credit_risk.rmd

# Credit Risk Measurement and Management

###  Measurement of Credit Risk
* Expected Loss

$$ EL=PD \times LGD \times EAD $$

* LGD and Recovery Rate

$$ LGD=1-RR $$

* Unexpected Loss

$$ UL = Credit VaR = WCL-EL $$

### Classification of Credit Risk
#### Default-mode Valuation
* Default Risk
* Recovery Risk
* Exposure Risk

#### Value-based Valuation
* Migration Risk -- 评级下调
* Spread Risk -- 信用价差增大
* Liquidity Risk

### Estimate PD

#### Risk-neutral

$$ P = \frac{100}{1+YTM}=(1-PD)\frac{100}{1+R_f}+PD\frac{100(1-LGD)}{1+R_f}  $$

#### Real-world
$$ P_* = \frac{100}{1+YTM}=(1-PD^*)\frac{100}{1+R_f+R_{premium}}+PD^*\frac{100(1-LGD)}{1+R_f+R_{premium}}  $$


$$ PD^* \times LGD \approx YTM-R_f-R_{premium} $$

#### Credit Spread
$$ Credit spread \approx YTM-R_f - R_{premium}   $$

$$ Credit spread = -\frac{1}{T-t}ln(\frac{D}{F})-r_f $$

$$ CS = PD \times LGD = PD(1-RR) \Rightarrow PD = \frac{CS}{1-RR}  $$

$$ PD=PD^*+ R_{premium} + Premium_{liquidity}$$

* Credit spreads are observable and can be used to infer hazard rate.

* 违约率+生存率=1

### Change on PD
#### 违约概率密度函数
* 投资级先低后高
* 投机级先高后低

#### Risk-neutral Probability

* Risk-neutral default probability = real-world default probability + Risk premium + Liquidity Premium


#### Credit Spread Curve Mapping
* 利率曲线期限相同
* **i-spread** 如果期限不同，则插值
* Credit spread = YTM-rf = PDxLGD
*

### **TODO** Models table here


### Merton Model
#### Assumption
* Only one issue of equity and debt, zero coupon.
* Default can only occur at the maturity date.
* The value of the firm follows a lognormal distribution.
* Risk free rate is constant
* No need to adjust for liquidity

#### Formula

* S 为公司价值，K为债务价值
* Call option 相当于股权
* Put option 相当于债权

$$ V_t = D_t+S_t  $$

$$ S_t=max(V_t-F, 0)  $$


$$ D_t=F-max(F-V_t,0)  $$

$$ Call=S_0N(d_1)-Ke^{-rT}N(d_2)  $$

* PV(EL): Long put option on firm's value

$$ Put=Ke^{-rT}[1-N(d2)]-S_0[1-N(d_1)]  $$


* With dividend

$$ S=S_0-PV(d) $$

$$ d_1=\frac{ln(\frac{S_0}{K})+(r+\frac{\sigma^2}{2})T}{\sigma\sqrt{T}}  $$

$$ d2=DtD = d1-\sigma\sqrt{T} =\frac{ln(\frac{S_0}{K})+(r-\frac{\sigma^2}{2})T}{\sigma\sqrt{T}} \approx \frac{lnS-lnK}{\sigma} $$

$$ PD= N(d2) $$
* 投资级Naive, 投机级别 Merton

#### Credit spread

$$ CS=-[\frac{1}{T-t}]ln(\frac{D}{F})-r_f  $$

* (T-t) --  remaining maturity
* D -- Current value of debt
* F -- Face value debt
* r_f == risk-free rate

#### Relationship
|                 | Firm Value | Face value of debt | Time to Maturity | r  | sigma |
|-----------------|------------|--------------------|------------------|----|-------|
| Value of debt   | \+         | \+                 | \-               | \- | \-    |
| Value of Equity | \+         | \-                 | \+               | \+ | \+    |

----

#### Limitation
* Only to liquid, publicly traded names
* A continuous need for calibration

#### Subordinated debt
**TODO**

### KMV Model vs Merton
* KMV不需要Lognormal假设
* KMV Only 2 debt
* KMV 模型需要真实的分布，需要查表

#### Distance to default

$$ DD=Z = \frac{A-K}{\sigma_A} $$


$$ Z_{0.9} = 1.65  $$

$$ Z_{0.95} = 1.96    $$

$$ Z_{0.98} = 2.33  $$


$$ Z_{0.99} = 2.58  $$


### Default Intensity Models
#### 伯努利分布

#### 二项分布

#### Poisson distribution

#### Exponential distribution
* Hazard rate, default intensity
$$ \lambda  $$


* Cumulative Default probability

$$ P(t^* < t) = F(t) = 1-e^{-\lambda t}  $$

* Cumulative survival probability

$$ P(t^* \geq t) = 1-F(t) = e^{-\lambda t}  $$

* Conditional default probability

$$ P(t^*<t+\tau | t^*>t) = 1-e^{-\lambda \tau}  $$

* Risk-neutral Hazard Rates

$$ \lambda^* \approx \frac{Z_\tau}{1-RR}  $$

#### 解题思路
* 第N年的违约概率为前N年的违约概率-前N-1年的违约概率

### Default Correlation

$$ \rho_{12} = \frac{\pi_{12} - \pi_1 \pi_2}{\sqrt{\pi_1(1-pi_1)}\sqrt{\pi_2(1-\pi_2)}}  $$

### Credit Scoring Models
* 针对个人和小企业
* A large number of small, low-value loans
* 关注 Expected Loss
* Cutoff score 使得False bad + false good 最小

### Other Models
* Structured approaches 关注自变量，Q宗
* Reduced Form approaches 关注因变量，P宗
* Logistic 回归不需要给定cutoff score, 直接输出概率

$$ log(\frac{\pi}{1-\pi}) = \beta_0 + \beta_1x_1 + ... + \beta_k x_k $$

$$ \pi = \frac{1}{1+e^{-( \beta_0 + \beta_1x_1 + ... + \beta_k x_k )}}$$

### Credit Exposure
* 未来有正现金流才有credit exposure

#### Credit Exposure Metrics

* **Expected Mark to Market(MtM)** is the expected value of a transaction at a given point in the future.
* **Expected Exposure(EE)** is the amount that is expected to be lost if there is positive MtM and the counterpart defaults.
* **Potential Future Exposure(PFE)** is an estimate of MtM value at a specific point in the future.
* **Maximum PFE** is the highest PFE value over a stated time frame.
* **Expected positive exposure(EPE)** is the average EE through time.
* **Negative exposure** is the exposure from the counterparty's point of view.
* **Effective EE** is equal to nondecreasing EE.
* **Effective EPE** is the average of the effective EE.

$$ Effective EE > Effective EPE > EPE  $$

* **EE**只计算MtM>0, 若MtM<0, 则EE=0

* **TODO** 表格及各种工具的exposure

----

### Counterparty Risk
#### Counterparty Risk VS Lending Risk
* Lending Risk 有固定的exposure， 单边风险
* Counterparty risk 的exposure是动态的， 双边风险

#### Wrong-way risk and Right-way Risk
* Wrong-way risk, PD与exposure同向变动
* Right-way risk, PD与expsure反向变动
* 从自己赚钱的方向开始分析
* 看主营业务

----

#### Netting

##### Netting factor

$$ NF=\frac{\sqrt{n+n(n-1)\rho}}{n}  $$

* 同一主协议下可以多边netting；若无主协议，只能双边netting
* netting factor取值（0，1), 越小说明组合netting效果越好
* 如果合约exposure都为正，则netting不起作用
* 有法律风险
* Payment netting, 债务抵消，合约继续
* Close-out netting, 合约终止，一般因为信用事件发生
* netting降低资源占用
* Walkaway features 一方出现信用风险，则另一方不支付欠款

#### The Impact of netting
* 降低exposure, 降低系统风险
* 增加集中度风险
* Improve RR

#### CCP Loss Waterfall
**TODO**

----

#### Collateral
* Exposure 

$$ Exposure = max(V_{portfolio}-V_{collateral}, 0) $$

* Benefit

$$ benefit = E_{no_collateral}-E_{with_collateral} $$

* Credit Support Annex (CSA) 约定担保相关细节
* 当超过Threshould + minimum transfer amount时，支付担保，担保按照rounding取整
* haircut

$$ haircut = \frac{V_{collateral}-V_{exposure}}{V_{collateral}}  $$

* Volatility 越高，credit risk越高，maturity越长，liquidity越差，则haircut越高
* 有rehypothecation风险，越特异的资产，再质押风险越高，因为不易获得
* overcollateral 时 Exposure=0

$$ round(max(V-threshould-MTA, 0))  $$



----

### Measurement of counterparty risk
#### CVA
* Credit Value Adjustment(CVA) is the expected value or price of counterparty credit risk. An adjustment to the risk-free value of a derivative to account for potential default.

* Risky value = risk-free value - CVA
* CVA 从头寸开始时计算一直到头寸终止
* CVA 对手风险减值准备
* CVA 有倾向于与少数优质对手交易的趋势，因此增加集中度风险

$$ CVA \approx LGD \sum{d(t_i)EE(t_i)PD(t_{i-1}, t_i)}  $$

$$ CVA \approx EPE \times Spread  $$
* EPE = average EE

#### Netting CVA
* 组合的Netting CVA 小于CVA之和

$$ CVA_{NS} \leq \sum{CVA_i}  $$

#### Incremental CVA
* 要不要新做一笔交易

$$ CVA^{incremental}_{i} = CVA_{NS+i} -CVA_{NS}  $$

#### Marginal CVA
* 存量资产中每个资产的CVA
* 组合CVA最小化
* **Formula**

#### DVA and Bilateral CVA
* DVA 按照本方的模型算出的给对方的期望损失
* DVA与对方算出的CVA不同，因为各自模型不同

$$ BCVA=CVA+DVA $$

### Portfolio Credit Risk
* **TODO** formula

#### Single-factor Model
$$ r_i=\beta_im+\sqrt{1-\beta_i^2}\epsilon_i  $$

$$ m \sim N(0,1) ,  \epsilon \sim N(0,1)  $$
$$ r_i \sim N(\beta_im, 1-\beta_i^2)  $$

$$ m=\frac{Z_{\alpha}(1-\sqrt{1-\beta^2})}{\beta}  $$

$$ Z_{\alpha} = \frac{Z_{\alpha}-m \beta}{\sqrt{1-\beta^2}} $$

* epsilong 公司个体风险

#### Credit Risk Portfolio Models
* **TODO** Table

### Management of Credit Risk`

### Credit Derivative
#### First-to-Default CDS
* 篮子中资产数量多，资产相关性低，则保费变贵

#### Nth-to-Default CDS
* 篮子中资产数量多，资产相关性高，则保费变贵

#### Total Return Swap(TRS)
* 通常是风险资产与无风险资产收益之间的互换

#### Asset-backed Credit-linked Notes
* 不出表，可以有杠杆，有部分担保
* 初始本金作为担保品
* 买CLN等于卖CDS，卖CLN等于买CDS

#### Single-tranche CDO
* Senior和Equity自持，junior卖给投资者

### CDOs
#### Synthetic CDO
未出表，卖出CDS

### Structured Credit Risk Product
#### Covered bonds
* YTM低，有担保，不出表

### Classification of Structured Credit Product
#### Static Pool
* 底层资产不变

#### Revolving Pools
* 底层资产滚动更新，如信用卡

#### Managed Pools
* 主动管理底层资产

* **TODO** Tree

#### 信用增强
* OC, overcollateralization account. 流入的正现金流先存入OC账户，等OC账户满了后再分给投资者。遇到信用事件， 则用OC账户支付。
* 锁定期内的收益用于信用增强
* Margin step-up 违约则利息越来越高
* Excess spread is the difference between the cash flows collected and the payment made to all bondholders.

#### Benefit

**TODO** Benefit to investors and originator



### 结构化产品信用特征

#### Mortgate pass-through securities
* Investors benefited from a new liquid asset class
* Lenders benefited by removing interest rate risk off the balance sheet.
* Investor receive cash flows based on the performance of pool
* Most are agency MBS
* Primary risk is prepayment risk
* Debt service coverage ratio(**DSCR**)

$$ DSCR=\frac{NetOperatingIncome}{TotalDebtPayment}  $$

* Weighted average coupon(**WAC**)

$$ WAC=\frac{\sum{F_ir_i}}{\sum{F_i}}  $$

* Weighted average maturity(**WAM**)

$$ WAM=\frac{\sum{F_it_i}}{V_i}  $$

* Weighte average life(**WAL**)
$$ WAL= \sum{\frac{a}{365}}PF(t)  $$


#### Collateralized mortgate obligations (CMOs)
* Different tranches have different maturity
* The remaining tranches will receive interest only until upper tranches are retired.


#### Structured credit products
#### Asset-backed securities
* ABS/MBS have diversity with thousands of obligors
* CDO typically consists of less than 200 loans.

##### Auto loan 
* Loss curve
* Absolute prepayment speed(APS)

##### Credit card
* Delinquency ratio
* Default ratio
* Monthly payment rate(MPR)

#### PD and correlation
* **TODO** Table !

##### Drawback of default correlation-based credit portfolio framework
* The number of required calculation
* Do not fit well to some credit positions such as guarantees
* Limited Data

$$ Events=2^n   $$

$$ Conditions=(n+1) + \frac{n(n-1)}{2}  $$

#### Convexity
| tranche | Convexity                        |
|---------|----------------------------------|
| Equity  | Convexity                        |
| Junior  | 先convexity, 再negatve convexity |
| Senior  | Negative convexity               |
----

#### Performance Measures for MBS
* Debt service coverage ratio(DSCR) Net operation income/Debt payments
* Weighted average maturity(WAM) ???
* Weighted average life(WAL)
* Prepayment Performance CRP 

$$ CPR=1-(1-SMM)^{12} $$

#### Payment Forecast
**TODO**

### Seven Frictions of Securitization Process
**TODO**


### Misc
* CMO 优先偿还短duration投资者