g: Compute the effect of transaction costs (spreads) on a foreign exchange arbitrage opportunity.

Example: What happens if transaction costs are 5% per trade?

1 dollar will buy 2.2529 (1 - .05) = DM 2.1403 in Germany.

DM 2.1403 will buy (2.1403)(0.8250)(1 - .05) = SF 1.6775 in Switzerland.

SF 1.6775 will buy (1.6775)(1 / 1.7799)(1 - .05) = $0.90

Arbitrage is not possible here. You turned $1 into 90 cents.

What happens if transaction costs fall to 1%?

1 dollar will buy 2.2529 (1 - .01) = DM 2.2304 in Germany.

DM 2.2304 will buy (2.2304)(0..8250)(1 - .01) = SF 2.8217 in Switzerland.

SF 1.8217 will buy (1.8217)(1 / 1.7799)(1 - .01) = $1.0132.

Arbitrage is possible here. You turned $1 into $1.0132.

Transaction costs do in fact hamper arbitrage opportunities and allow some market inefficiencies to exist. Thus, there will be a trading range around an exchange rate that exists within which arbitrage does not occur because of transaction costs.

h: Compute the no-arbitrage range of direct quotes for a currency.

For a single exchange rate, the arbitrage free range will be:

exchange rate (1 - TC)2 less than or equal to the exchange rate less than or equal to the exchange rate / (1 - TC)2.

Using the data for the DM relative to the dollar from the previous LOS example, (DM 2.2529 per dollar using transaction costs of 1 %), the no-arbitrage range would be:

2.2529 (1 - .01)2 is less than or equal to 2.2529 is less than or equal to [(2.2529)/(1 - .01)2 or, 2.20807 is less than or equal to 2.2529 is less than or equal to 2.2986. Hence, holding all else constant, the DM can trade within this range (i.e., the bid-ask spread can be within this range) relative to the dollar before arbitrage is available.

We can also compute this range in $/DM by inverting the quotes and reversing the upper and lower ranges as follows (why do we reverse the quotes? For the same reason that we switch the bid and ask when we invert):

1/(2.2986) is less than or equal to 1/(2.2529) is less than or equal to 1/(2.20807) or, $.43505 is less than or equal to $.44387 is less than or equal to $.45288. This is the widest bid-ask spread possible that will eat up 1% of transaction costs.

i: Define forward discount and forward premium.

A foreign currency is at a forward premium if the forward rate expressed in dollars is above the spot rate. Forward premium = forward rate - spot rate = positive number.

$/DM 0.4439 - 0.4315 = + 0.0124

A foreign currency is at a forward discount if the forward rate expressed in dollars is below the spot rate. Forward discount = forward rate - spot rate = negative number.

$/SF 0.5618 - 0.5705 = - 0.0087

j: Calculate a forward discount or premium and express either as an annualized rate.

The forward premium or discount is frequently stated as an annualized percentage using the following formula.

[(forward rate - spot rate) / (spot rate)] [(360)/(# of forward contract days)]

Example: If the 90-day forward rate for the DM is $.4439 and the spot rate is $.4315, then the annualized premium is: [(.4439 - .4315) / (.4315)] [(360) / (90)] = 11.49%.

Since it takes more dollars to buy a DM in the forward market relative to the spot, then the DM is trading at a premium to the dollar.

j: Calculate a forward discount or premium and express either as an annualized rate.

The forward premium or discount is frequently stated as an annualized percentage using the following formula.

[(forward rate - spot rate) / (spot rate)] [(360)/(# of forward contract days)]

Example: If the 90-day forward rate for the DM is $.4439 and the spot rate is $.4315, then the annualized premium is: [(.4439 - .4315) / (.4315)] [(360) / (90)] = 11.49%.

Since it takes more dollars to buy a DM in the forward market relative to the spot, then the DM is trading at a premium to the dollar.

k: Explain the interest rate parity theory.

Interest rate parity can be approximated by equating the difference between the domestic interest rate and the foreign interest rate to the forward premium or discount. Interest differential ? the forward differential. Restating this equation in more familiar terms gives:

r domestic - r foreign is approximately equal to (forward exchange rate - spot exchange rate / spot exchange rate)

When the condition above prevails, equilibrium will exist in the international money markets. You should also know that the exact interest rate parity equation, which appears below. Interest rate parity insures that the return on a hedged (covered) foreign investment will just equal the domestic interest rate of investments of identical risk. When this happens there are no arbitrage possibilities and the difference between the domestic interest rate and the hedged foreign rate (called the covered interest differential) is zero.

Forward (DC/FC) / Spot (DC/FC) = (1 + r domestic / 1 + r foreign)

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