Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Was there a Nasdaq bubble in the late 1990s?

Lubos Pastor and Pietro Veronesi, University of Chicago

IGIER Visiting Student Initiative

Daniele Imperiale and Song Zhang

28 March 2014

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The bubble hypothesis

The first example of a “bubble”

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The bubble hypothesis

Our focus is on the dot-com “bubble”

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The bubble hypothesis

The bubble as an unintelligible phenomenon

“Before we relegate a speculative event to the fundamentallyinexplicable or bubble category driven by crowd psychology,however, we should exhaust the reasonable economic explanations...“bubble” characterizations should be a last resort because they arenon-explanations of events, merely a name that we attach to afinancial phenomenon that we have not invested sufficiently inunderstanding”. Garber (2000)

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The bubble hypothesis

The role of uncertainty on fundamental value

An important determinant of a firm’s fundamental value isuncertainty about the firm’s average future profitability, whichcan also be thought of as uncertainty about the average futuregrowth rate of the firm’s book value

Pastor and Veronesi argue that the late 1990s witnessed highuncertainty about the average growth rates of technology firmsand that this uncertainty helps us understand the observedhigh level and volatility of technology stock prices

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Uncertainty in a dividend discount model

The Gordon growth model (I)

A Dividend Discount Model is a method of valuing a company’sstock price based on the theory that its stock is worth the sum ofall its future dividend payments discounted back to their presentvalue. The DDM most widely used is called the Gordon growthmodel (Gordon and Shapiro, 1956). In this very simple model thecurrent stock price over dividend ratio is given by:

P

D=

1r − g

Where P is the stock price, D is the value of next year’s dividend, ris the discount rate and g is the mean growth rate of dividends.

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Uncertainty in a dividend discount model

The Gordon growth model (II)

What happens if the mean growth rate of dividends g is uncertain?Under some conditions discussed in the technical appendix wehave:

P

D= E

(1

r − g

)Notice that this expectation increases with uncertainty aboutg because 1r−g is convex in gThis implies that uncertainty about g makes the distributionof future dividends right skewedAlthoug this uncertainty may increase or decrease r , it alwaysincreases expected future dividends and its overall effect onP/D is positive

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Uncertainty in a dividend discount model

A simple example with the Gordon model

Ofek and Richardson (2002) argue that the earnings of Internetfirms would have to grow at implausibly high rates to justify theInternet stock prices of the late 1990s. However, things change ifwe add uncertainty on the growth rate.

Example

Consider now a stock with r = 20% and P/D = 50.

To match the observed P/D in the Gordon formula with a knownvalue of g , the required dividend growth rate is g = 18%

On the other hand, if g is unknown and drawn from a uniformdistribution with standard deviation 4%, then the expected grequired to match the P/D drops to 13.06%

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Uncertainty in a dividend discount model

The role of Jensen’s inequality

Mathematically, Jensen’s inequality implies that:

P

D= E

(1

r − g

)>

1r − E (g)

That is, plugging the expected growth rate E (g) into the Gordonformula understates the P/D ratio and this understatement isespecially large when uncertainty about g is large.

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The research question

The research question

Was there a Nasdaq bubble in the late 1990s?

NOT NECESSARILY

We are going to show you a rational framework that could explainthe rise in the stock prices of Nasdaq traded firms and the suddenddecline of them in March 2000.

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The research question

The choice of the stock valuation model

The Gordon growth model is not well suited for pricingtechnology firms because many of these firms pay no dividends

They develop a stock valuation model that focus on the ratioof the market value to the book value of equity

M/B is an increasing function of the uncertainty about theaverage growth rate of the firm’s book value

The pricing formula can then be inverted to compute the“implied uncertainty”, i.e. the level of uncertainty that sets thefirm’s model-implied M/B equal to the observed M/B

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The research question

What do they do?

They calibrate the valuation model and compute the implieduncertainty of the Nasdaq traded firms on March 10, 2000

They argue that the uncertainty level they obtain is plausiblebecause it implies a return volatility that is close to thevolatility observed in the data

Nasdaq stock prices in the late 1990s were high and highlyvolatile, and both facts are consistent with the highuncertainty about average profitability

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The empirical evidence of a higher uncertainty

Why uncertainty increased in late 1990s? (I)

Some empirical evidence of the rise in uncertainty about averagefuture growth rates of technology firms in late 1990s:

1 Nasdaq return volatility increased dramatically (Schwert, 2002)

2 Dispersion of profitability across Nasdaq stocks increased

3 Stock price reaction to earnings announcements was unusuallystrong (Ahmed et al, 2003)

4 Tech firms went public unusually early in their lifecycle(Schultz and Zaman, 2001)

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The empirical evidence of a higher uncertainty

Why uncertainty increased in late 1990s? (II)

To sum up:

The period was characterized by rapid technological progressin Internet and Telecom industriesTechnological revolutions are likely to be accompained by highuncertainty about future growth

In addition to these, the turn of the millenium was alsocharacterized by a low equity premium (Welch, 2001) and thisamplified further the effect of uncertainty on stock prices in the late1990s.

ExampleWhen the discount rate is low, a large fraction of firm value comesfrom earnings in the distant future and those earnings are the mostaffected by uncertainty about the firm’s average future growth rate.

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The empirical evidence of a higher uncertainty

Why prices went down?

While an asset price bubble caused by investors’ irrationalitycan burst at any time for any reason, there was a fundamentalreason for Nasdaq prices to come down after the 1990s: anunprecedented decline in the profitability of Nasdaq tradedfirms in 2000 and 2001

In their model, low realized profitability induces investors torevise their expectations of future profitability downward,which in turn pushes prices down

They show that the model is capable of producing a post-peakNasdaq price decline that is comparable to that observed inthe data

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The novelty

Alternative models

Two recent studies provide different explanations for the highvaluations of technology stocks in the late 1990s:

Ofek and Richardson (2003) try to demonstrate that thesevaluations were high due in part to short-sale constraintsCochrane (2003) believes that tech stocks were valued highlybecause they offered high convenience yields

However, neither study demonstrates that the magnitudes of theseeffects could be sufficiently large to justify the observed valuationsof Nasdaq firms. In addition, they don’t explain why the prices oftech stocks were so volatile at that time.

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The novelty

A step ahead in the literature

The paper is related to the theoretical literature on asset bubbles,which can be separated in two different streams:

1 rational explanations to the bubble, like in Tirole (1985) andGarber (2000)

2 neural network models for dividend, like in Donaldson andKamstra (1996)

The noveltyPastor and Veronesi develop a calibration thanks to which theyargue that the effect of uncertainty can be strong enough torationalize the valuations of Nasdaq traded firms.

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The valuation framework

Instantaneous profitability of the firm (I)

The firms i ’s instantaneous profitability at time t is defined as thefirm’s instanteneous accounting return on equity, ρit =

Y itB it. Here Y it

is the earning rate and B it is the book value of equity. Profitabilityfollows a mean reverting process (Beaver, 1970):

dρit = φi (ρit − ρit)dt + σi ,0dW0,t + σi ,idWi ,t

where φi > 0, t < Ti . W0,t and Wi ,t are two uncorrelated Wienerprocesses that capture, respectively, systematic and firm-specificcomponents of the random shocks that drive the firm’s profitability.

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The valuation framework

Instantaneous profitability of the firm (II)

The firm’s average profitability ρit can be decomposed as:

ρit = ρt + ψit

where the second term is the firm’s average excess profitability andthe first one is the expected aggregate profitability, which exhibitsmean-reverting variation that reflects the business cycle in theaggregate economy:

dρt = kL(ρL − ρt)dt + σL,0dW0,t + σL,LdWL,t

where WL,t is uncorrelated with both W0,t and Wi ,t .

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The valuation framework

The firm’s average excess profitability

We can assume there are two channels through which the firm’saverage excess profitability changes over time:

1 It slowly decays to zero so to take into account the effect ofslow-moving competitive market forces:

dψit = −kψψ

itdt kψ > 0, t < T

i

2 Competition in the firm’s product market can also arrivesuddenly, at some random future time T i . We assume that T i

is distributed exponentially with density h(T i , p). The firm’smarket value of equity at time T i equals the book value,M i

T i= B i

T i, since the sudden entry of competition eliminates

the present value of the firm’s future abnormal earnings(Ohlson, 1995)

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The valuation framework

Main assumptions

We now need to introduce some assumptions. Notice that relaxingthem would only add complexity to the model with no new insights:

1 the firm pays out a constant fraction of its book equity individends

D it = ciB it

where c i ≥ 0 is the dividend yield. This is also known as thepolicy of smoothing dividends over time.

2 the firm is financed only by equity and it issues no new equity:the only driver of growth is retained earnings

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The valuation framework

The book equity and the market value of equity

Given these assumptions, the clean surplus relation impliesthat book equity grows at the rate equal to profitability minusthe dividend yield:

dB it =(Y it − D it

)dt =

(ρit − c i

)B itdt

The market value of the firm’s equity at any point in time isgiven by the sum of the discounted value of all futuredividends and the terminal value MT = BT :

M it = Et

[ˆ ∞t

(ˆ T it

πsπt

D isds +πT i

πtBiT i

)h(T i , p)dT i

]

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

The valuation framework

The stochastic discount factor

Notice that πt is the stochastic discount factor and it is assumed tobe given by:

πt = e−ηt−γ(st+εt)

Where:

st = a0 + a1yt + a2y2t

dyt = ky (y − yt) dt + σydW0,t

dεt = µεdt + σεdW0,t

SDF

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Main results

What happens when excess profitability is known?

Proposition 1

Suppose that ψit is known. Then the firm’s M/B ratio is:

M itB it

= G i (yt , ρt , ρit , ψ

it) =

(c i + p

) ˆ ∞0

Z i (yt , ρt , ρit , ψ

it , s)ds

Then we have:1 With respect to profitability, M/B increases with ρt , ψ

it and ρ

it

2 With respect to the discount rate, M/B increases with ytwhen it is high, the equity premium is low and M/B is high

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Main results

What happens when excess profitability is unknown?

In this case we can assume that investors’ beliefs about ψit can besummarized by the probability density function ft(ψ

it). Since

G i (yt , ρt , ρit , ψ

it) is a convex function of excess profitability, then

more uncertainty about ψit implies a higher expected M/B ratio.

Proposition 2

Suppose that ψit is unknown and that the market perceives anormal distribution for it, ft(ψ

it) = N(ψ̂

it , σ̂

2i ,t). The firm’s M/B

ratio is given by:

M itB it

=(c i + p

) ˆ ∞0

Z i (yt , ρt , ρit , ψ̂

it , s)e

12Q4(s)

2σ̂2i,tds

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Main results

The key relation of the paper

From the previous equation we can conclude that M/B increases if:

1 firm’s average profitability ρit increases2 firm’s expected excess profitability ψ̂it increases3 firm’s instantaneous profitability ρit increases4 the state variable yt increases (Campbell and Cochrane 1999)5 uncertainty about the excess profitability σ̂i ,t increases

We also have two important findings:

1 The effect of σ̂i ,t is stronger for firms that pay no dividends2 Firms that pay no dividends have higher return volatility

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Introduction

Old Economy and New Economy

In the calibration section of the paper, they calibrate their model tomatch some key features of the data on asset returns andprofitability, as obtained from Compustat and CRSP. They dividefirms into:

New Economy, which includes firms traded on NasdaqOld Economy, which includes firms traded on the NYSE andAmex

We first introduce you to the calibration of the Old Economy. Thecalibration of the New Economy will come next.

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the NYSE/Amex firms

Setting the framework

Once you shut down the excess profitability ψit , you candescribe the evolution rule of the profitability of the generici-th firm in the old economy as the aggregate profitability ofNYSE/Amex

The old economy pays aggregate dividends forever at a rate ofDOt = c

OBOt and they compute cO = 5.67% as the

time-series average of the old economy’s annual dividend yields

The old economy’s aggregate market value is given byequation (16):

MOtBOt

= Φ(ρt , yt) = cO

ˆ ∞0

Z (yt , ρt , s) ds

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the NYSE/Amex firms

Step I: calibration of aggregate profitability (I)

The first equation they calibrate is equation (4):

dρt = kL(ρL − ρt)dt + σL,0dW0,t + σL,LdWL,t

Equation (4) implies a normal likelihood function for ρt , asdescribed by the following Lemma (Duffie, 1996).

Lemma 4For any linear vector process zt that satisfies:

dzt = (At + Bzzt)dt +∑zdWt

we then have:

zt+τ |zt ∼ N(µz(zt , τ),Sz(τ))

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the NYSE/Amex firms

Step I: calibration of aggregate profitability (II)

To estimate the process for ρt , they compute the aggregateprofitability as the sum of the current-year earnings across allNYSE/Amex firms, divided by the sum of the book values ofequity at the previous year-end

Then they match this empirical distribution of aggregateprofitability with the theoretical distribution of profitability, asobtained by Lemma 4, using maximum likelihood

This procedure gives the parameter values of equation (4) thatwill be used for the calibration of the model

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the NYSE/Amex firms

Step I: calibration of aggregate profitability (III)

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the NYSE/Amex firms

Step II: calibration of the M/B ratio (I)

They construct the 1962-2002 annual time series of the oldeconomy’s M/B by computing the ratio of the sums of themarket values and the most recent book values of equityacross all NYSE/Amex firms

From the pricing equation MOt /BOt = Φ(ρt , yt) they want to

obtain the time series of yt thanks to which you can match theobserved M/B to the M/B implied by the model for eachperiod

The pricing equation is actually misleading since we need theparameters of the SDF system in order to invert it:Π = (η, γ, ky , y , σy , a0, a1, a2, µε, σε)

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the NYSE/Amex firms

Step II: calibration of the M/B ratio (II)

They construct the moment conditions from the stationarydistribution of (ρt , yt), obtained by substituting it for zt inLemma 4

They also impose additional moment conditions to ensure thatthe average values of the estimated equity premium µR,t ,market return σR,t , and the real interest rate rf ,t are close tothe values observed in the data

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the Nasdaq firms

Step III: calibration of instantaneous profitability (I)

Then they move to the calibration of equation (2):

dρit = φi (ρit − ρit)dt + σi ,0dW0,t + σi ,idWi ,t

To do this, they use the same method we showed you before.In particular:

they estimate the parameters of that process by maximumlikelihood

here the maximum likelihood function is obtained bysubstituting (ρNt , ρt , yt) for zt in Lemma 4

we take as given the parameters of the ρt and yt processesdescribed before

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the Nasdaq firms

Step III: calibration of instantaneous profitability (II)

On March 10, 2000, on the Nasdaq:the current profitability was ρNt = 9.96% per yearthe M/B ratio was equal to 8.55the dividend yield was c = 1.35%

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the Nasdaq firms

Nasdaq’s valuation without uncertainty

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the Nasdaq firms

Discussion (I)

They compute the Nasdaq’s annualized daily return volatilityin March 2000 and they obtained 41.49% per year

They also compute the average of monthly volatilities in 2000and obtained 47.03%

Both values are far above the model-implied volatility values inPanel B. Their model is unable to match Nasdaq’s return volatilityunder the assumption of zero uncertainty.

M itB it

= G i (yt , ρt , ρit , ψ

it)

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the Nasdaq firms

Nasdaq valuation with uncertainty

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the Nasdaq firms

Discussion (II)

Acknowledging uncertainty about ψN leads to values of M/Band volatility that are closer to the observed values

Uncertainty about ψN has the greatest effect on prices andvolatilities when the equity premium is low

Using the same approach of Table 2 and 3, they identify threepairs of implied uncertainty and equity premium for whichimplied uncertainty matches Nasdaq’s M/B and its returnvolatility

One such pair is: ψ̂N = 3% and equity premium equal to 3%,which leads to implied uncertainty of 3.38%

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Calibration of the Nasdaq firms

Return volatility, uncertainty and expected profitability

σiR =1

Φi

(∂Φi

∂ytσy +

∂Φi

∂ρtσL +

∂Φi

∂ρitσi +

∂Φi

∂ψ̂itσψ̂,t

)

This equation predicts a linear positive relation betweensquared implied uncertainty and idiosyncratic return volatility,as showed in Table 2 and Table 3

The return volatility increases with average excess profitability:higher values of ψ̂t

Nimplies that firm profits lie farther in the

future, which makes the stock price more sensitive to revisionsin ψ̂t

N

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Why did the bubble burst?

The time series of M/B ratios

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Why did the bubble burst?

The time series of realized profitability

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Why did the bubble burst?

Updating of the beliefs (I)

In the model, investors update their beliefs about the Nasdaq’saverage excess profitability ψN by observing the realizedprofitability of Nasdaq and NYSE/Amex

Given the unprecedented fall in Nasdaq’s ROE, ψ̂N must havebeen revised downward and this revision is likely to have beensubstantial, because of the high uncertainty at that time andthe properties of Bayesian updating

In conclusion, they attribute the bursting of the bubble tounexpected negative news about Nasdaq’s average futureprofitability

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Why did the bubble burst?

Updating of the beliefs (II)

They assume that investor’s prior beliefs about ψ̂N in March2000 are summarized by the normal distribution with mean 3%and standard deviation of 3.38% per year

These beliefs are then revised by the Bayesian investors whoobserve realized profitability subsequent to March 2000

Given the resulting posterior beliefs about ψN , they computethe model-implied M/B and return volatility at the year-endsof 2000,2001 and 2002 and compare them with observed data

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Why did the bubble burst?

Quantitative predictions on the M/B ratio

Because of the poor realized profitability and uncertainty, theposterior mean of ψN drops from 3% at the end of 1999 to0.4% at the end of 2000, to −3% in 2001 and −2% in 2002

Posterior uncertainty about ψN declines slowly due to learning,from 3.38% in March 2000 to 2.98% in 2002

Given the large negative revision to ψN in 2000, themodel-implied M/B of Nasdaq drops from 8.6 to 3, while theactual M/B exhibits a decline from 8.6 to 3.5 over the sameperiod

In 2001, the Nasdaq’s model-implied M/B drops to 0.9, whilethe actual M/B dropped only to 3.3

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Why did the bubble burst?

Quantitative predictions on return volatility

Nasdaq’s model-implied volatility declines from 46.8% inMarch 2000 to 39.3% at the end of 2000 and 27.7% at theend of 2001, before rising to 29.4% at the end of 2002

Nasdaq’s actual year-end volatility falls from 47.2% in 2000 to32% in 2001, before rising to 37.4% in 2002 and thisdifference is on the margin of statistical significance

Possibly the post-peak downward revision in ψN was not aslarge as predicted by the model and higher perceived ψN wouldimply higher volatility

In fact, higher ψN implies that firm profits lie farther in thefuture, which makes the stock price more sensitive to revisionsin ψN

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Wrap up

Rationalization of Nasdaq prices

They argue that Nasdaq valuations were not necessarilyirrational ex ante because uncertainty about averageprofitability was unusually high in the late 1990s

After calibrating a stock valuation model that incorporatessuch uncertainty, they compute the level of uncertainty thatrationalizes the observed Nasdaq valuations at the peak of the“bubble”

They find the implied uncertainty plausible because it matchesnot only the high level but also the high volatility of Nasdaqstock prices

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Wrap up

The role of expected profitability

While they argue that uncertainty had a role in the rise ofNasdaq prices in the late 1990s, they do not claim that pricesfell in 2000 due to a decline in uncertainty

They argue that Nasdaq’s expected profitability wassubstantially revised downward when Nasdaq’s profitabilityplummeted in 2000 and 2001

The model they have developed is capable of explaining largeprice declines on Nasdaq after March 2000

The model also produces a post-peak pattern in returnvolatility that is comparable to the empirical pattern

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Concluding remarks

Strengths and Weaknesses

Strengths:Calibrate a rational valuation model to match the level andvolatility of stock prices both for Nasdaq and NYSE/Amexindexes for the last 30 years and this is an additional novelty inthe financial literatureThey argue that the level and volatility of stock prices arepositively linked through firm-specific uncertainty aboutaverage future profitability

Weaknesses:Even if stock prices in March 2000 appear to be consistentwith a rational model, maybe investors were not fully rationaland behavioral finance actually plays a roleThe model produces a price decline in 2001 that is too largerelative to the data

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

Concluding remarks

Thank you!

Questions & Answers

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

How do they construct the SDF? (I)

SDF

The SDF is derived in PV 2005 from a habit utilty modelintroduced by Campbell and Cochrane in 1999. In particularwe have:

U(C kt ,Xt , t) = e−ηt(C kt − Xt

)1−γ1− γ

where Xt is an external habit index, γ regulates the local curvatureof the utility function and η is the time discount parameter.

Then they let Ct =∑

k Ckt denote aggregate consumption and

St =Ct−XtCt

denote the surplus consumption ratio

They then define εt = log(Ct) and st = log(St)

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

How do they construct the SDF? (II)

The log of the surplus consumption ratio st follows amean-reverting process with time-varying volatility and perfectcorrelation with unexpected consumption growth.To obtain their version of the analytical solutions for prices,PV 2005 assume that:

st = a0 + a1yt + a2y2t

Here yt is a state variable driven by the followingmean-reverting process:

dyt = ky (y − yt) dt + σydW0,t

Introduction The role of uncertainty The Model Calibration Discussion Conclusions

How do they construct the SDF? (III)

In PV 2005 they derive the unique stochastic discount factorfrom a model in which there are only two types of individuals(investors and inventors) and the markets are complete, so toensure that inventors and investors can perfectly insure eachother’s consumptionThey will chose identical consumption plans and the SDF isthen given by:πt = Uc(Ct ,Xt , t) = e

−ηt (CtSt)−γ = e−ηt−γ(εt+st)

The resulting process for the stochastic discount factor is givenby:

dπt = −rf ,tπtdt − πtσπ,tdW0,tThey assume that the log aggregate consumtpion follows thefollowing process:

dεt = (b0 + b1ρt) dt + σεdW0,t

IntroductionThe bubble hypothesis

The role of uncertaintyUncertainty in a dividend discount modelThe research questionThe empirical evidence of a higher uncertaintyThe novelty

The ModelThe valuation frameworkMain results

CalibrationIntroductionCalibration of the NYSE/Amex firmsCalibration of the Nasdaq firms

DiscussionWhy did the bubble burst?

ConclusionsWrap upConcluding remarks