A Literature Review of the Efficiency of the Foreign Exchange Market

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Lecture 7.*

Lecture 7

Lecture 7.*

Revision

Lecture 7.*

Chapter 11

Optimum currency areas

and

monetary union

Lecture 7.*

Figure 11.1 Exchange rate regimes

The Spectrum of Exchange Rate Regimes

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Mundell: Optimal Currency Area (OCA)

OCA defined by labour mobility e.g. fall in demand for labour in one US region causes migration to high-demand region, so migration even if wages inflexible (contrast with Europe)
Conclusion: optimal currency zone is area small enough for labour to be mobile

Lecture 7.*

McKinnon: OCA

Devaluation ineffective if labour suppliers respond to price level rise by raising money wage
In large country with relatively small open sector, imports have small weight in price index, so workers barely notice price rise caused by devaluation
OCA should be big enough for labour suppliers to behave as if in a closed economy (USA, Europe, China?)

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Costs of Transition

The costs of joining will depend crucially on how similar the economies are before the start

Micro cost of changing the currency (and benefits e.g. reduce fraud)

Need for convergence

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11.3.4 Costs of Transition

Table 11.1 EMU convergence criteria
Fiscal Policy
Budget deficit	< 3% of GDP
Government debt	< 60% of GDP
Monetary Policy
Inflation (RPI)	< 1.5% above 3 best performing countries
Long-term interest rates	< 2% above 3 best performing countries

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Chapter 12

Market Efficiency and Rational Expectations

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12.2 Rational Expectations Hypothesis (REH)

Problem: agents’ expectations are key to financial market behaviour. But how does trader/investor forecast?
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+1e as the agent’s expectation of the future value, xt+1.

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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.

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12.3 Forward Market Efficiency

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

- publicly available information summarised in the RE, ;

- market’s attitude to risk, as embodied in the risk premium, .

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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’)

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12.4 Unbiasedness

When the forward market is efficient and investors are risk neutral:

▪ 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’)

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Testing for efficiency, a problem

Neither RE nor market efficiency is usually testable on its own.

We almost always test a joint hypothesis, market efficiency AND rational expectation

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Chapter 14

The Risk Premium

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Introduction

What actually determines the risk premium?

Define, risk aversion, risk neutrality, risk premium

All about constrained optimisation under uncertainty

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14.1 Assumptions

Perfect capital market, no transaction costs, forward margin requirements etc
2 periods: current period, 0, future period, 1
Agents maximise expected utility of future consumption, E[U(C1)] with usual properties
No inflation

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Current wealth, W0 given. Future consumption determined by forward purchase of number of contracts, A, each giving time 1 payoff of S1 – F0.

So total amount available for consumption at time 1 is:

(14.1)

Forward purchases of dollars:

- increase future consumption if the price of dollars rises

- decrease future consumption if the price of dollars falls

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Figure 14.1 Speculator’s Equilibrium

14.2 A Simple Model Of The Risk Premium: Mean-variance Analysis

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14.2.2 Constraint: the speculative opportunity line

Expected value of consumption:

(14.2)

Variance of consumption (using (14.1) and (14.2)):

(14.3)

where is the variance of the spot rate. Solving (14.3) for A

(14.4)

as can be seen in Figure 14.1

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14.2.2 Constraint: the speculative opportunity line (continued)

Substitute for A from Equation (14.4) into Equation (14.2):

(14.5)

where

The normalised risk premium i.e the reward for forward speculation (numerator) per unit of (spot) exchange rate risk carried by the speculator (denominator)

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14.2.3 Equilibrium

A rise in the risk premium changes the slope in 14.1 to move from H to J say

But ambiguous sign because of the income effect

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14.3 General Model

Allow for utility to depend more generally on future consumption (i.e. not simply mean and variance of return to speculation)
Mean variance as special case where either utility depends only on mean and variance or where currency returns normally distributed (which is improbable)

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14.3.1 The speculator’s decision problem

Speculator solves problem

subject to constraint:

(14.1)

Differentiate utility function with respect to A

(14.6)

For the utility-maximising value of A, derivative in Equation (14.6) = 0:

(14.7)

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14.3.1 The speculator’s decision problem (continued)

Rewrite Equation (14.7) in terms of the forward rate:

(14.8)

Using the definition of the covariance of two random variables:

(14.9)

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Conclusion of general model

What drives things is the covariance of expected utility and the spot rate and the covariance of expected utility
So the risk premium will vary if either of these change

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14.3.3 The simple model revisited: mean-variance assumptions

Quadratic utility function for an agent:

(14.10)

When maximizing the utility function with respect to A:

(14.11)

(14.12)

where

- coefficient of relative risk aversion.

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14.3.4 Risk premium and portfolio balance models

Solve Equation (14.12) for the desired holding of foreign assets

relative to wealth:

(14.13)

which gives the optimal number of forward contracts relative to consumption as a decreasing function of the risk of FX and the speculator’s level of risk aversion, and an increasing function of the expected return

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14.4 Evidence on risk premia

Evidence from efficiency tests and survey data suggest there is a large and volatile forward bias but this may not be due to a risk premia

Evidence suggests that the risk premia may be too small to explain the volatility in most financial markets

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14.5 Conclusion

Risk premia emerge as a natural consequence of risk aversion

But the risk premia seems to be too small

Either we face non rational behaviour or market inefficiency

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Chapter 16

Target Zones

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Introduction

In practise exchange rates are rarely fixed, a target zone is much more realistic
Also exchange rates must be expected to move smoothly, otherwise there is an opportunity for infinite arbitrage
Given this how can we expect discreet changes in fundamentals but continuous changes in the exchange rate?

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What is a Target Zone (TZ)?

Authorities announce a fixed rate system
Par or central value, x , with fluctuation limits b%

(sometimes, b unannounced e.g. Hong Kong in 1990’s)

Assume: central bank allows free float as long as exchange rate between limits x + b% and x - b%
Intervenes to buy (sell) domestic currency if/when exchange rate reaches x + b% (x - b%)
Most “fixed” exchange rate systems permit floating over some range e.g Bretton Woods System 1% fluctuation bands, European Exchange Rate Mechanism, 2.25%

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Q. What happens when exchange rate is allowed to float over limited range (Target Zone)?

Assume: underlying exchange rate process (in absence of TZ) is random walk (pure float)
Then obvious answer: exchange rate follows random walk inside TZ, till it hits upper (lower) bound, then stays at that level until a negative (positive) shock brings it back inside TZ (Figure 16.2)
This is not correct because it does not allow for the way expectations are formed near the barrier of the zone

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Figure 16.2 Target zone: the way it isn’t

16.2 Effects of target zone

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Why Figure 16.2 is impossible

Assumes agents take no account of existence of TZ – ignore prospect of hitting bounds till they hit them
Leaves opportunity for unlimited arbitrage profits

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Preliminary

Path of expected exchange rate must be continuous (i.e. no jumps up or down as in Figure 16.1). Any step changes must be unanticipated.
Reason: anticipated jump implies arbitrage opportunity – buy (sell) the instant before a step-appreciation (depreciation), to earn profit at infinite rate

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16.2 Effects of target zone

Rewrite Equation (13.7) by replacing by :

(16.1)

is a random walk as long as the exchange rate is not actually at the

boundary of the target zone:

(16.2)

where is a zero-mean error term or innovation.

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16.2 Effects of target zone (continued)

At the upper intervention point, , the fundamental is given by:

whenever ,

for

(16.3) for

At the lower intervention point, , the fundamental is given by:

whenever ,

for

(16.4) for

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16.2 Effects of target zone (continued)

In the middle of the band:

(16.5)

At the upper boundary, as implied in Equation (16.3):

(16.6)

At the lower boundary, as implied in Equation (16.4):

(16.7)

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16.3 smooth pasting

So the exchange rate must be expected to never quite reach the barriers
The shape of the curve must be S shaped
This is called smooth pasting

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Figure 16.3 Target zone – the S curve

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What would happen without this?

If we were very close to the band the fundamentals might push us outside the band
The authorities would intervene to stop this.
So we can never expect to be on the band

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Figure 16.4 Target zone without smooth pasting

16.3 Smooth pasting

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16.4 An options interpretation

When we are close to the upper bound buyers of the currency know that they can switch back at the band price or lower
So they are buying a call on foreign currency with an exercise price of the upper bound
Given this they will be willing to overpay for the currency which drives them away from the bound

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16.4 An option interpretation

Figure 16.5 Effect of Target zone on the exchange rate

Even though the fundamental would drive the exchange rate above its barrier, the value of the option keeps it away from this level.

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16.5 Implications of Basic TZ Model

“Honeymoon effect” as speculators’ expectations damped by prospect of hitting bounds
Importance of bounds being:
Publicly announced
Credible
Extra degree of freedom for policymakers (lower exchange rate volatility for any given monetary policy) ?
Yes, but at cost of greater interest rate volatility

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16.6 Beauty and the beast:

the target zone model meets the facts

TZ Model prediction: exchange rate will spend more time near bounds (where it tends to “stick”) than in interior
Fact: exchange rates in TZ spend most time in interior
Possible explanations of failure:
intramarginal intervention. Authorities try to pre-empt crises by intervening before bounds reached e.g leaning against the wind
TZ lacks credibility, so agents give nonzero probability to bound being breached

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In fact authorities do not wait to hit the barriers before acting
Instead they try and keep the exchange rate in the center of the zone
They ‘lean into the wind’

16.7 Intramarginal interventions:
leaning against the wind

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If target level of fundamental = 0, within the band the authorities

follow the rule:

for (16.8)

- the change in k between t-1 and t would be zero whenever the previous

period’s fundamental is spot on its target level of zero.

Equation (16.8) implies:

(16.9)

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16.7 Intramarginal interventions:
leaning against the wind (continued)

At the borders of the zone, the rule becomes as follows:

whenever ,

for

(16.10) for

At the lower intervention point, , the fundamental is given by:

whenever ,

for

(16.11) for

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Outcome

S-curve honeymoon effect reduced even further
S-curve effect generated by intramarginal intervention itself, even without bounds
Imposing bounds adds little extra curvature (Figure 16.7)

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Figure 16.7 Target zone with intramarginal intervention

16.7 Intramarginal interventions: leaning against the wind

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16.8 Credibility and realignment prospects

Everything above assumes perfect credibility and no chance of realigning the bands.
But if realignments are possible then we tend to stay near the centre of the band except when a crises hits.
Then we may get volatile realignments

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16.9 conclusion

Smooth pasting shows what happens near to zone boundaries under full credibility
When we relax this we get clustering in the exchange rate near the centre of the band and sudden crises periods
This seems to fit the data reasonably well

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