Transcript Asset Price Bubbles

MSc Financial Economics
The Banking System in
February 2011
an Age of Turbulence
Professor Anne Sibert
Professor Willem H. Buiter, London
School of Economics
Professor Anne Sibert, Birkbeck,
University of London
Asset Price Bubbles
Rational Bubbles
In this talk I am going to consider rational
bubbles, so it is worth digressing and
reminding you what an economist means
by the term rational.
Expectations
• Suppose that market participants’ behaviour
depends on their forecast of some future variable.
• If an economist wants to model the behaviour of
market participants, then he must make some
assumption about how the they form their
forecasts.
• It is typical to assume that the market participants
have rational expectations.
• Rational expectations is a modelling technique.
We use it because it is a good, if not perfect,
description of reality and it is relatively
straightforward to implement.
Rational expectations
• (technical definition) If market participants
have rational expectations then their
forecast of a future variable is the
statistical expectation conditional on all
available information.
• (intuitive) Market participants with rational
expectations make the best possible
guess using all available information.
If market participants have
rational expectations
•
•
•
•
They are correct on average.
They do not make systematic errors.
They cannot be systematically fooled.
“You can fool some of the people all of the time,
and all of the people some of the time, but you
cannot fool all of the people all of the time.”
• In a scenario where nothing is random, market
participants with rational expectations have
perfect foresight. That is, their expectation of a
future variable is the actual value of that variable.
History
• The theory of rational expectations was proposed
by John Muth, a graduate student at Carnegie
Mellon, in the early 1960s.
• It pretty much rendered obsolete in macroeconomics the earlier theories based on adaptive
expectations. If market participants have adaptive
expectations then their beliefs about the future
depend upon the past.
• Rational expectations theory was developed in the
late 1960s and 1970s. The most important
contributors were Robert Lucas, Edward Prescott,
Thomas Sargent and Neil Wallace.
Non-Rational expectations
• "Most, probably, of our decisions to do something positive, the
full consequences of which will be drawn out over many days
to come, can only be taken as the result of animal spirits - a
spontaneous urge to action rather than inaction, and not as
the outcome of a weighted average of quantitative benefits
multiplied by quantitative probabilities." (John Maynard
Keyes,The General Theory of Employment Interest and
Money, 161-162.)
• “But how do we know when irrational exuberance has
unduly escalated asset values, which then become subject to
unexpected and prolonged contractions as they have in Japan
over the past decade?” (Alan Greenspan, speech at the
American Enterprise Institute, 5 Dec 1996)
A model of house prices
• Let R(t) be the amount it costs to rent one
unit of housing at time t.
• Let P(t) be the amount it costs to buy one
unit of housing at time t.
• Let I(t) be one plus the interest rate at on
bonds held between time t and time t + 1
Simplifying Assumptions
• Assume that people are risk neutral. That
is, they only care about the expected
return on investments.
• Assume that there is no depreciation, no
transactions costs, no repairs, no tax
considerations.
• Suppose that R(t) = R and I(t) = I: the
rental price of housing and the interest
rate are constants.
Investing in bonds vs. rental property
• People must be indifferent between buying
houses as an investment and renting them
out and investing their money in bonds.
Otherwise, they would only do one of these
things. Thus, the expected (forecasted)
returns on these two options must be the
same.
• If you invest one unit of money in a bond at
time t, then you get I units of money at the
start of time t + 1.
Returns to different investments
• If you take one unit of money at time t, you
can buy 1/P(t) units of housing. You can
rent the house in period t (suppose the rent
is paid at the start of the period) and you
can sell the house at the start of period t +
1. At the start of time t + 1, you have [RI +
P(t+1)] / P(t) units of money.
• There is no randomness in our model so we
assume that market participants have
perfect foresight. Their forecasted value of
P(t+1) is P(t+1).
Arbitrage condition
• If the returns on these two investments are
equal we have: [RI + P(t+1)] / P(t) = I
• This implies: P(t+1) = IP(t) – RI.
• Note that this tells us how the house price
changes over time, but it does not pin
down the level and this causes rational
bubbles to be possible.
Example:
• Suppose that R = £1,000 and I = 1.10.
• Then we have P(t+1) = 1.1P(t) – 1,100.
• Suppose that P(0) = 20,000. Then P(1) =
20,900, P(2) = 21,890, P(3) = 22,979, P(4) =
24,177, …
• Suppose that P(0) = 30,000. Then P(1) =
31,900, P(2) = 33,990, P(3) = 36,289, P(4) =
38,818,…
• Suppose that P(0) = 5,000. Then P(1) = 4,400,
P(2) =3,740, P(3) =3,014, P(4)=2,215, …
Multiple outcomes are possible
• This scenario is similar to being told that
someone has driven 100 kilometers per
hour down the highway for the last hour
and then being asked where they are.
• We know where the person is relative to
where they were an hour ago, but to
answer the question we need to know
where they started out.
Some Possible Outcomes:
The horizontal axis is time; the vertical axis is the house price.
There are many outcomes,
depending on P(0)
• Some of them have prices becoming
negative: clearly these are not equilibria.
• Some of the have prices going to infinity:
these are possible equilibria: bubbles.
• One of them has a constant price.
Note that the environment is constant
• In this model of house prices, we take the
(gross) interest rate I and the rental price R as
given. They are the dependent, or exogenous,
variables.
• We want to use these exogenous variables to
solve the model for the independent, or
endogenous, variable, the house price.
• Note that exogenous variable are constant. This
suggests that it is reasonable for the
endogenous variable to be constant. Guess that
the house price is a constant: P(t) = P.
Substitute this into the arbitrage
condition:
• P = IP – RI
• Solve for P: P = RI / (I – 1) = 11,000
• It turns out that this has an economic
interpretation: It says that the price of a
house should be equal to the discounted
present value of the rental payments.
Boundary Conditions
• Equations that relate the value of a variable at time t + 1 to its value
at time t are called difference equations.
• The problem I gave with the car is similar. There, we wanted to find
the location of a car given its speed (how fast its location is
changing). This type of equation is a differential equation.
• Both types of equations have an infinite number of solutions. With
the car example, we pick out the solution we want by appealing to
some additional information: where the car started. Mathematicians
call this a boundary condition.
• Here, we are not going to use P(0) (where we started) to pick out the
right solution. Instead we are going to reason that if the
fundamentals are constant, the solution we want is the only one
where the price is constant.
• More generally, when the fundamentals are changing we will pick
out the single solution that depends just on the fundamentals. All of
the others will have the price going to infinity because of self-fulfilling
expectations.
Another way to see this
• Consider the equilibrium condition: P(t+1)
= IP(t) – RI.
• I am going to graph this and use the result
to show that a bubble can exist.
P(t+1)
P(t+1)=IP(t)-RI
P(t+1)=P(t)
P(t)
P(0)
P(1)
P(2)
This is related to
coordination games
• Suppose that two players are confronted
with four tiles
• They simultaneously select a tile.
• If they select the same tile, they each get
£1,000. Otherwise, they get nothing.
Player 1 sits here
Player 2 sits here
Nash equilibrium
• Both players select the same tile.
• But, how would a Nash equilibrium result
in this game?
Try the game again
• Suppose that two players are confronted
with four tiles
• They simultaneously select a tile.
• If they select the same tile, they each get
£1,000. Otherwise, they get nothing.
Player 1 sits here
Player 2 sits here
Choosing the snazzy tile is focal
• I conjecture that the likely outcome is a
Nash equilibrium where the players pick
the tile that is different.
• Perhaps the equilibrium that depends
solely on the fundamentals is focal too.
• How would any of the uncountably infinite
number of equilibria arise?
The existence of bubbles is a generic
property of financial asset markets
• Consider the arbitrage condition for stock and
bonds.
• Let Q(t) be the time-t price of stock and let D be
the dividend, which I assume is constant.
• Then, you can take one unit of money and buy
1/Q(t) units of stock. At the end of the period you
get a dividend D/Q(t) and you can sell the stock for
Q(t+1)/Q(t).
• For investors to be indifferent between stock and
bonds we need I = [Q(t+1) + D] / Q(t) or Q(t+1) =
IQ(t) – D. If you try some examples you will see
that there are bubbles in this case as well.
Exchange Rates
• Demand for a particular currency depends
positively upon how much it is expected to
appreciate.
• All of the fundamentals (the exogenous
variables) may be constant, but if people
think that the currency will appreciate, they
buy more of it than they otherwise would.
• As a result, the currency appreciates:
beliefs are self fulfilling.
The problem with financial asset
prices
• The price of a financial asset today typically
depends upon what market participants
believe the price will be tomorrow.
• If they believe the price will go up (even
though the fundamentals are constant), then
they demand more than they otherwise would
and, as a result, the price goes up. Their
expectations are self-fulfilling. The result is a
bubble.
Market participants do not need to
believe a bubble will last forever
• The bubbles I have described are a bit
unrealistic. Market participants believe that they
will last forever and they do.
• Most historical episodes that have been
described as bubbles have the price collapsing
at some point.
• The theoretical model works if people correctly
believe that there is some chance that the price
will collapse in each period and that as time
goes to infinity the bubble will certainly have
collapsed.
Canonical Bubble
• In Feb 1637 tulip
prices in the
Netherlands soared
until the price of a
tulip could be 10
times the annual
earnings of a skilled
craftsman. They
quickly plummeted.
Source: Wikipedia
Racehorses
• The price of yearlings
seems far to high
relative to their future
earnings.
• In 1985 a yearling
named Seattle Dancer
sold for $13.1 million.
• Only half of all good
quality yearlings win a
race. They are sold as
parents of future
yearlings.
Other supposed examples:
• Famous early bubbles: The South Sea
and Mississippi Company Bubbles of
1720.
• Japanese real estate and stock market
(1986 – 1990)
• Dot-com bubble (mid 1990s – 2001)
• Various residential housing bubbles
Can monetary policy makers tell
if a bubble exists?
• “Some economists, reluctant to let go of the
comforting world of rational expectations, still tell
us it is impossible for a central bank – or anyone
else, for that matter – to call a bubble. This is
baloney. When looking at house prices, just look
at price-to-rent and the price-to-income ratios,
sales volumes and credit statistics, and you
know everything you need to know. Almost
everything else central bankers do is more
difficult than calling a housing bubble.”
• Wolfgang Münchau, Why Central Banks should
Prick Bubbles,” Eurointelligence, 27 Oct 2009
Do bubbles exist?
Can we tell when?
• Testing for bubbles is difficult. An
econometrician must first specify a model. If a
rise in the price of a financial assest cannot be
explained by the model, the econometrician
might claim that it is a bubble. But, it could just
be that the model is misspecified.
• Apparent bubbles might be due to nonstationary fundamentals. Examples are
hyperinflations that be explained by money
growth.
There is no conclusive
evidence of a bubble
• Consider our model of house prices. If we
let rents and the interest rate be time
varying we could solve it to find that the
non-bubble equilibrium price is equal to
the present discounted value of rental
payments.
• We could test to see whether discounted
rental prices explain house prices.
The null hypothesis is then
• No bubbles
• The simple model of fundamentals is
correct.
Recall that the simple model assumed no
transactions costs, no maintenance costs,
no depreciation, no tax considerations, no
uncertainty, risk neutrality
Rejecting the null hypothesis:
• There is a bubble.
• Or, the model is not specified properly.
• Clearly adding uncertainty and risk
aversion is necessary to have a really
sensible model and this makes the model
way more complicated: I can’t use a
simple arbitrage condition.
Was the tulip mania a bubble?
• The tulip mania was popularized by Charles
Mackay in 1841 in his the book Extraordinary
Popular Delusions and the Madness of Crowds.
It is the classic example of a bubble: if it wasn’t a
bubble, what was? But was it a bubble?
• At the time, tulips had just been introduced into
Europe from the Ottoman Empire. The valuable
ones owed their beauty to a mosaic virus.
Garber (1990) argues that the tulip
mania was not a bubble
• Cultivating tulips with the virus took years and
could not be done from seeds. This suggests
that the few early bulbs with the virus should
have been very expensive and that as bulbs
accumulated, the price should have fallen.
• The data is not good
• There was also a spike in the prices of common
tulips, but Garber dismisses this as a winter
drinking game of the lower classes.
Sun spots
• Sun spots are variables that are not fundamentals but affect
prices anyway because of self-fulfilling expectations.
• Example: Henry Kaufman: an economist who used to work for
Solomon Brothers and was famed for his interest rate
forecasts.
• The name comes from a study by a 19th-century British
economist William Stanley Jevons: he thought that sun spots
really might affect agriculture.
• Sun spots could also be fundamental variables that are given
too much weight, and because of self-fulfilling expectations,
are more important than they ought to be in determining
prices: an example might be money supply numbers in the
United States when Paul Volcker was the Fed chairman.
Overshooting
• Suppose that goods prices are changed
infrequently but that financial asset prices,
such as exchange rates, are changed
instantaneously.
• Then, because prices are sticky, a shock
can cause the exchange rate to overshoot
its equilibrium value.
A non-linear adjustment process
• Our very simple model of house prices yielded a
linear solution. That is, when we graphed P(t+1)
= IP(t) – RI it turned out to be an upward sloping
straight line. The result was the prices go
monotonically off to infinity.
• But more complicated outcomes are possible.
We can get prices oscillating, either off to infinity
or toward a steady state – even when the
fundamentals are constant.
Chaos
• It is even possible for macroeconomic
models with no uncertainty and perfect to
exhibit chaos.
• Chaos is an adjustment process that is so
“disorderly” as to appear random.
• The importance of this is controversial.
Example
• Suppose that equilibrium for some price
were given by x(t+1) = ax(t)[1 – x(t)] and
consider strictly positive starting values.
• If 2 < a < 3, the equilibrium is a cobweb: it
is stable with damped oscillations.
• If 3 < a < 3.68: the price can oscillate with
ever greater swings until the amplitude
settles down to a cycle over two or more
periods.
When a becomes large
• Chaos may emerge: there is no regular
cycle or pattern.
• The outcome is highly sensitive to the
starting value or the precise value of a.
Some stylised facts about booms
and busts (from an IMF study)
• Data from 19 industrialised countries
• Housing data from 1970 – 2002; equity
data from 1959 – 2002
• Booms defined as a trough-to-peak rise in
the top quartile of rises; busts defined as a
peak-to-trough fall in the top quartile of all
falls
Results
• Equity price busts occurred about every 13
years, lasted for 2-1/2 years and where
associated with price declines of about 45
percent.
• House price busts occurred about every
20 years, lasted for 4 years and were
associated with price declines of about 30
percent.
Results
• A quarter of equity price booms were
followed by busts.
• Forty percent of house price booms were
followed by busts.