SUGGESTED READINGS
4.1 THE LAW OF ONE PRICE
4
The International Parity Conditions and Their Consequences
Though this be madness, yet there is method in it.
— William Shakespeare
T
his chapter describes how prices in the currency and Eurocurrency markets are linked through a set of international parity conditions that relate forward premiums and expected spot exchange rate changes to cross-currency differentials in nominal interest rates and inflation. These parity relations are then used to develop a measure of a currency’s purchasing power relative to other currencies, called thereal exchange rate.The chapter concludes with a discussion of exchange rate forecasting from the international parity conditions and other predictors.Arbitrage Profit
Arbitrage profit has no net investment or risk.
Although the popular press often uses the term ‘‘arbitrage’’ or ‘‘risk arbitrage’’
to refer to speculative positions, arbitrage is more strictly defined as a profitable position obtained with
■ No net investment
■ No risk
This ‘‘no money down and no risk’’ opportunity sounds too good to be true. In the high-stakes interbank currency and Eurocurrency markets, it usually is too good to be true once transaction costs are included. Arbitrage opportunities are quickly exploited, and just as quickly disappear as arbitrageurs drive prices back toward equilibrium.
Let Pddenote the domestic currency price of an asset and Pf denote the foreign currency price of the same asset. The law of one price requires that the value of an asset be the same whether value is measured in the foreign or in the domestic currency. This means that the spot rate of exchange must equate the value in the foreign currency to the value in the domestic currency.
Pd
Pf =Sd/f⇔Pd=PfSd/f (4.1) If this equality does not hold within the bounds of transaction costs, then there may be an opportunity for an arbitrage profit.1
As an example, suppose gold sells for P$ = $1508.00/oz in New York and P£ = £942.50/oz in London. The no-arbitrage condition requires that the value of gold in dollars must equal the value of gold in pounds, so S$/£ = P$/P£ = ($1508.00/oz)/(£942.50/oz) = $1.6000/£, or S£/$ = 1/S$/£ = £0.6250/$. If this relation does not hold within the bounds of transaction costs, then there is an opportunity to lock in a riskless arbitrage profit in cross-currency gold transactions.
Transaction costs are relatively small for actively traded financial assets, such as currencies in the interbank market. PPP nearly always holds in these markets, because the potential for arbitrage ensures that prices are in equilibrium. PPP is less likely to hold in illiquid markets, or in markets where high transaction costs or financial market controls prevent arbitrage from enforcing the law of one price.
Transaction Costs and the No-Arbitrage Condition
For there to be no arbitrage opportunities, PPP must hold within the bounds of transaction costs for identical assets bought or sold simultaneously in two or more locations. Thisno-arbitrage conditionis the foundation upon which the law of one
price is built. Whether PPP holds depends on the extent to which market frictions restrain arbitrage from working its magic. Some barriers to the cross-border flow of capital are generated in the normal course of business, as fees are charged for making a market, providing information, or transporting and delivering an asset.
Other barriers are imposed by governmental authorities, including trade barriers, taxes, and financial market controls.
The no-arbitrage condition ensures PPP holds within the bounds of transac- tion costs.
Buying or selling real assets usually entails higher costs than trading a financial claim on the real asset. As an example, gold is costly to transport because of its weight, but a financial asset representing ownership of gold is easily transferred from one party to another and can be as simple as a piece of paper or a credit in an account. Although large amounts of gold are a nuisance to store, currency can be stored conveniently in the Eurocurrency market at a competitive interest rate.
Because of this difference between financial and real assets, actively traded financial assets are more likely than similar real assets to conform to the law of one price.
Figure 4.1 illustrates how transaction costs influence the analysis. Suppose gold is quoted at ‘‘£930/oz bid and £940/oz ask’’ in London and ‘‘$1,500/oz bid and
$1,516/oz ask’’ in New York. A forex (FX) dealer quotes pounds in the spot market as ‘‘$1.599/£ bid and $1.601/£ ask.’’ Translated into pounds at the $1.600/£ mid- rate, the New York dealer’s mid-price is ($1,508/oz)/($1.600/£)=£942.50/oz. This is slightly higher than the London dealer’s mid-price of £935/oz, so if there is an arbitrage opportunity it would likely be to buy gold from the London dealer and sell gold to the New York dealer. Suppose you buy 1,000 ounces of gold for £940,000 at the London dealer’s £940/oz ask price for gold. The FX dealer will sell £940,000 to you at the $1.601/£ ask price for pounds for a payment of (£940,000)($1.601/£)
= $1,504,940. Selling the gold in New York yields only $1,500,000 at the New Yorker dealer’s bid price for gold. This leaves you with a net loss of $4,940 (i.e., a cash inflow of $1,500,000 and an outflow of $1,504,940). Even though PPP does not hold exactly, it does hold within the bounds of transaction costs in this example.
Unfortunately for your dreams of wealth, the dealers’ bid-ask prices overlap each other and an arbitrage profit is not possible.
Buy gold in London with £s
“£930/oz bid & £940/oz ask”
Buy £s & sell $s
“$1.599/£ bid & $1.601/£ ask”
Sell gold in New York for $s
“$1,500/oz bid & $1,516/oz ask”
+(1,000 oz gold) +£940,000 +$1,500,000
–£940,000 –$1,504,940 –(1,000 oz gold)
Net result is a loss of $4,940
FIGURE 4.1 The No-Arbitrage Condition in the Gold Market with Transaction Costs.