INDIAN INSTITUTE OF CAPITAL MARKETS_CONFERENCE_18DEC2014

-- TANMOY GANGULI

Assistant Manager,

Financial Service Analytics, Genpact (Kolkata)

A DETERMINISTIC DESCRIPTION OF STOCK MARKET BUBBLES IN ASSET MARKETS

TABLE OF CONTENTS

• Why study asset price bubbles?

• Major strands of thought in bubble theories

• Need for deterministic bubble theories

• A review of literature in Deterministic asset price bubbles.

• A deterministic representation of asset price bubble

• Major characteristics of deterministic representation

WHY STUDY ASSET PRICE BUBBLES?

• Loosely defined, bubbles in asset pricesrefer to a sudden increase in the price of anasset over and above its fundamental value,followed by a crash.

• Bursting stock market bubbles lead to hugelosses of wealth. Several instances from thedistant and the recent past have re-iteratedthe severity of such bursts.

• The review of two most recent stock marketcrashes show the type of financial damagethat the burst of bubbles can bring about -Dotcom bubble burst (1999-2002) and theSub-prime bubble burst (2007-2009).

• Following the dotcom crash NASDAQ lost78% of its value in a span of 2 years (2000-2002). The markets lost $7 trillion inmarket value.

• Sub-prime crisis (2007-2009) resulted inS&P 500 to decline 57% from October 2007to March 2009.


• Bursts of asset bubbles abruptly increasethe riskiness associated with the economicsystem, as they result in huge losses ofsavings of households and corporations,who invested in the asset .

• The frequency of such bubble bursts haveincreased over time. The frequent numberof such bursts have increased the riskinessof the financial systems of countries.

• Data suggests that from 2 crashes in 1600-1700, there has been 10 crashes between2000-2014. The average span of marketcrashes have reduced from 14 years in theseventeenth century to 1.4 years in the firsttwo decades of the 21st century.


Time Period No . of Crashes Avg. Gap of time

between two

crashes

1600 - 1700 2 14 years

1700-1800 7 10 years

1800-1900 14 5 years

1900-2000 12 3 years

2000-2014 10 1.4 years


• Given enhanced financial integration amongthe nations in 21st century, the impacts ofbubble spill overs can be severe.

• Bubble bursts cause stock markets to crash.The impacts of these crashes in one countrypasses on to other integrated.

• The Indian experience during the mostrecent financial crisis following the crash ofthe U.S. stock markets, during 2007-2009,following the burst of the U.S. HousingBubble has been significantly tough.

• Subbarao (2009) defined financialintegration as the ratio of total externaltransactions to the GDP. This had doubledfrom 46.8% in 1997-98 to 117.4% in 2007-2008.

• Following the crisis, India was affectedthrough three main channels: (i) TradeChannel (ii) Financial Channel (iii)Confidence Channel


• India was affected through three mainchannels: Trade channel (expected shockto GDP up to 1.5%), the financial channeland confidence channel (Subbarao,2009)

• The enhanced financial integrationaffected India in three related ways: (i)Reducing India’s access to overseasfinance, hence, lower or no externalborrowings (ii) Lower domesticliquidity, given lower accretion ofreserves to reserve ratio (iii) fallingstock index, as a result of net sales byFIIs. (Ram Mohan T.T; 2010)

• The chances of loss of wealth from stockmarket crashes have become very high.And, given the financial integration ofinternational markets, economies aresusceptible to periods of downturn in achain like fashion.

MAJOR STRANDS OF THOUGHT IN BUBBLE THEORIES

• Bubble: Bubble exists for a stock if itsprice deviates from its fundamental pricefor a significant period. A bubble in a stockprice can be defined as :

𝑃𝑡 = 𝐹𝑡 + 𝐵𝑡 + 𝑒𝑡 (1)

Where, 𝑃𝑡 = Price of the asset at time t.

𝐵𝑡 = Bubble component at time t.

𝑒𝑡 = White noise error term.

• Fundamental value: The fundamentalvalue of an asset is defined as the presentdiscounted value of expected futuredividends, where the discount rate is therate of return of the risk-free asset. Thisdefinition has been used primarily in theempirical works Taylor (1977), Shiller(1978), Blanchard and Watson (1982),Diba and Grossman (1984), West(1987), Phillips, Wu and Yu (2011) etc.


• Rationale behind the definition of thefundamental value: Dividend yieldprovides a compact measure of how stocksare valued vis-à-vis their fundamentals.Low dividend yield indicates that stocks areover-priced relative to their earningcapacity. So, if price of a stock increases at agreater rate relative to the increase in thedividends then the increase is not driven byfundamentals. So for stable growth, thestock prices must grow at a slower ratethan the growth of the dividend (Campbelland Shiller, 1988).

• Literature describes the existence of thefollowing main strand of thoughts inbubble theorization:

Rationalist strand of Bubble Theory

Irrational Bubble Theory

Semi-Rational or Quasi-Rational Bubble


• Rationalist strand of bubble theorisation (Rational Bubbles):

Brock (1974), Taylor (1977) and Shiller

(1978) explains that the rational expectation

models of stock pricing are characterised by

an indeterminate aspect, which arises when

the current decision of agents depends both

on the market price of the stock as well as

their expectation of future prices. As current

stock price is determined by the expectations

of the future price and future price on the

current price, any simple demand-supply

model cannot calculate the equilibrium

market price of a stock. At best, a sequence

of prices can be obtained. Only one

sequence is the market fundamental price and

all the other have price bubbles.


Blanchard (1970) pointed out that the

equilibrium amount of stock purchased by a

risk-neutral individual is the result of an

arbitrage between stocks and riskless asset, the

arbitrage equation being:

𝐸 𝑝𝑡+1 𝐼𝑡 −𝑝𝑡

𝑝𝑡+𝑑𝑡

𝑝𝑡= r

The arbitrage stops when the expected return

from stock equals the return on the riskless

asset:

𝑝𝑡 = a E 𝑝𝑡+1 𝐼𝑡 + a 𝑑𝑡

Where a =1

1+𝑟. Solving recursively up to T

periods:

𝑝𝑡 = a. 𝑖=0𝑇 𝑎𝑖E 𝑑𝑡+𝑖 𝐼𝑡 +𝑎𝑇+1 E[ 𝑝𝑡+𝑇+1 𝐼𝑡]


• Asset prices are at its fundamental when 𝑎𝑇+1

E[ 𝑝𝑡+𝑇+1 𝐼𝑡] 0 as time tends to infinity. For

the convergence of the asset prices to occur we

must have the condition that expectation of

dividends should not grow faster than the rate of

returns on the safe asset, else bubbles occur.

• Irrational bubble theory:

Minsky (1972), Kindleberger (1989) have

explained bubbles as a manifestation of the irrational

mob psychology. Camerer (1989) attributed the

occurrence of bubbles to the presence of

heterogeneous expectations of agents, which caused

some investors to believe that the bubble will last

longer than their contemporaries believe and hence

they run high chances of getting stuck with the risky

asset. Shiller (1984) attributed the occurrence of

bubbles to the epidemic and contagious nature of

‘fads’. Phillips, Wu, Yu (2011) identified the

existence of cognitive bias in the irrational dot-com

bubble.


• Semi-rational bubbles: Closely relatedto irrational bubbles, is another class ofbubbles, called semi-rational bubbles.These models of stock market contain amix of well-informed agents and ill-informed (noise) agents. These modelsmainly comprise of a ‘fundamentalist’who trade on fundamentals and ‘noisetraders’ who chase trendsGoodman(1968), Day and Huang(1990). Some models formulated by DeLong, Schleifer, Summers andWaldmann (1990), Gennotte andLeland (1990) include a third agentwho influences the market price of astock through speculation. This agent isalso known by names as ‘rationalspeculators’, ‘supply-informed investors’etc.

NEED FOR A DETERMINISTIC BUBBLE THEORY

1. The econometric models of bubbleidentification use the existence of therandom error in empirical exercise. Itbasically estimates the fundamentalcomponent and the bubble componentunder the assumption of RationalExpectations. It does not consider the roleof the endogenous turbulence that can begenerated as a result of the interaction ofagents with heterogenous expectations.

2. Samuelson (1957) pointed out thatrational bubbles are a theoreticalpossibility. Tirole (1982) established thatthere can be no bubbles if there are afinite numbers of risk-averse, infinitelylived agents, with common priorinformation and beliefs, trading a finitenumber of assets with real returns indiscrete time periods. Bubbles are mostlyirrational. It is easy to model irrationalbubbles using a deterministic model.

REVIEW OF LITERATURE ON DETERMINISTIC BUBBLES:

• The deterministic models of asset pricebubble tries to explain the turbulence inthe asset price generated endogenouslyas a result of the interaction of traderswith heterogeneous agents.

• Day and Huang (1990), Gennotte andLeland (1990) defined three types ofagents who interact in purchasing andselling assets :

• (1) The Rational Speculators – Thesetraders use the available information inmaking their trading decisions

• (2) The Noisy Traders – These tradersare the ones who believe thatinformation collection is expensive andhence they follow the trends. If pricesare going up, they believe that this wouldcontinue and they can make a profit outof it by selling, once prices rise in thefuture.

REVIEW OF LITERATURE ON DETERMINISTIC BUBBLES

(3) The Market Makers: These areinstitutions which adjust the price of theasset, depending on the excess demand forthe asset.

• The interaction of these agents create aturbulence in the dynamics of the assetprice. The turbulence is observed asswitching over from a bull market to abear market and vice-versa. Suddenswitches explain the points where abubble burst has occurred

A DETERMINISTIC REPRESENTATION

OF ASSET PRICE BUBBLE:

• OBJECTIVE: To provide a representation ofan irrational bubble in a deterministicframework. By a deterministic frame work,we imply that there are no exogeneouswhite noise in this representation. Anystochasticity within the model is generatedendogenously.

• THE FRAMEWORK: The framework of anasset price bubble that is referred here isthe one provided by Rosser (1994). Asynopsis is provided below:

(a) Initial displacement from the fundamentalvalues of the stocks, usually upwardswhere the displacements were broughtabout by system behaviour or humanerror.

(b) Speculation develops as buyers startmaking money and many such buyersfollow suit. Prices rise and ‘euphoria’emerges.

A DETERMINISTIC REPRESENTATION OF ASSET PRICE BUBBLE:

(c) The presence of Cognitive bias in taking

investment decisions has been observed in all

of these bubbles. A cognitive bias refers to a

pattern of deviation in judgement which

occurs in particular situations which may

lead to perceptual distortions, inaccurate

judgement, or what is called irrationality.

They mainly arise from lack of appropriate

mental mechanisms (such as bounded

rationality) or from limited capacity of

information processing on part of the

economic agents.

(d) The euphoria is followed by a periodof distress where expectations undergomajor change. ‘Bad news’ follows and thebubble crashes where the stocks return tothe fundamental value. This change is very

discreet in nature.


ASSUMPTIONS: This representation is based on the following assumptions:

1. The financial asset is assumed to have afundamental value of zero. So anyincrease in the price of the asset wouldbe due to non- fundamental factors.

2. Investors form their expectations aboutthe future price of the asset based onthe decisions of their fellow investorswhom they assume to be relativelywell-informed.

3. There are two types of investors in thismodel : (i) Rationalists (ii) Noisytraders. Rationalists buy because theyknow that they would make a profit byselling their holdings in the immediatefuture. The noisy traders observe thebehaviour of the rationalists and alsostart purchasing the share. Noisytraders behave in a naïve fashion in thisrepresentation.

A DETERMINISTIC REPRESENTATION

OF ASSET PRICE BUBBLE:

MAJOR EQUATIONS USED IN THE REPRESENTATION:

1. P(0) = 0 – A major challenge in identifyingbubbles in the existing literature is todistinguish between price rises caused byfundamentals and those that are caused bynon-fundamental factors. Shiller (2000)pointed out that pointing out explosivebehaviour in the price data does not implythe existence of bubble. To represent anirrational bubble the response of the noisyagents. So, fundamental price can beassumed off.

2. 𝑬𝑹 𝑷(𝟏) 𝑰𝟎 = 𝑬𝑵 𝑷(𝟏) 𝑰𝟎 = Ԑ - R implies‘Rational Speculators’ and N implies ‘NoisyTraders’. The Rational Speculators knowthe actual price of the asset, and they knowthat they can make a profit by selling it off.As they start buying based on theirexpectations of the future price, the priceincreases. Noisy traders thus buy it.


3. P(t+1) = α E 𝑷(𝒕 + 𝟏) 𝑰𝒕 - Investors buyassets in the current period in the hope ofselling it in the future for a higher price. Thisacts as a self fulfilling process- the expectedasset price movements in the present periodresult in actual price movement in the nextperiod (t+1). Excess demand for the assetwould be generated if E 𝑃(𝑡 + 1) 𝐼𝑡 > P (t).Hence α > 1.

4. E 𝑷 (𝒕 + 𝟏) 𝑰𝒕 = 𝜹𝒕 P (t) -- Investors are

driven most strongly by the price increases in the

most recent period than in the relatively distant past,

since the latter is remembered less distinctly

compared to the former. This is a simple expectation

formation rule for the investors on aggregate. This

says that every increase in the asset prices causes

the investors to scale up their expectations at an

increasing rate.


5. Necessary substitutions show that :

P (t+1) = P (t)/ a , where a = (1/ 𝜹𝒕). Thisshows the positive feedback pattern, whichwould continue as long as investors wouldexpect the prices to rise at an increasingrate. After a terminal time (𝑡𝑐 ) the assetprices stop rising and the prices come backto its fundamental value.

6. A reverse feedback pattern replaces thepositive pattern once a maximumsustainable price is obtained. As ‘Insiders’start bailing out and the ‘outsiders’ startbuying the asset. When the bail off occursfaster than the buying activities of the‘outsiders’ the asset price starts falling. Aftersome calculations we obtained the followingequation for the negative feedback pattern:

P (t+1) = [1 – P (t)]/ [1 – a] a < P (t) ≤ 1,

a є (0, 1)

MAJOR CHARACTERISTICS OF THE REPRESENTATION:

• The pair of equations that were obtainedto represent an irrational bubble isknown as Tent- Map Recursions. Thesemaps are Chaotic in nature.

• A dynamic variable is Chaotic in nature if:

(1) If the path generates trajectories thatlocally diverge away from each other

(2) If small changes in the initialconditions build up exponentially fastinto large changes.

(3) If the error of the dataset eitherremains small or grows exponentially.This distinguishes the determinism fromany stochastic models where the errorterm is random.

So in the presence of chaos smalldifference in the value of initialconditions or parameters build upthrough the model.

MAJOR CHARACTERISTICS OF THE REPRESENTATION:

• Some major characteristics of the equation

were checked previously which showed:

1. It was checked that this equation

generates the same correlation

coefficients as the first order

autoregressive process:

V (t+1) = (ka-1) V (t) + u (t+1), k is a

constant

2. Despite an initial displacement from its

fundamental value, the asset price

converges to the latter over a finite period

of time. The average displacement in the

price of the asset from its fundamental

value over a sufficiently long period of

time is zero.

THANK YOU

APPENDIX

APPENDIX

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INDIAN INSTITUTE OF CAPITAL MARKETS_CONFERENCE_18DEC2014