1. Credit Loss Distribution
Credit Loss Distribution，描述在给定的时间段内（通常为一年）可能的信用损失及其发生的概率。
- 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.【收益有限， 比如利息；损失比较大，比如本金】
- 信用损失的期望 $\rightarrow$ 预期损失(Expected Loss, EL)
- 信用损失的标准差 $\rightarrow$ 非预期损失(Unexpected Loss, UL)
可以用 beta distribution 建模信用风险损失【关于 beta 分布,无需了解太深】
2. EL、UL、ULC(重要的不能再重要了)
(1) Expected Loss (EL)
指在未来一段时间（通常是一年）可能发生的平均损失，也就是损失的期望值。
| 类型 | 公式 | 备注 |
|---|---|---|
| 单个资产 | $EL(%) = PD \times LGD$ $EL($) = PD \times LGD \times EAD$ [主要用这个公式] | PD→信用评级、CS、莫顿模型计算PD LGD=loss rate=1-回收率 EAD，也写作exposure amount(EA) |
| 组合 | $$EL_p = \sum_{i=1}^n PD_i \times LGD_i \times EAD_i$$ | 和资产之间的相关性ρ无关 |
| 两资产组合 | $$EL_p = PD_1 \times LGD_1 \times EAD_1 + PD_2 \times LGD_2 \times EAD_2$$ |
(2) Unexpected Loss (UL)
UL is the standard deviation of credit losses, that is, the standard deviation of actual credit losses around the expected loss average (EL).
切记：在这个科目中，UL的概念，与一级以及二级其他科目中的Unexpected Loss概念不同。
| 类型 | 公式 | 备注 |
|---|---|---|
| 单个资产 | $UL = EAD \times \sqrt{PD \times \sigma_{LGD}2 + LGD2 \times \sigma_{PD}2}$Bernoulli伯努利分布→$\sigma_{PD}2 = PD \times (1-PD)$ | - $\sigma_{LGD}2$$：LGD的方差$ $\sigma_{PD}2$：PD的方差 |
| 组合 | $UL_p = \sum_i\sum_j \rho_{i,j} \times UL_i \times UL_j$ | $\rho_{i,j}$：资产i和j的违约相关性 |
| 两资产组合 | $UL_p = \sqrt{UL_12 + UL_22 + 2 \times \rho_{1,2} \times UL_1 \times UL_2}$ | |
| 若单个资产的UL相等，相关系数相等 | $UL_p = UL \times \sqrt{n + n \times (n-1) \times \rho}$ |
(3) Unexpected Loss Contribution (ULC)
ULC是指单个资产（如loan）对整个UL的边际贡献。即衡量一个特定资产在整个组合可能出现的损失中占了多大的份额。【重点掌握两资产】
| 类型 | 公式 | 备注 |
|---|---|---|
| 一般公式 | $ULC_i = \frac{UL_i \times \sum_j \rho_{i,j} \times UL_j}{UL_p}$ | |
| 两资产 | $ULC_1 = \frac{UL_12 + \rho_{11} \times UL_1 \times UL_2}{UL_p}$ $ULC_2 = \frac{UL_22 + \rho_{11} \times UL_1 \times UL_2}{UL_p}$ $ULC_1 + ULC_2 = UL_p$ | |
| 前提：n笔loan， 每笔loan规模相同、风险特征相同 | $ULC_i = \frac{UL_p}{n} = UL \times \sqrt{\frac{1}{n} + \rho \times (1-\frac{1}{n})}$ | the correlation between assets increases, the bank suffers from concentration risk集中度风险。 |
| 前提：n比较大时 | $ULC_i = UL \times \sqrt{\rho}$ |
Measures the tendency of multiple entities to default simultaneously
$\bullet$ 一家公司的违约可能引发另一家公司的违约，这被称为信用传染效应（creditcontagion effect）。
$\bullet$ joint default probability( $\pi_{12})$ :在时间范围 T 内两者都违约的概率
$$ \rho = \frac{Cov(X_1X_2)}{\sigma_{X_1}\sigma_{X_2}} = \frac{\pi_{12} - \pi_1\pi_2}{\sqrt{\pi_1(1-\pi_1)\sqrt{\pi_2(1-\pi_2)}}} $$
$\pi_{1}$ ：资产1的违约概率
$\pi_{2}$ ：资产 2 的违约概率
拓展推导过程: 一级数量科目，相关系数 $\textstyle\cdot\rho={\frac{\operatorname{Cov}(X_{1}X_{2})}{\sigma_{X_{1}}\sigma_{X_{2}}}}$
$\mathrm{E(X_{i})=\pi_{i};E(X_{1}X_{2}){=}\pi_{12}}$
$\mathrm{Var(X_{i}){=}E(X_{i}){2}-[E(X_{i})]{2}{=}{\pi}{i}(1-{\pi}{i})\Delta{i}{=}1,2}$
$\mathrm{Cov(X_{1},X_{2}){=}E(X_{1}X_{2}){-}E(X_{1})E(X_{2}){=}\pi_{12}-\pi_{1}\pi_{2}}$ $\mathrm{E}(\mathrm{X}{1}\mathrm{X}{2}){=}\mathrm{\pi}_{12}$
，见下表:
| 结果 | X₁ | X₂ | X₁X₂ | 概率 |
|---|---|---|---|---|
| No default | 0 | 0 | 0 | $1 - \pi_1 - \pi_2 + \pi_{12}$ |
| Firm 1 only defaults | 1 | 0 | 0 | $\pi_1 - \pi_{12}$ |
| Firm 2 only defaults | 0 | 1 | 0 | $\pi_2 - \pi_{12}$ |
| Both firms default | 1 | 1 | 1 | $\pi_{12}$ |
$\mathrm{E(X_{1}X_{2})}{=}0{*}(1-,\pi_{1}-,\pi_{2}+\pi_{12})+0{}(\pi_{1}-,\pi_{12})+0^{}(\pi_{2}-,\pi_{12})+1^{*}\pi_{12}{=}\pi_{12}$
所以： $\begin{array}{r}{\rho=\frac{\mathrm{Cov}(X_{1}X_{2})}{\sigma_{X_{1}}\sigma_{X_{2}}}!=!\frac{\pi_{12}-\pi_{1}\pi_{2}}{\sqrt{\pi_{1}\left(1-\pi_{1}\right)}\sqrt{\pi_{2}\left(1-\pi_{2}\right)}}}\end{array}$
If default correlation in a portfolio of credits is equal to 1, then the portfolio behaves as ifit consisted ofjust one credit.【违约相关性 $=!1$ ，可以把这个组合看成一个资产】
- No credit diversification is achieved.没有分散化效果
- If default correlation is equal to O, then the number of defaults in the portfolio is a binomial ly distributed random variable. Significant credit diversification may be achieved.
Effect of Granularity on Credit VaR
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“Granular颗粒度”指的是增加组合中独立信贷的数量，并减少每个信贷在整个组合中所占的比例。
-
组合越“granular”，就意味包含更多的独立小额信贷，每个信贷占组合的比重越小。
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Granularity is a measurement of the portfolio's concentration risk;
- For a given default probability, Credit VaR decreases as the credit portfolio becomes more granular. 【granular↑，Credit VaR+】
- The convergence is more drastic with a high default probability. 【违 约概率越高，CreditVaR下降越多】It is harder to reduce VaR by making the portfolio more granular, if the default probability is low．【违约概率越低，Credit VaR 较少，甚至很难下降】
其他计量违约相关性的模型
1. Reduced FormModels
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Reduced form models assume that the hazard rates for different companies follow stochastic processes and are correlated with macroeconomic variables，
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优势：数学上具有吸引力，反映了经济周期产生违约相关性的趋势。
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劣势：可实现的违约相关性范围有限，即使两家公司的危险率完美相关，它们在同一短时间内同时违约的概率通常也很低。
2) Structural Models
- 类似于Merton模型，公司在其资产价值低于某一水平时算违约
- Default correlation between companies A and B is introduced into the model by 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.
- 优势：可以设定任意高的违约相关性。
- 劣势：计算速度较慢。
4.Credit VaR
Credit VaR = WCL-EL
Worst case loss (WCL): Loss at the confidence level
Credit risk VaR & Market risk VaR
| Credit risk VaR | Market risk VaR | |
| Time horizon | One-year time horizon | One-day time horizon |
| Tools | More elaborate model 更复杂的 模型， such as the Vasicek model | Main tool: historical simulation |
5．量化信用风险的挑战
Credits are illiquid assets.【以前量化信用风险。但是随着金融市场的发展，信贷资产的流动性也在慢慢变好，会导致对实际风险的低估】
- 精确计算风险比较困难。【对信贷的多期特性( multi-period nature of credits) 进行详细建模，包括借款人信用质量预期和非预期变化以及它们之间的相关性。尽管这些因素可以纳入分析方法 analytical approach 中，但非常复杂和繁琐.】
- 相关性考虑不足。尽管信用跟某些风险评估方法会在同一类型的风险内考虑相关性，但在实际测量时，它们假设所有其他风险成分（如市场风险和操作风险）是相互独立的，并由银行的不同部门单独进行测量和管理。【在评估信用风险时，可能已经考虑了信用风险内的相关性，但往往与其他风险(如市场风险、操作风险)相关性考虑不足。在银行的实际风险管理中，往往各different departments。】