CHAPTER 5 - INTERNATIONAL PARITY RELATIONSHIPS

We now look at Int'l. Parity Relationships, starting with the Law of One Price, extended to: Purchasing Power Parity (PPP) and Interest Rate Parity (IRP).  These parity relationships help us to understand: 1) how ex-rates are determined, and 2) how to forecast ex-rates.

Int'l. Parity is based on EMH (Efficient Market Hypothesis).  FX markets are efficient when: 1) securities/FX are priced efficiently reflecting all currently available information, and 2) no arbitrage opportunities exist.

Arbitrage: Riskless, certain profit opportunities by exploiting price discrepancies.  Simultaneously buying and selling mispriced securities/FX to make a guaranteed, riskless profit without any investment.  "Picking up dimes with a bulldozer."  Example: triangular arbitrage.

Int'l. parity conditions exist when there are no arbitrage opportunities and markets are in equilibrium.  "No $100 bills laying on the sidewalk."   

Law of One Price (LOP): PD = S ($/£)  PF, where
PD = Domestic Price ($)
PF = Foreign Price (£)
S ($/£) = spot ex-rate.

Example: Gold in US is $405/oz., and gold in UK =  £225  x $1.80/£ = $405  

If (LOP) Law of One Price (Price Equalization Principle) did not hold, arbitrage would be possible, and would restore parity.
 

INTEREST RATE PARITY (IRP)

IRP: (No) Arbitrage condition when int'l. financial markets are in equilibrium.  Assuming free movement of capital, int'l. financial markets should be efficient.  "Smell of profits" eliminates any discrepancies.  Covered Interest Rate Parity = Parity conditions in fin. mkts., when forward mkts. are used to eliminate or "cover" any FX risk.

Example: U.S. investor has $1 to invest for one year.  You consider two strategies: 1) Invest in U.S. treasury securities at i$, the domestic interest rate, for one year; or 2) Invest in foreign U.K. treasury securities at i£, and hedge FX risk by selling maturity value of £s forward one year.

In U.S., your payoff (maturity value) in one year will be: $1(1 +  i$)

In equilibrium this should be the same as your payoff in U.K.

In U.K., your investment strategy involves:

1. Sell $1 for £s to get 1 / S($/£) pounds.  (We assume that S = S($/£)).
2. Invest £s at U.K. int. rate ( i£) with payoff = 1/S x (1 +  i£ )
3. Sell £s forward at F360 ($/£) for the maturity value of the UK investment, to get a guaranteed amount of $s.

For either investment, you start and end with U.S. dollars.  For Strategy #2, you have completely hedged ("covered") FX risk with the forward contract.

The Interest Rate Parity (IRP) condition would be:

(1 +  i$) = (F / S) (1 +  i£)

IRP is an application of the Law of One Price (LOP) to financial securities, says that two identical securities (e.g. Treasury securities) should have the same return, after accounting for the ex-rates (S and F).  We need the F rate here because we have added the time dimension, in this case one year in the future.

Example: i$ = 5%; i£  = 8%; F = $1.4583/£ and S = $1.50

IRP Holds: (1.05) = ($1.4583 / $1.50) x 1.08

Invest $1000 in U.S.: $1000 x 1.05 = $1050

Invest $1000 in U.K.: $1000 / ($1.50/£) = £666.6667 x 1.08 = £720 x $1.4583333 = $1050

One of the reasons IRP should hold is because of Covered Interest Arbitrage (CIA), no risk, no net investment arbitrage when IRP does not hold.  Covered Interest Arbitrage (CIA) involves: 1) Borrow $s in U.S. at i$ and buy UK pounds, 2) Invest (lend) in UK at  i£, 3) Sell pounds forward at F, to cover ex-rate risk.

See Example 5.1 (page 101).  i£ = 8%  and  i$ = 5%.  S = $1.50/£ and F = $1.48/£.  IRP does not hold and can be exploited by CIA. 

Logic: Nominal interest rates are 3% higher in UK than U.S..  If IRP holds, what would we expect will happen to the £?  Should depreciate by approx. 3% if IRP holds. 

%CHG = (F - S) / S  x   100.
($1.50 - 1.48) / $1.50  x   100 = -1.333%

British Pound is expected to depreciate by only -1.33% instead of 3%.  Therefore, expected covered return in U.K. to a U.S. investor would be 8% - 1.33% ≈ 6.667% (U.K.) vs. 5% (U.S.).

Effective Return to U.S. Investor = iF  +  % Appreciation Foreign Currency

 

Effective Return to U.S. Investor = iF  -  % Depreciation Foreign Currency
 

Logic: When investing in a foreign market you are making 2 simultaneous investments: 1) the foreign security, and 2) the foreign currency.   


We can also check IRP formula:

1.05 ?=? (1.48/1.50) (1.08) = 1.0656
1.05 < 1.0656
5% < 6.56%  

Returns to a U.S. investor in U.K. (6.56%) are higher than in U.S. (5%) by more than 1.5%.  

 

Arbitrage Strategy:

1. Borrow $1m in U.S. at 5%, promise to pay $1.05m back in one year. 

2) Buy $1m worth of BP in spot market at S($1.50/£) for £666,667.

3. Invest £666,667 in U.K. at 8% to get guaranteed £720,000 payoff in one year. 

4. Enter into a 1 yr. forward contract to sell £720,000s at $1.48/£, for $1,065,600 guaranteed in one year.

5. Pay back $1,050,000 on the loan in U.S., and make $15,600 profit. 

 

No risk, no investment, arbitrage strategy,  see CF diagram, p. 103.

 

What will happen over time?

1. Int. rates will rise in U.S. due to borrowing pressure.  Demand for Credit goes up.   
2. Int. rates will fall in U.K. due to buying pressure for bonds.  Bond prices rise, int. rates fall.
3. £ will appreciate in spot market due to buying pressure, S will rise.  
4. £ will depreciate in the forward market, due to selling pressure, F will fall.

The difference between the two int. rates (3%) will narrow, and the difference between the S and F will widen (the forward discount for £ will increase from 1.33%), until IRP is restored, possibly at a forward discount of 2% for the £, until the int. rate spread is EXACTLY equal to the %CHG in £.  For example, suppose interest rates end up at 5.5% in U.S. and 7.5% in U.K., and the £ sells at a forward discount of 2% in the Forward Mkt. (S = $1.505, F = $1.4749).  In that case, your effective return is 5.5% in EITHER country, and IRP is restored, partly by: a) a decrease in the interest rate differential and partly by: b) an increase in the forward premium.  

Another way to view IRP:

 i$ =   i£ + (F - S) / S

(i$ -  i£) = (F - S) / S

Shows that int. rates (bond prices) are directly linked to S and F ex-rates, and says that the difference in interest rates should be equal to the forward discount or premium for FX.   

The above equality can be represented graphically, IRP line on page 103.  Note: Units for both axes are %.   

Point A represents the previous example.  Int. rates are 3% higher in U.K. than U.S., so that the £ should depreciate by 3% and be selling at a 3% forward discount according to IRP, however it is actually selling at a 1.33% forward discount, representing profit opportunities in U.K.  Anything above the IRP line represents profits by either: a) investing in U.K. instead of U.S., or by b) borrowing in U.S. and lending in UK (arbitrage).

Anything below the line, represents profit opportunities by either: a) investing in U.S., or b) borrowing in U.K. and lending in U.S.  Point B: U.S. interest rates are 4% higher than U.K., so U.S. dollar should depreciate by 4% and pound appreciate by 4%.  However, if the dollar was actually selling at a 2% discount (pound at 2% premium) in the forward market, U.S. investment would be very attractive.  You could borrow in U.K., invest in U.S. and make money.

See CIA Example 5.2 (p. 103) for a 3-month period using interest rates in Germany and the €.  Interest rates are usually quoted at annual rates, so an adjustment must be made.  Also, IRP formula (5.1) is based on American terms ($1.50/£), and the € is quoted here in European terms (€/$). 

3-month Interest Rates: i$ = 2% (U.S.) and i€ = 1.25% (Germany)

Convert ex-rates on p. 103 to American terms to check IRP:

S = $0.9887/€ and F3 = $0.9900/€, dollar selling at  

1.02 ?=? (.9900/.9887) (1.0125)

IRP: 1.02 > 1.0138, return is higher in U.S. than Germany.  Invest in US, or use CIA by borrowing in Germany, invest in U.S. to make money.

According to the difference in interest rates (.75%), the $ should be selling at a .75% forward discount for IRP to hold.  However, the dollar is selling in the forward market at only about a .1315% discount [(F - S) / S].  A German investor can get 1.25% yield in Germany vs. a 2% - .1315% = 1.8685% in the U.S., selling the $s forward at F3 to get €s in 3 months.

 

CIA:
1. Borrow €1,011,400 (equal to $1m) in Germany @ 1.25%, promise to pay €1,024,042.5 in 3 months.  

2. Sell €1,011,400 for $ at $.9887285/€ = $1m

3. Invest $1m for three months @ 2% to get $1,020,000 in three months.

4. Sell $1,020,00 forward @ €1.0101/$ = €1,030,302

5. Pay back loan of €1,024,042.5 and end up with €6,259.50 (or $6196 @ F = $0.99).   

 

Note: This would represent a point below the line on Exhibit 5.3.  Adjustment would take place by a decrease in the interest rate differential and an increase in the forward discount for the dollar.  Example: i$ = 1.75% and i€ = 1.50%, and $ (€) sells at forward discount (premium) of .25%.  German investor gets effective return of 1.5% in either country, U.S. investor gets 1.75% effective return in either country. 

 

US: 1.75% - .25% = 1.50%

 

 

IRP and EX-RATE DETERMINATION

 

IRP helps explain ex-rate determination, by linking interest rates and ex-rates.  We can rearrange the IRP formula:

 

S =    (1 +  i£)   x  F

         (1 + i$) 

Spot ex-rates ($/£) are partly determined by relative interest rates, i.e., the interest rate in U.K. (and other countries) relative to interest rates in U.S.  From the formula above, we can see that if interest rates increase in the U.K. (U.S.), the £ ($) will appreciate (holding F constant) as capital flows to the countries where interest rates are increasing, especially if the real interest rate is increasing.  

Forward rates also influence ex-rates.  We can express F as:

F  = E (St+1 | It )

which says that the Forward Rate (F) is the Expected (E) future Spot Rate when the forward contract matures in the future at time t+1, conditional upon all current and relevant information (It).  What relevant information??

Combining the two equations above, we have:

S     =      (1 +  i£)   x   E (St+1 | It )

                (1 + i$) 

We conclude that: 1) Expectations about the future spot rate, influence today's spot rate.  If the expected spot rate goes up, the current spot rate (S) goes up now.  Expectations drive the spot and forward rates. 

2) Information (I) drives ex-rates.  Information changes daily, and forward and spot rates change daily as information changes.  Ex-rates are dynamic and volatile.  EMH says that prices reflect ALL currently available information, and the only thing that changes prices is NEW information.

Conclusion: Expectations and Information are important determinants of ex-rates, Spot and Forward markets, which are extremely dynamic markets.  

We can also say that:  (i$ -  i£) =  E(e),

where E(e) is the expected rate of change in S, or the expected percentage (%) change.  The relative interest rate differential between two countries should reflect the expected percentage change in S.  If one-year interest rates are 3% higher (lower) in UK, we expect the Pound to depreciate (appreciate) by 3%.
 

(5% - 8%) = -3%  Interest rates are higher in U.K. than U.S., £ is expected to depreciate by 3%.

 

(7% - 4%) = +3% Interest rates are higher in U.S., £ is expected to appreciate by 3%. 

 

 

WHY MIGHT IRP NOT ALWAYS HOLD?

 

Although it should and does hold most of the time, why might IRP not always hold, i.e., how to explain deviations from IRP.

 

1. Transaction costs.  We have so far assumed that transaction costs = 0.  Deviations from IRP could be explained by transaction costs.  For example, a) the borrowing and lending interest rate are NOT usually the same, as we assumed in the previous examples of CIA.  Interest rates are usually higher for borrowing than for lending, e.g., commercial banks.  Therefore, higher interest rates for borrowing may eliminate or exceed any potential arbitrage profits.

 
b) Also, there are transaction costs (commissions, bid-ask spreads, fees, etc.) to buy/sell currency and forward contracts and securities, which we have ignored, these may reduce or eliminate arbitrage profits.

Therefore, the IRP line should have a band around it to reflect transactions costs, see Exhibit 5.4, p. 106.  Deviations within the shaded area (Point D) do not represent arbitrage opportunities, only deviations outside the shaded area (Point C).  

2. Capital controls that limit, restrict or ban cross border capital flows (in or out of a country), can create significant barriers to intl. arbitrage, resulting in possible deviations from IRP.  See Japan story on pages 106-107.  Between 1978-1979, Japan restricted capital inflows to prevent Yen from appreciating.  Why?  Deviations from IRP resulted in 1978 and 1980, but did not reflect unexploited arbitrage opportunities.  See Exhibit 5.5 on p. 107. 
 

PURCHASING POWER PARITY (PPP)

PPP is the Law of One Price (LOP) applied to a standard commodity basket.

P$ =  S  x  P£

where P$ is the domestic price level (CPI), P£ is the foreign price level (CPI), and S = $/£ ex-rate.  PPP says that the dollar price of the commodity basket in the U.S. should be equal to the dollar price of the same commodity basket purchased in U.K.  PPP is the LOP applied to a standard commodity basket, which should be priced the same in all countries, when measured in a common currency.  What if prices were higher on average in U.S. than U.K.?:

P$  >  S  x  P£

Commodities would be purchased in U.K. and sold in the U.S., which would put upward pressure on: a) the £ (S goes up) and b) British prices (go up), restoring PPP.   

Or we can say:

S = P$ / P£, where the spot rate (S) is the ratio between the two country's price levels.  If domestic prices rise (P$ goes up), S should go up, meaning that the £ is appreciating and the $ is depreciating.  Absolute PPP links ex-rates to the relative price levels in two countries.

Relative PPP looks at Relative Inflation Rates (%):

e =  %INFus -  %INFF ,

where e is the % change in the ex-rate.

If   INFus >  INFF , then the dollar will depreciate and the pound will appreciate (S will get bigger).

If   INFus <  INFF, then the dollar will appreciate and the pound will depreciate (S will get smaller).
 
Therefore, ex-rates are linked to relative prices and relative inflation rates.  If annual inflation in the U.S. is 6% and only 4% in U.K., then the dollar should depreciate by 2% annually and the pound will appreciate by 2%.  If inflation in the U.S. is 6% and 9% in U.K., the pound should depreciate by 3% and dollar appreciate by 3%.

 

Whichever country has the higher (lower) inflation rate will experience a currency depreciation (appreciation).
 

BIG MAC PPP,  see p. 110-111, annual report by The Economist to test for PPP/LOP.  If LOP holds, the price of a Big Mac should sell for the same price around the world, measured in dollars using the actual spot ex-rate.  Procedure: Big Mac sells for $2.49 in U.S., take $2.49 to countries around the world, convert to local currency, go buy a Big Mac.  If Big Macs are cheap (expensive), the dollar is overvalued (undervalued), according to PPP or LOP. 

 

Example: Take $2.49 to Argentina, convert to 7.79 pesos (2.49 x 3.13), compare 7.79 pesos to Big Mac price in pesos: 2.50 pesos, you could buy 3, dollar is overvalued by 68% (7.79 vs. 2.50).  Take $2.49 to Switzerland, convert to SF4.134 (2.49 x 1.66), compare to Big Mac price of SF6.30, dollar is undervalued by +53% (4.134 vs. 6.30).      

 

EVIDENCE on PPP:

e =  %INFus -  %INFF , or

(INFUS -  INFF) - e = 0 if PPP holds

Example: Inflation in U.S. is 5%, inflation in U.K. is 3.5%, we would expect e = +1.5%.  The dollar should depreciate by 1.5% and the pound should appreciate by 1.5%, according to PPP.  If the dollar actually depreciates by more than 1.5%, it depreciates by more than predicted by PPP, and strengthens the competitiveness of U.S. industries in world markets (below line in Exhibit 5.6 on p. 109).  Assume that U.S. inflation is 2% and U.K. inflation is 5%, dollar is expected to appreciate by +3%.  If dollar appreciates by more than 3%, it weakens our competitiveness in world markets (mid-1980s), above the line in Exhibit 5.6.  

See page 109.  Ex-rates often deviate from PPP, are above and below the PPP rate.  Example: US dollar was much stronger in mid-1980s than justified by PPP.  Our inflation rate was not that much lower than other countries, and the $ still got much stronger than PPP would explain.  (Appreciation of the dollar was not explained by low U.S. inflation compared to inflation in other countries, see example above.) 

PPP is just the LOP extended to a standard commodity basket.  For PPP to hold, the commodity basket would have to be: a) consistent across countries, b) all goods in basket would have to be tradable so that arbitrage would be possible, and c) prices would have to be perfectly flexible (no price controls).

Not all of these conditions are met, so PPP does not always hold.  For example, see world prices on p. 112.  The standard CPI commodity basket is not the same in U.S. vs. Japan vs. U.K., and not all goods are tradable (e.g. haircuts) and not all prices are perfectly flexible.  Also, there are transportation costs, transactions  and possible trade barriers (tariffs, quotas) that would prevent PPP from strictly holding.

PPP would hold with "frictionless trading."  Holds most closely with homogenous, traded commodities, (Treasury bills, gold, oil, wheat, steel, bank CDs, etc.)

SUMMARY OF PPP:
1. More of a long run than short run.
2. Can be used to tell if a currency is over or under-valued.
3. Int'l. comparisons are more accurate using PPP-ex-rates vs. mkt ex-rates.  See page 121.

To compare GDP across countries, we have to convert to a common currency, like dollars.  However using market rates can distort the comparison.  If the dollar (pound) is under/over valued, then converting into dollars may be distorted.  For example, according to PPP, China's currency is artificially undervalued (and $ is artificially overvalued) at the market ex-rate, lowering the value of China's GDP in the second column ($1.16T), ranking 6th.  Adjusting for PPP value of China's currency, they are second ($5.51T).  India, Brazil and Russia also rise in the rankings because their currencies are undervalued according to PPP, and Japan, Germany, France, U.K. and Italy fall.  The ¥ and € are OVERVALUED according to PPP.  Also, prices are cheaper in China and India than in U.S. for the same goods and services.   
 

FISHER EFFECTS (PARITY CONDITION)

Nominal ius =  r (real rate) +  INFe (Expected inflation)  Fisher Equation

Example: One-year nominal interest rates in U.S. are 5%, the one-year real rate is 2%, and the one-year expected rate of U.S. inflation is 3%.  This would hold for other countries as well.   

If real interest rate (r) is constant, changes in nominal interest rates reflect changes in Expected inflation (Δi = ΔПe).  Fisher Effect: One-to-one relationship between changes in inflation expectations and changes in nominal interest rates.  Assume that the real interest rate is the same internationally, due to free capital flows.  In that case the relative difference in nominal interest rates reflects relative differences in expected inflation:

Fisher Effect:

( i$ - i£ )  =  E (П$ -  П£)        

Assuming a constant r, the differences in nominal interest rates reflect differences in expected inflation. 

With unrestricted capital flows, we can assume that the real rate of interest is constant internationally. We would have the following additional parity conditions (see page 114):

 

International Fisher Effect (IFE):

 

E (e)  =   ( i$ - i£ )

 

The difference in nominal interest rates (i$ - i£) between U.S. and U.K. reflects the expected change in the ex-rate.  For a one year period, if i$ = 5% and i£ = 7%, the dollar (pound) is expected to appreciate (depreciate) by 2% over the next year.

 

Forward Expectations Parity (FEP):

(F - S) / S  =   E (e)

 

Any forward premium or discount will be equal to the expected change in the ex-rate.  If the pound is expected to depreciate by 2% over the next year, it will be priced at a 2% forward discount in the forward market. 

 

SUMMARY OF INTERNATIONAL PARITY CONDITIONS

See Exhibit 5.9, p. 115: The difference in nominal interest rates (%) reflects the difference in inflation rates (%), which is also equal to the expected change in S (%), which is also reflected in the forward premium or discount (%). 

 

EXAMPLE 1:  If T-bills = 8% in U.S. and T-bills are 6% in U.K., the +2% difference reflects the expectation of 2% higher inflation in U.S.  In addition, the BP (USD) is expected to appreciate (depreciate) by +2%, which is reflected in a forward premium (discount) of +2% for the BP (USD).

EXAMPLE 2: Assume that real rate r = 2%, Expected inflation is 2% in U.S. and 4% in U.K.  Nominal interest rates will be 6% in U.K. and 4% in the U.S. for one-year T-bills, and (i$ - i£) = (4% - 6%) = -2%, which reflects E (П$ -  П£) = -2%, which reflects a -2% depreciation of the £, E(e) = -2%, and the pound should be selling at a -2% forward discount in the forward markets [(F - S) / S)] = -2%.  

The following parity conditions exist:
IFE (Int'l. Fisher Effect)
FE (Fisher Effect)
IRP (Int. Rate Parity)
FP (Forward Parity)
PPP (Purchasing power parity)
FPPP (Forward PPP)
 

 

PPP AND EX-RATE DETERMINATION

 

Monetary approach, combining PPP and Quantity Theory of Money.  Remember that:

M * V  =  P * Y

where M is Money Supply (M1), V is Velocity (turnover rate) of money, P is Price Level (CPI) and Y is Real Output.  Rearranging:

PUS = MUS  VUS  /  YUS    (for US)

PUK = MUK VUK /  YUK    (for UK)

Divide U.S. / U.K. to get:

 

PUS / PUK = (MUS / MUK)  x  ( VUS /  VUK)  x   ( YUK  /  YUS ) or since S = PUS / PUK
 

 

S  =  (MUS / MUK)  x  ( VUS /  VUK)  x   ( YUK  /  YUS )

 
so we can see that ex-rates (IN LONG RUN) are influenced by a) Relative Money Supplies (MS), b) Relative Velocity, and c) Relative productivity.

 

Ceteris paribus:

1) An increase in U.S. MS (relative to UK), will depreciate the $ and appreciate the £ (S rises). 

2) an increase in U.S. V, relative to UK, will also depreciate the dollar.  V goes up, like MD going down, same as increase in MS.
3) an increase in U.K. productivity (output) relative to U.S. will appreciate the £.  Why?  Increased productivity translates into cheaper, better products, increases demand for U.K. products, appreciates the £.