FORWARD FOREIGN EXCHANGE RATES AND INTEREST RATE ARBITRAGE About a third of the foreign exchange market is what is called the Forward Market That is, the market for foreign exchange for delivery at some agreed future time. Like spot, this is an inter-bank 24 hour market. While the main players in the spot market are international short-term money managers, constantly moving money around the world (often for only a few hours in any one country) seeking higher returns on paper assets. End users of Forwards tend to be corporations planning on fixed long-term foreign investments (i.e., bricks and motar). The forward foreign exchange rate depends on the interest rate differential between two countries and today’s spot rate. Notation: Ft be the Forward foreign exchange rate for some time “t” in the future S0 is the spot rate (i.e., the exchange rate now) “t” measures time as a percentage of one year. For example: F1 F0.25 F1.5 = = = The one year forward rate The three month forward rate The eighteen month forward
1 if Ft S 0 1 id
Where And
t
if = the foreign interest rate id = the domestic interest rate
The domestic currency is always the one quoted first in the notation
For example:
$/¥ = ¥115.25 ₤/$ = 2.0850
Domestic = $ Domestic = ₤
Foreign = ¥ Foreign = $
€/$ = 1.4835 €/¥ = 170.97
Domestic = € Domestic = €
Foreign = $ Foreign = ¥
Thus, what is the nine-month $/¥ forward rate if spot $/¥ = ¥115.25 and interest rates are 4% in the US and 2% in Japan? 115.25(1.02/1.04)0.75 = ¥113.58 What is the six-month ₤/$ forward rate if spot ₤/$ = $2.0850 and interest rates are 4% in the US and 5% in the UK 2.0850(1.04/1.05)1/2 = $2.0750 What is the one and a half year €/¥ forward rate if spot €/¥ = ¥170.97 and interest rates are 6% in the Euro-zone and 2% in the Japan 170.97(1.02/1.06)1.5 = ¥161.38 What is the half year $/€ forward rate if spot $/€ = €0.6741 and interest rates are 6% in the Euro-zone and 4% in the US 0.6741(1.06/1.04)1/2 = €0.6806
Notice that in all the above examples, we are given four pieces of information: Spot rate (S0 ) Maturity (t), Foreign interest rate (iF) Domestic rate (iD); Which we use to fid the unknown Forward foreign exchange rate, F t What if you were given the Forward rate and three of the other four other pieces of information, and told to find the fourth? For example, You are given the following information. S0 = ₤/$ = $2.0660 F0.25 = $2.0562 (t =0.25) iUS = 4% What is i UK?
Solution
1.04 2.0562 2.0660. UK 1 i
0.25
1.04 0.9953 UK 1 i
0.25
1.04 Ln(0.9953) 0.25.Ln UK 1 i 0.0190 Ln(1.04) Ln(1 i UK ) A Because Ln Ln( A ) Ln(B ) B
0.0582 Ln 1 i UK Ln 1 i UK 0.0582
UK e Ln 1 i e 0.0582
1 i UK 1.06 i UK 0.06 or 6%
Suppose instead that in the previous example you had been given i been asked to find “t” (which we know to be 0.25)
UK
= 6%, but you had
1.04 2.0562 2.0660. 1.06 1.04 0.9953 1.06
t
t
1.04 Ln(0.9953) t .Ln 1.06 0.0048 0.0190.t t 0.25
Interest Rate Arbitrage. How do we know that the formula for forward rates is correct? The easiest way is to show that if it was incorrect, we would have the opportunity for a risk-less arbitrage (free money). Suppose that spot €/$ = $1.4765 ($/€ = €0.6773) and that interest rates are 4% in the US and 6% in the Euro-zone. The correct one-year forward rate would be
1.04 Ft 1.4765. $1.4486 , or $/€ = €0.6903 1.06
However, let us suppose, arguendo, that the quoted forward rate was actually: €/$ = $1.4900 , or $/€ = €0.6711 How can we profit from this miss-pricing?
1
First we notice that the Forward Euro is artificially high, €/$ = $1.49 > €/$ = $1.4486. Thus, on the forward leg of the arbitrage we want to be buying Dollars with Euros. Therefore, if we are going to end up with dollars, we must start with dollars. 1) 2) 3) Borrow $1,000,000.00 Buy Spot Euros @ $/€ = 0.6773 Invest the Euros in the Euro-zone For one year @ 6%. Buy Forward dollar @ €/$ = $1.4900 In a year you will owe €677,277.35 €717,913.99 $1,040,000.00
4)
$1,069,691.84
5)
Repay the loan and interest with the proceeds, and trouser the risk free profit of $29,691.84
Had we used the correct forward rate of €/$ = $ 1.4486 ($/€ = €0.6903) , there would have been no arbitrage profit. 1) Borrow $1,000,000.00 In a year you will owe $1,040,000.00
2) 3)
Buy Spot Euros @ $/€ = 0.6773 Invest the Euros in the Euro-zone For one year @ 6%. Buy Forward dollar @ €/$ = 1.4486 Hence, no arbitrage profit
€677,277.35 €717,913.99 $1,0399,70.20 ≈ $1,040,000 with rounding error.
4)
5)
In the above example, the forward interest rate was incorrect, but risk-free arbitrages can occur if the interest rate differential is incorrect relative to correct spot and forward exchange rates. Suppose €/$ = $1.4765 ($/€ = €0.6773) and, the one-year forward rate is €/$ = $1.4486. The above is correct if interest rates are 4% in the US and 6% in the Euro-zone. However, suppose that, for whatever reason, interest rates are still 4% in the US, but 7% in the Euro-zone. A risk-free arbitrage now exists whereby we can profit from the artificially high Euro-zone interest rate.
1) 2) 3)
Borrow $1,000,000.00 Buy Spot Euros @ $/€ = 0.6773 Invest the Euros in the Euro-zone For one year @ 7%. Buy Forward dollar @ €/$ = 1.4486
In a year you will owe €677,277.35.00 €724,686.76
$1,040,000.00
4)
$1,049,781.24
5)
Repay loan + interest with the proceeds, and trouser the risk free profit of $9,781.24