Topic 3: The Relationship Between Forward and Spot Exchange Rates


To this point we have analyzed the reasons why spot and forward foreign exchange markets exist and explored some details of how those markets function. The question now is: What is the relationship between the forward and spot exchange rates? In general, of course, the forward exchange rates on contracts of different lengths to maturity will differ from each other and from the spot rate at the time those forward contracts are entered into.

The forward discount on domestic currency is defined as the excess of the forward price of the foreign currency in terms of the domestic currency over its spot price, taken as a fraction or percentage of the spot price. That is,

Ψ = (Π′ - Π) ⁄ Π

where Ψ is the forward discount, Π′ is the forward price of foreign currency in terms of domestic currency (for contracts of a particular length to maturity) and Π is the spot price of foreign currency in terms of domestic currency. A positive forward discount means that the purchaser of the foreign currency forward has to pay more domestic currency than he would have to pay to buy a unit of that foreign currency spot.

Let's take an example. Suppose that the forward rate is 360 yen per dollar and the spot rate is 350 yen per dollar. The forward discount on the yen will then be (360 - 350)/350 = .028, or 2.8 percent. Treating the dollar as the domestic currency, Π′ is .0027777 and Π is .0028571. The forward discount on the dollar is thus

Ψ = (.0027777 - .0028571)/.0028571 = -.028.

The dollar is thus at a negative forward discount, or forward premium, in terms of the yen. A forward premium on the foreign currency (or forward discount on the domestic currency) signifies that the foreign currency can be purchased forward with domestic currency for less than it can be purchased spot or, alternatively, the domestic currency is worth less forward in foreign currency than it is worth spot.

Let us suppose now that the 3 month forward exchange rate of the yen in terms of the dollar is 355 yen per dollar while the current spot rate is 360 yen per dollar. Suppose further that you think that the spot exchange rate will not change over the next three months. If you are correct, you can make a profit by selling yen forward in return for dollars. A forward sale of 355 yen will net you one dollar. In 90 days when you have to fulfill the forward contract, you can use that dollar to purchase 360 yen for a net profit of 5 yen which, converted into dollars, amounts to 1.4 cents or 1.4 percent. Note that today's spot rate is really irrelevant---all that matters is the forward rate and the future spot rate.

This profit is, of course, a speculative one in that it depends on your being right about what the price of the yen in terms of the dollar is going to be in 3 months. If it turns out that a dollar was worth only 355 yen in three months when the forward contract matures, you will make nothing. If the dollar happens to be worth only 350 yen at that time you will lose---you will have to pay more than one dollar to obtain the 355 yen you have agreed to sell for one dollar under the forward contract.

You expected to gain by selling yen forward in the above situation because you believed that the yen would depreciate in relation to its current 90-day forward value during the next 90 days. If you think that the yen is going to appreciate in relation to its forward value, you would expect to gain by buying yen forward. In general, letting the expected future spot price of foreign currency be   Πe,  if   Πe > Π′  there is an expected gain from selling the domestic currency forward and if   Πe < Π′   there is an expected gain to purchasing the domestic currency forward.

It is thus clear that when foreign exchange traders and dealers expect that the future spot price of a currency in terms of another currency will be above the price at which that currency can be purchased forward they will all try to buy it forward. The current forward price of the currency will be bid up until such an expected gain is no longer possible. And when the market expects that the future spot price of a currency in terms of another currency is going to be below the current forward price, everyone will sell that currency forward in the expectation that they will be able to purchase it when the contract matures at a spot price below the agreed-upon forward sale-price. The current forward price of the currency will be bid down until an expected profit from doing this is no longer possible.

Any dealer or trader who is willing to ignore the risk can gain from foreign exchange market speculation whenever the expected future spot rate differs from the current forward rate. Given enough of such risk-neutral traders in the market, the forward exchange rate will be bid into equality with the expected future spot rate. The forward rate would then equal the market's estimate of where the spot rate will be when the contract matures.

Unfortunately, we cannot assume that foreign exchange speculators are indifferent to risk. In the Lesson entitled Asset Markets we concluded that the risk from holding an asset depends on the extent to which its return is positively correlated with the return on the entire bundle of assets held in people's portfolios and on how widely the its return fluctuates when the return on the entire portfolio fluctuates. Otherwise, the risk can be diversified away by holding the asset in a portfolio with many other assets whose returns are not correlated with its return. If holding three month forward yen is more risky than holding the average asset, a person with a contract to receive yen in, say, 30 days will require the forward price to be below the price she can expect to sell those yen for in 30 days when they are delivered under the contract. Similarly, a person holding a short position in yen will insist that the forward price of those yen be above the price at which she can expect to buy the yen 30 days hence for delivery under the contract. The forward discount can thus be expressed as

      Ψ = (Π′ - Π) ⁄ Π
          = (Π′ - Πe + Πe - Π) ⁄ Π
          = (Π′ - Πe) ⁄ Π + (Πe - Π) ⁄ Π

or

      Ψ = ρx + Eπ

where   ρx = (Π′ - Πe) ⁄ Π   is the risk premium on the domestic currency and   Eπ = (Πe - Π) ⁄ Π  is the expected rate of change of the spot price of foreign currency in terms of domestic currency---that is the expected rate of depreciation of the domestic currency---over the contract period. This is known as the efficient markets condition.

The forward discount on domestic currency in terms of a foreign currency is equal to the expected rate of depreciation of the domestic currency in terms of foreign currency over the life of the forward contract plus a risk premium for holding it forward. New information that leads market participants to believe that the domestic currency will depreciate in terms of the foreign currency in the future will cause a forward discount to emerge as the forward rate gets bid into line with the expected future spot rate, assuming that risk factors do not change. This is another way of saying that expectations are rational---that all information about the future course of the exchange rate is used by market participants in the market behavior that establishes the forward exchange rate. This will imply that the foreign exchange market is efficient.

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