文章目录

- 2. Measuring Credit Risk
- - 2.1 Default Probability
 - 2.2 Estimate PD
 - - 2.2.1 Estimate PD - Default Intensity Model
 - 2.2.2 Estimate PD - KMV Model(Merton Model)
 - 2.3 Exposure
 - 2.4 Loss Given Default
 - 2.5 Expected Loss vs. Unexpected Loss
 - - 2.5.1 Expected Loss
- 2.2 Unexpected Loss
- 2.3 Credit Loss Distribution(信用损失分布)
3. Capital for Bank's Credit Risk
- 3.1 Capital
- 3.2 The Basel Committee
- 3.3 Economic Capital v.s. Regulatory Capital
- 3.4 Regulatory Capital — Vasicek Model
- 3.5 Economic Capital — CreditMetric
- 3.6 Risk Allocation — Euler's Theorem
- 3.7 Challenges to Measuring Credit Risk for Derivatives
- 3.8 Challenges to Quantifying Credit Risk

2. Measuring Credit Risk

2.1 Default Probability

Issuer default rate is the number of bonds that have defaulted in a given year divided by the number of issues outstanding.

Dollar default rate is the total par value of bonds that have defaulted in a given year divided by the total par value of all bonds outstanding.

Cumulative default probability: an issuer with a certain rating will default within one year, within two years, within three years and so on.
Cumulative survival rate=1−cumulative PD\text{Cumulative survival rate} = 1-\text{cumulative PD}Cumulative survival rate=1−cumulative PD

Unconditional default probability: the probability of a bond defaulting between time t1t_1t1 and t2t_2t2
PDKUncond=PDt+kcumulated−PDtcumulatedPD_K^{\text{Uncond}}=PD_{t+k}^{\text{cumulated}}-PD_t^{\text{cumulated}}PDKUncond=PDt+kcumulated−PDtcumulated

Conditional default probability: if the firm survives to the end of year nnn, what is the probability that it will default during year n+1n+1n+1
PDt:kCond=(Deft+k−Deft)Cumulative survivaltPD^{\text{Cond}}_{t:k}=\frac{(Def_{t+k}-Def_t)}{\text{Cumulative survival}_t}PDt:kCond=Cumulative survivalt(Deft+k−Deft)

What is the cumulative survival rate within four years for a B-rates Bond?

For the same rate, what is the unconditional PD that the bond will default during the fifth year? And the conditional PD during the fifth year assume no earlier default?

Cumulative survival rate: 100%−15.87%=84.13%100\%-15.87\%=84.13\%100%−15.87%=84.13%

Unconditional PD: 18.32%−15.87%=2.45%18.32\%-15.87\%=2.45\%18.32%−15.87%=2.45%

Conditional PD: 2.45%/84.13%=2.91%2.45\%/84.13\%=2.91\%2.45%/84.13%=2.91%

2.2 Estimate PD

2.2.1 Estimate PD - Default Intensity Model

Poisson Distribution is used to model number of default events over time
f(x)=P(X=x)=(λt)xe−λtx!f(x)=P(X=x)=\frac{(\lambda t)^x e^{-\lambda t}}{x!}f(x)=P(X=x)=x!(λt)xe−λt

No default within TTT years
P(X=0)=(λt)0e−λt0!=e−λtP(X=0)=\frac{(\lambda t)^0 e^{-\lambda t}}{0!}=e^{-\lambda t}P(X=0)=0!(λt)0e−λt=e−λt

λ\lambdaλ is hazard rate, which is the rate at which default are happening. We can use it to calculate unconditional default probabilities.

Cumulative survival rate=e−λt;Cumulative PD=1−e−λt\text{Cumulative survival rate}=e^{-\lambda t};\;\text{Cumulative PD}=1-e^{-\lambda t}Cumulative survival rate=e−λt;Cumulative PD=1−e−λt

PDKUncond=PDt+kcumulated−PDtcumulated=e−λt−e−λ(t+k)PD_K^{\text{Uncond}}=PD_{t+k}^{\text{cumulated}}-PD_t^{\text{cumulated}}=e^{-\lambda t}-e^{-\lambda (t+k)}PDKUncond=PDt+kcumulated−PDtcumulated=e−λt−e−λ(t+k)

Suppose that the hazard rate is constant at 1%1\%1% per year. Please calculate the probability of a default by the end of the third year and unconditional probability of a default occurring during the fourth year.

Cumulative PD=1−e−λt=1−e−1%×3=2.9554%\text{Cumulative PD}=1-e^{-\lambda t}=1-e^{-1\%\times3}=2.9554\%Cumulative PD=1−e−λt=1−e−1%×3=2.9554%

Unconditional PD=(1−e−λ(t+k))−(1−e−λt)=(1−e−1%×4)−(1−e−1%×3)=0.9656%\text{Unconditional PD}=(1-e^{-\lambda (t+k)})-(1-e^{-\lambda t})=(1-e^{-1\%\times4})-(1-e^{-1\%\times3})=0.9656\%Unconditional PD=(1−e−λ(t+k))−(1−e−λt)=(1−e−1%×4)−(1−e−1%×3)=0.9656%

2.2.2 Estimate PD - KMV Model(Merton Model)

We can take credit risk as an option. Consider a firm with total value VVV that has one bond due in one year with face value D=100D=100D=100.

One year later	Total value of firm (V)	Face value of bond (D)	Equity
Scenario 1	500>D500>D500>D	100100100	400400400
Scenario 2	300>D300>D300>D	100100100	200200200
Scenario 3	100=D100=D100=D	100100100	000
Scenario 4	70<D70<D70<D	707070	000

Equity is a call option on the assets of the firm with a strike price equal to the face value of the debt.

Model includes factors

The amount of debt in the firm’s capital structure
The market value of the firm’s equity
The volatility of the firm’s equity

ct=StN(d1)−Xe−rtN(d2)c_t=S_tN(d_1)-Xe^{-rt}N(d_2)ct=StN(d1)−Xe−rtN(d2)

St=VtN(d1)−De−rtN(d2)S_t=V_tN(d_1)-De^{-rt}N(d_2)St=VtN(d1)−De−rtN(d2)

PD=1−N(d2)PD =1-N(d_2)PD=1−N(d2)

2.3 Exposure

Exposure is the amount at risk during the life of the financial instrument.

Exposure at default(EAD) is the amount of money lender can lose in the event of a borrower’s default.

2.4 Loss Given Default

Loss given default(LGD) is the amount of creditor loss in the event of a default.

The recovery rate(RR) for a bond defined as the value of the bond shortly after default and it is expressed as a percentage of its face value.

The loss given default provides the same information of loss give default, and it it the percentage recovery rate subtracted from 100%100\%100%.

LGD=1−RRLGD = 1-RRLGD=1−RR

RR=RecoveryamountExposure=1−LGDExposureRR=\frac{\text{Recovery\;amount}}{\text{Exposure}}=1-\frac{LGD}{\text{Exposure}}RR=ExposureRecoveryamount=1−ExposureLGD

Recovery rates are negatively correlated with default rates.

Recessionary period: default rates on bonds are high and recovery rates are low.
Economy is doing well: default rates on bonds are low and recovery rates are high.

2.5 Expected Loss vs. Unexpected Loss

2.5.1 Expected Loss

Expected loss(EL) is the amount a bank can expect to lose over a given period of time as a result of credit events.
EL=EAD×PD×LGDEL=EAD\times PD\times LGDEL=EAD×PD×LGD

Expected Loss(%)=PD×LGD\text{Expected Loss}(\%)=PD\times LGDExpected Loss(%)=PD×LGD

Management of expected loss(price in 通过定价来覆盖预期损失)
Bank loans is a bank allows for in the way it sets interest rates on its loans.

Expected default rate = 1.5%	Expected loss(%)=0.9%
Recovery rate = 40%	Expected loss(%)=0.9%
Margin to cover its expenses	1.6%
Average funding cost	1%
Interest rate it charges on its loans	0.9% + 1.6% + 1% = 3.5%

2.2 Unexpected Loss

Unexpected loss(非预期损失) is the amount a bank cannot anticipates as a result of credit events.

The unexpected loss is high percentile of the loss distribution minus the expected loss.
Management of unexpected loss: the bank’s capital(银行资本金) is a cushion(安全垫) that covers the unexpected loss.

2.3 Credit Loss Distribution(信用损失分布)

EADiEAD_iEADi: The amount borrowed in the ithi_{th}ith loan (assumed constant throughout the year).
PDiPD_iPDi: The probability of default for the ithi_{th}ith loan.
LGDiLGD_iLGDi: The loss rate in the event of default by the ithi_{th}ith loan (assumed known with certainty)
ρi,j\rho_{i,j}ρi,j: The correlation between losses on the ithi_{th}ith and jthj_{th}jth loan.
σi\sigma_iσi: The standard deviation of loss from the ithi_{th}ith loan.
σp\sigma_pσp: The standard deviation of loss from the portfolio.

For an individual loan:
The mean loss (expected loss) is
EL=PDi×EADi×LGDiEL = PD_i\times EAD_i \times LGD_iEL=PDi×EADi×LGDi
The standard deviation of the credit loss is
σi2=E(loss2)−[E(loss)]2=(PDi−PDi2)(LGDi×EADi)2\sigma_i^2=E(\text{loss}^2)-[E(\text{loss})]^2=(PD_i-PD^2_i)(LGD_i\times EAD_i)^2σi2=E(loss2)−[E(loss)]2=(PDi−PDi2)(LGDi×EADi)2

→σi=PDi−PDi2(LGDi×EADi)\to \sigma_i=\sqrt{PD_i-PD^2_i}(LGD_i\times EAD_i)→σi=PDi−PDi2(LGDi×EADi)

For a loan portfolio:
The mean loss(expected loss) is

ELp=∑i=1nELi=∑i=1nPDi×EADi×LGDiEL_p=\sum^n_{i=1}EL_i=\sum^n_{i=1} PD_i\times EAD_i \times LGD_iELp=i=1∑nELi=i=1∑nPDi×EADi×LGDi

The variance of the credit loss is

σp2=∑i∑jρi,jσiσj\sigma^2_p=\sum_i\sum_j\rho_{i,j} \sigma_i \sigma_jσp2=i∑j∑ρi,jσiσj

We assume all PDPDPD, EADEADEAD, LGDLGDLGD and ρ\rhoρ are the same and constant for all loans:

σn2=nσi2+n(n−1)ρσi2\sigma^2_n=n\sigma^2_i+n(n-1)\rho\sigma^2_iσn2=nσi2+n(n−1)ρσi2

Example: suppose a bank has a portfolio with 100,000100,000100,000 loans, and each loan is USD 1 million and has a 1%1\%1% probability of default in a year. The recovery rate is 40%40\%40% and correlation between loans is 0.10.10.1. What is the standard deviate of individual loan credit loss, and the mean and standard deviate of portfolio credit loss?

Standard deviate of individual loan credit loss:
σi=1%×99%(1×60%)=0.0597million\sigma_i=\sqrt{1\% \times 99\%}(1\times 60\%)=0.0597\;\text{million}σi=1%×99%(1×60%)=0.0597million

Mean of portfolio credit loss:
EL=100,000×1×1%×60%=60millionEL=100,000\times1\times 1\%\times 60\%=60\;\text{million}EL=100,000×1×1%×60%=60million

Standard deviate of portfolio credit loss:
σp2=100,00×0.05972+100,000×99,999×0.1×0.05972=3,564,41→σp=1.888million\sigma_p^2=100,00\times0.0597^2+100,000\times99,999\times 0.1\times 0.0597^2=3,564,41\to \sigma_p= 1.888\;\text{million}σp2=100,00×0.05972+100,000×99,999×0.1×0.05972=3,564,41→σp=1.888million

3. Capital for Bank’s Credit Risk

3.1 Capital

A bank must keep capital for the risks it takes.

If a bank has equity capital of USD 444 billion and incurs losses of USD 1.51.51.5 billion, its equity capital is reduced to USD 3.53.53.5 billion.

3.2 The Basel Committee

In 1974, the central banks of the G10 countries formed the Basel Committee to harmonize global bank regulation.

Global bank regulatory requirements are determined by the Basel Committee on Banking Supervision in Switzerland. The requirements are then implemented by bank supervisors in each member country.

3.3 Economic Capital v.s. Regulatory Capital

Regulatory capital(监管资本) is the capital bank regulators (also known as bank supervisors) require a bank to keep.

Separate capital calculations are added to give the total capital requirements.
Internal ratings-based (IRB, Basel II) - Vasicek Model
The Basel Committee sets one year X=99.9%X = 99.9\%X=99.9% for regulatory capital in the internal ratings-based approach. It occur only once every thousand years.

Economic capital(经济资本) is a bank’s own estimate of the capital it requires.

Correlations between the risks are often considered.
CreditMetrics model.
When banks determine economic capital, they tend to be even more conservative.
An AA-rated corporation with PD=0.02%PD=0.02\%PD=0.02% → setting X as high as 99.98%99.98\%99.98%

Example: Consider a bank rated as AA. One of its key objectives will almost certainly be to maintain its AA credit rating. If an AA-rated corporation has a default probability of about 0.02%0.02\%0.02% in one year. The 99.999.999.9 percentile of the default rate distribution is therefore around 14.89%14.89\%14.89%. The 99.9899.9899.98 percentile of the distribution is around 22.31%22.31\%22.31%. The expected default rate for all rated companies is 1.305%1.305\%1.305%. The recovery rate is 25%25\%25%. Please calculate the regulatory capital and economic capital.

EL=(1−25%)×1.305%=0.98%EL=(1-25\%)\times1.305\%=0.98\%EL=(1−25%)×1.305%=0.98%

Regulatory capital=0.75×14.89%−0.98%=10.19%\text{Regulatory capital}=0.75\times14.89\%-0.98\%=10.19\%Regulatory capital=0.75×14.89%−0.98%=10.19%

Economic capital=0.75×22.31%−0.98%=15.75%\text{Economic capital}=0.75\times22.31\%-0.98\%=15.75\%Economic capital=0.75×22.31%−0.98%=15.75%

3.4 Regulatory Capital — Vasicek Model

Vasicek model is used by regulators to determine capital for loan portfolios. It uses the Gaussian copula model to define the correlation between defaults.

The Basel Committee sets X=99.9%X = 99.9\%X=99.9% for regulatory capital in the internal ratings-based approach.
(WCDR−PD)×EAD×LGD(WCDR - PD) \times EAD\times LGD(WCDR−PD)×EAD×LGD
WCDR (worst case default rate)

Gaussian copula model: a Gaussian copula creates a joint probability distribution between two or more variables which are both normal distributed variables.

One-factor correlation model: Now suppose we have many variables, Vi(i=1,2,...)V_i(i = 1, 2,...)Vi(i=1,2,...). Each ViV_iVi can be mapped to a standard normal distribution UiU_iUi in the way we have described.
Ui=aiF+1−ai2ZiUi = a_iF +\sqrt{1- a_i^2}Z_iUi=aiF+1−ai2Zi

FFF is a factor common to all the UiU_iUi
ZiZ_iZi is the component of UiU_iUi that is unrelated to the common factor FFF (idiosyncratic). The ZiZ_iZi corresponding to the different UiU_iUi are uncorrelated with each other.
F∼N(0,1),Zi∼N(0,1)F\sim N(0,1),\;Z_i\sim N(0,1)F∼N(0,1),Zi∼N(0,1)
aia_iai are parameters with values between −1-1−1 and +1+1+1.
Ui∼N(0,1)U_i \sim N(0,1)Ui∼N(0,1)
ρ=E(UiUj)−E(Ui)E(Uj)SD(Ui)SD(Uj)→ρ=aiaj\rho=\frac{E(U_iU_j)-E(U_i)E(U_j)}{SD(U_i)SD(U_j)} \to \rho=a_ia_jρ=SD(Ui)SD(Uj)E(UiUj)−E(Ui)E(Uj)→ρ=aiaj
- SD(Ui)=0,SD(Uj)=0SD(U_i)=0,\;SD(U_j)=0SD(Ui)=0,SD(Uj)=0
- E(Ui)=0,E(Uj)=0E(U_i)=0,\;E(U_j)=0E(Ui)=0,E(Uj)=0

Vasicek model - Unconditional default distribution
Assume the probability of default( PDPDPD ) is the same for all companies in a large portfolio.
The aia_iai are assumed to be the same for all iii. Setting ai=aa_i=aai=a
Ui=aF+1−a2ZiU_i=aF+\sqrt{1- a^2}Z_iUi=aF+1−a2Zi

The binary probability of the default distribution for company iii for one year is mapped to a standard normal distribution UiU_iUi. Company iii defaults if:
Ui≤N−1(PD)U_i \leq N^{-1}(PD)Ui≤N−1(PD)

PD=1%PD=1\%PD=1%, company iii default if Ui≤N−1(0.01)=−2.33U_i \leq N^{-1}(0.01)=-2.33Ui≤N−1(0.01)=−2.33

Vasicek model - Conditional default distribution
Ui=aF+1−a2ZiU_i=aF+\sqrt{1- a^2}Z_iUi=aF+1−a2Zi

The factor FFF can be thought of as an index of the recent health of the economy.

If FFF is high, the economy is doing well and all the UiU_iUi will tend to be high (making defaults unlikely).
If FFF is low, all the UiU_iUi will tend to be low so that defaults are relatively likely.
Ui∼N(aF,1−a2)⟶ρ=a2Ui∼N(ρF,1−ρ)U_i \sim N(aF,\;1-a^2) \stackrel{\rho=a^2}{\longrightarrow}U_i \sim N(\sqrt{\rho}F,\;1-\rho)Ui∼N(aF,1−a2)⟶ρ=a2Ui∼N(ρF,1−ρ)

Vasicek model
The default rate conditional nnn the factor FFF:

99.9%99.9\%99.9% percentile worst case default rate(WCDR)

WCDR=N(N−1(PD)−ρN−1(0.001)1−ρ)WCDR=N\left( \frac{N^{-1}(PD)-\sqrt{\rho}N^{-1}(0.001)}{\sqrt{1-\rho}} \right)WCDR=N(1−ρN−1(PD)−ρN−1(0.001))

Capital requirement=(WCDR−PD)×EAD×LGD\text{Capital requirement}=(WCDR-PD)\times EAD \times LGDCapital requirement=(WCDR−PD)×EAD×LGD

Example: A bank has a USD 100100100 million portfolio of loans with a PD of 0.75%0.75\%0.75%. Assume a correlation parameter of 0.20.20.2. The recovery rate in the event of a default is 30%30\%30%. What is the required regulatory capital using 99.999.999.9 percentile of the
default rate given by the Vasicek model?

N−1(0.001)=−3.0902andN−1(0.0075)=−2.432N^{-1}(0.001)=-3.0902\;and\;N^{-1}(0.0075)=-2.432N−1(0.001)=−3.0902andN−1(0.0075)=−2.432

WCDR=N(N−1(PD)−ρN−1(0.001)1−ρ)=N(−2.4324+0.2×3.09021−0.2)=N(−1.17)=12.1%WCDR=N\left( \frac{N^{-1}(PD)-\sqrt{\rho}N^{-1}(0.001)}{\sqrt{1-\rho}} \right)=N\left(\frac{-2.4324+\sqrt{0.2}\times 3.0902}{\sqrt{1-0.2}}\right)=N(-1.17)=12.1\%WCDR=N(1−ρN−1(PD)−ρN−1(0.001))=N(1−0.2−2.4324+0.2×3.0902)=N(−1.17)=12.1%

Capital requirement=(WCDR−PD)×EAD×LGD=(12.1%−0.75%)×100×70%=7.9million\text{Capital requirement}=(WCDR-PD)\times EAD \times LGD=(12.1\%-0.75\%)\times 100 \times 70\%=7.9\;\text{million}Capital requirement=(WCDR−PD)×EAD×LGD=(12.1%−0.75%)×100×70%=7.9million

3.5 Economic Capital — CreditMetric

CreditMetrics is the model banks often use to determine economic capital. Under this model, each borrower is assigned an external or internal credit rating.

Step 1: The bank’s portfolio of loans is valued at the beginning of a one-year period.
Step 2: Use Monte Carlo simulation to model how ratings change during the year.
Step 3: The portfolio is revalued.
Step 4: The credit loss is calculated as the value of the portfolio at the beginning of the year minus the value of the portfolio at the end of the year.

CreditMetrics considers the impact of rating changes as well as defaults.

3.6 Risk Allocation — Euler’s Theorem

Euler’s theorem: can be used to divide many of the risk measures used by risk managers into their component parts.

If a risk measure meets homogeneity
Qi=ΔFΔXi/Xi(Decomposition)→F=∑i=1nQi(Combination)Q_i=\frac{\Delta F}{\Delta X_i/X_i}(\text{Decomposition})\to F=\sum^n_{i=1}Q_i(\text{Combination})Qi=ΔXi/XiΔF(Decomposition)→F=i=1∑nQi(Combination)

ΔXi\Delta X_iΔXi is a small change in variable iii, ΔX/Xi\Delta X/ X_iΔX/Xi is a proportional change.
ΔF\Delta FΔF is the resultant small change in FFF.
QiQ_iQi is the risk component decomposition.

Example: Suppose that the losses from loans A and B have standard deviations of $ 222 and $ 666. The correlation between two loans is 0.50.50.5. The standard deviations of portfolio is $ 7.21117.21117.2111. Please calculate the dollar contribution of loan A and loan B to the whole portfolio risk.

Loan A:
If size of loan A is increase by 1%1\%1% (ΔX/Xi=1%\Delta X/ X_i = 1\%ΔX/Xi=1%). The SD of loan A is 2×1.01=2.022\times1.01=2.022×1.01=2.02

Δσp=2.022+62+2×2.02×6×0.5−7.2111=0.01388\Delta \sigma_p=\sqrt{2.02^2+6^2+2\times2.02\times 6\times 0.5}-7.2111=0.01388Δσp=2.022+62+2×2.02×6×0.5−7.2111=0.01388

Contribution Lona A =ΔFΔXi/Xi=0.01388/1%=1.388=\frac{\Delta F}{\Delta X_i/X_i}=0.01388/1\%=1.388=ΔXi/XiΔF=0.01388/1%=1.388

Loan B:
If size of loan B is increase by 1%1\%1% (ΔX/Xi=1%\Delta X/ X_i = 1\%ΔX/Xi=1%). The SD of loan B is 6×1.01=6.066\times1.01=6.066×1.01=6.06

Δσp=22+6.062+2×2×6.06×0.5−7.2111=0.05826\Delta \sigma_p=\sqrt{2^2+6.06^2+2\times2\times 6.06\times 0.5}-7.2111=0.05826Δσp=22+6.062+2×2×6.06×0.5−7.2111=0.05826

Contribution Lona A =ΔFΔXi/Xi=0.05826/1%=5.826=\frac{\Delta F}{\Delta X_i/X_i}=0.05826/1\%=5.826=ΔXi/XiΔF=0.05826/1%=5.826

3.7 Challenges to Measuring Credit Risk for Derivatives

Derivatives also give rise to credit 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.
Netting agreements: all outstanding derivatives with a counterparty may be considered a single derivative in the event that the counterparty defaults.

3.8 Challenges to Quantifying Credit Risk

PD: Banks are faced with the problem of making both through-the-cycle estimates (to satisfy regulators) and point- in-time estimates (to satisfy their auditors).
LGD: the recovery rate is negatively correlated with the default rate.
EAD: derivative transactions need a relatively complex calculation (may contain wrong-way risk).
Correlations: are difficult to estimate.

Credit risk is only one of many risks facing a bank.

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