†
Abstract
Several reputable banks have recently paid substantial penalties for their mortgage securitization practices that helped lead to the 2008 financial crisis. The fact that such prestigious banks misbehaved is puzzling, however, since theory suggests that financial intermediaries should carefully maintain their credibility and such reputation concerns should police the industry. The natural question is what incentives were misaligned in the asset-backed securities market that led to such a debacle. This paper proposes an infinite horizon dynamic model in which the reputation of a securitizer is defined as the market’s belief about his ability to screen loans and issue high quality securitized products. The securitizer’s ex ante screening effort, however, creates an ex post adverse selection in secondary markets. To build up his reputation and signal quality, the securitizer retains a fraction of his securitized products. My model shows that the securitizer’s reputational premium significantly decreases when the retention serves as a signal of product quality. For example, during boom years, when loan default rates are generally low and screening only confers a small benefit, the securitizer screens the loans and retains a part of his products only when his reputation is less than a threshold level, and then he chooses to shirk and sell all when his reputation is high. At the end of paper I discuss the “skin in the game” policy and characterize the optimal retention.
Key Words: Securitization, Asset-Backed Securities, Reputation, Moral hazard,
Adverse selection
∗
I would like to thank Ennio Stacchetti and Alberto Bisin for their advising and support. I would also like to thank Douglas Gale, Thomas Philippon and Jaroslav Borovicka for their valuable comments, and thank the participants of the NYU microeconomics student seminar and financial economics workshop for helpful discussion. I am deeply indebted to Andrew Schotter for his constant guidance and support for this project.
All remaining errors are mine.
For the latest version, please visit my website at https://sites.google.com/a/nyu.edu/tding/papers .
†
Department of Economics, New York University, 19 West 4th St., New York, NY, 10012. Email: tingting.ding@nyu.edu
.

Securitization played an important role in providing liquidity for the U.S. private sector prior to the 2008 financial crisis. By 2002 the amount of securitized bonds issued had reached $662.4 billion, exceeding the amount of corporate bonds issued in the same year
( Gorton and Metrick , 2013 ). However, the burst of the housing bubble led to the collapse
of subprime mortgage-backed securities in 2007, and investors soon lost their confidence in the whole securitization market, even though many of the securitized products were unrelated to subprime mortgages. The total issuance of asset-backed securities for 2008 shrank to $168.9 billion of which the private label mortgage-backed securities totaled only
$31.7 billion ( SIFMA , 2009 ). Seven years later the recovery of the asset-backed securities
market still seems remote.
Investors blame moral hazard and adverse selection problems in securitization practices and claim that loan originators and securitization underwriters lost their incentives for due diligence as they were able to pass all risks to the capital market. On August
21st 2014, Bank of America agreed to pay $16.65 billion, the largest civil settlement with a single entity in American history, to settle the federal and state claims for fraud in mortgage-backed securities issued by Bank of America and its former and current subsidiaries (including Countrywide Financial Corporation and Merrill Lynch.) The investigations have found that those banks either did not conduct their due diligence or ignored the advice from their due diligence vendors at the time when they packaged the loans and sold them off in securities. In an internal email that discussed due diligence on one particular pool of loans, a consultant in Merrill Lynch’s due diligence department wrote: “[H]ow much time do you want me to spend looking at these [loans] if [the cohead of Merrill Lynch’s RMBS business] is going to keep them regardless of issues?...
Makes you wonder why we have due diligence performed other than making sure the loan
closed.” 1
Bank of America is not the only bank that was found to misbehave in mortgage securitization practices. During 2013-2014, several other reputable banks such as J.P.
Morgan, Morgan Stanley and Citibank have also paid the penalties for frauds in their mortgage securitization practices. Goldman Sachs, which has avoided a penalty so far, has agreed to buy back $3.15 billion in mortgage bonds from Fannie Mae and Freddie
Mac as those bonds do not satisfy the underwriting standards.
The fact that such prestigious banks misbehaved is puzzling. It has been conventionally believed that financial intermediaries would make an effort to maintaining their
1
See the U.S. Department of Justice, “Annex 1: Bank of America Corporation Statement of
Facts,” http://www.justice.gov/iso/opa/resources/4312014829141220799708.pdf
, Last retrieved:
Sep 25, 2015.
1

creditability as they are afraid to lose future investors, and such reputational concerns should police the industry. There are many empirical studies demonstrating that reputation provides strong incentives for financial intermediaries to make an effort in diminishing
information asymmetry in IPO, bond, loan and acquisition markets ( Griffin, Lowery, and
Saretto , 2014 ). Investors appreciate the reputation of financial intermediaries and would
like to pay the reputational premium. The natural question then arises: What role does reputation play in asset-backed securities markets? What incentives in securitization are different from other financial practices?
In this paper I build an infinite horizon model to investigate the reputation effect in the asset-backed securities market. In the model the securitizer is a long-run player, who packages a pool of loans and sells the securitized product at the start of each period, while the investors are short-run players who only participate in the market for one period. The payoff of the securitized product is realized at the end of each period; it depends on a costly screening effort of the securitizer, which cannot be observed by the investors, as well as an aggregate economic shock which follows a stochastic process and is persistent over time. A reputation concern arises as there are two types of securitizers. One type is called the normal type who has the screening technology, and the other type is called the inept type who has no such ability. The reputation is then defined as the probability that the investors place on the securitizer being of the normal type.
The adverse selection problem endogenously arises in the secondary market as the securitizer’s unobservable screening technology determines the quality of the securitized product. To signal that he has made the screening and that his product has a high quality, the securitizer of the normal type can choose to send a signal to the market by retaining a fraction of his securitized products. The retention is costly as the securitizer has an intra-period discount on the cash flow realized by the end of each period. The securitizer of the inept type, who is also a strategic player, chooses between mimicking the normal type by retaining the same fraction as the normal type and revealing his own type by retaining a different fraction. At the time of transaction, the investors pay the expected payoff of the products given the retention that the securitizer holds, the securitizer’s reputation and their knowledge of aggregate shock in the previous period.
By the end of the period, the economic aggregate shock is realized and then the payoff of the product is realized. Given this information the investors update their belief about the securitizer’s type.
In this model the reputation effect and signaling mechanism are linked. If the securitizer could not signal via retention, there was no screening made in any equilibrium. The intuition is similar to the reputation model constructed by
Ely and V¨ aki ( 2003 ) and
Ely, Fudenberg, and Levine ( 2008 ): When his reputation approaches one, the securitizer
2

of the normal type no longer has an incentive to make the screening as the reputational premium is not great enough to compensate his cost. Given such an expectation, the investors will only pay the price for the products with high default rates, and then the screening equilibrium will collapse. The signaling mechanism, therefore, makes it possible that the securitizer of the normal type will choose the screening in the equilibrium.
However, the signal via retention is costly and it may decrease the securitizer’s incentive to screen the loans. My model shows that as long as the securitizer is patient and the screening confers enough benefit, the securitizer of the normal type always chooses to screen and sends the market a signal via retention. On the other hand, during boom years, when loan default rates are generally low and screening only confers a small benefit, the securitizer of the normal type screens the loans and retains a part of his products only when his reputation is less than a threshold level, and then he chooses to pool with the inept type to shirk and sell all when his reputation is high. The investors stop updating the reputation of the securitizer as the outcome of the project reveals no information about the securitizer’s type. The securitizer believes that there is no reputational risk because the aggregate shock is persistent.
My model captures several important features of the asset-backed securities market. First of all, securitization is usually a repeated process and the securitizers are not anonymous. The investors understand whom they have been working with and make their investment decisions after observing a history of the securitizer’s performance. Therefore a reputation model is suitable to explain the interaction between investors and securitizers. Second, my model assumes the adverse selection problem arises endogenously in the secondary market. The securitizer of the normal type has the option whether to make the screening to improve the quality of his products. The ex ante screening effort, nevertheless, may be discouraged by the ex post “lemon discount” due to the securitizer’s hidden information. In other words, there exists a tension between trade volume and the quality of products, which captures the fact that in the securitization boom securitizers intentionally ignored their due diligence. Third, I allow the risk retention as a tool of credit enhancement during securitization. Though the risk retention was not mandatory prior to the financial crisis, securitizers often retained a fraction of their securitized product
(usually equity tranches) while marketing their products ( Chen, Liu, and Ryan , 2008 ).
Since the financial crisis the Dodd-Frank Act has mandated that the securitizers retain not less than 5 percent of the assets as a part of the Wall Street reform. My model, which investigates the voluntary retention, sets up a benchmark for the policy.
Several implications of risk retention requirement are provided by my model. First, the risk retention is more likely to change the securitizer’s incentives during boom years when they care more for trade volume. When the mandatory risk retention is high enough
3

the securitizer of the normal type makes the screening regardless of the aggregate shock.
However, the excessive requirement of risk retention increases the cost of securitization and decreases the provision of liquidity. In particular, the securitization market may completely freezes when the macroeconomic environment is in depression. Second, my model provides a quantitative way to calibrate the optimal mandatory risk retention. It has been widely discussed that the universal 5 percent risk retention for all the securitized products is inefficient. My model argues that the optimal retention should be smaller if the performance of securitized products is less likely to be influenced by business cycles.
The rest of my paper is organized as follows. Section 2 provides a related literature review. Section 3 characterizes the equilibrium in the one-period model when the securitizer has no reputational concern, which provides a benchmark. Section 4 presents the formal infinite horizon model. Section 5 discusses comparative statics while Section 6 provides empirical implications. Section 7 analyzes the benefit and cost of mandatory risk retention and characterizes the retention level that guarantees the securitizer always makes the screening. Section 8 concludes. All the lengthy proofs are presented in the
Appendix.
My model is motivated by the observations of moral hazard and adverse selection problems in securitization practices prior to the 2008 financial crisis. In addition to the evidence revealed by recent investigations that reputable banks did not conduct due diligence when structuring and selling securities, there are many empirical studies demonstrating that during the housing boom securitization reduced the incentives of financial interme-
diaries to carefully screen borrowers ( Elul , 2011 ;
Keys, Mukherjee, Seru, and Vig , 2010 ;
Keys, Seru, and Vig , 2012 ;
Purnanandam , 2011 ;
Keys, Piskorski, Seru, and Vig , 2013 ),
and there were sizable misrepresentations when underwriters reported the asset quality
to investors ( Piskorski, Seru, and Witkin , 2013 ;
Griffin and Maturana , 2014 ).
This paper is related to the literature of signaling mechanisms in security design.
Demarzo and Duffie ( 1999 ) and DeMarzo ( 2005 ) consider the design and sale of asset-backed
securities when there exist exogenous shocks on the asset quality. The securitizer has private information about the quality of assets, which induces a downward-sloping demand curve for securities. Their models characterize the optimal security design under several circumstance and analyzes the benefits of pooling and tranching in the asset-backed securities market.
Vanasco ( 2014 ) assumes that the quality of securitized products depends
on how much soft information the bank has acquired when pooling assets. However, there exists a trade-off between the gain from information acquisition and the gain from trade
4

in secondary markets. The investors cannot observe the balance sheet information of the bank but form their belief about the type of bank by observing the securities sold to them.
Given such a setup, Vanasco concludes that in the optimal mechanism the securitizers should commit to a predesigned securities tranching structure and the secondary markets should have transfers across banks with different loan quality.
There are several theoretical papers investigate the role of reputation in securitization practices.
Hartman-Glaser ( forthcoming ) assumes that there are two types of securitizers:
one is the committed type who always truthfully reports the asset quality, while the other is the opportunistic type who chooses a reporting strategy to maximize his payoff.
His model predicts that the opportunistic type is more likely to tell the truth when his reputation is low but eventually will cash out his reputation.
Chari, Shourideh, and
Zetlin-Jones ( forthcoming ) claim that without reputational concerns the adverse selection
problem between securitizers and investors can be quickly resolved, but when there exists a reputational effect the bank with low-quality assets always has an incentive to pool with the bank with high-quality assets by selling the same structured products. As a result, the adverse selection problem will persist over time in secondary markets and a small fluctuation of collateral values may lead to a significant fall in trade volume.
Griffin,
Lowery, and Saretto ( 2014 ) argue that there exists an internal conflict between banks and
their specialists. Even though the bank cares about its reputation, the specialists, who work for the bank in only limited periods of time, may sacrifice the long-run objective of the bank to maximize their own payoffs. In particular, specialists within reputable banks are more likely to sell securities that make excess profits during boom periods but severely underperform during market downturns.
Winton and Yerramilli ( 2011 ) measures
the reputation of a bank according to the number of recent defaults on loans it has originated. The bank can decrease the probability of defaults by monitoring loans. Their model predicts that monitoring is more likely to sustain when the bank has a greater liquidity need or monitoring confers a great benefit.
Kawai ( forthcoming ) builds a three-
period model in which the reputation of a securitizer is defined as the market belief about his monitoring cost. His model suggests that when it is costly to monitor the quality of asset the bank who has made the monitoring effort will hold all of its portfolios at the interim period to boost his reputation. Then there is a great loss in allocation efficiency.
Several papers empirically examine whether the investors recognize the differences among securitizers or the difference between securitized products.
Demiroglu and James
( 2012 ) find that when loan originators and security underwriters are affiliated, the secu-
ritized products have less subordination for senior tranches (which means a less cushions to AAA-rated securities agains losses) and lower initial yield spreads (which are implied by lower yield spreads between securities and 10-year Treasury Bond).
Begley and
5

Purnanandam ( 2014 ) find that among the products with the same credit rating the in-
vestors pay higher prices for the tranches backed by higher equity tranches. By examining mortgage loans originated by Italian banks from 1995 to 2006,
Albertazzi, Eramo, Gambacorta, and Salleo ( 2011 ) offer a similar observation.
Jiang, Nelson, and Vytlacil ( 2014 )
provide evidence that investors are able to selectively purchase relatively higher quality loans by exploiting new information predicting loan performance between the time of loan origination and the time of loan sale.
Say a long-lived securitizer faces a continuum of short-run competitive investors. Time is discrete and the time horizon is infinite. At each period there is a spot market. The securitizer packages a continuum of assets and creates one unit of securitized product backed by this asset pool at the beginning of period, and the securitized product delivers one unit of cash at the end of period. The securitized product is risky. If too many assets in the asset pool default, the securitizer will have no cash to support his product.
Denote the outcome of product y t
1, or simply call it a success, if the securitizer is able to support his product, and denote y t
0, or call it a a failure, otherwise. The realization of y t depends on two factors. First, it depends on an aggregate shock S t
, which can be interpreted as the economic environment. Assume the aggregate shock is a binary variable such that S t that is, Prob( S t
∈ {
= s | S t − 1
G, B
= s )
} . It follows a stochastic process and it is persistent;
> Prob( S t
= s | S t − 1
= s
0
). Second, the realization of y t depends on whether the securitizer has made a screening when packaging assets and creating the securitized product. By screening the assets the securitizer ensures that the assets conform the underwriting standards and the default risk is within expectation.
However, the game is of imperfect public monitoring. If the screening is made, y t is 1 with probability q
H,S t and 0 with a complementary probability. If the screening is not made, y t is 1 with probability q
L,S t and 0 with a complementary probability. Under any aggregate shock S t
, q
H,S t
> q
L,S t
; in this sense the screening technology increases the quality of the product. Given the same screening condition, the product is more likely to succeed when the aggregate shock is G ; therefore q
L,G
> q
L,B
, q
H,G
> q
H,B
. In addition, screening confers fewer benefits when the aggregate shock is G such that q
H,B
− q
L,B
> q
H,G
− q
L,G
.
It is because the default rate is generally low when the economic environment is good.
Even without a proper screening, the product is more likely to survive. The investors cannot observe whether the securitizer has made the screening.
6

A reputational concern arises because there are two types of securitizer. One is called the normal type who has the screening technology as described above, and the screening costs him a constant c . The other type is called the inept type. I assume the inept type has an extremely high screening cost and therefore he simply forgoes the screening. The reputation of the securitizer at period t is then summarized by the probability µ t that the investors believe he is of the normal type.
Because the screening is unobservable, the investors are not sure about the quality of the securitized product and an adverse selection problem endogenously arises when the securitizer sells his product to the market. In line with the literature of security design,
I assume that the securitizer can signal the quality of his product by retaining a fraction of it. Such a retention signal is costly, because the securitizer has a discount factor β over the cash flow delivered at the end of period, while the investors do not discount any cash flow within the period. The difference in the intra-period discount factor between the securitizer and the investors represents their different liquidity preferences, which is often assumed in adverse selection models in financial markets. It is commonly believed that financial intermediaries prefer immediate cash flow so that they can raise the capital for new investment opportunities or to meet the mandatory capital requirements.
The timeline is shown in Figure 1. It proceeds as follows: At the start of period t , both the securitizer and the period t investors observe the aggregate shock S t − 1 and reputation of the securitizer µ t − 1 from the last period. The securitizer of the normal type decides whether to make the screening s t when packaging the asset pool and creating the securitized product. He then sells a fraction α t
∈ [0 , 1] of his product, which is perceived as a signal of the quality of his product. The investors earn the zero expected profit and pay the price equal to the expected quality conditional on the signal α t and their knowledge about the aggregate shock S t − 1 and the reputation µ t − 1 in the last period.
The aggregate shock in the current period S t is then realized and so is the outcome of the product y t
. Given the realized aggregate shock and the product performance, the investors update their belief µ t about the type of securitizer, pass this information to the next generation and then retire.
The information history of the normal type until period t is h t
N
= { s t
, α t
, y t
, S t
, µ t
} t
0 and he chooses a strategy { s t
( h t − 1
) , α t
( h t − 1
, s t
) } ∞ t =0
. For the inept type his information
, history until period t is h t
I
= { α t
, y t
, S t
, µ t
} t
0 and he chooses { α t
( h t − 1
) } ∞ t =0
. Notice in my model there is only one securitizer and he is of the normal type consistently across periods.
In this sense, the inept type is a hypothetical player. Different from the conventional reputation models, my model assumes that the hypothetical player is able to choose
his strategy to optimize the payoff instead of using a mechanic strategy.
2
Finally, the
2
Another model that allows the hypothetical player to optimize the payoff rather than using a me-
7

Figure 1: Timeline information history of investors until period t is h t i
= { α t
, y t
, S t
, µ t
} t
0
. The strategy of the investors is the price p t
( h t − 1 i
, α t
) that they choose to pay for the product.
At period t , given the aggregate shock S t − 1
, the reputation µ t − 1 and the investors’ price function p t
, when the normal type makes the screening and sells a fraction α t of his product, his period payoff is v
N t
( µ t − 1
, S t − 1
; s t
= 1) = α t p t
+ β (1 − α t
) Q
H,S t − 1
− c (1) where Q
H,S t − 1
=
E
( q
H,S t
| S t − 1
) is the expected quality of the product conditional on S t − 1 when the screening is made.
On the other hand, if the normal type chooses not to screen at period t and sells a fraction α t
, his period payoff is v
N t
( µ t − 1
, S t − 1
; s t
= 0) = α t p t
+ β (1 − α t
) Q
L,S t − 1
(2) where Q
L,S t − 1
=
E
( q
L,S t
| S t − 1
) is the expected quality of the product conditional on S t − 1 when the effort is not made.
When the inept type chooses to sell the fraction α t of his product at period t , his period payoff is v t
I
( µ t − 1
, S t − 1
) = α t p t
+ β (1 − α t
) Q
L,S t − 1
(3) which yields the same value as v N t
( µ t − 1
, S t − 1
; s t
= 0).
Assumption 1.
(Screening is Socially Beneficial) For any aggregate shock S t − 1 previous period, Q
H,S t − 1
− c > Q
L,S t − 1
.
in the
Assumption
1
implies that given any aggregate shock S t − 1 in the previous period, the first-best outcome is that the securitizer of normal type always makes the screening.
Investors buy the whole unit of securitized product and pay the price Q
H,S t − 1
.
chanic strategy is
Hartman-Glaser ( forthcoming ).
8

To set up a benchmark, first consider the one-period model in which the securitizer does not worry about how his behavior in the current period influences his future payoff. For the ease of exposition, I drop all the time subscripts and denote S t − 1 as S , S t as S
0
, µ t − 1 as µ and etc.
If no inept type exists in the market, the problem boils down to a classical moral
S of the fraction sold by the securitizer: Given any aggregate shock S in the previous period, the investors believe that the securitizer has made the screening if he sells the fraction no greater than
α
S
; otherwise the investors believe that no screening has ever been made. The threshold
α
S is determined by
S p + β (1 − ˜
S
) Q
H,S
− c = ˜
S p + β (1 − ˜
S
) Q
L,S c
α
S
= 1 −
β ( Q
H,S
− Q
L,S
)
Figure 2: Game Tree
S is well-defined.
(4)
The investors face an adverse selection problem when the inept type exists in the market. As the game tree in Figure 2 shows, at the time when the investors observe the fraction sold α they do not know whether this signal is sent out by the normal type or the inept type, or even if this is a securitizer of the normal type whether he has made the screening. However, the investors may infer some information from the signal α . For example, the investors believe that there is no screening ever been made if the securitizer
9

α
S
defined in ( 4 ). Moreover, given the aggregate
shock S in the previous period, if the investors observe a signal α satisfying the following two inequalities:
αQ
H,S
+ β (1 − α ) Q
H,S
− c ≥ Q
L,S
Q
L,S
≥ αQ
H,S
+ β (1 − α ) Q
L,S
(5)
(6) the investors will believe that the securitizer is of the normal type and has made the screening with probability 1, because the signal α completely separates the normal type
from the inept type; the inequality ( 5 ) implies that the normal type would prefer screening
and selling the fraction α than shirking and selling the whole unit of his product, while
the inequality ( 6 ) implies that the inept type would prefer revealing his own type rather
than mimicking the normal type to sell the fraction α . Define ˆ
S as the maximal fraction that the normal type can separate himself from the inept type, and it is determined by:
α
S
=
Q
L,S
(1 − β )
Q
H,S
− βQ
L,S
(7) which is derived by having the inept type indifferent between mimicking the normal type and revealing his own type.
As in many signaling games this model has multiple equilibria.
In the following analysis I focus on the equilibrium which provides the maximum trade volume, because such an equilibrium Pareto dominates any other equilibria. In the equilibrium, given the reputation µ and S in the last period the investors pay the following price function after observing the signal α .

Q
H,S p ( α, µ, S ) =


µQ
H,S
+ (1 − µ ) Q
L,S

 Q
L,S if α ≤ ˆ
S
α
S
< α ≤ ˜
S otherwise
(8)
α
S
α
S
are defined in ( 4
) and ( 7 ), respectively.
I then assume that the securitizer of the normal type would only like to separate himself from the inept type when the aggregate shock in the previous period is B . When the aggregate shock in the previous period is G , the difference between the expected quality Q
H,G and Q
L,G is small. Therefore the securitizer has to retain a large fraction of his product to persuade the investors that he has screened the assets. Such a signal is so costly that the normal type would like to shirk and sell the whole unit of his product.
This is implied by Assumption
2 .
Assumption 2.
(No Separating Equilibrium When Aggregate Shock is G )
10

β ( Q
H,G
− Q
L,G
) 2
Q
H,G
− βQ
L,G
< c <
β ( Q
H,B
− Q
L,B
) 2
Q
H,B
− βQ
L,B
(9)
Technically, to ensure the existence of separating equilibria there must exist a α ∈
α
S
, ˜
S
] for any aggregate shock S in the previous period. Such α need to be less or equal
α
S to persuade the investors that the screening has already been made, while be greater
α
S to provide the normal type enough profits to screen. However, Assumption
2
implies that [ ˆ
G
, α
G
] is an empty set and therefore no separating equilibrium exists when the aggregate shock is G in the previous period.
The equilibrium of the one-period model is summarized in Proposition 1.
Proposition 1.
Given Assumption
2 and the price function defined in ( 8 ),
(i) When the aggregate shock in the previous period is G , the normal type never makes the screening and sells the whole unit of his product.
(ii) When the aggregate shock in the previous period is B , the normal type always screens when packaging the assets and creating the securitized product. When µ ≤ µ
∗
, the normal type sells the fraction ˆ
B
. When µ > µ
∗
, the normal type sells the fraction
α
B
. The threshold µ
∗ is determined by:
α
B
Q
H,B
+ β (1 − ˆ
B
) Q
H,B
α
B p ( ˜
B
, µ
∗
, B ) + β (1 − ˜
B
) Q
H,B
(10) where the normal type is indifferent between separating himself and pooling with the inept type.
Proof.
See Appendix.
Proposition 1 states that, as discussed above, when the aggregate shock is G in the previous period the normal type never screens. In contrast, when the aggregate shock is B in the previous period the normal type is always better off revealing that he is the normal type who has made the screening. When the fraction sold by the normal type
α
B
, the investors believe that the inept type will mimic the normal type by selling the same fraction and then the investors pay the price µQ
H,B
+ (1 − µ ) Q
L,B
; that is, the normal type has to face a “lemon discount,” due to the adverse selection in the market, and this discount is decreasing in reputation µ . As a result, the normal type
α
B when his reputation is low and
α
B reputation is high and the lemon discount is not severe.
when his
11

I now turn to an infinite horizon model in which the securitizer has a reputational concern. Again, I drop all the time subscripts. Given the state variables ( µ, S ), the value functions of the normal and inept types are denoted by V
N
( µ, S ) and V
I
( µ, S ), respectively. Moreover, I denote V s
N as the value function of the normal type when he chooses the screening variable s . Given the strategy of the inept type and the price function of the investors, if the normal type chooses to screen, his value function is:
V
1
N
( µ, S ) = max
α
αp + β (1 − α ) Q
H,S
− c
+ δ
E q
H,S
0
V
N
( µ h
, S
0
) + (1 − q
H,S
0
) V
N
( µ l
, S
0
) | µ, S
(11) where µ h and µ l are the belief updating functions given the realized outcomes of the products are 1 and 0, respectively, and they are functions of the aggregate shock S
0 in the current period. Assume the belief is updated according to Bayes’ rule whenever possible.
If the securitizer of the normal type chooses not to screen, his value function is
V
0
N
( µ, S ) = max
α
αp + β (1 − α ) Q
L,S
+ δ
E q
L,S
0
V
N
( µ h
, S
0
) + (1 − q
L,S
0
) V
N
( µ l
, S
0
) | µ, S
Given any state variable ( µ, S ), the value function of the normal type is defined by:
(12)
V
N
( µ, S ) = max s ∈{ 0 , 1 }
V s
N
( µ, S )
Finally, the value function of the inept type is defined by:
(13)
V
I
( µ, S ) = max
α
αp + β (1 − α ) Q
L,S
+ δ
E q
L,S
0
V
I
( µ h
, S
0
) + (1 − q
L,S
0
) V
I
( µ l
, S
0
) | µ, S
(14)
The equilibrium conception that I focus on is the perfect Bayesian equilibrium. The equilibrium is defined as the following:
Definition 1.
An equilibrium in the infinite horizon model is: given the state variables
( µ, S ), a pair of value functions ( V N , V I ) , a pair of strategies (( s, α N ) , α I ) , a price function p , and the belief updating rules ( µ h
, µ l
) such that:
(i) given the state variables, the price function, the belief updating rules and the strategy of the inept type, the strategy ( s, α
N
) maximizes the value function V
N
;
12

(ii) given the state variables, the price function, the belief updating rules and the strategy of the normal type, the strategy α I maximizes the value function V I ( µ, S ) ;
(iii) the investors earn the zero expected profit;
(iv) the belief is updated according to Bayes’ rule whenever possible.
Moreover, as discussed in the one-period model, I focus on the equilibrium which maximizes the trade volume α . In the following analysis I assume that the aggregate shock S follows a Markov transition such that P
GG
= P
BB
= θ and P
GB
= P
BG
= 1 − θ , where θ > 0 .
5. The symmetric transition between states is assumed for the simplification of analysis. However, my results should apply to a more general setup as long as the aggregate shock is persistent.
Proposition 2.
The value functions V N ( µ, S ) and V I ( µ, S ) are continuous and increasing in reputation µ .
Proof.
The proposition can be derived by the standard argument of dynamic programming.
To solve the equilibrium, I will first find the optimal fraction to sell and derive the corresponding value function V s
N ( µ, S ) for each s ∈ { 0 , 1 } , and then determine the optimal
effort level.
3
First consider the case when the normal type chooses not to screen.
Proposition 3.
If the normal type chooses not to screen, in the equilibrium the investors pay the price Q
L,S for any fraction that the securitizer sells. There is no belief updating; i.e., µ h,S
0
= µ l,S
0
= function is given by:
µ . The normal type sells the whole unit of product and his value
V
0
N
( µ, S ) = Q
L,S
+ δ
E
V
N
( µ, S
0
) | µ, S (15)
Proof.
Straightforward.
Then consider the optimal fraction to sell if the normal type chooses to screen. In the following analysis I restrict myself to the situation where the reputational premium is high enough that no complete separating equilibrium exists. Such a situation is also considered in
Hartman-Glaser ( forthcoming ),
Chari, Shourideh, and Zetlin-Jones ( forthcoming ) and
Vanasco ( 2014 ). The following assumption is imposed.
3
Here I exclude the equilibrium in which the securitizer of the normal type mixes between screening and shirking.
13

Assumption 3.
(Nonexistence of Complete Separating Equilibria) For any aggregate shock S in the previous period,
Q
L,S
(1 − β ) < δ v
I
(1 , B ) − v
I
(0 , B )] .
Q
L,S
+ δ
E
V
I
(0 , S
0
) | µ, S ≥ αQ
H,S
+ β (1 − α ) Q
L,S
+ δ
E
V
I
(1 , S
0
) | µ, S
(16) where v
I
(1 , B ) and v
I
(0 , B ) are period payoffs of the inept type when the his reputation is respectively 1 and 0 and the aggregate shock is B in the previous period.
Assumption
3
implies that if the inept type reveals his own type by selling the whole unit of his product, the benefit is always less than the gain of mimicking the normal type and being paid as the normal type next period when the aggregate shock is B , even if the normal type currently chooses to hold all of his product.
Proposition 4.
Given Assumption
3 , there is no complete separating equilibrium at any
state ( µ, S ) .
Proof.
Suppose there exists a separating equilibrium at states ( µ, S ). Then it implies that the inept type prefers revealing his own type over mimicking the signal α of the normal type such that
(17) which requires
α ≤
Q
L,S
(1 − β ) − δ
E
V I (1 , S
0
) − V I (0 , S
0
) | µ, S
Q
H,S
− βQ
L,S
(18)
With some tedious algebra shown in Appendix, I can show that as long as Assumption
3
holds, Q
L,S
(1 − β ) ≤ δ
E
V
I
(1 , S
0
) − V
I
(0 , S
0
) | µ, S for any state ( µ, S ) and hence no such α exists.
Because the retention is a costly and credible signal, the investors believe that the more a securitizer retains, the more likely the securitizer is of the normal type and has already made the screening. I impose the following assumption.
Assumption 4.
(Retention Increases Credibility) For any state ( µ, S ) , the belief updating functions µ h and µ l and the price function are monotonically decreasing in α .
The following lemma characterizes the condition when the investors believe that the normal type has made the screening.
Lemma 1.
The investors believe that the screening is made with a positive probability if the securitizer sells a fraction less than
α
S
+
δ
E
( q
H,S
0
− q
L,S
0
) V N ( µ h
, S
0
) − V N ( µ l
, S
0
) | µ, S
β ( Q
H,S
− Q
L,S
)
14
(19)

for any state ( µ, S ) , where the first part ˜
S is the fraction needed to persuade the investors that the normal type has already made the screening in the one-period model, and the second part is the reputational premium.
Proof.
Given the price function and the belief updating rules, if the normal type chooses to screen, his payoff is
αp + β (1 − α ) Q
H,S
− c + δ
E q
H,S 0
V
N
( µ h
, S
0
) + (1 − q
H,S 0
) V
N
( µ l
, S
0
) | µ, S (20)
If he chooses to shirk, his payoff is
αp + β (1 − α ) Q
L,S
+ δ
E q
L,S
0
V
N
( µ h
, S
0
) + (1 − q
L,S
0
) V
N
( µ l
, S
0
) | µ, S (21)
To make sure that the normal type makes the screening in the equilibrium, I need ( 20 )
>
( 21 ) which implies:
β (1 − α )( Q
H,S
− Q
L,S
) ≥ c − δ
E
( q
H,S
0
− q
L,S
0
) V
N
( µ h
, S
0
) − V
N
( µ l
, S
0
) | µ, S
The desired result is obtained after substituting equation ( 4 ) into the above.
The following lemma is needed when I construct the equilibrium.
Lemma 2.
In the class of equilibrium that I consider, the securitizer of the normal type never uses the mixed strategy.
Proof.
See Appendix.
Given Lemma
2 , when the securitizer of the normal type chooses to screen, in the
equilibrium the investors believe that the normal type signals the fraction α , while the inept type chooses either to completely pool with the normal type to signal α or to randomize between pooling with the normal type and revealing his own type. In the equilibrium in which the inept chooses to completely pooling with the normal type, I call the equilibrium a complete pooling, otherwise I call it a partial pooling.
The following proposition characterizes the equilibrium when the normal type chooses to screen.
Proposition 5.
When the normal type has already made the screening, the equilibrium is: given the state variables ( µ, S ) , a pair of value functions ( V
1
N
, V
I
) , a pair of strategies
( α, γ ) , a price function p , and the belief updating rules such that:
(i) given the price function, the belief updating rules and the strategy of the inept type, the normal type sells the fraction α to maximize his value function V
1
N
( µ, S ) ;
15

(ii) given the price function, the belief updating rules and the strategy of the normal type, the inept type chooses a strategy γ ∈ [0 , 1] such that with probability γ he pools with the normal type to signal α and with probability 1 − γ he reveals his own type by selling the whole unit of his product. When pooling with the normal type to sell the fraction α , the inept type yields
V
I
( µ, S ; γ ) = αp ( α, µ, S ) + β (1 − α ) Q
L,S
+ δ
E q
L,S
0
V
I
( µ h
( α ) , S
0
) + (1 − q
L,S
0
) V
I
( µ l
( α ) , S
0
) | µ, S
(22) where V
I
( µ, S ; γ ) ≥ Q
L,S
+ δ
E
V
I
(0 , S
0
) | µ, S , and the equality holds when γ ∈ (0 , 1) .
(iii) as long as the fraction that the securitizer sells is less than the threshold defined in
( 19 )
, the investors believe that there exists a positive probability that the screening is made and they would like to pay the price as follows: p ( α, µ, S ) =
E
"
µQ
H,S
0
µ
+ (1
+ γ (1
−
−
µ )
µ
γQ
)
L,S
0
| µ, S
#
(iv) When the investors believe that there exists a positive probability that the securitizer is of the normal type, they update their belief as follows:
µ h
( α, µ, S
0
) =
µ l
( α, µ, S
0
) =
µq
H,S
0
µq
H,S 0
+ (1 − µ ) γq
L,S 0
µ (1 − q
H,S
0
)
µ (1 − q
H,S 0
) + (1 − µ ) γ (1 − q
L,S 0
)
(23)
(24)
(25)
Theorem 1 characterizes the equilibrium behavior of the normal type. The proof of
Theorem is lengthy and therefore I leave it in the Appendix.
Theorem 1.
In the equilibrium, for any aggregate shock S in the previous period, the normal type makes the screening and holds all of his product until his reputation reaches
µ
S
. Then, when the aggregate shock in the previous period is G , the normal type shirks when his reputation is greater than ˜
G
. When the aggregate shock in the previous period is B , the normal type still makes the screening when his reputation is greater than ˜
B and starts selling his product when his reputation is greater than µ
∗
B
.
Figure 3 (a) presents the value functions of the normal type for each aggregate shock
S in the previous period and Figure 3 (b) presents the retention. Several points are worth mentioning in Figure 3. First, for any aggregate shock S in the previous period, the value function is a constant in the partial pooling equilibrium, and then the value
µ
S when the equilibrium is complete pooling.
Second, when the reputation is in the range [˜
B
, µ
∗
B
], the normal type still retains all his
16

0.6
0.4
0.2
0
0
27.1
27
26.9
26.8
0
27.3
Figure 3: Equilibrium in the Infinite-Horizon Model
(a) Value Functions of Normal Type
V
N
(
µ
,B)
V
N
(
µ
,G)
27.2
˜
B µ
∗
G
Reputation
µ
(b) Retention in Infinite Horizon Model
µ 1
1
1−
α
(
µ
,B)
1−
α
(
µ
,G)
0.8
µ
∗
G
Reputation
µ
µ 1
1
−
˜
B
17

products because the market price is too low. Third, to persuade the investors that the product may be of high quality the optimal fraction that the securitizer is able to sell is not monotonic in reputation. That is because when the reputation is high, the realized outcome of the product cannot shake the investors’ belief significantly, and so the incentive within the period plays a more important role. In particular, when the investors believe that the securitizer is of the normal type with probability one, the securitizer can only
α
B
, the fraction to persuade the investors to pay Q
H,B in the one-period static model.
Finally, it is worth pointing out that in the infinite horizon model the endogenously adverse selection will eventually be resolved because the reputation of the normal type will go to 1 with probability one. Though there is no belief updating when the reputation is above µ
∗
G and the aggregate shock in the previous period is G , there exists a positive probability that the aggregate shock will change to B and then the normal type will be able to push his reputation by making the screening. Figure
4
presents an example of reputation dynamics when the normal type and the inept type have the same initial reputation and face the same aggregate shock over periods. The reputation of the normal type will eventually converge to 1 while the reputation of the inept type will converge to
0.
Figure 4: Dynamics of Reputation
Dynamics of Reputation: Normal Type
1
0.8
0.6
0.4
0.2
0
2 4 6 8
Period
10
Dynamics of Reputation: Inept Type
12 14 16
1
0.8
0.6
0.4
0.2
0
2 4 6 8
Period
10 12 14 16
One important result of my model is that: Although it is socially optimal to produce the high-quality securitized products, the securitizer may shirk due diligence as he faces the
18

“lemon discount” in secondary markets. This result, as are many equilibria in dynamic signaling games, crucially depends on the parameter values specified.
In Figure
5 , I conduct an experiment by changing the benefit of screening assets when
the aggregate shock in the previous period is G , which is denoted by q D
G
= q
H,G
− q
L,G while having other parameter values fixed. The horizontal axis is the benefit of screening
, q
Q
D
G and the vertical axis is the reputation µ . The minimum of q D
G
H,G is chosen to satisfy
− c = Q
L,G
, the condition guaranteeing that the screening is socially optimal.
Figure
5
shows that when the benefit is less than q
1
, the normal type never screens regardless of his reputation level because the benefit of screening cannot compensate the
“lemon discount” which he faces in the market. When q
D
G is in the range [ q
1
, q
2
], the normal type makes the screening when his reputation is below the threshold function
µ
∗
G
( q D
G
) but shirks otherwise. The threshold function µ
∗
G is increasing in the benefit of screening q D
G
. When the screening benefit is above q
2
, there is no qualitative difference in behavior whether the aggregate shock in the previous round is G or B : The normal type makes the screening all the time, and he retains all when his reputation is less than
µ
∗
S and starts selling otherwise.
Figure 5: Equilibrium Screening Choice When S t − 1
= G
1
No Effort
µ
∗
G
( q D
G
)
Effort
0 q1 q
D
G
= q
HG
− q
LG q2
Another parameter of interest is the intra-period discount β . The trade of assetbacked securities is beneficial because the securitizer discounts future cash flow at a higher rate than the investors. Several factors may affect the securitizer’s intra-period discount. For example, if the policy maker increases the minimum requirement of capital adequacy, the policy often discussed after the financial crisis, securitizers have a greater craving for immediate cash flow, and they are more discouraged by the adverse selection in the secondary market. When the intra-period discount decreases, the securitizer is more eager to trade and therefore both µ
∗
B and µ
∗
G decrease. Meanwhile, the reputational premium increases and so there is a chance that the securitizer is able to sell more when
19

his reputation is greater than µ
∗
B
.
Finally, the persistence of the aggregate shock is a crucial assumption in my model.
The securitizer of the normal type is less likely to shirk due diligence when the aggregate shock is less persistent.
My model explores two questions: (1) How is reputation related to the performance of securitized product? (2) How is reputation related to risk retention? It has the following empirical implications.
Prediction 1.
During boom years, securitized products issued by reputable securitizers have higher default rates, all else being equal.
The equilibrium in my model shows that during boom years the securitizer of the normal type only screens the asset pools underlying his securitized products when his reputation is low. When his reputation is above some threshold, he will forgo the screening. Notice, however, if the difference between Q
H,G and Q
L,G during boom years is very small, we may not find any significant difference in default rates between the products issued by high reputation and low reputation securitizers in empirical studies. The difference in default rates should be more evident when the economic environment changes from boom to bust. Prediction 1 is consistent with the findings in
Griffin, Lowery, and
Saretto ( 2014 ), in which they find that the reputable banks have higher ratios of default
senior tranches in the 2007-2008 financial crisis as well as the min-structured finance crisis
of 2002.
4
Prediction 2.
During non-boom years, there is no difference in default rates between products issued by securitizers with high and low reputation, all else bing equal.
My model shows that during non-boom years the securitizer of the normal type always screens the asset pools underlying his securitized product regardless of his reputation level.
To my best knowledge, I know none of empirical studies that have tested Prediction 2.
Empirical studies that use more recent data should reveal whether my prediction holds.
4
Griffin, Lowery, and Saretto ( 2014 ) also find that reputable banks have higher ratios of default senior
tranches in 2009. As the economy was in downturn during 2007 to 2009, my model predicts that all securitizers should have a more careful screening during that period and the defaults rates of securitized products should be indifferent between the securitizers with high and low reputation. Notice, however, the performance measure used in
Griffin et al.
( 2014 ) is not time specified. Instead, their measure is
constructed by calculating the ratio of the nominal value of tranches that are in default to the total nominal value of the deal in any time . Therefore, it is not clear when the senior tranches that defaulted in 2008 and 2009 were issued.
20

Prediction 3.
Securitizers with high reputation retain less than those with low reputation, all else being equal.
My model shows that securitizers with high reputation always retain less than those with low reputation, regardless of economic environment. By examining the retained interests of mortgage-backed securities from 2001 to 2006,
Scholz ( 2013 ) finds that the
securitizers with high reputation retain less while controlling total assets and credit risks.
Similar phenomenon is also found in loan syndication market: Reputable lead arrangers
retain smaller portions of the loan ( Dennis and Mullineaux , 2000 ;
Lee and Mullineaux ,
2004 ; Sufi , 2007 ).
Gopalan, Nanda, and Yerramilli ( 2012 );
Lin and Paravisini ( 2011 ) find
that when a lead arranger experienced large-scale bankruptcies among his borrowers, he is more likely to retain larger fractions of the loans in their subsequent syndication activity.
The empirical study on reputation in securitization market is limited, in particular on the part of securitizers. There are two empirical challenges. First, there is no dealspecified data about securitizers’ risk retention. Before the financial crisis, it was not a mandatory for securitizers to report their retained interests for each deal. Though the offering prospectuses my reveal the initial retained interests of securitizers when they market their securitized products, it is possible that securitizers eventually sell their re-
tained interests in the subsequent months. Some researchers ( Chen, Liu, and Ryan , 2008 ;
Scholz , 2013 ) collect retained interests of securitizers from FR Y9-C reports. Since 2001
the Federal Reserve has required bank holding companies with total consolidated assets of $500 million or more to report detailed information of their securitization activities
in FR Y9-C every quarter.
5
However, the retained interest information reported in FR
Y9-C is not deal-specified but cumulative sum. Suppose an securitizer initially retains x% of assets when he markets his securitized product but sells these retained assets in subsequent three months. He then creates another securitized product and retains x% by next report. We may observe the same x% retained interests on his FR Y9-C, although the securitizer actually retains no skin for any deal. Second, there lacks a consensus how to measure reputation in securitization markets.
Scholz ( 2013 ) uses the mortgage-backed
securities market share to classify reputable banks. He considers a bank to have a high reputation if it is among the top 25 securitizers of non-agency MBS from 2001 to 2006, and to have a low reputation otherwise.
Gopalan, Nanda, and Yerramilli ( 2012 ) argue
that the reputation of a financial intermediary is shared across divisions. A bank that has a large market share in MBS may be not well known outside the MBS market. Therefore they believe that reputation scores associated with IPO market are more appropriate.
Notice reputation measures used in both above papers are static.
Hartman-Glaser ( forth-
5
The initial threshold for the filling requirement was $150 million. Since 2006 the threshold has changed to $500 million.
21

coming ) suggests a dynamic measure of reputation by using downgrades on the senior
tranches in previously issued MBS.
As a part of the Wall Street reform, the Dodd-Frank Act requires securitizers of assetbacked securities to retain at least 5 percent of the credit risk of the assets underlying the securities. It has been believed that this risk retention, often called “skin in the game,” helps better align the interests of securitizers and investors by providing an incentive for securitizers to monitor the quality of securitized assets. In this section I investigate how the mandatory risk retention influences the equilibrium behavior and the efficiency.
Suppose there is a minimum retention requirement r , which implies that the maximum fraction that the securitizer can sell is ¯ = 1 − r . Notice, for any state ( µ, S ), this risk retention requirement has no impact on the normal type when the reputation is less than
µ
∗
S
, because at that state the normal type holds all of his product. Whenever the investors observe a sale, they believe that the securitizer is of the inept type. When the reputation is greater than µ
∗
S
, the belief of the investors is the same with the analysis provided in the previous section if ¯
is weakly greater than the fraction defined in ( 19
α
is strictly less than the fraction defined in ( 19 ), the investors believe that the securitizer
is of the normal type with probability µ when the securitizer sells the fraction ¯ , and update their belief according to Bayes’ rule.
Theorem 2 states the condition that the securitizer of the normal type always makes the screening and therefore the quality of the product is always high.
Theorem 2.
For any state ( µ, S ) , the securitizer of the normal type always makes the screening as long as ¯ satisfies
¯
L,S
+ β (1 − ¯ ) Q
L,S
≤ ˜
S
Q
H,S
+ β (1 − α
S
) Q
H,S
− c (26)
Equation ( 26 ) implies that to ensure that the normal type always chooses to screen,
he cannot make more profit by shirking and selling the fraction ¯ at any state.
Though the risk retention requirement helps improve the quality of securitized products, it increases the cost of securitization and decreases the provision of liquidity in the market. In particular, when the aggregate shock in the previous period is G the risk retention that ensures the securitizer makes the screening may be significantly greater than the retention required when the aggregate shock in the previous period is B . An excessive requirement in the pessimistic economic condition may further depress the credit market. Therefore, compared to the static retention requirement, a dynamic framework
22

tied to the business cycle should be more efficient.
Securitization is an important source of credit formation for the economy. However, recent experience shows that without proper regulation, securitization may lead to a deterioration of underwriting standards and an unstable financial system, because originators and securitizers do not internalize the social costs.
My model shows that reputation can play a positive role when the screening provides enough benefit. However, when the asset default rates are generally low and the screening only confers a small benefit, reputational concerns may not be strong enough for securitizers to make the screening and produce high-quality securitized products, especially when the market has a high demand for liquidity and investors game their opportunity in the good economic environment. As a result, the securitizer only makes the screening when his reputation is low and then chooses to shirk when his reputation is high. In such a situation, the risk retention requirement helps align the incentives between the securitizer and the investors.
The risk retention requirement, if appropriately designed, could promote the efficiency of capital allocation. The retention may decrease the provision of liquidity in the market, though this reduction may not be bad, as it may help counter business cycles and prevent predatory lending from the side of originators. In addition, it should be more efficient if the risk retention is designed in a dynamic manner; for example, the required amount of risk retention could be tied to the business cycle or financial market indicators.
Furthermore, policy makers should encourage a variety of credit enhancement tools to work in conjunction with the risk retention requirement, which includes but is not limited to: subordination, excess spread, shifting interest, performance triggers and interest rate swap.
In the real wold, securitization is a complex process involving multiple agents. My model focuses on the incentive of securitizers and investigates how the reputation of securitizers influences equilibrium behavior and asset allocation. My results provide insight for the design of the risk retention requirement. Nevertheless, there are still many open questions worth further pursuing. For example, does tranching structure help alleviate the endogenously adverse selection problem? Moreover, what format should the risk retention requirement take? The current policy permits that risk retention can be accomplished through one or a combination of methods: a vertical interest, a horizontal interest, or an equivalent exposure of the securitized pool. It is of interest to investigate how the incentive of securitizers changes according to the format of risk retention.
23

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26

Proof of Proposition 1.
To ensure the existence of separating equilibrium, the following condition of the fraction sold α must be satisfied:
αQ
H,S
+ β (1 − α ) Q
H,S
− c ≥ Q
L,S
Q
L,S
≥ αQ
H,S
+ β (1 − α ) Q
L,S where the first inequality ensures that the normal type would like to signal that he is of the normal type and has made the screening rather than shirking and selling the whole unit of product, and the second inequality ensures that the inept type would like to reveal his own type instead of mimicking the signal of the normal type.
The above two inequalities imply that in the separating equilibrium the normal type should signal a fraction α ∈ h
Q
L,S
− ( βQ
H,S
− c )
Q
H,S
(1 − β )
,
Q
L,S
(1 − β )
Q
H,S
− βQ
L,S i
. However, Assumption
2
implies that
Q
L,B
− ( βQ
H,B
− c )
<
Q
H,B
(1 − β )
Q
L,G
− ( βQ
H,G
− c )
>
Q
H,G
(1 − β )
Q
L,B
(1 − β )
Q
H,B
− βQ
L,B
Q
L,G
(1 − β )
Q
H,G
− βQ
L,G that is, there is no such α to ensure the separating equilibrium when the aggregate shock is G in the previous period and hence only the pooling equilibrium exists. Furthermore,
Assumption
2
implies that it is never optimal for the normal type to execute the screening in the pooling equilibrium, because he needs to retain at least 1 − ˜
G to persuade the investors that the screening has been made when pooling assets and creating the securitized product. Even if he is able to receive a price Q
H,G
, his payoff is still less than what he receives when he shirks and sells the whole unit of his product.
α
G
Q
H,G
+ β (1 − ˜
G
) Q
H,G
< Q
L,G
Therefore in the equilibrium the normal type always chooses to shirk and sell the the whole unit of his product regardless of his reputation level.
When the aggregate shock is B in the previous period, the price function defined in
( 8 ) implies that if the normal type chooses to screen and sell the fraction ˜
B his payoff is
α
B
µQ
H,B
+ (1 − µ ) Q
L,B
+ β (1 − ˜
B
) Q
H,B
− c which is strictly increasing in reputation µ . As Assumption
2
implies that ˜
B
> α
B
,
27

there must exist a µ
∗ such that
ˆ
B
Q
H,B
+ β (1 − ˆ
B
) Q
H,B
= ˜
B p α
B
, µ
∗
, B ) + β (1 − ˜
B
) Q
H,B
When µ ≤ µ
∗
α
B
;
When µ > µ
∗
α
B and receive the pooling price.
Proof of Proposition
4 .
Given any aggregate shock S in the previous round, the inept type always has the option to sell all his product with the price Q
L,S and receive a continuation value δ
E
V I (0 , S
0
) | µ, S . Denote this reservation payoff as V I (0 , S ). With the assumption that the aggregate shock follows a Markov transition progress such that
P
GG
= P
BB
= θ and P
GB
= P
GB
= 1 − θ , I yield
V
I
(0 , G ) = Q
L,G
+ δθV
I
(0 , G ) + δ (1 − θ ) V
I
(0 , B )
V
I
(0 , B ) = Q
L,B
+ δθV
I
(0 , B ) + δ (1 − θ ) V
I
(0 , G )
(27)
(28) which implies the value functions of the inept type at reputation 0:
V
I
(0 , B ) =
(1 − δθ ) Q
L,B
+ δ (1 − θ )
(1 − δ )(1 + δ − 2 δθ )
Q
L,G
V
I
(0 , G ) =
Q
L,G
1 − δθ
+
δ (1 − θ )
V
1 − δθ
I
(0 , B )
(29)
(30)
When the reputation reaches 1 the securitizer arrives at another absorbing state:
Whatever the product performance is, the investors believe that the securitizer is of the normal type with probability 1. However, the investors only pay the price Q
H,S when
α
S
. Otherwise the investors pay the price Q
L,S for any fraction of the product the securitizer sells. The equilibrium boils down to the complete pooling in the static game. Given Assumption
2 , to mimic the normal type, the inept type should
sell the fraction ˜
B when the aggregate shock is B in the previous period, while sell the whole unit of the product when the aggregate shock is G in the previous period. As a result, the value function of the inept type at reputation 1 is
V
I
(1 , G ) = Q
L,G
+ δθV
I
(1 , G ) + δ (1 − θ ) V
I
(1 , B )
V
I
(1 , B ) = ˜
B
Q
H,B
+ β (1 − ˜
B
) Q
L,B
+ δθV
I
(1 , B ) + δ (1 − θ ) V
I
(1 , G )
(31)
(32) which implies that
V
I
(1 , B ) =
(1 − δθ ) ˜
B
Q
H,B
+ β (1 − ˜
B
) Q
L,B
(1 − δ )(1 + δ − 2 δθ )
+ δ (1 − θ ) Q
L,G
28
(33)

Notice that
V
I
(1 , G ) =
Q
L,G
1 − δθ
+
δ (1 − θ )
V
1 − δθ
I
(1 , B )
V
I
(1 , G ) − V
I
(0 , G ) =
δ (1 − θ )
1 − δθ
V
I
(1 , B ) − V
I
(0 , B )
(34)
(35) and therefore
δ
E
V
I
(1 , S
0
) − V
I
(0 , S
0
) | µ, G = δθ V
I
(1 , G ) − V
I
(0 , G ) + δ (1 − θ ) V
I
(1 , B ) − V
I
(0 , B )
=
=
δ (1 − θ )
V
I
(1 , B ) − V
1 − δθ
δ (1 − θ ) v
I
(1 , B ) − v
I
(0 , B )
(1 − δ )(1 + δ − 2 δθ )
I
(0 , B ) where v I (1 , B ) = ˜
B
Q
H,B
+ β (1 − ˜
B
) Q
L,B and v I (0 , B ) = Q
L,B
. Because
1 − θ
(1 − δ )(1+ δ − 2 δθ )
>
1,
Q
L,G
(1 − β ) < δ
E
V
I
(1 , S
0
) − V
I
(0 , S
0
) | µ, G as long as Assumption
3
holds.
The condition for the previous aggregate shock B is immediately derived given the observation that δ
E
V
I
(1 , S
0
) − V
I
(0 , S
0
) | µ, B > δ
E
V
I
(1 , S
0
) − V
I
(0 , S
0
) | µ, G and
Q
L,B
(1 − β ) < Q
L,G
(1 − β ).
Proof of Lemma 2.
Suppose both the normal and the inept types randomize between two signals ( α
H
, α
L
), where α
H
> α
L
. Denote the strategy of the normal type is ( γ
N
, 1 −
γ
N
), while the strategy of the inept type is ( γ
I
, 1 − γ
I
). When he chooses to sell the fraction α k
, where k ∈ { H, L } , the payoff of the normal type is where
α k p ( α k
, µ, S ) + β (1 − α k
) Q
H,S
− c
+ δ
E q
H,S 0
V
N
( µ h,S 0
( k ) , S
0
) + (1 − q
H,S 0
) V
N
( µ l,S 0
( k ) , S
0
) | µ, S ] p ( α
H
, µ, S ) = p ( α
L
, µ, S ) =
µ h
( H ) =
µ l
( H ) =
µ h
( L ) =
µγ
N
Q
H,S
µγ
N
+ (1 − µ ) γ
I
Q
L,S
+ (1 − µ ) γ
I
µ (1 − γ
N
) Q
H,S
+ (1 − µ )(1 − γ
I
) Q
L,S
µ (1 − γ
N
) + (1 − µ )(1 − γ
I
)
µγ
N
Q
H,S
µγ
N
Q
H,S
+ (1 − µ ) γ
I
Q
L,S
µγ
N
(1 − Q
H,S
)
µγ
N
(1 − Q
H,S
) + (1 − µ ) γ
I
(1 − Q
L,S
)
µ (1 − γ
N
) Q
H,S
µ (1 − γ
N
) Q
H,S
+ (1 − µ )(1 − γ
I
) Q
L,S
(36)
(37)
(38)
(39)
(40)
(41)
29

µ l
( L ) =
µ (1 − γ
N
)(1 − Q
H,S
)
µ (1 − γ
N
)(1 − Q
H,S
) + (1 − µ )(1 − γ
I
)(1 − Q
L,S
)
(42)
Because the securitizers are indifferent between these two signals, in the equilibrium the normal type must signal α
H less frequently than the inept type does. As a result, p ( α
H
, µ, S ) < p ( α
L
, µ, S ). Suppose p ( α k
, µ, S ) is greater than βQ
H,S for any k ∈ { H, L } .
Then given the strategy of the inept type, the normal type has an incentive to sell the fraction α
H with probability 1, which means this equilibrium falls out of the class of equilibrium I consider to maximize the trade volume. Next, suppose p ( α
H
, µ, S ) is less than βQ
H,S
. Given Assumption
3 , the normal type would prefer holding all of his product.
The equilibrium collapses.
Proof of Theorem 1.
The equilibrium is derived based on the following lemmas.
Lemma 3.
In the equilibrium, at any state ( µ, S ) , the mimic probability γ of the inept type is increasing in the fraction α that the normal type sells.
Proof.
This result is intuitive. The more the normal type sells, the more likely the inept type to mimic him. To show it, first notice at any state ( µ, S ), V I ( µ, S ; γ ) is increasing in α when the mimic probability γ is fixed. If the normal type chooses the signal α high enough such that V I ( µ, S ; α, γ = 1) ≥ Q
L,S
+ δ
E
V I (0 , S
0
) | µ, S , the inept type will mimic the normal type with probability 1. Otherwise, if the fraction α is not great enough that
V
I
( µ, S ; α, γ = 1) < Q
L,S
+ δ
E
V
I
(0 , S
0
) | µ, S , the inept type needs to choose a γ ∈ (0 , 1) to make himself indifferent between revealing his own type and mimicking the normal type to signal α such that
V
1
I
( µ, S ; α, γ ) = Q
L,S
+ δ
E
V
I
(0 , S
0
) | µ, S (43)
Because the price function and the continuation value are decreasing in the mimic probability γ , in the equilibrium the mimic probability must be increasing in the fraction α
to ensure that the left hand of equation ( 43 ) is a constant.
Lemma 4.
When the equilibrium is partial pooling, the normal type retains all of his product, while the inept type randomizes between revealing his own type by selling the whole unit of his product and mimicking the normal type by holding all.
Proof.
In the partial equilibrium, the payoff of the inept type is a constant. Differentiating
V I ( µ, S ) with respect to α , I yield
∂p
( p − βQ
L,S
) + α
∂α
+
∂V c
I ( µ, S )
∂α
∂V c
I ( µ, S )
∂α
= 0
= βQ
L,S
∂p
− p − α
∂α
(44)
30

where V c
I = δ
E q
L,S
0
V I ( µ h
( α ) , S
0 | µ, S ) + (1 − q
L,S
0
) V I ( µ l
( α ) , S
0 | µ, S ) is the continuation value for the inept type. Moreover, differentiating V
1
N ( µ, S ) with α , I yield
∂V
1
N ( µ, S )
∂α
= ( p − βQ
H,S
∂p
) + α
∂α
+
∂V N
1 ,c
( µ, S )
∂α where V
N
1 ,c is the continuation value function for the normal type.
Substituting equation ( 44
) into equation ( 45 ), I have
(45)
∂V N ( µ, S )
= − β ( Q
H,S
− Q
L,S
) +
∂α
∂V
N
1 ,c
( µ, S )
−
∂α
∂V c
I ( µ, S )
∂α
(46)
Because in the partial equilibrium the continuation value function of the normal type is decreasing faster in α than that of the inept type, the derivative of V
1
N with respect to
α is less than zero. Therefore in the partial pooling equilibrium the normal type always chooses to hold all of his product.
Lemma 5.
For any aggregate shock S in the previous period, there exists a threshold ˜
S such that when the reputation is less than ˜
S the equilibrium is partial pooling. When the reputation is greater than ˜
S
, the equilibrium is complete pooling.
Proof.
At reputation 0, the inept type only mimics the normal type with probability 1 when the normal type sells the whole unit of his product. However, Assumption
3
implies that it is never optimal for the normal type to sell the whole unit of his product when his reputation is 0. Therefore when the reputation is 0 the equilibrium is partial pooling such that the normal type always holds all his product and the inept type randomizes between selling all and mimicking the normal type. Because the value functions are continuous,
I further conclude that the equilibrium should be partial pooling when the reputation is in the range [0 , ˜
S
µ
S is determined by:
V
I
µ
S
, S ; α = 0 , γ = 1) = Q
L,S
+ δ
E
V
I
(0 , S
0
) | µ, S (47)
As V I ( µ, S ; α = 0 , γ = 1) is increasing in reputation µ , when the reputation is greater
µ
S the inept type will always choose to pool with the normal type regardless what signal the normal type sends out.
Lemma 6.
When the aggregate shock in the previous period is B , the normal type always makes the screening. There exists a threshold µ
∗
B than µ
∗
B such that, when his reputation is less he holds all of his product. Otherwise, he sells a fraction defined in
( 19 )
. When the aggregate shock in the previous period is G , the normal type only makes the screening when his reputation is less than a threshold µ
∗
G
, and then he chooses no effort and sells all.
31

Proof.
Lemma 6 is derived from the following observations.
(i) When his reputation is 0, the securitizer of the normal type is always better off by making the screening and holding all of his product rather than shirking and selling all. If the normal type does not make the screening in the equilibrium, he will be stuck with 0 reputation forever. By making the screening and holding all, the securitizer has a chance to boost his reputation.
(ii) When the securitizer’s reputation is 1, the game degenerates into a similar structure with the one-period model. The investors believe there is a positive probability that the securitizer has made the screening if the seller sells a fraction less than ˜
S and pay a price Q
H,S
. As shown in the one-period model, when the aggregate shock in the previous period is B , the securitizer of the normal type would like to make the screening and signal
α
B
. However, when the aggregate shock is G in the previous period, the signal ˜
G is too costly for the normal type. Hence he chooses to make no effort and sells the whole unit of his product. Notice, there is no belief updating when the securitizer chooses no effort in the equilibrium. His reputation remains 1.
µ
(iii) The value functions are continuous and monotonic. Moreover, for any reputation
, the derivative of the continuation value for the normal type with reputation µ ,
∂V
N
1 ,c
∂µ
>
∂V
N
0 ,c , or if there exists a
∂µ
. Therefore, there is either no crossing between V
1
N and V
0
N crossing, there is only one crossing between V
1
N and V
0
N . Given the observation provided in (ii), the desired result is derived.
32