最近学习了 edX 上的一门课程 Credit Risk Management，做了一些笔记，中英文结合，供参考。
1. Introduction
1.1 Defining Credit Risk
Credit risk is one of the fundamental risks for banks and companies, together with market risk and operational risk.
Imagine we hold a portfolio（资产组合） of loans（贷款） or securities（证券）.
Credit risk is the risk that the value of our portfolio varies, because of the unexpected changes in the credit quality（信用质量）of trading partners or issuers.
Therefore credit risk can be divided into two sub-risks:
Default Risk（违约风险）: the risk of losing money because of the default of our counterparty（合同的一方； 对手方）.
Credit Deterioration（信用恶化）: It is linked to changes in the credit quality（信用质量） of a counterparty.
如何计算信用质量？How is this quality computed? Most used one is Credit Rating（信用排行） - we can rely on the ratings computed by third parties, namely rating agencies, or we can compute our own ratings.
例如：If a AAA bond is downgraded to BBB, this implies that the bond is becoming much riskier, and it also have effects on a series of quantities and measures.
What is market risk and operational risk?
Market risk is easily defined as the risk of losses arising from movements in market prices and other market quantities. 市场价格和其他市场因素变化导致的风险。
Example: Say that we hold a portfolio of securities, and that we observe changes in the prices and in the interest rates. All these changes naturally influence the value of our portfolio, which can increase or decrease. This is market risk.Operational risk is defined as the risk deriving from the internal and external activities of a bank
or a another financial institution. It includes the risk of fraud, people risk, cyber risk, terrorism, calamities, and so on. 银行或者金融机构内部或者外部活动导致的风险，例如欺诈风险，人员风险，信息技术风险，恐怖袭击等等。
1.2 Basel II
1.2.1 What is Basel II?
In 1999, the Basel Committee on Banking Supervision (BCBS) released Basel II, a set of rules for regulating the activities of banks, for example by defining new risk management practices, and by imposing certain capital requirements.
Basel II 是一系列的规则，用于监管银行或其他金融机构的活动，例如定义新的风险管理措施，引入 capital requirements 的概念。
1.2.2 Three pillars of Basel II 三个核心
Minimum Capital Requirements（最低资产要求） for three major components: market risk, credit risk and operational risk.
Supervisory Review（监督审查） define a framework for dealing with other types of risks: systemic risk, pension risk, concentration risk, strategic risk, liquidity risk and so on.
Market Discipline（市场纪律） aims at making the markets more efficient and transparent.
1.2.3 What are capital requirements? 什么是资产要求？
Capital requirements are simply the amount of capital that a bank or another financial institution has to hold as required by its financial regulator.
金融监管者所要求的一个银行或者一个金融机构应该持有的最少资产。
The requirements are meant to guarantee that these institutions do not become extremely leveraged and, as a possible consequence, insolvent.
目的是确保金融机构不会过度地举债，即过度地杠杆化，从而导致无力偿还债务，因此破产。
1.2.4 Assess and Hedge Credit Risk 计算和规避信用风险
即如何计算 Capital Requirements？
The computation of capital requirements is always dependent on a quantity called RWA, risk-weighted assets（风险加权资产）.
Once we have the RWA, capital requirements for credit risk are just 8% of it.
Capital Requirements 依赖于 RWA。
How to calculate RWA, there are three approaches:
Standardized Approach（STA 标准方式）：
In the Standardized approach, the RWA is computed using simple formulas. (Will cover in section [2.1])Foundation Internal Rating Based Approach（F-IRB）：
In this approach the RWA is computed after introducing an important quantity, the PD or Probability of Default（违约概率）.
This can be computed using credit ratings (internal or external) and other models.
Once you have it, you just plug the PD into some formulas given by the regulator, and you compute the RWA. （使用监管机构给出的公式） (Will cover in section [2.2])Advanced Internal Rating Based Approach（A-IRB）
In the advanced approach, banks are "free" to compute many different quantities, from the probability of default to the loss given default.
All these quantities are then used to obtain the RWA, according to internal formulas. （银行或金融机构可以使用自己开发的公式） (Will cover in section [2.2])
Complexity 复杂程度：A-IRB > F-IRB > Standardized Approach
1.3 Basel III
Basel III can be just seen as a modification of Basel II.
The key points of Basel III are new capital definitions (Tier 1 & Tier 2) and requirements, the introduction of the so-called capital buffers, a stronger attention for leverage ratio and liquidity risk, and a stricter definition and treatment of counterparty credit risk.
2. Approaching Credit Risk 计算信用风险
2.1 The Standardized Approach （STA 标准方式）
Formula to calculate RWA, risk-weighted assets（风险加权资产）:
2.1.1 On-balance sheet item & Off-balance sheet item
- On-balance sheet item: an asset or debt that does appear on a company's balance sheet（资产负债表）.
- Off-balance sheet item: an asset or debt that does not appear on a company's balance sheet（资产负债表）.
2.1.2 Credit equivalent amount Cj
The goal of the credit equivalent amount is therefore to translate the value of off-balance sheet items into risk equivalent credits.
It is computed as the current replacement cost plus an add-on factor, which varies from instrument to instrument, for example 0.5% for a 1-5 year interest rate swap.
The add-on factor is set by the regulator.
2.1.3 Risk weight 风险权重
In the standardized approach, all weights are provided by the regulator.
2.1.4 Example 计算示例
Assume we are a bank. Our assets include:
- 120 million euros of loans to A-rated corporations
- 10 million of AA-rated government bonds
- 60 million euros of residential mortgages.
What is the value of our RWA?
RWA = 120 * 50% + 10 * 0% + 60 * 35% = 81
What is the value of our Capital Requirements?
Capital Requirements = RWA * 8% = 81 * 8% = 6.48
2.2 Internal-Rating Based Approaches（IRB 基于内部排行的方式）
Under the IRB approaches, the RWA is generally computed using 3 different elements:
2.2.1 Risk parameters（风险参数）
the Probability of Default (PD 违约概率)：the likelihood of a default over a given time horizon.
the Exposure at Default (EAD 违约风险敞口)：the total value that a bank is exposed to at the time of a loan's default. 可能发生违约风险的资金额度。
the Loss Given Default (LGD 违约损失率)：the percentage (%) of loss over the total exposure, in the case in which a counterparty defaults. 债务人一旦违约将给债权人造成的损失数额，即损失的严重程度。
Hence LGD is a percentage of EAD.
Example: Assume that we are a Japanese bank, and that one of our clients goes bankrupt and defaults.
Say that the outstanding debt of our client is ¥150 million. ** So the EAD is ¥150 million**.
Let us assume that, when our client defaults, we can obtain ¥90 million, by selling some collateral. This means that we really lose "only" ¥150-¥90=¥60 million. So the LGD is 40% = 60 / 150.Maturity (M )： the final payment date of a loan or another financial instrument/security. For example a 2-year bond has a maturity of 2 years. A 5-year mortgage has simply a maturity of 5.
All these quantities above are computed by banks using some models - some proprietary（专有的） models.
2.2.2 Risk-weight functions（风险加权函数）
These are functions defined in the Basel II-III Accords, and they are meant to compute the RWA given the risk parameters of the previous point. 用于计算 RWA
-
Foundation Internal Rating Based Approach（F-IRB）：
- Use self model to calculate PD and then use it to calculate RWA based on formulas from regulator
-
Advanced Internal Rating Based Approach（A-IRB）
- Use self model to calculate PD, EAD, LGD, Maturity and then use them to calculate RWA based on formulas from self
2.2.3 Minimum requirements
Simply the minimum standards a bank must comply with, in order to be authorized to the use of the IRB methods.
3. The Value-at-Risk （VaR）
3.1 Introducing Value-at-Risk（VaR 介绍）
The VaR is a measure that tries to answer a simple but significant question: How bad can things get, in terms of losses, when we invest, we lend money, and so on?
In more probabilistic terms, we look for a measure that tells us:
With probability alpha we will not lose more than V euros (or dollars, or pounds, and so on) in time T.
- The quantity V is the VaR.
- Alpha is the so-called confidence level 置信水平 (置信水平是指总体参数值落在样本统计值某一区内的概率，一般用 1-α 表示)
- capital T is the time horizon over which the VaR is computed.
假设 T = 1 year, alpha = 99%, VaR = 100，表示在一年的时间范围内，有 99% 的概率损失不会超过 100 块钱。
3.1.1 Compute VaR
The VaR can be computed using two different distributions:
- the distribution of gains
- the distribution of losses（prefered）
The Value-at-Risk essentially depends on 2 elements:
-
the loss distribution：A loss distribution is always expressed over a time horizon T and it can be empirical or theoretical.
- In the first case, it is the so-called historical distribution, that is the distribution that emerges from the observation of reality, when we collect data about historical losses.
- In the second case, it can be whatever distribution and it is essentially used for modeling purposes.
- the alpha value：from a theoretical point of view, may be freely chosen by the risk manager. In reality, it is often determined by law or other prescriptions. Common values are 0.95, 0.99, 0.995 and 0.999.
3.1.2 Example VaR 计算示例 1
Suppose that, for a 1-year project, all the outcomes between a gain of 80 million euros and a loss of 20 million euros are considered equally likely. (This means that our loss distribution is represented by a uniform distribution over the support [-80,20].)
What is the VaR for alpha=0.9, that is to say at the 90% confidence level?
此时: T = 1，alpha = 90% 可以算出 VaR = -80 + (20 + 80) * 90% = 10
即在一年的时间内，有 90% 的概率损失不会超过 10 million，损失超过 10 million 的概率为 10%。
3.1.3 Example VaR 计算示例 2
A 1-year project has
- A 94% probability of leading to a gain of 5 million
- A 3% change of a gain of 2 million euros
- A 2% change of leading to a loss of 3 million
- A 1% chance of producing a loss of 8 million
The question is: what is the VaR at alpha level 0.98…so the 98% VaR? And what happens if alpha is 0.99?
So VaR(0.98) = 3, VaR(0.99) = 5.5
3.2 Special VaRs and the Expected Shortfall（亏空）
3.2.1 Mean-VaR 带均值的 VaR
3.2.2 Distribution-specific VaRs 特定分布对应的 VaR
高斯分布，指定均值和标准差。
**Example: **
The historical 1-year loss distribution of a portfolio of loans in € million is well approximated by a N(10,5).
What is the 95% VaR? And the 98%?
可以看出 均值为10，标准差为 5。带入公式：
3.2.3 The Expected Shortfall（亏空）
Now assume that things go bad, and that we can observe a loss which is greater than our VaR alpha. Now, a natural question we may want to answer is the following: what's the expected loss?
The expected shortfall is the statistical quantity that tries to answer this question.
**Example: **
3.3 Coherent Measures of Risk（一致的风险度量） and Back-testing
3.3.1 Coherent Measures of Risk（一致的风险度量）
A measure of risk is said coherent when, in mathematical terms, it possesses 4 important properties:
- positive homogeneity
- translation invariance
- sub-additivity
- monotonocity.
Value-at-Risk is not coherent!
Expected Shortfall is always a coherent measure of risk!
3.3.2 Back-testing
Back-testing is a statistical tool that risk managers use to verify the accuracy and the reliability of the estimated Value-at-Risk.
4. Default Probabilities（PD 违约概率）关注如何计算 PD
4.1 Introduction and Overview
4.1.1 What is a default? 什么是违约
The convention is that a debt obligation（债务欠款） is said to have defaulted when:
- our counterparty, the obligor, is more than 90 days past due on his/her credit obligation;
- it is considered unlikely that the obligor will repay his/her debt without giving up any pledged collateral（质押担保）.
4.1.2 How to determine PD? 如何计算 PD
There are different tools that we can use for this purpose.
-
ratings（评级）
- internal ratings
- external ratings
-
default models（模型）
-
structural models of default：a model in which default happens when the assets of a company reach a sufficiently low level with respect to liabilities（债务）.
例如：Merton's model, and proprietary models like Moody's KMV and JP Morgan's CreditMetrics. - non-structural models of default
-
structural models of default：a model in which default happens when the assets of a company reach a sufficiently low level with respect to liabilities（债务）.
4.2 External Credit Ratings
The goal of a credit rating is to provide reliable information about credit quality, about the credit worthiness of a counterparty, a company, a country, and so on.
The three major rating agencies are Moody's, Standard and Poor's and Fitch.
Rating agencies essentially provide two types of products, in terms of PD:
- Historical default probabilities
- equity-based predictions（预测）
4.3 Internal Credit Ratings and Recovery Rates
4.3.1 Internal Credit Ratings
The internal-rating based approach of Basel II and III allows banks to use internal methods to determine the probability of default of a counterparty.
Internal-rating approaches generally rely on: profitability and balance sheet ratios.（依赖于盈利状况和资产负债表）
The prototype of internal rating methods is represented by Altman's Z-score.
The Z-score is a financial distress index, extremely important in fundamental analysis.
It is obtained using discriminant analysis, a well-known tool in statistics.
There are different versions of Altman's Z-score, depending on the type of company/client under scrutiny: large company, small company, manufacturing company, and so on.
The typical time horizon is 1 year.
Example 例子：for publicly traded manufacturing companies, the Z-score reads：
This number Altman's Z-score must be compared with some specific thresholds, which are obtained by analyzing historical data about the financial distress and the default of companies.
These thresholds depend on the type of Z-score you use. 例如：
4.3.2 Recovery Rates（回收率）
The recovery rate is "the amount of credit recovered through foreclosure or bankruptcy procedures in event of a default, expressed as a percentage of face value". This is the definition.
For a bond it is typically the price at which it trades about 30 days after default, as a percent of the face value.
The average recovery rate for bonds is around 35-40%.
For loans and mortgages with first lien on assets, it is usually around 65%.
4.4 Merton's Model
Merton's model is not really a proprietary model, but rather an academic/scientific one, strictly linked to the famous Black-Scholes formula. However, it is the starting point for many proprietary models, hence we need it.
4.5 The KMV Model
Moody's KMV model, another model we can use to estimate the PD of a company under the IRB class.
Moody's KMV can be used both as a F-IRB and an A-IRB model.
A fundamental quantity in the KMV model is the so called Expected Default Frequency, or EDF.
4.6 CreditMetrics by JP Morgan
CreditMetrics is a structural model of default which derives from Merton's one. But there is a big difference: the default threshold is not given by liabilities, but computed through credit ratings.
Moreover, CreditMetrics not only takes into account the risk of default of a counterparty, but also the deterioration of its creditworthiness.
4.7 C-VaR and F-IRB Capital Requirements
Once we have the PD of a counterparty, how can we quantify the capital requirements for credit risk, with respect to that counterparty?
WCDR indicates the “worst case probability of default” and it is defined as the 99.9% quantile of the default rate distribution.（最坏的违约概率）
