Lectures 19-20

Revision
Revision
• A quick run through the assessed essay
• An overview of the course
– What are the main things you are expected to
learn and know about?
– I’ll try to highlight the most important topics we
have studied
– We won’t go through all of these slides point-by-
point, but they should help you revise and in
particular help you focus your revision
 Assessed essay

Why is it apparently so difficult to forecast

exchange rate movements? Discuss with
reference to the monetary model, the
Mundell-Fleming model and/or the
Dornbusch model and its extensions.
Key points
• Yes, the evidence (since Meese and Rogoff) suggests
it is hard to forecast the ER in the sense that
structural models, like the three listed, struggle to
beat an autoregressive model out-of-sample. They
can’t capture the “excess” volatility of the nominal
ER
• Model uncertainty: fact that there are many models
(not just these three), suggests there is uncertainty
about how best to model and forecast the ER
• These three models are all structural. They link the ER to
fundamental variables, like relative prices (PPP), incomes,
interest rates, trade balance and money
– But they make different assumptions about the shape of
the long-run AS curve, whether PPP holds, the degree of
flexibility of financial markets versus wages, the role of
expectations, etc.
– So difficulties in forecasting the ER stem from
uncertainty about which if any of these assumptions are
the correct ones; and which of the fundamental
variables matter; and what are the future values of
these fundamentals? Policy is also unpredictable
– So marks for listing these different assumptions and
questioning their plausibility
 6 Main Points
1. ER is volatile, especially at high frequency
– Distinguish short versus long run predictability
2. Different models (different fundamentals: relative
incomes, prices, money, interest rates, trade balance…)
= different forecasts
• Uncertainty about the right fundamentals
3. Different assumptions (PPP, fixed vs. flexible prices,
flexibility of financial markets, demand for money
equation, mobility of capital, slope of AS): which are
correct– depends on short run/long run?
4. Mention empirical/econometric testing of models
(Meese-Rogoff etc.) and performance relative to an AR
• problems measuring not just expectations but other
variables ; time-varying parameters; nonlinearity
5. Role of expectations (Rationality? Dornbusch model
makes a specific assumption about expectation formation,
not in monetary model)
• If ERs are like assets, their movements are based on
movements in expectations/news. Bubbles; speculation
6. Is “news” explaining the poor performance? Something is
happening which is not picked up in the model
• Meese-Rogoff rule out forecasting errors in the
explanatory variables
Revision
Basic concepts and definitions
• Floating versus Fixed Exchange rate
• Fixed exchange rate: monetary authority intervenes
by buying up excess supply of $ with £ (when £
strong, $ weak); or the reverse
• Balance of payments
• Current and capital accounts
the current account and the sum of the capital and
financial accounts should balance each other out
• Reserve change = 0 under a pure float
What determines the exchange rate?
• PPP as a long-run explanation
2.4 Purchasing Power Parity
Pi  SPi* (i = 1, . . ., N) (2.3)
where
Pi - domestic prices of good number i;
Pi* - foreign prices of good number i.

P  SP* (2.4)

where
P - home country’s price index,
*
P - foreign country’s price index.

The general level of prices , when converted to a common

currency, will be the same in every country
 PPP and the empirical evidence
• Real exchange rate should be flat under PPP. They are
not
• PPP as a long-run relationship?
• For traded goods only (so does PPP hold better for
producer prices than retail prices?)
• Candidate explanations for why PPP appears to fail:
– Balassa-Samuelson effect
– Trade costs and adjustment to PPP
– Trade costs: the iceberg model
– Incomplete pass-through
Uncovered Interest Rate Parity
r  r *  s e
• Investors will only carry the risk of expectations error
without requiring a premium if they are risk-neutral
• Covered IRP: no ER risk
– Use the forward rate F instead of the expected
spot rate
r  r*  f
 4.1.3 IS/LM and Aggregate demand

Figure 4.4 derivation of the aggregate demand curve; and the effects of a higher
price level
4.2.3 Aggregate Supply: alternative formulations

Figure 4.7 the sticky price short-run aggregate supply curve

5.1 The simple monetary model of a floating exchange rate

5.1.1 Setting
The Monetary model rests on three assumptions:
1. the aggregate supply curve is vertical

2. the demand for real money balances is a stable function of

only a few domestic macroeconomic variables – using the
Cambridge quantity equation (4.7), in equilibrium:

M s  M d  kPy  kY k 0
where y is real national income
3. PPP obtains at all times
 Left panel is locus of points that satisfy PPP
Right panel shows AD-AS given M, P* and y

Figure 5.2 effects of a money supply increase under floating rates

Leads to a depreciation (S↑) in proportion to the money supply increase. This depreciation
is needed to restore competitiveness, given that P↑
5.1.6 Two-country model of a floating exchange rate
(continued)
Under PPP, P / P*  S , and (5.5) can be rewritten as:

M 0s / M 0s*  k.Sy / k * y* (5.6)

Solving for S,

M /M*
S (5.7)
ky / k * y*

- the exchange rate equals the ratio of the relative money stocks
to the relative real money demands
Therefore, it is relative movements in both the money supply and real
incomes that matter
What about the monetary model with
fixed exchange rates?
• Important to clarify what is exogenous
• Money stock is endogenous, since monetary policy needs
to defend the fixed exchange rate, S
• Money stock contains FX component which increases
(decreases) when there is excess demand (supply) for
domestic currency, so domestic credit is the monetary
policy instrument – not the money supply as a whole
• Endogenous: P, FX
• Exogenous: y, P* (determined by foreign money supply)
and DC (domestic credit: recall FX+DC=money supply)
 s
Figure 5.5 domestic credit increase, DC0 to DC1, under fixed rates ( M 0  P0  1)
is neutralised by a fall in the foreign reserves FX due to the temporary BOP
deficit which progressively reduces the money supply and reduces P
Mundell-Fleming model
• The Mundell-Fleming model is the opposite extreme
to the monetary model
• It adheres to the Keynesian tradition
– with prices fixed (AS curve is horizontal), y is
determined by the AD curve; and we need only
consider the IS-LM model
• May be used for fixed and floating exchange rates
• Very popular in 1960s/70s
 6.1 Setting: Assumptions
1. The aggregate supply curve is flat (i.e. prices/wages fixed, as in
Keynesian paradigm). Set domestic and foreign price level = 1 (for
simplicity), so Q = SP*/P = S; and M denotes real and nominal
money stocks
2. Short run view: instead of PPP, current account equilibrium
condition:
B  B( y, Q)  B( y, S ) By  0 Bs  0 (6.1)

- the current account surplus (B) depends:

positively on the real exchange rate (competitiveness)
negatively on real income via consumption effect
(contrasts with the monetary model where if domestic income
rises the exchange rate appreciates as Md↑ causing P↓)
3. Exchange rate expectations are static
4. Capital mobility is less than perfect, so interest rate differential
in favour of domestic country causes finite capital inflow
- could be down to risk aversion
Figure 6.1 (real=nominal) monetary expansion under floating rates in the M-F
Model. LM shifts to the right, so r↓. But we don’t move all the way from A to C as the
lower r induces a current & capital a/c deficit. So S↑ and IS shifts to the right too. Current
a/c ends up at F, in surplus
6.3 Conclusion: Monetary expansion with a floating
exchange rate

Proposition 6.1 In the M-F model of a floating exchange rate,

money supply increase causes:

 a depreciation in the exchange rate

 an increase in income
 a fall in the interest rate, provided capital is not completely mobile
 an improvement in the current account of the balance of payments
 6.5 Monetary expansion with a fixed exchange rate

Figure 6.3 monetary expansion under fixed rates in the M-F model
DC↑ causes LM to shift down so current a/c worsens at F. So to maintain S, FX↓ so money
supply contracts and LM curve returns to initial position
 6.5 Monetary expansion with a fixed exchange rate: in sum

Proposition 6.3 In the M-F model of a fixed exchange rate,

a money supply increase causes:

In the short term (and provided capital is not completely mobile, if so SR=LR)
 a fall in the interest rate

 a rise in income

 a deterioration of the balance of payments on both current and capital

accounts

In the long term

 a fall in the foreign currency reserves and a rise in DC

 no change in income, the interest rate or the balance of payments

6.7 The Monetary model and the M-F model compared
The M-F model, in contrast to the monetary model

- emphasises the level of activity and interest rates

rather than the price level

In the monetary model changes in real income can only be exogenous events. Increases
in income increase Md and hence S↓. In the M-F model they are endogenous and (i)
raise Md; (ii) raise C and (iii) worsen the current a/c so S↑

In the monetary model r determined by loanable funds market and r*. So P does the
adjusting. In M-F, UIRP does not hold and r clears both the money and goods markets

- concentrates on flows of spending and capital rather than on

stocks of assets

- gives a central role to the crowding-out mechanism

 Dornbusch model
• Both the monetary and M-F models really ignore the
role of expectations in determining ER differentials
and this may be the main reason why they fail to
account for most of the variation in exchange rates
– And monetary model relies on PPP
– And M-F model on fixed prices
• The Dornbusch model is a hybrid of the monetary and
M-F models, with expectations as well
– Short run properties of Keynesian models
– Long run properties of the Monetary Model
• It rests on the idea that real markets adjust slowly,
while financial markets adjust very rapidly
 Empirical observation: financial markets adjust to
shocks far more rapidly than goods markets
Consequence for the model: in the short run, financial
markets have to overadjust in order to compensate for
sluggish goods markets (OVERSHOOTING)
With prices (and output) fixed in the short run, any
change in the nominal money supply changes real
balances, requiring the interest rate to adjust to clear
the money market (Liquidity Effect)
In the long run, prices adjust fully, returning all real
variables to their pre-shock levels, but leaving the
nominal exchange rate at the new equilibrium level
predicted by the simple Monetary Model
 7.1 Outline of the Dornbusch model

Assumptions
1. Small open economy, like the UK (so P*, r* exogenous)
2. At outset, time 0, inflation and exchange rate depreciation are zero
3. Aggregate demand is determined by the standard open
economy IS-LM mechanism (so AD determined by G and Q=SP*/P).
4. Financial markets adjust instantaneously.
Investors are risk neutral, so that UIRP holds always.
r  r *  s e (7.1)

5. Investors’ exchange rate expectations are formed adaptively; i.e. by

s e   (s  s)  0
(7.2)
 Fig 7.2 unanticipated money supply increase
in the Dornbusch model

Monetary policy had no real effects in the monetary model. In this model it
has short-run effects. Suppose M0 increases to M1…
 Long-run effect
• In the long-run, r=r* (so there’s no expectation of ER
change). Therefore, P must increase to offset the
increase in (nominal) M; and restore IS and LM
curves to their old positions at point A
– With equilibrium at point J on the RP (UIRP) line
• For P to rise, AD must have increased to keep output
at its long-run value, D
• This requires the ER to have depreciated to point G
(to ensure current a/c has a zero balance)
 Short-run effect
• LM curve shifts downwards and r falls (the liquidity
effect).
• To prevent money flooding out of the UK there’s an
instant depreciation so the £ is considered
undervalued relative to its long-run equilibrium and
there’s an expected appreciation
• This downward effect on r is mitigated somewhat by
rightward shift of the IS curve (as a falling s and
constant P increase competitiveness) and output
increases to point e and to temporary equilibrium at
k, given the short-run interest rate differential
Summary: ER overshooting
• The immediate effect of the monetary expansion is
to cause the ER to move from S0 beyond its long-run
value of S1 to S2, because prices are fixed
• The £ overshoots, with an ensuing exaggerated
effect on UK competitiveness
• Only in the long-run do prices increase (via
successively steeper AS curves) and the £ falls in
proportion to the increase in M, keeping the real ER
constant and shifting the IS and LM curves back to
their original positions
 Portfolio Balance
• Again we assume that financial markets adjust
quickly while real ones are sticky
• However now we assume that UIRP does not hold
due to large risk premia effects; which because
investors are risk averse do matter - and means UIRP
does not hold (unlike in Dornbusch model)
• We then have to think of an internationally
diversified portfolio with risk averse investors
– Investors diversify their holdings of risky assets;
with portfolio shares increasing in their returns
– i.e. assets are not perfect substitutes, esp. when
denominated in different currencies
 10% increase in UK money supply
• In long run might expect 10% increase in P and S
• But in Portfolio Balance model S depreciates <10%
• So Q (real ER) is lower; and UK more competitive
• This is because we’ve seen
– S↑ due to OMO (Case 1). But as P is fixed, as in Dornbusch,
there is a real depreciation (overshooting) too
– So is current a/c surplus and build-up in foreign current
assets. But, from Case 3, this build-up implies an ER
appreciation (reversing the initial depreciation). As time
passes P also ↑ helping restore real ER to where it was and
beyond as real ER continues to appreciate
– Although net exports=0, there’s a current account surplus
equal to investment income on these foreign stocks
Currency substitution: introduction
• This is about money (currency, not exchange rates),
which is a non-interest bearing store of value
• The question is, should you hold domestic or foreign
currency - and what determines this and the
proportions in which you hold the two currencies?
– e.g. the use of dollars in Russia is an extreme case
• Proportions hold depend on the rate of currency
depreciation
– Minor modification to portfolio balance model
with just two assets
• So again current account imbalances drive the flows
Currency substitution: intuition
• The stock of FX changes, via currency account
surpluses/deficits only
• Similar to portfolio balance model: instantaneous
equilibrium in the financial sector and a flow excess
supply/demand in the real sector changing the stock of
FX in the adjustment period (while prices adjust)
• But we now consider both traded (with prices
determined in world markets) and non-traded goods
(with prices determined nationally) in the real sector
→ Changes in the real ER affect the value of wealth
both via a real balance effect and in the LR due to a
change in the terms of trade on the current account
 Effects of a monetary acceleration
• Currency depreciation at a slower rate than domestic inflation,
hence an appreciating ER
• A current account surplus on a decreasing scale as the economy
adjusts
• A consequent rise in foreign currency balances and wealth
• Contrasts portfolio balance, as we now have a nominal depreciation
with a current account surplus
– Due to currency substitution per se. Higher inflation leads to an
increasing proportion of wealth held in foreign assets, which
requires a short-run a/c surplus
– Amounts to hedging against domestic inflation; dollar price of
traded goods is fixed, so by accumulating FX you’re gaining the
right to buy these goods in the future
General Equilibrium Models
• Earlier models may be too simple: aggregated and
partial equilibrium models
• Now we consider a model with micro foundations
and a complete description of all markets
– Consumers maximise utility; producers maximise
profits
– Consider implications for the ER
• Also we no longer make the small country
assumption. So what happens in the domestic
country affects the RoW
Redux model of Obstfeld and Rogoff
• Current account deficit tells us whether a country’s
international net asset position is changing
• Equally, the change in its wealth is its national saving
or dissaving
• Central to Redux model; implies household choice
between consumption and saving is the determinant
of the current account and the ER
• Model is far more complex. We’ll just sketch it and
give you an idea of how the model works
Optimum currency areas and monetary
union
• Costs and benefits
• ER cannot depreciate in response to a demand shock
(say P* falls relative to P) under a fixed ER. Onus of
adjustment falls on the domestic price level (and this
leads to more painful short-run changes in
unemployment etc.)
What affects how smooth this adjustment is?
• Price/wage flexibility critical
• Labour market important: either the wage or
employment changes after the shock
• With floating rates, fall in demand causes depreciation
and a (domestic, traded) price level rise, reducing real
wages, as required to reinstate equilibrium (“full
employment”), without the need for a reduction in
money wages
• With fixed rates, if money wages are not fully flexible
(and don’t fall by 10% too), real wages fail to adjust
(they rise due to the fall in foreign prices) →
unemployment, until nominal wages fall by 10%
• If labour is mobile, under fixed rates there will also be
no unemployment (Mundell, 1961 AER)
Market Efficiency and Rational
Expectations
• An efficient market is where prices fully reflect all
available information
• An efficient market exploits all profit opportunities
• The problem of assessing market efficiency is that it
is closely linked to expectations formation
12.2 Rational Expectations Hypothesis
(REH)
• Problem: agents’ expectations are key to financial
market behaviour. But how does a 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 subjective expectation of the future
value, xt+1
Q. How is xt+1e formed?
REH answer: a rational economic agent uses all information
available at t in best possible way, so:
xte1  E ( xt 1 | I t )
LHS is agent’s subjective expectation
RHS is (statistical, objective) expectation of xt+1 conditional on
information set, It
We often use abbreviated notation:
xte1  E ( xt 1 I t )  Et xt 1
where Et means expectation conditional on information at t
Definition: Agents hold RE if their subjective expectation is
the same as the (mathematical) expected value,
conditional on all publicly available relevant information
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:

ft t 1
 Et st 1  t (12.2)

which is efficient market equilibrium as the forward rate reflects

- publicly available information summarised in the RE, Et st 1
- market’s attitude to risk, as embodied in the risk premium, t .
Consider implications of (12.2) by subtracting from both LHS and
RHS
• That is, under non RW processes the return will be
non-zero at particular points in time but will be zero
on average
• For example, if the spot rate is generated as in
(12.10); subtracting equation (12.11) from (12.10):

st  Et 1 st   (Z t  Et 1 Z t )  u t

• we see that the expected return can be non-zero but

will be zero on average as, under RE (and quadratic
loss), the error made in forecasting Z is also zero-
mean and random
12.3 Market Efficiency (continued)
Rewrite Equation (12.2) by subtracting s t 1 from both sides:

f tt 1  st 1  [ Et st 1  st 1 ]   t  u t 1   t (12.3)

Equation (12.3) implies:

st 1    t  f tt 1  u t 1 (12.4)

Alternatively, stepping back one period:

st    t 1  f tt1  u t (12.4’)

Conclusion: the spot rate is the sum of three components:

1. The risk premium (which in general is unknown, so how do we estimate (12.4)?)
2. The previous period’s forward rate
3. An unpredictable zero-mean expectational error
Weak and strong rationality
• A weakly efficient market is one which reflects all the
information in its own past history
– Weak rationality implies subjective expectations
are given by xte1  E ( xt 1 xt , xt 1 , xt 2 ,...)
• A strongly efficient market reflects all publicly
available information
– Strong rationality implies subjective expectations
are given by xte1  E( xt 1 It , It 1 , It 2 ,...)
12.4 Unbiasedness…
… is 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

st  ft t1  ut (12.5)

Rewriting (12.5) in terms of rate of depreciation:

st  st 1  ( f tt1  st 1 )  u t (12.5’)

→ actual rate of change of ER is equal to the expected one, plus or minus a

random error
 Efficiency and the random walk
From (12.2) forward market efficiency for a RW implies:

f tt 1  st   t (12.12)
But when the spot rate is generated by (12.10)
efficiency requires that the forward rate is given by
(12.11) plus any risk premium
Conclusion: a random walk is neither necessary nor
sufficient for forward market efficiency
Processes other than a RW do not imply an opportunity
for profit, even though only under a RW is expected
change (return from holding FX) exactly equal to
zero
 Testing for efficiency: problems
• Neither RE nor market efficiency is usually
testable on its own

• We almost always to test a joint hypothesis:

market efficiency AND rational expectations
– Think back to (12.1): t  E (st It 1 )
e
s
e
– Even if we had subjective expectations data, t s
we’d still need to specify a model for st from
which to derive the objective conditional E
 Testing rationality and efficiency…
• So if the subjective expectations differed from the
objectives ones, we would not know whether this was
due to irrationality or use of the incorrect forecasting
model for st
• But as long as we have data on the subjective
expectations we can test RE by exploiting relationship
between any variable yt and its expectation under RE
E ( yt It 1 )  yt   t
• where the error Ɛt is iid mean zero
• To test efficiency requires an assumption on the risk
premium, t
12.8 Results

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

st  a  bft t1  vt (12.15)

1. estimate of the intercept a

- insignificantly different from zero?
2. estimate of the slope coefficient b
- insignificantly different from unity?
3. vt serially uncorrelated?
 News
• Examine the factors affecting the forward premium
and its apparent gap with the future spot rate
• In particular, we consider the random component in
the premium, ut, which are the forecasting errors
– Under RE these errors must be due to ‘news’ – the
arrival of new information
st    t 1  f tt1  u t (12.4’)
• The problem here is that, by definition, ‘news’ is
unexpected
– e.g. the latest UK trade balance data are not
‘news’, it is the deviation from what people
expected (for the trade balance) which is news
13.1 The ‘News’ Model: a simple example

Suppose the (log of the) spot exchange rate is given by:

st  zt (13.1)

where  is a slope coefficient and zt are the “fundamentals”

Assuming RE,

Et 1st  Et 1 zt (13.2)

Conclusion:
A (rational) expectation of the exchange rate involves
making a forecast of the fundamentals
13.1 The ‘News’ Model: a simple example (continued)
For the forecast errors, subtract Equation (13.2) from Equation (13.1):

st  Et 1st   ( zt  Et 1 zt ) (13.3)

* left-hand side - unexpected component of (“innovation in”) the spot rate

* RHS term in brackets - news (the surprise component or innovation in)
zt

Writing (13.3) in terms of the forward rate:

st  Et 1st   ( zt  Et 1 zt ) ft t1  t 1   ( zt  Et 1 zt ) (13.4)

since from (12.2) ft t 1  Et st 1  t

* So, the expectational error is written in terms of “news” regarding

fundamentals – the unforecastable component of the fundamentals
 13.2 The monetary model revisited - with
endogenous not exogenous interest rates

The monetary model takes the form:

~  c~
st  m y  b~
rt (13.5)
t t

where m~ - (log of) relative money stocks

t
~y - (log of) relative income
t
~
r - interest rate differential
t

Allowing for risk aversion, replace UIRP (which holds under neutrality only) by:
~
rt  rt  rt*  ste   t (13.6)

where  t - currency risk premium

13.2 The monetary model revisited (continued)

After N substitutions:

st   (1  b) 1 ( z t  Et z t 1   2 Et z t  2     N Et z t  N )   N 1 Et st  N 1

(13.14)
Remembering that  0  1 , we can rewrite (13.14):

 N

st   (1  b) 1 z t  k 1  k Et z t  k   (1  b) 1 k 0  k Et z t  k
N
(13.15)

Conclusion: the value of the ER at time t is determined by market

perceptions of the entire future path of the fundamentals,
starting with zt and ending with zt+N
 13.2 The monetary model revisited (continued)

What will be the value of Et 1 s t ?

Et 1 st   (1  b) 1 ( Et 1 z t  Et 1 z t 1     N Et 1 z t  N )   N 1 Et st  N 1

  (1  b) 1 k 0  k Et 1 z t  k
N
(13.16)

Subtracting (13.16) from (13.15):

st  Et 1 st   (1  b) 1 k 0  k ( Et z t  k  Et 1 z t  k )
N
(13.17)

st  Et 1st unexpected change in (news about) exchange rate

( Et zt  k  Et 1 zt  k ) unexpected change in (news about) fundamental, zt+k
= revision in expectations between t-1 and t
These revisions must be unpredictable under RE
 13.3 Testing the ‘news’
We need to know three things to test the ‘news’ model:
1. What is the expected exchange rate? How do we
measure expectations?
1. Most researchers use the forward rate; but replies
on the risk premium being zero or at least constant
2. What are the fundamentals?
3. What is the expected value for the fundamentals?
1. Most work has focused on this: many approaches
tried…
All three are very difficult to get right
Results
• Objective is to explain unexpected ER movements
– which we know is very volatile
• Work carried out since 1981
• Broad consensus that there is some explanatory
power from news: 5% to 20% of ER variation
• Conclusions are not, in fact, very sensitive to the way
news calculated (univariate, VAR model etc.)
• Interest differential seems most important
• However, news only explains a small part of exchange
rate movements (irrational expectations perhaps?)
13.5 Volatility tests and bubbles
• One explanation for the volatility may be rational bubbles,
rather than irrationality
• A rational bubble is where an asset price moves away from
its fundamental because of self fulfilling expectations
– You are willing to hold an asset which you know is overvalued
according to the fundamentals because the expected gain of
continuing to hold is > risk of the asset falling in value
– Higher the probability the bubble will burst (i.e. the more
overvalued the asset) the more rapidly it must rise to
compensate for the increased risk
– Mathematically, we can see this by noting that (13.15) is not a
unique solution to the RE model
Peso problem
• This is where there is a small possibility of a large
event occurring, especially an event which is not
repeated regularly and so can not be captured by a
normal fundamental variable
– e.g. the fall of a government, entry to the Euro,
nuclear war, Cuban missile crisis
• In other words, there is a lot more going on out
there than we can measure
Risk Premium
• Spoken a lot about it, but what actually determines
the risk premium?

• Define risk aversion, risk neutrality, risk premium

• All about constrained optimisation under uncertainty

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

Figure 15.1 Speculator’s Equilibrium

Expected consumption is μ and risk (the s.d. of consumption) is σ. You have
no risk when you have no forward contracts and your C is μ0. Indifference
curves are upward sloping. Aim is to get onto as high an indifference curve (I2,
rather than I1 rather than I0) as possible, s.t. the constraint
 15.5 Conclusion
• Risk premia emerge as a natural consequence of risk
aversion
– Risk premium is determined by the parameters of the
probability distribution of the future ER, attitudes to
risk and the quantity of assets in existence
• But risk premia seem to be too small to explain rejections
of efficiency (and unbiasedness)
• Either we face non-rational behaviour or market
inefficiency
16.1 Nonlinearity and Chaos
• Uncertainty and unpredictability are not the same
thing
• Unpredictability may come about from a certain
non-linear structure
– Even though there is no uncertainty
– We will explain this apparent paradox in the
context of the ER
– Technical literature; we’ll sketch the results
• This chapter looks at the possibilities for chaos in
financial markets
 16.1 Deterministic and Stochastic models
• Consider a conventional stochastic model
St = a0 + a1St-1 + a2Zt + ut
• To forecast with certainty we would need to know
– the parameters
– the values of the variables
– the value of the errors
• If all these were known we would have a deterministic
model
– Deterministic models contain no random terms ut
– In the presence of ut (‘news’) the future is uncertain
• Best we can do is take the conditional expectation:
• E(St )= a0 + a1St-1 + a2E(Zt )
• The bigger the variance of ut the more uncertain this is
Nonlinearity
• But we can have deterministic models (with no
stochastic error) which, in principle, are
unforecastable
• These models are nonlinear
16.2 An example of a simple non-linear model

Assume the (log of the) exchange rate change

- is proportional to the previous period’s gap between the actual
exchange rate and its long-run equilibrium level: i.e.,

st   (s  st 1 ) (16.1)

where  is an indicator of the speed of adjustment of the actual rate to

deviations from its equilibrium level
Suppose  is not simply a scalar (as before) but an increasing function of st
perhaps as exporters convert their foreign money more quickly when the
domestic ER is depreciating

  st  0 (16.2)
 Target Zones and Smooth Pasting
• In practice exchange rates are rarely fixed
– a target zone is much more realistic, whereby the
currency floats between an upper and lower bound
– To what extent does this restore monetary
independence to the central bank, lost with a fixed ER?
• We will discuss how exchange rates must be expected to
move smoothly (rather than experience discrete
changes), otherwise there is an opportunity for infinite
arbitrage to exploit the step-change in st
 Q. What happens when exchange rate is
allowed to float over a 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
17.3 Smooth Pasting
• The exchange rate must be expected to never quite
reach the upper and lower bounds
• These conditions are satisfied by the S curve in Fig.
17.3
• As the ER approaches the floor or ceiling it does so
smoothly
• This is called smooth pasting
 kt

Figure 17.3 Target zone – the S curve

Honeymoon?
• Inside the TZ the ER is not as weak (strong) as it
would be under a pure float –the effect of the
fundamentals on the ER is dampened
• i.e. the ER is less volatile than under a pure float
– Honeymoon effect is greater the closer to the
bounds the ER is. The ER is unaffected by the
fundamentals as it reaches the boundary: it
approaches very slowly (smooth pasting)
• So the TZ is stabilising
• And frees up policymakers to target other variables,
not exclusively the ER
 Currency crises
• We are again interested in fixed exchange rate
regimes
• But in how they collapse, rather than how they work
– They tend to collapse as a country’s FX reserves
are exhausted
• Real world markets are periodically hit by crises
• Why do they happen?
– And why does the end come with a bang?
– Why do the reserves not decline smoothly to
zero?
Investors’ actions are critical
• To understand crises we use a model which explains
crises by the actions of traders who speculate
against the currency in order to benefit from an
anticipated devaluation
– Leads to a reserves crisis but not a step
devaluation
Crisis Models
Why/when do fixed exchange rate regimes collapse?
• First-generation models: inconsistency (with fixed
exchange rate regime) of monetary policy (Krugman
(1979))
• Second generation model: credibility (i.e. perceived
inconsistency (with fixed exchange rate regime) of
monetary policy (Krugman (1996), Obstfeld (1996))
• Third generation model: structural imbalances and
perverse incentives (Asian crisis 1997-8)
 Conclusion
We started with a model which said crises were
caused by fundamental imbalances

Then we considered that a ER crisis might occur

largely because of speculation and multiple
equilibria

We ended with a model of banking crises, coming

from a failure in confidence