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Futures and Forwards on Currencies

  • Interest rate Parity

 

  • Formula to remember:
  • If Spot rate is given in USD/INR terms then take American Risk-free rate as the first rate
  • In other words, individual who is interested in USD/INR rates would be an American (Indian will always think in Rupees not dollars!!!!!), which implies foreign currency (rf) in his case would be rINR
 

Example

The forward rate of a 3-month EUR/USD foreign exchange contract is 1.1565 USD per EUR. USD LIBOR is 4% and EUR LIBOR is 2%. The spot USD per EUR exchange rate is?
 

Solution

F0 = S0 e(r-rf)t
1.1565 e-(.04- .02).25  = 1.1507
 


 

Example

Assume that the current 1-year forward exchange rate is 1.200 USD per EUR. An American bank pays 2.4% annual interest rate on a 1-year deposit and a 4.0% annual interest rate on a 3-year USD deposit. A European bank pays a 1.5% annual interest rate for a 1-year deposit and a 2.0% annual interest rate for a 3-year EUR deposit. The forward exchange rate in USD per EUR for exchange three years from today is closest to:
 

Solution

The 2 year forward rate in US = [(1.04)3 / 1.024] – 1 = 4.81%
The 2 year forward rate in Europe = [(1.02)3 / 1.015] – 1 = 2.25%
The forward exchange rate in USD per EUR for exchange three years: 1.2 *(1.04812) / (1.02252) = 1.261
 


 

Example

The two-year risk-free rate in the United Kingdom is 8% per annum, continuously compounded . The two-year risk-free rate in France is 5% per annum, continuously compounded. The current French Franc to the GBP currency exchange rate is 1GBP = 0.75 French Franc.
What is the two-year forward price of one unit of the GBP in terms of the French Franc so that no arbitrage opportunity exists?

  1. 0.578
  2. 0.706
  3. 0.796
  4. 0.973
     
Solution

Ans = 0.75*e(0.05-0.08)*2 = 0.706


 

Example

A bank has a USD50,000,000 portfolio available for investing. The cost of funds for the USD50,000,000 is 4.5%. The bank lends 50% of the assets to domestic customers at an average loan rate of 6.25%. The rest of the portfolio is lent to UK clients at 7%. The current exchange rate is USD1.642/GBP. At the same time, the bank sells a forward contract equal to the expected receipts one year from now. The forward rate is USD1.58/GBP. The weighted average return to the bank on its investments is closest to:
 

Solution

The return from UK customers, $25,000,000/1.642 = GBP 15,225,335* 1.07 = GBP 16,291,108
The bank sells a forward contract: GBP 16,291,108*1.58 = USD 25,739,951
Earnings (USD 25,739,951 – 25,000,000) / 25,000,000 = 2.96%
Weighted average return = 6.25%*.5 + 2.96%*0.5 = 4.61%
 


 

Example

Given the following:
Current spot CHF/USD rate: 1.3680 (CHF1.3680 = USD1)
3-month USD interest rates: 1.05% ; 3-month Swiss interest rates: 0.35%
A currency trader notices that the 3-month forward price is USD / CHF 0.7350. In order to arbitrage, the trader should
 

Solution
  • The spot is quoted in terms of Swiss Francs per USD. To convert this into USD per Swiss Franc, we get: 1/1.3680 = 0.7310. The theoretical futures price = 0.7310 * exp((0.0105 – 0.0035) * 0.25) = 0.7323. Therefore, the quoted futures price is too high. Thus, one should sell the overvalued CHF futures contract.
  • In order to arbitrage, one would do the following:
    • Borrow USD 0.7310 * exp((-0.0035)*0.25) = USD 0.7304 for 3 months
    • Buy spot exp((-0.0035)*0.25) = CHF0.9991, invest at 0.35% for 3 months
    • Short a futures contract on CHF1
  • At maturity,
    • Pay back 0.7304 * exp((0.0105) * 0.25) = USD 0.7323
    • Receive 0.9991 * exp((0.0035) * 0.25) = CHF 1
    • Delivers CHF 1 on the futures contract, receives USD 0.7350
    • An arbitrage profit of USD0.7350 – USD0.7323 = USD 0.0027 would be realized in 3 months’ time

 





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