ECON 8872: International Finance UIP Puzzle, Lecture 1 Rosen Valchev (Boston College)

September 16, 2017

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UIP Intro • We turn to studying exchange rates – the price of one currency in

terms of another • We’ll typically denote the (nominal) exchange rate by St , and quote it terms of home currency per foreign currency • St ↑ = depreciation (appreciation) of the home (foreign) currency

• How is it determined? • Asset market equilibrium: Demand and supply of currency deposits • Uncovered Interest Parity (UIP) is the key condition underlying

exchange rate determination in standard open economy models • Basic intuition: risk-free returns across countries are equalized • Standard in benchmark open economy models • Not just due to linearization

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UIP Intro

• The UIP condition is

E (s − s ) + it∗ = it | t t+1 {z t }

Return on Foreign Bonds

• The log-linearized version of

Et (

St+1 (1 + it∗ )) = 1 + it St

• Return on foreign default-free investment = return on home

default-free investment

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UIP Intro • It is standard to write it as

Et (st+1 − st ) = it − it∗ • Home currency depreciation = interest rate differential • High-interest rate currencies are expected to depreciate and offset

interest rate differentials • In a standard model, monetary policy sets the interest rates, and this determines the exchange rate • Solve forward st = lim Et (st+k ) + k→∞

∞ X

∗ Et (it+k − it+k )

k=0

• The asset view of the exchange rate • It equals long-run mean (typically non-stationary) plus future discounted sum of interest rate differentials Rosen Valchev (BC)

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Covered Interest Parity • Covered parity equalizes returns on positions where exchange rate risk

has been hedged • Forward contract – a time t agreement to buy currency at a future

date t + k at a price specified at time t • Let Ft,k be the time t price of a forward contract with value date t + k • Again expressed in terms of home currency per unit of foreign

currency • Most papers focus on the one-period ahead forward, Ft,1 and denote

it simply by Ft • You can use forward to sell exchange rate risk on foreign investments

and create risk-free investments

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Covered Interest Parity • Take $1 today and invest it in foreign currency at interest rate 1 + it∗ • Next period you have 1 + it∗ foreign currency. You can exchange it

back at the spot rate St+1 and obtain St+1 (1 + it∗ )

• But instead, you could have sold that future (1 + it∗ ) of foreign

currency at the rate Ft . Then your return next period is Ft (1 + it∗ ) • Equalizing risk-free returns at home and abroad you get the Covered

Interest Parity (CIP) condition Ft (1 + it∗ ) = 1 + it St • CIP in its most popular, log-linearized form

ft − st + it∗ = it Rosen Valchev (BC)

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UIP and CIP together • So we have two interest parity conditions • UIP

Et (st+1 − st ) = it − it∗ • CIP

ft − st = it − it∗ • Note that only the second one is a true arbitrage condition – it

equates risk-free USD return to risk-free USD return • The return is USD because all foreign currency exposure has been

removed at time t • The UIP condition abstracts from both risk and liquidity factors • You expose yourself to uncertainty in St+1 • You hold a long position in one currency, and short in another – exposed to differential liquidity

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UIP and CIP together • So it’s perhaps not surprising that the CIP actually holds in the data

but UIP does not • Although, post-2009, seemingly not quite as well as it used to

• Tests of the UIP are numerous and come in many varieties • A lot of papers test

Et (st+1 − st ) = it − it∗ and try to forecast exchange rates with interest rates. • Others put the CIP and UIP together and test

Et (st+1 − st ) = ft − st • RHS is called ”forward discount” and you test whether the forward

correctly predicts future exchange rates Rosen Valchev (BC)

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Excess currency returns • The excess return of foreign currency over home currency is given by

λt+1 = st+1 − st + it∗ − it • Substituting in

it − it∗ = ft − st • We get

λt+1 = st+1 − ft • You can construct the trade either through actual bond investments

or through forwards • Under UIP, there should be no forecastable excess returns

Et (λt+1 ) = 0

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General asset view of exchange rates

• If we don’t assume UIP, Et (λt+1 ) 6= 0, hence solving

λt+1 = st+1 − st + it∗ − it forward we get st = lim Et (st+k ) + k→∞

∞ X

∗ Et (it+k − it+k ) −

k=0

∞ X

Et (λt+1 )

k=0

• so the exchange rate is strong (appreciated) when • Interest rate differentials are expected to be positive • Excess returns are expected to be negative (low)

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Should it be obvious that UIP does not hold? • At first look it looks naive – “risk-free” foreign assets are not really

risk free because of the exchange rate uncertainty • So why would you ignore risk? • But not easy to find big role for simple risk-stories • Buying German bonds is risky for Americans, but buying American bonds is risky for German investors • St+1 bites both ways, if it was to be unconditionally risky to invest abroad, then both sides of the market would require compensation • Can’t be an equilibrium (up to Jensen’s inequality) • UIP holds (or holds up to very small numeric term) in standard

models with risk-aversion too • So no, it is not obvious that we should not expect UIP to hold • It’s more tricky than equity risk-premium, and as we’ll see it requires

more than simply large risk-aversion

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Two main ways of testing UIP 1

You can perform direct regression analysis • To see whether interest rates (forward discount) predict exchange rates

with slope of 1 • Or to see if Et (λt+1 ) is forecastable 2

You can look if there are any persistently profitable exchange rate trading strategies • Carry trade is the most famous, and most profitable one • Borrow in low interest, invest in high interest rate currencies • But there’s also momentum, value, etc.

• We start with one of the first and most famous papers – Fama(1984)

– which focuses on regressions using the forward discount • Then we look at one of the papers in the burgeoning literature

documenting profits of currency trades

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Fama(84) Intro

• At the time, it was already recognized that forward rates are bad

predictors of future exchange rates, i.e. • Corr (ft − st , st+1 − st ) is small and often insignificant

• However it was unclear if that was because • exchange rates are inherently volatile and hard to forecast or • forwards include significant time-varying premia • This paper shows that • Time-varying premia accounts for most of the variation in forwards • The premia are negatively correlated with the expected spot rate • Both serve to decrease Corr (ft − st , st+1 − st )

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Forward Rates

• From a purely statistical point of view, we can decompose

ft = Et (st+1 ) + pt • where pt is the “forward premium” • For the statistical analysis that is to follow it does not matter what is

the source of the premium – to give it an economic meaning we’ll need a model • For example, • Risk-premium – perhaps st tends to be high in ”bad” states of the

world. Then the excess return λt+1 is risky

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Forward Rates • Note that

ft − st = ft − st+1 + st+1 − st re-arrange ft − st+1 = pt + (Et (st+1 ) − st+1 ) | {z } =−λt+1

• Then consider the regressions

ft − st+1 = α1 + β1 (ft − st ) + ε1,t+1 st+1 − st = α2 + β2 (ft − st ) + ε2,t+1 • Non-zero β1 tells us that the forward premium is a significant

component of excess currency returns ft − st+1 = −λt+1 • If yes, then Et (st+1 − st ) does not offset interest rate differential

• Non-zero β2 tells us that the forward discount can predict future

exchange rate changes Rosen Valchev (BC)

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The regression coefficients • The coefficients are

β1 =

Var(pt ) + Cov(pt , Et (st+1 − st )) Var(pt ) + Var(Et (st+1 − st )) + 2 Cov(pt , Et (st+1 − st ))

β2 =

Var(Et (st+1 − st )) + Cov(pt , Et (st+1 − st )) Var(pt ) + Var(Et (st+1 − st )) + 2 Cov(pt , Et (st+1 − st ))

• Kind of split variation in terms of premium (pt ) and future expected

depreciation Et (st+1 − st ). • They are also very closely related

β1 = 1 − β2 • Or perhaps somewhat more intuitively

−β1 = β2 − 1 Rosen Valchev (BC)

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Data

• 1-month exchange rate changes and forward contracts • So

ft − s t • gives difference in 1 month interest rates

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• Exchange rate changes look like white noise – not serially correlated • Same for the excess return λt+1 = st+1 − ft • It is primarily driven by the exchange rate component • Notice that the exchange rate depreciation’s standard deviation is like 10 times larger than that of the interest rate differential • Interest rate differentials are quite persistent • Although generally speaking stationary

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• Notice first the negative β2 estimates • UIP implies β2 = 1 • So not only exchanger rate movements do not help close interest rate differentials, they tend to exacerbate them • You can see that in the high estimates of β1 – excess returns are quite forecastable with current interest rates • Recall

β2 =

Var(Et (st+1 − st )) + Cov(pt , Et (st+1 − st )) Var(pt ) + Var(Et (st+1 − st )) + 2 Cov(pt , Et (st+1 − st ))

• So the premium must be negatively correlated with exchange rate

changes • In particular,the negative β2 implies that

Var(Et (st+1 − st )) < | Cov(pt , Et (st+1 − st ))| < Var(pt )

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Main takeaway in simpler terms • The issue is that

Cov(it − it∗ , λt+1 ) < 0 Cov(it − it∗ , Et (st+1 − st )) ≤ 0 • Note that by definition

Et (st+1 − st ) = it − it∗ + λt+1 • Whenever you regress exchange rates on interest rate differentials there

is an omitted variable bias • So you need to explain why • Exchange rates and excess returns have volatility of about 10 times as high as it − it∗ • The covariance between excess returns (λt+1 ) and interest rates (it − it∗ ) is negative • Volatility of the expected excess return is greater than volatility of expected depreciation Rosen Valchev (BC)

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BER(2008) – Intro • We saw regression tests that show the UIP condition fails • Another way to test for it is to look for profitable trading strategies • If UIP holds, then returns are equalized • Most popular such strategy is the “carry trade” • Borrow in low interest rate currency, invest in high interest rate currency • This makes a lot of money – exchange rates do not close the interest rate gap • Return on carry is zt+1 = xt ( • where

( xt =

Rosen Valchev (BC)

St+1 (1 + it∗ ) − (1 + it )) St 1 −1

, if it∗ > it , if ifit∗ ≤ it

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BER(2008) – Intro • You can also implement the carry with forwards with a payoff

z˜t+1 = xt ( • where

( 1 xt = −1

St+1 − 1) Ft , if Ft < St , if Ft ≥ St

• Substitute the CIP to get

xt (

St+1 1 + it∗ − 1) St 1 + it

• The returns are equivalent up to first order

st+1 − st + it − it∗ = st+1 − ft Rosen Valchev (BC)

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BER (2008) – Intro

• You can implement this on single currencies or on portfolio of

currencies • All trades are trying to exploit the fact that exchange rates, on

average, do not offset interest rate differentials • The payoffs are correlated – a USD appreciation affects all USD carry

trades • Turns out, though, there are large gains from diversification • So idiosyncratic shocks play a large role in carry trade payoffs. • The Sharpe Ratios (risk-adjusted) returns rise by 50% when you

consider a portfolio of carry trades

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Data • More or less the standard data set in this literature • Datastream: 1976:M1 - today • Best to start in late 70s to avoid fixed exchange rates and other capital controls • Get 30-day forward and spot rates for OECD countries • They strangely define St in terms of foreign currency per home

currency • Typically people use home currency per foreign currency

• They are careful to account for bid-ask spreads, so as to construct

“feasible” payoffs • They can then compute payoffs actually available to someone trading

with market orders

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• Taking into account the bid/ask spreads the strategy is

  1 xt = −1   0

, if Fta < Stb , if Ftb ≥ Sta , otherwise

• Taking into account the bid-ask spread you may not always choose to

invest due to transaction costs • They consider three types of strategies 1 Individual currency carry 2 Equally weighted carry 3 High minus low

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• Sharpe Ratio

E (Ri − R f ) SR = p Var(Ri − R f ) • A measure of risk-adjusted return • It is the optimal measure for CARA utility and Gaussian returns • i.e. mean-variance portfolios are efficient • It is a standard, model-free measure of risk-adjusted returns • Only optimal in CARA-Normal setups, but even in other setups you

can show it is at least approximately optimal • e.g. CRRA - Log-Normal returns

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• Equally weighted portfolio has a Sharpe Ratio of 0.83 (1.23 in late

sample) • The median individual currency Sharpe Ratio is 0.5 (0.54) • The high-low strategy has Sharpe Ratio 0.54 (0.55) • Biggest difference with equally weighted carry is much larger volatility • Thus, there are apparent diversificaiton benefits – not all carries with

USD as one of the currencies are always positive or negative at the same time • Using single currency or high-low strategies you do not enjoy the

diversification benefits that this non-perfect correlation offers • With equally weighted carry, negative payoffs in a period are offset with

the positive payoffs in other currencies • This is an important insight – it can’t be simply something that is

unique to the USD

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• There are important diversification benefits to be had in terms of

reducing kurtosis as well • Equally weighted carry is much less negatively skewed • In fact, it is less skewed than the US stockmarket • That goes against the saying you hear some time that carry trades are

”very skewed” • The perception of “hugely” skewed carry payoffs is because of the fact

that carry traders typically use a lot of leverage • Here they report returns on carry with a total of $1 exposure, no

leverage • Leverage amplifies the negative skewness • Generally carry returns are a little lower than stock market – 3 − 4%

excess return, vs 5 − 7% on stock market • So if you want to match stock market return, you do need to lever up a

bit

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