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stopping criterion which is used in this paper, stops optimization when Variational lower bound changes between previous and current steps less than 〖10〗^(-4).
Based on what is explained above, the results from optimization using finite Variational Bayes are reported as following.
The number of features was inferred from sampled data via a posterior distribution. The discovered representation of latent features by means of mentioned model is as follows. First, it can be seen that the algorithm has found solutions with approximately six latent features. In other words, in making loans to applicants, six dimensions of underwriting standards have been exerted. MATLAB output for binary matrix Z with dimension 20×6, the expected posterior weight matrix, A, with dimension 6×16,are reported at Appendix. In these simulations the number of rows was set to 20.
The updated values for the parameters of Variational distribution are reported in the following.
The updated values for parameters of Beta distribution (the feature probabilities), π_k: τ_kn, ∀n=1,2 و∀k=1,…, 6

Table 1. The parameters of Beta distribution for π_k at optima
τ_61
τ_51
τ_41
τ_31
τ_21
τ_11
0.1667
0.1667
2.1667
3.1667
9.1667
14.1667
τ_26
τ_25
τ_24
τ_23
τ_22
τ_21
21.000
21.0000
19.0000
18.0000
12.0000
7.0000

The updated values for second parameter of Normal distribution (features), A_k0: Φ_k, ∀k=1,…, 6 are reported in the tables at Appendix.
Maximized lower bounds for each restart number are reported in following table.

Table 2. Maximum lower bounds for 5 restart numbers
Restart Num.5
Restart Num.4
Restart Num.3
Restart Num.2
Restart Num.1
28.6352
28.6352
-383.5820
16.3322
28.6352

As we know, the entries of binary latent feature matrix Z are representing the presence and non-presence of features in each observation; thus, they have a Bernoulli distribution with a parameter which represents the probability of presence of that feature. From number 3 above, and tables in Appendix, it is known that the probability of first four features is nonzero, and the probability of presence of fifth and sixth features is zero. Therefore, the expected posterior weight matrix for most frequent features which are exerted for most loan applicants, and are inferred from 1000th iterations, is shown only for these features and is corresponded to the first four columns of matrix A which are reported in below table.
As mentioned above, the weights in matrix A quantitatively determine the impact of intensity of each dimension of effort on the revenue streams of the underlying loan pools. A positive weight for each entry of matrix A corresponds to the more influence, a negative weight corresponds to fewer impacts, and a zero weight shows that the feature (the dimension of effort which is hidden information of originator) has any impact on the set of observations.
For example, for the first pool of loans in the first row of matrix X, the first row of matrix Z shows that only second dimension of effort influences the data.

Table 3. The expected posterior weight matrix: estimated values for most frequent features within data φ ̅_k, k=1,…,4
A_04
A_03
A_02
A_01
-0.0651
-0.0268
-0.0358
0.9695
-0.0394
-0.0474
0.0202
0.9894
1.0995
1.0653
-0.0729
0.9863
0.0246
1.0693
-0.0936
0.0241
0.0697
-0.0583
-0.0252
0.9817
0.0009
0.0210
-0.0242
-0.0169
0.9515
-0.0156
0.0363
-0.0304
-0..0261
0.9671
0.0120
0.0322
0.0029
0.1676
0.9269
-0.0362
-0.0295
-0.0162
0.9949
0.0046
1.0241
-0.0871
0.9802
-0.0019
-0.1118
1.0146
0.9870
0.0561
-0.0430
-0.0527
-0.0048
0.0062
-0.0695
-0.0418
0.0465
0.0313
0.9033
0.0027
0.1157
0.0336
-0.0235
-0.0093
-0.0158
0.0216

According to the first column of matrix A, one can see that the impact of intensity of this dimension of effort is 0.9894 which is approximately equal to one and is a significant number. Thus, the results from Variational Bayes inference show that for underlying pool of loans, originator has exerted only the second dimension of effort (second dimension of underwriting standards). Therefore, the cash flow streams of underlying pool of loans are influenced by this dimension of underwriting standards (this dimension of effort).
Notice that, according to results from optimization, for the best restart number, the number 5 has been obtained.

Bu-Ali Sina University
PhD Studies Thesis/Dissertation Information
Title:
Optimal Design of Securitization in the Principal-Agent Relationship Based on Bayesian Inference Approach for Moral Hazard
Author: Elham Farzanegan
Supervisor: Ezatollah Abbasian (Ph.D)
Advisor: Mohsen Ebrahimi (Ph.D)
Faculty: Economics and Social Science
Department: Economic
Degree: Ph.D
Field: Economic Science
Subject: Economic Science
Number of Pages:264
Defence Date: 17/09/2014
Approval Date: 26/12/2011

Abstract:
In this thesis, the moral hazard problem in the secondary mortgage market is considered. In fact, it is focused on the optimal designing of mortgage-backed securities (MBS) in the framework under moral hazard problem. Under this contract, an originator (mortgage underwriter) can engage in the optimal effort to screen for creditworthy borrowers. The investors in the secondary market cannot observe mortgage underwriter’s effort, but can condition underwriter’s payment on loan’s defaults. Therefore, the optimal contract between underwriter and investor is as a payment which will be paid after a period of time. Unlike static models where focused on risk retention policy, the model in this thesis shows that delaying the payments is a key incentive mechanism. The framework in this thesis is such that the investor learns the originator’s (underwriter’s) efforts, about undertaken underwriting standards, during a given period. In other words, it is supposed that the investor is a Bayesian decision masking who updates her posterior believes and by using it deal with designing the compensation scheme. In fact, in this thesis, in the securitization framework, we are seeking to model the explicit form of beliefs in optimal designing of contracts. In the following, by using the syndicated data about revenue streams of MBS securities, a deterministic Variational procedure is used to infer various dimensions of undertaken (unobservable) efforts in the IBP process, based on stick-breaking approximation. Then the investor can consider this additional information, as an informative signal, in her designing contract purposes.
Key Words: MBS Securities, Moral Hazard, Optimal Design of Compensation Scheme, IBP Process, Variational Bayesian Inference.

BU-Ali Sina University

Faculty of Economics and Social Sciences
Department of Economics

Thesis Submitted in Partial Fulfillments of the the Degree of Doctor of Philosophy (Ph.D) Economic Science

Title

Optimal Design of Securitization in the Principal-Agent Relationship Based on Bayesian Inference Approach for Moral Hazard

Supervisor:
Ezatollah Abbasian (Ph.D)

Advisor:
Mohsen Ebrahimi (Ph.D)

By:
Elham Farzanegan

September, 17, 2014

1- Subprime Mortgage Crisis
1- Asset-Backed Securitization
1- Mortgage-Backed Securities
1- Single Pooling
1- Grouped
1- Claims
1- Secondary Market
1- Originator
1- Originate
1- Special Purpose Vehicle
1- Issuer
1- Bankruptcy Remote
1- True Sale
1- Creditor
1- Off-Balance Sheet
1- Underlying Assets
1- Mortgage Loans
1- Underwriting
1- Repackage
1- Completing Markets
1- Improving Liquidity in Market
1- Transparency
1- Specializing in the Business
1- Organizational Costs
1- Mortgage Underwriting
1- Risky Mortgage Applicants
1- Ultimate Holder
1- First-Loss Position
1- Conflict of Interests
1- Hidden
1- Informative Signals
1- Hidden Effort
1- Second-Best
1- Compensation Scheme
1- Mortgage Bank
1- Funding
1- Update
1- Prior
1- Evidence
1- Posterior
1- Likelihood
1- Normalizing Constant
1- Evidence
1- Mrginal Likelihood
1- Subjective
1- Model Evidence
1- Unnormalized
1- Mixture Models
1- Exponentially Increasing
1- Intractable
1- Closed Form Solution
1- Approximate Inference
1- Personalistics
1- Generative Process
1- Machine Learning
1- Unsupervised Learning
1- Unlabeled
1- Clustering
1- Business Confidence
1- Happiness
1- Conservatisms
1- Non-Parametric Statistics
1- Distribution Free
1- Objective
1- Semi-Parametric
1- Paradigm
1- Variational Bayesian Inference
1- Approximate Inference
1- Deterministic Methods
1- Stochastic (or Monte Carlo) Methods
1- Ensemble Learning
1- Mean Field Approximation
1- Factorized Distributions
1- Numerical Sampling
1- Exact
1- Analytical Approximations
1- Sparse Matrices
1- Sparse Binary Matrices
1- Binary Relation
1- Variational Free Energy
1- Support
1- Shannon Entropy
1- Relative Entropy
1- Reference
1- Distance
1- Triangle Inequality
1- Conjugate
1- Raiffa & Schlaifer
1- Hyperparameters
1- Probability Model
1- Nodes
1- Equivalence Classes
1- Digamma Function
1- Exponential Family
1- Analytically
1- Probability Space
1- Outcomes
1- σ-Algebra
1- Measurable
1- Events
1- Probability Measure
1- Measure
1- Measurable Space
1- Countably Additive Function
1-

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