6.6. NOISE-TRADERS

194CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

of rational and irrational agents produce spot and forward exchange dynamics that is consistent with the Þndings from survey data. The model adapts the overlapping-generations noise trader model of De Long et. al. [38] to study the pricing of foreign currencies in an environment where heterogeneous beliefs across agents generate trading volume and excess currency returns.

The irrational ‘noise’ traders are motivated by Black’s [14] suggestion that the real world is so complex that some (noise) traders are unable to distinguish between pseudo-signals and news. These individuals think that the pseudo-signals contain information about asset returns. Their beliefs regarding prospective investment returns seem distorted by waves of excessive optimism and pessimism. The resulting trading dynamics produce transitory deviations of the exchange rate from its fundamental value. Short-horizon rational investors bear the risk that they may be required to liquidate their positions at a time when noise-traders have pushed asset prices even farther away from the fundamental value than they were when the investments were initiated.

The Model

We consider a two-country constant population partial equilibrium model. It is an overlapping generations model where people live for two periods. When people are born, they have no assets but they do have a full stomach and do not consume in the Þrst period of life. People make portfolio decisions to maximize expected utility of second period wealth which is used to Þnance consumption when old.

The home country currency unit is called the ‘dollar’ and the foreign country currency unit is called the ‘euro.’ In each country, there is a one-period nominally safe asset in terms of the local currency. Both assets are available in perfectly elastic supply so that in period t, people can borrow or lend any amount they desire at the gross dollar rate of interest Rt = (1+it), or at the gross euro rate of interest, Rt = (1+it ). The nominal interest rate di erential—and hence by covered interest parity, the forward premium—is exogenous.

In order for Þnancial wealth to have value, it must be denominated in the currency of the country in which the individual resides. Thus in the second period, the domestic agent must convert wealth to dollars

and the foreign agent must convert wealth to euros. We also assume that the price level in each country is Þxed at unity. Individuals therefore evaluate wealth in national currency units. The portfolio problem is to decide whether to borrow the local currency and to lend uncovered in the foreign currency or vice-versa in an attempt to exploit deviations from uncovered interest parity, as described in chapter 1.1.

The domestic young decide whether to borrow dollars and lend euros or vice versa. Let λt be the dollar value of the portfolio position taken. If the home agent borrows dollars and lends euros the individual has taken a long euro positions which we represent with positive values of λt. To take a long euro position, the young trader borrows λt dollars at the gross interest rate Rt and invests λt/St euros at the gross rate Rt . When old, the euro payo Rt (λt/St) is converted into (St+1/St)Rt λt dollars. If the agent borrows euros and lends dollars, the individual has taken a long dollar position which we represent with negative λt. A long position in dollars is achieved by borrowing −λt/St euros and investing the proceeds in the dollar asset at Rt. In the second period, the domestic agent sells −(St+1/St)Rt λt dollars in order to repay the euro debt −Rt (λt/St). In either case, the net payo is the number of dollars at stake multiplied by the deviation from uncovered interest parity, [(St+1/St)Rt − Rt]λt. We use the approximations (St+1/St) '

(1 + ∆st+1) and (Rt/Rt ) = (Ft/St) ' 1 + xt to express the net payo as12

The foreign agent’s portfolio position is denoted by λ t with positive values indicating long euro positions. To take a long euro position, the foreign young borrows λ t dollars and invests (λ t/St) euros at the gross interest rate Rt . Next period’s net euro payo is (Rt /St −Rt/St+1)λ t. A long dollar position is achieved by borrowing −(λ t/St) euros and investing −λ t dollars. The net euro payo in the second period is −(Rt/St+1 − Rt /St)λ t. Using the approximation (FtSt)/(StSt+1) '

12These approximations are necessary in order to avoid dealing with Jensen inequality terms when evaluating the foreign wealth position which render the model intractable. Jensen’s inequality is E(1/X) > 1/(EX). So we have [(St+1/St)Rt − Rt]λt = [(1 + ∆st+1)Rt − (1 + xt)Rt ]λt, which is (6.47).

196CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

1 + xt − ∆st+1, the net euro payo is13

The foreign exchange market clears when net dollar sales of the current young equals net dollar purchases of the current old

Fundamental and Noise Traders

A fraction µ of domestic and foreign traders are fundamentalists who have rational expectations. The remaining fraction 1 − µ are noise traders whose beliefs concerning future returns from their portfolio investments are distorted. Let the speculative positions of home fundamentalist and home noise traders be given by λft and λnt respectively. Similarly, let foreign fundamentalist and foreign noise trader positions be λft and λnt. The total portfolio position of domestic residents is λt = µλft + (1 −µ)λnt and of foreign residents is λ t = µλft + (1 −µ)λnt.

We denote subjective date—t conditional expectations generically as Et(·). When it is necessary to make a distinction we will denote the expectations of fundamentalists denoted by Et(·). Similarly, the conditional variance is generically denoted by Vt(·) with the conditional variance of fundamentalists denoted by Vt(·).

Utility displays constant absolute risk aversion with coe cient γ. The young construct a portfolio to maximize the expected utility of

Both fundamental and noise traders believe that conditional on time-t information, Wt+1 is normally distributed. As shown in chapter 1.1.1, maximizing (6.50) with (perceived) normally distributed Wt+1 is equivalent to maximizing

198CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

(6.57) displays the familiar property of constant absolute risk aversion utility in which portfolio positions are proportional to the expected asset payo . The factor of proportionality is inversely related to the individual’s absolute risk aversion coe cient. Recall that individuals undertake zero-net investment strategies. The portfolio position in our setup does not depend on wealth because traders are endowed with zero initial wealth.

The foreign fundamental trader faces an analogous problem. The second period euro-wealth of fundamentalist foreign agents is the payo

from portfolio investments plus an exogenous euro payment of ‘labor’

f

income c , Wft+1 = [∆st+1 − xt]λStt + c . The solution is to choose λft = Stλft . Because individuals at home and abroad have identical tastes but evaluate wealth in national currency units, they will pursue identical investment strategies by taking positions of the same size as measured in monetary units of the country of residence.

These portfolios combined with the market clearing condition (6.49)

where Γt ≡ [(St/St−1) + St−1Rt−1]/(1 + St). The level of the exchange rate is indeterminate but it is easily seen that a solution for the rate of depreciation is

The independence of vt and xt−1 implies Et(∆st+1) = xt and the fundamentals solution therefore does not generate a forward premium bias because uncovered interest parity holds in the fundamentals equilibrium even when agents are risk averse. The reason is that under homogeneous expectations and common knowledge, you demand the same risk premium as I do, and we want to do the same transaction. Since we cannot Þnd a counterparty to take the opposite side of the transaction, no trades take place. The only way that no trades will occur in equilibrium is for uncovered interest parity to hold.

15The left side of the market clearing condition (6.49) is λt + λ t = (1 + St)λt =

(1 + St)/(γσs)[Et∆st+1 − xt]. The right side is, (St/St−1)R λt−1 + Rt−1St−1λt−1 = [(St/St−1) + (1 + xt−1)St−1]λt−1. Finally, using λt−1 = [Et−1∆st − xt−1]/(γσs2), we get (6.58).

200CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

Using the method of undetermined coe cients, you can verify that

is a solution.

Properties of the Solution

First, fundamentalists and noise traders both believe, ex ante, that they will earn positive proÞts from their portfolio investments. It is the di erences in their beliefs that lead them to take opposite sides of the transaction. When noise traders are excessively pessimistic and take short positions in the dollar, fundamentalists take the o setting long position. In equilibrium, the expected payo of fundamentalists and noise-traders are respectively

On average, the forward premium is the subjective predictor of the future depreciation: µEt∆st+1 +(1−µ)Et∆st+1 = xt. As the measure of noise traders approaches 0 (µ → 1), the fundamentals solution with no trading is restored. Foreign exchange risk, excess currency movements, and trading volume are induced entirely by noise traders. Neither type of trader is guaranteed to earn proÞts or losses, however. The ex post proÞt depends on the sign of

which can be positive or negative.

Matching Fama’s regressions. To generate a negative forward premium bias, substitute (6.62) and (6.52) into (6.66) to get

where ξt+1 ≡ (1/ρ)[1 − k(1 − µ)]vt+1 − (1 − µ)/ρut+1 − (1 − µ)ut is an error term which is orthogonal to xt. If [1 − k(1 − µ)] < 0, the

of over-reaction k cannot be identiÞed here.

We can ‘back out’ the implied value of over-reaction k if we are willing to make an assumption about survey measurement error. If µˆ 6= µ, then ∆sˆet+1 = µˆEt(∆st+1) + (1 − µˆ)Et(∆st+1) = [1 + k(µ − µˆ)]xt + (µ − µˆ)ut, which implies, β2 = 1 + k(µ − µˆ), β1 = k(1 − µˆ), and β = 1 − k(1 − µ). For given values of µ,ˆ β1, and β, we have,

k= β1/(1 −µˆ), and µ = (β −1 + k)/k. For example, if we assume that µˆ = 0.5, the 3-month horizon BIC-US results in Table 6.4 imply that

k= 11.94 and µ = 0.579.

202CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

Foreign Exchange Market E ciency Summary

1.The Þnancial market is said to be e cient if there are no unexploited excess proÞt opportunities available. What is excessive depends on a model of market equilibrium. Violations of uncovered interest parity in and of themselves does not mean that the foreign exchange market is ine cient.

2.The Lucas model–perhaps the most celebrated asset pricing model of the last 20 years–provides a qualitative and elegant explanation for why uncovered interest parity doesn’t hold. The reason is that risk-averse agents must be compensated with a risk premium in order for them to hold forward contracts in a risky currency. The forward rate becomes a biased predictor of the future spot rate because this risk premium is impounded into the price of a forward contract. But the Lucas model requires what many people regard as an implausibly coe cient of relative risk aversion to generate su ciently large and variable risk premia to be consistent with the volatility of exchange rate returns data.

3.Analyses of survey data from professional foreign exchange market participants predictions of future exchange rates Þnd that the survey forecast error is systematic. If you believe the survey data, these systematic prediction errors may be the reason that uncovered interest parity doesn’t hold.

4.Market participant’s systematic forecast errors can be consistent with rationality. A class of models called ‘peso-problem’ models show how rational agents make systematic prediction errors when there is a positive probability that the underlying structure may undergo a regime shift.

5.On the other hand, it may be the case that some market participants are indeed irrational in the sense that they believe that pseudo signals are important determinants of asset returns. The presence of such noise traders generate equilibrium asset prices that deviate from their fundamental values.