最近學(xué)習(xí)了 edX 上的一門課程 Credit Risk Management狰右，做了一些筆記错敢，中英文結(jié)合痊臭，供參考踩萎。
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（資產(chǎn)組合） of loans（貸款） or securities（證券）.
Credit risk is the risk that the value of our portfolio varies, because of the unexpected changes in the credit quality（信用質(zhì)量）of trading partners or issuers.
Therefore credit risk can be divided into two sub-risks:
Default Risk（違約風(fēng)險）: the risk of losing money because of the default of our counterparty（合同的一方停局； 對手方）.
Credit Deterioration（信用惡化）: It is linked to changes in the credit quality（信用質(zhì)量） of a counterparty.
如何計算信用質(zhì)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. 市場價格和其他市場因素變化導(dǎo)致的風(fēng)險。
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. 銀行或者金融機構(gòu)內(nèi)部或者外部活動導(dǎo)致的風(fēng)險董栽，例如欺詐風(fēng)險履怯，人員風(fēng)險，信息技術(shù)風(fēng)險裆泳，恐怖襲擊等等。
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 是一系列的規(guī)則柠硕，用于監(jiān)管銀行或其他金融機構(gòu)的活動工禾，例如定義新的風(fēng)險管理措施，引入 capital requirements 的概念蝗柔。
1.2.2 Three pillars of Basel II 三個核心
Minimum Capital Requirements（最低資產(chǎn)要求） for three major components: market risk, credit risk and operational risk.
Supervisory Review（監(jiān)督審查） 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（市場紀律） aims at making the markets more efficient and transparent.
1.2.3 What are capital requirements? 什么是資產(chǎn)要求闻葵？
Capital requirements are simply the amount of capital that a bank or another financial institution has to hold as required by its financial regulator.
金融監(jiān)管者所要求的一個銀行或者一個金融機構(gòu)應(yīng)該持有的最少資產(chǎn)。
The requirements are meant to guarantee that these institutions do not become extremely leveraged and, as a possible consequence, insolvent.
目的是確保金融機構(gòu)不會過度地舉債癣丧，即過度地杠桿化槽畔，從而導(dǎo)致無力償還債務(wù)，因此破產(chǎn)胁编。
1.2.4 Assess and Hedge Credit Risk 計算和規(guī)避信用風(fēng)險
即如何計算 Capital Requirements厢钧？
The computation of capital requirements is always dependent on a quantity called RWA, risk-weighted assets（風(fēng)險加權(quán)資產(chǎn)）.
Once we have the RWA, capital requirements for credit risk are just 8% of it.
Capital Requirements 依賴于 RWA。
How to calculate RWA, there are three approaches:
Standardized Approach（STA 標準方式）：
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（違約概率）.
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. （使用監(jiān)管機構(gòu)給出的公式） (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. （銀行或金融機構(gòu)可以使用自己開發(fā)的公式） (Will cover in section [2.2])
Complexity 復(fù)雜程度：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 計算信用風(fēng)險
2.1 The Standardized Approach （STA 標準方式）
Formula to calculate RWA, risk-weighted assets（風(fēng)險加權(quán)資產(chǎn)）:
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（資產(chǎn)負債表）.
- Off-balance sheet item: an asset or debt that does not appear on a company's balance sheet（資產(chǎn)負債表）.
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 風(fēng)險權(quán)重
In the standardized approach, all weights are provided by the regulator.
2.1.4 Example 計算示例
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 基于內(nèi)部排行的方式）
Under the IRB approaches, the RWA is generally computed using 3 different elements:
2.2.1 Risk parameters（風(fēng)險參數(shù)）
the Probability of Default (PD 違約概率)：the likelihood of a default over a given time horizon.
the Exposure at Default (EAD 違約風(fēng)險敞口)：the total value that a bank is exposed to at the time of a loan's default. 可能發(fā)生違約風(fēng)險的資金額度嬉橙。
the Loss Given Default (LGD 違約損失率)：the percentage (%) of loss over the total exposure, in the case in which a counterparty defaults. 債務(wù)人一旦違約將給債權(quán)人造成的損失數(shù)額早直，即損失的嚴重程度。
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（專有的） models.
2.2.2 Risk-weight functions（風(fēng)險加權(quán)函數(shù)）
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. 用于計算 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 介紹）
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 (置信水平是指總體參數(shù)值落在樣本統(tǒng)計值某一區(qū)內(nèi)的概率市框，一般用 1-α 表示)
- capital T is the time horizon over which the VaR is computed.
假設(shè) T = 1 year, alpha = 99%, VaR = 100霞扬，表示在一年的時間范圍內(nèi)，有 99% 的概率損失不會超過 100 塊錢枫振。
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 計算示例 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?
此時: T = 1喻圃，alpha = 90% 可以算出 VaR = -80 + (20 + 80) * 90% = 10
即在一年的時間內(nèi)，有 90% 的概率損失不會超過 10 million粪滤，損失超過 10 million 的概率為 10%斧拍。
3.1.3 Example VaR 計算示例 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（虧空）
3.2.1 Mean-VaR 帶均值的 VaR
3.2.2 Distribution-specific VaRs 特定分布對應(yīng)的 VaR
高斯分布，指定均值和標準差杖小。
**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 均值為10饮焦，標準差為 5。帶入公式：
3.2.3 The Expected Shortfall（虧空）
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（一致的風(fēng)險度量） and Back-testing
3.3.1 Coherent Measures of Risk（一致的風(fēng)險度量）
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 違約概率）關(guān)注如何計算 PD
4.1 Introduction and Overview
4.1.1 What is a default? 什么是違約
The convention is that a debt obligation（債務(wù)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（質(zhì)押擔保）.
4.1.2 How to determine PD? 如何計算 PD
There are different tools that we can use for this purpose.
-
ratings（評級）
- 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（債務(wù)）.
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（債務(wù)）.
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（預(yù)測）
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.（依賴于盈利狀況和資產(chǎn)負債表）
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.（最壞的違約概率）
