3.1 Commodity arbitrage

In the case of commodity arbitrage, arbitragers buy a commodity in the market where it is cheap and sell it where it is more expensive, making profit as the difference between the selling price and the buying price. This activity leads to a rise in the price of the commodity in the market where it is cheap and a decline in price in the market where it is expensive until profit is eliminated and the equilibrium condition is restored.

Let P^j be the price of a commodity, j, in terms of x, Pyj the price of the same commodity in terms of y and S is the exchange rate measured as S(x/y). The no- arbitrage condition is

P^j= SPyj (3.1)

which says that the price of the commodity measured in the same currency must be equal. Notice that P_x^j is the price of the commodity in terms of currency x, whereas SPyj is the price in terms of y converted into x at the current spot rate. Equation (3.1) is also know as the law of one price (LOP), which generally says that the price of a commodity should be equal across countries when measured in the same currency.

Figure 3.1 shows how the process works, starting from a disequilibrium position described by the inequality P^ > SPyj . Given this situation, arbitragers buy the commodity where it is cheaper (in the country whose currency is y), leading to an increase in demand and a shift in the demand curve. They will also sell the commodity where it is more expensive (in the country whose currency is x), leading to an increase in supply. Thus, SP_y^j rises, while P_x^j falls, until they are equal. At this point, arbitrage profit disappears and the no-arbi­trage condition is restored.

Pj, P

Quantity (x)

Quantity (y)

FIGURE 3.1 The effect of commodity arbitrage.

The LOP can be generalised by assuming that the latter holds for each and every commodity. Assuming that there are n commodities, the LOP can be generalised to obtain

Pj= s£

j _p]

y ^Ly

£ ^w

j=i

(3.2)

w

j=1

where w{ and w^]y are the weights of commodity j in the general price level of the two countries, with the weights . presumably reflecting consumption patterns. This relationship is valid if wX = Wy for all j. Thus

(3.3)

^Px ^SPy

which is the same as equation (3.1) except that it is written in terms of the general price levels, P_x and Py, and not the prices of individual commodities. This relationship is purchasing power parity (PPP), which can be represented diagrammatically by a straight line passing through the origin in the S-P_x space, as shown in Figure 3.2. Changes in Py are represented by the movement of the PPP line downwards (when P_y falls) and upwards (when P_y rises).

Commodity arbitrage may be intertemporal in the sense that a time factor is involved. In this case the arbitrager does not buy and sell at the same time, but rather after some time. Suppose that the arbitrager buys a bundle of commodi­ties in country x at time t and sells it in the same country at time t + 1. The rate of return on this transaction will be the inflation rate between t and t + 1, which we shall denote P_x . If, on the other hand, the arbitrager chooses to sell the bundle in country y, then the rate of return (in terms of currency x) will consist of two components: the inflation rate in country y, Py and the percentage change in the spot exchange rate, S, which is the rate of appreciation or depreci­ation of y against x. Hence the no-arbitrage condition is

Px

Px= Py + S (3.4)

If, for example, P_x < P y + S, it will be profitable to buy commodities in country x and sell them in country y after some time. The no-arbitrage condition (3.4) can be written as

S = Px - Py (3.5)

which is PPP written in first differences (invariably, but perhaps inappropri­ately, known as relative PPP). If the expected values of the variables are used then equation (3.5) would represent the so-called ex ante PPP.

It is possible to show that (3.5) can be derived by assuming that (3.3) holds at the points in time t and t + 1. If this is so, then

Px,t = StPyt (3.6)

^Px,t + 1 = ^St + 1^Py,t + 1 ⁽³.7)

By dividing equation (3.7) by equation (3.6), we obtain

^Px,t+ 1 ^St+ 1 ^Py ,t+ 1

(3.8)

^Px,t ^St ^Py,t

which can be rewritten as

(1 + P _x ) = (1 + S )(1 + Py ) (3.9)

which, after simplification, reduces to (3.5). Moreover, equation (3.8) can be written in the following form that makes it something like an exchange rate determination model

P

^St+1 = ^St

(3.10)

x ,t+ 1^/Px ,t

^Py ,t+ 1^/Py ,t

If we allow for a base period, 0, to which prices are measured then this means that the exchange rate at any point in time t can be obtained by adjusting the exchange rate at 0 for differences in inflation rates. Hence we could define the exchange rate consistent with PPP, S, as

^Px ,t ^/Px ,0

1 + P.

x ,t

(3.11)

^St ^{- S}0

- S_r

1 + P.

y^,t

^Py ,t ^/Py ,0

Equation (3.11) is used to measure the extent of misalignment (overvaluation or undervaluation) of currencies, such that currency y would be deemed over­valued if S_t > S_t, and vice versa. Figure 3.3 shows the actual and PPP rates (S_t and S_t respectively) of the pound and the Canadian dollar against the US dollar. It indicates that the pound has been predominantly overvalued, whereas the Canadian dollar has been predominantly undervalued against the US dollar. Exchange rate misalignment would normally indicate the possi­bility of commodity arbitrage.