Exchange Rate Arithmetic: Cross Rates & Triangular Arbitrage

16.1: Cross Rates:

Cross rates for spot quotations have been discussed in the Session 13.2. However, at the cost of repetition, the cross rate calculation is given here, as it forms basis for calculation of forward cross rates.

A spot cross rate is a rate which can be calculated from two other spot rates. Suppose USDCAD (US dollar and Canadian Dollar) rate is given along with and an USDAUD (US Dollar and Aussie Dollar) quotation.

USDCAD = 1.1546 (CAD 1.1546 is equal to 1 USD)

CADAUD= 1.0421 (AUD of 1.3421 is equal to 1 CAD)

The USDAUD rate = (CAD1.1546/ USD) * (1.0421AUD/1 CAD) = AUD1.2032/USD However, in real life, the dealers give bid-ask spread for currency pair. With bid-ask spread, the cross calculation becomes little complex.

16.1:2: Cross Rate Calculation with bid-ask spared:

USDCAD bid-ask rate is given along with and USDAUD bid-rate is given in in Table 16.1. The rates for CADAUD (or AUDCAD) can be calculated from the above two quotes.

	Table 16.1: Spot r Bid-Ask Rates
USDCAD		USDAUD
Bid	Ask	Bid	Ask
1.1641	1.1646	1.2948	1.2956

From the above quotations, cross rate between AUD/CAD can be calculated as follows:

•      Bank buys 1USD and pays (sells) 1.1641 CAD

•      Bank sells 1 USD and receives (buys)1.1646 CAD

•      Bank buys 1 USD and pays (sells) 1.2948 AUD

•      Bank sells 1 USD and receives(buys) 1.2956 AUD

To get the bid rate for CADAUD (CAD as base currency and AUD as quote currency), the bank must sell AUD and buy CAD. This is achieved in two steps. That is

•      The bank must sell AUD and buy USD

•      Simultaneously sell USD and buy CAD.

This indicates that 1.1646 CAD = 1.2948 AUD. In other words, 1 CAD = 1.1118 AUD.

To get the ask rate for CADAUD, the bank must sell CAD and buy AUD. This is achieved in two steps ie. the bank must sell CAD buy USD and simultaneously sell USD and buy AUD.

This means that 1.1641 CAD = 1.2956 AUD. In other words, 1 CAD = 1.1129 AUD. Hence the cross rate, given in Table 16.2 is

Table 16.2:CADAUD cross rates

Bid                                        Ask

1.1118                              1.1129

16.2: Forward Cross Rates:

Cross rates for different maturities can be found out in the similar manner like the spot cross rates. The following table, Table 16.3 lists the actual rates(forward contracts rates for) different maturities. From these rates, the cross rates have been calculated and listed

in Table 16.4

Table 16.3 Outright Quotations for USD/INR and USD/ZAR(*)

	USDINR		USDRAND
	Bid Rate	Ask Rate	Bid Rate	Ask Rate
Spot	47.0725	47.0745	7.5937	7.5950
1 week	47.0750	47.0775	7.5918	7.5938
2 weeks	47.0795	47.0835	7.5887	7.5911
1 month	47.0840	47.0890	7.5812	7.5860
2 months	47.0900	47.0965	7.5775	7.5838

Table 16.4 INR/ZAR(*) Cross Rates for different forward periods

INRRAND	Bid	Ask
spot	0.16131	0.16135
1 week	0.16126	0.16131
2 weeks	0.16118	0.16124
1 month	0.16100	0.16112
2 months	0.16089	0.16105

(*): ZAR is the currency of South Africa. It is known as South African Rand but has a ISO code of ZAR.

Cross rate calculation becomes necessity when two currency pair exchange rate is not quoted by a dealer or bank. For example, an Indian company imports textile yarns from South Africa for which the payment has to be made in ZAR. As none of the Indian banks directly offer, INRZAR quotations, the exporter has to sell INR and buy USD and then sell USD and buy ZAR to make the payment. As most currencies of the world are quoted with either Euro/USD/Pound sterling, these currencies form one leg of trade in any cross currency transaction.

To summarize, for many infrequently traded currencies pairs, cross rates are the only sources of rate quotations as no dealer/bank directly offers quote.

16.3: Cross Rates and Triangular Arbitrage:

Cross rates are the exchange rates of 1 currency with other currencies, and those currencies with each other. Cross rates are equalized among all currencies through a process called triangular arbitrage. As different forex dealers quote different rates for a given currency pair at a given point of time, it provides forex traders with arbitrage opportunity. Hence this is known as “intermarket arbitrage”.

Cross rate calculations helps in identifying the intermarket arbitrage opportunity. The following table, Table 16.5, indicates the list of exchange rates for currency pair. From these pairs, cross rates can be calculated.

TABLE 16.5: Currency Rates

http://www.reuters.com/finance/currencies

		US $	¥en	Euro	Can $	UK £	Aust $	SFranc
1	US $	1.0	92.1700	0.7161	1.1633	0.6218	1.2924	1.0843
1	¥en	0.010843	1.0	0.007768	0.012614	0.006741	0.014014	0.011758
1	Euro	1.3958	128.6900	1.0	1.6244	0.8682	1.8052	1.5138
1	Can $	0.8592	79.2100	0.6153	1.0	0.5344	1.1106	0.9318
1	UK £	1.6072	148.1600	1.1512	1.8699	1.0	2.0779	1.7429
1	Aust $	0.7732	71.2900	0.5538	0.8997	0.4809	1.0	0.8383
1	SFranc	0.9218	85.0000	0.6602	1.0726	0.5735	1.1919	1.0

Now let us calculate the cross rate for Can$ and US$ by comparing Can$-Euro and Euro-US$ rates.

•      1 Can $ = 0.8592 US $.

•      1 Euro = 1.3958 US$.

•        From these two rates, the cross rate for Can $/Euro can be calculated.

Box 16.1: Implied Cross Rates

1 Euro = 1.3958 US$
1US$ =	1	Can$	= 1.16387Can$
	0.8592

1 Euro = 1.3958 ∗ 1.16387 Can$ = 1.6245 Can $ Can$1.6245/Euro is known as the implied cross rate.

The implied cross rate calculated in this manner is compared with the actual rate quoted by another dealer or bank. If these two are rates are significantly different, then arbitrage opportunity arises.

The actual Euro/Can$ rate given in Table 16.5 is Can$ 1.6244/Euro. The implied cross rate is very close to the actual rate. The minor difference arises as the transaction cost and bid-ask spread have not been factored into the calculation of implied cross rates,

Suppose there is a significant difference between the implied cross rate and the actual rate. This will lead to arbitrage profit.

Let us take a numerical example, to understand how triangular arbitrage happens. Three different banks are quoting spot rates for three currency pairs given below.

Bank of Japan  quotes : St = 100 JPY/USD

Bank of America quotes: St = 1.60 USD/GBP

Bank of England quotes St = 140 JPY/GBP

If we consider the first two quotes, then JPY/GBP quote should be at 160 JPY/GBP. However bank of England quotes 140 JPY/GBP. It indicates that Bank of England is undervaluing GBP.

The triangular arbitrage happens:

•      Borrow 100 USD

•      Convert it to 10000 JPY at Bank of Japan.

•      Convert 10,000 JPY to GBP at Bank of England. Receive 71.428 GBP.

•      Convert 71.428 GBP to USD at Bank of America. Receive 114.285 USD.

Profit of 14.285 USD before adjusting for USD borrowing cost

In real life, the rates are quoted with bid-ask spread. The triangular arbitrage with bid and ask spread is given in Table 16.6

Table 16.6: Bid-ask rates offered by three banks

	Bid	Ask
Bank A (GBP/USD)	1.60	1.61
Bank B (MYR/USD)	0.2	0.202
Bank C (GBP/MYR)	8.10	8.20

MYR : Malaysian Ringgit

With the rate given in Table 16.6, the triangular arbitrage happens in three steps:

•      Sell USD 1.61 and Receive 1 GBP at Bank A ask rate

•      Sell 1 GBP and Receive 8.10 MYR at Bank C bid rate

•      Sell 8.10 MYR and receive $ 1.62 at Bank B bid rate.

Arbitrage profit is $0.01 for every 1 USD investments. The investor takes full circle – sell USD to receive USD to get the benefit of triangular arbitrage.

The triangular arbitrage can also happen

•      Sell 1 MYR, receive 0.2 USD

•      Sell 0.2 USD and buy 0.124224 GBP (Sell 1.61USD and receive 1 GBP)

•      Sell 0.124224 GBP and receive 1.0062 MYR.

Hence for every 1 MYR investment, the trader would be bale to make a profit of MYR 0.006.

Can arbitrage happen, if the investors starts from GBP? This needs to be found out!

Exploiting Triangular arbitrage opportunity is a favourite hunting ground for forex traders. In fact, forex traders buy software packages which analyses real time bid-ask quotations offered by forex dealers, identify arbitrage opportunity and place order to benefit from the opportunity without any human intervention.

16.4: Interest rate arbitrage.

Arbitrage opportunities available to forex traders as discussed in Section16.3 are known as the inte rmarket arbitrage. Forex traders regular make arbitrage profit though interest rate differential in two countries. This is known as “interest rate arbitrage”.

Interest rate arbitrage works like this:

Spot rate £1 = €1.6140. Interest rate for coming 12 months is 5.5% for Pound Sterling and 3.75% for Euro. Suppose a bank quotes a 3 month forward rate as £1 = €1.5970. Now let us see whether there exist an arbitrage opportunity or not.

For example, a trader borrows £100,000 for 3 months. He has to pay £101,375 after 3-months. He converts £100,000 to € at the spot rate. He receives €161400. Invests €161400 at 3.75% interest rate for 3 months. He earns € 162913. He converts euro proceeding to Pound sterling at the 3 month forward rate of £1 = €1.5970. He earns £102,012. He returns £101,375and makes a arbitrage profit of £636.

This profit opportunity will entice many traders to borrow Pound Sterling, sell Pound sterling to buy Euro, invest in Euro and sell Euro forward to buy Pound Sterling. This will ensure that the arbitrage opportunity vanishes quickly. In fact, with spot bid £1 = €1.6140, interest rate for coming 12 months is 5.5% for Pound Sterling and 3.75% for Euro, the 3month forward rate should have been £1 = €1.6070 and not £1 = €1.5970.

Interest rate arbitrage opportunity forms the core of interest rate parity and covered interest rate arbitrage. This aspect will be discussed in greater detail in later sessions.