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
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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 :
Costless access to information
Lack of government controls
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
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 ($/£) spurts up in New York. Here is a chance to make some money!
To exploit the situation, this arbitrager does two things:
Consider the following quotes in New York, Frankfurt,
and London. (Assume no transaction costs)
Frankfurt ($/€ = 1.2471)
London (€/£ = 1.4544)
New York ($/£ = 1.8590)
$/€ = 1.2471
$/£ = 1.8590
€/£ = 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. € £ $ €
1,000,000 * 1 * 1.8590 * 1 = 1,024,930
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.
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.
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
¥/U$ = 106.50
C$/U$ = 1.3215
¥/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$
1 * 82.905 * 1 * 1.3215 = C$1.02772
Profit ≈ 2.8%
The CIA strategy proceeds as follows:
A US investor Converts Dollars to Pounds
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:
AP = A (1 + iuk )F - A (1 + ius)
This is the additional (or incremental) return obtainable from undertaking CIA.
The Interest Rate Parity:
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,
And subtracting 1 from both LHS & RHS gives:
[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.
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%
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)
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.
i.e., it is just as likely to guess too high as it is to guess too low.(See Appendix)
predictor of the future spot or that the forward differential
equals the expected change in exchange rates as follows:
t = 0 t = n
[ Expected Δ in Ex Rate ≈ Fwd Prem (disc)]
Question: Under what conditions would the forward rate be a biased predictor of future spot? What would happen?
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.
et = Pd ----- 5(a)
Pd = et Pf -----5(b)
Equation 5 can be interpreted as follows:
common currency, will be the same in every country.
Exercise: The Big Mac Exchange Rate (handout)
The Relative Version: is more meaningful in practice.
It modifies the absolute version as follows:
et = Ptd / P0d = Ptd/ Ptf --- (6)
eo Ptf / P0f P0d / P0f
where Pod, Pof, Ptd, Ptf are price indices.
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
et - eo = d - fd - f
eo 1 + f ---- (7)
(assuming that 1 + f 1, for low values of f )
t = 0 and t = t should equal the relative change in price indices of the two countries between t = 0 and t = t .
(See Appendix … handout)
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.
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.
Given that r = 5%, = 8%, Compute:(r=10%, = 20%)
The following experiences have been recorded
in the US in recent years:
Note that an increase (decrease) in the expected rate of inflation will cause a proportionate increase (decrease) in nominal interest rates of a country.
Algebraically, the IFE states that :
et = 1 + id(11)
eo 1 + if
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
et = 1 + id
eo 1 + if
which reduces to :
et - eo id - if(12)*
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
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)
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.