Advisors and Asset Prices: A Model of the Origins of Bubbles

Harrison Hong, Jos ́e Scheinkman, and Wei Xiong

Journal of Financial Economics 89 (2008) 268– 287

Introduction:

1. What are the origins of speculative asset price bubbles? 
2. That differences of opinion among investors and short sales constraints are sufficient to generate a price bubble 
3. Once a bubble begins, it is difficult for smart money to eliminate the mispricing 
4. All these studies take as given - investors disagree about asset values 
5. But where does this divergence of opinion come from? 
6. Two sets of stylized facts motivate the analysis 
7. Asset price bubbles tend to occur during periods of excitement about new technologies 
8. (1) railroads, (2) electricity, (3) automobiles, (4) radio, (5) micro-electronics, (6) personal computers, (7) biotechnology, (8) the Internet 
9. Aftermath of the bubble, the media and regulators placed much of the blame on biased advisors for manipulating the expectations of naive investors 

analyst incentives to generate biased, optimistic forecasts 

naive individual investors who do not recognize that these biased recommendations are motivated by incentives to sell stocks 

analysts’ optimistic forecasts have an impact on prices

This paper:

1. focus on the role of advisors and their communication process with investors in generating divergence of opinion and asset price bubbles 
2. two types of investors - smart and naive (short sales constrained) 
3. smart ones - recognize the heterogeneity in advisors 
4. naive ones - take whatever recommendations they receive at face value 
5. all advisors are well-intentioned 
6. only some advisors understand the new technology - the tech-savvies) 
7. only make a downward-biased recommendation - the old-fogies 
8. only issue upward-biased recommendations - the dreamers
9. Over-valued - it may not be due to investors overreacting to news nor to sell-side bias 
10. such bias can explain bubbles that have occurred during earlier periods? 
11. other causes of upward biased forecasts by advisors aside from the sell-side incentives of analysts 
12. An exploration of an alternative and theoretically interesting mechanism for generating divergence of opinion as opposed to simply assuming investors overreact to news

Model:

• There are three dates, denoted by t = 0, 1, 2 
• The stock pays a liquidating dividend at t = 2 given by v= θ + ε 
• θ is uniformly distributed on the interval [0, 1], and ε is normally distributed with a mean of zero and a variance of σ2 
• There are two types of advisors in the economy 
• tech-savvy (with a mass of π0 ∈ [0, 1] in the population) 
• old-fogies (with a remaining mass of 1 − π0)

• two types of investors at t = 0 
• smart ones (with a mass of ρ ∈ [0, 1] in the population) 
• naive ones (with a remaining mass of 1 − ρ) 
• assume that investors cannot short sell shares and there is an upper bound to the number of shares an investor can hold, which we denote by k 
• At t = 1, 
• The fixed cost of the project is I, which is a constant between 0 and 1 
• The payoff of the project is f, which is uniformly distributed on the interval [0, 1] 
• f is independent of θ 
• Tech-savvy advisors observe f and send a report to investors at t = 1, denoted by s1TS 
• incur a dishonesty cost per advisee of c(s1TS −f)2 
• Old-fogies again do not understand this new technology, and they send a signal at t = 1 given by s1OF =af (a parameter restriction that a ≥ I)

• The motivation for the t = 1 set-up is that it is a reduced-form model meant to capture a stream of future advising engagements in an advisor’s career 
• More specifically, one can think of the advisor as a sell-side analyst at date 0 who becomes a consultant to hedge funds or corporations on other projects later in his career (date 1) 
• Those client institutions at date 1 have information regarding his track record as a sell-side analyst 
• More realistically, advisors with better reputations, i.e. higher π1’s, would attract a larger number of advisees at date 1

Proposition 1
Smart investors have perfect information about advisor type - Suppose that π1 = 1 or π1 = 0
The tech-savvy advisor truthfully reports his information s1TS = f
The old-fogey reports s1OF = af

Proposition 2

Proposition 3

Verify the optimality of the smart investor’s inference rule

If s0 ≥ a, the signal must come from a tech-savvy advisor, since old-fogies would never report such a signal π1 = 1

If s0 ∈ (θ∗,a), the signal must come from an old-fogey, since tech-savvies would never report signals in this region

If s0 ≤ θ∗, the signal could come from either a tech-savvy or an old-fogey

The cut-off value θ∗ captures the degree to which the tech-savvy advisor biases his report in order to build a better reputation 

The lower is θ∗, the greater the bias 

a − θ∗ can be interpreted as a measure of the upward bias in the tech-savvy’s reporting strategy

Proposition 4
The upward bias of the tech-savvy advisor’s reporting strategy (as measured by a − θ∗)
increases with the number of advisees at t = 1 (n)
and the fraction of smart investors (ρ)
and decreases with the fraction of tech-savvy advisors (π0)
Since the upward bias in the initial report is driven by the incentive to gain a better reputation

Asset price at t = 0
An individual investor i, who observes a signal si,0 from his advisor and takes the asset price p as given, chooses his asset holding xi (shares) to maximize his expected final wealth

investor cannot short sell the asset and can only take a position smaller than k
so investor’s optimal position :

types of investor- advisor pairs
the equilibrium asset price is determined by the highest belief among these classes of investors 

Case 1: θ > a

the asset price is θ, which is unbiased 

Case 2: θ ∈ [θ∗,a]

the asset price is a, with an upward bias that equals a − θ 

Case 3: θ < θ∗

the equilibrium asset price is unbiased—given by θ 

the smart investor would not bid a price equal to the expected asset fundamental πLθ + (1 − πL)θ/a

because of the winner’s curse (receive the asset that imply that no one is bidding θ/a)

Aware of this curse, any smart investor would only bid a price θ

Theorem 2
When there is a sufficient number of naive investors advised by tech-savvy advisors
the equilibrium stock price is identical to the tech-savvy advisors’ signal


tech-savvy advisors’ message inflation would directly lead to a price bubble
i.e., the asset price is upward biased by a − θ for asset fundamental θ between θ∗ and a

Proposition 5

The price bubble (bias) is maximized when there is a mix of naive and smart investors
The recommendation bias on the part of the tech-savvy advisors increases with the proportion of smart investors

When there are only smart investors in the market, tech-savvy advisors would have the greatest incentive to signal their type by inflating their signals, but smart investors understand this and will de-bias the signals accordingly. Thus, there is no price bias on average. 

When there are only naive investors in the market, tech-savvy advisors have no incentive to inflate their signals. As a result, there is no price bias either. 

Taken together, the price bias is maximized when there is a mix of naive and smart investors.

Proposition 6

The reporting strategy of a tech-savvy advisor with a reputation of π ∈ [0, 1] of being tech-savvy


where fTS is the advisor’s belief about the project fundamental f.
The parameters f∗ and d are determined by the following equations

The reporting strategy of an old-fogey advisor with reputation π is the same as that of a tech-savvy advisor with the same reputation

where fOF is the advisor’s belief about the project fundamental f

After observing the signal, a naive investor invests if and only if the signal is above I.
A smart investor does not invest if the signal is equal to or below dI and invests otherwise.

for a benevolent tech-savvy advisor with a reputation π
the expected inefficiency - equal to his dishonesty cost plus the investment loss by his advisees

For a benevolent old-fogey advisor with a reputation π, the expected inefficiency is

Proposition 7

Both KTS(π) and KOF (π) decrease with π and are zero when π = 1
at t = 0, tech-savvy advisors have incentives to separate themselves from old-fogey advisors by reporting an optimistic signal
At the same time, old-fogey advisors also have the incentive to mix with tech-savvy advisors by inflating their signals as well

(1) a separating outcome in which tech-savvy advisors report an extremely optimistic signal that is too costly for old-fogey advisors to match, when the fundamental is sufficiently high 

(2) a pooling outcome, in which tech-savvy advisors truthfully report their belief, and old-fogey advisors match such a recommendation, when the fundamental is not too high

Proposition 8
Under certain sufficient conditions

Given a tech-savvy advisor’s belief θTS, which is equal to the true value θ, his reporting strategy

Given an old-fogey advisor’s belief θOF , which is equal to aθ, his reporting strategy

When θ < θ∗,the naive investor’s belief turns out to be correct
When θ ≥ θ∗, his belief is

upward biased when he is matched with a tech-savvy advisor 

downward biased when matched with an old-fogey advisor 

technology fundamental θ is sufficiently high (θ ≥ θ∗) 

tech-savvy advisors are able to separate themselves from old-fogey advisors by inflating their signal 

When θ is small (θ < θ∗) 

tech-savvy and old-fogey advisors’ beliefs are close enough that it becomes too costly for tech-savvy advisors to separate themselves 

there is a pooling equilibrium in which old-fogies inflate their signal to match the tech-savvies’ truthful report

Proposition 9
Because of the short sales constraints, the asset price at t = 0 is determined by the highest belief in the market
When there is a sufficient number of naive investors advised by tech-savvy advisors, the asset price at t = 0 is determined by

The asset price is upward-biased when θ ≥ θ∗ and is unbiased otherwise

Extensions

• both dreamer and old- fogey advisors in the economy 
• assume that dreamers can only send an upward-biased signal about the new technology at t = 0: 
• s0DR =b+(1−b)θ 
• as b increases, the dreamers’ signal becomes more optimistic 
• initial distribution of the three types of advisors, dreamers, old-fogies, and tech-savvies, by πDR, πOF, and πTS, respectively (πDR + πOF +πTS =1)

Theorem 3
The reporting strategy of a tech-savvy advisor is

if s0 ≥ θ2∗, the advisor can either be

tech-savvy with a probability of πTS/(πTS+ πDR/(1−b)) 

dreamer with a probability of (πDR/(1−b))/(πTS+ πDR/(1−b)) 

if b < s0 < θ2∗,

the advisor is a dreamer for sure 

if a ≤ s0 ≤ b,

the advisor is a tech-savvy for sure 

if θ1∗ ≤ s0 < a,

the advisor is an old-fogey for sure 

if s0 ≤ θ1∗, the advisor can either be

tech-savvy with a probability of πTS/(πTS+ πOF/a) 

old-fogey with a probability of (πOF/a)/(πTS+ πOF/a)

Note that for high (but not too high) realizations of the fundamental, θ ∈ (b,θ2∗), 

the tech-savvy advisor deflates his report to b 

worried about being pooled with dreamers in this region 

And for low (but not too low) realizations of the fundamental, θ ∈ (θ1∗, a), 

the tech-savvy advisor inflates his report to a 

worried about being pooled with old-fogies in this region

Proposition 10
when investors are more concerned about their advisors being an old-fogey (i.e. πOF is higher),

the tech-savvy inflates his report for a larger range of fundamental values (i.e. a−θ1∗ is bigger) 

when investors are more concerned about their advisors being a dreamer (i.e. πDR is higher),

the tech-savvy deflates his reports for a larger range of fundamental values (i.e. θ2∗ − b is bigger) 

Keeping πTS constant,

an increase in πOF (which corresponds to a decrease in πDR for the probabilities to sum up to one) 

would cause a − θ1∗ to rise and θ2∗ − b to fall 

Since there are short- sales constraints

there will be an upward price bias and the bias is greater when there is more concern about old-fogies

Conclusion

• Asset price bubbles based on the communication process between advisors and investors 
• Advisors - the tech-savvies , the old-fogies (, and the dreamers) 

well-intentioned & maximize the welfare of their advisees 

• Investors - smart ones & naive ones 
• Tech-savvies inflate their forecasts to signal that they are not old-fogies 
• A bubble is maximized - when there is a mix of smart and naive investors in the economy 
• Suggesting an alternative source for stock over-valuation in addition to investor overreaction to news and sell-side bias