# ARBITRAGE AND PARITY CONDITIONS - IRP, PPP, IFE, EH & RW

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ARBITRAGE AND PARITY CONDITIONS IRP, PPP, IFE, EH & RW Arbitrage in FX Markets Arbitrage Definition It is an activity that takes advantages of pricing mistakes in financial assets in one or more markets. It involves no risk and no capital of your own. • There are 3 types of arbitrage (1) Local (sets uniform rates across banks) (2) Triangular (sets cross rates) (3) Covered (sets forward rates) Note: The definition presents the ideal view of (riskless) arbitrage. “Arbitrage,” in the real world, involves some risk. We will call this arbitrage pseudo arbitrage. 1

Local Arbitrage (One good, one market) Example: Suppose two banks have the following bid-ask FX quotes: Bank A Bank B USD/GBP 1.50 1.51 1.53 1.55 Sketch of Local Arbitrage strategy: (1) Borrow USD 1.51 (2) Buy GBP 1 from Bank A (3) Sell GBP 1 to Bank B (4) Return USD 1.51 & make a USD .02 profit (1.31% per USD borrowed) Note I: All steps should be done simultaneously. Otherwise, there is risk! (Prices might change). Note II: Bank A and Bank B will notice a book imbalance. Bank A will see all activity at SA,ask (traders placing buy GBP orders) and Bank B will see all the activity at SB,bid (sell GBP orders). They will adjust the quotes. Say, Bank A increases the ask quote to 1.54 USD/GBP (SA,ask ↑). ¶ Triangular Arbitrage (Two related goods, one market) Triangular arbitrage is a process where two related goods set a third price. • In the FX Markets, triangular arbitrage sets FX cross rates. • Cross rates are exchange rates that do not involve the USD. Most currencies are quoted against the USD. Thus, cross-rates are calculated from USD quotations. • For example, a JPY/GBP quote is derived from SJPY/USD,t (say, 100 JPY/USD) & SUSD/GBP,t (say, 1.60 USD/GBP) • The cross-rates are calculated in such a way that arbitrageurs cannot take advantage of the quoted prices. Otherwise, triangular arbitrage strategies would be possible. For example, using above quotes: SJPY/GBP,t = SJPY/USD,t * SUSD/GBP,t = 160 JPY/GBP 2

Example: Suppose Bank One gives the following quotes: SJPY/USD,t = 100 JPY/USD SUSD/GBP,t = 1.60 USD/GBP SJPY/GBP,t = 140 JPY/GBP Take the first two quotes ⇒ Implied (no-arbitrage) JPY/GBP: SIJPY/GBP,t = SJPY/USD,t * SUSD/GBP,t = 160 JPY/GBP > SJPY/GBP,t At St = 140 JPY/GBP, Bank One undervalues the GBP against the JPY (with respect to the first two quotes).

Example (continuation): Note: It does not matter which currency you borrow in step (1). Recall the pricing mistake: Bank One undervalues the GBP against the JPY (with respect to the first two quotes), SIJPY/GBP,t = 160 JPY/GBP > SJPY/GBP,t = 140 JPY/GBP Sketch of Triangular Arbitrage (Key: Buy undervalued GBP with the overvalued JPY): (1) Borrow JPY 100 (2) Sell the JPY for GBP (at St = 140 JPY/GBP). Get GBP 0.7143 (3) Sell the GPB for USD (at St = 1.60 USD/GBP). Get USD 1.1429 (4) Sell the USD for JPY (at St = 100 JPY/USD). Get JPY 114.29 ⇒ Π = JPY 14.29 (14.29% per USD borrowed). Covered Interest Arbitrage (4 instruments: 2 goods per market and 2 markets) Open the third section of the WSJ: Brazilian bonds yield 10% and Japanese bonds 1%. Q: Why wouldn't capital flow to Brazil from Japan? A: FX risk: Once JPY are exchanged for BRL (Brazilian reals), there is no guarantee that the BRL will not depreciate against the JPY. ⇒ The only way to avoid this FX risk is to be covered with a forward FX contract. 4

Intuition: Suppose today, at t=0, we have the following data: iJPY = 1% for 1 year (T=1 year) iBRL = 10% for 1 year (T=1 year) St = .025 BRL/JPY Carry Trade: A strategy to take “advantage” of interest rate differentials: Today (time t=0), we do the following: (1) Borrow JPY 1000 at 1% for 1 year. (At T=1 year, we will need to repay JPY 1010.) (2) Convert to BRL at .025 BRL/JPY. Get BRL 25. (3) Deposit BRL 25 at 10% for 1 year. (At T=1 year, we will receive BRL 27.50.) At time T=1 year, we do the final step: (4) Exchange BRL 27.50 for JPY at ST=1-year ⇒ Π = BRL 27.50 / ST=1-year – JPY 1010 Problem with this strategy: It is risky ⇒ today (t=0), ST=1-year is unknown Suppose at t=0, a bank offers Ft,1-year = .026 BRL/JPY. Then, at time T=1 year, we do the final step: (4’) Exchange BRL 27.50 for JPY at .026 BRL/JPY. ⇒ We get JPY 1057.6923 (= BRL 27.50/.026 BRL/JPY). ⇒ П = JPY 1057.6923 – JPY 1010 = JPY 47.8 or 4.78% per JPY borrowed. Now, instead of borrowing JPY 1000, we will try to borrow JPY 10 billion (and make a JPY 478M profit) or more. Obviously, no bank will offer a .026 BRL/JPY forward contract! ⇒ Banks will offer Ft,1-year contracts that produce П ≤ 0. 5

Interest Rate Parity Theorem Q: How do banks price FX forward contracts? A: In such a way that arbitrageurs cannot take advantage of their quotes. To price a forward contract, banks consider covered arbitrage strategies. Notation: id = domestic nominal T days interest rate (annualized). if = foreign nominal T days interest rate (annualized). St = time t spot rate (direct quote, for example USD/GBP). Ft,T = forward rate for delivery at date T, at time t. Note: In developed markets (like the US), all interest rates are quoted on annualized basis. Now, consider the following (covered) strategy: 1. At t=0, borrow from a foreign bank 1 unit of a FC for T days. At time T, We pay the foreign bank FC: (1+if * T/360). 2. At t=0, exchange FC 1 = DC St. 3. Deposit DC St in a domestic bank for T days. At time T, we receive DC: St(1+id * T/360). 4. At t=0, buy a T-day forward contract to exchange DC for FC at a Ft,T. At time T, we exchange (in DC) St(1+id * T/360) for FC, using Ft,T. We get FC: St(1+id * T/360)/Ft,T. This strategy will not be profitable if, at time T, what we receive in FC is less or equal to what we have to pay in FC. That is, arbitrage will force: St (1 + id * T/360)/Ft,T = (1 + if * T/360). S t (1 i d x T/360) Solving for Ft,T, we get: Ft, T (1 i f x T/360) 6

(1 i d x T/360) Ft,T S t (1 i f x T/360) This equation represents the Interest Rate Parity Theorem (IRPT or just IRP). It is common to use the following linear IRPT approximation: Ft,T St [1 + (id – if) * T/360]. This linear approximation is quite accurate for small differences in (id – if). Example: Using IRPT. St = 106 JPY/USD. id=JPY = .034. if=USD = .050. T = 1 year Ft,1-year = 106 JPY/USD * (1+.034)/(1+.050) = 104.384 JPY/USD. Using the linear approximation: Ft,1-year 106 JPY/USD * (1 – .016) = 104.304 JPY/USD. Example 1: Violation of IRPT at work. St = 106 JPY/USD. id=JPY = .034. if=USD = .050. Ft,1-year-IRP = 106 JPY/USD x (1 – .016) = 104.304 JPY/USD. Suppose Bank A offers: FAt,1-year= 100 JPY/USD. FAt,1-year= 100 JPY/USD < Ft,1-year-IRP (a pricing mistake!) The forward USD is undervalued against the JPY. Let’s take advantage of Bank A’s mistake: Buy USD forward. Sketch of a covered arbitrage strategy: (1) Borrow USD 1 from a U.S. bank for one year at 5%. (2) Exchange the USD for JPY at St = 106 JPY/USD. (3) Deposit the JPY in a Japanese bank at 3.4%. (4) Cover. Buy USD forward (Sell forward JPY) at FAt,1-yr= 100 JPY/USD 7

Example 1 (continuation): t=today T = 1 year Borrow 1 USD 5% USD 1.05 Deposit JPY 106 3.4% JPY 109.6 Cash flows at time T = 1 year, (i) We get: JPY 106 * (1+.034)/(100 JPY/USD) = USD 1.096 (ii) We pay: USD 1 * (1+.05) = USD 1.05 П = USD 1.096 – USD 1.05 = USD .046 That is, after one year, the U.S. investor realizes a risk-free profit of USD .046 per USD borrowed (4.6% per unit borrowed). Note: Arbitrage will force Bank A’s quote to quickly converge to Ft,1-yr-IRP = 104.3 JPY/USD. ¶ Example 2: Violation of IRPT 2. Now, suppose Bank X offers: FXt,1-year= 110 JPY/USD. FXt,1-year= 110 JPY/USD > Ft,1-year-IRP (a pricing mistake!) The forward USD is overvalued against the JPY. Let’s take advantage of Bank X’s overvaluation: Sell USD forward. Sketch of a covered arbitrage strategy: (1) Borrow JPY 1 for one year at 3.4%. (2) Exchange the JPY for USD at St = 106 JPY/USD (3) Deposit the USD at 5% for one year. (4) Cover. Sell USD forward (Buy forward JPY) at FXt,1-yr= 110 JPY/USD. Cash flows at T=1 year: (i) We get: USD 1/106 * (1+.05) * (110 JPY/USD) = JPY 1.0896 (ii) We pay: JPY 1 * (1+.034) = JPY 1.034 П = JPY 1.0896 – JPY 1.034 = JPY .0556 (or 5.56% per JPY borrowed) 8

The Forward Premium and the IRPT Reconsider the linearized IRPT. That is, Ft,T St [1 + (id – if) * T/360]. A little algebra gives us: (Ft,T - St)/St * 360/T (id – if) Let T=360. Then, p id – if. Note: p measures the annualized % gain/loss of buying FC spot and selling it forward. We think of p + if as the annualized return from borrowing DC and investing in FC (covered) for T days. The opportunity cost of doing this is id. Equilibrium: p exactly compensates (id – if) → No arbitrage → No capital flows. Equilibrium: p id - if. IPR Line IRP Line id -if (id - if) > p (Capital inflows) 45º B p (forward premium) Example: Go back to Example 1 p = [(Ft,T - St)/St] * 360/T = [(100 – 106)/106] * 360/360 = - 0.0566 p = - 0.0566 < (id - if) = - 0.016 Arbitrage (pricing mistake!) Capital flows to DC country 9

Under the linear approximation, we have the IRP Line: IRP Line i d - if B (Capital inflows) - Example 1 (Capital outflows) A - Example 2 45º p (forward premium) Consider point A (like in Example 2): p > id – if (or p + if > id), Borrow at id & invest at if: Capital fly to the foreign country! Intuition: What an investor pays to finance the foreign investment, id, is more than compensated by the high forward premium, p, plus if . IRPT: Assumptions Behind steps (1) to (4), we have implicitly assumed: (1) Funding is available. Step (1) can be executed. (2) Free capital mobility. Step (2) and later (4) can be implemented. (3) No default/country risk. Step (3) and (4) are safe. (4) No 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). 10

IRPT with Bid-Ask Spreads Exchange rates and interest rates are quoted with bid-ask spreads. Consider a trader in the interbank market: She buys FC (Sask,t, Fask,t) or borrows at the other party's ask quote (iask). She sells FC (Sbid,t, Fbid,t) or lends at the bid price (ibid). There are two roads to take for arbitrageurs: (1) Borrow domestic currency (at iask,d). (2) Borrow foreign currency (at iask,f). • Bid’s Bound: Borrow Domestic Currency (1) A trader borrows DC 1 at time t=0, and repays 1+iask,d at time=T. (2) Using the borrowed DC 1, she buys FC spot at Sask,t , getting (1/Sask,t). (3) She deposits the FC at the foreign interest rate, ibid,f. (4) She sells the FC forward for T days at Fbid,t,T This strategy would yield, in terms of DC: (1/Sask,t) (1+ibid,f) Fbid,t,T. In equilibrium, this strategy should yield no profit. That is, (1/Sask,t) (1+ibid,f) Fbid,t,T (1+iask,d). Solving for Fbid,t,T, Fbid,t,T Sask,t [(1+iask,d)/(1+ibid,f)] = Ubid. 11

• Ask’s Bound: Borrow Foreign Currency (1) The trader borrows FC 1 at time t=0, and repay 1+iask,f. (2) Using the borrowed FC 1, she sells the FC spot for Sbif,t units of DC. (3) She deposits the DC at the domestic interest rate, ibid,d. (4) She buys the FC forward for T days at Fask,t,T Following a similar procedure as the one detailed above, we get: Fask,t,T Sbid,t [(1+ibid,d)/(1+iask,f)] = Lask. Example: IRPT bounds at work. Data: St = 1.6540 - 1.6620 USD/GBP iUSD = 7¼-½, iGBP = 8⅛ – ⅜, Ft,one-year= 1.6400 - 1.6450 USD/GBP. Check if there is an arbitrage opportunity (we need to check the bid’s bound and ask’s bound). i) Bid’s bound covered arbitrage strategy: 1) Borrow USD 1 at 7.50% for 1 year Repay USD 1.07500 in 1 year. 2) Convert to GBP & get GBP 1/1.6620 = GBP 0.6017 3) Deposit GBP 0.6017 at 8.125% 4) Sell GBP forward at 1.64 USD/GBP we get (1/1.6620) * (1 + .08125) * 1.64 = USD 1.06694 No arbitrage: For each USD borrowed, we lose USD .00806. 12

Example (continuation): ii) Ask’s bound covered arbitrage strategy: 1) Borrow GBP 1 at 8.375% for 1 year we will repay GBP 1.08375. 2) Convert to USD & get USD 1.6540 3) Deposit USD 1.6540 at 7.250% 4) Buy GBP forward at 1.645 USD/GBP we get 1.6540 * (1 + .07250) * (1/1.6450) = GBP 1.07837 No arbitrage: For each GBP borrowed, we lose GBP 0.0054. Note: The bid-ask forward quote is consistent with no arbitrage. That is, the forward quote is within the IRPT bounds. Check: Ubid = Sask,t[(1+iask,d)/(1+ibid,f)] = 1.6620 * [1.0750/1.08125] = 1.6524 USD/GBP Fbid,t,T = 1.6400 USD/GBP. Lask = Sbid,t[(1+ibid,d)/(1+iask,f)] = 1.6540 * [1.0725/1.08375] = 1.6368 USD/GBP Fask,t,T = 1.6450 USD/GBP. ¶ Synthetic Forward Rates A trader is not able to find a specific forward currency contract. This trader might be able to replicate the forward contract using a spot currency contract combined with borrowing and lending. This replication is done using the IRP equation. Example: Replicating a USD/GBP 10-year forward contract. iUSD,10-yr = 6% iGBP,10-yr = 8% St = 1.60 USD/GBP T = 10 years. Ignoring transactions costs, she creates a 10-year (implicit quote) forward quote: 1) Borrow USD 1 at 6% for 10 years 2) Convert to GBP at 1.60 USD/GBP 3) Invest in GBP at 8% for 10 years 13

Transactions to create a 10-year (implicit) forward quote: 1) Borrow USD 1 at 6% 2) Convert to GBP at 1.60 USD/GBP (GBP 0.625) 3) Invest in GBP at 8% Cash flows in 10 years: (1) Trader will receive GBP 1.34933 (=1.0810/1.60) (2) Trader will have to repay USD 1.79085 (= 1.0610) We have created an implicit forward quote: USD 1.79085/ GBP 1.34933 = 1.3272 USD/GBP. ¶ Or Ft,10-year = St [(1+id,10-year)/(1+if,10-year)]10 = 1.60 USD/GBP [1.06/1.08]10 = 1.3272 USD/GBP. ¶ Synthetic forward contracts are very useful for exotic currencies. IRPT: Evidence Starting from Frenkel and Levich (1975), there is a lot of evidence that supports IRPT. Taylor (1989): Strong support for IRPT using 10’ intervals. Akram, Rice and Sarno (2008, 2009): Using tick-by-tick data, 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. Overall, the short-lived nature and small profit range point out to a fairly efficient market, with the data close to the IRPT line. But, there are situations where we see significant deviations from the IRPT line. These situations reflect violations of IRPT’s assumptions For example, during the 2007-2008 financial crisis there were violations of IRPT. Probable cause: funding constraints –Step (1) in trouble! 14

IRPT: Evidence May 2009: (-.0154,-.0005). The Behavior of FX Rates • Fundamentals that affect FX Rates: Formal Theories - Inflation rates differentials (IUSD - IFC) PPP - Interest rate differentials (iUSD - iFC) IFE - Income growth rates (yUSD - yFC) Monetary Approach - Trade flows Balance of Trade - Other: trade barriers, expectations, taxes, etc. • Goal 1: Explain St with a theory, say T1. Then, StT1 = f(.) Different theories can produce different f(.)’s. Evaluation: How well a theory match the observed behavior of St. • Goal 2: Eventually, produce a formula to forecast St+T = f(Xt) E[St+T]. 15

• We want to have a theory that can match the observed St. It is not realistic to expect a perfect match, so we ask the question: On average, is St ≈ StT1 ? Or, alternatively, is E[St] = E[StT1]? MXN/USD Level: 1989-2018 25 20 15 10 5 0 12/1/1989 12/1/1990 12/1/1991 12/1/1992 12/1/1993 12/1/1994 12/1/1995 12/1/1996 12/1/1997 12/1/1998 12/1/1999 12/1/2000 12/1/2001 12/1/2002 12/1/2003 12/1/2004 12/1/2005 12/1/2006 12/1/2007 12/1/2008 12/1/2009 12/1/2010 12/1/2011 12/1/2012 12/1/2013 12/1/2014 12/1/2015 12/1/2016 12/1/2017 12/1/2018 • Like many macroeconomic series, exchange rates have a trend –in statistics the trends in macroeconomic series are called stochastic trends. It is better to try to match changes, not levels. • Now, the trend is gone. Our goal is to explain st, the percentage change in St. (Notation: Many times st = ef,t). MXN/USD Changes: 1990-2018 0.5 0.4 0.3 0.2 0.1 0 ‐0.1 ‐0.2 1/1/1990 3/1/1991 5/1/1992 7/1/1993 9/1/1994 11/1/1995 1/1/1997 3/1/1998 5/1/1999 7/1/2000 9/1/2001 11/1/2002 1/1/2004 3/1/2005 5/1/2006 7/1/2007 9/1/2008 11/1/2009 1/1/2011 3/1/2012 5/1/2013 7/1/2014 9/1/2015 11/1/2016 1/1/2018 • Our goal is to explain st, the percentage change in St. Again, we will try to see if the model we are using, say T1, matches, on average, the observed behavior of st. For example, is E[st] = E[stT1]? 16

• We will use statistics to formally tests theories. • Let’s look at the distribution of st for the MXN/USD –in this case, we look at monthly percentage changes from 1990-2018. MXN/USD Changes: Frequency 140 120 100 80 60 40 20 0 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.16 More • The average (“usual”) monthly percentage change is a 0.64% appreciation of the USD (annualized 8% change). The SD is 3.92% (13.6% annualized). • These numbers are the ones to match with our theories for St. A good theory should predict an average annualized change close to 8% for st. • Descriptive stats for st for the JPY/USD and the MXN/USD. JPY/USD USD/MXN Mean -0.0026 0.0064 Standard Error 0.0014 0.0026 Median -0.0004 0.0025 Mode 0 0 Standard Deviation 0.0318 0.0392 Sample Variance 0.0010 0.0021 Kurtosis 1.6088 49.8443 Skewness -0.2606 4.8432 Range 0.2566 0.5812 Minimum -0.1474 -0.1282 Maximum 0.1092 0.4530 Sum -1.2831 -2.7354 Count 491 349 • Developed currencies: less volatile, with smaller means/medians. 17

Purchasing Power Parity (PPP) Purchasing Power Parity (PPP) PPP is based on the law of one price (LOOP): Goods, once denominated in the same currency, should have the same price. If they are not, then some form of arbitrage is possible. Example: LOOP for Oil. Poil-USA = USD 80. Poil-SWIT = CHF 160. StLOOP = USD 80 / CHF 160 = 0.50 USD/CHF. If St = 0.75 USD/CHF Oil in Switzerland is more expensive –once denominated in USD- than in the US: Poil-SWIT (USD) = CHF 160 * 0.75 USD/CHF = USD 120 > Poil-USA Example (continuation): St = 0.75 > StLOOP (LOOP is not holding) Trading strategy: (1) Buy oil in the US at Poil-USA = USD 80. (2) Export oil to Switzerland (3) Sell US oil in Switzerland at Poil-SWIT = CHF 160. (4) Sell CHF/buy USD at then St. This trading strategy, exporting US of oil to Switzerland, will affect prices: Poil-USA↑; Poil-SWIT↓; & St↓ StLOOP ↑ (= Poil-USA↑/Poil-SWIT↓) St ⟺ StLOOP (convergence). ¶ 18

Example (continuation): LOOP Notes : ⋄ LOOP gives an equilibrium exchange rate. Equilibrium will be reached when there is no trade in oil (because of pricing mistakes). That is, when the LOOP holds for oil. ⋄ LOOP is telling what St should be (in equilibrium). Not what St is in the market today. ⋄ Using the LOOP we have generated a model for St. We’ll call this model, when applied to many goods, the PPP model. Problem: There are many traded goods in the economy. Solution: Use baskets of goods. PPP: The price of a basket of goods should be the same across countries, once denominated in the same currency. That is, USD 1 should buy the same amounts of goods here (in the U.S.) or in Colombia. 19

• A popular basket: The CPI basket. In the US, the basket typically reported is the CPI-U, which represents the spending patterns of all urban consumers and urban wage earners and clerical workers. (87% of the total U.S. population). • U.S. basket weights: Food US: CPI-U Weights Energy 12% 14% Food Household Furnishings Apparel 7% New vehicles 10% Used cars and trucks 7% Recreation 3% Housing 4% Health care 3% 2% Housing 6% 32% • A potential problem with the CPI basket: The composition of the index (the weights and the composition of each category) may be very different. • For example, the weight of the food category changes substantially as the income level increases. 20

Absolute version of PPP: The FX rate between two currencies is simply the ratio of the two countries' general price levels: StPPP = Domestic Price level / Foreign Price level = Pd / Pf Example: Law of one price for CPIs. CPI-basketUSA = PUSA = USD 755.3 CPI-basketSWIT = PSWIT = CHF 1,241.2 StPPP = USD 755.3/CHF 1,241.2 = 0.6085 USD/CHF. If St 0.6085 USD/CHF, there will be trade of the goods in the basket between Switzerland and US. Suppose St = 0.70 USD/CHF > StPPP. Then, PSWIT (in USD) = CHF 1,241.2 * 0.70 USD/CHF = USD 868.70 > PUSA = USD 755.3 Example (continuation): (disequilibrium: St = 0.70 USD/CHF > StPPP) PSWIT (in USD) = CHF 1241.2 * 0.70 USD/CHF = USD 868.70 > PUSA = USD 755.3 Potential profit: USD 868.70 – USD 755.3 = USD 93.40 Traders will do the following pseudo-arbitrage strategy: 1) Borrow USD 2) Buy the CPI-basket in the US 3) Sell the CPI-basket, purchased in the US, in Switzerland. 4) Sell the CHF/Buy USD 5) Repay the USD loan, keep the profits. Note: “Equilibrium forces” at work: 2) PUSA ↑ & 3) PSWIT ↓ (=> StPPP↑ = PUSA ↑ / PSWIT ↓) 4) St ↓. St ⟺ StPPP (converge) ¶ 21

• Real v. Nominal Exchange Rates The absolute version of the PPP theory is expressed in terms of St, the nominal exchange rate. We can modify the absolute version of the PPP relationship in terms of the real exchange rate, Rt. That is, R t = St P f / P d = 1 Rt allows us to compare prices, translated to DC: If Rt > 1, foreign prices (translated to DC) are more expensive If Rt = 1, prices are equal in both countries –i.e., PPP holds! If Rt < 1, foreign prices are cheaper Economists associate Rt > 1 with a more efficient domestic economy. Example: Suppose a basket’s –the Big Mac– cost in Switzerland and in the U.S. is CHF 6.50 and USD 4.93, respectively. Pf = CHF 6.50 Pd = USD 4.93 St = 0.9908 USD/CHF => Pf (in USD) = USD 6.44 > Pd Rt = St PSWIT / PUS = 0.9908 USD/CHF * CHF 6.50/USD 4.93 = 1.3065. Taking the Big Mac as our basket, the U.S. is more competitive than Switzerland. Swiss prices are 30.65% higher than U.S. prices, after taking into account the nominal exchange rate. To bring the economy to equilibrium –no trade in Big Macs-, we expect the USD to appreciate against the CHF. According to PPP, the USD is undervalued against the CHF. => Trading Signal: Buy USD/Sell CHF. ¶ 22

• The Big Mac (“Burgernomics,” popularized by The Economist) has become a popular basket for PPP calculations. Why? 1) It is a standardized, common basket: beef, cheese, onion, lettuce, bread, pickles and special sauce. It is sold in over 120 countries. Big Mac (Sydney) Big Mac (Tokyo) 2) It is very easy to find out the price. 3) It turns out, it is correlated with more complicated common baskets, like the PWT (Penn World Tables) based baskets. Using CPI baskets may not work well for absolute PPP. CPI baskets can be very different. In theory, traders can exploit the price differentials in BMs. The Economist's Big Mac Index • In the previous example, Swiss traders can import US BMs. From UH (US) to Rapperswill (CH) • Not realistic. But, the components of a BM are internationally traded. LOOP suggests that prices of components should be similar in all markets. The Economist reports the real exchange rate: Rt = StPBigMac,f/PBigMac,d. For example, in 2020, for the British pound (GBP): Rt = [GBP 3.39 * 1.2987 USD/GBP] / USD 5.67 = 0.7764 (22.36% overvaluation) 23

Example: (The Economist’s) Big Mac Index in 2020. StPPP = PBigMac,d / PBigMac,f (The Economist reports Rt – 1 = StPBigMac,f/PBigMac,d – 1). Rt > 1 Rt < 1 Example: (The Economist’s) Big Mac Index in January 2020. 24

Example: Big Mac Index - Rt Changes over time in 2000-2016. CHF/USD BRL/USD But, Rt departures from 1, can be very persistent. Example: Iphone 6 (March 2015, taken from seekingalpha.com). Rt = StPIPhone,f/PIPhone,d (d=US) Rt=1 under Absolute PPP 25

• Empirical Evidence: Simple informal test: Test: If Absolute PPP holds Rt = 1. In the Big Mac example, PPP does not hold for the majority of countries. Several tests of the absolute version have been performed: Absolute version of PPP, in general, fails (especially, in the short run). • Absolute PPP: Qualifications (1) PPP emphasizes only trade and price levels. Political/social factors (instability, wars), financial problems (debt crisis), etc. are ignored. (2) Implicit assumption: Absence of trade frictions (tariffs, quotas, transactions costs, taxes, etc.). Q: Realistic? On average, transportation costs add 7% to the price of U.S. imports of meat and 16% to the import price of vegetables. Many products are heavily protected, even in the U.S. For example, peanut imports are subject to a tariff as high as 163.8%. Also, in the U.S., tobacco usage and excise taxes add USD 5.85 per pack. • Absolute PPP: Qualifications Some everyday goods protected in the U.S.: - European Roquefort Cheese, cured ham, mineral water (100%) - Paper Clips (as high as 126.94%) - Canned Tuna (as high as 35%) - Synthetic fabrics (32%) - Sneakers (48% on certain sneakers) - Japanese leather (40%) - Peanuts (shelled 131.8%, and unshelled 163.8%). - Brooms (quotas and/or tariff of up to 32%) - Chinese tires (35%) - Trucks (25%) & cars (2.5%) Some Japanese protected goods: - Rice (778%) - Beef (38.5%, but can jump to 50% depending on volume). - Sugar (328%) - Powdered Milk (218%) 26

• Absolute PPP: Qualifications (3) PPP is unlikely to hold if Pf and Pd represent different baskets. This is why the Big Mac is a popular choice. (4) Trade takes time (contracts, information problems, etc.). (5) Internationally non-traded (NT) goods –i.e. haircuts, home and car repairs, hotels, restaurants, medical services, real estate. The NT good sector is big: 50%-60% of GDP (big weight in CPI basket). Then, in countries where NT goods are relatively high, the CPI basket will also be relatively expensive. Thus, PPP will find these countries' currencies overvalued relative to currencies in low NT cost countries. Note: In the short-run, we will not take our cars to Mexico to be repaired, but in the long-run, resources (capital, labor) will move. We can think of the over-/under-valuation as an indicator of movement of resources. • Absolute PPP: Qualifications The NT sector also has an effect on the price of traded goods. For example, rent and utilities costs affect the price of a Big Mac: 25% of Big Mac due to NT goods. • Empirical Fact Price levels in richer countries are consistently higher than in poorer ones. This fact is called the Penn effect. Many explanations, the most popular: The Balassa-Samuelson (BS) effect. • 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). 27

But, borders play a role, not just distance! Engel and Rogers (1996) computed the variance of LOOP deviations for city pairs within the U.S., within Canada, and across the border.: Distance between cities within a country matter, but the border effect is 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 has been revised downward in subsequent studies, but a large positive border effect remains. • Balassa-Samuelson Effect Labor costs affect all prices. We expect average prices to be cheaper in poor countries than in rich ones because labor costs are lower. This is the so-called Balassa-Samuelson effect: Rich countries have higher productivity and, thus, higher wages in the traded-goods sector than poor countries do. But, firms compete for workers. Then, wages in NT goods and services are also higher Overall prices are lower in poor countries. 28

• For example, in 2000, a typical McDonald’s worker in the U.S. made USD 6.50/hour, while in China made USD 0.42/hour. • The Balassa-Samuelson effect implies a positive correlation between PPP exchange rates (overvaluation) and high productivity countries. • Incorporating the Balassa-Samuelson effect into PPP: 1) Estimate a regression: Big Mac Prices against GDP per capita. BigMac_Price_(in USD)t = α + β GDP_per_capitat + εt Switzerland Brazil + β GDP_per_capitat Hong Kong 29

• Incorporating the Balassa-Samuelson effect into PPP: Same regression in July 2011. • Incorporating the Balassa-Samuelson effect into PPP: 2) Compute fitted Big Mac Prices (GDP-adjusted Big Mac Prices), along the regression (red) line. Use the difference between GDP-adjusted Big Mac Prices and actual prices (the white/blue dots) to estimate GDP-adjusted PPP over/under-valuation. 30

Example: Raw vs GDP-Adjusted Big Mac Index in 2019. • Pricing-to-Market Krugman (1987) offers an alternative explanation for the strong positive relationship between GDP and price levels: Pricing-to-market –i.e., price discrimination. Based on price elasticities, producers discriminate: the same exact good is sold to rich countries (lower price elasticity) at higher prices than to poorer countries (higher price elasticity). Alessandria and Kaboski (2008) report that U.S. exporters, on average, charge the richest country a 48% higher price than the poorest country. Again, pricing-to-market struggles to explain why PPP does not hold among developed countries with similar incomes. For example, Baxter and Landry (2012) report that IKEA prices deviate 16% from the LOOP in Canada, but only 1% in the U.S. 31

Main PPP criticism Absolute PPP does not incorporate transaction costs and frictions. Relative PPP allows for fixed transaction costs and frictions (say, a fixed USD amount). Relative PPP 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): S tT S t (1 I d ) s tPPP ,T 1 (Relative PPP) St (1 I f ) where, If = foreign inflation rate from t to t+T; Id = domestic inflation rate from t to t+T. Note: st,TPPP is an expectation; what we expect to happen in equilibrium. • Linear approximation: st,TPPP (Id - If) one-to-one relation Relative PPP • Linear approximation: st,TPPP (Id - If) one-to-one relation Example: From t=0 to t=1, prices increase 10% in Mexico relative to prices in Switzerland. Then, St should also increase 10%. If S0 = 9 MXN/CHF S1PPP = E[S1] = 9.9 MXN/CHF. Suppose S1= 10.2 MXN/CHF > 9.9 MXN/CHF, According to Relative PPP, the CHF is overvalued. ¶ Notation: E[S1] = Expected value of S1 (according to a model), a forecast. 32

Example: Forecasting St (USD/ZAR) using PPP (ZAR=South Africa). It’s 2013. You have the following information: CPIUS,2013 = 104.5, CPISA,2013 = 100.0, S2013 = .2035 USD/ZAR. You are given the 2014 CPI’s forecast for the U.S. and SA: E[CPIUS,2014] = 110.8 E[CPISA,2014] = 102.5. You want to forecast S2014 using the relative (linearized) version of PPP. E[IUS-2014] = (110.8/104.5) - 1 = .06029 E[ISA-2014] = (102.5/100) - 1 = .025 E[S2014] = S2013 * (1 + st,TPPP ) = S2013 * (1 + E[IUS]- E[ISA]) = .2035 USD/ZAR * (1 + .06029 - .025) = .2107 USD/ZAR. Under the linear approximation, we have PPP Line Id - If PPP Line B (FC appreciates) A (FC depreciates) 45º sT (DC/FC) Look at point A: sT > Id - If, Priced in FC, the domestic basket is cheaper pseudo-arbitrage (trade) against foreign basket FC depreciates 33

• Relative PPP: Implications (1) Under relative PPP, Rt remains constant (it can be different from 1!). (2) Relative PPP does not imply that St is easy to forecast. (3) Without relative price changes, an MNC faces no real operating FX risk (as long as the firm avoids fixed contracts denominated in FC). • Relative PPP: Absolute versus Relative - Absolute PPP compares price levels. Under Absolute PPP, prices are equalized across countries: "A mattress costs GBP 200 (= USD 320) in the U.K. and BRL 800 (=USD 320) in Brazil.“ - Relative PPP compares price changes. Under Relative PPP, exchange rates change by the same amount as the inflation rate differential (original prices can be different): “U.K. inflation was 2% while Brazilian inflation was 8%. Meanwhile, the BRL depreciated 6% against the GBP. Then, relative cost comparison remains the same.” • Relative PPP is a weaker condition than Absolute PPP: Rt can be different from 1. • Relative PPP: Testing Key: On average, what we expect to happen, st,TPPP, should happen, st,T. On average: st,T st,TPPP Id – If or E[st,T] = E[st,TPPP ] E[ Id – If ] A linear regression is a good framework to test theories. Recall, st,T = (St+T - St)/St = α + β (Id - If )t+T + εt+T, where ε: regression error. That is, E[εt+T]=0. Then, E[st,T ] = α + β E[(Id - If )t+T] + E[εt+T] = α + β E[st,TPPP ] E[st,T] = α + β E[st,TPPP ] For Relative PPP to hold, on average, we need α=0 & β=1. 34

• Relative PPP: General Evidence Under Relative PPP: st,T Id – If 1. Visual Evidence Plot (IJPY-IUSD) against st(JPY/USD), using monthly data 1971-2015. Test: Is there a 45° line? st,T IJPY-IUSD No 45° line Visual evidence rejects PPP. • Relative PPP: General Evidence Under Relative PPP: st,T Id – If 1. Visual Evidence Plot (IGBP-IUSD) against st(GBP/USD), using monthly data 1971-2017. Test: Is there a 45° line? Relative PPP (GBP/USD) 0.04 0.03 0.02 (Igbp - Iusd) 0.01 0 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -0.01 -0.02 s_t (GBP/USD) No 45° line Visual evidence rejects PPP. 35

• Relative PPP: General Evidence 1. Visual Evidence Test: Is Rt 1? (Actually, constant, under Relative PPP) Real Exchange Rate: JPY/USD 3.5 3 2.5 2 1.5 1 0.5 0 1/1/1971 6/1/1972 11/1/1973 4/1/1975 9/1/1976 2/1/1978 7/1/1979 12/1/1980 5/1/1982 10/1/1983 3/1/1985 8/1/1986 1/1/1988 6/1/1989 11/1/1990 4/1/1992 9/1/1993 2/1/1995 7/1/1996 12/1/1997 5/1/1999 10/1/2000 3/1/2002 8/1/2003 1/1/2005 6/1/2006 11/1/2007 4/1/2009 9/1/2010 2/1/2012 7/1/2013 12/1/2014 5/1/2016 10/1/2017 Some evidence for mean reversion, though slow, for Rt (average = 1.94). • Relative PPP: General Evidence (continuation) In general, we have some evidence for mean reversion for Rt. Loosely speaking, in the long run, Rt moves around some mean number, which we associate with the long-run PPP parity. But, the deviations from the long- run parity are very persistent –i.e., very slow to adjust. 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! • Descriptive Stats IJPY IUSD IJPY-IUSD st,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 36

2. Statistical Evidence More formal tests: Regression st,T = (St+T – St)/St = α + β (Id – If )t+T + εt+T, ε: regression error, E[εt+T]=0. The null hypothesis is: H0 (Relative PPP true): α=0 and β=1 H1 (Relative PPP not true): α≠0 and/or β≠1 • Tests: t-test (individual tests on α and β) & F-test (joint test) (1) Individual test: t-test t-test = tθ = [θ – θ0]/S.E.(θ) Statistical distribution: tθ ~ tv (v = N – K =degrees of freedom) where θ represents α or β ( θ0 = α or β evaluated under H0). Rule: If |t-test| > |tv,α/2|, reject H0 at the α level. When v = N – K > 30, t30+,.025 ≈ 1.96 2-sided C.I. α = .05 (5 %) 2. Statistical Evidence (2) Joint Test: F-test F-test = {[RSS(H0) – RSS(H1)]/J}/{RSS(H1)/(N-K)} Statistical distribution: F-test ~ FJ,N-K J = # of restrictions in H0, (under PPP, J=2: α=0 & β=1) K = # parameters in model, (under PPP model, K=2: α & β) N = # of observations, RSS = Residuals Sum of Squared, εt = et = st – [α β (Id – If )t ]. RSS(H0) = ∑ t (Id,t − If,t) ] 2 RSS(H1) = ∑ (εt)2 Rule: If F-test > FJ,N-K,α, reject at the α level. Usually, α = .05 (5 %) When N > 300, FJ=2,300+,α=.05 ≈ 3. 37

Example: Using monthly Japanese and U.S. data (1/1971-9/2007), we fit the following regression: st (JPY/USD) = (St – St-1)/St-1 = α + β (IJAP – IUS) t + εt. R2 = 0.00525 Standard Error (σ) = .0326 F-stat (slopes=0 –i.e., β=0) = 2.305399 (p-value=0.130) Observations (N) = 439 Coefficient Stand Err t-Stat P-value Intercept (α ) 0.00246 0.001587 -1.55214 0.121352 (IJAP – IUS) (β ) -0.36421 0.239873 -1.51835 0.129648 We will test the H0 (Relative PPP true): α=0 & β=1 Two tests: (1) t-tests (individual tests) (2) F-test (joint test) Example: Using monthly Japanese and U.S. data (1/1971-9/2007), we fit the following regression (Observations = 439): st (JPY/USD) = (St – St-1)/St-1 = α + β (IJAP – IUS) t + εt. R2 = 0.00525 Standard Error (σ) = .0326 F-stat (slopes=0 –i.e., β=0) = 2.305399 (p-value = 0.130) F-test (H0: α=0 and β=1): 16.289 (p-value: lower than 0.0001) reject H0 at 5% level (F2,467,.05= 3.015) Coefficient Stand Err t-Stat P-value Intercept (α) -0.00246 0.001587 -1.55214 0.121352 (IJAP – IUS) (β) -0.36421 0.239873 -1.51835 0.129648 Test H0, using t-tests (t437.05=1.96 – Note: when N-K>30, t.05 = 1.96): tα=0: (-0.00246–0)/0.001587 = -1.55214 (p-value=.12) cannot reject H0. tβ=1: (-0.36421-1)/0.239873 = -5.6872 (p-value:.00001) reject H0. ¶ 38

• PPP Evidence: ⋄ Relative PPP tends to be rejected in the short-run (see example above). In the long-run, there is debate about its validity. Research shows that currencies with high inflation rate differentials tend to depreciate. Real FX (GBP/USD): 1971-2017 1 0.9 0.8 GBP/USD 0.7 0.6 0.5 0.4 0.3 0.2 Feb-71 Feb-73 Feb-75 Feb-77 Feb-79 Feb-81 Feb-83 Feb-85 Feb-87 Feb-89 Feb-91 Feb-93 Feb-95 Feb-97 Feb-99 Feb-01 Feb-03 Feb-05 Feb-07 Feb-09 Feb-11 Feb-13 Feb-15 Feb-17 Some evidence for a mean reverting Rt (average Rt = 0.61). But deviations can last for years! • PPP: Rt and St Mussa (1986) and others shows that Rt is much more variable under a free float. Rt variability tends to be highly correlated with St variability. Since 1973, when floating exchange rates were adopted, Rt moves like St. Nominal vs. Real FX (GBP/USD): 1971-2017 1 0.9 0.8 GBP/USD 0.7 0.6 0.5 0.4 Real FX 0.3 Nominal 0.2 Feb-71 Feb-73 Feb-75 Feb-77 Feb-79 Feb-81 Feb-83 Feb-85 Feb-87 Feb-89 Feb-91 Feb-93 Feb-95 Feb-97 Feb-99 Feb-01 Feb-03 Feb-05 Feb-07 Feb-09 Feb-11 Feb-13 Feb-15 Feb-17 As a check to the visual evidence: Volatility(changes in Rt) = 2.94% & Volatility(changes in St) = 2.90 (correlation = .98). Almost the same! 39

Implications: Price levels play a minor role in explaining the movements of Rt (prices are sticky). Engel (1999) reports that prices seem to be sticky also for traded-goods. Possible explanations: (a) Contracts: Prices cannot be continuously adjusted due to contracts. In a stable economy, contracts have longer durations. In high inflation countries (contracts with shorter duration) PPP deviations are not very persistent. (b) Mark-up adjustments: Manufacturers and retailers tend to moderate any increase in their prices in order to preserve market share. Changes in St are only partially transmitted or pass-through to import/export prices. Average ERPT (exchange rate pass-through) is around 50% over one quarter and 64% over the long run for OECD countries (for the U.S., 25% in the short-run and 40% over the long run). (c) Repricing costs (menu costs) It is expensive to adjust continuously prices; in a restaurant, the repricing cost is re-doing the menu. For example, Goldberg and Hallerstein (2007) estimate that the cost of repricing in the imported beer market is 0.4% of firm revenue for manufacturers and 0.1% of firm revenue for retailers. (d) Aggregation Q: Is price rigidity a result of aggregation –i.e., the use of price index? Empirical work using micro level data –say, same good (exact UPC!) in Canadian and U.S. grocery stores– show that on average product-level Rt move closely with St. But, micro level prices show idiosyncratic movements, mainly unrelated to St: 10% of the deviations from PPP are accounted by St. • PPP: Puzzle The fact that no single model of exchange rate determination can accommodate both the high persistent of PPP deviations and the high correlation between Rt and St has been called the “PPP puzzle.” 40

• PPP: Summary of Empirical Evidence ⋄ Rt and St are highly correlated, Pd (even for traded-goods) tend to be sticky. ⋄ In the short run, PPP is a very poor model to explain short-term St movements. ⋄ PPP deviation are very persistent. It takes a long time (years!) to disappear. ⋄ In the long run, there is some evidence of mean reversion, though very slow, for Rt. That is, StPPP has long-run information: Currencies that consistently have high inflation rate differentials –i.e., (Id – If) > 0– tend to depreciate. • The long-run interpretation for PPP is the one that economist like and use. PPP is seen as a benchmark, a figure towards which the current exchange rate should move. • Calculating StPPP (Long-Run FX Rate) Let’s look at the MXN/USD case. We want to calculate StPPP = Pd,t / Pf,t over time. (1) Divide StPPP by SoPPP (t=0 is our starting point). (2) After some algebra, StPPP = SoPPP * [Pd,t / Pd,o] * [Pf,o/Pf,t] By assuming SoPPP = So, we plot StPPP over time. (Note: SoPPP = So assumes that at t=0, the economy was in equilibrium. This may not be true: Be careful when selecting a base year.) 41

Let’s look at the MXN/USD case. Actual vs Long Run PPP: MXN/USD 30 25 20 15 10 5 PPP MX/US MXN/USD 0 12/1/1987 5/1/1989 10/1/1990 3/1/1992 8/1/1993 1/1/1995 6/1/1996 11/1/1997 4/1/1999 9/1/2000 2/1/2002 7/1/2003 12/1/2004 5/1/2006 10/1/2007 3/1/2009 8/1/2010 1/1/2012 6/1/2013 11/1/2014 4/1/2016 9/1/2017 - In the short-run, StPPP is missing the target, St. - But, in the long-run, StPPP gets the trend right. (As predicted by PPP, the high Mexican inflation rates differentials against the U.S., depreciate the MXN against the USD.) Another example, let’s look at the JPY/USD case. Actual vs Long Run PPP: JPY/USD 500 450 400 350 300 250 200 150 100 50 PPP JPY/USD JPY/USD 0 1/1/1971 1/1/1973 1/1/1975 1/1/1977 1/1/1979 1/1/1981 1/1/1983 1/1/1985 1/1/1987 1/1/1989 1/1/1991 1/1/1993 1/1/1995 1/1/1997 1/1/1999 1/1/2001 1/1/2003 1/1/2005 1/1/2007 1/1/2009 1/1/2011 1/1/2013 1/1/2015 1/1/2017 As predicted by PPP, since U.S. inflation rates have been consistently higher than the Japanese ones, in the long-run, the USD depreciates against the JPY. 42

• PPP Summary of Applications: ⋄ Equilibrium (“long-run”) exchange rates. A CB can use StPPP to determine intervention bands. ⋄ Explanation of St movements (“currencies with high inflation rate differentials tend to depreciate”). ⋄ Indicator of competitiveness or under/over-valuation: Rt > 1 FC is overvalued (& Foreign prices are not competitive). ⋄ International GDP comparisons: Instead of using St, StPPP is used to translate local currencies to USD. For example, Chinese per capita GDP (World Bank figures, in 2017): Nominal GDP per capita: CNY 59,670.52; St = 0.14792 USD/CNY; - Nominal GDP_cap (USD)= CNY 59,670.52 * 0.1479 USD/CNY= USD 8,827 StPPP = 0.2817 USD/CNY ⇒ “U.S. is 90% more expensive” - PPP GDP_cap (USD)= CNY 59,670.52 * 0.2817 USD/CNY = USD 16,807. GDP per capita (in USD) - 2017 Country Nominal PPP Luxembourg 104,103 103,745 USA 59,532 59,532 Japan 38,428 43,279 Italy 31,953 39,427 Czech Republic 20,368 36,504 Costa Rica 11,631 17,044 Brazil 9,821 15,484 China 8,827 16,807 Lebanon 8,524 14,676 Algeria 4,123 15,275 India 1,937 7,056 Ethiopia 767 1,899 Mozambique 416 1,247 Note: PPP GDP/Nominal GDP = USD 16,807/ USD 8,827 = 1.9040 ⇒ “U.S. is 90% more expensive.” ¶ 43

International Fisher Effect (IFE) • IFE builds on the law of one price, but for financial transactions. • Idea: The return to international investors who invest in money markets in their home country should be equal to the return they would get if they invest in foreign money markets once adjusted for currency fluctuations. • Exchange rates will be set in such a way that international investors cannot profit from interest rate differentials --i.e., no profits from carry trades. The "effective" T-day return on a foreign bank deposit is: rf (in DC) = (1 + if * T/360) (1 + st,T) – 1. • While, the effective T-day return on a home bank deposit is: rd (in DC) = id * T/360. • Setting rf (in DC) = rd and solving for st,T = (St+T/St - 1) we get: (1 i d x T/360) s IFE T -1 (This is the IFE) (1 i f x T/360) • Using a linear approximation: s , (id – if) * T/360. • s , represents an expectation. It is the expected change in St from t to t+T that makes looking for the “extra yield” in international money markets not profitable. 44

• Since IFE gives us an expectation for a future exchange rate –St+T-, if we believe in IFE we can use this expectation as a forecast. Example: Forecasting St using IFE. It’s 2015:I. You have the following information: S2015:I = 1 .0659 USD/EUR. iUSD,2015:I = 0.5% iEUR,2015:I = 1.0%. T = 1 semester = 180 days. s, : = [1+ iUSD,2015:I * (T/360)]/[1+ iEUR,2015:I * (T/360)] – 1 = = [1+.005*(180/360))/[1+.01*(180/360)] – 1 = -0.0024875 E[S2015:II] = S2015:I * (1+ s , : ) = 1.0659 USD/EUR *(1 – 0.0024875) = 1.06325 USD/EUR We expect St to change to 1.06325 USD/EUR to compensate for the lower US interest rates. ¶ Example (continuation): E[S2015:II] = S2015:I * (1+ s , : ) = 1.0659 USD/EUR *(1 – 0.0024875) = 1.06325 USD/EUR Suppose S2015:II = 1.08 USD/EUR > E[S2015:II] = 1.06325 USD/EUR According to IFE, EUR is overvalued. Trading signal: Sell EUR/Buy USD. Note: we can get to the same result by looking at the changes: s2015:II = 1.08/1.0659 – 1 = 0.01329 > s , : = -0.0024875. According to IFE, EUR appreciated more than expected. That is, EUR is overvalued. ¶ 45

• Note: Like PPP, IFE also gives an equilibrium exchange rate. Equilibrium will be reached when there is no capital flows from one country to another to take advantage of interest rate differentials. IFE: Implications If IFE holds, the expected cost of borrowing funds is identical across currencies. Also, the expected return of lending is identical across currencies. Carry trades –i.e., borrowing a low interest currency to invest in a high interest currency- should not be profitable. If departures from IFE are consistent, investors can profit from them. Example: Mexican peso depreciated 5% a year during the early 90s. Annual interest rate differential (iMXN – iUSD) were between 7% and 16%. The E[st,T] = -5% > sIFEt,T = -7% Pseudo-arbitrage is possible (The MXN at t+T is overvalued!) IFE Suppose we expect Et[st,T] > s t,T to occur in the next T days. Carry Trade Strategy (USD = DC; we invest in the overvalued currency): 1) Borrow USD funds (at iUSD) 2) Convert to MXN at St 3) Invest in Mexican funds (at iMXN) 4) Wait until T. Convert to USD at St+T –expect: E[St+T]=St*(1+ Et[st,T]). Expected FX loss = 5% (Et[st,T] = -5%) Assume (iUSD – iMXN) = -7%. (Say, iUSD = 6%; iMXN = 13%.) Et[st,T]= -5% > sIFEt= -7% “On average,” strategy (1)-(4) should work. 46

Example (continuation): Expected USD return from MXN investment: rf (in DC) =(1+ iMXN*T/360)(1+ Et[st,T]) – 1 = (1.13)*(1 –.05) – 1 = 0.074 Payment for USD borrowing: rd = id * T/360 = .06 Expected Profit = E[Π] = 0.074 – .06 = .014 per year • Overall expected profits ranged from: 1.4% to 11%. ¶ Note: A carry trade strategy is based on an expectation: Et[st,T] = -5%. It may or may not occur every time. This is risky! Example: Risk at work. Fidelity used this uncovered strategy during the early 90s. In Dec. 94, after the Tequila devaluation of the MXN against the USD (40% in a month), it lost everything it gained before. • An IFE driven carry trade 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). • UIRP is difficult to test since it involves an expectation (an unobservable). In general, we test UIRP assuming that on average what we expect occurs. • Test: UIRP true (no carry trade profits) if st,T (id – if) * T/360. B (FC undervalued) IFE Line id - if A FC overvalued (Carry trade: Borrow DC) 45º sT (DC/FC) 47

1. Visual evidence. Based on linearized IFE: st,T (id – if) * T/360 Expect a 45 degree line in a plot of ef,T against (id – if) Example: Plot for the monthly USD/EUR exchange rate (1999-2017) IFE: USD/EUR 0.05 st 0.04 0.03 0.02 0.01 0 -0.12 -0.07 -0.02 0.03 0.08 -0.01 -0.02 -0.03 -0.04 -0.05 iUSD – iEUR No 45° line Visual evidence rejects IFE. ¶ 2. Regression evidence st,T = (St+T – St)/St = α + β (id – if )t + εt, (εt error term, E[εt]=0). • The null hypothesis is: H0 (IFE true): α=0 and β=1 H0 (IFE not true): α≠0 and/or β≠1 Example: Testing IFE for the USD/EUR with monthly data (1999-2017). R2 = 0.01331 Standard Error = 0.01815 F-statistic (slopes=0) = 2.6034 (p-value=0.1083) 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 = 195 Coefficients Standard Error t Stat P-value Intercept (α ) 0.000588 0.001935 0.303996 0.76141 (id - if )t (β) -0.05477 0.143501 -0.38169 0.70305 48

Let’s test H0, using t-tets (t104,.05 = 1.96) : tα=0 (t-test for α = 0): (0.000588 – 0)/0.00194 = 0.304 cannot reject H0 at the 5% level. tβ=1 (t-test for β = 1): (-0.05477 – 1)/0.1435 = -8.045 reject H0 at the 5% level. Formally, IFE is rejected in the short-run (both the joint test and the t-tests reject H0). Also, note that β is negative, not positive as IFE expects. ¶ • IFE is rejected. Then, Q: Is a “carry trade” strategy profitable? During the 1999-2017 period, the average monthly (iUSD – iEUR) was: -0.00164/12= -.00015 stIFE = -0.015% per month (≠0, statistically) Actual average monthly st(USD/EUR) was .0007 (≈0, statistically speaking) Et[st] = 0.07% > stIFE = -0.015% (EUR overvalued!) If we use the regression to derive an expectation, then: E[sReg,t] = 0.000588 – 0.05477 *(-.00164) = 0.0006 (≈0!) E[st] = E[sReg,t] = 0.06% > stIFE = -0.015% (EUR overvalued) Note: Consistent deviations from IFE make carry trades profitable. During the 1999-2017 period, USD-EUR carry trades should have been profitable. Carry trade strategy: 1) Borrow USD at iUSD 2) Convert to EUR 3) Deposit EUR at iEUR 4) Wait 30 days and convert back to USD (on average, 0% monthly change) From 1) + 3), we make 0.015% per month. From 2) + 3), we lose 0% per month. Total gain over the whole period: 3.3% (very low!). ¶ 49

• IFE: Evidence No short-run evidence Carry trades work (on average). Burnside (2008): The average excess return of an equally weighted carry trade strategy, executed monthly, over the period 1976–2007, was about 5% per year. (Sharpe ratio twice as big as the S&P500, since annualized volatility of carry trade returns is much less than that for stocks). Some long-run support: “Currencies with high interest rate differentials tend to depreciate.” (For example, the Mexican peso finally depreciated in Dec. 1994.) Expectations Hypothesis (EH) • According to the Expectations hypothesis (EH) of exchange rates: Et[St+T] = Ft,T. That is, on average, the future spot rate is equal to the forward rate. Since expectations are involved, many times the equality will not hold. It will only hold on average. Q: Why should this equality hold? Suppose it does not hold. That means, what people expect to happen at time T is consistently different from the rate you can set for time T. A potential profit strategy can be developed that works on average. 50

Example: Suppose that over time, investors violate EH. Data: Ft,180 = 5.17 ZAR/USD. An investor expects: Et[St+180] = 5.34 ZAR/USD. (A potential profit!) Strategy for this investor: 1. Buy USD forward at ZAR 5.17 2. In 180 days, sell the USD for ZAR 5.34. Now, suppose everybody expects Et [St+180] = 5.34 ZAR/USD Disequilibrium: Everybody buys USD forward (nobody sells USD forward). And in 180 days, everybody will be selling USD. Prices should adjust until EH holds. Since an expectation is involved, sometimes you will have a loss, but, on average, you will make a profit every time Et[St+T] ≠ Ft,T. ¶ Expectations Hypothesis: Implications EH: Et[St+T] = Ft,T → On average, Ft,T is an unbiased predictor of St+T. Example: Today, it is 2014:II. A firm wants to forecast the quarterly St USD/GBP. You are given the interest rate differential (in %) and St. Using IRP you calculate Ft,90: Ft,90 = St [1 + (iUS – iUK)t * T/360]. Data available: St=2014:II = 1.6883 USD/GBP (iUS – iUK)t=2014:II = -0.304%. Then, Ft,90 = 1.6883 USD/GBP * [1 – 0.00304 * 90/360] = 1.68702 USD/GBP SFt:2014:III = 1.68702 USD/GBP According to EH, if a firm forecasts St+T using the forward rate, over time, will be right on average. average forecast error Et[St+T - Ft,T] = 0. 51

Expectations Hypothesis: Implications Doing this forecasting exercise each period generates the following quarterly forecasts and forecasting errors, εt: Quarter (iUS-iUK) St SFt+90 = Ft,90 εt = St - SFt 2014:II -0.304 1.6883 2014:III -0.395 1.6889 1.68702 0.0019 2014:IV -0.350 1.5999 1.68723 -0.0873 2015:I -0.312 1.5026 1.59850 -0.0959 2015:II -0.415 1.5328 1.50143 0.0314 2015:III -0.495 1.5634 1.53121 0.0322 2015:IV 1.5445 1.56146 -0.0170 Calculation of the forecasting error for 2014:III: εt=2014:III = 1.6889 – 1.68702 = 0.0019. ¶ Note: Since (St+T – Ft,T) is unpredictable, expected cash flows associated with hedging or not hedging currency risk are the same. Expectations Hypothesis: Evidence Under EH, Et[St+T] = Ft,T → Et[St+T – Ft,T] = 0 Empirical tests of the EH are based on a regression: (St+T – Ft,T)/St = α + β Zt + εt,(where E[εt]=0) where Zt represents any economic variable that might have power to explain St, for example, (id – if). H0 (EH true): α = 0 and β = 0. ((St+T – Ft) should be unpredictable!) H1 (EH not true): α ≠ 0 and/or β ≠ 0. Usual result: β < 0 (and significant) when Zt= (id – if). But, the R2 is very low. 52

Expectations Hypothesis: IFE (UIRP) Revisited EH: Et[St+T] = Ft,T. Replace Ft,T by IRP, say, linearized version: Et[St+T] ≈ St [1+ (id – if) * T/360]. A little bit of algebra gives: (E[St+T] – St)/St ≈ (id – if) * T/360

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