Detecting Asset Price Bubbles

Fulcrum Research Notes – January 2014

Detecting bubbles in asset prices Ziad Daoud* Juan Antolin-Diaz** This note describes recent advances in econometric methodology which allow us to detect bubbles in asset prices. The method

identifies bubbles as periods in which prices have explosive dynamics, and proposes a procedure to date the start and the end of

bubbles in real-time. When applied to historical S&P 500 data, the method identifies nine episodes in which prices deviated from

fundamentals; around half of these were booms, the rest were crashes. While the S&P 500 might seem over-valued at the

present time according to some metrics, the method suggests that it is not experiencing explosive dynamics and therefore is not

currently in a bubble. It is intuitively easy to see why: over the last couple of years, the S&P 500 Index returned 40%, broadly in

line with its dividends which were up by 34% over the same period. This is not the behaviour of an asset whose price is greatly

detached from fundamentals.

The recent award of the 2013 Nobel Prize in economics

to three academics may have surprised some. Few

would question that, individually, the three laureates

(Eugene Fama, Lars Peter Hansen and Robert Shiller)

are fully deserving of the award as their work on

understanding asset prices has been hugely influential

in academia and beyond. What was surprising,

however, was that they won it jointly given that two of

them, Fama and Shiller, hold diametrically opposing

views on asset pricing.

Take for example their views on asset return

predictability. In a series of papers in the 1960s Fama

showed that returns are essentially unpredictable and

any attempt to exploit market predictability is unlikely

to overcome transaction costs. Asset prices, according

to Fama, follow a random walk, which makes sense in

an environment with competitive trading among

rational investors. Two decades later, however, Shiller

showed that in some circumstances stock returns are

in fact predictable: variables such as price-dividend

ratio can predict a significant portion of equity

variation.

One can reconcile the two views by concluding that

both are right, and that returns are unpredictable in

the short-run (ie at a monthly frequency or higher) but

they are predictable over the longer-run of several

years. It is harder, however, to reconcile the views of

the two laureates on whether asset markets can

experience bubbles, defined as periods in which prices

significantly deviate from their fundamental drivers.

Fama denies the possibility that bubbles exist, and

suggests that large swings in asset prices can be

explained by variations in risk premia.

Shiller, on the other hand, made his name outside the

confines of academia by correctly calling two recent

bubbles: in the equity market in the 1990s and in the

housing market in the 2000s. He views psychological

factors such as animal spirits and fads as significant

drivers of variation in the economy and financial

markets.

Until recently, empirical evidence on whether bubbles

exist in asset prices has not been conclusive, partly due

to the lack of a well-developed econometric

methodology to test this hypothesis. However, recent

advances made by Peter Phillips, Jun Yu and their co-

authors allow us to test this hypothesis with greater

accuracy. These authors empirically characterise

bubbles as periods in which asset prices experience

explosive dynamics, and develop statistical tools to

detect and time-stamp the presence of such behaviour.

Updating and extending the application of their

method to the monthly S&P 500 price-dividend ratio

series, nine episodes of bubbles are identified since the

beginning of the 20th century. When the method is

applied to other valuation metrics, such as Shiller’s

cyclically-adjusted price-earnings ratio, similar dates

for bubbles are detected.

*[email protected]; **[email protected]

Detecting bubbles in asset prices

Fulcrum Research Notes – January 2014 2

More generally, the presence of bubbles in historical

equity prices, together with evidence from survey data

on expected returns, suggests that variations in risk

premia cannot be the main driver of these explosive

episodes in asset prices, and that psychological or

informational factors may be at play.

The method also highlights that there is very little

evidence in favour of the presence of a bubble in

current equity prices despite elevated levels of

valuation metrics relative to the their historical

averages. It is intuitively easy to see why the method

reaches such a conclusion: over the last couple of

years, the S&P 500 Index returned around 40%,

broadly in line with its dividends which were up by

34% over the same period. This is not the behaviour of

an asset whose price is greatly detached from

fundamentals.

That said, failure to detect a bubble does not

necessarily mean that a large price drop cannot take

place. For example, a high price-to-earnings ratio may

reflect over-optimistic views about future earnings

growth because investors may expect the recent

increase in the profit share in the economy to persist.

Prices, then, can still experience significant corrections

if these optimistic forecasts fail to materialise and the

profit share reverts to its past historical average. The

fact that we do not detect a bubble does not therefore

mean that there is no risk of low returns going forward.

Such sharp conclusions could not be reached by the

early literature based on standard econometric tools.

The traditional approach to test for bubbles and its

drawbacks will be outlined later but first the next

section explains how bubbles can be empirically

identified.

Identifying bubbles in asset prices

The starting point for identifying bubbles is the

standard pricing equation widely used in financial

economics:

∑ [(

)

( )]

The equation says that the price-dividend ratio has a

fundamental value driven by expected future dividend

growth (the

term): the higher the expected future

dividend growth, the higher today’s stock price should

be. The fundamental component is discounted by the

required rate of return, . This rate of return is related

to investors’ risk appetite and impatience so variation

in these factors can also induce variation in the price-

dividend ratio.

In addition, today’s price can be high without a

commensurate increase in dividends if investors have

self-fulfilling beliefs that they will make capital gains

from ever higher future prices. This introduces the

possibility of prices deviating from their fundamental

value due to the presence of bubbles represented by

the term. This can be an outcome of extrapolative

price behaviour where past price increases induce

people to buy more of the asset driving its current price

higher.

The important implication of this discussion is that the

presence of the bubble term in the pricing equation

changes the dynamics of the observed price process in

a way that can be tested econometrically.

Generally speaking, we deal with three classes of

processes in this note: stationary processes which tend

to revert to their mean; random walks (or unit root

processes) which do not mean revert but have a

stochastic time trend instead (although their trend

evolves in a gradual manner) 1; and explosive processes

which are non-mean reverting and grow at exponential

speed.

Looking at the components of the pricing equation,

dividend growth is likely to be stationary but the price-

1 A random walk is a process with the property that the best forecast for the value of the process in the next period is its value in the current period. It is popular in modelling economic time series because it conveniently leads to trends which are stochastic, ie ones that can slow down or accelerate randomly over time.



dividend ratio can still be non-stationary if the

discount factor is time-varying. However, the degree of

non-stationarity is different when bubbles are present

because bubbles have explosive dynamics – a higher

degree of non-stationarity than the random walks

typically assumed for the fundamental component.

Therefore, detecting explosive dynamics in the price-

dividend ratio is a clue that bubbles might be present.

To summarise: deviation of prices from their

fundamental value introduces explosive dynamics in

the price-dividend ratio. This forms the basis of the

empirical bubble-detection method presented in this

note. A version of this empirical characterisation of

bubbles was also exploited by the earlier literature,

which is reviewed in the next section.

The early literature

A significant body of empirical literature on detecting

bubbles was developed in the 1980s and early 1990s

with mixed results. Because the econometrics for

dealing with explosive time series was not well-

developed, the typical approach in the early literature

began by log-linearly approximating the pricing

equation presented in the previous section. Doing this,

Craine (1993) showed that the presence of bubbles

changes the statistical behaviour of the logarithm of

the price-dividend ratio from a mean-reverting process

to a random walk. Standard econometric unit root

tests, such as that of Dickey and Fuller, could then be

applied to discriminate between the two hypotheses.

The old approach suffered from three shortcomings:

1. Even if the procedure worked perfectly, it would

not be able to date the beginning and end of

bubbles. At best, it would merely be able to state

whether bubbles are likely to be present in the

data.

2. Working with the logarithm of the price-dividend

ratio requires approximating a relationship around

constants (usually long-term averages or steady

states). With non-stationary processes, the validity

of the approximation is questionable as these

constants may not exist. It is therefore better to

work directly with the level of price-dividend ratio

rather than its logarithm. But this necessitates the

need to develop econometric tools for explosive

processes, which have not been available until

recently because of their analytical complexity.

3. Evans (1991) showed in a simulation study that

when bubbles are present but followed by periodic

crashes, standard unit-root procedures fail to

detect them, even when they are substantial in

magnitude and volatility like the ones shown in

Figure 1, because collapsing bubbles make the time

series appear mean-reverting.

Figure 1. The Evans Critique

Time series which contain collapsing bubbles pass as stationary

by standard unit root tests. The data shown here are simulated.

The Evans critique dealt a blow to the literature as it

showed the deficiency of the techniques available at the

time. Subsequent research developed in different

directions without converging to a consensus on either

methodology or conclusion. In his survey of the

literature, Gurkaynak (2005) summarised the state of

research by concluding that “[f]or each paper that finds

evidence of bubbles, there is another one that fits the

data equally well without allowing for a bubble”.

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Collapsing bubbles



However, in a series of recent papers, Peter Phillips,

Jun Yu and their co-authors developed a new battery of

methods that managed to overcome the shortcomings

of the traditional approach. Their test—explained in

detail in the next section—also provides a real-time

dating procedure for bubbles and crashes in asset

prices. It is designed to deal with levels directly as well

as logarithmic transformations and, unlike standard

tools, it successfully detects bubbles even when they

are subsequently followed by crashes.

A new method for detecting bubbles

The method of Phillips, Shi and Yao (2013), referred to

as PSY henceforth, relies on the characterisation of

bubbles as episodes in which the level of the price-

dividend ratio experience explosive dynamics.

In order to date the start and end of bubbles, PSY use a

procedure which tests for the presence of explosive

behaviour at each point in time, instead of running a

global test over the full sample.

How does this procedure work? We have classified

time series into three types of processes: stationary,

unit-root and explosive. These processes have differing

degrees of persistence. A stationary process is not very

persistent: the impact of a shock today on the forecasts

of its future values tends to die out eventually. Non-

stationary processes, such as unit-root and explosive

processes are very persistent: shocks to today’s value of

the process changes our forecast of all future values,

even those in the distant future. Explosive processes

are the most persistent of all, since any shock tends to

get compounded exponentially. Therefore, the degree

of persistence of the price process at the point of

interest is indicative of whether a bubble is present or

not.

But if we measure persistence using all the data up to

that point, then we may fail to detect the true

explosiveness of the most recent observations. As in

the Evans critique, past collapsing bubbles may make

the data look rather stationary.

To overcome this, PSY propose measuring the degree

of persistence over all backward-looking intervals of

variable sizes—varying the start point of each interval

but keeping the end fixed at the point of interest. The

final test statistic is the maximum of these different

persistence measures.

The intuition behind the procedure is that while some

subsamples which contain collapsing bubbles might

understate the degree of persistence in the process

(Subsample 1 in Figure 2), the procedure would also

give rise to other subsamples (such as Subsample 2 in

Figure 2) which are very persistent because they do not

contain past bubbles and their subsequent crashes. The

maximum persistence statistic would then be

determined by the latter group, and a large value is

evidence for the presence of a bubble.

The method has many advantages: First, the backward

subsampling procedure is a clever way to handle the

Evans critique, giving PSY’s method more power to

detect bubbles than older approaches.

Second, it provides a real-time tool for detecting

bubbles. To determine whether an asset is going

through a bubble or not, only current and past

information is needed and there is no look-ahead bias

involved.

Third, in addition to answering the question of

whether bubbles in asset prices exist, the method also

provides a live dating mechanism determining when

they begin and end.



Box 1. Testing for persistence and explosiveness in time series

The building block of both the old literature and the new method is unit-root tests, the most well-known of which is

the one by Dickey and Fuller. Specifically, if one assumes that the time series follows an autoregressive model, then

the Dickey-Fuller method tests if is a random walk by examining whether in the regression

(1)

This is done by calculating a standard t-statistic (the estimate of β divided by its standard error), but rather than

comparing it to a quantile of the t-distribution as in standard regression analysis, it is compared to a quantile from the

Dickey-Fuller distribution.

The early literature on testing for bubbles relied on such unit-root testing procedure, where represented the

logarithm of the price-dividend ratio. When bubbles are present, the logarithm of the price-dividend ratio would

behave like unit-root process and ; while if there are no bubbles, the series should be stationary and .

The method of Phillips, Shi and Yao (2013) differs from the old literature in the way it characterises bubbles and

implements unit-root testing. Under their procedure, represents the level of the price-dividend ratio and when

bubbles are present, is greater than one, while when they are not, is at most equal to one.

Moreover, as explained in the main text, instead of running one unit-root test over the full sample, PSY test for unit

root at each point in time, which allows them to date the entry into and exit from bubbles. How does testing at each

point work?

Suppose we want to test whether there is a bubble at the 100th observation. Rather than running a Dickey-Fuller test

using all the 100 available observations, PSY calculate t-statistics for each of the intervals that end with the 100th

observation such as {1, 2, …, 100}, {2, 3, …, 100} up to {65, 66, …, 100} (a minimum number of observations is needed

to run the regression (1) and calculate the resulting t-statistic. In this paper, this minimum number is set to 36).

Finally, PSY take the maximum over all these 65 t-statistics and compare its value to a quantile from their distribution.



Figure 2. Illustration of the bubble detection method on the S&P 500 real price-dividend ratio

Source: Fulcrum Asset Management, Price-dividend data from Robert Shiller’s website.

Finally, PSY deal directly with levels of price-dividend

ratio as opposed to logarithmic approximations which

may or may not be valid with non-stationary time

series, overcoming another of the shortcomings of the

old approach.

Having described the procedure, the next section

updates and extends PSY’s application to identify

historical bubbles in the S&P 500, and to answer the

question whether we are living through an equity

bubble now.

Bubbles and crashes in the S&P 500

The data used in this section are the monthly real price

to dividend ratio for the S&P 500 Index obtained from

Robert Shiller’s website. The sample spans the period

from January 1871 to December 2013.

Applying PSY’s method to the data, nine episodes of

stock market bubbles are identified since the beginning

of the 20th century (shaded areas in Figure 3). Five of

these were of the irrational exuberance variety when

investors’ optimism drove equity markets higher than

justified by fundamentals, including:

The stock market boom in the 1920s. The bubble

started in September 1928 and ended with the

great crash of October 1929.

The boom which started immediately after World

War 2 in October 1945 and ended in June 1946.

The second post-war boom from September 1954

to April 1956.

The mid-1980s bubble starting in March 1986 and

ending with Black Monday in October 1987.

The dot-com bubble of the second half of the

1990s. This is by far the longest episode of

exuberance lasting around six years from July 1995

to August 2001.

The other four episodes when prices were divorced

from their fundamental value were stock market

crashes, possibly driven by excessive pessimism,

including:

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[2] Subsample 1: The persistence of this interval is diluted by the crash of 1974 and the subsequent recovery ...

[3] Subsample 2: The persistence of the process over this subsample is high due to its recent explosive trajectory ...

[4] The Phillips, Shi and Yu (2013) procedure uses the maximum degree of persistence over all backward subsamples to identify bubbles, so will be determined by Subsample 2

[1] We want to test for the presence of bubbles at this point ...



Figure 3. Historical bubbles in the S&P 500 real price-dividend ratio

(Shaded areas = bubble)

Note: Last observation is December 2013. Source: Fulcrum Asset Management, Price-dividend data from Robert Shiller’s website.

Figure 4. Historical bubbles in the S&P 500 real cyclically-adjusted price-earnings ratio (CAPE)


Note: Last observation is October 2013. Source: Fulcrum Asset Management, CAPE data from Robert Shiller’s website.

Figure 5. Degree of support for the bubble hypothesis


Notes: Higher values indicate stronger support from the data for the presence of bubbles. Shaded regions are constructed as the periods in which the support metric exceeds the 0.95 threshold. Last observation is December 2013. Source: Fulcrum Asset Management, CAPE data from Robert Shiller’s website.

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The banking panic which started in August 1907

and ended in February 1908.

The stock market crash which began in August

1917 and ended in April 1918.

The brief stock market crash of 1974, which lasted

from July to December.

The financial crisis originating in October 2008

and ending April 2009.

Figure 4 shows bubbles in S&P 500 using an

alternative metric: Shiller’s cyclically-adjusted real

price-to-earnings ratio (CAPE). The chart highlights

the same episodes as the price-dividend ratio with one

notable difference in the 1920s where CAPE started

behaving explosively before the price-dividend ratio.

What does the method say about whether or not the

equity market is currently in a bubble?

Carrying out the test using either of the two metrics

suggests that we are not going through an equity

bubble, and it is fairly clear why this is the case. Since

2009, both the price-dividend ratio and the CAPE have

been moving sideways (albeit at a higher level than

historical average) rather than upwards in an explosive

manner. Over the last couple of years, the price of the

S&P 500 Index moved up by around 40%, broadly in

line with its dividends which were up by 34% over the

same period. This is not the stuff of bubbles where

price movements are completely divorced from

fundamentals.

One way to quantify this observation is given in Figure

5 which plots the strength of evidence in favour of the

bubble hypothesis. In statistical terms, the metric is

the complement of the test’s p-value, and the larger it

is (within a range from zero to one), the stronger is the

evidence for being in a bubble. The latest reading of the

support metric is quite low at less than 0.2, suggesting

that there is little support in the data for the bubble

hypothesis2.

Delving a little deeper into what the 0.2 figure means,

recall that the logic of statistical hypothesis testing is

such that the null hypothesis of no bubbles is

maintained unless there is strong evidence against it. A

threshold for the degree of support for bubbles is then

set so that a bubble is called if its degree of support

exceeds the threshold. The lower the threshold, the

easier it is to detect bubbles, but also the higher is the

probability of incorrectly discovering a bubble when

there is not one. This note uses the conventional

threshold of 0.95 in order to limit the false discovery

rate to 5%.

To put the latest reading in context, it would require a

big drop in our standards of controlling the likelihood

of erroneously calling a bubble when one does not in

fact exist (from 5% to over 80%) to accept the presence

of bubbles in current equity prices. Evidence in

support of bubbles is so weak that in order for us to call

a bubble now, we would have to conclude that equities

have been in a virtually continuous bubble over the

whole sample period!

Figure 6. Actual and simulated Cyclically-Adjusted Price to Earnings ratio (CAPE) under a bubble

scenario

Source: Fulcrum Asset Management, CAPE data from Robert

Shiller’s website.

2 Data for the CAPE ratio is only available up to October 2013 as of the writing of this note. If we extrapolate earnings the metric would rise slightly to 0.33

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Actual CAPE

Simulated CAPE ratio under a bubble scenario



To illustrate this point further, it may be useful to

consider how the S&P 500 Index would have behaved

if it had been in a bubble. Given the 34% move in

dividends over the last couple of years, it would have

taken at least a 100% move in the S&P 500 over the

same period for its behaviour to be considered

explosive and for a bubble to be detected. This would

have taken the index beyond the 2,500 mark by

September 2013 (when its actual value was around

1,700). Such a move would have also taken valuation

metrics to high levels only exceeded by the peak of the

dot-com bubble. As Figure 6 illustrates, had such a

bubble occurred, Shiller’s CAPE would have been

around 33 (compared to an actual value of 23.5),

almost double its historical average, at which point the

real price-dividend ratio would have been as high as 73

when its realised level was 49.

In contrast, The Economist (2013) has recently said

that

“[Investors] should be wary of stock markets when

they look expensive relative to the long-term trend

in profits. And that is the case with Wall Street at

the moment; the cyclically adjusted ratio is 23.5,

well above the long-term average.”

Proponents of the “bubble in equities” hypothesis tend

to point out the current elevated levels of valuation

metrics (the price-to-dividend ratio or CAPE) relative

to their historical averages. The method used in this

note, on the other hand, detects rapid changes in

levels. The two things can give rise to different

conclusions since bubbles, as defined in this paper, are

not synonymous with overvaluation as defined by

traditional metrics.

So why are current valuation metrics such as price-to-

earnings ratio high? In this note we have ruled out the

possibility that there is a bubble in the sense that

prices are high today just because investors expect

future prices to be even higher, regardless of

fundamentals. As argued earlier, this would have

introduced explosive price dynamics which would have

been detected by the econometric method used in this

note.

A second possibility for the high price relative to

(trailing) earnings is that investors expect high future

earnings growth, perhaps because of a change in the

structure of the economy and the way economic gains

are split between firms and workers. Such structural

change would prevent valuation metrics from reverting

to their historical averages, and would not be detected

by the method of this note because these changes tend

to occur smoothly and gradually over time rather than

in an explosive manner.

Does it matter which possibility is more likely? Yes,

because one would expect sharper price correction in

the case of bubbles than in the case of optimistic

earnings forecasts. However, it is worth emphasising

that while recent price dynamics in the equity market

do not seem to reveal bubble-like behaviour, it does

not follow that prices cannot experience abrupt

declines if over-optimistic earnings forecasts fail to

materialise.

It is also worth noting that the switch from a non-

bubble state to a bubble can happen over a short

period of time if the equity price evolves in a manner

that is inconsistent with fundamentals. For example,

we have conducted another simulation experiment in

which over the course of 2014, the S&P 500 Index

returns 30% and both its dividends and earnings grow

by 10% — a behaviour that mimics their performance

in 2013. In that hypothetical scenario the method

concludes that the S&P 500 will be entering a bubble.

Whether we are currently in a bubble or not, the

existence of bubbles in the first place poses theoretical

challenges for economists and different theoretical

frameworks to account for the presence of explosive

behaviour in asset prices have been developed. The

next section outlines three views and shows that

empirical evidence may be inconsistent with one of

them.



What is the method actually capturing?

The previous section has shown that explosive

dynamics are recurrent in equity prices. There are at

least three theories on the underlying forces which give

rise to such behaviour.

The first theory is psychological and is due to Akerlof

and Shiller (2010). These Nobel Prize winning

economists believe that variations in confidence drive

variations in asset prices. The econometric method of

PSY can be interpreted as empirically capturing the

change in animal spirits, identifying episodes of over-

confidence and panic by market participants which

drive the swings in the price-dividend ratio.

An alternative theory, due to Adam, Beutel and Marcet

(2013), uses informational restrictions on rational

investors rather than a behavioural explanation as the

generator of bubbles. In this framework, investors do

not know the true fundamentals of the economy, but

use variations in asset prices to learn about them. As

asset prices rise, investors revise their estimates of the

economy’s fundamentals upwards leading them to

invest more, driving prices further up. This continues

until the wealth effect (investors are richer, and

therefore want to consume more and invest less) starts

to dominate the substitution effect (expected return on

investment is higher, making investment more

appealing relative to consumption), at which point

investors revise down their estimates of the

fundamentals of the economy. From then onwards,

both income and substitution effects lead to lower

investment in the asset causing a sharp crash in its

price.

The third theory, due to Fama and elucidated in

Cochrane (2013), explains bubbles and crashes by

swings in investors’ risk premia. Sharp rises in prices

can occur if investors suddenly become more risk

tolerant and require a lower expected return for

holding the asset. Conversely, as risk premia sharply

reverse course, markets can experience crashes. Prices

according to this theory are always rational, and they

reflect investors’ attitude to risk. One might detect

explosive behaviour in asset prices, but one should not

interpret that as deviation from fundamentals. Indeed,

Phillips and Yu (2011) show that explosive behaviour

could be a result of time-varying risk premia.

Fama’s view, however, implies that investors’

expectations of future returns should be negatively

correlated with the price-dividend ratio, which is not

what we see in survey data. These show a positive and

significant positive correlation as documented in

Adam, Beutel and Marcet (2013). Surveys of expected

returns therefore suggest – assuming that investors are

responding truthfully to the surveys – that the detected

explosive dynamics in equity prices are unlikely to be

merely driven by business cycle related variation in

risk premia. Instead, they are likely to represent

bubble episodes in which prices are divorced from

underlying fundamentals which could arise because of

irrational investors driven by psychological factors of

fear and greed, or because of rational investors

learning in an environment with imperfect

information.

Conclusion

This note has described a new econometric technique

for determining the beginning and the end of bubbles

in real-time. The method does not suffer from the

disadvantages of traditional techniques, and shows

that bubbles and subsequent corrections are a

recurrent feature of equity prices.

The method exploits different features of the price

series compared with standard valuation methods such

as the price-dividend ratio and Shiller’s CAPE, which

focus on deviations of current valuations from their

historical averages. Instead, the method used here

relies on measuring the speed at which valuations

increase or decline over time. In this sense, bubbles as

characterised in this note are conceptually different

from over-valuations.



Since valuation metrics seem to have been stable in

recent years, the bubble-detection method used in this

note concludes that there is little evidence in favour of

a bubble in the S&P 500. Meanwhile, the levels of these

valuation metrics are elevated relative to their

historical averages, possibly indicating that prices are

driven primarily by the expectation that the high

earnings growth observed in recent years (compared

with their long-term rate) will persist into the future.

Despite the absence of any trace of bubbles, prices can

still experience significant declines if these earnings

forecasts turned out to be over-optimistic.

For asset managers, being able to identify periods

when asset prices are in a bubble in real-time has

important implications for portfolio optimisation and

hedging. Furthermore, successfully dating periods of

excessive exuberance and panic is useful for testing

how different strategies perform in these episodes

relative to more moderate periods of market

behaviour. We are currently undergoing such

assessment.

References

Adam, K., J. Beutel and A. Marcet (2013), “Stock Price Booms

and Expected Capital Gains,” Working Paper.

Akerlof, G. and R. Shiller (2010), Animal Spirits: How Human

Psychology Drives the Economy, and Why It Matters for Global

Capitalism. Princeton University Press.

Cochrane, J. (2013), “Bob Shiller’s Nobel,” available at bit.ly/1aEgYcN

Craine, R., (1993), “Rational Bubbles – A Test,” Journal of

Economic Dynamics and Control, 17, 829-846.

The Economist (2013), “A Very Rational Award,” 19 October

2013.

Evans, G. W. (1991), “Pitfalls in Testing for Explosive Bubbles in

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Detecting Asset Price Bubbles

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