5 Volatility tests, bubbles and the peso problem
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A world of uncertainty
which is a condition satisfied by a number of commonly encountered economic series,19 it
can be shown that Equation 12.22 satisfies the original difference equation (Equation 12.11)
every bit as well as the pure market fundamentals solution does.20
The bubble term, Bt, can be defined as simply the extent of the deviation from the market
fundamental equilibrium. Unfortunately, the theory has nothing to say about how or why a
bubble develops. There are two ways of looking at the role it plays. First, it is sufficient that
agents perceive the bubble factor to be important for it actually to be important – if it figures
in their subjective model, then it will find its way into the objective model driving the
exchange rate. Second, we could be said to be dealing with a variable that is unobservable
to economists but observable to market agents.21
At any point in time, there must be some perceived probability that the bubble will
burst next period. If it does burst, then the exchange rate will return to the level dictated
by the fundamentals. Otherwise, its movements will continue to reflect the behaviour of
the bubble.
The simplest model would take the following form. Suppose the probability that the
bubble, Bt, will last another period is Π and the probability it will burst is (1 − Π). We then
have the following possible outcomes for the next period, t + 1:
Bt+1 = (βΠ)−1Bt with probability Π
= 0 with probability (1 − Π)
The reader can easily verify that this is consistent with the restriction in Equation 12.23.
In this simple case, as long as the bubble persists, the exchange rate will appreciate
sufficiently to compensate a risk neutral currency holder for the possibility of loss when the
bubble bursts. What this means is that in addition to any rise justified by changes in the
fundamentals, there will need to be an explosive bubble superimposed, because the greater
the current divergence from equilibrium, the further the currency has left to fall, and hence
the greater the prospective capital gain needs to be if the process is to be sustained.
Notice that this is more a detailed description of the phenomenon than an explanation.
The literature has nothing to say about what causes bubbles to start or end.22 On the basis
of casual observation, it seems that some bubbles (or apparent bubbles) are triggered by
movements in the fundamentals, actual or perceived. Others seem to be spontaneous. In
this respect, the theory offers little enlightenment since it treats the bubble as simply a fact
of life, exogenous not only to the behaviour of the market but also to the fundamentals
themselves. Indeed, as we can see from Equation 12.22, as long as the bubble persists, it will
cause the exchange rate to move, even when the fundamentals are unchanged.23
How will the presence of bubbles show up in the data? Obviously, the effect will be
to weaken the overall link between the exchange rate and the fundamentals, even supposing that one can identify them. In terms of RE models, it can be shown that ignoring
the bubble term in standard econometric work may well produce results that apparently
contradict rationality – intuitively because a persistent positive bubble will generate a series
of underpredictions in forecasts based on the fundamentals, and vice versa for negative
bubbles.
The presence of bubbles could possibly account for the failure of ‘news’ models to explain
the variability of exchange rates. Indirect support for the view that bubbles have been at
work in currency markets can be found in formal comparisons of the variability of the
exchange rate, on the one hand, and the fundamentals, on the other, excess volatility or
variance bounds tests, as they are called.
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The ‘news’ model, exchange rate volatility and forecasting
To understand how these tests are formulated, compare the following equation with
Equation 12.17:
N
s*t = γ (1 + b)−1 ∑ βkzt+k
(12.17′)
k = 0
The right-hand side is simply Equation 12.15, with actual values of zt+k replacing expected
values. In other words, s*t is the level at which the exchange rate would settle if the market
participants had perfect foresight. Since they cannot know the future, and have to rely on
expectations, the actual exchange rate is given, assuming rationality, by:
st = Et s*t
(12.24)
which is simply a tidy way of rewriting Equation 12.17, using our newly introduced definition of the perfect foresight exchange rate, s*t .
Now the only reason why s*t differs from its expected value, st, is because of ‘news’ about
future fundamentals. Since, under RE, ‘news’ items are all random, their total impact, as
measured by the weighted sum on the right-hand side of Equation 12.17, must equally be
random. It follows that we can write:
s*t = Et s*t + ut
= s t + u t
(12.25)
where, as we saw in Chapter 11, ut not only has an average value of zero but also is
uncorrelated with the expected value itself. It other words, if we take the variances24 of the
right- and left-hand sides of Equation 12.25, we get:
var(s*t ) = var(st) + var(ut)
(12.26)
since we know that the covariance between st and ut must be zero. As variances are always
positive, this implies the following inequality test:
var(s*t ) ≥ var(st)
(12.27)
which states that, under RE, the variance of the actual exchange rate must be no greater
than the variance of the fundamentals that drive it.
Notice that this test relies on the obvious point that, although we can never have perfect
foresight at any time, t, so we can never know the true current value of s*t , we do have access
to it retrospectively, at time t + N. By then, as researchers, we can have the benefit of perfect
hindsight, which allows us to know past values of s*t because, looking back, we know what
actually happened to the fundamentals.25
Note also that the variance bounds test is very general, in so far as it places no restriction
on the variables to be included among the market fundamentals. In other words, it can serve
as a test of almost any model which defines the set of fundamentals relevant to determining
the exchange rate.
For technical reasons, many researchers tested slightly modified versions of Equation
12.27. In any case, the result was a near-unanimous conclusion: exchange rate volatility is
excessive relative to the volatility of the fundamentals. In other words, the right-hand side of
Equation 12.27 is unambiguously greater than the left.
To put this conclusion in perspective, it is worth pointing out that variance bounds tests
were originally used in research on stock market prices and dividends, with similar results:
share price volatility is too great to be consistent with the degree of variation in dividends.
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Notice the crucial distinction between tests in these two cases. In the case of stock prices,
we can identify the fundamentals with some confidence. Under these circumstances, inequality
tests not only are more informative but also can legitimately be interpreted as tests of stock
market efficiency.
In the case of the exchange rate, we can have far less confidence in our choice of fundamental variables. It follows that these tests can only throw light on the question of how
far the facts are consistent with the view that exchange rates are (1) driven by a particular
set of fundamentals and (2) not subject to bubbles. They are not very illuminating on the
more general question of spot market efficiency.
There are a number of possible explanations of the apparent excess volatility of the major
exchange rates: currency markets are irrational, or there are significant (rational) bubbles
or important and highly volatile fundamental variables have been omitted. The latter
explanation seems improbable, at least in so far as it relates to economic variables. The only
variables that fluctuate anywhere near enough to overturn the main findings are financial
series, such as stock market prices, which are themselves excessively volatile, as has already
been mentioned.26
However, at least one other conceivable explanation has been offered: the so-called
peso problem.27 This relates to the inherent difficulties that arise in sampling economic
events, which are by their very nature once-and-for-all experiments, incapable of ever being
replicated. To see what is involved, take an example from 1999, the year the euro was
launched.
At the time, Britain was facing mounting pressure from its EU neighbours (not to
mention all the usual suspects inside the country) to join the eurozone.28 Under those
circumstances, it would have been quite rational for any trader currency market to allow for
the possibility that at some stage in the future, a British government would succumb to that
pressure and agree to join the eurozone. Suppose the market believed that accession, if and
when it occurred, would be at the rate of £1.00 = €1.50.
Clearly, the event would have fed into expectations in 1999 and succeeding years. It may
be the case, for example, that the market – quite rationally – attached only a tiny probability
of, say, 0.01 to the event of British accession within twelve months, but with the probability
rising to 10% for accession within two years, 40% within three years, and so on.
If this was actually the case, it implies that the spot rate of £1.00 = €1.56 actually
observed at the time in 1999 was lower than it otherwise would have been because of
the probability, however small, that sterling would be devalued to the accession level
of £1.00 = €1.50 during the holding period of the currency trader. Moreover, this effect
would have been felt in the forward market too, where the premium or discount would have
been less favourable to the pound than seemed warranted on the basis of the observable
fundamentals alone.
As can be readily seen, in this sort of situation, models based on RE will seem to fail. Back
in 1999, the otherwise correct model (that is, the one that appeared to fit the facts before the
inception of the eurozone) would have appeared to overestimate the value of the pound.
There are two ways of looking at the reason for the apparent breakdown of the relationship
between the fundamentals and the exchange rate in this type of scenario.
The first would be to say that there was a variable omitted from the list of market fundamentals: the probability of a once-and-for-all discrete change in the currency regime. The
omission was not simply an oversight. Unfulfilled possibilities are inherently difficult to
measure – but that certainly does not mean that they are unimportant. Indeed, it may be the
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The ‘news’ model, exchange rate volatility and forecasting
case that many of the apparent departures from rationality are in reality due to unexpected
shifts in unobservable or immeasurable variables.
The second interpretation is that the peso problem can also be viewed as a sampling
problem. In principle, just as we should expect to find approximately 50 heads in 100 tosses
of a coin, so we should expect to find one example of an event like eurozone accession in a
100-year sample if, as we suppose, the probability of its occurring is really 0.01. However,
if our sample period is only 50 years, then its frequency in our sample may well be zero,
leading us, mistakenly, to discount the event altogether as impossible. In fact, even if we
have a hundred years of data, there is more than a one in three chance of us never observing
a single occurrence.
Indeed, the event may never occur. But that fact alone does not mean that it was
irrational to take the possibility into account – any more than we can say a man who survives a game of Russian roulette was irrational to make out his will before starting to play.
Note that there are similarities between the peso problem and the phenomenon of
rational bubbles. Both, it must be emphasised, are entirely consistent with rational beha
viour displayed by individual market participants. The difference is that whereas bubbles
represent a divergence from the equilibrium associated with the market fundamentals,
the peso problem arises out of the small probability of a large, discrete shift in the value
of one or more of the fundamental variables themselves.
As a consequence, it follows that the interpretation of variance bounds tests is or, rather,
could be different in the case of the peso problem from what it would be in the presence of
bubbles. As we saw, the common finding that the volatility of exchange rates is greater than
that of the fundamentals could well be indicative of the presence of price bubbles. However,
if over some part of our data period there were a widespread feeling that a major shift in the
fundamentals could not be ruled out, then it will follow that our estimate of the variance
of the fundamentals will be an underestimate of the market’s (rational) perception of the
variance at the time.
This possibility may well invalidate our conclusion altogether: the fact that the variance
bounds are breached could be a result of neither irrationality nor bubbles but simply of
the failure to take account of a significant event to which the market allocated a non-zero
probability, even if it never subsequently materialised.
The peso problem is a particularly intractable obstacle to research, at least where standard econometric methods are concerned. The most hopeful approach would seem to be
direct measurement of market expectations, although the relevant data have only recently
become available and, as was mentioned in Chapter 11, survey data present researchers
with a new set of methodological problems. Alternatively, there is one other way in which
we could directly observe the market’s assessment of exchange rate volatility: by extracting
implied variances from traded options data.
While the idea of variance bounds testing helped to focus attention on volatility as an
important aspect of the failure of exchange rate determination models, research tailed off
fairly quickly, as it became clear that this approach could tell us no more than the standard
regressions – in fact, the two could be shown to be more or less equivalent for all practical
purposes.
The rational bubbles model failed even more comprehensively for a number of reasons.
First, it was shown that the mathematics of rational bubbles had some very unattractive
features. For example, bubbles could only ever be positive, otherwise they would imply that
agents rationally expected negative prices. This made it hard to imagine an exchange rate
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A world of uncertainty
bubble – after all, if the value of a currency is 25% too high, the value of the other currencies
must be 25% too low. Secondly, once burst, a bubble could never re-inflate, which seemed
to conflict with historical experience in equities, for example. Thirdly, direct empirical testing on the whole did not support the idea of rational bubbles in any market. The outcome
has been that, while bubbles are as hot a topic as ever, they are more or less unanimously
viewed as reflecting some form of irrational fad, frenzy or mania.
12.6
Conclusions
In a sense, the conclusions reached in this chapter have been negative: the ‘news’ approach
can explain some of the variation in exchange rates but is ultimately defeated by their sheer
volatility. The impression that the task is hopeless is reinforced by direct comparisons of
the variance of exchange rates with the variance of the elements that are supposed to
explain it.
Moreover, the Meese and Rogoff results have dealt a body blow to traditional exchange
rate models. Since their conclusion that there is no link between the exchange rate and
fundamentals has proved more or less unshakeable under the standard assumptions of
risk-neutrality, rationality and homogeneous information, subsequent chapters will examine
whether relaxing these assumptions can provide a way out of the impasse.
The next chapter will deal with models of the risk premium, and Chapters 17 and 18
will explicitly address the question of what happens when we allow for the highly realistic
possibility that some agents in the currency market are better informed than others.
Summary
●
The ‘news’ approach involves relating unexpected movements in the exchange rate
to revisions in the market’s rational forecast of the fundamental variables. In a simple
RE model this relationship will be the same as the one between the level of the
exchange rate and the level of the fundamentals.
●
In general, the value of a currency will depend crucially on the prospective capital
gains or losses (that is, the expected rate of appreciation or depreciation) that the
market expects to see accruing to holders.
●
When the value of a currency depends on its expected rate of change as well as on
the market fundamentals, its behaviour will follow a difference equation, the solution
of which relates the current level of the exchange rate to a weighted sum of current
and expected future values of the fundamental variables. The weights will decline
geometrically as we go forward in time, starting with a weight of unity on the current
value of the fundamentals. Furthermore, the identical relationship will hold between
the unexpected component of the exchange rate and the innovations (or ‘news’) in the
fundamentals – the formulation known as the ‘news’ approach.
●
There are a number of problems to be overcome in making the ‘news’ model operational, notably the problem of how to estimate the ‘news’ variables themselves. Most
➨
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The ‘news’ model, exchange rate volatility and forecasting
published work has used univariate or multivariate time series or econometric methods
in order to model the fundamentals, taking the residuals as estimates of the ‘news’.
●
The results of implementing the ‘news’ approach are mixed. On the positive side, the
equations tend to fit reasonably well, frequently explaining a non-negligible proportion
of the substantial degree of exchange rate fluctuation left undiscounted by the forward
premium. On the negative side, the volatility of (unanticipated) exchange rate movements remains largely unexplained by the ‘news’.
●
Under RE, it ought to be the case that the variance of the observed spot rate is
no greater than that of the fundamentals determine it. Most tests suggest that the
opposite is the case. In other words, the volatility of the exchange rate is greater than
can be rationalised by reference to the market fundamentals.
●
One possible explanation of the observed volatility could be rational price bubbles,
which are said to occur when a gap opens up between the level of the spot rate and
its equilibrium value, as determined by the fundamentals. Another explanation could
be that agents attached a (possibly small) probability to the chance of a discrete, step
change in the fundamentals, in which case the apparent, measured variance taken
from the data may be an underestimate of the variance as actually perceived by
rational traders (the ‘peso problem’).
Reading guide
The paper that started the ‘news’ literature was Frenkel (1981), though the approach was derived in
some respects from one already well established in closed economy macroeconomics (for example,
Barro 1977). Both papers are relatively non-technical. Other influential papers have been by
Edwards (1983) and Hartley (1983). Among the few recent papers on this topic, see Fatum and
Scholnick (2007).
A fairly technical survey is to be found in MacDonald and Taylor (1989), which gives a particularly
good treatment of the subject matter of Section 12.5. For a discussion of some of the outstanding
issues in this area, particularly the estimation problems, see Copeland (1989) and references therein.
Note that the literature on the simple ‘news’ model petered out as the implications of Meese and Rogoff
sank in, and in a sense it dissolved into other branches of the literature, especially those covered in
Chapters 15, 17 and 18 of this book.
The peso problem was first discussed by Krasker (1980). Most of the original work on variance bounds
focused on stock markets (for example, Grossman and Shiller, 1981), although researchers have
tended to develop specialised versions of the tests to deal with exchange rates (for example, Meese
and Singleton, 1983). For dissenting views on volatility see West (1987) on the Deutschmark–dollar
rate and Honohan and Peruga (1986), who claim the bounds are breached only if PPP is imposed.
Many of the theories covered in the second half of this book have been imported from the stock market
literature. The peso problem is the only example of an exchange rate model being exported. Almost
all the recent work in this spirit has been on stock markets, directed at solving the so-called equity
premium puzzle (see, for example, Barro (2006) and subsequent work by the same author).
Both bubbles and the peso problem are discussed briefly and in non-technical fashion by Dornbusch
(1982). Two important early papers on the theory of bubbles (and the related question of collapsing
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fixed rate regimes) are to be found in Chapters 10 and 11 of Wachtel (1982), written by Flood and
Garber and Blanchard and Watson. The paper mentioned at the end of the chapter which examined
the unattractive mathematical properties of bubbles was by Diba and Grossman (1988).
As far as empirical work is concerned Evans (1986) is of interest, claiming to find a bubble in the
returns to sterling holders in the early 1980s. Meese (1986) is accessible only to readers with a
background in time-series econometrics. A good survey of the bubble literature can be found in
Brunnermeier and Oehmke (2012).
For a historical perspective on bubbles in financial markets, see Kindleberger (1978) and Reinhart and
Rogoff (2009), both of which incorporate high-quality scholarship in an entertaining history of
economic folly through the ages.
Notes
1 The fact that ut in Equation 11.4′ appears with a negative sign can be ignored, of course, since the
positive sign that it was given in Equation 11.3 was purely arbitrary in any case.
2 Also often called ‘surprises’ and ‘innovations’.
3 Realistically, zt will be a vector of values of the fundamental variables at t, possibly with a 1 as the first
element, and γ a coefficient vector, so that Equation 12.1 relates the exchange rate to a constant plus a
linear combination of a number of fundamentals.
4 The expression comes from stock exchange jargon, where it has a long pedigree.
5 As we shall see in Section 12.5, this solution is not unique.
6 Making use of the fact that Et−1(Et zt) = Et−1zt, the Law of Iterated Expectations again.
7 Almost all published work has involved linear time series, although nonlinear models would appear a
promising avenue for future exploration.
8 The example given here is known as pth order autoregressive. Along the same lines it is possible to
include among the regressors on the right-hand side lagged values of the residual: vt−1, vt−2, . . . vt−q,
making the model autoregressive (of order p)-moving average (of order q), or ARMA(p, q) for short.
Although the statistical properties are changed somewhat by this extension, its implications for RE are
identical.
9 It should be noted that, in principle at least, this modelling procedure needs to be repeated for each
exchange rate observation in the dataset. In other words, if we are trying to explain monthly exchange
rate movements starting, say, in January 1980, then our ‘news’ variables for the first month will need to
have been conditioned on data from the 1970s. For obvious reasons, market anticipations cannot have
been predicated on information dated later than January 1980. By the same token, by February more
information will have arrived and there is no reason why market agents could not have updated their
forecasting model. Hence, a new VAR with newly estimated parameters ought to be fitted, and so on,
throughout the data period.
Although this updating approach is undoubtedly correct, it is not only tedious, even with the latest
computer technology, but it also appears to yield results that are little if any improvement on a VAR
estimated once over the entire period. A cynic might say that this is because the relationships between
the variables are in any case so weak. More positively, if there really is a stable relationship between
the (current) fundamentals and their past values – if they are ‘stationary’, in the jargon of time-series
analysis – then the distinction will not matter.
10 See Reading guide for references.
11 This approach was actually pioneered by researchers looking at the relationship between interest rates
and the money stock in the context of domestic macroeconomic policy.
12 Although market commentary often produces, at first sight, convincing evidence of inconsistent beha
viour in the two markets – for example, the episodes in spring 1988 when the pound’s international value
rose on the perception in currency markets that UK fiscal, and presumably monetary policy, was tight,
while share prices fell in London, apparently on fears that monetary policy was far too loose.
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The ‘news’ model, exchange rate volatility and forecasting
13 It has already been noted that the ‘news’ approach is consistent with many possible models of the
exchange rate. The monetary model was selected in Section 12.2 as a convenient example only. If,
instead of tying ourselves down to a particular model relating the exchange rate to the fundamentals, we
take a more agnostic view, then we have no reason, a priori, to expect the signs on the ‘news’ variables
to be positive or negative.
14 The contrast is between the extent to which rates vary over the business day and the variance overnight
– that is, between closing rates in the evening and the opening rate on the following day (or the next
Monday, for a weekend). Thus, if ‘news’ arrives evenly over 24 hours and the exchange is open, say,
for 12 hours, then one would expect to find the variance of the overnight change equal to that of the
opening-to-closing change.
15 Also known as ‘bootstraps’, ‘sunspots’ and ‘will o’ the wisp’ equilibria. Note that this is another case where
research on exchange rates has accompanied research on other financial assets, particularly share prices.
16 See Section 2.5.
17 In other words, the critical question facing the investor in this type of situation is not the direction of the
next major price movement, but its timing. As this author found out to his cost in 1985, even when you
are right in judging a currency to be mispriced, you can still lose money if it continues to be mispriced for
longer than you expected.
18 History records a number of spectacular events that were regarded as bubbles either at the time or fairly
soon after they burst, for example the Dutch Tulip Bubble, the Mississippi Bubble, the South Sea Bubble
in the seventeenth and eighteenth centuries and, more debatably, the bull markets that preceded the
Wall Street crashes of 1929 and 1987, the 1990s boom in internet stocks and, of course, the insane
bubble in housing and real-estate-related securities in the run up to the 2008 banking crisis.
In some cases, the bubbles were initiated by fraudsters who successfully duped irrational, or at least
ill-informed, traders. However, that fact alone does not rule out the possibility that at some point a
rational bubble mechanism may well have taken over.
19 E.g. a so-called first-order autoregressive process:
Bt = β −1Bt−1 + ut
20 To prove that this is true, follow the same procedure as the one outlined for Equation 12.15, making use
of Equation 12.21 en route.
Note that the analysis of bubbles exploits the well-known property of difference (and differential)
equations that they permit an infinite number of solutions, each being the sum of a general and a particular component.
21 This is not quite as implausible as it sounds, if only because there are many factors affecting exchange
rates which are not included in economists’ datasets because they are inherently difficult to quantify, for
example political factors.
22 Although in the example given here, it can be shown the bubble will have an expected duration of
(1 − Π)−1.
23 There is no reason, in principle, why we could not respecify the simple model given here to make the
probabilities depend on the market fundamentals; however, not only does this complicate the analysis,
but also it is completely arbitrary.
There are a number of other possible extensions to the model which would make it more realistic,
inevitably at the cost of some considerable complication. For example, allowing for the fact that investors
may be risk-averse will exacerbate the explosive behaviour of the exchange rate, because an additional
capital gain will be required to compensate currency holders, over and above the capital gain needed
under risk neutrality. Also, the market might well perceive the chance of a crash to depend on the length
of time the bubble has already run. Since the greater the probability of a crash, the more rapidly the
exchange rate must increase, this modification makes the model even more explosive.
24 The variance of a random variable, X, is a measure of its dispersion, computed by taking the average
of the squared deviations from its mean value. In general, for two random variables, X and Y, if we
define:
Z = aX + bY
then it follows that:
var(Z ) = a2 var(X ) + b2 var(Y ) + 2ab.cov(XY )
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where cov(XY) denotes the covariance of X and Y, defined as the average of the product of the deviations
of X and Y from their respective means. Obviously, if cov(XY ) is zero, we can conclude that:
var(Z ) = a2 var(X ) + b2var(Y )
For details, see any elementary statistics textbook.
25 Strictly, this is only true of course as far as observable fundamentals are concerned, and it also assumes
that the N periods which have elapsed between t and t + N are sufficient for us to be able to ignore the
impact on s*t of news from t + N + 1, t + N + 2 . . .
26 An under-researched question is how far rational or irrational bubbles in one market are associated with
bubbles in another.
27 The term was coined by a researcher who took as his classic case the behaviour of the Mexican peso,
which, although notionally on a fixed exchange rate, traded consistently at a forward discount to the US
dollar in the mid-1970s, in anticipation of a devaluation that duly materialised in 1976.
28 Officially, the UK will accede to membership when a number of (fairly vague) tests of the British eco
nomy’s readiness have been satisfied. The first edition of this book used the example of UK accession to
the ERM to explain the peso problem. The fact that eurozone membership serves as an equally good
example speaks volumes about the wearyingly repetitious nature of British policy dilemmas.
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Chapter
13
The risk premium
Contents
Introduction
13.1
13.2
13.3
13.4
13.5
333
Assumptions 334
A simple model of the risk premium: mean–variance analysis
A more general model of the risk premium* 338
The evidence on the risk premium 346
Conclusions 348
Summary 349
Reading guide 350
Notes 350
Appendix 13.1: Derivation of Equation 13.16
335
353
Introduction
We have referred to the risk premium associated with international speculation on numerous
occasions throughout this book, without making any attempt to say what factors determine
its size. It is now time to rectify this omission.
Unfortunately, the subject is a difficult one and involves different analytical tools from
those used in the rest of the book. In particular, it relies on microeconomics – the theory of
constrained choice – as well as on mathematical statistics. To make the material in this chapter as accessible as possible to those who have no background in financial economics, many
complicating issues will be sidestepped. For the most part, the simplifications introduced
have little effect on the central question of what determines the size of the compensation
required by a risk-averse economic agent to persuade him to speculate.
The chapter takes the following form. We start in the first section by listing a number of
basic assumptions that allow us to focus on the issue at hand, without getting sidetracked
into consideration of extraneous questions. Then, in Section 13.2, a simplified model is
analysed using the indifference curve techniques familiar from basic microeconomics. The
next section, which some readers may choose to omit, contains a more formal analysis of a
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sequence of more general models. Finally, as usual, the chapter closes with an overview of
the evidence, followed by some conclusions.
Before continuing, there are a number of preliminaries that the reader is advised to undertake.
13.1
●
If necessary, check the definitions of risk aversion, risk neutrality, the risk premium and so
on, introduced initially in Section 3.1.3.
●
Where necessary, readers should refresh their understanding of what is meant by an
expected value (Section 11.1) and a variance. The concept of covariance also plays an
important part in Section 13.3. A brief definition will be given there to refresh the reader’s
memory. Explanations of all three concepts can be found in any elementary statistics
textbook. (Note that the standard deviation is defined simply as the positive square root of
the variance.)
●
Readers with no previous acquaintance with indifference curve analysis will find Section 13.2
heavy going. Unless, as an alternative, they can take the material in Section 13.3 in their
stride, they would be best to read the chapter on indifference curves in an introductory
economics textbook before proceeding.
●
Section 13.3 takes the theory of expected utility for granted. The Reading guide provides
references for those who wish to investigate these foundations further. However, an
understanding of expected utility theory is certainly not required in order to cope with the
material in this chapter.
Assumptions
The analysis in this chapter will focus on a representative economic agent (‘the speculator’),
whose environment is characterised by the following assumptions.1
●
There is a perfect capital market, with no transaction costs of any kind, and in particular
no margin requirements for forward purchases or sales.2
●
Until Section 13.3.6, we simplify matters by assuming that only two periods are relevant
to the decision: ‘the present’ (period 0) and ‘the future’ (period 1) and there is no consumption in period 0 (the agent has already consumed as much as he or she wanted in
the current period). The speculator/consumer seeks to maximise the expected value of
his utility, which depends only on the amount of consumption he can enjoy in period 1,
C1. Marginal utility diminishes as consumption increases, a condition equivalent to
assuming risk aversion.
●
By assumption, we start with no inflation. We shall examine briefly the implications of
allowing for inflation in Section 13.3.5.
●
Other than a given quantity of wealth, W0 (a fixed ‘endowment’), the resources available
for consumption in the future can be increased only by the device of speculation in
forward contracts. (In other words, the possibility of buying other assets in period 0 is
ruled out by assumption.)
A warning: in this chapter, we have no choice but to work with natural numbers, not
logarithms as in the rest of this book.
Every pound spent on buying dollars forward costs (that is, reduces future consumable
resources by) £F0, which is the current price of a dollar for delivery in period 1. On the other
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