Question

The present exchange rate between US dollars and Euros is 1.34 $/Euro. The price of a domestic 180-day Treasury bill is $99.50 per $100 face value. The price of the analogous Euro instrument is 98.50 Euros per 100 Euro face value.

a. What is the theoretical 180-day forward exchange rate?

b.Suppose the 180-day forward exchange rate available in the marketplace is 1.31 $/Euro. This is less than the theoretical forward exchange rate, so an arbitrage is possible. Describe a risk-free strategy for making money in this market. How much does it gain (in dollars), for a contract size of 100 Euro?

Answer 1

a.

According to Interest rate parity theorem(IRPT), Forward rate or theoretical rate can be calculated as below -

F/S = (1+rh)/(1+rf)

Where,

F = Forward exchange rate

S = Spot rate

rh = home currency interest rate

rf = foreign currency interest rate

Further, If Forward rate is not at equilibrium then there will arbitrage opportunity which means gain without making any investment and taking risk.

Please refer below spread sheet for calculation.

b.

Strategy for Arbitrage

Interest rate differential (rf-rh) < domestic currency forward premium     

Borrow in foreign currency and bring borrowed money to home country at spot rate and invest it with home currency interest rate and cover this with forward contract. On Maturity, sell the home currency at forward rate and pay the loan liability of foreign currency.

The difference between investment proceeds in home country and home currency required to sell at forward rate to pay foreign currency loan is Arbitrage gain.

Please refer below spread sheet for calculation.

U18 2$/Euro 1.34 180-Days T-Bill Interest rate 100 100 4 99.5 0.5025%Interest rate Differential 1.0203% 5 Euro 98.5 1.5228% 7 Equilibrium Forward Rare Market Forward rare $/Euro S/Euro 1.33|home currency Premuim 1.31 2.239% Arbitrage strategy 11 Borrow 98.5 euro and convert to $ at spot 12 Invest in $ for 180 da 13 Forward contract to buy euro at $/euro 14 Loan Liability of Euro 15 Proceeds from $ investment 16 $required to buy 101.52 euro at Forward rate 17 Arbitrage Gain 18 131.99 1.31 100.00 132.65 131.00 1.65

formula reference -

U18 1.34 180-Days T-Bill Interest rate (D4-CA)/CA (D5-C5)/C5 99.5 100 Interest rate Differential -E5-E4 Euro 98.5 100 Equilibrium Forward Rare Market Forward rare S/Euro /Euro (C2-E8)/C2 home currency Premuim 1.31 10 Arbitrage strategy Borrow 98.5 euro and convert to $at s Invest in S for 180 days Forward contract to buy euro at S/euro Loan Liability of Euro Proceeds from $investment $required to buy 101.52 euro at Forward rate Arbitrage Gain 98.5 C 12 13 14 15 16 17 18 19 20 -E8 100 E11 (1+E4 -E14 E13 E15-E16

Thus,

(a) Theoretical 180-day forward exchange rate is 1.33$/euro

(b) Arbitrage gain is $ 1.65

Please note-

Contract size is 100 euro, that's why Arbitrager borrow 98.5 euro so that after 180-day he require to pay 100 euro at prescribed interest rate.