# Evidence: Typically country by country time series regressions, following Fama s original

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1 FX We regoingtoreviewthecurrentstateoffxwork. Inmyopinionit sagoodbit behind the stock and bond work above, which means low-hanging fruit for you to apply the same ideas. (Note Lustig et. al. as a low hanging fruit example. ) The basic idea: Suppose the UK interest rate = 5%, US interest rate = 2%. Should you invest in UK? The naive view: Yes, you ll make 3% more. The traditional view: No, the pound will depreciate 3% (on average) The fact: The pound seems to go up! As with D/P the adjustment goes the wrong way. Evidence: Typically country by country time series regressions, following Fama s original in \$Return +1 = + ( ) Not even zero. Small R 2. This is the basis for the carry trade, borrow in low interest rate countries and lend in high interest rate countries. (Note the analogy to ride the yield curve also called a carry trade, borrow in low interest rate maturities andlendinhighmaturities. I long for a unifying view!) Variations: You can do this for two common choices of right hand variables, forwardspot spread or interest differential, and you can do this for two choices of left hand variable, exchange rate or excess return. As usual, everything is related by identities. The identies (or arbitrage) say that two ways of getting money to the same place give the same result. f t s t+1 i t i t * s t Figure 9: 1. Covered interest parity and a right hand variable identity. = i.e. going around the box gets zero + =0 195

2 Thus, regressions with on the right are the same as regressions with. (My units. Dollar per foreign currency. If and +1 are bigger numbers, that means you get more foriegn currency per dollar in the future. That s the same as earning a big interest rate abroad. 1Euro/Dollar today and 2Euro/dollar next year dollar appreciates, +1,:Holdaeuroforayearatzero is the same as losing 50% on dollars. Euro rates have to rise to compensate. If you want Dollars/Euro add negatives appropriately) 2. Left hand hand variable identity: You can look at exchange rate changes ( ) or expected returns. By an identity, +1 = ( +1 ) Thus, if you regress on either right hand variable, + =0 Either exchange rates are predictable or excess returns are predictable. Like bonds the first question was does expectations work? Is = ( +1 )? Fama figured out to do this by a) running +1 on b) much better, running +1 on as a much more powerful test c) once you see that isn t working, it s interesting to note +1 on d) I like expressing it in terms of which is more intuitive to me than forward rates. Preview: The CP common factor has not been done, covariance with what is only beginning, the present value relation hasn t been done, etc. Fact 1: Expectations is about right in levels (433). flat on average) (Just as the yield curve is pretty Fact 2: We whould see =+1in regressions. In fact: it s negative. Some numbers. From Asset Pricing 196

3 An update from Burnside et al. 197

4 On average vs. for countries. +1 = + ( )+ +1 We should see this as a panel data regression. The issue, just like credit spreads above. a) country dummies? b) time dummies c) common factors on the right hand side? Can we see it in a graph just like we did for bonds? 198

5 8 UK US TB \$ / Lb Higher interest rates are associated with stronger exchange. \$/LB goes up when UK rate goes up. There is something to the standard story! 2.IfyouinvestinhigherUKrates,youmakemoneyuntilpoundweakens,until \$/UK goes down. 3. A higher exchange rate goes on for many years at a time. These are 3 month rates. You see the same sluggish adjustment as in yields. Comments: 1. Carry trade by 0 and ( ) are very different! (See hoizontal lines in the graph). The right hand variable is very slow moving. 2. The R 2 is low (monthly data). It s economically large: All interest differential (and more?) is expected return, none expected depreciation ( 1 year). Again, read the regression as what is the information in the price not how do I start my hedge fund? 3. NB though, many hedge funds do essentially this. As in CP1 they usually trade enhanced carry, they have some idea of when to get out. They also form portfolios (something like Σ 1 ( )) 4. Economics? Low interest rate episodes are recessions, so this has the usual business cycle pattern. When the US risk premium is high, so is the premium for holding currency risk. 5. Wait, that was too quick. (a) Why does ( 0 ( +1) euro +1 \$ +1?)(AndtheoppositefortheEuroinvestor.) Models have some work to do. 199

6 (b) If so, we should not have separate regressors P for each country, there should be a common factor (maybe 1 ) that forecasts all currencies. Do like CP1 for currencies? It hasn t been done yet really. (c) If so, should forecast stock and bond returns, as 0 forecasts stock returns. And DP should forecast bond and FX returns. and 0 should forecast FX...Or there should be a reduced factor structure in which a common component of, ;, forecasts a common component of all returns. 6. is important. \$ Return +1 = + ( )+ +1; =1 2 There are country dummies in the regression. If you leave out you get Turkey or Brazil perpetually high (40%) matched by 40% inflation and devaluation. Thefactis morethanusual interest differential corresponds to a high return. Does this matter? Does slow moving right hand variable mean the estimate biases up? If there is a unit root in inflation, maybe the is meaningless, there is no usual differential. Paper topic! (We ll find that portfolio dummies vs. time dummies makes a big difference in looking at stocks too, see Discount rates discussion of Fama and French. 7. The graph suggests expectations works in longer run regressions. I m not aware of papers that do the right hand panel of the Fama-Bliss table well to document this well. (more low hanging fruit.) 8. What about the present value identity? Does this link to longer term regressions? Here is a stab at the question. +1 = +1 + = ( ) X = =0 This isn t much help, I m not getting any discounting and nominal exchange rates can go anywhere. But how about real exchange rates? Let =inflation, so real exchange rate change is +1 = Then, +1 = = ( ) = ( ) +1 where = ex-post real rates. = ( ) X = =0 200

7 So, if the expected real exchange rate must approach one in the long run, the current real exchange rate must be matched by real interest differentials or excess returns. I wonder which it is Big crashes the peso problem was invented precisely for this regression! Many governments do soft interventions leading to big left tails and long samples that don t include the left tails. 10. Strategies that involve small constant gain and occasional big crashes are ubiquitous in hedge funds etc. Dynamic trading can synthesize options. Earth quake insurance. Put options. This is very hard to tell by statistical measures. 11. The last two comments motivate Jurek, Burnside Eichenbaum Rebelo crash neutral currency trades. Jurek (April 2009 draft) Good: Updates, it investigates the peso problem, and the claim that UIP was profitable even if you bought crash insurance against peso problems. It extend pricing questions to currency options. I like to read a recent paper for the latest data, and som reassurance on the state of the art, so if you do better than this you re doing better than a big literature. Bad: In many ways it s an example of how not to write a paper. It s a train of thought and travelogue of experiments. Table IX is the paper. If the data kill you in revision, you have to rewrite the paper, not treat it as an update. Writing papers is the art of throwing things out. 1. Table 1: The standard regressions. Note the much better performance in the later priod. Note the small R2. 2. Table II carry trade portfolio returns. At least it s good to look at some portfolios across currencies rather than currency by currency regressions. The basic portfolio one is just long/short depending on the sign of, ignoring the amount. SPR is proportional to the amount of. (Why not a real portfolio, Σ 1 = Σ 1 ( + ( ))?) Note the high Sharpe ratios. Note that Portfolio returns can look good with low 2, like momentum! Portfolios are a way to look at ( ) ( ) not ( ) ( ), and the former is more interesting. 3. Carry = average interest differential. This is large warning! This suggests that there is one data point! How much of these profits are just sitting in one currency through the whole sample = 1 data point? See Figure 3 4. Figure 1: returns are scaled to same volatility. Note the big crash! Is it over? 5. Crash-neutral construction. P. 12 bottom use out of the money puts, but scale up the portfolio so at the money it has the same sensitivity to exchange rate variations. See Figure 5. Thus, simultaneously decreasing exposure to depreciations of the high interest rate currency and increasing exposure to its appreications (p 12) 201

9 Table 3: FF style portfolios HML and EW market. This is a cross sectional regression of = with prespecified factors. 1. Betas panel II: Market betas are all about 1 and hml betas rise. We knew this from the covariance matrix 2. It s not a tautology. The high interest rate countries will all fall or rise together. It didn t have to happen 3. Factor risk premiums. Of course, we account for cross sectional variation with a big on. You see the pattern that is higher in the high portfolios, and the betas are high there as well. Table5. Thisisbetter. Itusesthefirst two principal components as factors. is the first component, is the second. UIP risk premium is earned for covariance with the slope factor. This had to be. We have a pattern of - to + in the expected returns, so a successful factorhadtobe-to+aswell. Is this different from Cochrane Piazzesi who find covariance with the level factor? 1. No. CP had b 0 the b were all positive. So of course it had to be a level factor. 2. No, because CP are examining bonds, and here they re examining portfolios. CP model ( ) +1 = Thus, when is positive, CP will put long bonds in portfolio 1 and short bonds in portoflio 5. However, when is negative, CP will put short bonds in portoflio 1 and long bonds in portfolio 5. Covariance of bond returns with a level shock to yields is not the same thing as covariance of portfolio returns (which change composition) with a level shock to those portfolio returns. I suspect if we do the LRV procedure on CP bond data we would get exactly LRV s results. And vice versa? A good problem set/ paper question! Preview on portfolios and time series regressions: 1. Now, rthink about what they re doing relation between portfolio sorts and regressions. 2. Asset pricing is in the end about ( +1) = = tells you about ( +1 ), and forming portfolios based on also tells you ( +1 ). It s really a non-parametric forecasting regression with a rather inefficient kernel! 3. Similarly, the FF procedure amounts to = FF portfolios are a brilliant way to reduce a time-varying conditional problem to an unconditional problem. 4. Of course, we ve done this many times before, i.e. 0 = ( +1 +1); 0 = ( )= ( ) 203

10 5. Basically, but the details really matter. Is time series or cross sectional variation more important? Is the value of or the portfolio rank more important? The cp common factor investigation? Closing FX/predictability thoughts A common pattern across all assets: 1. Dividend yield forecasts stock returns 2. Long yield - short yield forecasts long-short bond returns 3. Foreign - domestic yield forecasts foreign - domestic returns 4. (Cross section B/M forecasts returns. In this case, both returns and earnings) Morefactsincommonwithstocks,bonds 1. Follow yield, All price variation = Expected returns 2. Missing adjustment (short run, i.e. 1year) 3. Expected returns are high in Bad times, P/D is low, is low relative to, and is low relative to. 4. Is there a single recession related factor?, term spread, bond forecast factor also forecast stock excess returns. Do they forecast fx? does forecast stock returns? Etc. More puzzles in international finance 1. News flow and price correlations 2. Fact: Net order flow is associated with price changes. ( order flow not trades ) 3. Don t jump to: Any order causes price changes. (Brandt and Kavaiecz coming up) 204

11 Brandt, Cochrane, Santaclara Puzzles in international finance pretty much define the field. 1. UIP (Just studied): seems to imply appreciation, not depreciation, at least for a while, and corresponding profits. 2. The volatility of exchange rates; correlation of real/nominal exchange rates. µ ln +1 =15% This is not matched 1-1 by inflation, so real exchange rates vary a lot. (a) (Mussa) Real relative prices across borders change when exchange rates change, suggesting sticky nominal prices. (b) When countries move from floating to fixed, relative prices across countries (sausage in Munich/Pizza in rome) become more stable, and relative prices of tradeables/nontradeables (Pizza in Rome/Oil in Rome) become more stable. Stock prices are a relative price too installed vs. uninstalled capital. This is more dramatic. (c) Put bluntly, why did this happen? (See pound plot) Figure 10: 3. Savings = investment and poor risk sharing across countries. One is about allocation across time, the other about allocation across states. (a) Permanent income logic means that temporary high income should be exported and then returned later. Also good news about future output (China opening) should lead to a consumption boom and huge imports of capital. Instead, China finances investment from domestic savings and exported the whole time. (This is an open economy facing world interest rates. If the whole world sees a boom, interest rates rise.) ( Feldstein-Horioka puzzle ) 205

12 (b) Complete markets, Pareto-Optimum means h X X i max 1 ( 1 )+ 2 ( 2 ) s.t = 1 0 ( 1 ) = 2 0 ( 2 ) 0 ( 1 ) = 0 ( 2 ) 0 ( 1 1 ) 0 ( 2 1 ) µ µ 1 2 = In fact, ( ) is small (numbers follow). Worse, consumption correlations are less than output correlations. (c) In both cases, I find it hilarous that as the world starts to look more like our models, people think this is a problem. Global imbalances is the buzzword for theideathatweneedtoslow down trade surpluses and deficits. Mortgage backed securities did a great job of sharing risk around the world. 4. Home bias in portfolios. US people hold mostly US equities, UK people hold more UK equities and so forth. This is only a puzzle however relative to a world capm model, in which the investor has no job, cares equally about consumption from all countries, etc. There are lots of easy reasons it s optimal for portfolios to focus on your own country, as in my Earth vs. Mars example. 5. Currency crashes, panics, etc. (Much silliniess; contagion capital flight. etc.) 6. International is RIPE for work, as witnessed by Lustig et al and this paper. Simple models are making big progress. This paper preview: We can connect domestic and foreign discount factors by a simple change of units. Equivalently, +1 = = +1 \$ \$ +1 \$ +1 = +1 +ln +1 1 = ( ) = ( ) +1 = ( +1 +1) 206

13 where = any return (domestic or foreign) expressed in foreign currency. This is cool! Exchange rates let you see mrs, directly, ex post! Well, they let you see differences in mrs, but that s something. Important distinguish the discount factor for returns expressed in domestic currency from discount factor that only prices domestic returns. The latter makes no sense unless markets are segmented somehow. All we re doing here is saying that we can find a discount factor that predicts a given set of returns converted to Euros from a discount factor that prices the same returns expressed in dollars. (Which is, when you see it, rather trivial.) We are not constructing a discount factor that prices Euro stocks from a discount factor that (only) prices dollar stocks. You can do the same thing with real vs. nominal discount factors. Just mulitply and divide by. Now, ln +1 = ( +1 ) = ³ ( ) ( ) What does it take to fit the facts? ( +1 )=15% 1. Asset pricing, risk sharing is better than you think. From asset markets, ( ) ( ) ( ) we need at least ( ) =50%. Since ( ) is much bigger than ( ) we need a lot of positive corrleation = = 0 50 (1 ) = (1 ) = Risk sharing is better than you think meaning marginal utility growth is very correlated across countires. 2. Asset pricing, or exchange rates are too smooth. Imposing =0, ( +1 )= 2 ( ) = =0 71 We should see 70% variation in exchange rates! 3. Consumption. If we use consumption data,, small risk aversion, and =0 as suggested by the data, no matter what we do with on the right hand side ( ) is just not enough to add up to the observed ( ). ( ) = ( ) = 2 2 ( ) 2 2 ( ) = 2 2 ( ) (1 ) = = ( )(1 ) 207

14 This isn t as bad as the equity premium (that s the whole point), where we needed ( ) =0 5. It sstillnoteasy.with =0and ( ) =0 01 we need =10. ( ) =2%gets us down to =5. And surely there s some positive correlation. In sum, you can see that models (especially models with risk sharing) have trouble producing enough exchage rate movement. The paper details: We compute a risk sharing index 1 2 (ln ln ) 2 (ln )+ 2 ( ) =1 2 (ln +1 ) 2 (ln )+ 2 ( ) Why? ln =2ln also violates risk sharing, but correlation is one. Procedure: Just like Hansen-Jagannathan. Find the minimum variance discount factors to price both domestic and foreign assets, expressed in dollars, and vice versa. We use continuous time so we can do logs vs. levels ("E(log) = log(e) theorem" is true in continuous time, with 1/2 2 terms) Continuous time Λ = Λ ln Λ = ln + ln Λ = ( + ) Λ Λ = 0 Σ 1 ln Λ = + 1 Λ 2 0 Σ 1 0 Σ 1 µ ln Λ 2 = 0 Σ 1 Λ so the regular calculation works in logs in continuous time. Table 2,3 gives the basic calculation. Strongly recommended reading: The introduction on transport costs (Earth vs. Mars) and incomplete markets, and reconciliation p. 692 i.e. apples and oranges. 208

15 Earth vs. Mars: Justread673. Betterinprint. Supposetherearecompletefinancial assets and communication but no goods may flow. If mars gets a good shock,mars stock goes up. The exchange rate must go down. and must be uncorrelated in the end. Knowing this, there is no advantage to Mars stock in the first place, so your portfolio should be completely home biased. Complete financial assets do not imply "perfect risk sharing" nor constant exchange rates, nor absence of home bias. Money, capital can t flow. International is about transport costs, not markets. (Example, crashes investors pulled capital out. They can t They can sell to locals at a cheap price; trade claim on US govt for claim to factory, but it needs a ship to remove capital.) Incomplete markets. Now = + again. 1. Reminder: Risk sharing in incomplete markets: = ( )= = ( ) = + we should use asset markets to share as much as possible. 2. It is not true that = + for arbitrary pairs. It is true that for any that prices assets expressed in dollars, we can construct an that prices assets expressed in Euros. Again, this is just a change of units. But that discount factor may not equal foreign consumption growth. It is true that = +. Thus minimum-variance discount factors in the payoff space do obey the identity. In this sense, what we are learning is do transport costs mean that we are not able to use asset markets to share as many risks as possible? The paper also asks if incomplete markets are quantitatively plausible. ( ) rises as becomes less correlated, so incomplete markets makes the equity premium puzzle that much worse (as in the correlation puzzle refinement of Hansen-Jagannathan that lower correlation of with implies a higher ( ) bound). How much international finance uses vs. does not use complete markets?? Be very careful here. if = and then you are assuming complete markets. If then you re not. = = ( ) ( ) Misconceptions: 209

16 1. Each investor is allowed to invest in all assets the HJ, minimum variance discount factor for all assets as viewed by each investor. Our equation only applies as a change of units. These are NOT the minimum variance discount factor for domestic assets and the minimum discount factor for foreign assets. Why not? You can compute such quantities, but they are not connected by the exchange rate. The paper does nothing about what if there are asset market frictions so you can t trade each other s assets? We allow markets to be incomplete, but once D can buy them, so can F. Would it be interesting to give the countries fundamentally different spaces, or (the same thing) prices that are different by shadow costs or transactions costs as well as by exchange rates? Yes, but we didn t do it. 2. The discount factor is not the optimal portfolio the market portfolio, or a portfolio anyone actually holds. The correlation of discount factors means nothing about correlation of portfolios. For example, if our income shocks are the same, we can have very correlated discount factors, very correlated but utterly different portfolios. 3. The correlation of stock markets, which underlie the usual benefits of international diversification, is not really in the calculation at all. (Technically, it is reflected in the Σ part of 0 Σ 1, but higher asset correlation does not imply higher discount factor correlation.) Again, we allow trade in both assets by both investors. 4. Converesly, the discount factor is not a portfolio that anyone holds, so highly correlated discount factors do not mean portfolios are highly correlated. Thus, also, home bias does not contradict our calculations. 210

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