Question

Suppose the 6-month risk free spot rate in HKD is 1% continuously compounded, and the 6-month risk free rate in NZD is 3% continuously compounded. The current exchange rate is 5 HKD/NZD.

1. Suppose our usual assumptions hold, i.e., no constraints or other frictions. What is the forward exchange rate with 6 months to maturity such that there is no arbitrage?

2. Suppose again that our usual assumptions hold, i.e., no constraints or other frictions. Suppose you can enter a forward contract to buy or sell NZD 1 for HKD 5. Is there an arbitrage? If yes, describe an arbitrage strategy. If no, briefly explain why not.

3. Suppose now that there are transaction costs is spot and forward exchange rates. That is, to buy NZD 1 you have to pay HKD 5.01 in the spot market today or HKD 5.03 in the 6-month forward contract, and to sell NZD 1 you receive HKD 4.99 in the spot market today or HKD 4.97 in the 6-month forward contract. Is there an arbitrage? If yes, describe an arbitrage strategy. If no, briefly explain why not.

4. Consider again the prices in (c) and further assume that your borrowing costs are 0.5% higher than the risk free rate, while the income from lending is equal to the risk free rate. That is, if you borrow HKD for 6 months then you have to pay a 1.5% continuous compounded interest rate, and similarly, if you borrow NZD for 6 months then you have to pay a 3.5% continuous compounded interest rate. Is there an arbitrage? If yes, describe an arbitrage strategy. If no, briefly explain why not.

Answer 1

A ) As per Interest rate parity,

Arbitrage free 6 month Forward Exchange rate

= Spot exchange rate * exp( (interest rate of HKD - interest rate in NZD) * 6/12)

= 5* exp((0.01-0.03)*6/12)

=4.950249 HKD/NZD

Forward rate for No Arbitrage is 4.950249 HKD/NZD

B) Since the forward contract is assumed to be 5 HKD/NZD and it is different from the No arbitrage forward price , Arbitrage is possible in the following way:

a) Borrow 5 HKD at 1% for 6 months, Amount payable at maturity =5*exp(0.01*6/12),=HKD 5.025

b) convert 5HKD to get 1 NZD in the spot market today

c) Invest 1 NZD at 3% for 6 months. Amount at maturity = 1*exp(0.03*6/12) = NZD 1.0151

d) Sell 1.0151 NZD in the forward contract after 6 months at the rate of 1NZD = 5HKD

e) After 6 months, get NZD 1.0151 and sell this using forward contract to get HKD 1.0151*5 = HKD 5.075565

Pay the maturity amount of borrowing = HKD 5.025 and take the remaining amount = 5.075565-5.025 = HKD 0.0505 as arbitrage profit

C) In presence of Transaction costs,  

Arbitrage can work in two ways

i) Borrow HKD, convert to NZD at spot rate and then sell NZD forward. After 6 months, you may have more HKD than you owe.

ii) Borrow NZD, convert to HKD at spot rate and then sell HKD forward. After 6 months, you may have more NZD than you owe.  

Strategy i)

a) Borrow 5.01 HKD at 1% continuously compounded (cc) rate for 6 months. Amount payable after 6 months

=5.01*exp(0.01*6/12) = HKD 5.0351

b) Convert the amount to NZD 1 at spot rate

c) Lend the amount at 3% cc rate in NZD . Maturity amount = 1*exp(0.03*6/12) = NZD 1.0151

d) Sell NZD 1.0151 forward after 6 months

e) After 6 months, get NZD 1.0151 and sell it using forward to get 1.0151*4.97 = HKD 5.0451

, pay the maturity amount and take the remaining amount = 5.0451-5.0351 =HKD 0.01 as arbitrage profit

There is no need to evaluate strategy ii) as strategy i) has arbitrage possibility

D) Arbitrage can work in two ways

i) Borrow HKD, convert to NZD at spot rate and then sell NZD forward. After 6 months, you may have more HKD than you owe.

ii) Borrow NZD, convert to HKD at spot rate and then sell HKD forward. After 6 months, you may have more NZD than you owe.  

Strategy i)

a) Borrow 5.01 HKD at 1.5% continuously compounded (cc) rate for 6 months. Amount payable after 6 months

=5.01*exp(0.015*6/12) = HKD 5.0477

b) Convert the amount to NZD 1 at spot rate plus transaction charges

c) Lend the amount at 3% cc rate in NZD . Maturity amount = 1*exp(0.03*6/12) = NZD 1.0151

d) Sell NZD 1.0151 forward after 6 months

e) After 6 months, get NZD 1.0151 and sell it using forward to get 1.0151*4.97 = HKD 5.0451

which is lesser than the amount owed. Hence Arbitrage is not possible this way

Strategy ii)

a) Borrow 1 NZD at 3.5% continuously compounded (cc) rate for 6 months. Amount payable after 6 months

=1*exp(0.035*6/12) = NZD 1.017654

b) Convert the amount to HKD to get HKD 4.99

c) lend the amount at 1% cc rate in HKD . Maturity amount = 4.99*exp(0.01*6/12) = HKD 5.015

d) Sell HKD 5.015 forward after 6 months

e) After 6 months, get HKD 5.015 and sell it using forward to get 5.015/5.03 = HKD 0.9970

which is lesser than the amount owed. Hence Arbitrage is not possible this way also

Thus, There is no arbitrage