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### AJA4604.07 Parity Conditions in International Finance

These are economic theories linking exchange rates, price levels, and interest rates.

International Arbitrage and the Law of One Price:

In a competitive market characterized by :

Many Buyers

Many Sellers

Costless access to information

Lack of government controls

Free Transportation

The price of identical tradeable goods and financial assets must be equalized. This is the “law of one price.“

- All goods and financial assets obey the law of one price or free trade will equalize the price of any particular product in all countries.
- This law is enforced by the international arbitragers who follow the dictum of “buy low" and “sell high" to generate (riskless) profits for themselves.

Arbitrage In Foreign Exchange Market:

- Arbitrage is one of the basic forces at work in a market economy.
- The basic idea of arbitrage is simple.

Buy a product where its price is relatively low and sell where its price is higher- provided that the price spread more than offsets transaction costs

- The effect of arbitrage is to reduce or eliminate price differentials on identical products.

Locational Arbitrage:

- The prices charged by different banks for foreign exchange cannot vary significantly. Prices are kept more or less in line with one another through a process called “Locational Arbitrage”
- If the price of a currency varies from one bank to another, an arbitrager will be able to "buy low" and "sell high." Such an activity should lead to an increase in the rate at the low-priced bank and decrease in the rate at the high-priced bank.
- Arbitrage activity will continue as long as the difference in prices is large enough to generate a profit.

Example:Bank ABank B

Bid price for SF $ .50 $ .52

Ask price for SF $ .51 $ .53

In this example profitable arbitrage opportunity exists. Given that the arbitrager has $1m.Buy SF from bank A at .51 and simultaneously sell them to Bank B at .52. so that :

$1,000,000 SF = 1,000,000/.51 = SF 1,960,784.

Sell to Bank B for $1,960,784 * .52 = $1,019,608 for a profit of $19,608.

The high demand for SF in Bank A leads to an increase in price and increased supply of SF in Bank B leads to a decrease in price.

Such a situation usually disappears before most firms even become aware of it.

Example of Locational Arbitrage: Consider a large bank in London. The manager of the FOREX department observes over the telex that the $ price of Pound ($/£) spurts up in New York. Here is a chance to make some money!

To exploit the situation, this arbitrager does two things:

- First he/she contacts a broker or a correspondent bank in New York and sells, say £1m (sells high).
- At the same time he/she buys the same amount of £s in London (buys low). If the arbitrager does not buy and sell at almost the same instant the spread appears, there is a risk of missing the opportunity.
- The sale of pounds in New York where the price is high and its purchase in London, where the price is low, contributes to eliminating the price deferential.
- As other market participants take similar actions, the price difference disappears.

Triangular Arbitrage:

Example 1:

Consider the following quotes in New York, Frankfurt,

and London. (Assume no transaction costs)

Frankfurt ($/€ = 1.2471)

London (€/£ = 1.4544)

New York ($/£ = 1.8590)

- Is Triangular Arbitrage feasible? Show Why/Why not.
- Describe a strategy to profit from triangular arbitrage.
- What percentage profit is possible?

FRANKFURT

$/€ = 1.2471

NEW YORK

$/£ = 1.8590

London

€/£ = 1.4544

On the Bases of New York and Frankfurt quotes:

€ = € * $ = 1 . 1.8590 = 1.4907

£ $ £ 1.2471

1.4544 ≠ 1.4907, triangular arbitrage exists.

The € is worth more in London (fewer € required to buy £ in London compared to cross rate)

Hence acquire € elsewhere and sell it in London.

e.g. € £ $ €

Given €1m:

1,000,000 * 1 * 1.8590 * 1 = 1,024,930

1.4544 1.2471

Profit = €24,930 or 2.5.

Arbitrage activity causes £ to appreciate against € in

London, the $ to appreciate against the £ in New York

and the € against the $ in Frankfort.

Example 2:

Assume no transaction cost. Suppose £1 = $1.8095 in NY,

$1 = C$1.3215 in Toronto and C$1 = £ 0.4342 in London.

Show whether or not triangular arbitrage opportunities exist.

How could a trader profit from triangular arbitrage?

Compute the percentage profit possible.

Solution:

Toronto

C$/$=1.3215

NY

$/£=1.8095

London

£/C$= 0.4342

Given New York and Toronto quotes:

£ = £ . $ = 1 * 1 = 0.4182 ≠ 0.4342

C$ $ C$ 1.8095 1.3215

The C$ is worth more in London than implied by cross rates from NY and Toronto.

It will be profitable to acquire C$ and sell it for £ in London.

i.e.: C$ £ $ C$

Given C$ 1000:

1000* 0.4342 * 1.8095 * 1.3215 = C$1,038.28

Profit = 3.828%

Example 3: Assume zero transaction costs:

A: ¥/U$ = 106.50, B: C$/U$ = 1.3215 , C: ¥/C$ = 82.905

- Determine if triangular arbitrage is feasible.
- State what you would do to profit from arbitrage.
- Obtain the percentage profit possible.

Solution:

A

¥/U$ = 106.50

B

C$/U$ = 1.3215

C

¥/C$ = 82.905

By cross rates from A & B,

¥= ¥ . U$ = 106.50 . 1 = 80.590 ≠ 82.905

C$ U$ C$ 1.3215

The C$ is more valuable at C:

Borrow C$ and sell it for ¥ at C, Sell ¥ for U$ at A and the U$ for C$ at B.

e.g.C$ ¥ U$ C$

Given 1C$:

1 * 82.905 * 1 * 1.3215 = C$1.02772

106.50

Profit ≈ 2.8%

Covered Interest Arbitrage (CIA):

- CIA is a process of capitalizing on interest rate differential between two countries while covering for exchange risk.
- Suppose a U.S. investor decides to capitalize on a relatively higher British interest rates. The spot exchange rate is known and there exists a forward market. The only uncertainty is the future spot exchange rate.
- A forward sale can be used to lock-in the rate at which pounds could be exchanged for dollars.

The CIA strategy proceeds as follows:

A US investor Converts Dollars to Pounds

- Deposits Pounds in the U.K.
- Sells Pound proceeds forward for term to Maturity
- At Maturity converts pounds proceeds to dollars at the agreed upon forward rate.

Algebraically,

Let A = Amount to be invested

S = Spot exchange rate (direct); ius = US interest rate

F = Forward exchange rate ; iuk = UK interest rate

then a U.S investor can earn either A (1 + ius) at home or

A (1 + iuk )F from CIA. Arbitrage profit is given by:

S

AP = A (1 + iuk )F - A (1 + ius)

S

This is the additional (or incremental) return obtainable from undertaking CIA.

CIA is profitable if AP > 0. However, market forces resulting from CIA will cause price realignments so that excess profits from arbitrage are no longer possible.

- For example, as $s are used to purchase £s in the spot market, the spot rate, S, increases, netting fewer £s. The forward sale of £s puts a downward pressure on the forward rate, F, netting fewer $s. In addition, as U.S. investors transfer funds to UK, there will be a decrease in iuk and an increase in ius.
- As a result of Market forces from CIA a relationship exists between the forward rate premium (discount) and interest rate differentials. This is the subject matter of Interest Rate Parity Theory

Consider,

ius = Interest rate in US

iuk = Interest in UK

S = Spot exchange rate (Dollar/ Pound)

F = Forward exchange rate ( Dollar/Pound)

A US investor can earn (1+ ius) in the US, or (1/S)(1 + iuk) in the UK Since investment proceeds will ultimately be converted to dollars, but future spot exchange rates are not known with certainty, the investor can eliminate the uncertainty regarding the dollar value of the proceeds by covering with a forward contract.

The covered return is therefore given by: (1/S)(1 + iuk)F.

Arbitrage between the two investments (the domestic

and the foreign) results in parity so that,

(1)

or

(2)

And subtracting 1 from both LHS & RHS gives:

(3)

[assuming (1 + iuk) ≈ 1, especially with small values of iuk].

Equation 3 says that the interest rate differential between a comparable US and UK investment is equal to the forward premium (discount) on the pound.

- CIA tends to force a relationship between forward rate premiums/discounts and interest rate differentials.

Example: Given the following rates:

Spot $/C$ = .80 [try for $/C$ = .7505]

90 day forward $/C$ = .79 [try for $/C$ =.7430]

90 day iCan = 4%

90 day iUS = 2.5%

- Show whether or not CIA is profitable
- Find the yield to a US investor who executes a CIA.
- Find the 90 day forward rate on C$ if IRP holds.

Solution:

- The no arbitrage 90-day fwd rate on CS is given by:

From which solving for F gives F = .78846

The currency of the country with a lower interest rate

should be at a forward premium with respect to that of the

higher interest rate country.

The interest differential should be approximately equal to

the forward differential when the parity exists. (See Appendix)

Interpret points A,B,X,Y.

In reality the IRP line is a band because transaction costs arising from the spread on spot and forward contracts and brokerage fees on security purchases and sales cause effective yields to be lower than nominal yields.

- Note that if IRP exists, it does not mean that domestic and foreign investors earn the same return. Rather existence of IRP means that investors cannot use CIAto achieve higher returns than those possible at home. Effective returns are equalized for domestic investors if IRP holds regardless of where they invest - - domestic or foreign market.

Empirical Evidence:

- It is difficult to get quotations that reflect the same point in time for interest rates, and the forward rates.
- Nevertheless IRP theory is well supported empirically in International Finance.
- Indeed. in the Euro-Currency markets the forward rate is frequently calculated from interest rate difference between two countries using the no arbitrage condition.
- Eurocurrency markets are relatively unregulated.
- Deviations from IRP may occur due to capital controls, taxes, transaction costs, and political risks.
- Note that "default risks" could exit on loan contracts for which IRP is supposed to apply. This would create deviations from parity.

Forward Rate Parity (FRP)(The fwd rate as an unbiased predictor of future spot)

- The forward exchange rate must be equal to the expected future spot exchange rate at maturity otherwise riskless arbitrage will take place.
- One may question whether the forward rate should equal the expected future spot rate or whether there is a premium incorporated in the forward rate that serves as reward for bearing risk in which case the forward would differ from the expected future spot by this premium.
- Empirical work has dealt with the issue of whether the forward rate is an unbiased predicator of future spot rate.
- An unbiased predictor is one that is correct on average,

i.e., it is just as likely to guess too high as it is to guess too low.(See Appendix)

Pressure from the forward market is transmitted into the spot market and vice-versa.

- Equilibrium is achieved only when the forward differential equals the expected change in the future exchange rate.
- At I, parity prevails as an expected 2% depreciation of the pound is matched by a 2% discount on the Pound. Point J is a position of disequilibrium because a 3% forward discount on the Pound is more than offset by a 4% expected depreciation of the Pound.
- Speculators are expected to sell Pound forward and replenish or cover their commitments with 4% fewer units of domestic currency (short sale). Points L, X, K are in disequilibrium.

Formally we can state that the forward rate is an unbiased

predictor of the future spot or that the forward differential

equals the expected change in exchange rates as follows:

S0 Sn

|-----------------------------------|

t = 0 t = n

Fn

(4)

[ Expected Δ in Ex Rate ≈ Fwd Prem (disc)]

Empirical Evidence:

- There are pros and cons for the notion that the forward rate is an unbiased predictor of the future spot.
- It is probably unrealistic to expect a perfect correlation between forward rates and the realized future spot rates since future spot rates are influenced by events that cannot be perfectly forecast.
- The rationale for the “unbiased hypothesis” is that the foreign exchange market is reasonably efficient (semi-strong).

The forward market can be said to be efficient if the forward rate ruling at anytime is equal to the rational expectation of the future spot when the contract matures, plus the risk premium that speculators require to in order to compensate them for the additional risk that they bear in the forward market.

Question: Under what conditions would the forward rate be a biased predictor of future spot? What would happen?

Note:

- If Forward Rate Parity holds the expected return from speculating in a forward contract is zero.
- An unbiased forward rate need not be a very good predictor in terms of accuracy.

The Purchasing Power Parity (PPP)

- The PPP was first stated by Gustav Cassel (1918).
- He used it as a basis for a new set of official exchange rates at the end of World War I so that normal trade relation might resume.

The absolute form: States that equilibrium exchange rate between the domestic and the foreign currencies equals the ratio between the domestic and foreign price levels.

The absolute PPP postulates that perfect commodity arbitrage [in the absence of transaction or information costs or other restrictions] ensures that PPP relationship holds at each point in time.

Formally, consider the time t = 0 to t = t, and let

|--------|---------|----------------------------|

0 1 2 t

Let S0 = e0 and St = et ,

(Where spot exchange rate = domestic currency units per unit of a foreign currency ex = d/f)

Pod, Pof = initial aggregate price levels respectively for domestic and foreign.

Ptd, Ptf = aggregate price levels at time t respectively for domestic and foreign.

Then:

et = Pd ----- 5(a)

Pf

Pd = et Pf ----- 5(b)

Equation 5 can be interpreted as follows:

- The general level of prices, when converted to a

common currency, will be the same in every country.

- In other words, a unit of domestic currency should command the same purchasing power around the world.

This theory rests on the law of one price which states that free trade will equalize the price of any identical good (or asset) in all countries.

- The theory, however, assumes away transportation cost, tariffs, quotas, product differentiation, and other restrictions.

Exercise: The Big Mac Exchange Rate (handout)

The Relative Version: is more meaningful in practice.

It modifies the absolute version as follows:

- In comparison to a period when equilibrium rates prevailed, changes in the ratio of domestic and foreign prices will indicate the necessary adjustment in exchange rate between domestic and the foreign currencies.

Formally,

et = Ptd / P0d = Ptd/ Ptf --- (6)

eo Ptf / P0f P0d / P0f

where Pod, Pof, Ptd, Ptf are price indices.

Let,

Ptd / Pod = 1 + d ; d = Domestic inflation rate

Ptf / Pof = 1 + f ; f = Foreign inflation rate

then, et = 1 + d

eo 1 + f

et - 1 = 1 + d - 1

eo 1 + f

So that,

et - eo = d - fd - f

eo 1 + f ---- (7)

(assuming that 1 + f 1, for low values of f )

Equation (7) states that the relative (expected) exchange rate change for two currencies between

t = 0 and t = t should equal the relative change in price indices of the two countries between t = 0 and t = t .

Alternatively,

- For parity in the purchasing power of two currencies to obtain over a period of time, the rate of change of the exchange rate must equal the rate of change of relative prices.

(See Appendix … handout)

At A or B inflation differentials are offset by corresponding appreciation (depreciation) of the foreign currency. At A, there exists a 3% more domestic inflation than foreign. This is matched by a 3% increase in ex (d/f), (i.e. a 3% depreciation of domestic currency). At B, 1% more f is matched by a 1% decrease in ex (d/f).

- At D, f > d by 3%, but ex (d/f) has reduced by only 1%. This means that domestic residents have a reducedpurchasingpower over foreign goods. They therefore reduce their purchase of foreign goods but foreigners continue to purchase domestic goods. Foreign currency depreciates in value relative to domestic currency, so that D approaches the PPP line.
- Similarly at C, d > f by 2% but ex (d/f) has risen by only 1%. There is a higher purchasing power on foreign goods for domestic residents.

Note that if changes in nominal exchange rates are fully offset by relative price level changes between two countries, then the real exchange rates remains unchanged.

- Alternatively, a change in the real exchange rate is a deviation from PPP.

Empirical Evidence:

- Relative PPP tends to hold in the long-run especially in high inflation countries. The PPP does not hold consistently for many reasons:

01. Other factors maybe at work

02. No substitute for traded goods

03. Existence of internationally non-traded goods in

the national price indexes.

04. Changes in taste.

05. Technological Progress

06. Differently constructed price indexes

07. Different "Market Baskets"

08. Different weighing formula for Market Basket

09. Relative price changes (Vs changes in general price level)

10. The "best" index cannot be a basis for a perfect

representation of theoretical parity.

- PPP-determined exchange rates provide a valuable benchmark especially for policy makers.

The Fisher Equation

- There is a distinction between the real and nominal interest rate.
- The nominal interest rate is the rate quoted or observed in the market.
- The real rate measures the return after adjusting for inflation.
- It is the rate at which current goods are being converted into future goods.
- It is the net increase in wealth that people expect to achieve when they save and invest their current income.
- It is the added future consumption promised by a borrower to a lender.

Since virtually all financial contracts are stated in nominal terms, nominal interest rates will tend to incorporateinflation expectation in order to provide lenders with a real return.

- The Fisher Equation (named after Irving Fisher) states that the nominal interest rate, i is made up of a real required rate of return, r and inflation premium (the expected rate of inflation), . This equation is given approximately by:

i r + (10)

Theexactrelationship is given by:

1 + i = (1 + r) ( 1 + )

= 1 + + r + r

i r + (as r0)

Thus an increase in will tend to increase i.

Examples:

Given that r = 5%, = 8%, Compute:(r=10%, = 20%)

- The exact nominal interest rate.
- The approximate nominal interest rate.

The following experiences have been recorded

in the US in recent years:

- In the1950\'s and the 60\'s low inflation was accompanied by low i.
- In the 1970\'s and 80\'s high inflation was accompanied by high i.
- In the 90\'s the experience was low inflation with low interest rates.
- So far in 2000’s it has been low inflation and low interest rates.

Note that an increase (decrease) in the expected rate of inflation will cause a proportionate increase (decrease) in nominal interest rates of a country.

The International Fisher Equation (IFE)

- The IFE expresses the relationship between interest rate differentials and the expected changes in exchange rates. This relationship can be derived by combining the PPP with the GFE or from UIRP.
- The IFE is based on nominal interest differentials which are influenced by expected inflation.

Algebraically, the IFE states that :

et = 1 + id (11)

eo 1 + if

and,

et - eo = id - if id - if (12)

eo 1 + if

i.e., the spot exchange rates will change in accordance with the difference in interest rates between countries. Hence the return on foreign uncovered money market securities will, on an average, be no higher than return on domestic money marketfor a domestic resident.

i.e., 1 + id = 1 (1 + if) et

eo

or,

et = 1 + id

eo 1 + if

which reduces to :

et - eo id - if(12)*

eo

Essentially, IFE says that arbitrage between financial

markets in the form of capital flows should ensure

that nominal interest rate differential, (id - if ), is an

unbiased predictor of the future change in spot

exchange rates between domestic and foreign

currencies.

Note: If International Fisher Equation holds, no

superior investments or low cost source of funds are

possible between countries (ignoring currency risks)

(See Appendix … on my website)

At E, if > id by 3%, foreign currency depreciates by 3% to offset the interest difference so that parity holds.

- At F parity also obtains as a 2% higher domestic interest is offset by a 2% increase in exchange rates.
- Points along IFE reflect exchange rates adjusting to offset interest rate differentials.
- Points below the IFE line implies an increase in return on foreign assets owned by domestic investors.
- Points above the IFE line implies a reduction in return on foreign assets owned by domestic investors.

Empirical Evidence:

Evidence exists that currencies with high interest rates like the Peso, Lira, Real, etc tend to depreciate, while those with low interest rates like the Yen, Swiss Franc, tend to appreciate. In the long-run interest rate differentials tend to offset exchange rate changes.

Problems with IFE:

Determining the effects of a change in ion e becomes complicated if Fisher Equation is stated as:

i r + .

Given this, it is clear that a change in i can be due to a change in r or a change in .

These two possibilities have opposite effects on e.

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