Take logs,* log(1 + d t,t+1 ) = log( x t+1 x t ) = logx t+1 logx t. The approximation is: log(1 + d t,t+1 ) d t,t+1, which is a good approximation as long as d t,t+1 isn t too big (check it on your calculator). The result is that changes in logarithms are a good approximation for the rate of change. d t,t+1 logx t+1 logx t. * NOTE: By log(), I mean natural logarithm. Note that in Excel you must use the function LN(), not LOG(). 56 / 57

57 Cont. Compounding CIP (cont-compounding and f t logf t, s t logs t ): F t S t = e i t i t = f t s t = i t i t Logs Cont. Compounding UIP: i t i t = E t s t+1 s t Combining the two: f t s t = E t s t+1 s t = f t = E t s t+1 (5) Equation (5) is the continuously-compounded version of the forward-rate expectations hypothesis. It says that the log forward rate equals the conditional expectation of the log of the future spot rate. 57 / 57