1

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.

2

Local Arbitrage (One good, one market)Example: Suppose two banks have the following bid-ask FX quotes:

Bank A Bank BUSD/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

3

Example: Suppose Bank One gives the following quotes:SJPY/USD,t = 100 JPY/USDSUSD/GBP,t = 1.60 USD/GBPSJPY/GBP,t = 140 JPY/GBP

Take the first two quotes ⇒ Implied (no-arbitrage) JPY/GBP:SI

JPY/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). <= This is the pricing mistake!

Sketch of Triangular Arbitrage (Key: Buy undervalued GBP with theovervalued JPY):(1) Borrow USD 1(2) Sell the USD/Buy JPY at St = 100 JPY/USD. Get JPY 100.(3) Sell JPY/Buy GBP (at St = 140 JPY/GBP). Get GBP 0.7143(4) Sell GPB/Buy USD (at St = 1.60 USD/GBP). Get USD 1.1429

⇒ Profit: Π = USD 0.1429 (14.29% per USD borrowed).

Example (continuation):

Note: Bank One will notice a book imbalance: All the activity involvesselling USD for JPY, buying GBP for JPY, & selling GPB for USD.

Bank One will adjust the quotes:SJPY/USD,t ↓ (say, SJPY/USD,t = 93 JPY/USD).SJPY/GBP,t ↑ (say, SJPY/GBP,t = 145 JPY/GBP).SUSD/GBP,t ↓ (say, SUSD/GBP,t = 1.56 USD/GBP).

There is convergence between SIJPY/GBP,t & SJPY/GBP,t :

SIJPY/GBP,t ↓ (= SJPY/USD,t ↓ * SUSD/GBP,t ↓ ⇔ SJPY/GBP,t ↑ ¶

Remark: Again, all the steps should be done at the same time.

JPY ⋄ Sell USD at Sell JPY at St = 100 JPY/USD St = 140 JPY/GBP USD ⋄ ⋄ GBP Sell GBP at St = 1.60 USD/GBP

4

Example (continuation):Note: It does not matter which currency you borrow in step (1). Recall thepricing mistake: Bank One undervalues the GBP against the JPY (withrespect 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 theovervalued 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 2markets)

Open the third section of the WSJ: Brazilian bonds yield 10% and Japanesebonds 1%.

Q: Why wouldn't capital flow to Brazil from Japan?A: FX risk: Once JPY are exchanged for BRL (Brazilian reals), there is noguarantee that the BRL will not depreciate against the JPY.

⇒ The only way to avoid this FX risk is to be covered with aforward FX contract.

5

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.

6

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 onannualized 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 isless 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).

Solving for Ft,T, we get: T/360) x i (1

T/360) x i (1S F

f

dtTt,

7

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.

T/360) x i (1

T/360) x i (1S F

f

dtTt,

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 FA

t,1-yr= 100 JPY/USD

8

Example 1 (continuation): t=today T = 1 yearBorrow 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: FX

t,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 FX

t,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 1.034П = JPY 1.0896 – JPY 1.034 = JPY .0556 (or 5.56% per JPY borrowed)

9

The Forward Premium and the IRPTReconsider 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 sellingit forward. We think of p + if as the annualized return from borrowing DCand investing in FC (covered) for T days. The opportunity cost of doingthis is id.

Equilibrium: p exactly compensates (id – if) → No arbitrage→ No capital flows.

Equilibrium: p id - if. IPR Line

Example: Go back to Example 1p = [(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

id -if

p (forward premium)

(id - if) > p(Capital inflows)

B45º

IRP Line

10

IRP Line

Under the linear approximation, we have the IRP Line:

id - if

p (forward premium)

B (Capital inflows)- Example 1

(Capital outflows)- Example 2

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, ismore than compensated by the high forward premium, p, plus if .

45º

A

IRPT: AssumptionsBehind 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).

11

IRPT with Bid-Ask SpreadsExchange 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.

12

• 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’sbound 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 0.60173) 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.

13

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 1.65403) 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 RatesA 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/GBPT = 10 years.Ignoring transactions costs, she creates a 10-year (implicit quote) forwardquote:1) Borrow USD 1 at 6% for 10 years2) Convert to GBP at 1.60 USD/GBP3) Invest in GBP at 8% for 10 years

14

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. ¶

OrFt,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: EvidenceStarting 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!

15

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].

16

• We want to have a theory that can match the observed St. It is not realisticto expect a perfect match, so we ask the question: On average, is St ≈ St

T1 ?Or, alternatively, is E[St] = E[St

T1]?

• Like many macroeconomic series, exchange rates have a trend –instatistics the trends in macroeconomic series are called stochastic trends. It isbetter to try to match changes, not levels.

0

5

10

15

20

2512/1/1989

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

MXN/USD Level: 1989-2018

• Now, the trend is gone. Our goal is to explain st, the percentage changein St. (Notation: Many times st = ef,t).

• Our goal is to explain st, the percentage change in St. Again, we will try tosee if the model we are using, say T1, matches, on average, the observedbehavior of st. For example, is E[st] = E[st

T1]?

‐0.2

‐0.1

0

0.1

0.2

0.3

0.4

0.5

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

MXN/USD Changes: 1990-2018

17

• 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.

• 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.

0

20

40

60

80

100

120

140

-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

MXN/USD Changes: Frequency

• 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.

18

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 –oncedenominated 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). ¶

19

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 themarket 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 thesame amounts of goods here (in the U.S.) or in Colombia.

20

• A popular basket: The CPI basket. In the US, the basket typically reportedis the CPI-U, which represents the spending patterns of all urban consumersand urban wage earners and clerical workers. (87% of the total U.S. population).

• U.S. basket weights:

US: CPI-U Weights

14%

10%

3%

4%

3%

2%

6%32%

7%

7%

12%

Food

Energy

Household Furnishings

Apparel

New vehicles

Used cars and trucks

Recreation

Housing

Health care

Housing

Food

• 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 theincome level increases.

21

Absolute version of PPP: The FX rate between two currencies is simplythe 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 basketbetween 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) ¶

22

• Real v. Nominal Exchange Rates

The absolute version of the PPP theory is expressed in terms of St, thenominal exchange rate.

We can modify the absolute version of the PPP relationship in terms of thereal exchange rate, Rt. That is,

Rt = St Pf / Pd = 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 theU.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 thanSwitzerland. Swiss prices are 30.65% higher than U.S. prices, after takinginto account the nominal exchange rate.

To bring the economy to equilibrium –no trade in Big Macs-, we expect theUSD to appreciate against the CHF.

According to PPP, the USD is undervalued against the CHF.

=> Trading Signal: Buy USD/Sell CHF. ¶

23

• The Big Mac (“Burgernomics,” popularized by The Economist) has becomea 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, likethe PWT (Penn World Tables) based baskets.

Using CPI baskets may not work well for absolute PPP. CPI baskets can bevery 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) toRapperswill (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)

24

Example: (The Economist’s) Big Mac Index in 2020. St

PPP = 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.

25

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

26

• 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: Absoluteversion 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, transactionscosts, 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 productsare heavily protected, even in the U.S. For example, peanut imports aresubject to a tariff as high as 163.8%. Also, in the U.S., tobacco usage andexcise taxes add USD 5.85 per pack.

• Absolute PPP: QualificationsSome 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%)

27

• 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 willalso be relatively expensive. Thus, PPP will find these countries' currenciesovervalued 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 ofthe over-/under-valuation as an indicator of movement of resources.

• Absolute PPP: QualificationsThe NT sector also has an effect on the price of traded goods. Forexample, rent and utilities costs affect the price of a Big Mac: 25% of BigMac due to NT goods.

• Empirical FactPrice levels in richer countries are consistently higher than in poorer ones.This fact is called the Penn effect. Many explanations, the most popular: TheBalassa-Samuelson (BS) effect.

• Borders MatterYou may look at the Big Mac Index and think: “No big deal: there is also abig dispersion in prices within the U.S., within Texas, and, even, withinHouston!” 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).

28

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 EffectLabor costs affect all prices. We expect average prices to be cheaper in poorcountries than in rich ones because labor costs are lower.

This is the so-called Balassa-Samuelson effect: Rich countries have higherproductivity and, thus, higher wages in the traded-goods sector than poorcountries do. But, firms compete for workers.

Then, wages in NT goods and services are also higher Overall prices arelower in poor countries.

29

• For example, in 2000, a typical McDonald’s worker in the U.S. made USD6.50/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

Hong Kong

Switzerland

Brazil

+ β GDP_per_capitat

30

• 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 theregression (red) line. Use the difference between GDP-adjusted Big Mac Pricesand actual prices (the white/blue dots) to estimate GDP-adjusted PPPover/under-valuation.

31

Example: Raw vs GDP-Adjusted Big Mac Index in 2019.

• Pricing-to-MarketKrugman (1987) offers an alternative explanation for the strong positiverelationship between GDP and price levels: Pricing-to-market –i.e., pricediscrimination.

Based on price elasticities, producers discriminate: the same exact good issold to rich countries (lower price elasticity) at higher prices than to poorercountries (higher price elasticity).

Alessandria and Kaboski (2008) report that U.S. exporters, on average, chargethe richest country a 48% higher price than the poorest country.

Again, pricing-to-market struggles to explain why PPP does not hold amongdeveloped countries with similar incomes. For example, Baxter and Landry(2012) report that IKEA prices deviate 16% from the LOOP in Canada, butonly 1% in the U.S.

32

Main PPP criticismAbsolute PPP does not incorporate transaction costs and frictions. Relative PPP allows for fixed transaction costs and frictions (say, a fixed USD amount).

Relative PPPThe rate of change in the prices of products should be similar when measured in a common currency (as long as trade frictions are unchanged):

(Relative PPP)

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

1)1(

)1(,

f

d

t

tTtPPPTt I

I

S

SSs

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 toprices 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.

33

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.

PPP Line

Under the linear approximation, we have PPP Line

Id - If

sT (DC/FC)

B (FC appreciates)

A

Look at point A: sT > Id - If, Priced in FC, the domestic basket is cheaper pseudo-arbitrage (trade) against foreign basket FC depreciates

45º

(FC depreciates)

34

• 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 costsGBP 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 theinflation rate differential (original prices can be different): “U.K. inflation was2% while Brazilian inflation was 8%. Meanwhile, the BRL depreciated 6% against theGBP. Then, relative cost comparison remains the same.”

• Relative PPP is a weaker condition than Absolute PPP: Rt can bedifferent 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.

35

No 45° line Visual evidence rejects PPP.

• Relative PPP: General Evidence

Under Relative PPP: st,T Id – If

1. Visual EvidencePlot (IJPY-IUSD) against st(JPY/USD), using monthly data 1971-2015.

Test: Is there a 45° line?

IJPY-IUSD

st,T

No 45° line Visual evidence rejects PPP.

• Relative PPP: General Evidence

Under Relative PPP: st,T Id – If

1. Visual EvidencePlot (IGBP-IUSD) against st(GBP/USD), using monthly data 1971-2017.

Test: Is there a 45° line?

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

(Igb

p -

Iusd

)

s_t (GBP/USD)

Relative PPP (GBP/USD)

36

Some evidence for mean reversion, though slow, for Rt (average = 1.94).

• Relative PPP: General Evidence

1. Visual EvidenceTest: Is Rt 1? (Actually, constant, under Relative PPP)

0

0.5

1

1.5

2

2.5

3

3.5

1/1/

1971

6/1/

1972

11/1

/197

34/

1/19

759/

1/19

762/

1/19

787/

1/19

7912

/1/1

980

5/1/

1982

10/1

/198

33/

1/19

858/

1/19

861/

1/19

886/

1/19

8911

/1/1

990

4/1/

1992

9/1/

1993

2/1/

1995

7/1/

1996

12/1

/199

75/

1/19

9910

/1/2

000

3/1/

2002

8/1/

2003

1/1/

2005

6/1/

2006

11/1

/200

74/

1/20

099/

1/20

102/

1/20

127/

1/20

1312

/1/2

014

5/1/

2016

10/1

/201

7

Real Exchange Rate: JPY/USD

• Relative PPP: General Evidence (continuation)

In general, we have some evidence for mean reversion for Rt. Looselyspeaking, in the long run, Rt moves around some mean number, which weassociate 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 -0.0015SD 0.0063 0.0038 0.0061 0.0316Min -0.0107 -0.0191 -0.0192 -0.1474Median 0.0010 0.0030 -0.0019 -0.0001Max 0.0431 0.0177 0.0346 0.1092

37

2. Statistical EvidenceMore 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=1H1 (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-testt-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-testF-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.

38

Example: Using monthly Japanese and U.S. data (1/1971-9/2007), we fitthe 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 fitthe 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. ¶

39

• 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.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fe

b-71

Fe

b-73

Fe

b-75

Fe

b-77

Fe

b-79

Fe

b-81

Fe

b-83

Fe

b-85

Fe

b-87

Fe

b-89

Fe

b-91

Fe

b-93

Fe

b-95

Fe

b-97

Fe

b-99

Fe

b-01

Fe

b-03

Fe

b-05

Fe

b-07

Fe

b-09

Fe

b-11

Fe

b-13

Fe

b-15

Fe

b-17

GB

P/U

SD

Real FX (GBP/USD): 1971-2017

Some evidence for a mean reverting Rt (average Rt = 0.61). But deviations can last for years!

As a check to the visual evidence: Volatility(changes in Rt) = 2.94% &Volatility(changes in St) = 2.90 (correlation = .98). Almost the same!

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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

GB

P/U

SD

Nominal vs. Real FX (GBP/USD): 1971-2017

Real FXNominal

• 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.

40

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.”

41

• 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, St

PPP 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 So

PPP (t=0 is our starting point).

(2) After some algebra,

StPPP = So

PPP * [Pd,t / Pd,o] * [Pf,o/Pf,t]

By assuming SoPPP = So, we plot St

PPP 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.)

42

Let’s look at the MXN/USD case.

- 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 theMXN against the USD.)

0

5

10

15

20

25

30

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

Actual vs Long Run PPP: MXN/USD

PPP MX/US MXN/USD

Another example, let’s look at the JPY/USD case.

As predicted by PPP, since U.S. inflation rates have been consistently higherthan the Japanese ones, in the long-run, the USD depreciates against theJPY.

0

50

100

150

200

250

300

350

400

450

500

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

Actual vs Long Run PPP: JPY/USD

PPP JPY/USD JPY/USD

43

• 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.

CountryGDP per capita (in USD) - 2017

Nominal PPP

Luxembourg 104,103 103,745USA 59,532 59,532Japan 38,428 43,279Italy 31,953 39,427Czech Republic 20,368 36,504Costa Rica 11,631 17,044Brazil 9,821 15,484China 8,827 16,807Lebanon 8,524 14,676Algeria 4,123 15,275India 1,937 7,056Ethiopia 767 1,899Mozambique 416 1,247

Note: PPP GDP/Nominal GDP = USD 16,807/ USD 8,827 = 1.9040⇒ “U.S. is 90% more expensive.” ¶

44

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 marketsin their home country should be equal to the return they would get ifthey invest in foreign money markets once adjusted for currencyfluctuations.

• Exchange rates will be set in such a way that international investorscannot profit from interest rate differentials --i.e., no profits from carrytrades.

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:

(This is the IFE)

• Using a linear approximation: s , (id – if) * T/360.

• s , represents an expectation. It is the expected change in St from t tot+T that makes looking for the “extra yield” in international moneymarkets not profitable.

1- T/360)x i (1

T/360)x i (1 s

f

dIFET

45

• Since IFE gives us an expectation for a future exchange rate –St+T-, if webelieve 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 thelower 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. ¶

46

• 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 acrosscurrencies. Also, the expected return of lending is identical acrosscurrencies.

Carry trades –i.e., borrowing a low interest currency to invest in a highinterest 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!)

Suppose we expect Et[st,T] > sIFEt,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.

47

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%. Itmay or may not occur every time. This is risky!

Example: Risk at work. Fidelity used this uncovered strategy during theearly 90s. In Dec. 94, after the Tequila devaluation of the MXN against theUSD (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 calledUncovered 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.

IFE Lineid - if

sT (DC/FC)

45º

B (FC undervalued)

A FC overvalued(Carry trade: Borrow DC)

48

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)

No 45° line Visual evidence rejects IFE. ¶

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

-0.12 -0.07 -0.02 0.03 0.08

IFE: USD/EUR

iUSD – iEUR

st

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

49

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-testsreject 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!). ¶

50

• IFE: Evidence

No short-run evidence Carry trades work (on average).

Burnside (2008): The average excess return of an equally weighted carrytrade strategy, executed monthly, over the period 1976–2007, was about 5%per year. (Sharpe ratio twice as big as the S&P500, since annualizedvolatility 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.

51

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 USDforward). And in 180 days, everybody will be selling USD. Prices shouldadjust until EH holds.

Since an expectation is involved, sometimes you will have a loss, but, onaverage, 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.

52

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 - SF

t

2014:II -0.304 1.68832014:III -0.395 1.6889 1.68702 0.00192014:IV -0.350 1.5999 1.68723 -0.08732015:I -0.312 1.5026 1.59850 -0.09592015:II -0.415 1.5328 1.50143 0.03142015:III -0.495 1.5634 1.53121 0.03222015:IV 1.5445 1.56146 -0.0170

Note: Since (St+T – Ft,T) is unpredictable, expected cash flows associatedwith hedging or not hedging currency risk are the same.

Calculation of the forecasting error for 2014:III:εt=2014:III = 1.6889 – 1.68702 = 0.0019. ¶

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 toexplain 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.

53

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 <= IFE linearized!

• EH can also be tested based on the Uncovered IRP (IFE) formulation:

(St+T – St)/St = st = α + β (id – if) + εt.

The null hypothesis is H0: α=0 and β=1.

Usual Result: β < 0 when (id – if)=2%, the exchange rate appreciatesby (β * .02), instead of depreciating by 2% aspredicted by UIRPT!

• Risk Premium

The risk premium of a given security is defined as the return on thissecurity, over and above the risk-free return.

• Q: Is a risk premium justified in the FX market?

A: Only if exchange rate risk is not diversifiable.

After some simple algebra, we find that the expected excess return on theFX market is given by:

(Et[St+T] – Ft,T)/St = Pt,t+T.

A risk premium, P, in FX markets implies

Et[St+T] = Ft,T + St Pt,t+T.

If Pt,t+T is consistently different from zero, markets will display a forwardbias.

54

• Example: Understanding the meaning of the FX Risk Premium.

Data: St = 1.58 USD/GBP

Et[St+6-mo] = 1.60 USD/GBP

Ft,6-mo= 1.62 USD/GBP.

• Expected change in St:

(Et[St+6-mo] – St)/St = (1.60 – 1.58)/1.58 = 0.0127.

• 6-mo FX premium

p6-mo = (Ft,6-mo – St)/St= (1.62 – 1.58)/1.58 = 0.0253.

• In the next 6-month period:

The GBP is expected to appreciate against the USD by 1.27%

The forward premium suggests a GBP appreciation of 2.53%.

E[st+6-mo] < p6-mo

Higher USD return from a USD deposit, than from a

GBP deposit.

• In the next 6-month period: E[st+6-mo] ≠ p6-mo

• Discrepancy: The presence of a FX risk premium, Pt,t+6-mo, makes theforward rate a biased predictor of St+6-mo.

• The expected (USD) return from holding a GBP deposit will be less(different) than the USD return from holding a USD deposit.

• Rational Investor: The lower return from holding a GBP deposit isnecessary to induce investors to hold the riskier USD denominatedinvestments. ¶

55

Structural Models

• We will go over two models that incorporate different views of the FX market:

(1) BOP approach treats exchange rates as determined in flow markets.

(2) Monetarist approach treats exchange rates as any other asset price.

(1) BOP Approach• Balance Of Payments (BOP)BOP divides the flow of foreign currency to the domestic country into:

Current account (CA): measures the movement of good and services + unilateral transfers.

Capital Account (KA): measures financial transactions associated with trade + changes in the composition of international portfolios.

Official Account (OR): measures changes in official reserves.

BOP = CA + KA + OR.

• As long as the country is not bankrupt, BOP = 0.

In general OR is small CA – KA.

56

• The Balance of Trade as a determinant of exchange rates• BOP Approach: Supply and demand for a currency arise from the flowsrelated to the BOP. That is, trade, portfolio investment, and directinvestment.

BOP approach views exchange rates as determined in flow markets.

• The Balance of Trade theory simplifies the BOP approach: It postulates arelation between CA and Rt (Real exchange rate = StPf/Pd):

CA = X – M = f(Rt, Yd,Yf).

In general, St moves to compensate a CA imbalance:- A country experiences a persistent CA deficit. Then,

Rt (St) X and M (CA)

- A country experiences a persistent CA suplus. Then, Rt (St) X and M (CA)

• US Trade Balance: 1960-2016

Note: The USD has not depreciated according to Trade Balance Approach!

57

• The Macroeconomics of the BOP: Absorption ApproachIn equilibrium, we can write:CA = S – [I + (G – T)],where S: after-tax private savings.I: private investment. G: government spending. T: national taxes.

To reduce a CA deficit, one of the following must happen in equilibrium:i. S , for a given level of I and (G – T).ii. I , for a given level of S and (G – T).iii. G – T , for a given level of S and I.

Example: Japan has a (relative) high savings rate. It is argued that this is the reason behind the persistent Japanese CA surpluses. ¶

•

• The Monetary Approach to the BOPConsider the capital account (KA): When the OR is small, KA provides the other side of the CA.

KA is assumed to depend on the interest rate differential and St:KA = f(id – if, St).

• Look at the KA:When KA > 0 (CA < 0) the country is either accumulating debt or selling its current stock of foreign assets. When KA < 0 (CA > 0) the country is either reducing debt or increasing its current stock of foreign assets.

Example: Japan may run a CA>0 without changes in the real JPY, as long as the Japanese continue to accumulate foreign assets. ¶

58

• BOP Approach: ImplicationsFirst, we ignored financial flows when we analyzed the BOP. We said:

A CA deficit (surplus) tends to be corrected with a depreciation (appreciation) of St.

But once we consider the capital account, this depreciation (appreciation) might not occur. Foreigners might finance the CA imbalance.

Now, the exchange rate depreciation (appreciation) might not occur.

Example: Japan may have CA>0 without any change in the real JPY because the Japanese continue to accumulate foreign assets. ¶

In the long-run the BOP approach has more precise predictions: “Countries will not be able to finance CA imbalances forever. Then, in the long-run

persistent CA deficits will create KA outflows and depreciation pressures.”

(2) Monetary ApproachExchange rates are asset prices traded in efficient markets.

Like other asset prices, exchange rates are determined by expectations.Current trade flows are irrelevant (flow markets are not important).

• Asset approach models assume a high degree of capital mobility betweenassets denominated in different currencies.

We need to specify the assets an investor considers.

For example, a simple monetary model considers domestic and foreign money.Only news related to these assets will move exchange rates.

59

• A Simple Monetary Approach ModelAssets: Domestic money and foreign money.

We start from the equilibrium QTM relation:Mj Vj = Pj Yj j = domestic country, foreign country.

To get St, we use Absolute PPP, where St is a ratio of prices (= Pd/Pf):

St = (Vd/Vf) * (Yf/Yd) * (MSd/MSf),

Vj: velocity of money of country j,Yj: real output of country j,MS,j: supply for money of country j (in equilibrium, MS=LD).

Recall that xt,T = (Xt+T – Xt)/Xt ≈ log(Xt+T) – log(Xt) ( a growth rate).

Assuming V is constant, we express the model in changes: st,T = yf,T – yd,T + mSd,T – mSf,T.

• A Simple Monetary Approach ModelThe simple Monetary Model produces the following model for st,T :

st,T = yf,T – yd,T + mSd,T – mSf,T.

• Monetary Approach: Implications– A stable monetary policy –i.e., low mSd– tends to appreciate the DC.– Economic growth –i.e., yd,T>0– tends to appreciate the DC.

But, keep in mind that all variables are in relative terms.

Note: St behaves like any other speculative asset price: St changes wheneverrelevant information –in this case, money growth and GDP growth– isreleased.

60

• Monetary Approach: Application Example: Forecasting St with the simple monetarist model.The money supply in the U.S. market increases by 2% and all the othervariables remain unchanged.yf,T = yd,T = mSf,T = 0.mSd,T = .02 st,T = .02.MSd incresaes 2% St increases 2% (depreciation of the USD).

Now, if investors expect the U.S. Fed to quickly increase U.S. interest ratesto avoid inflationary pressures, then the USD may appreciate instead ofdepreciate. ¶

• Monetary Approach: Sticky Prices There are many variations of the monetary model.

• One popular variation takes into account that PPP is not a good short-runtheory and proposes sticky prices (in the short-run).

The sticky prices monetary model incorporates the fact that exchange rates are a lot more volatile than prices of goods and services.

• Since prices do not react instantaneously to “shocks” –say, a monetary shock–, financial prices do. In this case, St overreacts (depreciates more than it should in a non-sticky prices world) to bring prices and quantities immediately to equilibrium.

In the long-run, prices will adjust and the exchange rate will appreciate accordingly. This is called overshooting.

This is a popular model to explain the behavior of St after financial crisis.

61

• Monetary Approach: Sticky Prices • There is a shock in the economy (say, a monetary shock).

• Prices do not react instantaneously. We move from A to B. Id stays at Id,0, but St overreacts to bring prices and quantities immediately to equilibrium.

• In the long-run, prices will adjust (with higher inflation, Id,1) and the exchange rate will move back to S1.

A

Id

st

B

C

Id,0

Id,1

s0 s2s1

BB

(3) Portfolio-Balance ApproachIt is also part of the Asset Approach of FX determination. Under thisapproach the financial assets to be considered by investors are: Money (DC& FC) and Bonds (domestic and foreign).

Investors’ asset preferences may be similar across countries (uniformpreference model), or investors may prefer assets of their home country(preferred local habitat model).

• St is determined to bring equilibrium to the investors’ portfolios.

• The balance between domestic and foreign bonds in a portfolio is positively related to expected excess return on domestic bonds over foreign bonds.

For example, an increase in the supply of DC-denominated bonds or foreign interest rates depreciates the DC.

62

• Structural Models: Evidence

• Standard tests of structural models are based on a regression:

st = α + β Zt + εtwhere Zt represents a structural explanatory variable: money growth, incomegrowth rates, (id-if), etc.

Usual results:

– The null hypothesis: H0: β=0, is difficult to reject.

– The R2 tends to be small.

• Many economists suggest that structural models are misspecified, because of the so-called structural change problem: The parameters change with changes in economic policy!

For example, a new Chairman of the Fed may have an effect on the coefficient β. Then, St may become more sensitive to (id-if).

• Structural Models: Evidence

• But, studies analyzing the movement of St around news announcements have found some support for structural model .

• These event studies find that news about:- Greater than expected U.S. CA deficits tends to depreciate the USD, as predicted by the BOP approach. - Unexpected U.S. economic growth tends to appreciate the USD, as predicted by the monetary approach.- Positive MS surprises tends to appreciate the USD. (Consistent with the monetary approach if agents expect the US Fed to quickly change interest rates to offset the increase in money supply.)- Unexpected increases (decreases) of interest rate differentials tends to depreciate (appreciate) the domestic currency, as predicted by the monetary approach.

• Regression-based structural models do poorly. But, the variables used in structural models tend to have power to explain changes in St.

63

• Summary • Parity Conditions- PPP Rejected in the short-run, some long-run support.- IFE Rejected in the short-run, some long-run support.- EH Rejected in the short-run. Puzzle!

• Structural models- BOP & Monetary Approach Rejected in the short-run, some support through event studies.

• Q: Why is st so (statistically) difficult to explain?

-0.15

-0.05

0.05

0.15

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

GB

P/U

SD

FX Changes (GBP/USD): 1971-2017

Martingale-RW Model

The Martingale-Random Walk Model

A random walk is a time series independent of its own history. Your last step has no influence in your next step. The past does not help to explain the future.

64

Motivation: Drunk walking in a park. (Problem posted in Nature. Solved byKarl Pearson. July, 1905 issue.)

Very difficult to predict where the drunk will end up after T steps.

Intuitive notion: The FX market is a “fair game.” (Unpredictable!)

• Martingale-Random Walk Model: ImplicationsThe Random Walk Model (RWM) implies:

Et[St+T] = St.

Powerful theory: At time t, all the info about St+T is summarized by St.

Theoretical Justification: Efficient Markets (all available info is incorporatedinto today’s St.)

Example: Forecasting with RWMSt = 1.60 USD/GBPEt[St+7-day] = 1.60 USD/GBPEt[St+180-day] = 1.60 USD/GBPEt[St+10-year] = 1.60 USD/GBP. ¶

Note: If St follows a RW, a firm should spend no resources to forecast St+T.

65

• Martingale-Random Walk Model: EvidenceMeese and Rogoff (1983, Journal of International Economics) tested the short-run forecasting performance of different models for the four most tradedexchange rates. They considered economic models (PPP, IFE, MonetaryApproach, etc.) and the RWM.

The metric used in the comparison: MSE (mean squared error)

MSE =∑ ε

=∑

where εt = – St+T = forecasting error at horizon T.

The RWM performed as well as any other model. Big surprise!

Cheung, Chinn & Pascual (2005) checked Meese & Rogoff ’s results with 20more years of data. RWM still the best model in the short-run.

M&R started a big literature. In general, M&R’s results hold in the short-run (say, up to 6-months), but for longer horizons (say, 1-4 years), modelscan do better (PPP, IFE and Taylor rule models, individually or combined).

Quarter (iUS - iUK) St Forward Rate Random Walk SF

t+90 = Ft,90 εt-FR = St - SFt SF

t+90=St εt-RW = St - SFt

2014:II -0.304 1.68832014:III -0.395 1.6889 1.6870 0.0019 1.6883 0.00062014:IV -0.350 1.5999 1.6872 -0.0873 1.6889 -0.08902015:I -0.312 1.5026 1.5985 -0.0959 1.5999 -0.09732015:II -0.415 1.5328 1.5014 0.0314 1.5026 0.03022015:III -0.495 1.5634 1.5312 0.0322 1.5328 0.03062015:IV 1.5445 1.5615 -0.0170 1.5634 -0.0189

MSE 0.04427 0.04443

Example: MSE - Forecasting St (USD/GBP) with Forwards and the RWMData: interest rate differential (in %) and St from 2014:II on.Using IRP, you calculate the forward rate, Ft,90, and, then, to forecast Et[St+90] = .

Using the RWM you forecast Et[St+90]= St. Then, to check the accuracy of the forecasts, you calculate the MSE.

Both MSEs are similar, though the Ft,T’s MSE is a bit smaller (.4% lower). ¶

66

• Martingale-Random Walk Model: Empirical Models Trying to CompeteModels of FX rates determination based on economic fundamentals have problems explaining the short-run behavior of St. This is not good news if the aim of the model is to forecast St.

As a result of this failure, a lot of empirical models, modifying the traditional fundamental-driven models, have been developed to better explain equilibrium exchange rates (EERs).

Some models are built to explain the medium- or long-run behavior of St, others are built to beat (or get closer to) the forecasting performance of the RWM.

A short list of the new models includes CHEERs, ITMEERs, BEERs, PEERs, FEERs, APEERs, PEERs, and NATREX. Below, I include a Table, taken from Driver and Westaway (2003, Bank of England), describing the main models used to explain EERs.

67