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starting from Arrow (1963) and Pauly (1974), economists have debated two solutions for moral hazard problem: (1) Incomplete coverage against occurrence the loss, and (2) observing behavior of informed party by uninformed one, in order to avoiding a loss from occurring. Incomplete coverage, by exposing an individual to financial risk, engages him to prevent the occurrence of loss; and observing the behavior give incentive to individual to prevent loss. Such informative systems, inform principal (uninformed party) about what is done by agent (informed party), therefore, they more likely prevent from agent’s opportunism because agent will recognize that cannot fraud the principal. Thus, when principal has information that reveals the behavior of agent, the agent is expected to act in the best interests of the principal. In this paper, it is concentrated on second solution for moral hazard problem. But it is clear that investor cannot monitor how originator exercises care and controls over borrowers default risk. Yet the investor can observe outcomes of these actions, which in the securitization framework, are cash flows from underlying loan pools in MBS securities. Also, investors can try to infer from this information (observations), which efforts are undertaken by originator.
In such circumstances, attentions are focused on using estimates of actions and gathering additional information from this way. Evidence indicates that additional information has widespread application in practice, for reducing the moral hazard problem. Thus, additional information can be considered as an appropriate tool in order to mitigate asymmetric information between originator and investor, and reducing conflict of interests in managing loans. Generally, the Informativeness Principal says that in order to give incentive to the underwriter to prevent his from shrinking of assigned tasks, the contract between investor and originator should be based on any variables which have information about originator’s actions.
Here, two questions arise: first, when one can use additional information regarded to actions to mitigate conflicts of interests? Second, how such additional information should be used optimally?
Harris et al (1979), have discussed these questions in the Principal-Agent relationship. In their theoretical framework, agent provides a productive input which cannot directly be observed by principal. They conclude that monitoring provides information which can help the principal to discover any shrinking by agent with positive probability. Accordingly, monitoring is of value.
Holmstrom (1979) in his paper made simpler the analysis of Harris et al, and generalized their results. He showed that any kind of additional information about agent’s action, even imperfect, can be used for improving both welfare of principal and agent. Such results explain the widespread application of imperfect information in designing the contracts.
Given the importance of the problem and necessity of acquiring additional information for uninformed party about behavior of informed one in any transactions, particularly in MBS transactions, in this paper we are seeking to generalize and apply Bayesian inference framework in to securitization literature, in order to infer hidden efforts chosen by originator (informed party) within this process.

Principal-Agent Problem
For investor (whom we refer to as she), at the end of securitization process and only bearing the risks, especially credit risks, the PA topic, faced with, arises mainly due to the delegation of decision-making about originating loans (to implement the underwriting standards) to the originator (whom we refer to as he) on behalf of investor (Bolton & Dewatripont, 2005). Securitization process strengthens the relationship between investors’ cash payments and lender’s performance. Hence, this agency relationship is as a contract between them. In fact, it is the information gap between investor and lender which is the main reason for designing bilateral contract, (namely designing MBS) in the PA arrangement. In this setting, the investor is uninformed principal and lender, namely originator, is informed agent. The investor’s gains are dependent on whether originator implements what is agreed between them (namely, implementing the underwriting standards when lending).
We saw that, securitization process is a credit risk transfer mechanism, so originator is not exposed to performance risk of created loans; thus, there is a disconnection or conflict between his objectives, and that of investor, seeking to maximize benefits from investing in mortgage-backed securities. Indeed originator’s decision, regarding shirking or working, is based on comparison of his benefit and cost, which anyway has skew to shrink the agreed duties; as a result, makes loans with low level quality.
On the other hand, for investor complete observation of undertaken efforts, if possible, is very costly. Hence, mitigating credit risk needs to align the incentives of the originator with those of the investor. This is exactly what is concentrated on in PA relationship.
In this paper, the main goal is designing the optimal contract in the securitization framework with moral hazard. Here asymmetric information means that, contrary to what is agreed and obligated, after a contract is signed, (Hart & Holmstrom, 1987), the originator implements actions which will have an effect on the loans’ credit quality, and as a result on the payments to investor. These costly actions are hidden from view of investor, raised awareness of investor concerning about the securities’ credit risk, which may caused by failure of the mortgage borrowers. In this setting, the agency problem takes to be considered, because one cannot align diverse interests. (Barnard, 1938; March & Simon, 1958). In fact, the most important thing that prevents costlessly aligning different interests (minimize incentive conflicts), is asymmetric information. Hence, if the agent’s effort levels are unobservable one cannot have any solution for issues of conflict of interests, because any implementable contract cannot guarantees which obligated action in practice, is implemented by agent.
This challenge is very important in the structured finance because the main reason for optimal designing of securitization is controlling incentive problems. However, in the agency theory, conflict of interests is achieved with a compensation scheme, similar to the price mechanism in microeconomics.

Compensation Scheme
In the economic models of behavior of agents, individuals hate working; hence inducing effort requires principal provides them with outside motivation. In the literature, incentive systems of PA model are used to optimal allocation of risks and optimal compensation of productive work.
Compensation model affects degree of moral hazard via offering an action-award pair; such that principal rewards good performance, and/or penalties for bad performance (Laffont & Martimort, 2002).When agent’s effort is unobservable, one of the options for principal is designing the outcomes-based contingent contracts which are based on the outcomes of agents’ behavior. But contracting based on outcome aligns preferences of the originator (agent) with those of the investor (principal) if accomplished with transferring risk to agent (originator) (Mas-Colell et al., 1995). Therefore, resolving agency problem needs analysis for the design of optimal compensation plan, which is also includes a premium for agent’s actions. In the following next sections the model, considered in this paper, will be introduces.

Model Assumptions
A problem, arising here, is to determine optimal allocation of proceeds between principal and agent, namely to determine the solution for agency problem, which is designing the optimal compensation plans. For these purposes, it is assumed that both principal and agent are maximizing the expected utility. Principal’s utility function G(w) is defined over wealth domain. The utility function G is continuously differentiable, and over the domain of wealth is strictly concave or linear.
Agent’s utility function H(w, e) is defined over wealth and action domain. Notice that in this setting, the effort is considered as a nonmonetary concept. Effort level e is arrived in the agent’s utility function as an argument to incorporate agent’s hate of hard working. Additional restriction is put in the model: following what is convention in agency literature, we assume that agent’s utility function is von Neumann-Morgenstern additively separable utility function in terms of its two components, and has the net utility form H(w, e)=U(w)-V(e). Agent’s preferences over domain of income lotteries are independent of his actions and show his risk aversion. The reverse is also true: if agent’s preferences over income lotteries are independent of e, thus one can write agent’s utility function H as above (Keeney, 1973).
It is assumed that U(w) is a continuous, strictly increasing and concave real-valued function. It is also assumed that V(e) is a continuous, strictly increasing and convex real-valued function. There is also a disutility of implementing the effort,V(e), for the agent. Furthermore, we have V^’ (e)0. One interpretation is that the effort e is individually costly for the agent and thus, has direct disutility for the agent; this is what just causes the main difference between incentive of principal (investor) and agent (originator) about selecting effort. Notice that here agent’s actions are considered as the effort levels exerting in the various activities. For model developed in this paper, another assumption is that investor (principal) is risk-neutral (G^” (w)=0) and the agent (originator) is risk-averse (V^” (e)<0). It can argue that if an agent who is unable to effectively diversify his income, should be risk-averse, and a principal, who are able to diversify his investments, should be risk-neutral. Hence, expected revenues will be of interest of the principal.
Therefore, investor and originator may prefer different actions because they have different risk attitudes.
When we apply the PA theory to the optimal design of securitization of mortgage loans (MBS), risk sharing is desirable for investor and originator. In this agency model, because information asymmetry exists, the investor cannot observe which actions chosen by originator. Notice that, agent’s behavior has effect on him and their common welfare. In the securitization framework, the owner of outcomes is investor, and outcomes are default and non-default states of underlying pool of loans. In order to resolve the agency problem in such setting, it is needed to design compensation scheme to offer to the originator (agent). But if his actions, e, are not observable, investor cannot directly design contracts in terms of effort levels. To solve this problem, principal can put compensations based on outcomes which here are default status of underlying pool of loans. Because outcome is an observable and verifiable variable by investor, thus she can link the agent’s compensation function s(X) to the outcome X. Notice that, the agency problem is arises here because the two parties have different information regarding chosen efforts (agent have more information) not because of difficulties in monitoring the agent’s actions (Demougin, et al., 2001).
Shavell (1979) demonstrated that a compensation scheme should be dependent on extra information, in addition to observable outcomes. His conclusion is as follows: suppose that compensation plan is only dependent on outcomes. Now, change the compensation plan through slightly depending it on imperfect information about agent’s actions. According to the envelop theorem, any changes in effort will have no first-order impact on expected utility of agent, because initially, for a given outcome, their wealth were fixed. However, if the plan appropriately is altered, changes in the effort will have a positive first-order impact on principal’s expected utility, and some of her benefit can be given to the principal which also makes better her situation.
When securitization transactions are closed, lump-sum payment is transferred from the buyer to the originator. But investor can delay the payments to the originator for some periods, and by using the information collected during these periods regarding the credit performance of loans, attempts to infer the efforts exerted by originator regarding implemented underwriting standards. Thus investor in the one side of PA relationship, seeking to mitigate the moral hazard in the agency problem, can use this additional information in designing compensation planning process.
Common opinion in the economics literature proposes that the less the principal (investor) knows about the agent’s actions, the more agent will shirk. (Calvo and Wellisz, 1978). Indeed Informativeness Principle argued that the incentive contract should be based on all variables which have information about agent’s actions. (Holmstrom, 1979; Kim, 1995)
Harris and Raviv (1976) considered a PA relationship in which agent provides a productive input (effort) that cannot be directly observed by principal. In this setting, monitoring is valuable because provides information which is independent from state of nature, and provides the possibility for principal to discover shirking by the agent, with positive probability.
Holmstrom (1979) demonstrated that each type of additional information regarding agent’s action, even imperfect, can be used to improve the welfare of both of the principal and the agent.
Notice that in the real world, originator (e.g. bank) for lending to applicants is responsible to implement underwriting standards, and determine the creditworthiness of applicants which encompasses various items. For this purpose, the model, which is considered in this paper, is a multi-action model. Let E denote the set of all possible actions for agent, and e=(e_1, …, e_K) denote a K-dimensional vector of efforts e∈E={(e_1, e_2,…,e_K )}. Each dimension of effort vector, e_k, is binary and takes two possible values, which we normalize them into zero and one e_k∈{0,1 }, ∀k. Therefore, an effort vector indicates that which dimensions of it are exerted by agent and from which of them he is shirked.
As was illustrated in the introduction, the opinion is that securitization causes moral hazard in implementing mortgage underwriting standards. In order to mitigate associated credit risk, this paper analysis designing and structuring PA relationship in the securitization process in order to align the interests of the originator with those of the investor.
Generally, contract theory is considered as a constrained optimization problem. In particular, it is assumed that the uninformed party designs a contract that maximizes her objective function, is acceptable for the informed party, and induces him to implement actions, which despite being to the best interests of the investor, are also pursuing his own self-interest.
Notice that exerting each effort dimension e_k is costly; if agent accepts the offered principal’s contract, chooses the effort vector e=(e_1, …, e_K ): e_k∈{0,1}, ∀k with his personal cost C.e_k, such that the chosen effort is unobservable. In such circumstance, the information is the credit state of loans, X, which is identified for investor during some given periods. The outcome X necessarily doesn’t contain any perfect information about originator’s performance. Therefore, relative to first-order solution there is welfare loss.
This means that other measures of performance, beside the outcomes, can potentially improve expected utility of both parties, if can be used for rising incentives or improving risk-sharing contract.
This is a sufficient condition for additional information to be valuable in an agency framework. Therefore, if there is an indicator Y of the efforts, correlated to the data observations (here credit states of underlying pool of loans), and its realization is observable by the principal, the compensation should be put based on both the outcomes, and the signal Y.

Optimal Risk-Sharing Rule
In such setting, having additional information Y from various dimensions of chosen efforts, what is optimal contract for implementing the efforts e=(e_1, …, e_K )∈E? This question can be answered through the following optimization problem.
Let f(├ X,Y┤|e) denote the joint probability density of the outcome X and the signal Y (joint distribution of X and Y), given the effort vector e. Assume that f_e and f_ee exist. If investor knows X and Y (observes X and based on her observation, infers Y) her problem (PP), namely choosing a payment s(X,Y) for all possible realization of stochastic variables (X,Y) and an effort vector e for the agent, is represented as following
■(Minimize@s(X,Y), e)∬▒(s(X,Y))f(X,Y├|e┤)dXdY (PP)
Subject to ∬▒U(s(X,Y))f(X,Y├|e┤)dXdY-∑_(k=1)^K▒〖C.e_k 〗≥▁H, (IR)
e_k∈■(argmax@e ̃_k∈E)∬▒U(s(X,Y))f(X,Y├|e ̃_k, e_(-k) ┤)dXdY-C.e_k, ∀k=1,…,K (IC_k)
In choosing a compensation function, the principal must be ensure that he is offering an acceptable level of expected utility to the agent which is attractive enough to him. This minimum level is often determined based on the agent’s expected utility in his next best opportunity, and is interpreted as his reservation level of utility. Such interpretation indicates that it is the principal who has all the bargaining power; she can hold the agent to this minimal acceptance level while keeping the surplus. Thus, individual rationality constraint (IR) for the investor’s choice s(X), means that maximized expected utility of the agent should not be less than the reservation utility level ▁H. In other words, this constraint says that the originator must prefer working for the investor (Salanier, 1997).
The second constraint is the incentive compatibility constraint (IC) for the effort, and reflects that the principal can observe X while cannot observe e. Indeed, in this PA setting, the investor designs a contract which induces the originator, in addition to considering his own self-interest, simultaneously undertakes the actions which are desirable to the investor (Salanier, 1997).
In fact, the investor through compensation plan can, in addition to determining permitted tasks for the originator, control the incentives (Salanier, 1997).
If there are K possible actions in the set E, the above incentive constraints include (K-1) constraints which should be estimated. Grossman and Hart (1983) considered the PA problem in a different way. They separately calculated costs and benefits of the various actions undertaken by the principal. For each action, a specified incentive scheme is considered, which minimizes (expected) costs of giving incentive to the agent to choose that effort. Under the assumption that agent preferences over income lotteries are independent of actions, cost minimizing problem will be a convex programming problem.
Local first-order conditions (F.O.Cs) of the originator for each dimension of effort e_k, is derived as follows
∬▒〖U(s(X,Y)) f_(e_k ) (X,Y├|e┤)dXdY〗-C=0 ∀k=1,…,K (1)
In the agency theory, the first-order conditions for each dimension k of effort, which is a simple equally constraint, are put in place of incentive compatibility constraint. Let λ denotes the Lagrange multiplier for incentive compatibility constraint, and μ_k, ∀k=1,…K denotes Lagrange multipliers for originator’s FOCs, ∀e_k. By differentiating from investor’s problem (PP) with respect to s(X,Y) for each value of (X,Y), the optimal contract is derived. The FOC for optimal contract is as follows
-f(X,Y├|e┤)+λU^’ (s(X,Y))f(X,Y├|e┤)+∑_(k=1)^K▒〖(μ_k f_(e_k ) (X,Y├|e┤) U^’ (s(X,Y)))=0〗
With simplifying will have
1/(U^’ (s(X,Y)))=λ+(∑_(k=1)^K▒〖μ_k f_(e_k ) (X,Y├|e┤) 〗)/f(X,Y├|e┤) (2)
Notice that, because E is a finite set, the Kuhn-Tucker theorem gives both necessary and sufficient conditions for optimality. Thus if s(X,Y) satisfies PC and ICk, ∀k conditions, we say the contract (s,e) will implement the actions (Jewitt, 1988).
The question is that at the second-best solution, the IR constraint satisfies as an equality or is binding? When agent’s utility function is additively separable in action, and reward, this constraint at second-best solution is binding (Grossman and Hart, 1983). However, if V is strictly concave, there is an optimal incentive scheme which implements every special second-best action.

Nonparametric Bayesian Inference
In order to reduce moral hazard problem and as a result minimize credit risk, the investor should seek to improve originator’s incentive to that of her. This is done through mitigating the information gap between them which, by using verifiable signals, is possible. But collecting information via monitoring in practice is very costly. Conversely, investor can inference various dimensions of efforts in the form of informative signal vectors; such that investor, prior designing contract, by using Bayes rule, calculates her posterior beliefs about whether originator’s operation is efficient for all value of this signal,
P(e├|X┤)=(P(X,e))/(∫▒〖P(X,e)de〗)=(P(X├|e┤)P(e))/(P(X)) (3)
As is observed from the optimization scheme of the principal and solution to (2), to calculate the optimal compensation scheme, the investor needs the joint probability distribution of observation X related to credit performance of pool loans, and signal Y which contains the information about various dimensions of chosen efforts, for this purpose, she can use the equation (3).
According to the Bayesian paradigm, in this paper, we consider an unobservable effort e as latent variable and infer it from observable data vectors X. This part of our methodology can be cast in to Machine Learning theory. Based on Machine Learning theory, in this paper our purpose is to seek a set of hidden efforts (the latent variables) undertaken by lender in the form of various underwriting standards, during the lending to applicants.
In the previous approaches in machine learning, the number of latent variables as an input were given; but in this paper, the nonparametric Bayesian approach is used for inferring the hidden variables, because in this approach the number of latent variable (which we refer to as latent feature) is operated as a stochastic quantity that must be determined as a part of posterior inference. In this research, using a nonparametric latent feature model, we seek to explain observed outcomes vector X in terms of latent features which are the different dimensions of chosen efforts e_k ,∀k by the originator, during the loan origination process (loans lending).

Latent Feature Models
Assume that we have N objects, showed by a matrix X with N rows and D columns such that the nth row of X, denoted X_n0, contains measurements of D observable properties of nth object. In a latent feature model, each object is represented by a vector of latent feature values Z_n (corresponding columns of matrix F) and X_n0 are drown from a distribution whose properties are determined by the values of the latent features. Using the matrix F, the hidden latent feature values for all N objects are characterized; therefore, this model is specified through a prior over features p(F), and a distribution over matrices of observed data (regarding the quality state of MBSʼs underlying pool), given these features p(X|F┤). By concentrating on p(F), we show that how this prior can be defined without placing any upper bound on the number of features.
The matrix F can be separated into the two matrix: a binary matrix Z which each its entity z_nk shows that if kth feature is owned by nth object, if so then z_nk=1.
In the securitization framework, the objects denote observations related to credit quality of different pool loans, and features denote various dimensions of underwriting standards which affect credit quality of the underlying pools in MBS. Notice that in specifying the matrix Z, sum of rows can be more than one.
Second matrix V denotes values for each object. Therefore, matrix F can be considered in the form of Kroncker product F=Z⊗V.
By Specifying priors over Z and V, a prior over F is defined p(F)=P(Z)p(V). Because effective dimension of our count data model is determined by Z, we focus on defining a prior over Z. Given the assumption that Z is sparse, one can define a prior over infinite latent feature models, through defining a distribution over infinite binary matrices.
We consider a setting in which each row of matrix Z represents a 1×D vector of pool of loans’ credit states in the mortgage-backed securities, where dimension D denotes data collection periods (here denotes 12 months).
Then, using Bayesian nonparametric approach this matrix is separated into two matrices. We concentrated on a matrix, which demonstrates what latent feature is owned by each observation of loan’s default corresponding to each row. In the securitization literature these means that the entries of matrix Z indicate which dimension of effort for this row, (a pool of loans) is implemented by originator.
The most common nonparametric prior for latent feature models is Indian Buffet Process (IBP). The Indian buffet process is a stochastic process with a nonparametric prior distribution over infinite binary matrices Z, which does not limit the number of features K, and allows inferring how many features exist for each observation of underlying pool loans’ credit quality. However, for a finite number of observations N, this distribution guarantees that the number of features K is finite. (Griffiths & Ghahramani, 2006).
IBP has extensive applications in various fields including models for Choice Behavior (Göür & Jakel & Rasmussen, 2006), and the structure of causal graphs (Wood & Griffiths & Ghahramani, 2006). Notice that, any studies have not yet been done about application of IBP in the securitization literature in particular mortgage backed securities. Hence, there is no extension of IBP toward these subjects. In this literature, the IBP models the structure connecting hidden efforts to the observed default status of MBS during time.
Based on the above mentioned theoretical framework, we assume that investor (principal) is a Bayesian decision maker who, using the observations about new information arriving in the interval of length 12 months regarding the number of the underlying pool loans’ defaults, deals with making Bayesian inference about what are undertaken by originator and from inferred results decides to design and pays the incentive compatibility compensation to the agent (originator). For this purpose, here, the investor during the interval of 12 months gathers revealed information regarding the default status of underlying pool loans; and using our nonparametric Bayesian approach infers and discovers the different dimensions of efforts (latent variables) undertaken by the originator when mortgages were made to the borrowers.

Data Modeling
To clear that how IBP can be used as a prior in unsupervised learning models, we derive a Poisson-Gamma latent feature model to explain important latent features of the data. In this model, the feature matrix is binary matrix Z. For a given observation matrix X, we seek to find the posterior distribution over Z and A. From Bayes rule, we have (Reverend Thomas Bayes, 1740)
p(Z,A├|X┤)∝p(X├|Z,A┤)p(Z)p(A)
We know that count data in observation matrix X, often follows the Poisson distribution, as a result the Poisson likelihood model is considered. Therefore, features A will have Gamma priors. We now cannot put a prior on Z. We want to consider a prior on it which allows the number of latent features is determined within the inference. The IBP process is one of the several options can be considered.

Indian Buffet Process
IBP process places the following prior on the equivalence classes Z, [Z] (Griffiths & Ghahramani, 2006). [Z] is a standard form of binary matrix Z, which is invariant to the features’ ordering (the concept of Exchangeability748)
P([Z]├|α┤)=α^K/(∏_(h∈〖{0,1}〗^N∖0)▒〖K_h !〗) exp⁡{-αH_N}∏_(k=1)^K▒((N-m_k )!(m_k-1)!)/N!
In the above equation, P([Z]) is a distribution on equivalence classes Z, conditioned on α. Also K is the number of nonzero columns in Z, for which m_k>0, and m_k is the number of ones in the column k of matrix Z, K_h is the number

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