Modern Financial Engineering

The following sections are included:

• Dedication
• Foreword
• Preface
• Acknowledgments
• Contents
• Useful Notations

The following sections are included:

• Common Distributions Families
• Discrete Distributions
  • Bernoulli Distribution
  • Binomial Distribution
  • Geometric Distribution
  • Negative Binomial Distribution
  • Poisson Distribution
• Continuous Distributions
  • Uniform Distribution
  • Normal Distribution
  • Log-Normal Distribution
  • Exponential Distribution
  • Gamma Distribution
  • Beta Distribution
• Indicator Function
• Multivariate Distributions
  • Multivariate Gaussian Distribution

In this chapter, we provide some useful notions about Poisson processes which will be later used in Chapter 19 to explain how defaults are modeled by the first jump of a Poisson process. A detailed analysis of the Poisson process can be found in [Ross (2014)].

The following sections are included:

• Estimator Finite Sample Properties
  • Estimator Selection Criteria
  • Cramer-Rao Inequality
• Probability Generating Functions (PGF)
  • PGF Properties
• Monte Carlo Methods
  • Integral Evaluation
  • Estimator Variance
• Variance Reduction
  • Antithetic Variates
• Copulas
  • Definitions
  • Sklar Theorem
  • Properties
  • Popular Families
  • Examples
• Dependence Measures
  • Rank Correlation
  • Dependence Measurement with Copulas
  • Tail Dependence

The following sections are included:

• Risk Types
• Credit Risk Ratios
  • Loan Loss Provision (LLP)
  • Coverage Ratio (CovR)
  • Cost of Risk (CoR)
  • Non-Performing Loans (NPL)
• Write-Offs
• Prepayment Risk

The following sections are included:

• Basel Committee and Basel I
• Basel II
  • Standardized Approach (ST)
  • Internal Ratings-Based (IRB) Approach

The following sections are included:

• Stock Market Turmoil
• Fuel for the Crisis
• Complexity Tree of Securitizations
• Lehman Brothers Collapse and the Change in Deposit Insurance
• Wider Macroeconomic Implications of the Great Recession

The following sections are included:

• From Basel II to Basel III
  • Basel III Credit Risk Amendments
• Basel III Liquidity Ratios
  • Liquidity Coverage Ratio (LCR)
  • Net Stable Funding Ratio (NSFR)
• Basel III Capital Definition
  • Additional Capital Instruments
  • Capital Conservation Buffer
  • Countercyclical Buffer
  • Systemically Important Financial Institutions (SIFI) Buffer
  • Minimum Capital Ratio Level
  • Capital Ratio for Islamic Banks
• Basel III Leverage Ratio
• Remuneration Regulation

The following sections are included:

• Default Definition
• Rating System
  • Rating System Concept
  • Rating Grades
• PD Estimation Methods
  • Probit and Logit Models
  • Calibration
  • Bernoulli Distribution and Binomial Process
• Time Scaled Default Probabilities
• Time Scaled Rating Transitions
• Modeling Tails
• Additional Thoughts on PD

The following sections are included:

• Basic Definition
• Recovery Rating Scale
• LGD Modeling Approaches
• LGD Proxy Indicators
• Additional Thoughts on LGD

The following sections are included:

• Exposure at Default (EAD)
  • Modeling the CCF
  • CCF Estimation Models
• Maturity (M)
• Expected and Unexpected Losses
• Portfolio Loss
• Economic Capital

The Basel Committee has released recommendations on how to validate the IRB models right after Basel II adoption (see [BCBS (2005c)]). These recommendations cover PD, LGD, EAD properties validation. They involve the description of the applicable statistical procedures. However, there are no thresholds to decide whether a risk model is compliant or not. Another shortcoming is the absence of guidance regarding the validation of the predicted total (gross) risk. Below we present the most important findings to overcome the shortcomings of the published regulatory guidelines.

The following sections are included:

• Counterparty Credit Risk for OTC Instruments
• Current Exposure Method (CEM)
• Exposure at Contract Level
• Exposure at Counterparty Level
• Exposure with Collateral Payments
• Potential Future Exposure (PFE)
• Regulatory Methodology
• From the Add-on Factors to the Parametric Approach
• Parametric Potential Future Exposure

The following sections are included:

• Credit Loss
• Discounted Contractual Cash Flow (DCCF)
• Risk-Neutral Valuation (RNV)
• Parametric Approach
• Unexpected Loss
• Economic Capital
• Central Counterparty (CCP)
• Cox Process
• Business Cycles
• Potential Collateral Requirements

This chapter describes a method to include the correlation between different positions within a netting group at the first stage and between netting groups at a second stage. A change in the value of a derivative might impact the value of other derivatives with this counterparty. This effect can be properly captured on a netting group level, as we cannot net positions from different netting groups. Therefore, the next sections explain how to include the correlation into the future net replacement value and the credit loss…

In Section 10.3 we defined the loss variables as indicators of default events. A very common approach in actuarial mathematics is modeling the defaults by Bernoulli or Poisson random variables. This approach is preferred by the following credit risk models:

• CreditMetrics^TM, an original model proposed by the US bank J.P. Morgan, under which the default probabilities of debtors belong to different rating classes;
• CreditPortfolioView^TM, a model developed by McKinsey, based on an econometric analysis of the relationships between default probabilities or default migrations and the state of the macroeconomic cycle;
• CreditRisk +, a model proposed by Credit Suisse Financial Products, based on some actuarial mathematical models;
• PortfolioManager^TM, a model developed by Moody’s KMV, based on the default probability estimation methods through the Merton model. The original name of the company is an acronym for its founders’ family names: Stephen Kealhofer, John McQuown, and Oldrich Vasicek (see [Vasicek (2012), p. 1]). It is sometimes mistakenly called Kealhofer-Merton-Vasicek (see [Tetereva (2012), p. 1]) most probably because the [Vasicek (1987)] paper is an extension of the [Merton (1974)] one

Generally, the CreditRisk + assumes that the default probability (PD) of each debtor is random and its variability depends on some factors, present in the portfolio, named risk factors. The debtors may be divided into sectors, i.e. sets of debtors that are affected by the influence of a particular (systemic) risk factor. Any factor influences the sector through the total PD, which is modeled as a random variable Λ_[s], with mean λ[s] and standard deviation σ_[s]. If we have k sectors, the Credit-Risk + assumes that Λ_[s] ∼ Gamma(α_s, β_s), so that the default number in each sector s follows a Gamma-Poisson distribution with parameter Λ_[s]. Furthermore, it assumes that the variables Λ_[1], …, Λ_[k] are two by two independent. Analogously to the procedure adopted in Chapter 15, first, we compute the PGF of the default amount in the sector s. Second, we derive the probability distribution of the default events in s. Third, we obtain the PGF for the portfolio loss.

The following sections are included:

• Introduction
• Materials and Methods
  • Non-Performing Exposure and Regulation
  • Asymptotic Methods and Data Separation
• Experimental Results on a Portfolio of Italian Companies
  • Results for the PD
  • Results for the LGD
• Discussion

In Chapter 15 we have analyzed the CreditRisk+, i.e. a credit risk model able to derive the loss probability distribution, the default probabilities and all the counterparty weights in a given portfolio, once the exposures are known (see also Chapter 16). More generally, this model splits the counterparties into independent sectors (i.e. independent groups of counterparties each of them being influenced by the same systemic risk factor)…

Credit derivatives are financial contracts that have been widely adopted by the credit market participants as a tool for exposure management or credit investments. Upon their introduction in the mid-1990s, the credit derivatives market expanded rapidly during the period of loose monetary policy and expanding credit from 2002 through 2007…

The following sections are included:

• Introduction
• Quantitative Analysis and the Exposure to Oil
  • Data
  • Cluster Analysis
  • Correlation
  • Econometric Analysis
  • Panel Analysis
• Discussion

In the previous chapter, we considered an oil-exporting country where Islamic finance is prevailing. A reader may challenge us that our finding may have a limited scope for other countries with the western banking system. What about systemic risk and regulation in those counties? The current chapter closes this gap.

The following sections are included:

• Concluding Remarks
• Appendices
  • Appendix A Financial Engineering: Coding in R
    • R Code for Chapter 3
      • R Code 1
      • R Code 2
      • R Code 3
      • R Code 4
      • R Code 5
      • R Code 6
      • R Code 7
    • R Code for Chapter 18
      • R Code for Section 18.3.2
      • R Code for Section 18.3.3
  • Appendix B Financial Engineering: Coding in Matlab
    • Asymptotic Single Risk Factor (ASRF) Model
      • Matlab Code for the ASRF Model in Chapter 5
    • Matlab Code for Chapter 15
      • Matlab Code 1
      • Matlab Code 2
    • Matlab Code for Chapter 16
      • Matlab Code 3
      • Matlab Code 4
    • Matlab Code for Chapter 18
      • Code 5
    • Matlab Code for Chapter 19
      • Matlab Code 6
      • Matlab Code 7
  • Appendix C Dataset Used for Modeling Non-Performing Loans
• Bibliography
• Index
• About the Authors