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5 Volatility tests, bubbles and the peso problem

# 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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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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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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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

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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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

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

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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