CFA Level I: Foreign Exchange Topics: Real Rates, Cross-Rates and Arbitrage

Currency Exchange Calculations

The Foreign Exchange market is one of the largest in the world. Foreign currencies are bought and sold in long/short pairs. There are both nominal exchange rates and real exchange rates. Nominal rates are the familiar rates that are quoted on exchanges, indicating the amount of the numerator or price currency that can be bought per denominator or base currency.

The real exchange rate uses an idea called purchasing power parity in order to establish an exchange rate that takes into account the price levels differences of goods between two countries. It is a rate that attempts to equate the exchange in terms purchasing power by multiplying the nominal rate by a ratio of consumer price indices.

  • Real exchange ratesd/f : Spot exchange rate­(d/f) x (CPIf/CPId)

We can use a variation of this equation to determine the real change value between two currencies, given the change in spot rates and changes in CPIs.

  • Percent change in real rate: [(1 + change in S (d/f)) x (1+ change in CPIf/1 + change in CPId)] – 1
  • Approx Percent change in real rate: Change in S (d/f) + change in CPIf – change in CPId

Note that we can use any measure of price levels in place of CPIs, such as inflation or baskets of currency. We also derived an approximate function for the calculation, but this equation works best when changes are all appreciations.

Finally, note that when calculation nominal appreciation between a currency pair, the amount of appreciation is not equal to the amount of depreciation.

  • Change in EUR/CAD ≠ Change in CAD/EUR

Cross Rate Exchange Calculations

Given three exchange rates, it is possible to calculate a non-quoted exchange rate using cross multiplication. If necessary, some spot rates may need to be inverted for the cross rate calculation.


Foreign Exchange Derivatives

There are a variety of tools used in foreign markets which create flexibility in hedging and speculation options. The most basic derivatives contracts used are FX forwards and futures, which guarantee an exchange rate at a future date. FX call and put options are also traded, allowing the choice to buy or sell at a specified exchange rate at the contract maturity date. A more complicated FX derivative transaction called the FX swap involves a simultaneous spot and forward transaction, to close out a previously existing forward contract. In this manner, the FX swap will roll forward the existing position to a future date and creates a cash flow on the settlement date, depending on the spot rate used to close out the forward.

  • Plan to sell EUR by acquiring a USA/EUR forward contract at forward rate. Close out sale at maturity by purchasing EUR at spot USA/EUR rate. This creates a cash flow unless the forward and spot rates have become equivalent. Initiate new USA/EUR forward contract and position is rolled forward.

Using these contracts, we can create an arbitrage equation involving the risk free rate.

  • Covered interest rate arbitrage: (1 + i­d) = Sf/d(1 + if)(1/Ff/d)

This equation states that the domestic risk free rate should be equal to the foreign risk free rate, when the currency is converted today at the spot rate, and reconverted at maturity using a predetermined forward rate. In other words, given a USD, the risk-free rates of investing in the US should be the same as converting to a GBP, then investing at the risk-free rate in the UK, as long as we secure the reconversions with a forward contract. If this is not the case, then we have a riskless arbitrage opportunity.

This equation enables us to solve for the forward rate.

  • Ff/d = Sf/d [(1 + if)/(1 + i­d)]

To take advantage of the arbitrage opportunity, you would borrow in the currency with the lower real return, invest in the higher return, and hedge the reconversion with a forward contract.

Countries with the higher interest rates tend to create a forward premium in the pair.

  • Ff/d/ Sf/d = [(1 + if)/(1 + i­d)]

It is tempting to read the forward rate as predictors of the futures rate, but the CFA prefers that the forward rate be understood solely as a derived from this equation. Thus Ft+n ≠ St+n, but merely Ff/d + n = Sf/d [((1 + if)(n/360))/(1 + i­d(n/360))]. In other words, the forward rate does not equate to the future Spot rates after n days. That value is simply a fraction of the interest rates multiplied by the fraction of the investment horizon.


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1 thought on “CFA Level I: Foreign Exchange Topics: Real Rates, Cross-Rates and Arbitrage”

  1. the percent change in real rate formula is wronggggg!!!! Please correct it. The original formula is,
    1+%change in nominal spot rate = (1+%change in real rate) x (1+CPIforeign)/(1+CPIdomestic). I see that you have rearranged this equation. However, when rearranging it looks like you have forgotten to switch the (1+CPI foreign)/(1+CPIdomestic) part. Please recheck your rearranging of the formula. Cause when you bring it to the other side of the equation CPIdomestic should be the numerator and CPIforeign should be denominator. Does that make sense?


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