Question

Suppose the spot $/Yen exchange rate is 0.008, the 1-year continuously compounded dollardenominated rate is 5% and the 1-year continuously compounded yen-denominated rate is 1%. Suppose the 1-year forward exchange rate is 0.0084. Explain precisely the transactions you could use (being careful about currency of denomination) to make money with zero initial investment and no risk. What is such a strategy being referred to in the markets?

Answer 1

Given: Spot $/Yen=0.008 i.e. 1 Yen= 0.008 $.

$ interest rate=5%. Yen interest rate=1% Forward Rate=0.0084 i.e. after 1 year, 1 Yen= 0.0084$.

To make money with zero initial investment and risk is called arbitrage. For arbitrage, you borrow at one currency, buy a spot, invest in the other currency, and also sell a forward. After the term (1 year in this case), we get our principal plus interest., converted to the borrowing currency using the forward, and use the proceeds to repay the loan. The amount we are left with is riskless profit with no investment.

Lets us assume we are borrowing in $. We borrow 1 $ (at 5 % continuously compounding rate). We buy equivalent Yen, given by 1/0.008=125 Yen.

Now we invest 125 Yen in Yen denominated rate (1%). After 1 year we will get 125*(e^^RT) Yen. (This is the formulae for continous compounding, where e=2.718).

R=1%=0.01 (We use the decimal rate for this formula)

T=1 year (Given)

Therefore Yen maturity amount= 125*e^^0.01 = 126.2563

We will use this amount to buy back dollars after 1 year using the forward rate. (the transaction will be booked right now, though). $ equivalent of 126.2563 Yen @0.0084 = 125.2563*0.0084 =1.060053

Dollar amount to be repaid= 1* (e^^RT) , where R =5% or 0.05 ($ rate) and T=1 year.

Amount =1* e ^^0.05*1 = 1.051271

Amount left with us after repaying, = 1.060053-1.051271=0.009282.

This $0.009282 is our profit.

The order of transactions are:1) Borrow in $, 2) Buy Yen spot, 3) Lend/Invest in Yen, 4)Buy $ in Forward,

Notes:1) If we did the reverse by 1) borrowing in Yen 2) Buy $ Spot 3) Invest in $ 4) Buy Yen Spot, we would be losing money in a sure shot way. Any arbitrage strategy will have an opposite arm which gives sure shot loss.

2) The inherent assumptions in this question is that you can borrow and lend at the same rate (This normally does not happen).