chapter 12 copeland

Copeland, Exchange Rates and International Finance, 4th edition

© Pearson Education Limited 2006

Slide 12.1

Chapter 12

Market Efficiency and Rational

Expectations



Slide 12.2

Some key readings

1. Hansen, L.P. and Hodrick, R.J. 1980. Forward exchange

rates as optimal predictors of future spot rates: an

econometric analysis. Journal of Political Economy, vol.

88, pp. 829-853.

2. Meese, R.A. and Rogoff, K. 1983. Empirical exchange

rate models of the seventies: do they fit out of sample?

Journal of International Economics, vol. 14, pp. 3-24.

3. Hakkio, C.S. 1985. Expectations and the forward

exchange rate. International Economic Review. no. 22, p.

663-678.



Slide 12.3

• An efficient market was defined there as one in

which prices fully reflect all available information.

• Example: a share tipster

– If we are to make money, we need to know more than

simply that company X is going to make large profits.

We also need to be sure that those profits are

underestimated by the stock market.

• It is the application of the ‘no free lunch’ argument to the field of information

– As the market price fails to incorporate publicly

available information, there must exist unexploited

profit opportunities.



Slide 12.4

• The focus of this chapter is:

– The consequences of market efficiency for the

relationship between spot and forward

exchange rates.



Slide 12.5

Mathematical Expected Value

• Expected value of a (random) variable, E(X), is

the weighted average of all possible outcomes

– The weight on any outcome is equal to its probability.



Slide 12.6

• The expected value has considerable superficial

attractiveness as the best single number to use in

comparing alternative risky propositions.

– Corresponds to what we might intuitively feel to be the

best guess of a person’s likely winnings

– Paradoxical situation: the average prize is ‘the most likely outcome’

• Even in cases where the outcome is never equal

to (or anywhere near equal to) the expected

value, a forecast based on it will still be correct on

average.



Slide 12.7

• Decisions involving uncertainty are a lot more

complicated.

– More complicated prospects, with a greater number of

possible outcomes and consequently more difficult

computations

– In this respect, we have to deal with a genuine

forecasting problem.

• Use of whatever relevant information is

available.



Slide 12.8

12.2 Rational Expectations Hypothesis (REH)

• Problem: agents’ expectations are key to financial market behaviour. But how does trader/investor forecast? – Expectations have incredible power to shape the

reality (Dan Ariely).

• Let xt be the value at the current time, t, of the variable/asset/security in question (e.g. an exchange rate, share price, retail price index..)

• At t, xt is known, but xt+1 is still unknown. Write xt+1

e as the agent’s expectation of the future value, xt+1.



Slide 12.9

• An economic agent is said to hold a (fully)

rational expectation with respect to a variable if

his subjective expectation is the same as the

variable’s (mathematical) expected value, conditional on an information set containing all

publicly available information.

• The rational expectations (RE) hypothesis states

that the market’s (subjective) expectations are in

fact the same as the expected value, conditional

on the set of all available information.

12.2 Rational Expectations Hypothesis (REH)



Slide 12.10

Q. How is xt+1e formed?

REH answer: a rational economic agent uses all information available at t in best possible way, so:

LHS is agent’s subjective expectation

RHS is (statistical) expectation of xt+1 conditional on information set, It.

We often use abbreviated notation:

where Et means expectation conditional on information at t.

)|( 11 ttet IxEx

111 )( ttttet xEIxEx



Slide 12.11

Q. What defines relevant information? How

should it be rationally used?

REH answer: depends on model. Rational

expectation is consistent with structure of the

model it appears in. REH views any other

expectation as irrational, hence arbitrary

Note:

1. Since RE is optimal, it is unique i.e. all rational

investors share same expectation

2. RE is NOT the same as perfect foresight:



Slide 12.12

RE versus Perfect Foresight

11 tet xx

ttttttet uxxEIxEx 1111 )(

Perfect foresight implies:

RE implies:

Where, by the definition of a mathematical expectation, ut is a

zero-mean process Etut+1 = 0 .

So, whereas perfect foresight implies investors always right,

REH implies they may make mistakes (possibly large,

possibly frequent), but their average error is zero – if they

were systematically consistently wrong, they would be failing

to make full use of the information.



Slide 12.13

Exchange rates under RE

Why we use logs

Since is the FX price of domestic currency, we

would like it to be true that:

But this is not generally true under RE, since:

because is a nonlinear relationship.

Solution: use logs, so log of inverse exchange rate is

which is linear

eeee

SSor

S

S 111

1

XE

XE

1

)(

1

X

1

S1

S



Slide 12.14

12.3 Forward Market Efficiency

Suppose you think that future spot price is 10% higher in 1 year. You

can make profit by buying the currency forward and selling it at spot, in 1

year. When will the speculation stop?

The relationship between the forward and spot markets under the

assumptions of RE, adequate arbitrage funds, free movement of funds,

and negligible transactions costs:

(12.2)

which is efficient market equilibrium as the forward rate reflects

1. Publicly available information summarised in the RE, ;

2. Market’s attitude to risk, as embodied in the risk premium, .

tttt

t sEf

11

t1ttsE



Slide 12.15

12.3 Market Efficiency (continued)

Rewrite Equation (12.2) by subtracting from both sides:

(12.3)

Equation (12.3) implies:

(12.4)

Alternatively, stepping back one period:

(12.4’)

tttttttt

t ussEsf

11111 ][

1ts

11

1

ttt

tt ufs

ttt

tt ufs 11



Slide 12.16

12.4 Unbiasedness

When the forward market is efficient and investors are risk neutral, there

fore they require no risk premium to take risky transactions:

Forward rate = expectation of the spot rate at the time contract matures

Spot rate = forward rate set in the previous period, plus or minus a

random error

(12.5)

Rewriting (12.5) in terms of rate of depreciation:

(12.5’)

tt

tt ufs 1

ttt

ttt usfss )( 111



Slide 12.17

12.5 Random Walk Model

Random walk

Change in time series from one period to next is purely random:

(12.6)

Alternatively:

(12.6’)

where is completely random (no pattern over time).

Random walk model outperform sophisticated model using fundamental

variables—today’s exchange rate as the best guess of tomorrow’s.

ttt uXX 1

tttt uXXX 1

tX

tu



Slide 12.18

12.5 Random walk model (continued)

Random walk with drift d

Change in time series from one period to next is equal to drift factor

plus purely random component:

(12.7)

Alternatively:

(12.7’)

where is completely random (no pattern over time).

ttt udXX 1

tttt udXXX 1

tX

tu



Slide 12.19

12.5 Market Efficiency and the Random Walk Model

1. The random walk model is compatible with RE, efficiency and

unbiasedness.

2. However, efficiency does NOT require that the spot rate

follow a random walk. Deviation from a random walk may be

due to a risk premium or a nonzero expected return

(depreciation).

tttt uss 1

The first term on the RHS could be explained by a risk

premium (possibly nonconstant). Even in the absence of a

risk premium (i.e. risk-neutrality), we could well have (st+1 – st)>0 if there is long run anticipated depreciation –

compensated by the interest rate differential.



Slide 12.20

12.5 Random Walk Model (continued)

If spot rate follows a RW with drift:

(12.8)

Taking expectations in (12.8) conditional on

(12.9)

(Note Et-1ut= Etut+1=0 because the residual is zero-mean by definition,

and Et-1st-1= E(st-1|It-1)=st-1 because st-1 is in It-1

ttt udss 1

dsuEdEsEsE tttttttt 111111



Slide 12.21

12.5 Random Walk Model (continued)

If spot rate does not follow a RW with drift: for example,

(12.10)

RE forecast of the next period’s spot rate:

(12.11)

Forward market efficiency for a RW:

(12.12)

Subtracting equation (12.11) from (12.10):

(12.13)

Profit made by a speculator paying the rationally expected spot rate at

time t – 1 and selling on the spot in the next period—on average zero.

tttttt uZZsss 121

11211 ttttttt ZZEsssE

ttt

t sf 1

ttttttt uZEZsEs )( 11



Slide 12.22

12.8 Results

To test for unbiasedness, fit equations of the following form:

(12.15)

1. estimate of the intercept a

- insignificantly different from zero?

2. estimate of the slope coefficient b

- insignificantly different from unity?

3. serially uncorrelated?

tt

tt vbfas 1

tv



Slide 12.23

When market sentiment

changes it results in a

change of direction in both

spot and forward rates

simultaneously

Sudden wave of bullishness about the

pound pushes up both the spot price of

sterling and its price for 30-day delivery

Spot rate against the lagged one-month forward rate



Slide 12.24 Change in spot rate and the lagged forward premium

The premium is not only invariably smaller in

absolute terms, it is also far less volatile



Slide 12.25

12.8 Results (continued)

If RE is assumed, then UIRP implies:

(12.16)

Since by definition:

Then a test of RE + UIRP would involve testing:

(12.17)

Alternatively, if we have direct (survey) information on expectations, we can test:

(12.18)

*111 ttttttt

et rrssEsEs

ttttt urrss *

1

11 ttet vss

tttt ussE 11



Slide 12.26

Another topics?

1. Peso problem – A perennial discount (i.e. high money market rate) on

a currency which is officially pegged to the U.S. dollar or some

other reference currency. The discount exists because the market

perceives a small immediate probability of a large devaluation.

– 1955-76 Mexican peso was pegged against USD.

2. Excess volatility – The foreign exchange market "overreacts“ to events—prove that

the foreign exchange market is sending confusing signals to

traders and investors who base their decisions on exchange rates.

– Exchange rate should be volatile however are substantially more

volatile than the underlying factors that move them.

chapter 12 copeland

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