172CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

Mark and Wu [102] used the VAR method to get quarterly estimates of pt for the US dollar relative to the deutschemark, pound, and yen. Their estimates, shown in Figure 6.1, show that of E(∆st+1|Ht) are

ˆ

persistent for the pound and yen. Both pˆt and E(∆st+1|Ht) alternate between positive and negative values but they change sign infrequently. The cross-sectional correlation across the three exchange rates is also evident. Each of the series spikes in early 1980 and 1981, the pˆts are generally positive during the period of dollar strength from mid-1980 to 1985 and are generally negative from 1990 to late 1993. You can also

ˆ

see in the Þgures the negative covariance between pˆt and Et(∆st+1) deduced by Fama’s regressions.

Deviations from uncovered interest parity are a stylized fact of the foreign exchange market landscape. But whether the stochastic pt term ßoating around is the byproduct of an ine cient market is an unresolved issue. As per Fama’s deÞnition, we say that the foreign exchange market is e cient if the relevant prices are determined in accordance

(110) with a model of market equilibrium. One possibility is that pt is a risk premium. At this point, we revisit the Lucas model and use it to place some structure on pt.

6.2Rational Risk Premia

Hodrick [75] and Engel [44] show how to use the Lucas model to price forward foreign exchange. We follow their use of Lucas model to understand deviations from uncovered interest parity.

Recall that forward foreign exchange contracts are like nominal bonds in the Lucas model in that they are not actually traded. We are calculating shadow prices that keep them o the market. Let St is the nominal spot exchange rate expressed as the home currency price of a unit of foreign currency and Ft be the price the foreign currency for one-period ahead delivery.

The intertemporal marginal rate of substitution will play a key role. In aggregate asset-pricing applications, it is common to work with per capita consumption data. One way to justify using such data in the utility function in Lucas’s two-country model is to assume that the period utility function is homothetic and that the relative price between the home good and the foreign good (the real exchange rate) is constant. This allows you to write the representative agent’s intertemporal marginal rate of substitution between t and t + 1 as

where u0(Ct) is the representative agent’s marginal utility evaluated at equilibrium consumption.3

Let Pt be the domestic price level and let β is the subjective discount factor. A speculative position in a forward contract requires no investment at time t. If the agent is behaving optimally, the expected marginal utility from the real payo from buying the foreign currency forward is Et [u0(ct+1)(Ft − St+1)/Pt+1] = 0. To express the Euler equation in terms of stationary random variables so that their unconditional variances and unconditional covariances between random variables exist, multiply both sides by β and divide by u0(ct) to get

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(6.11) is key to understanding the demand for forward foreign exchange risk-premia in the intertemporal asset pricing framework. Keep in mind that the intertemporal marginal rate of substitution varies inversely with consumption growth so that when the agent experiences the good state, consumption growth is high and the intertemporal marginal rate of substitution is low.

constant, the growth rate of consumption of X is the same as the growth rate of the consumption index and the intertemporal marginal rate of substitution becomes that in (6.10)

µt+1 is low when consumption growth is high. What it boils down to is this. Holding the euro forward pays o well in good states of the world but you don’t need an asset to pay o well in the good state. You want assets to pay o well in the bad state—when you really need it. But the forward euro will pay o poorly in the bad state and in that sense it is risky.

If the euro is risky the dollar is safe. If Et Ft−St+1 < 0 and you

Pt+1

like a strange idea to buy an asset with expected negative payo , but this is something that risk-averse individuals are willing to do if the asset provides consumption insurance by providing high payo s in bad (low growth) consumption states. The expected negative payo can be viewed as an insurance premium.

To summarize, in Lucas’s intertemporal asset pricing model, the risk of an asset lies in the covariance of its payo with something that individuals care about—namely consumption. Assets that generate high payo s in the bad state o er insurance against these bad states and are considered safe. A high payo during the good state is less valuable to the individual than it is during the bad state due to the concavity of the utility function. Risk-averse individuals require compensation by way of a risk premium to hold the risky assets.

Risk-neutral forward exchange. If individuals are risk neutral, the intertemporal marginal rate of substitution µt+1 is constant. Since the covariance of any random variable with a constant is 0, (6.13) becomes

So even under risk-neutrality the forward rate is not the rationally expected future spot rate because you need to divide by the future and stochastic price level. To see more clearly how covariance risk is related to the fundamentals, it is useful to take a look at expected nominal speculative proÞts.

176CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

The Nominal Risk Premium.

Multiply (6.11) by Pt and divide through by St to get

poral marginal rate of substitution of money. In chapter 4, (equation (4.62)) we found that the price of a one-period riskless domestic cur-

Perhaps now you can see more clearly why the foreign currency (euro) is risky when the forward euro contract o ers an expected proÞt. If

ated with a weakening of the euro (low values of St+1 ). The euro is risky

St

because its value is positively correlated with consumption. Agents consume both the domestic and the foreign goods but the foreign currency buys fewer foreign goods in the bad state of nature and is therefore a bad hedge against low consumption states.

Pitfalls in pricing nominal contracts. Suppose that individuals are risk neutral. Then µmt+1 = and the covariance in (6.16) need not be 0 and again you can see that the forward rate is not necessarily the rationally expected future spot rate. Agents care about real proÞts, not nominal proÞts. Under risk neutrality, equilibrium expected real proÞts are 0, but in order to achieve zero expected real proÞts, the forward rate may have to be a biased predictor of the future spot.