Who should bet on rare events?

Preliminary and Incomplete

November 12, 2014

This paper studies incentives for trader specialization in assets paying off in rare or

common events. A trader specializing in a rare event asset will typically receive a negative

signal and take a short position. This “nickels in front of a steamroller” strategy results from

traders’ optimal specialization. Highly skilled traders specialize in common-event assets,

while less-skilled traders specialize in rare-event assets. We also study long-term careers

where agents learn about ability. Rare-event traders trade immediately, while common-

event traders trade significant positions only after they have succeeded with small positions.

The highest expected returns accrue to traders who have successfully predicted the rare

event; however, the aggregate importance of learning in this asset is small.

1 Introduction

What kinds of assets will traders choose to specialize in? Financial market efficiency re-

quires agents to produce information. Information production, however, is costly, requiring

time and effort that could have expended on other economic activities. In particular, the

opportunity cost of producing information about one asset may be the foregone production

of information about another asset. If there is there is some noise in the financial market

price system then in equilibrium the price will contain some but not all of the informa-

tion produced, allowing agents to recoup the opportunity costs of information production

with trading profits (Grossman and Stiglitz, 1980, Kyle, 1985). Plausibly, specialization is

a long-term decision. For example, someone might specialize in valuing mortgage backed

securities, stock prices of potential takeover targets, foreign exchange, or out of the money

options. Some types of information production might be more profitable than others, for

example, information that will be revealed in the short term may be more profitable than

long term information (Hirshleifer, 1971), and information that is closely correlated with

asset payoffs may be more attractive than information that can only be traded with basis

risk (Dow, 1998). This would cause agents to choose to specialize in predicting certain kinds

of information.

This paper studies the incentives for trader specialization in assets that pay off in rare or

in common events. For example, buying CDS or investing in start-ups are trades that pay off

rarely, while a foreign exchange dealer makes trades that are almost equally likely to win or

lose. The question is important in understanding the possibility of biases in financial market

information production, which would have implications for the distortion of the real economy.

Agents can take both long and short positions, so a trader specializing in an asset which

rarely pays off will typically receive a negative signal and take a short position (depending

on trading costs such as the bid-ask spread). Therefore, data on such a trader’s position

will show frequent trades that are low profitability when they succeed and large losses when

they occasionally fail. Such trades are short positions in a rare-event asset. Occasionally, the

1

trader will take a long position with the opposite characteristics: a large payoff if it succeeds

and a small loss if it fails, as is characteristic of a long position in the rare-event asset.

To an outside observer, given the high frequency of what we describe as short positions in

the rare event asset, it will appear that such a trader is typically engaged in “picking up

nickels in front of steamroller.” This type of trading strategy has been frequently criticized

as symptomatic of distortions caused by agency problems, but in our model is simply the

result of traders optimizing their information production.

In this paper, we analyze these questions in a benchmark equilibrium model, in which

we simultaneously study equilibrium in financial and labor markets. The interaction of

equilibrium conditions in these two markets is important. If assets simply trade at their

expected value, or with an exogenous bid–ask spread around their expected value, one could

hypothesize that certain types of assets would be more attractive to trade based on the

ability to scale up position size. For example, the rare event asset, being cheap, allows

traders with limited capital to buy large quantities. This would seem attractive to traders

who are confident in their ability, but to analyze this choice properly one needs to allow for

the larger bid-ask spread that would result from an influx of traders with good prediction

ability. Further, since the cost of trading in one asset is the opportunity cost of foregone

trading profits in another asset, the equilibrium allocation of traders to assets depends on

the relative marginal value of trading skill on the two assets.

We show that, in equilibrium, the most skilled traders specialize in trading common-event

assets, while less-skilled traders specialize in rare-event assets. Since most trades in the rare-

event asset are short positions, our model predicts that “picking up nickels” strategies are

pursued by lower-skilled traders. As a consequence of this predicted allocation of skill, our

model predicts that rare-event assets are less subject to adverse selection, in the sense that

the absolute bid-ask spread is smaller. Nonetheless, rare-event assets are expensive, in the

sense that the unconditional expected return on rare-event assets is low. Empirically, this is

consistent with “smirks” in option markets, with the value premium in equity markets, and

2

with the favourite-long shot bias.

If specialization in trading a particular type of asset is a career choice, we also need to

study the long-term considerations that affect specialization. An agent who specializes in a

particular type of trading will learn over time. Suppose that different agents have different

abilities; then their trading records will reveal their skill levels. It seems reasonable to

assume that prior to trading, agents have rather imprecise information about their potential

ability as traders. An asset that pays off half the time, after one observation, will give a

small amount of information about underlying ability. On the other hand an asset that pays

off rarely will either give almost no information (if the non-occurrence of the rare event is

predicted) or a large amount of information (if the agent correctly predicts the rare event).

For an agent with low ex-ante expectations of ability, this offers the possibility (albeit rarely

realized) of learning enough to have an accurate signal.

In equilibrium, the career paths of traders in the two assets are very different: rare-event

traders trade as soon as they enter the labor market, while common-event traders trade

significant positions only after they have successfully traded small positions. The highest

expected returns accrue to traders who have previously successfully predicted the rare event;

however, the aggregate importance of learning in the rare-event asset is very small.

In our analysis, traders are allowed to take either short or long positions. The results are

robust, however, to restricting the agents to trade in only one direction. Restricting traders

to long-only positions is a natural restriction in some circumstances, as the transaction costs

of short positions may be higher than for long positions. Restricting to short positions only

is relevant if we redefine assets to be the negative of the assets considered here; then the

results on short positions on rare-event assets can be interpreted as results on long positions

of asset with very high probability of generating a small profit and occasional large losses.

In other words, the above results on “picking up nickels” results carry through even if agents

are not allowed to do the reverse of such trades.

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1.1 Related literature

[To be added]

2 Model

2.1 Assets

There are two independent random variables, ψr and ψc, each distributed uniformly over

[0, 1]. There is a single financial asset associated with each of the random variables ψr and

ψc, and we denote these assets as the r-asset and the c-asset. The r-asset pays 1 if ψr ≤ qr,

where qr is a constant. Likewise, the c-asset pays 1 if ψc ≤ qc, where qc is a constant. Thus,

the r-asset pays off with probability qr and the c-asset pays off with probability qc. The

r-asset is the rare-event asset: we assume qr < qc, and moreover, we focus primarily on the

case in which qr is very small. For j = r, c, we write ωj1 for the event{

ψj ≤ qj}

, and write

ωj2 for the complementary event{

ψj > qj}

, so Pr(

ωj1)

= qj and Pr(

ωj2)

= 1− qj.

2.2 Skilled traders

There is a continuum of “skilled” traders. Each trader either observes an informative signal,

or observes noise, as described below. No-one, include traders themselves, know which

type they are. However, there is publicly observable heterogeneity in the likelihood that a

trader will observe an informative signal. Formally, each trader is associated with a publicly

observed probability α that he observes informative signals. The population distribution of

α is given by a measure µ̄, which we assume to be atomless. We refer to each trader’s α as

his perceived skill, or often simply as “skill.”

Collecting information takes time. To capture this, we assume that each trader must

choose between receiving signals about ψr or signals about ψc. Conditional on (ψr, ψc),

signals are independent across traders. A trader with skill α who chooses to observe a signal

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about ψj observes the true realization with probability α, and observes the realization of a

noise term uniformly distributed over [0, 1] with complementary probability 1− α.

After observing his signal, a trader chooses whether to trade. He can trade either asset,

and can take both long and short positions. If he takes a long position in the j-asset, he

trades at the ask price, which we denote by PAj. Likewise, if he takes a short position, he

trades at the bid price, denoted by PBj. These prices are determined in equilibrium, as

described below.

Skilled traders are risk-neutral. However, they face position limits. A potential attraction

of trading the r-asset is that its price is low, and so a trader can adopt large positions. To

allow for this, we model position limits as potentially depending on the price. For example,

position limits would arise naturally if traders have limited wealth W . Suppose that traders

have to hold cash as margin that is sufficient to close out their positions in all outcomes.

This is a natural assumption when the asset payoff is a binary random variable as we have

assumed, or more generally has a finite support. In other words for a long position, the

lowest realization of asset value should be sufficient to pay off any loan that was taken out

to buy the asset, in addition to the trader’s initial weath. For a short position, the initial

wealth plus proceeds from the short sale should be enough to buy back the borrowed stock

at its highest possible value. In this case, the largest long position would be W/PAj, and

the largest short position would be W/(

1− PBj)

.1

More generally, when asset supports are not finite, as well as in reality, position limits

could be determined differently. For example, traders could be required to hold enough

collateral to close out their positions with a given high probability. We therefore generalize

these position limits, and assume the largest long position a trader can take is hj(

PAj)

,

and the largest short position he can take is hjS(

PBj)

, where both hj and hjS are continuous

functions over (0,∞), and take strictly positive values over this interval.2 We impose:

1Note that the short position limit arises as follows. If he short sells x units, he has total wealthW+xPBj .This is sufficient collateral for x = W + xPBj short positions. Solving for x gives the short position limit.

2For our results, it is enough for just one of hj and hjS to be strictly positive.

5

Assumption 1 limP→0 PhjS (P ) = 0.

Intuitively, as the rare event becomes rarer its price (presumably) falls. The assumption

says that a trader’s short position cannot grow fast enough to dominate this effect. Note that

in the collateral motivation of position limits, a trader’s maximum short position actually

shrinks as the price falls, since the short proceeds fall and so he can collateralize fewer

short positions. In this sense, the assumption is therefore very weak. At the same time,

it is important for our analysis to make an assumption limiting the size of short positions,

otherwise trading the rare-event asset would become overwhelmingly attractive due purely

to traders’ ability to take implausibly large positions.

We initially assume traders live for just one period. We extend this to two periods in

Section 6 where we consider learning.

2.3 Liquidity (or noise) traders

In addition to skilled traders, there is a continuum of uninformed traders who trade for non-

informational reasons. We refer to these traders as liquidity traders. Each liquidity trader

receives an endowment shock that gives him a strong desire for resources in a particular

state. A measure λr of liquidity traders are r-liquidity traders, and each receives a shock

χr ∼ U [0, 1], meaning that a liquidity trader wants resources in the state ψr = χr. Liquidity

trader shocks are correlated across traders, so the aggregate trade of liquidity traders is

stochastic. Similarly, a measure λc of liquidity traders are c-liquidity traders, and each

receive a shock χc ∼ U [0, 1], meaning that a liquidity trader wants resources in the state

ψc = χc. For simplicity, we assume that j-liquidity trader preferences for resources in state

χj are lexicographic, so that each j-liquidity trader takes as large a long position as possible

in the j-asset as possible if χj ≤ qj , and as large a short position as possible if χj > qj . The

long and short position limits for j-liquidity traders are the same as for skilled traders, i.e.

wjhj and wjhjS for some constant wj. It may seem natural to fix λr = λc and wj = wj;

however our results do not require this.

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Given the above, a j-liquidity trader buys wjhj(

PAj)

units of the j-asset if he experiences

shock χj ≤ qj , and short sells wjhjS(

PBj)

units of the j-asset if he experiences shock χj > qj.

Consequently, the expected number of buy orders for the j-asset from liquidity traders equals

qjλjwjhj(

PAj)

, while the expected number of sell orders equals (1− qj) λjwjhjS(

PBj)

.

Note that we have motivated non-informational trades as liquidity trades. Two alter-

native motivations are that these trades are simply “noise” trades, or alternatively, these

trades come from traders who have zero skill but mistakenly believe they have high skill.

Our analysis below is independent of which of these three assumptions is adopted.

The liquidity traders in our model are rational in the sense of making choices that are

consistent with a complete and transitive preference ordering, although their demands for

the asset are price inelastic. Allowing price elastic demands for the liquidity traders would

make the derivation of equilibrium more complex.

2.4 Financial market structure

Both skilled and liquidity traders submit orders to a competitive set of (uninformed) market-

makers. Skilled traders can split their trades so that market-makers cannot distinguish orders

from skilled and liquidity traders. This pricing rule is similar to that in Glosten and Milgrom

(1985). Given competition, the following zero-profit conditions hold, and determine ask and

bid prices (it is helpful here to remember that “buys” are market maker sales):

Pr(

ωj1)

E[

buys|ωj1] (

PAj − 1)

+ Pr(

ωj2)

E[

buys|ωj2]

PAj = 0

Pr(

ωj1)

E[

sales|ωj1] (

1− PBj)

+ Pr(

ωj2)

E[

sales|ωj2] (

−PBj)

= 0.

Rearranging gives

PAj = Pr(

ωj1) E

[

buys|ωj1]

E [buys](1)

PBj = Pr(

ωj1) E

[

sales|ωj1]

E [sales]. (2)

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2.5 Minimum skill levels for profitable trading

Consider a skilled trader of skill α who specializes in the j-asset. If he observes a signal

sj ∈ [0, 1] and buys, his expected profits on each unit bought are

Pr(

ψj ≤ qj|sj)

− PAj, (3)

while if he sells, his expected profits on each unit sold are

PBj − Pr(

ψj ≤ qj|sj)

. (4)

Evaluating,

Pr(

ψj ≤ qj |sj)

= α1sj≤qj + (1− α) Pr(

ψj ≤ qj)

= α1sj≤qj + (1− α) qj . (5)

Consequently, a skilled trader buys after seeing signal sj ≤ qj if and only if his skill α exceeds

PAj − qj

1− qj, (6)

while he buys after seeing signal sj > qj if and only if his skill is below

1−PAj

qj. (7)

Likewise, a skilled trader sells after seeing signal sj ≤ qj if and only if his skill is below

PBj − qj

1− qj, (8)

while he sells after seeing signal sj > qj if and only if his skill exceeds

1−PBj

qj. (9)

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2.6 Equilibrium definition

An equilibrium consists of prices(

PAr, PBr, PAc, PBc)

and an allocation of skilled traders

(µr, µc, µ0) across r-events, c-events and doing nothing, such that:

1. Labor market equilibrium:

(a) A trader of skill α specializes in the event j that delivers maximal profits, given

prices.

(b) Labor markets clear: µr (α) + µc (α) + µ0 (α) ≤ µ̄ (α) for almost all skill levels α.

2. Financial market equilibrium: Given profit-maximizing trading by skilled traders,

prices satisfy (1) and (2).

Throughout, we follow the convention that if a trader of skill level α never trades, then

he belongs to the do-nothing group, i.e., µ0 (α) = µ̄ (α).

3 Benchmark: No financial market equilibrium

As a benchmark, we briefly consider an economy in which equilibrium conditions are not

imposed in the financial market. Instead, in this benchmark model, both the ask and bid

prices of the j-asset match the unconditional value, i.e.,

PAj = PBj = qj.

With prob 1 − α, traders have an uninformative signal and make zero profits. With proba-

bility α, they make qj Wqj(1− qj)+ (1− qj) W

1−qjqj = W . Hence in expectation they make αW .

So in the benchmark model, skilled traders are indifferent between trading the two assets.

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4 Prices conditional on skill allocation

In this section we solve for the financial market equilibrium given the allocation of skill.

Given a labor market allocation (µr, µc, µ0), write Aj for the aggregate skill in asset j, i.e.,

Aj ≡

ˆ

αµj (dα) ,

and N j for the number of skilled traders in asset j, i.e.,

N j ≡

ˆ

µj (dα) .

Lemma 1 Given (Ar, N r, Ac, N c), prices for assets j = r, c are:

PAj = qj(

1 +Aj

λjwj +N j

1− qj

qj

)

(10)

PBj = qj(

1−Aj

λjwj +N j

)

, (11)

and the minimum skill required both to profitably buy the j-asset after observing signal sj ≤ qj

and to profitably sell the j-asset after observing signal sj > qj is

Xj(

Aj , N j)

≡Aj

λjwj +N j. (12)

The quantity Xj (Aj , N j) defined by (12) ties together two distinct (though related)

concepts. First, it is the minimum skill required to profitably trade the j-asset. Second,

from (10) and (11), the ask and bid prices can be written succinctly as

PAj = qj +(

1− qj)

Xj (13)

PBj = qj − qjXj, (14)

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and related, the bid-ask spread is

PAj − PBj = Xj. (15)

Note that Aj ≤ N j , and hence Xj (Aj, N j) ∈ [0, 1].

5 Equilibrium analysis

Using Lemma 1, it is straightforward to establish equilibrium existence using standard con-

tinuity arguments.

Proposition 1 An equilibrium exists.

5.1 Skill allocation

A skilled trader who specializes in the j-asset anticipates observing a signal sj ≤ qj with

probability qj , and a signal sj > qj with probability 1−qj . We write V j (α) for the expected

payoff of a skilled trader with skill α ≥ Xj who specializes in the j-asset. By Lemma 1,

combined with (3), (4) and (5), this payoff is

V j (α) = qjhj(

PAj) (

α + (1− α) qj − PAj)

+(

1− qj)

hjS(

PBj) (

PBj − (1− α) qj)

.

The first term corresponds to long positions, and the second term corresponds to short

positions. Substituting in (13) and (14), this payoff can be expressed solely in terms of Xj:

V j (α) = qj(

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj)) (

α−Xj)

. (16)

We first show that the combination of equilibrium in financial and labor markets implies

that the minimum skill level Xr required to trade the r-asset is bounded away from 0, even

as the r-event grows very rare (qj → 0). This result is central to many of the results derived

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in the rest of the paper, and the underlying intuition is as follows. Suppose there is just one

asset in the economy, and the chance of its payoff approaches zero. One might guess that the

price of this asset would also approach zero, because that is the expected value of the payoff.

Even a fixed percentage markup over the expected value would imply a price of zero in the

limit. But then, all agents, however low their chance of receiving an informative signal about

the asset payoff, would start to trade and buy the asset when they received a “buy” signal.

But whenever nearly all agents buy, the ask price is informative and cannot be close to zero;

hence in the limit the ask price would have to be bounded away from zero. Hence, a zero

ask price in the limit would violate a very basic equilibrium condition. More affirmatively,

we can see what will happen in the limit: as the chance of the payoff approaches zero, the

price approaches a limit that is higher than zero. At this price, the better-informed agents

trade and the less well informed agents do not trade. In between, there is a marginal type

of agent whose signal is just informative enough that he is indifferent between trading and

not trading. Given this, the ask price is higher than the expected value by a premium that

reflects the average informativeness of signals of all types that are higher than this marginal

type. Informally, this premium reflects the cumulative “brainpower” of traders who buy

when they receive a positive signal. In equilibrium, this premium in turn implies that the

marginal type is indeed indifferent between buying and not buying.

Lemma 2 formalizes this argument, and accounts for the fact skilled traders choose be-

tween the r-asset and c-asset:

Lemma 2 There exists some x > 0 such that Xr ≥ x for all qr small.

An immediate but important consequence is:

Corollary 1 The ask price P r stays bounded away from 0 even as the unconditional expected

value of the asset, i.e., qr, approaches zero.

Moreover:

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Corollary 2 Aggregate skill in the r-asset, Ar, is bounded away from 0 even as qr approaches

0.

From (16), the marginal value of skill for a trader specialized in the j-asset is

∂V j (α)

∂α= qj

(

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj))

. (17)

From this expression, and using Corollary 1 and Assumption 1, the marginal value of skill

is very low in the r-asset because qj is low. Formally:

Lemma 3 As qr → 0, the marginal value of skill in the r-asset (17) approaches 0.

There are two economic effects behind Lemma 3. First, as qr → 0, traders only rarely buy

the r-asset. Consequently, the expected profit from long positions also becomes small unless

traders are able to make enormous profits from long positions—which would in turn require

traders to take enormous long positions. This is what happens in the benchmark model of

Section 3 without financial market equilibrium. But by Corollary 1, the dual requirement

of equilibrium in financial and labor markets means that the ask price of the r-asset stays

bounded away from 0. Consequently, traders’ long positions cannot grow arbitrarily large,

which implies that the expected profit from long positions indeed approaches 0.

Turning to short positions, as qr → 0, traders specializing in the r-asset adopt short

positions with very high probability. A trader with skill α has an expected profit on each

short position of PBj − (1− α) qj = qj (α−Xj), which converges to 0 as qr → 0. It might

still be possible for a trader to make non-negligible expected profits on the short position if

he could take a large enough short position: however, Assumption 1 stops the short position

from growing large at a rate that dominates shrinking profits on each unit shorted.

Since the marginal value of skill grows small as the r-event grows rare, skilled traders

prefer to trade the c-asset (see Figure 1). However, by Corollary 2, we know some traders

trade the r-asset. So these traders must be relatively low skilled, as we next show:

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Proposition 2 For all qr sufficiently small, the minimum skill required to profitably trade

the r-asset is below the minimum skill to profitably trade the c-asset, i.e., Xr < Xc. More-

over, there exists α̂ ∈ (Xr, Xc) such that traders with skill α ∈ (Xr, α̂) trade the r-asset and

traders with skill α0 > α̂ trade the c-asset.

Proposition 2, which is illustrated by Figure 1, predicts that the least skilled traders

specialize in the r-asset. As noted, most of the time, they take a short position in the this

asset. The short position nets a small immediate profit, but exposes the trader to a small

risk of a much larger loss in the future (if the rare event is realized). Hence, our model

predicts that the least skilled traders pursue what are often described as “picking-up nickels

in front of a steamroller” strategies, such as writing out-of-the-money puts, or entering the

carry trade in currency markets.

Because traders in the r-asset are relatively unskilled, only a few them manage to success-

fully predict the rare event when it actually occurs. Hence our model rationalizes the fact

that rare events are foreseen by few people, even though the payoff to successfully forecasting

such events might seem very large. Nonetheless, a few traders do successfully predict the

rare event. As we show in Section 6 below, the posterior estimate of these traders’ skill is

very high.

5.2 Prices and returns

Recall that not only does Xj equal the minimum skill for profitably trading the j-asset, but

that it also equals the bid-ask spread. So an immediate implication of Proposition 2 is:

Corollary 3 The bid-ask price is smaller for the r-asset than the c-asset, PAr − PBr <

PAc − PBc.

Economically, this is consequence of the fact that the r-asset attracts less skilled people

than the c-asset, and so the adverse selection cost of trading the r-asset is lower. It is

consistent with the empirical finding that bid-ask spreads are increasing in the absolute

14

value of the option’s delta (see Muravyev and Pearson, 2014). For example, low-delta options

correspond to puts and calls that are very out of the money, and hence to the r-asset; while

higher-delta option correspond to puts and calls that are closer to being in the money, and

hence to the c-asset. Empirically, the bid-ask spread is larger in the latter case.

Nonetheless, the unconditional return on buying the r-asset, i.e., qr

PAr , is below the un-

conditional return on the c-asset, i.e., qc

PAc . Moreover, both returns are lower than those from

simultaneously buying a risk free bond and shorting the j-asset, i.e., 1−qj

1−PBj :

Lemma 4 For qr sufficiently small, the unconditional expected return from buying an asset

is increasing in the probability that the asset pays off:

qr

PAr<

qc

PAc≤

1− qc

1− PBc<

1− qr

1− PBr. (18)

Lemma 4 is consistent with the empirical observation that, in a various settings, the

expected return on assets that pay off rarely are lower than the expected returns on assets

that pay off more frequently.3 Examples include “smirks” in option markets, the value

premium in equity markets, and the favourite-long shot bias in betting markets.

Note that Lemma 4 evaluates trades using the ask prices, i.e., at the price it would cost

a trader to enter them.4 Next, we evaluate returns using bid rather than ask prices. This is

useful not only to obtain returns from shorting these positions, but also because prices and

returns reported in the data may use a combination of ask and bid prices. (Note that the

bid price of a security paying off in event ωj2 is 1 − PAj.) Switching from ask to bid prices

preserves the first two inequalities in Lemma 4, but reverses the third inequality:

Lemma 5 For qr sufficiently small, when evaluated at bid prices, the unconditional expected

3Moreover, one can show the analogues of the first two inequalities in (18) are also satisfied when uncon-

ditional returns are evaluated using the bid-ask midpoint, i.e., qr

1

2PAr+ 1

2PBr < qc

1

2PAc+ 1

2PBc ≤ 1−qc

1−( 1

2PAr+ 1

2PBr)

.

4It also assumes the positions are held until maturity, so the bid-ask spread at maturity is irrelevant.

15

returns are ordered according to

qr

PBr=

1− qr

1− PAr<

qc

PBc=

1− qc

1− PAc. (19)

Recall qj was normalized to lie below 12. Hence the asset that pays off in event ωc2 resembles

a bond with high-default risk, while the asset that pays off in event ωr2 resembles a bond

with low-default risk. As such, the result in Lemma 5 that 1−qr

1−PAr <1−qc

1−PAc is consistent with

the credit-spread puzzle of bond pricing, in which bonds with a low credit rating have lower

prices than their empirical default probabilities predict. Combined, Lemmas 4 and 5 suggest

that an interesting empirical exercise would be to measure how much of the credit-spread

puzzle is attributable to using bid as opposed to ask prices.

6 Dynamics: learning about skill

Thus far, we have restricted attention to a static model in which individuals live only one

period. This restriction obscures an important difference between rare- and common-event

assets, namely that successful prediction of a rare event is a very strong signal that a trader

is talented.

In this section, we extend our basic model to study the effect of learning. We show that,

perhaps surprisingly, learning plays a very limited role in trading the r-asset. Moreover, all

traders who specialize in the r-asset trade from the start of their careers, i.e., do not wait

until they have learned that they are skilled. Instead, it is in the c-asset that traders trade

only after having demonstrated success—even though successfully buying the c-asset is a

much weaker signal than is successfully buying the r-asset.

6.1 Dynamic model

In the dynamic model, each skilled trader has a career lasting for two periods. The decision

of which type of asset to specialize in is made prior to the first period. In each period,

16

a trader has the option of making a prediction without trading—in practice, this can be

interpreting as making only a small investment. To keep the economy stationary, we assume

that each period a new generation of skilled traders enters the economy, with skill distributed

according to the measure µ̄

2.

Also to ensure stationarity, we assume there are a continuum of r-assets, and a contin-

uum of c-assets, and traders who specialize in r-assets (respectively, c-assets) are randomly

allocated across the continuum of different r-assets (respectively, c-assets) at the start of

each period.

6.2 Updating

The unconditional probabilities that an individual who is skilled with probability α success-

fully predicts states ωj1 and ωj2 respectively are denoted pj1 (α) and pj2 (α):

pj1 (α) = qj(

(1− α)qj + α)

= qj(

qj + α(

1− qj))

pj2 (α) =(

1− qj) (

(1− α)(

1− qj)

+ α)

=(

1− qj) (

1− qj + αqj)

,

hence the posteriors that the trader has high skill, given successful prediction of ωj1 and ωj2

are respectively

αj′1 (α) =αqj

pj1 (α)=

α

qj + α (1− qj).

αj′2 (α) =α (1− qj)

pj2 (α)=

α

1− qj + αqj.

Notice that as qj → 0, the posterior belief approaches 1: a trader who has successfully

predicted an extremely unlikely event has almost certainly done so because of skill, not by

chance. On the other hand the posteriors that the trader has high skill, given unsuccessful

prediction of ωj1 and ωj2, are zero because skilled traders always predict correctly by assump-

tion. This means that a trader who makes a bad prediction has no value in the second

17

period.

Below, we make use of the inverses of the updating functions αj′1 and αj′2 . Evaluating,

α =

(

αj′1)−1

(α)

qj +(

αj′1)−1

(α) (1− qj),

so that

(α′1)

−1(α)(

α(

1− qj)

− 1)

+ αqj = 0,

and hence

(

αj′1)−1

(α) =αqj

1− α (1− qj)(20)

(

αj′2)−1

(α) =α (1− qj)

1− αqj. (21)

Note that αj′1 (α) ≥ αj′2 (α) ≥ α and(

αj′1)−1

(α) ≤(

αj′2)−1

(α) ≤ α, by the normalization

qj ≤ 12. In other words, successful prediction of ωj1 leads to more updating than successful

prediction of the complementary event ωj2.

6.3 Expected profits from specializing in the j-asset

A skilled trader’s decision to specialize in one asset over another reflects his expected lifetime

trading profits in each asset. From (16), expected lifetime trading profits in the j-asset are

V j (α) = qj(

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj))

×(

max{

0, α−Xj}

+ pj1 (α)max{

0, αj′1 (α)−Xj}

+ pj2 (α)max{

0, αj′2 (α)−Xj})

.

Here, the last two terms in parentheses correspond to an old-trader’s profits after, respec-

tively, successfully predicting ωj1 and ωj2.

When traders live two periods, the payoff function V j is convex, and piecewise linear with

three kinks (see Figure 2), at(

αj′1)−1

(Xj), then(

αj′2)−1

(Xj), and then Xj. Economically, for

18

skill levels below(

αj′1)−1

(Xj), even the posterior assessment of skill after successful predic-

tion of ωj1 is too low to justify trading. For skill levels in(

(

αj′1)−1

(Xj) ,(

αj′2)−1

(Xj))

a trader

trades only after successful prediction of ωj1 when young. For skill levels in(

(

αj′2)−1

(Xj) , Xj)

a trader trades after both successful prediction of ωj1 and ωj2 when young, but not after

unsuccessful prediction. Finally, for skill levels above Xj , a trader traders when young,

and continues trading when old provided he made profits (i.e., predicted successfully) when

young.

6.4 Basic equilibrium analysis

We start by reproducing several results from the one-period economy. First, and as in

Lemma 2, the minimum skill required to profitably trade the r-asset is bounded away from

zero. The basic economic force is the same as before: the only case in which traders without

skill (α = 0) can profitably trade the r-asset is if either the bid or ask price coincides with

the unconditional fair price of qj , but in this case, the r-asset would attract a non-trivial

amount of skilled trading, and both the equilibrium bid and ask prices would diverge from

the fair price of qj. The only new elements in the proof are associated with the need to

handle updating about skill levels.

Lemma 6 There exists some x > 0 such that Xr ≥ x for all qr small.

As before, an immediate but important corollary of Lemma 6 is that the ask price remains

bounded away from the fair price qj.

Next, we reproduce Lemma 3 from the one-period economy: even taking the value of

learning into account, the marginal value of skill in the r-asset still approaches 0.

Lemma 7 As qr → 0, the marginal value of skill in the r-asset approaches 0.

Given Lemma 7, similar arguments as in the one-period economy imply that it is the

lowest skill traders who actively trade who specialize in the r-asset (see Proposition 2). As

19

with other results, the only new elements in the proof are those associated with learning

about a trader’s skill:

Proposition 3 For all qr small enough, 0 < (αr′1 )−1 (Xr) < (αc′1 )

−1 (Xc).

6.5 Career dynamics

The results above are all direct analogues of results from the one-period economy. In this

subsection, we turn to the distinctive implications of the dynamic model. Most of these

implications relate to career dynamics.

First, although Proposition 3 shows that the lowest skill traders who actively trade spe-

cialize in the r-asset, it is nonetheless the case that the traders in the economy with the

highest identified skill trade the r-asset. This is a consequence of the power of updating from

successfully predicting event ωr1:

Corollary 4 As qr → 0, the posterior skill of some people who trade r-asset approaches 1,

i.e., there exists some α who trades the r-asset such that αr′1 (α) → 1.

Somewhat anecdotally, this is consistent with the descriptions in Lewis (2011), in which

the fund managers who predicted the housing crisis were unheralded prior to the crisis, but

attracted large fund inflows after the crisis.

At the same time, in spite of this powerful updating, in the aggregate there is only limited

updating from successful prediction of the rare event, in the following sense. For j = r, c,

define ALj as

ALj = aggregate skill trading asset jthat previously successfully predicted ωr1.

Formally, define µjy as the measure of young traders who trade asset j. Then

ALj =

ˆ

pj1 (α)αj′1 (α)µjy (dα) .

20

An immediate consequence of the rare event being rare is that:

Lemma 8 limqr→0ALr = 0.

A natural conjecture might be that the attraction of trading the r-asset is that it provides

individuals who are unsure about their ability with a very powerful signal, enabling them

to trade later in their careers. However, this conjecture is incorrect, in the sense that at

least some traders trade the r-asset when young. This implication follows from existing

results. From Lemma 8, learning from successful prediction of ωr1 is very limited. Moreover,

as qr → 0, learning from successful prediction of ωr2 is likewise very limited, for the simple

reason that there is only a very small update to an assessment of a trader’s skill from him

successfully predicting an event that is almost certain to occur. Since some people trade the

r-asset (see Lemma 6), it follows that some people must trade the r-asset when young.

Proposition 4 For all qr small enough, some people trade the r-asset when young.

Consequently:

Corollary 5 Learning from successful prediction of ωr1 plays a very small role in total trade

in the r-asset, in the sense that ALr/Ar → 0 as qr → 0.

In the one-period economy, the fact that more skill is required to specialize in the c-asset

than the r-asset immediately implies that the bid-ask spread is larger in the c-asset (see

Corollary 3). Because of learning, this implication is no longer immediate in the dynamic

economy: the minimum skill required to specialize in asset j is(

αj′1)−1

(Xj), which no longer

coincides with the bid-ask spread Xj. Nonetheless, the bid-ask comparison carries over to

the dynamic economy:

Proposition 5 For all qr small enough, the bid-ask spread in the c-asset is larger, Xc > Xr.

Above, we noted that at least some traders trade the r-asset when young, without spend-

ing any time learning about their prediction abilities—even though the r-event is potentially

21

very revealing about a trader’s skill. We next use the bid-ask comparison Xc > Xr to estab-

lish the converse of this: prices of the c-asset are sufficiently far from the fair price qj that

only traders who have established their skill when young trade the c-asset—even though

predicting the c-event does not provide as much information as predicting the r-event:

Proposition 6 For qr sufficiently small, there is a skill level α̂ ∈ (Xr, Xc) such that all

traders with initial skill in[

(αr′1 )−1 (Xr) , α̂

)

specialize in the r-asset, and all traders with

initial skill above α̂ specialize in the c-asset. The marginal-skill trader α̂ faces a choice

between: trading the r-asset immediately, and trading the c-asset only in the second period,

after successful prediction in the the first period.

The higher bid-ask spread in the c-asset reflects the fact that skilled traders flow dis-

proportionately into the c-asset. Despite the power of learning in the r-asset, numerical

simulations suggest that learning only aggravates this effect:

Conjecture 1 [Unproved] The bid-ask spread ratio Xc

Xr is higher in the dynamic economy.

Very rough sketch proof of conjecture: Suppose to the contrary thatXc

2

Xr2

≤Xc

1

Xr1

.

Note that V c and V r fully parameterized by Xc and Xr. Write V c (·;Xc) etc. Observe that

V c1 (·;Xc

2) < V c2 (·;Xc

2), because learning matters for the c-asset, while V r1 (·;Xr

2) ≈ V r2 (·;Xr

2),

because learning doesn’t affect payoff much for r-asset. Consequently [need work here], for

using (Xr2 , X

c2) in the one-period economy, there is less skill devoted to c-asset and more skill

devoted to the r-asset, relative to the two-period economy. By the suppositionXr

2

Xc2

≥Xr

1

Xc1

, this

is true a fortiori using (Xr1 , X

c1). [again, more work here. In particular, need care in ratios

vs differences.] Since prices are linked to skill, this impliesXc

1

Xr1

<Xc

2

Xr2

, which would contradict

the supposition and complete the proof.

7 Why did no-one predict the financial crisis?

Rare events are often unforeseen, as exemplified by the financial crisis of 2008. Our model

generates several implications that speak to this.

22

As a first step, note that ask and bid prices in our model are independent of the true

state, and hence uninformative. In contrast, the aggregate order flow is typically informative.

Accordingly, in this section we consider what an outside observer who observes the total

number of buy and sell orders for asset j can infer about the likelihood of events ωj1 and

ωj2. Note that the total buy and sell orders can, alternatively, be inferred from seeing a

combination of any two of the average transaction price, aggregate volume, and order flow

imbalance.

Write buyj and sellj for total buy and sell orders for asset j. Write φjbuy and φjsell for the

mass of liquidity traders who buy and sell asset j, along with φj ≡ φjbuy + φjsell. Hence the

expected number of buy and sell orders for asset j, conditional on the event, are

E[

buyj|ψj]

=(

E[

φjbuy]

+ Aj(

1ψj≤qj − qj)

+ qjN j)

hj(

PAj)

E[

sellj |ψj]

=(

E[

φjsell]

+ Aj(

1ψj>qj −(

1− qj))

+(

1− qj)

N j)

hjS(

PBj)

.

The first and most basic prediction of our model is that the difference in the expected number

of both buy and sell orders of the r-asset across states is proportional to Ar, the aggregate

skill deployed in trading the r-asset. So when Ar is small, the expected numbers of buys

and sells are insensitive to the true underlying state. Our model predicts that Xr is small

relative to Xc, which—loosely speaking—is related to Ar being small relative to Ac.

Nonetheless, under relatively mild assumptions, the order flow of the r-asset is a very

accurate predictor of event ωr2, in the sense that when the true event is ωr2, an outside observor

is extremely likely to be able to infer this from the order flow. The main assumption that

drives this is that skilled trader predictions average out by the law of large numbers, so that

buyj =(

φjbuy + Aj(

1ωr1− qj

)

+ qjN j)

hj(

PAj)

sellj =(

φjsell + Aj(

1ωr2−(

1− qj))

+(

1− qj)

N j)

hjS(

PBj)

.

Note that there is no assumption here about correlation among liquidity traders. For liquidity

23

traders, we make no assumption beyond that already contained in subsection 2.3, namely

that E[

φrbuy]

/E [φrsell] = qr/ (1− qr). This is equivalent to E[

φrbuy]

= qrE [φr].

Even though combined liquidity trade φr potentially fluctuates, it can be inferred from

buyj

hj (PAj)+

sellj

hjS (PBj)

= φj +N j .

Consequently, the information content of observing (buyj, sellj) is the same as the informa-

tion content of observing(

buyj, φj)

, and hence is the same as the information content of

observing(

φjbuy + Aj1ωj1

, φj)

. We write ηj ≡ φjbuy + Aj1ωj1

.

We now show that there is very little chance that the order flow falsely predicts event ωr1

when the true event is ωr2. The key driving force is Corollary 2’s observation that aggregate

skill Ar in the r-asset is bounded away from 0, by A say. Fix C ∈ (0, A). Suppose the true

event is ωr2. Realizations of φrbuy below Ar generate realizations of ηr that are consistent only

with ωr2. By Markov’s inequality,

Pr(

φrbuy ≤ Ar)

≥ 1−E[

φrbuy]

Ar≥ 1−

qrE [φr]

C.

So as qr → 0, Pr(

φrbuy < Ar)

→ 1, establishing that the order flow reveals the true event in

ωr2.

Our third result is that the analagous statement about prediction of event ωr1 is not true.

Instead, it is very possible that when the true event is ωr1, the observor sees—with very high

probability—an order flow that leaves him very unsure about the event. To see this, it is

useful to start by seeing why the argument based on Markov’s inequality fails in this case.

Suppose the true event is ωr1. In this case, all realizations of ηr are above Ar. Evaluating

24

the posterior conditional on seeing ηr ≥ Ar,

Pr (ωr1|ηr ≥ Ar) =

qr Pr (ηr ≥ Ar|ωr1)

qr Pr (ηr ≥ Ar|ωr1) + (1− qr) Pr (ηr ≥ Ar|ωr2)

=qr

qr + (1− qr) Pr(

φrbuy ≥ Ar)

≥qr

qr + (1− qr) Pr(

φrbuy ≥ C)

≥qr

qr + (1− qr) qrE[φr ]C

=C

C + (1− qr)E [φr]

≥C

C + E [φr].

Although this bound is above zero, it still leaves open the possibility that when the true

event is ωr1, an observor who sees order flows is very unsure about the event.5

Of course, the above discussion only shows that a particular proof does not work. How-

ever, here is a simple example that makes the same point. Suppose the total liquidity trade φr

is deterministic. The number of liquidity buys, φrbuy, is either qr (φr − Ar) or qr (φr −Ar)+Ar,

with probabilities 1 − qr and qr, respectively. Observe that E[

φrbuy]

= qrφr. Suppose the

true event is ωr1. So with probability qr, φrbuy is high, and the order flow reveals the event

ωr1. But with probability 1− qr, φrbuy is low, and the posterior probability of event ωr1 is

qr (1− qr)

qr (1− qr) + (1− qr) qr=

1

2.

Consequently, conditional on the true event being ωr1, with probability 1 − qr the observer

5Instead of using the Markov inequality Pr(

φrbuy ≥ C

)

≤ qrE[φr]C

, one might consider using Chebyshev’s

inequality: Pr(

φrbuy ≥ C|φr

)

= Pr(

φrbuy − qrφr ≥ C − qrφr|φr

)

≤ Pr(∣

∣φrbuy − qrφr

∣

∣ ≥ C − qrφr|φr)

≤var[φr

buy]C−qrφr . The Bhatia–Davis inequality gives var

[

φrbuy |φ

r]

≤ (φr − qrφr) qrφr. Hence we obtain

Pr(

φrbuy ≥ C|φr

)

≤ (φr)2qr(1−qr)C−qrφr . Substituting in, we have

Pr (ωr1|η ≥ Ar) ≥

qr

qr + (1− qr) (φr)2qr(1−qr)C−qrφr

=C − qrφr

C − qrφr + (1− qr)2(φr)

2 .

25

sees an order flow that still leaves him very unsure as to whether the event ωr1 will occur.

In summary:

• If aggregate skill in the r-event,Ar , is small, then expected order flow has low sensitivity

to the true event.

• Nonetheless, if the true event is ωr2, then with very high probability an observor can

infer this from interpreting the order flow.

• In contrast, if the true event is ωr1, an observor’s ability to infer this depends critically

on the variance of the aggregate order flow from liquidity traders. In particular, an

observor may remain very unsure about the true state even after observing aggregate

order flow.

• On the other hand, even when the true event is ωr1, observing the order flow leads to

an update in the probability an observor attaches to ωr1 from qr to at least CC+E[φr ]

.

8 Conclusion

[To be added]

References

[To be added]

26

Appendix: Proofs

Proof of Lemma 1: Given prices (10) and (11), from (6) the minimum skill level required

to profitably buy the j-asset after observing signal sj ≤ qj is

qj(

1 + Aj

λjwj+Nj

1−qj

qj

)

− qj

1− qj=

Aj

λjwj +N j,

while from (9) the minimum skill level required to profitably sell the j-asset after observing

signal sj > qj is

qj − qj(

1− Aj

λjwj+Nj

)

qj=

Aj

λjwj +N j.

Moreover, it is easy to see from (7) and (8) that profitable long positions are impossible after

signals sj > qj and profitable short positions are impossible after sj ≤ qj .

Given that the minimum skill levels for long and short positions coincide,

E[

buys|ψj]

= qjλjwjhj(

PAj)

+

ˆ

(

α1ψj≤qj + (1− α) qj)

µj (dα)hj(

PAj)

=(

qjλjwj + Aj(

1ψj≤qj − qj)

+ qjN j)

hj(

PAj)

E[

sells|ψj]

= (1− q)j λjwjhjS(

PBj)

+

ˆ

(

α1ψj>qj + (1− α)(

1− qj))

µj (dα)hjS(

PBj)

.

=(

(1− q)j λjwj + Aj(

1ψj>qj −(

1− qj))

+(

1− qj)

N j)

hjS(

PBj)

.

Hence from (1) and (2),

PAj = qjqjλjwj + Aj (1− qj) + qjN j

qjλjwj + qjN j= qj

(

1 +Aj

λjwj +N j

1− qj

qj

)

(22)

PBj = qj(1− q)j λjwj − Aj (1− qj) + (1− qj)N j

(1− q)j λjwj + (1− qj)N j= qj

(

1−Aj

λjwj +N j

)

. (23)

QED

Proof of Proposition 1: Construct a correspondence ξ : [0, 1]2 → [0, 1]2 as follows. For

any (Xr, Xc) ∈ [0, 1]2, construct prices by (13) and (refbid-price-X). Given prices, allocate

27

skilled traders to the asset where their expected profit is higher. For the case of indifference,

allow for all randomizations between the two assets. Finally, given the allocation of skilled

traders, evaluate new values of (Xr, Xc) according to (12). The resulting correspondence is

upper-hemicontinuous, and closed- and compact valued. By Kakutani’s fixed point theorem

it has a fixed point, which corresponds to an equilibrium of the economy. QED

Proof of Lemma 2: Suppose to the contrary that there exists some sequence {qr} such

that qr → 0 and the associated Xr → 0.

First, consider the case in which Xc stays bounded away from 0, by xc say. But then

Xr → 0 implies that skilled traders in the skill interval [Xr, xc] certainly trade the rare asset.

But then from (12), Xr 6→ 0, a contradiction.

Second, consider the case in which Xc → 0 for some subsequence. So all skilled traders

trade something. But by (12), this contradicts Xc +Xr → 0, completing the proof. QED

Proof of Corollary 2: From Lemma 2, there exists x > 0 such that Xr ≥ x even as as

qr → 0. Since Ar ≤ N r, it follows that Ar

λrwr+Ar ≥ Ar

λrwr+Nr ≥ x, and hence that there exists

A such that Ar ≥ A even as as qr → 0. QED

Proof of Lemma 3: By Lemma 2, as qr → 0, the term qr (1− qr) hr (qr + (1− qr)Xr)

approaches 0. The remaining term qr (1− qr)hrS (qr − qrXr) equals 1−qr

1−Xr (qr − qrXr)hrS (q

r − qrXr),

and approaches 0 as qr by Assumption 1 combined with the fact thatXr ≤ 1λjwj+1

< 1. QED

Proof of Proposition 2: First, we show Xr < Xc. Suppose to the contrary that

Xr ≥ Xc even as qr grows small. Observe that the bid and ask prices of the c-asset are

bounded, i.e., PAc ∈ [qc, 1] and PBc ∈ [0, qc]. So even as qr → 0, the position limits h(

PAc)

and hS(

PBc)

remain bounded away from 0, and so the marginal value of skill in the c-asset,

namely qc (1− qc)(

h(

PAc)

+ hS(

PBc))

, likewise remains bounded away from 0. In contrast,

from Lemma 3 we know the marginal value of skill in the r-asset approaches 0. It follows

that no-one trades the r-asset for qr sufficiently small. But then Xr = 0, contradicting

Lemma 2, and completing the proof of Xr < Xc. The existence of a cutoff skill level α̂ is

them immediate. QED

28

Proof of Lemma 4: First, as qr → 0, note that qr

PAr → 0 by Corollary 1, while

1−qr

1−PBr = 1−qr

1−qr+qrXr → 1. This establishes the first and third inequalities in (18). The second

inequality qc

PAc ≤ 1−qc

1−PBc is equivalent to (qc)2Xc ≤ (1− qc)2Xc, which holds since qc ≤ 12.

QED

Proof of Lemma 5: Evaluating,

qj

PBj=

qj

qj − qjXj=

1

1−Xj

1− qj

1− PAj=

1− qj

1− qj − (1− qj)Xj=

1

1−Xj.

The result then follows from Proposition 2. QED

Proof of Lemma 6: Suppose to the contrary that there exists some sequence {qr} such

that qr → 0 and the associated Xr → 0.

First, consider the case in which (αc′1 )−1 (Xc) stays bounded away from 0, by a say. But

then Xr → 0 implies that people in the skill interval [Xr, a] trade the rare asset immediately.

But then Xr9 0, contradicting the original supposition.

Second, consider the case in which (αc′1 )−1 (Xc) → 0 for some subsequence. From (20),

Xc → 0. Hence the total skill Ac in the c-asset must approach 0. Likewise, by supposition,

Xr → 0, so the total skill Ar in the r-asset must approach 0. But the combination of these

two statements is impossible, since because both Xc, Xj → 0, the fraction of skilled traders

who trades at least one of the two assets when young approaches 1. The contradiction

completes the proof. QED

Proof of Lemma 7: By the same argument as in the proof of Lemma 3, in the two-

period profit function V r (1) → 0, even when evaluated at the maximum skill level α = 0.

Since V r is weakly increasing and convex, it follows that the slope of V r must approach 0 at

all skill levels. QED

Lemma 9 The righthand derivatives of the two-period profit function V j, denoted by V j+,

29

satisfy

V j+(

(

αj′1)−1 (

Xj)

)

=(

qj)2 (

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj)) (

1−Xj(

1− qj))

V j+(

Xj)

= 2qj(

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj)) (

1−Xjqj(

1− qj))

.

Proof of Lemma 9: Note that

dpj1 (α)

dα=

dpj2 (α)

dα= qj

(

1− qj)

d

dααj′1 (α) pj1 (α) = 1−

d

dααj′2 (α) pj2 (α) = qj.

So

V j+(

(

αj′1)−1 (

Xj)

)

= qj(

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj))

×

(

dpj1 (α0)αj′1 (α)

dα−Xj dp

j1 (α)

dα

)

= qj(

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj)) (

qj −Xjqj(

1− qj))

=(

qj)2 (

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj)) (

1−Xj(

1− qj))

.

Likewise,

V j+ (X) = qj(

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj))

×

(

1 +d(

pj1 (α)αj′1 (α) + pj2 (α)α

j′2 (α)

)

dα−Xj

d(

pj1 (α) + pj2 (α))

dα

)

= qj(

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj)) (

1 + 1− 2Xjqj(

1− qj))

= 2qj(

1− qj) (

hj(

qj +(

1− qj)

Xj)

+ hjS(

qj − qjXj)) (

1−Xjqj(

1− qj))

.

QED

Proof of Proposition 3: First, we show (αr′1 )−1 (Xr) < (αc′1 )

−1 (Xc). Suppose to

the contrary that (αr′1 )−1 (Xr) ≥ (αc′1 )

−1 (Xc) even as qr grows small. From Lemma 9, the

30

righthand derivative V c+(

(αc′1 )−1 (Xc)

)

is bounded away from 0. In contrast, from Lemma

7 we know V r+ (Xr) → 0. It follows that no-one trades the r-asset for qr sufficiently small.

But then Xr = 0. If (αc′1 )−1 (Xc) > 0 this gives a contradiction, since people in the interval

[

0, (αc′1 )−1 (Xc)

]

would trade the r-asset in the first period. If instead (αc′1 )−1 (Xc) = 0, then

on the one hand, Xc = 0 by (20); but on the other hand, since Xc = Xr = 0, we must have

trade in at least one of the assets, implying Xr +Xc > 0, and contradicting Xr +Xc = 0.

This completes the proof of (αr′1 )−1 (Xr) < (αc′1 )

−1 (Xc).

Given (αr′1 )−1 (Xr) < (αc′1 )

−1 (Xc), we show (αr′1 )−1 (Xr) > 0. Suppose to the contrary

that (αr′1 )−1 (Xr) = 0. So by (20), Xr = 0. But then traders with skill in

[

0, (αc′1 )−1 (Xc)

]

trade the r-asset immediately, contradicting Xr = 0. QED

Proof of Proposition 4: Suppose otherwise. Then the only people trading the r-asset

are traders who successfully predicted either ωr1 or ωr2 when young. By Lemma 8, ALr → 0

as qr → 0. In addition, (αr′2 )−1 (Xr) → Xr, so the interval of skill types

[

(αr′2 )−1 (Xr) , Xr

]

who trade when old after successful prediction of ωr2 when young, grows arbitrarily small.

Hence aggregate skill in trading the r-asset approaches 0, so that Xr → 0, contradicting

Lemma 6, and completing the proof. QED

Proof of Corollary 5: Immediate from Lemma 8, and the fact that Lemma 6 implies

that Ar is bounded away from 0. QED

Proof of Proposition 5: First, note from (12) that Xr ≤ 1λjwj+1

since Ar ≤ N r, and

hence Xr stays bounded away from 1.6

Second, note that Xc stays bounded away from 0, as follows. Suppose to the contrary

that Xc → 0. From Lemma 9, the slope of V c is bounded away from 0 for all skill values

above (αc′1 )−1 (Xc). In contrast, from Lemma 7 the slope of V r approaches 0 as qr → 0.

Hence skill in the c-asset, Ac, is bounded away from 0 as qr → 0, contradicting Xc → 0.

Suppose that, contrary to the claimed result, Xc ≤ Xr even as qr → 0. So by above, Xc

6An alternative argument for why Xr is bounded away from 1 is as follows. Suppose instead that Xr → 1as qr → 0. Then the fraction of skilled traders who can trade approaches 0, implying Ar → 0, contradictingXr → 1.

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is bounded away from both 0 and 1. Hence

Xc − (αc′1 )−1

(Xc) = Xc −Xcqc

1−Xc (1− qc)=

(1−Xc (1− qc))Xc − qcXc

1−Xc (1− qc)

=(1− qc)Xc − (1− qc) (Xc)2

1−Xc (1− qc)=

(1− qc)Xc (1−Xc)

1−Xc (1− qc). (24)

is bounded away from 0. By Lemma 7, the slope of V r approaches 0 at all skill levels.

On the other hand, from Lemma 9, the slope of V c is bounded away from 0 for all skill

values above (αc′1 )−1 (Xc). Since Xc − (αc′1 )

−1 (Xc) is bounded away from 0, it follows that

V c (Xc) > maxα Vr (α). So no-one with initial skill above Xc specializes in the r-asset.

By the supposition Xc ≤ Xr, a fortiori no-one with initial skill above Xc specializes in

the r-asset. Hence no-one trades the r-asset when young, contradicting Proposition 4 and

completing the proof. QED

Proof of Proposition 6: First, note from (12) that Xc ≤ 1λcwc+1

since Ac ≤ N c, and

hence Xc stays bounded away from 1.7

By Proposition 5 and Lemma 6, Xc must also remain bounded away from 0.

Given that Xc is bounded away from both 0 and 1, the same argument as in the proof of

Proposition 5 implies that V c (Xc) > maxα Vr (α). So V c (α) > V r (α) for all α ≥ Xc. Since

some skilled traders trade the r-asset (Proposition 4), the curves V c and V r must intersect

at a skill level strictly below Xc. Moreover, from Lemma 9 and Lemma 7, for qr small the

slope of V c is steeper than the slope of V r for all skill levels above (αc′1 )−1 (Xc). Hence V c

and V r intersect exactly once above minj=r,c

{

(

αj′1)−1

(Xj)}

, and the intersection point is

below Xc. Finally, by Proposition 4, the intersection point is above Xr. This completes the

proof. QED

7An alternative argument for why Xc is bounded away from 1 is as follows. Suppose instead that Xc → 1as qr → 0. Then (by (20)) (αc′

1 )−1

(Xc) → 1, but then no-one trades the c-asset, contradicting Xc → 1.

32

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

Figure 1: The horizontal axis represents a trader’s perceived skill α. The steeper (flatter)line represents a trader’s expected profits from specializing in the c-asset (r-asset), andintersects the horizontal axis at the minimum skill level required for profitable trading, i.e.,Xc (respectively, Xr). The lines are drawn for equilibrium values, and illiustrate Proposition2: lower-skilled traders specialize in the r-asset, and higher skilled traders specialize in thec-asset.

33

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

Figure 2: The figure plots equilibrium profit functions for the dynamic economy, and isanalogous to Figure 1. The profit function for the c-asset has three kinks: first at the skilllevel that is enough for trading after successful prediction of ωc1, then at the skill level thatis enough for trading after successful prediction of either ωc1 or ωc2, and finally at the skilllevel Xc that is enough for immediate trading. (The profit function for the r-asset containsthese same three kinks, but the scale is too small to see them.)

34