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Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
Slide 12.1
Chapter 12
Market Efficiency and Rational
Expectations
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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.
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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.
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
Slide 12.4
• The focus of this chapter is:
– The consequences of market efficiency for the
relationship between spot and forward
exchange rates.
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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.
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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.
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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.
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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.
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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)
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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:
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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.
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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.
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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
Copeland, Exchange Rates and International Finance, 4th edition
© Pearson Education Limited 2006
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.