FRM Ⅱ Credit Risk Measurement and Management

Identification of Credit Risk

Fundamentals of Credit Risk

Definition of credit risk

1. Credit risk:is the possibility of losing money due to the inability, unwillingness, or nontimeliness of a counterparty to honor a financial obligation.不能、不愿、不及时履行金融义务
   • The borrower (issuer of debt) failing to make full and timely payments of interest and/or principal.
   • Credit migrations which bond issuer's creditworthiness deteriorates.
2. Credit risk components
   
   • Obligor债务人:Counterparties that have the responsibility of making good on an obligation
   • The obligations: a legal liability(written word or oral form)
3. Insolvency资不抵债: describes the financial state of an obligor whose liabilities exceed its assets.
   • It is common to use insolvency as a synonym for bankruptcy but these are different events.资不抵债不代表就破产了
4. Default: is failure to meet a contractual obligation Due either to insolvency or illiquidity.
5. Bankruptcy破产: occurs when a court steps in upon default after a company files for protection under either Chapter 11or Chapter 7 of the bankruptcy laws (in the United States).
   • Chapter 7清算 Chapter 11破产重组

Ability and willingness

1. Ability还款能力: an obligor's inability to pay a financial obligation.
   • A company funds a rapid expansion plan by borrowing and later finds itself with insufficient cash flows from operations to repay the lender.扩展太快还不了钱
   • Businesses whose products or services become obsolete or whose revenues simply no longer cover operating and financing costs.企业的产品或服务过时或收入减少
   • Inability to pay follows an unexpected and uninsured event that destroys an entity in a short time.意外和未投保事件
2. Willingness还款意愿: Credit losses can also stem from the unwillingness of an obligor to pay.
   • In instances in which unwillingness is at issue, if the dispute ends up in litigation and the lender prevails 胜诉,there is recovery of the amount owed, and ultimate losses are lessened or even avoided entirely because the borrower has the ability to pay.在不还款意愿成为争议焦点的情况下,如果争议最终诉诸诉讼,且出借人胜诉,可收回欠款,损失会减少甚至完全避免

Credit loss

1. Credit losses can arise in the form of timing.
2. Credit risk can be coupled with political risk.
3. The longer the term of a contract, the riskier that contract is.

Transactions and entities

Who is exposed to credit risk?

1. Financial Institutions
   • Banks:
     Extend credit
     Asset-based lending (repurchase agreements; securities lending)
     Derivative counterparty exposure
   • Asset Managers: face credit risk exposures on behalf of their clients
   • Hedge Funds: may purchase distressed loans, sell protection against a decline in a borrower's creditworthiness, or assume the riskiest positions in commercial real estate financing
     Unique: hedge funds also view the possibility of an entity defaulting as an opportunity to deploy capital 如通过CDS和short selling
   • Insurance Companies: underwriting activities, the investment portfolio, and reinsurance recoverables.承保活动、投资组合和再保险可收回款项
   • Pension Funds: a significant portion of these funds is invested in credit risky assets.
2. Corporates:
   • From customer: The bankruptcy of a customer creates negative publicity and can have a negative effect
   • Account receivables: corporates have effectively extended short-term credit to their customers.
   • Significant amounts of cash to invest
   • Derivative trading activities
   • From supply chain: manufacturing companies and service companies like retailers are dependent on their suppliers

Why manage credit risk?

1. Survival: It's a concern primarily for financial institutions for which large losses can lead to bankruptcy, but even a nonfinancial corporation can have credit losses that can cause bankruptcy.
2. Profitability: the less money one loses the more money one makes.
3. Return on equity: The key to long-term survival is a sufficiently high amount of equity capital complemented by prudent risk management.

Governance

The bankruptcy of a company

1. Serious problems that lead to bankruptcy occur when portfolios of toxic transactions are built. In the absence of fraud, what allows this to occur is a poor risk management framework and corporate governance failure.在没有欺诈的情况下,导致这种情况发生的原因是风险管理框架不完善和公司治理失效

The processes that lead to risk taking

1. The best way to avoid losses is to not enter into bad transactions to start with.避免损失的最佳方法是一开始就 不要进行不良交易
2. The processes that lead to risk taking:
   • Primarily origination 发起
   • Credit risk assessment信用风险评估
   • Approval processes审批流程

The Three Lines of Defense 三道防线

1. First line: business owners, who primarily own and manage risk
2. Second line: monitors and oversees risk performed by enterprise risk management, compliance, and legal. They establish policies and procedures and serve as management oversight for the first line.
3. Third line: provides independent assurance of the risk management and risk monitoring provided by first and second lines of defense: internal audit, external auditors, and special audit committees.

Four key principles for governance system

Guidelines

1. Guidelines: are a set of documents that explain the rules that must be complied with before a transaction is concluded. 有时也被称为"credit policies," "risk management standards,"或采用其他的叫法
   • Understandable 易于理解
   • Concise 简明扼要
   • Precise 准确
   • Accessible 易于获取
2. Creation and approval process 创建和审批流程
   • Guidelines must be sponsored by a senior executive such as the chief risk officer or the chief financial officer and the approval of guidelines must be done by the Board of Directors.
3. Promulgating and maintaining guidelines 颁布和维护准则
   • The chief risk officer's office will own the guidelines, and it is this department's responsibility to draft, seek approval for, promulgate, and maintain the guidelines.
4. Content of guidelines
   • Purpose of the guidelines 目的
   • Methodology for defining a transaction's key parameters确定交易参数的方法
   • Transaction approval and delegation of authority 交易的审批和授权
   • Process to deal with new products and new markets
   • Process to review and update the guidelines
   • Consequences_of failure to follow guidelines不遵守的后果
5. Breach of guidelines: a breach is a serious act, and in most financial firms, it leads to immediate termination of employment.违反准则会导致立即解聘

Skill

1. The authority granted by the Board of Directors to the executive management of a firm has to be delegated further.
2. The delegation of authority 权力下放
   • The assignment of fundamental parameters that characterize, from a risk management point of view, each and every transaction.从风险管理的角度,确定每笔交易的基本参数
   • The delegation of the approval authority based on these parameters. 根据这些参数下放审批权
3. Defining risk parameters of a credit-sensitive transaction
   • Amount of exposure 敞口
   • Credit quality 信用质量
   • Tenor期限
4. Delegation of authority: the simple rule is that the riskier the transaction, the higher the approval level must be.
   
   • Transactions with a high exposure or a low credit quality or a long tenor大额低质量长期限:necessitate senior-management attention and must be approved by committees staffed with people.
   • High-quality counterparty: can be approved at a lower level
   • Simple and straightforward transactions: single individual
5. Delegation of authority
   • Credit committees信贷委员会: the highest level of approval is often called the credit committee and is staffed with the firm's most senior executives
   • Authority delegation is cumulative, meaning each delegation level must approve the transaction.这种授权是累积性的,即每一级授权都必须批准交易
   • A transaction cannot go directly from the originator directly to the credit committee without first being endorsed by the intervening levels.一项交易如果没有经过中间层级的批准,就不能从发起人直接转到信贷委员会

Limits

1. Limits: represent the absolute dollar (or other currency) amount of risk that a company wants to take, or, in other words, the maximum loss that a company is prepared to withstand.
   • Are frequently called credit lines. 信用额度
   • Can be attached to counterparties, industries, countries
   • The existence and size of the credit lines are actually a frequent source of tension between front offices and risk management teams.

Oversight

1. Independence: there must not be any compromise affecting the independence of the risk management unit.风控独立性
   • It should never be located within a business unit with a profit center.
   • A risk manager's compensation should never be based on the profitability of the business.
   • CRO reports directly to the chief executive officer and has privileged access to the risk committee and or the audit committee of the Board of Directors
2. Qualifications: getting the respect of the business partner is a goal of risk managers.
3. Proximity to the business unit 靠近业务部门: risk managers must be located organizationally and physically near operations since they need to have a full understanding of the underlying business.
4. Open mind开放的心态: the art of risk management is not to refuse transactions, but to make suggestions that enable their acceptance.

Credit risk modeling and analysis

Introduction to Credit Risk Modeling and Assessment

Development

1. Credit risk management has changed dramatically over the past couple of decades, and it still evolves at all levels.
   • The first major development: regulatory framework: Capital adequacy ratio(CAR)
   • The second major development: the widespread use of analytical methods for credit risk modeling and management: CAMEL
2. Regulatory framework
   • Credit risk is a highly regulated topic in financial services, with a very broad set of rules and requirements imposed on the design, implementation, and monitoring of credit risk management practices.
     
   • Capital adequacy ratio(CAR)
     Higher capital adequacy standards(CAR>10.5%).
     Leverage and liquidity requirements are also imposed
     Emphasis on counterparty credit risk.
     Introduces a new framework to achieve better governance and more effective risk management
3. Analytical methods
   • CAMEL(骆驼评级法): has been the most widely used system for credit risk modeling.
     
   • Empirical systems cannot provide a solid and objective basis for credit risk management.

Quantitative measurement

1. Expected loss (EL): the amount a bank can expect to lose over a given period of time as a result of credit events.
   • Usually set to be one year: EL = PD × EAD × LGD
   • Probability of default(PD): represents the likelihood that the obligor will not meet scheduled loan payments during the period of the analysis (i.e., the likelihood of default).
     A payment delay by at least 90 days
   • Exposure at default(EAD): is the risk exposure at the time of default (i.e., the amount of capital that is due to be repaid).
     Depends on the characteristics of the loan, rather than the obligor.
   • Loss given default(LGD): is the loss in the case default occurs.(Recovery rate = 1-LGD)
     Characteristics of the obligor: the size of the firm, its business sector, and its financial status
     Characteristics of the loan: collateral and loan seniority(抵质押物和清偿顺序)
     The macroeconomic environment

Types of Credit Risk Assessment Approaches

1. Judgmental approaches(判断法)
2. Data-drivenempirical models(数据驱动的经验模型)
3. Financial models(金融模型)
4. Structural models(结构化模型)
5. Reduced form models(简约模型)

Judgmental approaches

1. Judgmental approaches: are based solely on the expert judgment of credit analysts of the fundamental qualitative and quantitative characteristics of the borrower
   • Also referred to as qualitative approaches or expert systems 专家系统
   • Considering a broad range of qualitative and quantitative factors
2. 5C analysis
   • Character: the personality of the borrower.
   • Capacity: the ability of the borrower to repay the loan,based on the existing income, expenses, and other debt obligations
   • Capital: the own capital of the borrower that is at risk
   • Collateral: the guarantees provided about the payment of the obligation
   • Conditions: general conditions that describe the status of the business environment and the characteristics of the loan (e.g., The interest rate).
3. Limitations
   • Assess the quality of the obtained results (validation)
   • Update the structure of the evaluation process and the actual results to changes in the decision environment and the available information
   • Examine scenarios that may affect the creditworthiness of a single borrower or the risk of a whole credit portfolio.

Data-driven empirical models

1. Data-driven empirical models: are based of models constructed based on historical data about loans accepted,rejected, paid as agreed, and cases in default.
   • Are applicable to both corporate and consumer loans
   • Are constructed to identify non-trivial patterns from the data, thus leading to the formulation of explicit relationships between the likelihood of default and the input variables/risk factors. 构建经验模型来从数据中识别出模式,找到违约和输入变量/风险因子的关系
2. Limitations
   • Reliance on historical data for predictive modeling. 依赖历史数据
   • Data used in empirical models are often considered as backward-looking and static. 实证模型中使用的数据通常被认为是后视的和静态的
   • Fail to incorporate the most recent facts about the status of theborrower.经验模型中考虑的信息并不是实时更新的

Financial Models

1. Financial models: are mostly based on theory.
   • A normative approach, which is founded on basic economic and financial principles that underlie the financial world.大多以理论为基础
   • Describe the mechanism of the default process.
2. Structural model: assumes that default is an endogenous process(内生过程) that is linked to a firm's structural characteristics, such as the value of its assets and debt.
3. Reduced form models: adopt a different approach assuming the default is a random event that may happen at any time driven by shift in an exogenous random variable(外生随机变量) (usually a Poisson jump process).

Credit Rating

Credit scoring system to the credit rating system

1. Data-driven approaches - credit scoring and credit rating: quantify the level of credit risk for a borrower (obligor), taking into consideration all the available information that is related to the borrower's ability and willingness to repay a debt obligation based on the agreed terms
2. Credit scoring(信用打分): models and systems that provide a numerical credit score for each borrower.The credit score is an indicator of the borrower's creditworthiness and probability of default
   • Automated analytical models
3. Credit rating(信用评级): is a risk rating expressed on an ordinal qualitative scale.Each riskgrade/is associated to an empirically estimated PD that describes all borrowers who are assigned to that grade.
   • Used for corporate loans, bond issues, and countries (e.g.,sovereign credit ratings), and often they are publicly available. 用于公司贷款、债券发行及主权评级
   • Involve a combination of analytical and judgmental assessments分析和判断相结合

External ratings

1. External models: are typically developed by credit rating agencies, credit bureaus, and consultancy companies, and their outputs are provided to users, including credit institutions, corporate clients, and investors.
   • Rating agencies such as Moody's, S&P, and Fitch provide ratings describing the creditworthiness of corporate bonds.
2. Through the cycle assessments(跨周期评估法): have a long term orientation covering at least one business cycle, rather than short-term estimates.
   • For major credit rating agencies
   • Less frequent updates
3. Point in time assessments(时点评估法): focus on the current and short-term condition of a borrower for major credit rating agencies
   • More volatile
   • Updated in a timely manner to reflect any new information
4. The choice: mostly on the scope and requirements of the analysis取决于分析的范围和要求

Development process

1. The development: is a complex and data intensive process that requires both judgmental expertise by the analysts as well as technical sophistication.评级是一个复杂和数据密集的过程,既需要分析人员的专业判断能力,也需要精湛的技术
2. Data collection and preprocessing: data for defaulted and non-defaulted cases are collected.
3. Model Fitting: the identification of the model's parameters that best describe (fit) the training data.确定最能描述(拟合)训练数据的模型参数
4. Validation: The model derived from the above training process should be validated against a sample different from the one used for model fitting.用训练集以外的样本验证
   • Validation or holdout sample验证样本或保留样本
5. Definition and Validation of Ratings: once the model is judged satisfactory through the above validation process, the derived credit scores should be mapped to risk rating classes.
6. Implementation, monitoring, review:
   • Used to derive real time credit risk estimates for all new loan applications
   • Under frequent monitoring and review, and updates

Credit rating agencies (CRAs)

1. Long-term credit ratings: medium and long-term ratings
2. Short-term credit ratings: measure credit risk over a short time period (typically one year) and reflect a debtor's ability to fulfill his short-term financial obligations.
3. Credit ratings issued by major CRAs combine analytical and judgmental approaches.
   • Data: proprietary databases, industry/market reports,onsite meetings of expert rating analysts with executives of the rated companies(受评公司高管的现场会议)

Criticism of credit rating agencies

1. Lack of transparency and accountability, conflict of interest 缺乏透明度和问责制,利益冲突
2. Promoting debt explosion 债务爆炸
3. Poor predictive ability 预测能力差
4. Pro-cyclicality 顺周期性

Internal ratings vs. External ratings

1. External ratings: a credit rating is designed to provide information about credit quality.
   • Usually available only for companies that have issued publicly traded debt.
2. Internal ratings: are used by credit institutions such as banks, who have access to the historical data needed for developing and calibrating the models to their specific credit portfolio.
   • Focus on cash: it is cash rather than profits that is necessary to repay a loan.

Altman Z-score model

1. Z-score model: using a statistical technique known as discriminant analysis(区分分析或判别分析), attempted to predict defaults from five accounting ratios:
   \text{Z}=1.21x_1+1.40x_2+3.30x_3+0.6x_4+0.999x_5
   • {x_1}: Working capital to total assets 营运资本/总资产
   • {x_2}: Retained earnings to total assets留存收益/总资产:
   • {x_3}: Earnings before interest and taxes to total assets息税前利润/总资产
   • {x_4}: Market value of equity to book value of total liabilities股票市值/负债账面总额
   • {x_5}: Sales to total assets 销售收入/总资产
2. Z-score
   • Above 3: the firm was unlikely to default
   • Between 2.7 and 3.0: on alert
   • Below 1.8: a firm had a very high probability of defaulting.
3. Model performance: proved to be fairly accurate when tested out of sample. Both Type I and Type ll errors were small.
   • Variations on the model have been developed for manufacturing companies that are not publicly traded and for non-manufacturing companies.

Definitions related to probability of default

1. Cumulative default probability(累积违约概率): an issuer with a certain rating will default within one year, within two years, within three years, and so on.
   • Cumulative survival rate(累积生存概率) = 1 - cumulative PD
   • 累积三年违约概率: \frac{10 + 12 +13}{1000}=3.5\%
   • 累积三年生存概率: 1 - 3.5\% = 96.5\%
2. Marginal default probability(边际违约概率,Joint PD,Unconditional default probability): the probability of a bond defaulting between time t, and t2
   P D_{t2-t1}^{\text {marginal }}=P D_{t2}^{\text {cumulated }}-P D_{t1}^{\text {cumulated }}
   • 第二年边际违约概率：\frac{10+12}{1000}-\frac{10}{1000}=1.2\%
   • 第三年边际违约概率：\frac{10+12+13}{1000}-\frac{10+12}{1000}=1.3\%
3. Conditional PD(条件违约概率): is contingent to the survival rate at time t1 is defined as:
   P D_{(t2-t1\mid t1)}^{\text {conditional }}=\frac{P D_{t2-t1}^{\text {marginal }}}{\text { CSR }_{t1}}
   • Conditional survival rate(条件生存概率)= 1 - Conditional PD
   • 第三年条件违约概率: \frac{1.3\%}{97.8\%}=1.33%
   • 第三年条件生存概率: 1-1.33\% = 98.67\%
4. 性质
   • 条件生存概率相乘等于累积生存概率
     \frac{965}{1000}=\frac{990}{1000} \times \frac{978}{990} \times \frac{965}{978}
   • 平均每年的违约概率
     \begin{aligned} &1-3.5\%=96.5\%=\left(1-d_1\right)\left(1-d_3\right)\left(1-d_3\right) \\ & =(1-\bar{d})^3\rightarrow \bar{d}=1.18\%\end{aligned}
     

Migration matrix

1. Interpretation:
   • A AAA-rated firm has an 87.05% chance of staying AAA
   • A BB rated firm only has a 76.98% chance of staying BB.

Recovery rate and default rate

1. Priorities(清偿顺序)
   
2. Mortgage: 2007-2008 is that the average recovery rate on mortgages is negatively related to the mortgage default rate.
3. Corporate bonds: the average recovery rate on corporate bonds exhibits a similar negative dependence on default rates.

Estimating Default Probabilities from Credit Spreads

Estimate PD - Default intensity model

1. Poisson Distribution is used to model number of default events over time.
   {f(x)}=P(X=x)=\frac{(\bar{\lambda} t)^x e^{-\bar{\lambda} t}}{x!}
2. No default within T years:
   P(X=0)=\frac{(\bar{\lambda} t)^0e^{-\bar{\lambda} t}}{0!}=e^{-\bar{\lambda} t}
   • \bar{\lambda}：Hazard rate 风险率(default intensity 违约强度)
     

Using spread to estimate hazard rate

1. 预期损失通过定价体现：利差等于平均违约概率乘以损失严重度
   \begin{gathered} \bar{\lambda}(T) \times(1-R)=s(T) \\ \rightarrow \bar{\lambda}(T)=\frac{s(T)}{1-R} \end{gathered}
   • \bar{\lambda}(T): T年平均风险率
   • s(T): T年利差
   • R: T回收率
2. T_1到T_2的\bar{\lambda}
   \bar{\lambda}(T_1-T_2)=\frac{T_2\times\bar{\lambda}(T_2)-T_1\times\bar{\lambda}(T_1)}{T_2-T_1}

Credit spreads

1. CDS spread
   • A CDS can be used to hedge a position in a corporate bond CDSs provide a direct estimate of the credit spread.
     
2. Bond yield spread
   • A spread over a Treasury bond of similar maturity
     Risky bond + CDS = Risk-free bond
   • The CDS-bond basis: CDS spread - Bond Yield Spread, should be close to zero but actually deviates from zero
     

     危机前: the basis tended to be positive.
     危机时: the basis was at times very negative(difficult to arbitrage because of a shortage of liquidity and other considerations.)
     危机后: has usually been small and negative.

3. Asset swap spread
   • Asset swaps provide a convenient reference point for traders in credit markets because they give direct estimates of the excess of bond yields over a floating reference rate.
     

Matching bond prices

1. Previous calculation works well for CDS spreads and for bond yield spreads and asset swap spreads when the underlying bond is selling for close to its par value.之前的计算方式更加适用于CDS利差,另外当债券价格趋近于面值时,收益率利差和资产互换利差也能使用.
2. 为了使计算更加精确,可以选择通过违约概率使其与债券价格匹配,倒求hazard rate.
   • 预期损失 = ∑损失概率×损失现值

PD from historical data vs. from credit spread

1. The table considers data on bond yields only up to the start of the 2008 Global Financial Crisis. During the crisis, credit spreads soared.
2. Example: for an A-rated company with 40% recovery rate
   • Average 7-year hazard rate from historical data:
     0.76\%=1-e^{-\lambda \times7} \rightarrow \lambda=0.1090\%
   • Average 7-year hazard rate from credit spread
     \lambda=\frac{s(T)}{1-R}=\frac{0.006868}{1-40\%} \rightarrow \lambda=1.1447\%
3. Difference
   
   • 利差得到的风险率更大 The hazard rates implied by credit spreads before the 2008 Global Financial Crisis are higher than those calculated from a long period of historical data.
   • 随着信用质量变差,两者差距增大The difference between the two hazard rates tends to increase as credit quality declines.
4. Reason for the Difference
   • 债券的非流动性 Corporate bonds are relatively illiquid and the returns on bonds are higher than they would otherwise be to compensate for this.
     perhaps 25 basis points of the excess return
   • 主观违约概率判断 Subjective default probabilities of bond traders are much higher
     Bond traders may be allowing for depression scenarios much worse than anything seen in the period covered by their data.
   • 违约相关性 bonds do not default independently of each other
     Systematic risk: bond traders earn an excess expected return for bearing this risk.
     Good macroeconomic conditions decrease the probability of default for all companies; bad macroeconomic conditions increase the probability of default for all companies.
     Credit contagion 信用危机传染效应
   • 债券的非系统性风险 Nonsystematic(or idiosyncratic) risk: bond returns are highly skewed, with limited upside. The nonsystematic component of this risk is difficult to "diversify away.

Real-world and risk-neutral default probabilities

1. Risk-neutral风险中性: for valuation it is common to make estimates in a risk-neutral world.
   • The estimates from credit spreads: can be used for valuation the same way the risk-neutral estimates of other variables are used.
2. Real-world真实世界: for scenario analysis it is more appropriate to use the "real-world" estimates obtained from historical data.

Estimating Default Probabilities from Equity Prices

Merton model

1. Merton model: is a structural model based on option pricing theory.
   • Equity prices can provide more up-to-date information for estimating default probabilities.
2. Assumption:
   • Assume a firm with a simple debt structure, consisting of a zero-coupon bond outstanding and that the bond matures at timeT.假设一个简化的债务结构,只发行一只零息债券
3. The equity of a company: is a call option on the value of the assets of the company with a strike price equal to the repayment required on the debt.
   • Example: Consider a firm with total value V that has one bond due in one year with face value F = 100.
     
   • 公司权益价值可看作是看涨期权
   • 标的资产为公司价值
   • 行权价格为零息债券面值(更确切为唯一现金流总值)
4. BSM
   • 原版BSM: c_0=S_0N\left(d_1\right)-X e^{-r T} N\left(d_2\right)
   • 公司场景: E_0=V_0N\left(d_1\right)-F e^{-r T} N\left(d_2\right)
     d_1=\frac{\ln \left(\frac{V_0}{F}\right)+\left(r_f+\frac{\sigma_v^2}{2}\right) T}{\sigma_v \sqrt{T}}, d_2=d_1-\sigma_v \sqrt{T}
     V_0: Value of company's assets today
     E_0: Value of company's equity today
     F: Face value of debt (Amount of debt interest and principal due to be repaid at time T )
     \sigma_v: Volatility of assets (assumed constant)
   • Under Merton's model, the company defaults when the option is not exercised.
     The risk-neutral probabilityof default can be shown to be N(-d2).
   • Distant to default (DtD,DD)
     d2\rightarrow D t D=\frac{\ln \left(V_0\right)-\ln (F)+\left(r_f-\frac{\sigma_v^2}{2}\right) T}{\sigma_v \sqrt{T}} \approx \frac{\ln V_0-\ln F}{\sigma_v}

Firm value and firm value volatility

1. Neither V_0 nor \sigma_v, is directly observable. However, if the company is publicly traded, we can observe E_0.
   • N(\mathrm{~d}1)为delta, delta=\frac{\Delta \text {期权}}{\Delta \text {标的}}
2. 解联立方程组
   \left\{\begin{array}{l}\sigma_E \times E_0=N\left(d_1\right) \times \sigma_v \times V_0\\E_0=V_0N\left(d_1\right)-F e^{-r T} N\left(d_2\right)\\d_1=\frac{\ln \left(\frac{V_0}{F}\right)+\left(r+\frac{\sigma_v^2}{2}\right) T}{\sigma_v \sqrt{T}}\\ d_2=d_1-\sigma_v \sqrt{T}\end{array}\right.

Merton model's Limitations

1. Applicable only to liquid, publicly traded names.用于上市公 司,有可观察的活跃交易的股票价格
2. Rely on a static model of the firm's capital and assume a simple debt structure.假设一个静态的资本结构以及极简单的债务端
3. 从BSM继承的假设:公司价值服从对数正态分布

Risk neutral vs. real-world

1. The default probabilities are higher in the risk-neutral world than in the real-world
   • the expected return on assets \mu replaces r_f
   • The growth rate of the company's assets is usually higher than risk-free rate (risk premium demanded by the market)
   • Produce a good ranking of default probabilities.用于排序

Extension of Merton model

1. 违约门槛 A default occurs whenever the value of the assets falls below a barrier level.
2. 违约时间 payments on debt instruments is required at more than one time.
3. Moody's KMV Expected Default Frequency model and Kamakura: provide a service that transforms a default probability produced by Merton's model into a real-world default probability.
   • 使用真实数据校准The DD is mapped to a probability of default scale. PDs are calibrated(校准) to actual default data (historical data).
   • 违约门槛 the sum of short-term debt and half of long-term debt
4. Estimate credit spreads: other organizations have used Merton's model to estimate credit spreads
   • The default probability, N(-d2), is in theory a risk-neutral default probability because it is calculated from an option pricing model.

Retail Credit Risk Management

Basel's definition of retail exposures

1. Definition:
   • Serves both small businesses and consumers and includes the business of accepting consumer deposits as well as the main consumer lending businesses.
   • Retail exposures as homogeneous portfolios that consist of:
     large number of small, low-value loans. 数量多金额少
     With either a consumer or business focus. 分散
     Where the incremental risk of any single exposure is small.增量风险低
2. Example:
   • Home mortgages: secured by the residential properties.
     Loan to value ratio(LTV).
   • Home equity loans(Home equity line of credit,HELOC房屋净值贷款): hybrid between a consumer loan and a mortgage loan, secured by residential properties.
   • Installment loans分期付款贷款: revolving loans, such as personal lines of credit that may be used repeatedly up to a specified limit.
     E.g., automobile and similar loans.
     Credit card revolving loans are unsecured loans.
   • Small business loans(SBL): are secured by the assets of the business or by the personal guarantees of the owners.

Retail credit risk vs. Corporate credit risk

1. Retail credit risk
   • Default by a single customer is never expensive enough to threaten a bank.
   • Expected loss can be treated like other costs of doing business and can be built into the price charged.通过提高价格应对风险
2. Corporate credit risk
   • Large exposures to single name; concentrations of exposures.
   • Dominated by the risk that credit losses will rise to some unexpected level.

Dark side of retail credit risk

1. Four prime causes
   • Not all innovative retail credit products can be associated with enough historical loss data to make risk assessments reliable.历史数据少
   • Even well-understood retail credit products might behave in unexpected fashion under change economic environment.经济环境变化可能会有意外的表现
   • The tendency of consumers to default is a product of a complex social and legal system that continually changes.客户违约的复杂性
   • Operational issue affects the credit assessment of customers can have a systematic effect on the whole consumer portfolio.贷款评估中的实操问题
2. Measures to avoid
   • Warning signal: a change in customer behavior.
     Alter the amount of money it lends to existing customers.
     Alter marketing strategies and customer acceptance rules.
     Price in the risk by raising interest rates for certain customers.

From default risk to customer value

1. Trade-off between creditworthiness and profitability
   • Often a trade-off to be made between the creditworthiness of customers and their profitability.
   • There's not much point in issuing costly credit cards to creditworthy customers who never use them.
   • Customers who are marginally more likely to default might still be more profitable than customers with higher scores.
2. Customer relationship cycle
   • Marketing initiatives: targeting new and existing customers for a new product
   • Screening applicants: decide which applications to accept or reject on the basis of scorecards.
   • Managing the account: dynamic process involves a series of decisions based on observed past behavior and activity.
   • Cross-selling: induce existing customers to buy additional retail products.
3. Risk based pricing基于风险的定价
   • Idea that customers with different risk profiles should pay different amounts for the same product.
   • Tiered pricing policy that sets price as an increasing function of riskier score bands can make risk-based pricing more effective.
   • Pricing is a key tool for retail bankers as they balance the goal of increasing market share against the goal of reducing the rate of bad accounts.

Definition of credit score

1. Credit scoring model: use a statistical procedure to convert information about a credit applicant into numbers that are then combined to form a score.
   • The higher the score, the lower the risk
   • Allows banks to avoid the riskiest customers.
   • Assess whether certain kinds of businesses are likely to be profitable by comparing the profit margin.
   • Also important for reasons of cost and consistency.

Types of credit score model

1. Credit bureau scores信用局评分
   • known as Flco scores, developed by Fair Isaac Corporation.
   • Can be tailored to the preferences of a financial institution.
   • Range from 300 to 850.
2. Pooled models
   • Using data collected from a wide range of lenders with similar credit portfolios.
   • Tailored to industry but not company specific.
3. Custom models
   • Developed in-house using data collected from the lender's own unique population of credit applications.
   • Tailored to screen for a specific applicant profile for a specific lender's product.
   • A strong competitive edge in selecting the best customers and offering the best risk-adjusted pricing.
4. Cost: Credit bureau scores < Pooled Models < Custom models.

Key variables in mortgage credit assessment

1. Risk Score = f(Doc Type, Transaction Type, FICO, LTV,DTI,Occup Type, Prop Type, Pmt,Economic Cycle)
   • FICO: Number score of the default risk.
   • DTI: Debt-to-income ratio.
   • LTV: the Loan-to-value ratio.
   • Payment type (Pmt): e.g., adjustable rate mortgage.

From cutoff scores to default rates and loss rates

1. Cutoff scores 临界值
   • Not to generate an absolute measure of default probability but to choose an appropriate cutoff score.
     
2. Cutoff scores: set the minimum acceptable score at 680 points.
   • Avoid lending money to the body of bad customers to the left of the vertical line but would forgo the smaller body of good accounts to the left of the line.
   • Moving the minimum score line to the right will cut off an even higher fraction of bad accounts but forgo a larger fraction of good accounts.
3. Default Rate and Loss Rate (Expected loss)
   • Each score band corresponding to a risk level.
   • The bank can estimate the loss rate using historical data.
     Given an estimate of the LGD, the bank can infer the implied PD.

Measuring the performance of scorecard

1. Cumulative accuracy profile(CAP)累积准确曲线
   • Goal: differentiate between the two by assigning high scores to good credits and low scores to poor ones.
     Minimize the overlapping area of the distribution of the good and band credits
2. On the horizontal axis are the population sorted by score from the highest risk score to the lowest risk score.
   3 On the vertical axis are the actual defaults in percentage terms taken from the bank's records.
   
   • The area under the perfect model is denoted Ap, while the area under the actual rating model is denoted Ar, AR (Accuracy ratio) = Ar/Ap
   • The closer this ratio is to 1, the more accurate is the model
   • The 45-degree line corresponds to a random model that cannot differentiate between good and bad customers.
3. Monitor the score model
   • Characteristics of the underlying population change overtime.
   • Bank has changed the nature of the products.

Country Risk

Life cycle

1. Risk exposure: A country that is still in the early stages of economic growth will generally have more risk exposure than a mature country, even it is well governed and has a solid legal system.一个国家处于早期发展阶段,会有更多的风险敞口暴露
2. Recession and recovery:
   • Mature markets:经济情况好3-4% GDP growth rate,经济 情况差1-2% drop
   • Emerging markets: double-digit growth in positive or negative terms
3. Equity markets: emerging market equity will often show much greater reactions.

Economic structure

1. Dependency 依赖性: some countries are dependent upon a specific commodity, product or service for their economic success.
   • That dependence can create additional risk for investors and businesses, since a drop in the commodity's price or demand for the product/service can create severe economic pain that spreads well beyond the companies immediately affected.
2. Why don't countries that derive a disproportionate amount of their economy from a single source diversify their economies?为何不改变对于单一资源的依赖?
   • 大国小国: it is feasible for larger countries to broaden their economic bases, it is much more difficult for smal countries to do the same.
   • 自然资源带来的财富:the wealth that can be created by exploiting the natural resource will usually be far greater than using the resources elsewhere in the economy.

Political risk

1. Continuous versus discontinuous risk:
   • Authoritarian governments: dictatorships create more discontinuous risk.
     The change may happen infrequently
     It is difficult to protect against
   • Democratic governments: the chaos of democracy does create more continuous risk(policies that change as governments shift)
2. Corruption and side cost: If those who enforce the rules are capricious(变化莫测的), inefficient or corrupt in their judgments, there is a cost imposed on all who operate under the system.
   • Corruption is an implicit tax on income: reduces the profitability and returns
3. Physigal violence: Countries that are in the midst of physical conflicts, either internal or external, will expose investors/businesses to the risks of these conflicts.
   • Not only economic: taking the form of higher costs for buying insurance or protecting business interests
   • Physical: with employees and managers of businesses facing harm).
4. Nationalization/Expropriation risk
   • Expropriation risk征收风险: arbitrary and specific taxes imposed
   • Nationalization risk国有化风险: business can be nationalized
   • Natural resource companies are more exposed to nationalization risk than others

Legal risk

1. 财产权和合同权是否获得保护Whether it pays heed to property and contract rights
   • Also affect potential investor
2. 法律系统的有效性 How efficiently the system operates.

Composite risk measure

1. Political Risk Services (PRS): uses 22 measures of political,financial, and economic risk to calculate its index.
2. Euromoney: bases its scores on a survey of 400 economists
3. The Economist: develops country risk scores internally based on currency risk, sovereign debt risk, and banking risk.
4. The World Bank: provides country risk data measuring corruption, government effectiveness, political stability,regulatory quality, the rule of law, and accountability.
5. Limitation:
   • Measurement models/methods: not business entities. the scores in some of these services are more directed at policy makers and macroeconomists than businesses. 某些打分更适合给政策制定者和宏观经济学家使用,并不是针对商业决策范畴
   • No standardization: each service uses its own protocol.
   • More rankings than scores

Sovereign debt

1. Foreign currency defaults:
   • A large proportion of sovereign defaults have occurred with foreign currency defaults
     Cannot repay the debt by simply printing more money
     Short of the foreign currency to meet its obligation
   • Countries have been more likely to default on bank debt owed than on sovereign bonds issued.
     Bank loans were the only recourse available to governments that wanted to borrow prior to the 1960s 上个世纪60年代前,银行贷款是政府借贷的唯一途径
   • In dollar value terms, Latin American countries have accounted for the biggest portion of sovereign defaulted debt in the last 50 years.
2. Local currency defaults: countries have shifted more towards defaults, under domestic law, rather than foreign law, in recent decades, reflecting both the willingness to default on local currency debt and the speedier restructurings that follow domestic-law defaults.反映了本币债务违约的意愿,也反映了国内法违约后的重组速度更快.
   • Gold standard 金本位: was a system prior to 1971 in which a country's currency was converted into fixed amounts of gold so that there was a limit on how much currency could be printed.
   • Shared currency共同货币: such as Eurozone.
   • Trade-off 权衡: printing more currency will debase and devalue the currency and cause inflation to increase exponentially - can cause the real economy to shrink.

The consequences of a country's default

1. Reputation loss 声誉损失:tagged with the "deadbeat" label
2. Capital Market turmoil 资本市场动荡:Investors withdraw from equity and bond markets
3. Real Output 实际产出: has ripple effects on real investment and consumption
4. Political Instability 政治不稳定: can put the leadership class at risk
5. Default has a negative impact on the economy, with real GDP dropping between 0.5% and 2%
6. Default does affect a country's long-term sovereign rating and borrowing costs.
7. Sovereign default can cause trade retaliation.贸易报复
8. Sovereign default can make banking systems more fragile. 会使银行系统更加脆弱

Factors determining sovereign default risk

Sovereign credit rating

1. Local currency rating: for domestic currency debt/ bonds
2. Foreign currency rating: for government borrowings in a foreign currency
3. Do sovereign ratings change over time?
   • While one of the critiques of these ratings is that they were sticky, the rate of change has increased over the last few years.
4. Rating measures评级衡量:
   • 服务对象 focused on the creditworthiness of the sovereign to private creditors (bondholders and private banks) and not to official creditors (which may include the World Bank, the IMF and other entities).
   • 反映内容 probability of default or also incorporates the expected severity 每个评级机构不一样
     违约定义相同: outright default-failure to pay interest or principal; rescheduling, exchange or other restructuring
5. Determinants of ratings评级的决定因素:Political risk,economic structure, economic growth prospects, fiscal flexibility, general government debt burden, offshore and contingent liabilities, monetary flexibility.
6. Rating process: draft report → debate each analytical category and vote → decided by a vote of the committee
7. Local versus foreign currency ratings:
   
   • The differential between foreign and local currency ratings is primarily a function of monetary policy independence.
     Floating rate exchange fund borrowing from deep domestic markets:本币外币评级差异大
     Dollarization or monetary union:本币评级向外币靠拢

Shortcomings of the sovereign rating systems

1. Ratings are upward biased
2. Herd behavior羊群行为:When one ratings agency lowers or raises a sovereign rating, other ratings agencies seem to follow suit.
3. Too little, too late: ratings agencies take too long to change ratings
4. Vicious Cycle恶性循环
5. Ratings failures: changes the rating for a sovereign multiple times in a short time period
6. Ratings failures
   • Information problems
   • Limited resources
   • Revenue Bias: ratings agencies offer sovereign ratings gratis(免费的) to most users
   • Other Incentive problems: other businesses, including market indices, portfolio performance evaluation and risk management services, which may be lucrative enough to influence sovereign ratings.

Sovereign default spread

1. Sovereign default spread: when a government issues bonds,denominated in a foreign currency, the interest rate on the bond can be compared to a rate on a riskless investment in that currency to get a market measure of the default spread for that country.
   • 外币国债收益率和此货币无风险利率相减 √E.g.:巴西发行了10年期以美元计价的国债,收益率为 6%,同期美国发行的10年期国债收益率为3.02%
   • Sovereign default spread: 6% -3.02% =2.09%
2. Sovereign default spread vs. sovereign ratings:
   • Strong correlation: between credit spreads and
   • More granular 同样评级,信用质量还是会有差别,主权违约利差给与更多信息
   • The market-based spreads are more dynamic than ratings,with changes occurring in real time
   • There has to be a default free security in the currency必须有发行债券的货币的无违约证券
   • Adjust quickly: provide earlier signals; more volatile
3. Credit default swap: The last decade has seen the evolution of the Credit Default Swap (CDS) market, where investors try to put a price on the default risk in an entity and trade at that price.
   
   • Trigger: the market price of the bond collapses → no compensation
   • Counterparty credit risk
4. CDs spread vs. sovereign ratings:
   • 反应新信息更快In contrast to ratings,that get updated infrequently, CDS prices should reflect adjust to reflect current information on default risk.
5. CDS spreads and default risk
   • Changes in CDS spreads lead changes in the sovereign bond yields and in sovereign ratings.
   • Evidence is emerging that sovereign CDS spread changes are better predictors of sovereign default events than sovereign ratings
   • Sovereign CDS spreads increases:(1)greater economic policy uncertainty; (2) currency depreciation
   • Clustering in CDS market
6. CDS spreads and default risk(limitations)
   • The exposure to counterparty and liquidity risk, endemic to the CDS market, can cause changes in CDS prices that have little to do with default risk.
   • The narrowness of the CDS market can make an individual CDS susceptible to illiquidity problems, with a concurrent effect on prices市场狭小,单个CDs易出现流动性不足
   • There is little to indicate that it is superior to market default spreads

Credit VaR and Capital

Credit Loss Distribution

Credit loss distribution

1. Credit VaR: defined as the difference between the maximum loss rate at a certain confidence level and the expected loss rate, in a given time horizon.
   • CreditVaR = WCL - EL
   • Worst case loss(WCL): Loss at the confidence level

Expected loss(Standalone)

1. Expected loss(EL): A bankcan expect to lose, on average, a certain amount of money over a predetermined period of time when extending credits to its customers.
2. Default event D is a Bernoulli variable.伯努利随机变量
   E L_i=P D_i \times E A D_i \times L G D_i
   • EAD_i: The amount borrowed in the ith loan (assumed constant throughout the year)
   • {P D}_i : The probability of default for the ith loan
   • LR/LGD: The loss rate in the event of default by the ith loan

Unexpected loss(Standalone)

1. Unexpected loss: is the standard deviation of credit losses Risk arises from the variation in loss levels
   • 注意:此处的信用非预期损失为信用损失的标准差, 和一级以及其他科目中表达的含义不同,请格外注意
   • 其他科目中的unexpected loss表示资本金需要覆盖的部分
   • 由于二级原版书选自不同教材, 学者的定义和命名会有所区别
2. Unexpected loss (UL)
   U L_i=E A D_i \times \sqrt{P D_i \times \sigma_{i, L G D}^2+L G D_i{ }^2\times \sigma_{i, P D}^2}
   • The default is a Bernoulli variable with a binomial distribution:
     \sigma_{P D}^2=P D \times(1-P D)

Portfolio EL and UL

1. Portfolio EL:
   E L_p=\sum_{i=1}^n E L_i=\sum_{i=1}^n E A D_i \cdot P D_i \cdot L G D_i
2. Portfolio UL:
   U L_p=\sqrt{\sum_i \sum_j \rho_{i j} U L_i U L_j} \leq \sum_i U L_i
   • \rho_{i j}: correlation that default or a credit migration (in the same direction) of asset i and asset j
   • A portfolio with two assets: \mathrm{UL}_{\text {two assets }}=\sqrt{U L_1^2+U L_2^2+2\rho_{12} U L_1U L_2}
3. Unexpected loss contribution (ULC)非预期损失贡献:measures the marginal impact of the involvement of an individual loan on the overall credit portfolio risk.
   U L C_i=\frac{U L_i \sum_j U L_j \rho_{i j}}{U L_p}
   • A portfolio with two assets:
     \begin{aligned} & U L C_1=\frac{U L_1^2+U L_1U L_2\rho_{12}}{U L_p} \\ & U L C_2=\frac{U L_2^2+U L_1U L_2\rho_{12}}{U L_p}\end{aligned}
4. The relationship between ULC, and UL;(assume that the portfolio consists of n loans that have the same characteristics and size and correlations between credits are the same)
   U L C_i=\frac{U L_P}{n}=\frac{1}{n} U L_i \sqrt{n+\rho\left(n^2-n\right)}=U L_i \sqrt{\frac{1}{n}+\rho\left(1-\frac{1}{n}\right)}
   • For large n: U L C_i=U L_i \sqrt{\rho}
   • The correlation is critical to measuring the potential portfolio loss. As the correlation increases, the bank suffers form concentrations risk.

Derive economic capital for credit risk

1. Economic capital is dependent on confidence level and riskiness of the bank's assets.
   • A bank will typically set the confidence level to be consistent with its target credit rating. 银行内部设置的置信水平和银行的目标评级相关
     Consider a bank that wants to target a very high credit rating, the bank must choose a very high confidence level (e.g.,99.97%), which corresponds to a higher capital multiplier(CM)资本乘数.
     
     \begin{aligned} & \text { Economic Capital }_p=U L_p \times C M \\ & \text { Economic Capital }_i=U L C_i \times C M\end{aligned}
   • The required economic capital at the single credit transaction level is directly proportional to its contribution to the overall portfolio credit risk.单个资产非预期损失贡献加总等于整体组合非预期损失,故单个资产的经济资本等于单个资产非预期损失贡献乘以资本乘数

Credit loss distribution

1. Credit risks are not normally distributed but highly skewed because the upward potential is limited to receiving at maximum the promised payments and only in very rare events to losing a lot of money. 信用损失分布是极度有偏的,债权类资产的收益是有上限的,但小概率下会面临损失极大的情况
2. The beta distribution can be recommended.
   • The tail-fitting exercise is best accomplished by combining the analytical (beta distribution) solution with a numerical procedure such as a Monte Carlo simulation.
   • Taking the inverse of the beta function at the chosen confidence level, we can determine CM, the capital multiplier.给定置信水平,拟合完分布,倒求出资本乘数

Problems with the Quantification of Credit Risk

1. 只考虑违约,没考虑价值变化This approach assumesthat credits are illiquid assets.
   • More liquid asset: a value approach would be more suitable.
2. 借款人信用质量会变 Multi-period nature of credits: the expected and unexpected changes in the credit quality of the borrowers (and their correlations).
3. 只考虑信用风险All other risk components(such as market and operational risk) are separated.

Default Correlation and Credit VaR

Definition of default correlation

1. Default correlation违约相关系数: measures the likelihood of having multiple defaults in a portfolio of debt issued by several obligors.
   • 联合违约概率Suppose two firms whose PD over the next time horizon t are π1 and π2. There is a joint probability that both firms will default over time t equal to π12.
     \rho_{12}=\frac{\pi_{12}-\pi_1 \pi_2}{\sqrt{\pi_1\left(1-\pi_1\right)} \sqrt{\pi_2\left(1-\pi_2\right)}}
2. Drawbacks of default correlation
   • Default is a relatively rare event 违约概率本就是小概率事件
     Default correlation is hard to measure or estimate using historical default data. Estimated correlations vary widely for different time periods, industry groups and are often negative.
     Default correlations are small in magnitude, small correlation can have a large impact.
   • Computationally intensive 计算量大
     We require N(N -1) correlations with N credits.

Reduced form model vs. Structural model

1. Reduced form models: the hazard rates for different companies follow stochastic processes and are correlated with macroeconomic variables.
   • Are mathematically attractive and reflect the tendency for economic cycles to generate default correlations. 数学形式吸引人,反应经济周期带来违约相关性的倾向
   • 缺点: the range of default correlations that can be achieved is limited.即便两家公司的违约率具有完美的相关性,在段时间区间内同时违约的概率会很低
2. Structural form models: similar to Merton's model.
   • A company defaults if the value of its assets is below a certain level.
   • Default correlation: assuming that the stochastic process followed by the assets of company A is correlated with the stochastic process followed by the assets of company B. 公司价值作为随机变量,两者的相关性
   • 优点: the correlation can be made as high as desired
   • 缺点:computationally quite slow

Credit VaR

1. Credit VaR: Defined as the difference between the maximum loss rate at a certain confidence level and the expected loss rate, in a given time horizon.
   \text{Credit VaR}=\text{WCL}-\text{EL}
   • Worst case loss(WCL): Loss at the confidence level
     
   • Some credit risk VaR models consider onlylosses from defaults只考虑违约
   • Others consider losses from downgrades or credit spread changes as well as from defaults考虑信用迁移
2. Credit VaR vs.Market VaR
   
   • A key aspect of any credit risk VaR model is credit correlation.Defaults for different companies do not happen independently of each other
3. Default correlation: has a tremendous impact on portfolio risk, and affects volatility and extreme quantiles of loss rather than EL.主要影响损失的波动以及高分位点损失
   • If default correlation in a portfolio of credits is equal to 1: then the portfolio behaves as if it consisted of just one credit.
   • If default correlation is equal to 0: the number of defaults in the portfolio is a binomially distributed random variable. Significant credit diversification may be achieved.

The effect of granularity on Credit VaR

1. The higher confidence level, the higher Credit VaR.
2. Granular: is contains more independent credits, each of which is a smaller fraction of the portfolio 颗粒度更高指的是内含更多笔相互独立的贷款(债权资产)
3. 颗粒度更高 Credit VaR更低 For a given default probability Credit VaR decreases as the credit portfolio becomes more granular.
   • 违约概率越高,这个降低效果越好 The convergence is more drastic with a high default probability.
   • 反之,违约概率越低,越难通过增加颗粒度降低VaR It is harder to reduce VaR by making the portfolio more granular, if the default probability is low.
4. For a credit portfolio containing a very large number of independent small positions, the probability converges to100 percent that the credit loss will equal the expected loss.
   • The portfolio then has zero volatility of credit loss, and the Credit VaR is zero.

Regulatory Capital

The Gaussian copula model for time to default

1. "Percentile-to-percentile" transformations 点对点转换
   • Define t_1 as the time to default of company 1 and t_2 as the time to default of company 2.
     a_1=N^{-1}\left[Q_1\left(t_1\right)\right], \quad a_2=N^{-1}\left[Q_2\left(t_2\right)\right]
   • Q: the cumulative probability distributions
   • N^{-1} : the inverse of the cumulative normal distribution
2. Gaussian copula: a_1 and a_2 have standard normal distributions. The model assumes that the joint distribution of ay and az is bivariate normal.二维正态分布
   

Single Factor Model

1. Single-factor model is used to examine the impact of varying default correlations based on a credit position's beta.
   • Now suppose we have many variables, X_i(i=1,2, \ldots). Each X_i can be mapped to a standard normal distribution a_i.
     a_i=\beta_i m+\sqrt{1-\beta_i^2} \varepsilon_i
     \varepsilon_i : firm's idiosyncratic shock
     m: market index
     \beta_i: own correlation between market value.
   • m \sim \mathrm{N}(0,1), \varepsilon_i \sim \mathrm{N}(0,1) \rightarrow a_i \sim \mathrm{N}(0,1)
   • The variable m and the variables \varepsilon_i have independent, standard normal distributions. \varepsilon_i, \varepsilon_j are independent
   • \beta_i are parameters with values between -1 and +1
   • \rho=\frac{E\left(a_i a_j\right)-E\left(a_i\right) E\left(a_j\right)}{S D\left(a_i\right) S D\left(a_j\right)} \rightarrow \rho=\beta_i \beta_j

Unconditional default distribution

1. Assume the probability of default (PD) is the same for all companies in a large portfolio.
2. The \beta_i are assumed to be the same for all i. Setting \beta_i=\beta
   a_i=\beta m+\sqrt{1-\beta^2} \varepsilon_i

Conditional default distribution

a_i=\beta \bar{m}+\sqrt{1-\beta^2} \varepsilon_i

1. The factor m can be thought of as an index of the recent health of the economy.
   • If \overline{\boldsymbol{m}} is high: the economy is doing well and all the ai will tend to be high (making defaults unlikely).
   • If \overline{\boldsymbol{m}} is low: all the ai will tend to be low so that defaults are relatively likely.
     a_i \sim\left(\beta \bar{m},1-\beta^2\right)
     
2. The conditional probability of default is greater or smaller than the unconditional probability of default, unless:
   • Market factor shock happens to be zero
   • The firm's returns are independent of the state of the economy.
3. The conditional variance is reduced from 1

Impact of correlation parameter

1. β → 1 (perfect correlation)
   • m < default threshold, nearly all the credits default
   • m > default threshold, almost none default
2. β → 0 (zero correlation): If there is no statistical relationship to the market factor, so idiosyncratic risk is nil, then the loss rate will very likely be very close to the default probability p.
3. In less extreme cases, a higher correlation leads to a higher probability of either very few or very many defaults, and a lower probability of intermediate outcomes.

Vasicek's Gaussian copula model

1. Vasicek's Gaussian copula model, Basel lIl internal-ratings-based (IRB): is a way of calculating high percentiles of the distribution of the default rate for a portfolio of loans.
   • 巴塞尔协议二内评法.模型的目标:找到一个糟糕市场情况下的条件违约概率
2. The Basel Committee sets X = 99.9% for regulatory capital in the internal ratings-based approach.千年一遇的糟糕市场
   (W C D R-P D) \times E A D \times L G D
   • WCDR (worst case default rate)

Worst case default rate

1. Conditional default distribution
   a_i=\beta \bar{m}+\sqrt{1-\beta^2} \varepsilon_i
   • 对表达形式做一些调整
     a_i \sim\left(\beta \bar{m},1-\beta^2\right) \xrightarrow{\rho=\beta^2} a_i \sim(\sqrt{\rho} \bar{m},1-\rho)
2. The default rate conditional n the factor M:
   N\left(\frac{N^{-1}(P D)-\sqrt{\rho} \bar{m}}{\sqrt{1-\rho}}\right)
3. 99.9% percentile worst case default rate
   W C D R=N\left(\frac{N^{-1}(P D)-\sqrt{\rho} N^{-1}(0.001)}{\sqrt{1-\rho}}\right)
4. Capital requirement/Credit VaR
   (W C D R-P D) \times E A D \times L G D

Economic Capital

CreditMetrics Model

1. Many banks have developed other procedures for calculating credit VaR. One popular approach is known as CreditMetrics. 除了上个章节介绍的巴塞尔协议二计算CreditVaR的模型,银行内部可以自行开发,建构损失分布,计算经济资本
   • CreditMetrics model: involves estimating a probability distribution of credit losses by carrying out a Monte Carlo simulation of the credit rating changes of all counterparties. 通过对所有交易对手的信用评级变化进行蒙特卡洛模拟,来估计信用损失分布
2. 模型特点
   • It can take account of downgrades as well as defaults. 不仅考虑违约,还考虑了信用质量变化导致的价值变化
   • It is based on a rating transition matrix基于评级转移矩阵
   • It provides an analytical methodology for risk quantification across a wide range of financial products. 可用于多种工具
   • Computationally quite time intensive

Rating transition matrices

1. These are matrices showing the probability of a company migrating from one rating category to another during a certain period of time.
   • Based on historical data.基于历史数据
   • The rating categories can be either those used internally by the financial institution or those produced by rating agencies such as Moody's, S&P, and Fitch.可以是金融机构内部的评级,也可以是外部评级机构的
2. 多年期评级转移矩阵: assume that the rating change in one period is independent of that in another period, a transition matrix for two years can be calculated by multiplying the matrix by itself. 假设一年期评级变化是独立于其他时间段的,两年期矩阵等于一年期矩阵相乘
   • 小于一年:求6个月矩阵,开根号taking the square root
3. Ratings momentum评级冲量:If a company has been downgraded recently, it is more likely to be downgraded again in the next short period of time.

Using CreditMetrics to estimate Credit VaR

1. Procedure: 确定现在, 估算变化, 考虑相关
   • Step 1 The profile of the exposure is determined.
   • Step 2 Estimate the volatility of the value of each financial product caused by revaluations, devaluations and defaults.
     On each simulation trial, the credit ratings of all counterparties at the end of one year are determined.每条模拟路径上,交易对手一年后的信用等级被确定.
     The credit loss for each counterparty is then calculated.This credit loss is likely to be negative.
   • Step 3 Considering the correlations between the above events, the volatility of individual financial products is added together to calculate the total volatility of the portfolio.考虑产品(或交易对手)之间的关系
     The copula correlation between the rating transitions for two companies is typically set equal to the correlation between their equity returns.

Sampling and the correlation model

1. In sampling to determine credit losses, the credit rating changes for different counterparties are not assumed to be independent.
2. A Gaussian copula model is used to construct a joint probability distribution of rating changes.

Credit Spread Risk

1. Credit-sensitive products in the trading book: calculating VaR or Es for portfolios including these products therefore involves examining potential credit spread changes.
   • Historical simulation: problem - If the company is alive today, it did not default in the past and the calculations carried out therefore assume no probability of default in the future.

Constant level of risk assumptions

1. 假设一家公司投资与BBB及债券,考虑两种投资策略
   • Buy and hold买入并持有: hold bonds for one year
   • Constant level of risk strategy恒定风险水平策略: rebalance(调仓) at the end of each month.If bonds are no longer BBB they are sold and replaced with BBB bonds 每月调仓,评级变化,就会卖出该资产,以BBB级别的新资产替代,始终维持BBB不变
2. The constant level of risk strategy: VaR and ES are generally smaller when the constant level of risk strategy is used.
3. The buy-and-hold strategy: will lead to greater losses from defaults and big downgrades than the constant level of risk strategy. The probabilities of losses from a small rating change are less.

Credit risk plus

1. 贷款数量n, 1年内违约概率均为q, 违约数量期望值qn, 假定违约事件之间互相独立
2. 二项分布 Binomial distribution: 其中m笔贷款发生违约的概率为
   C_n^m q^m(1-q)^{m-n}
3. 泊松分布 Poisson distribution: If q is small and n large,can be approximated by the Poisson distribution
   \operatorname{Prob}(\text{m defaults})=\frac{(q n)^m e^{-q n}}{m!}
   • 即便各个贷款违约概率不相等,式子仍然成立,只要违约概率都很小,q等于一年中贷款组合的平均违约概率
   • Uncertain about what the default rate q.但现实是对于平均违约概率是不确定的,可以假设预期违约数量qn服从gamma分布(均值\mu,标准差\sigma)
4. 负二项分布(Negative binomial distribution)
   • 泊松分布 → 负二项分布
   • \boldsymbol{\sigma} \rightarrow \mathbf{0}: the negative binomial distribution tends to the same probability distribution for the number of defaults as the Poisson distribution.
     There is no uncertainty about the default rate. Without this uncertainty, there is no default correlation
   • As σ increases: the probability of an extreme outcome involving a large number of defaults increases.
     Default correlation increases and a large number of defaults becomes more likely
     Credit VaR is considerably larger.
   • Without default correlation, the loss probability distribution is fairly symmetrical.
     With default correlation, it is positively skewed.

Comparison between models

1. Credit Risk Plus vs.CreditMetrics
   • 对于信用风险的定义不同,相关性假设不同
   • Credit Risk Plus:
     只考虑违约 Does not calculate credit ratings but only default events.
     假设相互独立,违约概率低 default rate is very small and independent of the remaining credit events.
   • Credit Metrics:考虑由于评级变化,信用资产价值的变 化,可通过copula考虑之间的相关性
2. Credit Risk Plus vs.KMV model
   • Credit Risk Plus: does not use the capital structure of the company 没有考虑公司的资本结构

Counterparty Credit Risk

Derivatives

Counterparty credit risk(CCR)

1. Derivatives transactions: represent contractual agreements either to make payments or to buy or sell an underlying security at a time or times in the future.
   • Financial institutions, mainly banks, provide derivative contracts to their end user clients and hedge their risks with one another.
   • are dominated by a relatively small number of large counterparties(dealers)许多市场都被数量相对较少的大型交易方所控制
2. Counterparty credit risk: the possibility that a counterparty may not meet its contractual requirements under the contract when they become due.

Exchange-traded vs.OTC

Clearing

1. Execution: buyer and seller enter into a legal obligation to buy/sell securities or another underlying
2. Clearing: Transaction is managed prior to settlement (margining, cashflow payments, etc.)
3. Settlement: Transaction is settled via exchange of securities and/or cash and legal obligations are therefore fulfilled.
4. Since it requires a certain amount of standardisation, not all OTC derivatives can be centrally cleared.
   • 主要的场外衍生品: interest rate (and inflation) derivatives,foreign exchange derivatives, equity derivatives, commodity derivatives, and credit derivatives.

Market participants

1. Large player: large global bank, often known as a dealer.
   • Have a vast number of derivatives trades on their books and have many clients and other counterparties
   • Trade across all asset classes and will collateralize positions
   • Will be a member of most or all exchanges and CCP to facilitate trading on their own account and for their clients.
2. Medium-sized player: a smaller bank or other financial institution
   • Smaller number of clients and counterparties
   • Will be a member of local exchanges and may be a member of a number of global ones
3. End user: a large corporate, sovereign, or smaller financial institution with derivatives requirements
   • Overall position will be very directional
   • Unwilling to commit to margining or posting collateral
4. Third parties: may offer settlement/margining/collateral management, software, trade compression, and clearing services.

Collateralization

1. Exchange-traded derivatives: are effectively settled on a daily
   • most simple, liquid, and short-dated derivatives. Margin must be posted against losses on a daily basis.
2. OTC derivatives: may be collateralized
   • OTC centrally cleared: Daily collateralization in cash (variation margin)
   • OTC collateralized: bilateral OTC derivatives. Parties post collateral (which may be cash or securities) to one another in order to mitigate the counterparty risk.
   • OTC uncollateralized: parties do not post collateral (or post less and/or lower quality collateral)
     One of the parties is an end user such as a corporate
3. Banks and end users
   
   • 头寸具有方向性,盯市价值及现金流波动大The overall portfolio will typically be directional. The value of the derivatives, or mark-to-market (MTM) volatility will be significant and any associated margin or collateral flows may vary substantially.
   • 一对一对冲敞口而非整体宏观对冲敞口End users may hedge risks on a one-for-one basis rather than macro basis.
   • The hedging needs may lead to certain imbalances

ISDA国际互换及衍生品协会

1. The International Swaps and Derivatives Association (ISDA) is a trade organization for OTC derivatives practitioners.
   • Standard legal documentation to increase efficiency and reduce aspects such as counterparty risk
   • The ISDA Master Agreement 主协议: the market standard for OTC derivative documentation
2. Master agreement
   • Bilateral framework which contains terms and conditions Multiple transactions will be covered under a general
   • Master Agreement to form a single legal contract of an indefinite term, covering many or all of the transactions.
   • Core section 核心部分+ a schedule containing adjustable terms 附表组(可调整条款)
   • 合约条款涉及:netting,collateral, termination events,the definition of default, and the close-out process
3. 从交易对手风险视角,ISDA主协议具有以下降低风险的特点:
   • The contractual terms regarding the posting of collateral 有 关抵押品的合同条款
   • Events of default and termination 违约事件和终止
   • All transactions referenced are combined into a single net obligation所有参考交易合并为单一净债务
   • The mechanics around the close-out process are defined 确定平仓机制
4. ISDA主协议涵盖的违约事件有:
   • Failure to pay or deliver 未付款或未交付
   • Breach of agreement 违背合约
   • Credit support default (collateral terms) 信用支持违约
   • Misrepresentation 错误陈述
   • Default under the specified transaction特定交易下的违约
   • Cross-default(default on another obligation)交叉违约
   • Bankruptcy
   • Merger without assumption 不承担义务的合并
5. May lead to the loss for the nondefaulting party:
   • The total value of the transactions to the nondefaulting party is positive and greater than the collateral (if any) posted by the defaulting party.非违约方价值为正,但违约方缴纳的抵质押物不足
   • The total value of the transactions is positive to the defaulting party and the collateral posted by the nondefaulting party is greater than this value.违约方价值为正,未为违约方缴纳抵质押物超过合约价值

Systemic risk

1. Systemic risk: A major concern with respect to OTC derivatives is systemic risk, which generally refers to an uncontained crisis which may cause the failure of an entire financial system or market.系统性风险一般是指可能导致整个金融系统或市场崩溃的无法控制的危机

Historical ways

1. SPV
2. Derivatives product companies
3. Monolines and CDPCs

Special purpose vehicle 特殊目的实体

1. Special purpose vehicle (SPV) or special purpose entity(SPE) is a legal entity(e.g. a company or limited partnership) created typically to isolate a firm from financial risk.
   • To isolate counterparty risk.
   • SPVs aim essentially to change bankruptcy rules
     Shift priorities 改变清偿顺序
   • An SPV transforms counterparty risk into legal risk.
     A bankruptcy court to combine the SPV assets with those of the originator

Derivatives product companies 衍生品公司

1. DPCs are generally triple-A-rated entities set up by one or more banks as a bankruptcy-remote subsidiary of a major dealer由一家或多家银行设立的三A级实体,作为主要交易商的破产远程子公司
   • Provides external counterparties with a degree of protection against counterparty risk by protecting against the failure of the DPC parent.通过防止 DPc 母公司倒闭
2. Maintained a triple-A rating
   • Minimizing market risk
     Mirror trades: matched book
   • Support from a parent: If the parent were to default, then the DPC would either pass to another well-capitalized institution or be terminated
   • Credit risk management and operational guidelines: Restrictions were also imposed on (external) counterparty credit quality and activities (position limits, margin, etc.)
3. However, was such a triple-A entity of a double-A or worse bank really a better counterparty than the bank itself?
   • The GFCessentially killed the already-declining world of DPCs.
   • A DPC's fate inextricably with that of its parent.DPCs 的命运与其母公司的命运密不可分地联系在一起

Monolines and CDPCs单一险种公司

1. Monoline insurance companies: were financial guarantee companies with strong credit ratings that were utilized to provide 'credit wraps', which are financial guarantees.
2. Credit derivative product companies (CDPCs): were an extension of the DPC concept that had business models similar to those of monolines.
3. When the GFC developed through 2007, monolines experienced major problems due to the MTM-based valuation losses on the insurance they had sold.

Central clearing of OTC derivatives

1. From the late 1990s, several major CCPs began to provide clearing and settlement services for OTC derivatives and other non-exchange-traded products(repo).
   • These OTC transactions are still negotiated privately and off-exchange, but are then novated into a CCP on a post-trade basis.这些场外交易仍然是私下和场外谈判的,但在交易后被更替到中央交易对等中心
2. The clearing mandate
   • All standardized OTC derivatives to be traded on exchanges or electronic platforms
   • Mandatory central clearing of standardized OTC derivatives 标准化场外衍生品的强制性中央结算
   • The reporting of OTC derivatives to trade repositories 向交易资料库报告场外衍生工具
   • Higher capital requirements for non-centrally cleared OTC derivatives.提高非集中清算场外衍生品的资本要求
3. The roles
   • To sets certain standards and rules for its clearing members制定一定的标准和规则
   • To take responsibility for closing out all the positions of a defaulting clearing member 负责平仓违约会员头寸
   • To maintain financial resources to cover losses in the event of a clearing member default:
     Variation margin: closely track market movements
     Initial margin: cover the worst-case liquidation or close out costs above the variation margin
     A default fund: mutualize losses in the event of a severe default
   • To have a documented plan for the very extreme situation when all their financial resources (initial margin and the default fund) are depleted 全部耗尽
     Additional calls to the default fund
     Variation margin haircutting
     Selective tear-up of positions
4. CCPs provide a number of benefits:
   • Allow netting of all trades executed through CCPs
   • CCPs manage margin requirements from their members to reduce the risk associated with the movement in the value of their underlying portfolios.
   • The CCP absorbs the 'domino effect' by acting as a sort offinancial shock absorber 充当金融冲击吸收器来吸收"多米诺骨牌效应"
   • Auctioning the defaulted members' positions

Bilateral margin requirements

1. Whilst clearing is mandatory, it can be avoided by trading contracts that are not sufficiently standardized to be cleared
   • Regulators introduced mandatory margin (collateral) rules for derivatives that would remain bilateral. 双边合约也进入了强制保证金制度

Derivatives risk modelling

1. VaR
   • VaR does not give an indication of the possible loss outside the confidence level chosen
   • VaR is that it is not a coherent risk measure (not behave in a sub-additive fashion)
2. ES: not ignoring completely the impact of large losses; coherent risk measure
3. Models: the financial markets have a somewhat love/hate relationship with mathematical models
   • The quantitative modelling of counterparty risk is complex and prone to misinterpretation and misuse.
   • Counterparty risk involves looking years into the future rather than just a few days, which creates further complexity that is not to be underestimated.交易对手风险涉及未来数年而非短短几天
4. Correlation and dependency: counterparty risk takes difficulties with correlation to another level.

Counterparty Risk and Beyond

Definition of counterparty risk

1. Counterparty credit risk is the risk that the entity with whom one has entered into a financial contract (the counterparty) will fail to fulfil their side of the contractual agreement.
   • Over-the-counter(OTC) derivatives
   • Bilaterally cleared
   • Uncollateralized

Lending risk vs. counterparty risk

1. Lending risk:
   • Notional amount at risk during lending period is usually known with a degree of certainty.
   • Only one party takeslending risk.(i.e. bondholder).
2. Counterparty risk:
   • The value of the contract in the future is uncertain.
   • Since the value of the contract can be positive or negative,counterparty risk is typically bilateral.

Settlement, pre-settlement, and margin period of risk

1. Pre-settlement risk: a counterparty will default prior to the final settlement of the transaction. (counterparty risk)
   • Long-dated OTC derivatives
2. Settlement risk: arises at settlement times due to timing differences between when each party performs its obligations under the contract.
   • Occurs for only a small amount of time (days or even hours)
   • A very large exposure (potentially 100% of the notional)
   • Spot contracts现货合同; Fx markets 外汇市场
     
3. Margin period of risk(MPoR): the risk horizon when collateralized/margined
   • Collateralized bilateral OTC derivatives: 10 business days
   • Centrally-cleared OTC derivatives: 5 business days
     

Institutions and transactions

1. Large global banks have exposure to all asset classes
2. Smaller banks may have more limited exposure
3. End users may also have limited exposure
4. Counterparty risk faced with CCPs may receive attention over the coming years

Mitigations of counterparty risk

1. Netting: Bilateral netting agreements allow cashflows to be offset and, in the event of default
2. Collateral: Collateral agreements specify the contractual posting of cash or securities against MTMlosses.
3. Hedging: hedging counterparty risk with instruments such as credit default swaps (CDSs).
4. Central counterparties: CCP act as intermediaries to centralize counterparty risk between market participants.
5. Other contractual clauses: resets or additional termination.

Netting, Close-out and Related Aspects

International Swaps and Derivatives Association (ISDA)

1. The market standard for OTC derivative documentation is the ISDA Master Agreement, which was first introduced in 1985 and is now used by the majority of market participants to document their OTC derivative transactions.
   • Aims to remove legal uncertainties and to provide mechanisms for mitigating counterparty risk.

Payment netting

1. Payment netting: involves the netting of different payments between two counterparties that are (generally denominated in the same currency).
   • Reduces the time and costs associated with making payments
   • Reduce operational risk, counterparty and liquidity risk
     

Currency netting and CLS

1. Settlement risk is also a major consideration in foreign exchange (FX) markets, where the settlement of a contract involves payment of one currency against receiving the other.
   • Herstatt risk赫斯塔特风险
2. Non-deliverable forward(NDF) transaction: currency exchange is cash-settled based on the difference between the NDF rate and the prevailing spot FX rate applied to the notional.
3. Continuous linked settlement (CLS): reduce settlement risk in FX transactions across a range of eligible currencies
   • Payment versus payment(PVP): Only after both currencies have arrived does CLS Bank make the outgoing payment to A and B.避免交易时差
     

Multilateral netting

1. Clearing rings: multilateral netting before the development of central clearing.
   • Three or more members who would agree to "ring out" offsetting positions.
   • Participants in the ring had to be willing to accept substitutes for their original counterparties.
     
2. Portfolio compression: achieves multilateral netting benefits via the cooperation of multiple counterparties that partially or entirely eliminate transactions.
   • Aims to minimize gross notional of positions in the market
   • preserve the market risk position of participants
   • reducing non-market risks such as settlement and counterparty risk.
   • reduce operational costs and may also lower systemic risk
3. Compression cycle works: an algorithm is run to determine changes to transactions that generate multilateral netting benefits, but keeping each participant's portfolio neutral in terms of value and risk.用算法来对交易的修改,产生多边净额结算利益,保持每个参与者在价值和风险方面的中性
   • Participants can specify constraints (internal credit limits of a participant).
   • Basic portfolio compression by its nature requires standard contracts

Benefits of cash flow netting

1. Operational costs: By reducing the number of cash flows and transactions, operational costs can be reduced.
2. Legal risks: Portfolio compression may reduce legal risk by contractually simplifying portfolios and not relying on future risk mitigation.
3. Regulatory: some simple regulatory methodologies are partially based on gross notional and will, therefore, appear more benefcial if these values are reduced.

Close-out netting 终止净额结算/净额交割

1. Close-out netting: comes into force in the event that a counterparty defaults, and aims to allow a timely termination and settlement of the net value of all transactions with that counterparty.对方出现一笔违约，立刻终止所有交易并净额结算
   • Close-out: the right to terminate transactions with the defaulted counterparty and cease any contractual payments
   • Netting: the right to offset the value across transactions and determine a net balance for the final close-out amount.
2. If the surviving party:
   • owes money: makes this payment
   • owed money: makes a bankruptcy claim for that amount.
     
   • Jump to the bankruptcy queue for all but its net exposure
   • It only depends on mark-to-market (MTM) values at the time of default and not matching cashflows

Close-out netting and ISDA

1. Payment netting reduces settlement risk, close-out netting is relevant to counterparty risk since it reduces pre-settlement risk.
2. Close-out netting reduces the complexity involved in the close-out of transactions in the event that a counterparty defaults.
3. The ISDA has obtained legal opinions supporting the close-out and netting provisions in their Master Agreements in most relevant jurisdictions.
4. Determining a close-out amount:
   • Firm or indicative quotations for replacement transactions supplied by one or more third parties由一个或多个第三方提供的替换交易的确定或指示性报价
   • Relevant market data supplied by one or more third parties
   • From internal sources

Set-off冲销

1. Netting across product categories(e.g. OTC derivatives and loans) is not definitely possible as they are documented differently.银行既给交易对手提供利率互换服务,同时又提供浮动利率贷款,但内部两个交易是分开的,在违约的情况下两者价值是否可以抵消
2. Set-off vs.netting:经济效果上相同,但含义不同
   • Set-off: recognizes the existence of cross-claims
   • Netting: results in a single contractual claim

The impact of netting

1. 风险降低 Risk reduction:
   • Close-out netting is risk mitigant for counterparty risk
   • Has been critical for the growth of the OTC derivatives market.
2. 波动大,可能有系统崩溃风险 Netted positions are inherently more volatile than their underlying gross positions,which can create systemic risk
3. 改变市场参与者对特定交易对手风险增加的看法If credit exposures were driven by gross positions, then all those trading with the troubled counterparty would have strong incentives to attempt to terminate existing positions and stop any new trading.

Netting impact on other creditors

1. Netting redistributes value to OTC derivatives creditors from other creditors.
   
   

Multilateral netting and bifurcation

1. Bifurcation 分歧: between cleared and bilateral transactions部分交易无法清算
   • Market participants executing offsetting transactions since one product may be clearable and the other not.
   • An increase in overall exposure caused by multilateral netting related to central clearing of only a subset of trades
     

Additional termination event (ATE)

1. Additional termination event(ATE): is designed to mitigate against counterparty risk by allowing a party to terminate transactions or apply other risk reducing actions when their counterparty's credit quality is deteriorating.
2. 触发事件
   • Rating downgrade(e.g. below investment grade)
   • For unrated parties such as hedge funds: market capitalisation, net asset value, or key person departure may be used.
3. 触发后果不一定是直接终止合约
   • Provide a replacement counterparty
   • Provide a counterparty offering a third-party guarantee
   • Terminate the transaction(s) at their current replacementvalue 按目前的重置价值终止交易
   • Post some sort of margin
4. Potential danger of ATE
   • Risk-reducing benefit: a systematic deterioration in credit quality is much harder to nullify.系统性恶化很难消除
   • Weaknesses in credit ratings 评级反应迟钝
   • Cliff-edge effects: single-notch rating downgrade may cause a dramatic consequence.下调一个等级,可能会造成巨大的后果
   • Modelling difficulty: rating transitions probability cannot be implied from market data

Margin(Collateral) and Settlement

Rationale for collateral

1. Counterparties
   • In the case of a positive value: a party will call for margin
   • In the case of negative value: a party will be required to post margin themselves.
     
2. Terminology
   • Collateral: used for OTC derivatives markets and contractual agreements such as ISDA Master Agreement 场外市场用更多
   • Margin: used for exchange-traded markets to represent a similar concept 场内市场用更多
3. The fundamental idea of OTC derivatives collateralization is that cash or securities are passed (with or without actual ownership changing) from one party to another primarily as a means to reduce counterparty risk.
   • 场外市场的抵质押物可以是现金、证券甚至是非金融资产
   • Counterparty risk, in theory, can be completely neutralized as long as a sufficient amount of margin is held against it.

Margins

1. Variation margin: members who have lost money pay their loss amount to the exchange CCP, while members who have gained receive their gain amount from the exchange CCP.
   • Reflects current exposure
   • MPoR (margin period of risk): effective delay in receiving margin
     The inherent delay in the process between the time the party last posted margin and the time they were declared in default 最后一次交保证金到违约间的延迟.
     The associated costs in replacing and/or rehedging defaulted transactions.替换和/或重新安排违约交易的相关费用
2. Initial margin: funds or marketable securities that must be deposited with the CCP in addition to variation margin.
   • Extra margin that must be posted independently from the value of the underlying portfolio.吸收额外损失
   • For OTC derivatives, use the term 'Independent amount'.
   • Reflects potential future exposure. 99% confidence interval over a 10-day horizon (incorporates a period of significant financial stress).
     

Margins - OTC derivatives market

1. Bilateral OTC derivatives market: has used margin almost entirely in the form of variation margin, and initial margin (independent amount) has been quite rare.场外市场,交换变动保证金,比较少交初始保证金
   • Initial margin is a much more common concept on derivative exchanges and central counterparties (CCPs).
2. Regulation covering margin-posting in bilateral markets is making initial margin more common.
3. Uncleared margin requirements(UMR): regulators have started to impose bilateral margin requirements
   • Reduction of systemic risk: reducing "contagion and spillover effects"减少传染和溢出效应
   • Promotion of central clearing: aligning costs (notably initial margin)双边清算成本和中央清算成本一直,促进中央清算的采用
4. Apply to all OTC derivatives (FX swaps and forwards)

Credit support annex (CSA)

1. Within an ISDA Master Agreement, it is possible to append a Credit Support Annex(CSA) where the parties agree to contractual margin-posting.
2. No CSA:最典型的是银行和end user(corporate)
3. Two-way CSA: For two financial counterparties,a two-way CSA is more typical, where both parties agree to post margin.
4. One-way CSA: only one party can receive collateral
   • A high-quality entity such as a triple-a sovereign or supranational trading with a bank.
   • Additional risk for the collateral giver

Margin term

1. Threshold: is the amount below which margin (collateral) is not required, leading to undercollateralization只给超过阈值的部分
   • If the MTM is above the threshold, only the incremental amount of collateral can be called for.
   • Threshold = 0: collateral would be posted under any circumstance
   • Threshold < 0: post initial margin
   • Threshold ∞: counterparty will not post collateral under any circumstance
2. Minimum transfer amount (MTA): is the smallest amount of margin that can be transferred.避免频繁缴纳
   • Represents a balance between risk mitigation versus operational workload.平衡风险缓释与业务工作量
3. Rounding取整:A collateral call or return amount may also be rounded to a multiple of a certain size to avoid dealing with awkward quantities.先取整再转，多交少退
4. Types of collateral
   • Partially collateralized: reduction of exposure is imperfect.
     Threshold can be seen as approximately capping the exposure.
   • Strongly collateralized: assume thresholds are zero and therefore the exposure is reduced significantly.
   • Overcollateralized: assume initial margin and therefore the exposure is reduced further.
5. example
   
   

Margin types and haircuts

1. Haircuts 折减率: acts as a discount to the value of the margin and provides a cushion in the event that the security is sold at a lower price.
   • Extra margin must be posted which acts as an overcollateralization to mitigate against a drop in the value of the asset
   • Cash: no haircuts
   • Defined as fixed percentage amounts
     

Valuation agent 估值代理

1. Valuation agent: the party calling for the delivery or return of margin and must handle all calculations.
2. The role of valuation agent:
   • Calculate the current value, market value of margin (adjusted by haircuts), uncollateralized exposure and credit support amount.
3. A dispute over a margin call is common.
   • Margin disputes are not relevant for centrally-cleared transactions since CCP is the valuation agent.有CCP相当于CCP是代理人
   • Reconciliations: aim to minimize the change of a dispute by agreeing on valuations and margin requirements across portfolio.
     Various third parties also offer services around valuation,exchange of margin, dispute management, and portfolio reconciliation.

Impact of margin on exposure

1. Why margin cannot perfectly mitigate exposure
   • The presence of a threshold means that a certain amount of exposure cannot be collateralized.阈值导致无法足值抵押
   • The delay in receiving margin and parameters such as the minimum transfer amount create a discrete effect 由于MTA保证金并非连续的,而是离散的
   • Initial margin can be thought of as making the threshold negative and can potentially reduce the exposure to zero.
2. Impact on other creditors: margin does not reduce risk, it merely redistributes it. Other creditors will be more exposed.
   

Rehypothecation and Segregation

1. Rehypothecation 再抵押: the reuse of securities or cash margin by the margin receiver (e.g. in another margin agreement or a repo transaction).
   • For variation margin: rehypothecation is important in OTC derivatives markets where many parties have multiple hedges and offsetting transactions. 导致抵押物被多次使用
     
2. Segregation 分离: aims to ensure that margin posted is legally protected in the event that the receiving party becomes insolvent确保初始保证金不被挪用
   • For initial margin: segregation can avoid there being additional counterparty risk in relation to the initial margin posted.
   • Whilst safer from the point of view of counterparty risk, segregation does create higher funding costs.
     
   • Segregated margin can be held:
     Held directly by the margin receiver
     Held by a third party acting on behalf of one party
     Held in tri-party custody where a third party holds the margin

Potential problems of margin

1. Market risk
   • Residual risk保证金数额不能完全覆盖: Threshold, Minimum transfer amounts
   • Margin period of risk(MPoR)保证金时间不能完全覆盖
2. Liquidity risk市场流动性风险: in the event that margin has to be liquidated following the default of a counterparty, the non-defaulting party faces transaction costs and market volatility over the liquidation period. 在交易对手违约后必须清算保证金的情况下,非违约方在清算期间将面临交易成本和市场波动.
3. FX risk外汇风险: margin posted in cash or securities denominated in a currency that does not match the currency or currencies of the portfolio 保证金与有关投资组合计价货币不一致
4. Legal risk法律风险: rehypothecation and segregation are subject to possible legal risks.
5. Operational risk操作风险: can be especially significant for the largest banks, which may have thousands of relatively non-standardized margin agreements with clients, requiring posting and receipt of billions of dollars of margin on a given day.它们可能与客户签订了数千份相对非标准化的保证金协议,需要在特定的一天缴纳和收取数十亿美元的保证金.
6. Funding liquidity risk融资风险: increasing collateralization may reduce counterparty risk, but it increases funding liquidity risk.
   • Potential risk arising from the difficulty to raise required funding in the future, especially when margin needs to be segregated and/or cannot be rehypothecated.
     

Central Clearing

Evolution of complete clearing

1. Direct clearing
2. Clearing rings: three or more counterparties could "ring out" offsetting positions
3. Complete clearing: a central counterparty(CCP) is placed centrally between all counterparties and assumes all such contractual rights and responsibilities

What is a CCP?

1. A CCP: is a particular financial market infrastructure(FMI) that represents a set of rules and operational arrangements designed to allocate, manage, and reduce counterparty risk in a bilateral market. 是一种特殊的金融市场基础设施(FMl),代表了一套旨在分配、管理和降低双边市场中交易对手风险的规则和操作安排.
   • Can reduce the interconnectedness within financial markets 相互关联性
   • Can provide more transparency on the positions
     
2. The true CCP landscape is much more complex than represented
   • Client clearing:不能成为CCP会员的交易对手需要通过CCP会员清算
   • Bilateral trades:不是所有衍生品都适合
   • Multiple CCPs:全球有许多不同的CCP
     Regional:为本地清算以本币计价的交易;监管
     Product:专注某些产品类型
3. Clearing relationships
   • CCP members: large global banks and some other very large financial institutions
     Smaller banks, buy-side and other financial firms, and other non-financial end users are unlikely to be direct clearing
   • Requirement: admission criteria; financial commitment; operational

Novation

1. Novation: means that the CCP essentially steps in between parties to a transaction and therefore acts as an insurer of counterparty risk in both directions.
   • Buyer to the seller and seller to the buyer
   • Matched book:
     No net market risk
     take the counterparty risk
     
2. Portfolio compression vs.CCP
   • Portfolio compression: requires actual transactions (or cash flows) to be eliminated
   • CCP can compress risk, whereas bilateral compression can only compress objective quantities such as cash flows.

Functions of CCP

1. Multilateral offset: provide netting and/or compression benefits
2. Lossabsorbency损失吸收能力
3. Default management: hedging, auctioning, allocating loss,transferring client trades

CCP membership requirements

1. Creditworthiness: the probability of default of the clearing member, as assessed by external or internal methods.
2. Liquidity: the ability of the member to meet liquidity requirements, such as the need to meet margin calls at short notice.可以短时间内满足追加保证金要求
3. Operationality: the ability to adhere to the CCP rules, such as those pertaining to auctions.

Margin and default funds

1. Variation margin: be transferred on a daily(and sometimes intradaily一天内) and must usually be in cash
   • requires timely and reliable price data for all cleared derivatives and, where price data is not directly available,market-standard valuation methods.可靠的价格数据或市场标准的估值方法
2. Initial margin: requirements may also frequently change with market conditions and must be provided in cash or liquid assets(e.g.treasury bonds).
   • Cover potential close-out losses平仓成本 in the event of a default by that clearing member.
   • is calculated using different scenarios for possible price movements over an assumed close-out period (MPoR or liquidation period), such as five days.
3. Default funds: CCP members all contribute to a CCP
   • Loss mutualization损失共担: losses above the resources contributed by the defaulter are shared between CCP members.
4. CCPs are typically required to calibrate the size of the total default fund qualitatively via pre-defined stress tests. CCP 通常需要通过预先确定的压力测试,定性地校准总违约基金的规模
   • Allocated to clearing members
     

Default scenarios and margin period of risk

1. In a default scenario, the CCP neutralizes its exposure through:
   • Macro-hedging: 宏观对冲
   • Auctions: 拍卖
     Sub-portfolio 拆分成子组合拍卖
     Surviving CCP members bid the best price 未违约会员还是有很强动力参与拍卖, 把不利后果压倒最低
   • Fire drills: 消防演习,定期进行违约管理程序演习,最大程度提高潜在拍卖效率
   • Driving test: 驾驶考试,对新成员进行测试
     

The Loss Waterfall

1. Skin-in-the-game: equity contribution from the CCP
   

   2.Losses wiping out a significant portion of the default fund

   • Rights of assessment评估权: an unfunded obligation to contribute additionally to a default fund that has been depleted by losses继续要求其它会员补交
   • Variation margin gains haircutting(VMGH) 变动保证金扣减
   • Tear-up强行终止合约,打破原有的matched book
   • Forced allocation把反向交易强行给到未违约会员

Bilateral and central clearing

Impact of central clearing

1. Advantages
   • Transparency
   • Multilateral offset
   • Loss mutualization
   • Legal and operational efficiency
   • Liquidity
   • Default management
2. Disdvantages
   • Moral hazard
   • Adverse selection
   • Bifurcations清算标准产品的要求可能会在清算交易和非清算交易之间造成分歧
   • Procyclicality 市场波动或危机的时候提高保证金要求

Future Value and Exposure

Credit Exposure

1. Definition: effective value of the relevant contracts at the default
2. Credit Exposure = max(value, 0)
   • Loans: the credit exposure is the notional amount and is fixed.
   • Swap, option and forward: the credit exposure is the value of the asset if it is positive.
   • Futures and short options: the credit exposure is zero.
3. Characteristics affects credit exposure:
   
   • Volatility
   • Time horizon
   • Risk mitigants

Nature of exposure

1. Asymmetric risk profile
   • Exposure is similar to an option payoff, a key aspect will be volatility (of the value of the relevant contracts and margin)
   • Options are relatively complex to price 敞口量化也有其复杂性
2. VaR vs. quantifying exposure
   • Time horizon: VaR-short horizon
     The "ageing" of transaction (cash flows, termination events, exercise decisions, and margin postings)
     When looking at longer time horizons, the trend (also known as the 'drift') of market variables is relevant
   • Risk mitigants 考虑净额结算和保证金的影响
   • Application: exposure must be defind for both riskmanagement and pricing(i.e.xVA)传统方法偏向风控

Metrics for Exposure

1. Expected future value(EFV):对敞口分布求期望
   • Represents the expected (average) of the future value calculated with some probability distribution.
   • At time zero: the current value
   • EFV may vary significantly from current value
     Cash flow differential 比如利率互换交叉货币互换
     Forward rates: can differ significantly from current spot variables
     Asymmetric margin agreements: one-way CSA
2. Potential future exposure(PFE): at a confidence level of x%未来某个时期出现较坏的风险敞口是多少
   • Is an equivalent metric to VaR
   • The highest (or lowest) exposure calculated to some underlying confidence level
3. Expected positive exposure(EPE): the average of all of the positive exposure.正敞口的平均数(负的取零)
   • Also referred to as "expected exposure (EE)"
4. Expected negative exposure(ENE): the average of all of the negative exposures.负敞口的平均数(正的取零)
   • Represents the positive exposure from the counterparty's point of view.
   • Also referred to as "negative expected exposure (NEE)"
5. Average EPE: the average exposure across all time horizons.
   • The weighted average of the EPE across time.
   • Also called a "loan equivalent": the average amount effectively lent to the counterparty.

Exposure profile of various securities

1. Bond exposure: can be considered almost deterministic and if interest rates decline, the exposure may increase slightly and vice versa.
2. Loan exposure: loans are typically floating-rate instruments,but the exposure may decline over time due to the possibility of prepayments.
3. Forward exposure: as time passes, there is increasing uncertainty bout the value of the final exchange
   • Follow a "square root of time" rule.
     
4. Interest rate swap exposure: is characterized by a peaked shape.峰值一般发生在1/3处
   • 利率不确定性增加，现金流不确定性减少
     
   • Unequal payment frequencies
     
   • Yield curve is upward sloping:最初浮动端现金流小,后期浮动端现金流大(收固定付浮动)
     
5. Cross-currency swap exposure: is a combination of an interest rate swap and an FX forward transaction.
   • Its exposures due to the high FX volatility driving the risk,coupled with long maturities and final exchanges of notional.
   • The contribution of the interest rate swap is typically smaller.
     
6. Swaption互换期权 exposure:
   • Before the exercise point: the swaption must always have a greater exposure than the forward starting swap
   • After the exercise point: this trend will reverse.There will be scenarios where the forward starting swap has positive value but the swaption would not have been exercised.
   • Optionality complexities: after the exercise date the underlying transaction will have a certain probability of being "alive" or not.
     
     
7. CDS exposure:
   • EPE: shows atypical swap-like profile.
   • PFE: has a jump due to the default of the reference entity.
     

Driving factor in exposure

1. Future uncertainty
2. Cash flow frequency
3. Moneyness
4. Combination of profiles
5. Optionality
6. Credit derivatives

The impact of aggregation on exposure

1. Reducing effect: the exposure profiles are partly offsetting (one is positive whilst the other is negative)
   • Negative correlations are clearly more beneficial
     

Impact of margin

1. Exposure
   • Positive exposure: max{(value-margin,0)
   • Negative exposure: min(value -margin, 0)
     
2. Collateralization is not a perfect form of risk mitigation
   • Granularity effect: margin parameters such as thresholds and minimum transfer amounts
   • Delay in receiving margin: MPoR
   • Potential variation in the value of the margin itself

Funding, rehypothecation, and segregation

1. Funding benefits and costs
   Funding = value - margin
   • Funding cost: term is positive没收齐保证金
   • Funding benefit: term is negative保证金收多了
   • Posted margin: create a positive funding cost.
   • Received margin: represent a benefit.
2. Funding liquidity risk
   
3. Various types of margin under certain situations
   
4. Impact of margin on exposure and funding
   \begin{gathered}\text { Positive exposure }=\max \left(\text { Value }-V M-I M^R,0\right) \\ \text { Funding }=\text { value }-V M+I M^p\end{gathered}
   • 信用敞口:收到保证金降低敞口,缴纳保证金增加敞口Funding:
   • Funding:
     初始保证金: segregation, 无法提供funding benefit
     变动保证金:假设可以使用,可提供funding benefit

CVA

Credit valuation adjustment

1. Credit valuation adjustment (CVA): was originally introduced as an adjustment to the risk-free value of a derivative to account for potential default.
   • A dealer calculates CVA for each counterparty with which it has bilaterally cleared OTC derivatives.
   • CVA is one's expected loss from a default by the counterparty.在无风险价值上做调整
   • Risky value = Risk-free value - CVA
2. Motivation: CVA has become a key topic for banks due to the volatility of credit spreads, the associated accounting and capital requirements(Basel IlI)
3. Challenge: cash flow not fully at risk due to partial offset with opposing cash flows
   

Credit limit vs. CVA

1. Credit limits (or credit lines): are a traditional tool to control the amount of risk to a given counterparty over time.
   • Counterparty risk can be diversified by limiting exposure to any given counterparty, sector, or region.
   • Characterize the potential future exposure (PFE) over time and compare this with a pre-determined limit.
   • Effectively favoring short-term exposures
   • A transaction with low(high) profitability may be accepted (rejected) because the existing credit limit is small (large).
   • encourage maximizing the number of trading to encourage smaller exposures and diversification
2. CVA: By pricing counterparty risk through CVA,the question becomes whether it is profitable once the counterparty risk component has been "priced in"
   • CVA directly incorporates the credit risk of the counterparty and so defines a minimum revenue that should be achieved.
   • encourages minimizing the number of trading counterparties since this maximizes the benefits of netting
3. Three levels to assessing the counterparty risk of a transaction
   
4. CVA and credit limits are typically used together as complementary ways to quantify and manage counterparty risk.

xVA

1. Economic costs of a derivative
   • Funding
   • Collateral
   • Initial margin
   • Regulatory capital
2. Components of xVA Terms
   
3. Valuation adjustments
   
   • Replacement cost and credit exposure
   • Default probability, credit migration, and credit spreads
   • Recovery and loss given default
   • Funding, collateral, and capital costs

Unilateral CVA (UCVA)

1. Unilateral CVA(UCVA): assumes the party making the calculation cannot default
   
   \text{UCVA} \approx-\text{LGD} \times \sum_{i=1}^m d\left(t_i\right) \times\text{EPE}\left(t_i\right) \times \text{PD}\left(t_{i-1}, t_i\right)
   • d\left(t_i\right): discount factor
   • P D\left(t_{i-1}, t_i\right): marginal default probability
   • assumes no wrong-way risk
     
2. CVA as a spread: adjust a rate paid or received as a running spread (per annum charge)
   \text{UCVA} \approx-\text{average EPE} \times \text{spread}
   • Credit component: the credit spread of the counterparty
   • Market component: the exposure, or average EPE
     

Bilateral CVA(BCVA)

1. Debt valuation adjustment(DVA): is the mirror image of CVA (own credit risk).我违约的“收益”
   • DVA is the expected cost to the counterparty
   • DVA will oppose the CVA as a benefit
   • DVA term corresponds to fact that in cases where party themselves default, they will make a "gain" if a negative exposure.
   • DVA creates "price symmetry". Parties can agree on prices.
     
2. Bilateral CVA
   \begin{aligned} & C V A=-L G D_C \times \sum_{i=1}^m d\left(t_i\right) \times E P E\left(t, t_i\right) \times P D_c\left(t_{i-1}, t_i\right) \times\left[1-P D_P\left(0, t_{i-1}\right)\right] \\ & D V A=-L G D_P \times \sum_{i=1}^m d\left(t_i\right) \times E N E\left(t, t_i\right) \times P D_P\left(t_{i-1}, t_i\right) \times\left[1-P D_C\left(0, t_{i-1}\right)\right]\end{aligned}
   • P is the party making the calculation
   • C is its counterparty
   • 注意加入考虑生存概率(有关协会PE与原版书生存概率处理的不同说明)
     
3. As spread
   \text{BCVA} \approx- \text{average EPE} \times \text{Spread}_C - \text{average ENE}\times \text{Spread}_P
   • Assuming that average EPE = -average ENE
     \text{BCVA} \approx- \text{EPE} \times\left(\text{Spread}_C- \text{Spread}_P\right)
   • Weaker counterparties would pay stronger counterparties in order to trade with them based on the differential in credit quality.

关于CVA和DVA符号问题的说明

Netting CVA

1. Netting CVA: Risk mitigants such as netting reduce the value of CVA.
   • Under one netting agreement (netting set)
     C V A_{N S} \leq \sum_{i=1}^n C V A_{\mathrm{i}}
   • where C V A_{N S} is the total CVA of all transactions under netting agreement and C V A_{\mathrm{i}} is the stand-alone CVA for transaction i.
   • The above effect can be significant.
2. Incremental CVA: the CVA of a transaction i is calculated based on incremental effect this transaction has on the netting set新增交易对原有CVA的影响
   \begin{gathered}C V A^{N S \rightarrow N S^*}=C V A^{N S^*}-C V A^{N S} \\ =-L G D \times \sum_{i=1}^m E P E^{N S \rightarrow N S *}\left(t, t_i\right) \times P D\left(t_{i-1}, t_i\right)\end{gathered}
   • Incremental EPE can be negative, due to beneficial netting effects.
   • The incremental CVA is never higher than the standalone CVA without netting.
   • The benefit of netting seen in incremental CVA of a new transaction also depends on the relative size of the new transaction.净额结算的好处还和交易本身规模有关.
     As the transaction size increases, the netting benefit is lost, and CVA (or DVA) will approach the standalone value.
3. Marginal CVA: break down a CVA for any number of netted transactions into transaction-level contributions that sum to total CVA
   • Example: analyze the incremental CVA and marginal CVA for an interest rate swap(IRS) and cross-currency swap (XCCY).
     

Impacts on CVA

1. Spread curve: if upwards-sloping, flat and inverted credit curves all have the same cumulative PD at the five-year point,the marginal default probabilities differ substantially.
   • Upwards sloping: largest CVA
   • Inverted sloping: smallest CVA
     
2. Credit spread: CVA increases with increasing credit spread.
   • In default, CVA may be zero.
3. Impact of recovery rate: increase the recovery rate will
   • reduce the value of CVA.
4. Initial margin and threshold: a high coverage of initial margin is held, CVA can be assumed to be zero.
5. Variation margin:
   • MPoR/Cure period: It is typically 10 or 20 days.
   • Example: The cure period is 20 days. The value of outstanding transactions to the bank at time t is 50 and their value 20 days earlier is 45. The bank's exposure is the uncollateralized value it has in the derivatives transactions, or 5.

Definition of WWR and RWR

1. Wrong-way risk(WWR): indicate an unfavorable dependence between exposure and counterparty credit quality.
   • The exposure is high when counterparty is more likely to default and vice versa.
2. Right-way risk(RWR): cases where the dependence between exposure and credit quality is a favourable one.
   • Right-way situations will reduce counterparty risk.

Examples of WWR and RWR

1. Put option: Buying a put option on a stock where the underlying in question highly correlated to those of the counterparty.
   • E.g. : buying a put on one bank's stock from another bank.
   • Wrong-way risk(WWR): The put option will be valuable if the stock goes down, in which case counterparty's credit quality will be deteriorating.
   • Equity call options: right-way risk(RWR)
2. FX forward: potential weakening of the currency and simultaneous deterioration in credit quality of the counterparty.
3. Interest Rate Swap:
   • Fixed receiver: wrong-way risk(WWR)
     When the economy is weak may represent WWR, since interest rates would be likely to be cut in a recession.
   • Fixed payer: right-way risk(RWR)
     Interest rates may rise during an economic recovery suggesting that a payer swap may have RWR.
4. Commodity swaps:
   • The counterparty is a commodity producer
     e.g.a mining company, Right-way risk(RWR)
   • The counterparty is a commodity user
     e.g. an airline company, Wrong-way risk(WWR)
5. Credit Default Swap:
   • Buying protection in a CDS contract:
     an exposure will be the result of the reference entity's credit spread widening.
     Wrong-way risk(WWR): If a strong relationship between the credit quality of the reference entity and counterparty

WWR modelling

1. Hazard rate (intensity) approaches: introduce a stochastic process for the credit spread.Conditional EPE will be calculated.(easy to use)
   • Correlation: directly via historical time series of credit spreads and other relevant market variables.
   • Weak dependence between exposure and default
2. Structural approaches: mapping the default distribution and exposure distribution onto a bivariate distribution.
   • WWR: Positive dependency will lead to an early default time being coupled with a higher exposure
     
3. Parametric approaches: linking the default probability parametrically to the exposure using a simple, functional relationship
   • If the portfolio has historically shown high values together with larger than average credit spreads, then this will indicate WWR.
4. Jump approaches: jump at default.
   • E.g.: counterparty defaults with FX rate. Implied jump is larger for better-rated sovereigns
   • A default of a large corporation should be expected to have quite a significant impact on the local currency
     

Collateralization and WWR

1. WWR may also be present in terms of the relationship between the value of margin and the underlying exposure. 可以定义为敞口和抵质押物价值的关系
   • 敞口上升,抵质押物价值反倒下降
2. Example:
   • Payer interest rate swap collateralized with a high-quality government bond
     WWR:利率上升,敞口上升,抵质押物价格下降
   • Cross-currency swap collateralized by cash in one of the two underlying currencies. Margin is held in the currency being paid.(如:付美元利息,收日元利息,抵质押物为美元)
     WWR:美元贬值,敞口上升,抵质押物价值下降

CCP and WWR

1. CCPs tend to disassociate credit quality and exposure. CCPs are in danger of implicitly ignoring WWR.
2. WWR increases with increasing credit quality
   • A large dealer represents more WWR than a smaller and/or weaker credit quality counterparty.
   • CCPs should require greater initial margin and default fund contributions from better-credit-quality and more-systemically-important members.信用质量较好的成员违约的可能性较小,但如果违约,影响可能会更严重.
3. CCPs face WWR on the margin they receive.
   • Under pressure to accept a wide range of eligible securities for initial margin purposes.
   • Adverse selection: clearing members (and clients) will naturally choose to post margin that has the greatest risk (relative to its haircut) and may also present the greatest WWR to a CCP

Stress Testing

Evolution of CCR management

1. The treatment of CCR (counterparty credit risk) as credit risk was predominant framework and was established as the basis for regulatory capital as part of Basel ll.
   • Risk mitigants such as netting and margining were incorporated into modeling of CCR.
   • WWR associated with CCR has been identified.
   • The treatment of CCR as a market risk was developing, but was largely relegated to pricing in CVA, prior to the financial crisis of 2007.
2. The view of CCR as market risk allows that its counterparty credit risk can be hedged.
   • Instead of waiting until the counterparty defaults to replace the contracts, financial institution will replace the trades with a counterparty in the market before it defaults.
   • At default, the financial institution will have already replaced the trades and the default itself would be a non-event

Stress testing

1. CCR: CCR manager would have to run this stress at the portfolio level and at the counterparty level, and have to consider CCR as both a credit risk and a market risk.
2. Loan portfolio: A stress test could take EAD and LGD as deterministic and focus on stresses where the PD is subject to a stress.
   \begin{gathered}\text { Expected loss }(E L)=\sum_{i=1}^N P D_i \times L G D_i \times E A D_i \\ \text { Stressed Expected loss }\left(E L_S\right)=\sum_{i=1}^N P D_i^S \times L G D_i \times E A D_i \\ \text { Stress loss }=E L_S-E L\end{gathered}
   • PD is taken to be a function of other variables. E.g.: exchange rate or an unemployment rate
3. Derivative portfolio: Average EPE multiplied by an alpha factor: allows CCR exposures to be placed in a portfolio credit model at a high-percentile loss of the portfolio of exposures
   \begin{gathered}\text { Expected loss }(E L)=\sum_{i=1}^N P D_i \times L G D_i \times \alpha \times \text { average } E P E_i \\ \text { Stressed Expected loss }\left(E L_S\right)=\sum_{i=1}^N P D_i^S \times L G D_i \times \alpha \times \text { average } E P E_i^S \\ \text { Stress loss }=E L_S-E L\end{gathered}
4. CVA: market value of the counterparty credit risk and the losses that could result due to changes in market variables.
   • Aggregating across N counterparties; apply an instantaneous shock to some of these market variables.
   • Unilateral CVA
     \begin{gathered}C V A=\sum_{n=1}^N L G D_n \times \sum_{i=1}^T d\left(t_i\right) \times E P E_n\left(t_i\right) \times P D_n\left(t_{i-1}, t_i\right) \\ C V A^S=\sum_{n=1}^N L G D_n \times \sum_{i=1}^T d\left(t_i\right) \times E P E_n^S\left(t_i\right) \times P D_n^S\left(t_{i-1}, t_i\right) \\ \text { Stressloss }=C V A^S-C V A\end{gathered}
5. DVA: The survival probabilities depend on CDS spreads and the losses depend on the firm's own credit spread.
   • lead to counterintuitive results such as losses occurring because the firm's own credit quality improves.
     \begin{aligned} & C V A \approx \sum_{n=1}^N L G D_n \times \sum_{i=1}^m d\left(t_i\right) \times E P E_n\left(t_i\right) \times P D_n\left(t_{i-1}, t_i\right) \times\left[1-P D_P\left(0, t_{i-1}\right)\right] \\ & D V A \approx \sum_{n=1}^N L G D_p \times \sum_{i=1}^m d\left(t_i\right) \times E N E_n\left(t_i\right) \times P D_P\left(t_{i-1}, t_i\right) \times\left[1-P D_n\left(0, t_{i-1}\right)\right]\end{aligned}

Pitfalls in stress testing CCR

1. 很难将单个对手方的结果有效汇总: It is also rare for CCR to be aggregated with either stress tests of the loan portfolio or with trading position stress-testing results in a consistent framework
2. 线性转换: Use of delta sensitivities to calculate changes in exposures is also especially problematic for CCR, since it is highly nonlinear.

Management of credit risk

Credit Risk Management

Establishing credit risk management policies

1. Limit or reduce credit risk
   • Policies on concentration and large exposures,diversification, lending to connected parties, and overexposure
2. Assess the credit risk exposure through asset classification
   • Periodic evaluation of the collectability of the portfolio of credit instruments
3. Make provisions for potential loss
   • Absorb the anticipated loss

Regulatory policies to limit credit risk

1. Large-exposure and concentration limits: the maximum permitted exposure to a single client, connected group, or sector of economic activity
   • Especially important for small, regionally oriented, or specialized banks
   • Most countries impose a single-customer exposure limit of 10-25 percent of capital.
   • Challenge: defining exposure; single client 对于单一客户的界定; identify common or related ownership
2. Related-Party Financing关联方融资: include a bank's parent,major shareholders, subsidiaries, affiliate companies,directors, and executive officers.
   • Most regulators establish limits for related parties
   • Stipulating that total credit to related parties cannot exceed a certain percentage of Tier 1 or total qualifying capital 通常规定关联方的信贷总额不得超过一级资本或合格资本总额的一定比例

Policies and actions

1. Lending authority 贷款权限
2. Type of loans and distribution by category 贷款类型和分布
3. Appraisal process 评估程序
4. Loan pricing贷款定价
5. Maturities
6. Overexposure to a geographic area or economic sector暴露于地理区域或经济部门
   • International lending: country(or sovereign) risk
7. Insistence on availability of current financial information 坚持提供最新财务信息
8. Collections monitoring收款监督
9. Limit on total outstanding loans 对未偿贷款总额的限制
10. Maximum ratio of loan amount to the market value of
11. pledged securities 贷款额与抵押证券市值的最高比率
12. Impairment recognition 减值确认
13. Renegotiated debt treatment 重新谈判债务处理
14. Written internal guidelines 书面内部指引
15. Loan portfolio review
16. 除贷款外的其他资产
    • Interbank deposits同业存放
    • Off-balance-sheet commitments表外贷款承诺
    • Overdue interest 逾期利息

Traditional Classification Categories

1. Standard,or pass标准/及格
2. Specially mentioned,or watch 特别提及/观察
   • Potential weaknesses
3. Substandard 低于标准
   • Well-defined credit weaknesses
4. Doubtful可疑
   • Collection in full is questionable
5. Loss
6. Nonperforming Loans analysis不良贷款分析
   • Aging of past-due loans 账龄
   • Reasons for the deterioration of loan portfolio quality恶化程度
   • Case-by-case assessment of a list of nonperforming loans评估不良贷款清单(细节)
   • Provision levels拨备水平

Loan loss provisioning

1. Bank's capacity to absorb losses
   • Provisions for possible loan losses
   • General loss reserves (Tier 2 capital and are not assigned to specific assets)
2. IFRS 9 implications
   
   • Probability of default (PD)
     Historical databases of actual defaults
     Estimates from the observable degradation of prices of credit default swaps, bonds, and options on common stock and other tangible security
     Data from external ratings agencies for estimating PDs from historical default experience
     Credit scorecards.
   • Loss given default (LGD)
   • Exposure at default (EAD)
   • Expected loss (EL): is the basis for performing the book provisions.
   • Unexpected loss (UL): The losses incurred, in the financial statement currency, under a high-stress scenario. It is typically identified as a point in the tail of a credit loss distribution
3. IFRs 9 implications - three stages
   
   • All performing assets (not in arrears): carry provisions calculated on a 12-month expected loss
   • Assets in arrears, or where a significant change in the credit environment has occurred: lifetime expected losses
   • Nonperforming assets: lifetime expected losses for the asset

Workout procedure for loss assets

1. Retaining loss assets: retain loss assets on the books until all remedies for collection have been exhausted 将损失资产保留在账簿上,直到用尽所有补救办法
2. Writing off loss assets: all loss assets be promptly written off against the reserve (removed from the books). 将所有损失资产立即从准备金中注销,即从账簿中删除
3. Workout strategies
   • Reducing the bank's credit risk exposure
     additional capital, funds, collateral, or guarantees
   • Working with the borrower to assess problems and find solutions
   • Arranging for a borrower to be bought or taken over
   • Liquidating exposure through out-of-court settlement or other legal action 通过庭外和解或其他法律行动

Credit risk management capacity

1. Credit risk analysis: what, whom, how long
   • Summary of the major loan types
   • Distribution of the loan portfolio
   • Loans with government or other guarantees由政府或其他机构担保的贷款
   • Review of loans by risk classification
   • Analysis of nonperforming loans
2. Board of directors must ensure:
   • Loans should be granted on a sound and collectible basis.贷款应在稳健和可收回的基础上发放
   • Funds should be invested profitably for the benefit of shareholders and the protection of depositors.为了股东的 利益和储户的保护,资金的投资应该是有利可图的
   • The legitimate credit needs of economic agents and households should be satisfied.应满足经济主体和家庭的 合法信贷需求
   • Lending process贷款流程的审查
   • Staffing人员配备
   • Information flows信息流
     Internal review and reporting system

Credit Derivatives

CDS Mechanics

1. Credit default swap (CDS): is a bilateral derivative contract between a protection buyer and a protection seller.
   
   • The CDS is similar to an insurance contract (put option) whereby the protection buyer makes premium payments to the protection seller in exchange for a contingent payoff in default or other credit event.
   • Single-name CDS单实体CDS: remain the most popular instrument type of credit derivative.
   • Reference entity 参考实体
     Companies and countries
   • Long risky bond = Long risk-free bond + Sell CD
   • Long risky bond + Buy CDS = Long risk-free bond
2. Settlement
   • Cash-settled CDS: The credit protection seller makes a single cash payment to the credit protection buyer
     Par minus current market price
     An ISDA-organized auction process is used to determine the mid-market value of the cheapest deliverable bond several days after the credit event
   • Physical settled CDS: Protection buyer delivers underlying reference to protection seller and receives a cash payment equal to par value.

Credit derivatives risk

1. The core credit derivative instrument, the CDS, is simple and has transformed the trading of credit risk.
   • CDSs themselves can prove highly toxic since, whilst they can be used to hedge counterparty risk in other products,there is counterparty risk embedded within the CDS contract itself.CDS虽然可以对冲交易对手的风险,但交易CDS本身就会带来新的交易对手风险
   • 推动CDS标准化

The CDS spread

CDS Indices

1. CDS indices: participants in credit markets have developed indices to track credit default swap spreads.
   • CDX NA IG, a portfolio of 125 investment grade companies in North America
   • iTraxx Europe, a portfolio of 125 investment grade names in Europe
   • On March 20 and September 20 each year: companies that are no longer investment grade are dropped from the portfolios and new investment grade companies are added.

The use of fixed coupons

1. CDS合约的标准化,支付标准的CDS coupons而非CDS spreads
   • 投资级债券的CDS coupon为1%,非投资级债券的CDS coupon为5%
2. CDS spread vs.CDS coupon
   • CDS spread:市场为参考实体信用风险的定价
   • CDS coupon:标准化合约下催生的标准化保费支付
3. Upfront premium 预付保费:
   • (CDS spread - CDS coupon) × Duration
   • CDS price: Upfront preimum × 100 + CDS price = 100
4. 标准化CDS coupon 5%,CDS spread 6%,意味着期初保护购买方要向保护出售方额外支付一笔预付保费,之后定时(比如每季度)支付5%

CDS forwards and options

1. Forward credit default swap: is the obligation to buy or sell a particular credit default swap on a particular reference entity at a particular future time T.
   • 如果在T时刻前违约,合约自动失效
2. Credit default swap option: is an option to buy or sell a particular credit default swap on a particular reference entity at a particular future time T.(Call买入CDS, Put 卖出CDS)
   • 签订期权,一年后以280 bps买入CDS保护.一年后CDS spread高于280 bps,选择行权

Basket credit default swaps

1. Basket credit default swap: there are a number of reference entities.
2. Add-up basket CDS: provides a payoff when any of the reference entities default.
3. First-to-default CDS: provides a payoff only when the first default occurs.
4. kth-to-default CDS: provides a payoff only when the kth default occurs.

Total Return Swap (TRS)

1. Total return swap: is an agreement to exchange the total return on a bond (or any portfolio of assets) for a floating rate plus a spread.
   • Includes coupons, interest, and the gain or loss on the asset
     
2. Financing tool:
   • The receiver:获得了一个债券的收益,付出了floating rate+25bps融资对价
   • The payer: 仍拥有债券,这样做比直接借钱给receiver买入债券交易对手方风险少很多
3. Counterparty risk: the spread over the floating rate received by the payer is compensation for bearing the risk that the receiver will default.当债券价值下降,receiver要支付价值下降的部分,可能出现违约

Collateralized debt obligations

1. Cash CDO
   
2. Synthetic合成 CDO
   • 合成的中心思想:CDS空头 = 债券多头(风险承担)
   • 找到目标债券组合,做空对应公司CDS
     Chooses a portfolio of companies and a maturity(e.g.,5 years) for the structure
     Sells CDS protection on each company in the portfolio with the CDS maturities equaling the maturity of the structure
3. Single-Tranche Trading 单份额交易: an imaginary reference portfolio is used to define the cash flows on tranches.
   • One tranche can be traded without there being any trading in other tranches.
   • Standard synthetic CDO tranches
     The six standard tranches of iTraxx Europe cover losses inthe ranges 0-3%,3-6%,6-9%,9-12%,12-22%,and 22-100%.
     The recovery rate R is usually assumed to be 40%

Valuation of a synthetic CDO

1. 核心思路和CDS定价估值类似,合成CDO本质就是卖出CDS
   
2. The Gaussian copula model of time to default
   • 引入有关违约事件的单因子高斯Copula模型,假定所有公司在时间t内违约的概为Q(t)
   • The probability of default by time t conditional on the factor m
     Q(t \mid m)=N\left(\frac{N^{-1}[Q(t)]-\sqrt{\rho} m}{\sqrt{1-\rho}}\right)
   • \rho is the copula correlation, assumed to be the same for any pair of companies.
3. The standard market model gives the probability of exactly k defaults by time t, conditional on m:
   Q(k, t \mid m)=C_n^k Q(t \mid m)^k[1-Q(t \mid m)]^{n-k}

Implied correlation

1. Applying the model to observed market prices of structured credit products to estimate a default correlation.
2. 类似于观察市场期权价格,通过BSM反求隐含波动率的步骤,观察市场合成CDO价格,反推隐含相关系数
3. Compound (tranche) correlation复合（层级）相关系数: For a tranche \left\{\alpha_{q-1}, \alpha_q\right\}, leads to the spread calculated from the model being the same as the spread in the market \rightarrow \rho.找到某一层级的市场价格，使得模型得到的利差等于市场上的利差,反推出的相关系数
4. Base correlation基础相关系数: the value of \rho that leads to the \left\{0, \alpha_q\right\} tranche being priced consistently with the market.
   • Calculate the compound correlation for each tranche.
   • Use the compound correlation to calculate the present value of the expected loss on each tranche算出每个层级的预期损失
   • Calculate the present value of the expected loss on the \left\{0, \alpha_q\right\}

Alternative approaches to estimate default correlation

1. Homogeneous model同质化模型: are assumed to be the same for all companies and the Copula correlations for any pair of companies are the same.
2. Heterogeneous Model 异质模型: is more complicated to implement because each company has a different probability of defaulting by any given time and Q(k, t|m) can no longer be calculated using the binomial formula in equation.
3. Other Copulas: Student t Copula,Clayton Copula

Structured credit products

Definition of Securitization

1. Securitization refers to the sale of assets, which generate cash flows from the institution that owns the assets, to another company that has been specifically set up for purpose of acquiring them, and issuing of notes by this second company.
   • These notes are hacked by cash flows from original assets.
   • SPV is bankruptcy-remote from the sponsor. 破产隔离
   • Investors have credit risk exposure only to the underlying assets of the SPV.
     

Key participants

1. Originator (sponsor, seller)发起人: the entity whose assets are being securitized. The loan originator is the original lender who creates the debt obligations in the collateral pool, often a bank.
   • Banks move away from the traditional "buy and hold" business model toward an "originated to distribute" (OTD).
2. Issuer(underwriter, arranger, SPV)发行人: is set up specially for the purpose of securitization.
   • Aggregates the underlying loans, designs the securitization structure and markets the liabilities.
   • Warehousing risk库存风险: risk that deal will not be completed and value of accumulated collateral falls.
3. Rating agencies评级机构: are engaged to assess the credit quality of the liabilities and assign ratings to them.
   • Rating agencies are compensated by issuers, creating a potential conflict
4. Servicers: collects principal and interest from the loans in the collateral pool and disburses principal and interest to liability holders, as well as fees to the underwriter and itself.
5. Managers
6. Trustee and custodian

SPV Structures

1. Amortizing structures (pass-through structure): pay principal and interest to investors on a coupon-by-coupon basis throughout the life of the security.
2. Revolving Structures: revolve the principal of the assets.
   • During the revolving period, principal collections are used to purchase new receivables that fulfil necessary criteria.
   • Used for short-dated assets with a relatively high pre-payment speed
3. Master trust: allow multiple securitizations to be issued from the same SPV.
   • used by MBS and credit card ABS originators.

Reasons for Undertaking Securitization

1. Funding assets
   • Banks aim to optimize their funding between a mix of retail,inter-bank and wholesale sources.
   • Most of notes issued by SPVs will be higher rated than the bonds issued directly by the originating bank itself (provide a lower cost of funding).
   • The management of maturity mismatching.
2. Balance sheet capital management
3. Risk management and credit risk transfer: can be used to remove nonperforming assets from banks' balance sheets.
4. Investors: are attracted to investing in diversified loan pools that they would not otherwise have access to without securitization, such as mortgage loans and auto loans.

Classifications of asset pools

1. By assets
   • Collateral
     Auto loans
     Credit card
     Mortgage
     Debt obligation
   • Loan pool收益权: even non-debt assets such as recurring fee income, can also be packaged into securitizations.
2. By type of pools
   • Static pools: are amortizing pools in which a fixed loans is placed in the trust
     E.g.: auto loans and residential mortgages.
   • Revolving pools: specify an overall level of assets to be maintained during a revolving period. As underlying loans are repaid, the size is maintained by introducing additional loans from the balance sheet of the originator.
     E.g.: creditcard debt.
   • Managed pools: manager of the structured product has discretion to remove individual loans from the pool, and replace them with others.
     E.g.: CDOs.

Auto loan

1. Auto Loan: it offers an easily sellable, tangible asset in the case of obligor default.
   • Prepayment speed is extremely stable
   • Losses are relatively low.
   • Time horizon is a short 3-5 years.
2. Performance measures:
   • The loss curve: shows expected cumulative loss through the life of a pool
   • The absolute prepayment speed: Prepayments / Outstanding pool balance

Credit card

1. Credit card: loans have no predetermined term, include a revolving period.
2. Performance measures:
   • Delinquency ratio拖欠(逾期90天)率: Delinquents (overdue for more than 90 days) / Receivables
   • Default ratio违约率: Defaults (written off) / Receivables
   • Monthly payment rate: Collections / Receivables

Commercial mortgage

1. Debt service coverage ratio(DSCR):
   =\frac{\text { Net operating income }}{\text { Debt payments }}
   • It indicates a borrower's ability to repay a loan.来源/总额
   • A DSCR of less than 1 means that there is insufficient cash flow generated by the property to cover required debt payments.

Mortgage

1. Weighted average coupon(WAC):is the weighted coupon of the pool that is obtained by multiplying the mortgage rate on each loan by its balance.
2. Weighted average maturity(WAM):is the average weighted of the remaining terms to maturity of the underlying pool of mortgage loans in the MBS.
3. Weighted average life(WAL): priced and traded based on expected maturity and weighted average life(WAL)考虑了提前还款
   • WAL includes various pre-payment assumptions to estimate the change of the rate at which principal is repaid toinvestors.
   • PF: repayment weighting adjustment to the notional value outstanding (O/S).
     W A L=\sum \frac{a}{365} \times P {F(t)}
     
4. Prepayment
   • Single monthly mortality(SMM): is the single-month proportional prepayment.
   • The constant prepayment rate(CPR):annualized SMM
     C P R=1-(1-S M M)^{12}
   • The Public Securities Association (PSA): a pool of mortgages has a 100% PSA if its CPR starts at 0 and increases by 0.2% each month until it reaches 6% in month30, and then it keeps constant 6% afterwards.
     

Structured products

1. Mortgage pass-through securities(MPS): are backed by a pool of mortgage loans, removed from the mortgage originators' balance sheets, and administered by a servicer,who collects principal and interest from the underlying loans and distributes them to the bondholders.
   • Most pass-throughs are agency MBS, so there is little default risk.
   • Bondholders are exposed to prepayment risk.
2. Collateralized mortgage obligations(CMO) are "sliced," or tranched into bonds or tranches, that are paid down on a specified schedule.
   • The simplest structure is sequential pay顺序支付, in which the tranches are ordered, with "Class A" receiving all principal repayments from the loan until it is retired, then "Class B," and so on.
3. Collateralized debt obligations(CDOs): can be thought of as customized baskets of debt instruments segmented broadly into senior, mezzanine, and equity tranches.

Tranching

1. Equity (First-loss piece): typically receives no fixed coupon payment, but fully exposed to defaults in the collateral pool. The notional value is typically small.最先吸收损失
   • Is the lowest rated or non-rated.
2. Junior debt: earns a relatively high fixed coupon or spread,also called mezzanine tranches and are typically also thin.
3. Senior debt: earns a relatively low fixed coupon or spread,but is protected by both the equity and mezzanine.最安全
4. Attachment point起赔点 and detachment point止赔点: the boundary between two tranches, expressed as a percentage of total of the liabilities
   • The equity tranche only has a detachment point, and the most senior only has an attachment point.
     

Credit enhancement

1. Credit enhancement: the group of measures that can improve the credit rating for ABS and MBS issues.
   • The lower the quality of the assets being securitized, the greater the need for credit enhancement.
2. Overcollateralization(hard)超额抵押: selling a par amount of bonds that is smaller than the par amount of underlying collateral.
   • Provides credit enhancement for all of the bond tranches of a securitization.
3. Excess spread(Reserves) (soft)超额利差储备: must be filled and kept at certain levels before equity notes can receive money.
4. Pool insurance
5. Senior/Junior note classes
6. Margin step-up:保护投资者不会出现potential extension

Impact of PD and Default Correlation

1. Assume default correlation constant
   
   • Mean value convexity effect at low correlation
     Equity tranche: positively convex.Equity tranche loses value rapidly as default rates increase from a low level.
     Mezzanine tranche: has negative convexity for low default rates, but is positively convex for high default rates.
     Senior tranche: negatively convex. Its losses accelerate as defaults rise.
2. Assume default rate constant
   
   • Equity tranches: benefits from high correlation
   • Mezzanine tranche:
     At low default rates, an increase in correlation increases losses (behaves more like the senior tranche).
     At high default rates, an increase in correlation decreases losses (behaves more like the equity tranche)
   • Senior tranches: is hurt by high correlation

Default sensitivities of the tranches

1. Default 01: measures the impact of an increase of 1 basis point in the default probability.
   • Each default O1 is expressed as a positive number and expresses the decline in value or increase in loss resulting from a 1-basis point rise in default probability.
   • The default 01 sensitivity converges to zero for all the tranches for very high default rates.
   • The default O1 varies most as a function of default probability when correlation is low.
   • The largest-magnitude default O1, begins at a default rate that brings losses in the collateral pool near that tranche's attachment point.

Risk factors impacting the structured products

1. Systematic risk: Structured credit products can have great systematic risks, even when the pools are well-diversified.
2. Tranche thinness: if the equity and mezzanine tranche is relatively thin, senior tranche is exposed to systematic risk
3. Granularity: can significantly diminish securitization risks. Lumpy pools数量少 of collateral have greater risk of extreme outliers than granular ones.

Cash waterfall

1. Definition: a process that all the cash that is generated by the asset pool is paid in order of payment priority.
   • The senior debt receives all of its promised payments before any lower tranche receives any monies

Tracking annual cash flows

1. Cash Inflows:
   • L_t(Aggregate loan interest received)+R_t(Recovery amount)
     L_t: Aggregate loan interest received at the end of year t
     R_t: Recovery amount deposited into the overcollateralization account at the end of year t
2. Cash outflows:
   • B_t Bond coupon interest due to both the junior and senior bonds前两层的资金流入
3. Overcollateralization account(OC)相当于蓄水池:
   • Excess spread is diverted up to the OC account, where it will earn the financing money market rate of r.
     K: Maximum amount diverted annually from excess spread into the overcollateralization account
   • If L_t - B \gt 0, the next test determines whether the excess spread is great enough to cover K
     If L_t - B \gt K, then K flows into overcollateralization account, and there may be some excess spread left over for equity第三层的资金流入.
     If L_t - B \lt K, then the entire amount L_t - B flows into overcollateralization account, no excess spread left.
     If L_t - B \lt 0, then use OC to compensate junior and senior tranch用OC顺序偿还缺口
   • OC account will be used to pay interest on bonds if interest flow from loans in collateral pool isn't enough.上一期OC终值再投资变成下一期OC初值
   • Recovery amount is paid into the overcollateralization account. Interest is paid at the end of the year by surviving loans only.违约回收的钱直接进入OC
     Value of the account at the end of the year t is:
     R_t+O C_{{t}}+\sum_{\tau=1}^{t-1}(1+r)^{t-\tau} O C_t
     最后一期OC按顺序偿还本息