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
t
1
t_1
t1 and
t
2
t_2
t2
P
D
K
Uncond
=
P
D
t
+
k
cumulated
−
P
D
t
cumulated
PD_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
n
n
n, what is the probability that it will default during year
n
+
1
n+1
n+1
P
D
t
:
k
Cond
=
(
D
e
f
t
+
k
−
D
e
f
t
)
Cumulative survival
t
PD^{\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
)
=
(
λ
t
)
x
e
−
λ
t
x
!
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
T
T
T years
P
(
X
=
0
)
=
(
λ
t
)
0
e
−
λ
t
0
!
=
e
−
λ
t
P(X=0)=\frac{(\lambda t)^0 e^{-\lambda t}}{0!}=e^{-\lambda t}
P(X=0)=0!(λt)0e−λt=e−λt
λ \lambda λ 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 − λ t ; Cumulative PD = 1 − e − λ t \text{Cumulative survival rate}=e^{-\lambda t};\;\text{Cumulative PD}=1-e^{-\lambda t} Cumulative survival rate=e−λt;Cumulative PD=1−e−λt
P D K Uncond = P D t + k cumulated − P D t cumulated = e − λ t − 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 − λ t = 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 − λ t ) = ( 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 V V V that has one bond due in one year with face value D = 100 D=100 D=100.
One year later | Total value of firm (V) | Face value of bond (D) | Equity |
---|---|---|---|
Scenario 1 | 500 > D 500>D 500>D | 100 100 100 | 400 400 400 |
Scenario 2 | 300 > D 300>D 300>D | 100 100 100 | 200 200 200 |
Scenario 3 | 100 = D 100=D 100=D | 100 100 100 | 0 0 0 |
Scenario 4 | 70 < D 70<D 70<D | 70 70 70 | 0 0 0 |
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
c t = S t N ( d 1 ) − X e − r t N ( d 2 ) c_t=S_tN(d_1)-Xe^{-rt}N(d_2) ct=StN(d1)−Xe−rtN(d2)
S t = V t N ( d 1 ) − D e − r t N ( d 2 ) S_t=V_tN(d_1)-De^{-rt}N(d_2) St=VtN(d1)−De−rtN(d2)
P D = 1 − N ( d 2 ) 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%.
L G D = 1 − R R LGD = 1-RR LGD=1−RR
R R = Recovery amount Exposure = 1 − L G D Exposure RR=\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
Expected loss(EL) is the amount a bank can expect to lose over a given period of time as a result of credit events.
A bank can manage expected loss by setting lending rates.
E L = E A D × P D × L G D EL=EAD\times PD\times LGD EL=EAD×PD×LGD
Expected Loss ( % ) = P D × L G D \text{Expected Loss}(\%)=PD\times LGD Expected Loss(%)=PD×LGD
Expected default rate = 1.5% | Expected loss(%)=0.9% |
Recovery rate = 40% | |
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% |
Unexpected loss 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.
The bank’s capital is a cushion that covers the unexpected loss.
2.6 Credit Loss Distribution
E
A
D
i
EAD_i
EADi: The amount borrowed in the
i
t
h
i_{th}
ith loan (assumed constant throughout the year).
P
D
i
PD_i
PDi: The probability of default for the
i
t
h
i_{th}
ith loan.
L
G
D
i
LGD_i
LGDi: The loss rate in the event of default by the
i
t
h
i_{th}
ith loan (assumed known with certainty)
ρ
i
,
j
\rho_{i,j}
ρi,j: The correlation between losses on the
i
t
h
i_{th}
ith and
j
t
h
j_{th}
jth loan.
σ
i
\sigma_i
σi: The standard deviation of loss from the
i
t
h
i_{th}
ith loan.
σ
p
\sigma_p
σp: The standard deviation of loss from the portfolio.
For an individual loan:
The mean loss (expected loss) is
E
L
=
P
D
i
×
E
A
D
i
×
L
G
D
i
EL = PD_i\times EAD_i \times LGD_i
EL=PDi×EADi×LGDi
The standard deviation of the credit loss is
σ
i
2
=
E
(
loss
2
)
−
[
E
(
loss
)
]
2
=
(
P
D
i
−
P
D
i
2
)
(
L
G
D
i
×
E
A
D
i
)
2
\sigma_i^2=E(\text{loss}^2)-[E(\text{loss})]^2=(PD_i-PD^2_i)(LGD_i\times EAD_i)^2
σi2=E(loss2)−[E(loss)]2=(PDi−PDi2)(LGDi×EADi)2
→
σ
i
=
P
D
i
−
P
D
i
2
(
L
G
D
i
×
E
A
D
i
)
\to \sigma_i=\sqrt{PD_i-PD^2_i}(LGD_i\times EAD_i)
→σi=PDi−PDi2
For a loan portfolio:
The mean loss(expected loss) is
E L p = ∑ i = 1 n E L i = ∑ i = 1 n P D i × E A D i × L G D i EL_p=\sum^n_{i=1}EL_i=\sum^n_{i=1} PD_i\times EAD_i \times LGD_i ELp=i=1∑nELi=i=1∑nPDi×EADi×LGDi
The variance of the credit loss is
σ p 2 = ∑ i ∑ j ρ i , j σ i σ j \sigma^2_p=\sum_i\sum_j\rho_{i,j} \sigma_i \sigma_j σp2=i∑j∑ρi,jσiσj
We assume all P D PD PD, E A D EAD EAD, L G D LGD LGD and ρ \rho ρ are the same and constant for all loans:
σ n 2 = n σ i 2 + n ( n − 1 ) ρ σ i 2 \sigma^2_n=n\sigma^2_i+n(n-1)\rho\sigma^2_i σn2=nσi2+n(n−1)ρσi2
Suppose a bank has a portfolio with 100 , 000 100,000 100,000 loans, and each loan is USD 1 1 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 0.1 0.1 0.1. 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:
σ
i
=
1
%
×
99
%
(
1
×
60
%
)
=
0.0597
million
\sigma_i=\sqrt{1\% \times 99\%}(1\times 60\%)=0.0597\;\text{million}
σi=1%×99%
Mean of portfolio credit loss:
E
L
=
100
,
000
×
1
×
1
%
×
60
%
=
60
million
EL=100,000\times1\times 1\%\times 60\%=60\;\text{million}
EL=100,000×1×1%×60%=60million
Standard deviate of portfolio credit loss:
σ
p
2
=
100
,
00
×
0.059
7
2
+
100
,
000
×
99
,
999
×
0.1
×
0.059
7
2
=
3
,
564
,
41
→
σ
p
=
1.888
million
\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 Economic Capital v.s. Regulatory Capital
Regulatory capital 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 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
P
D
=
0.02
%
PD=0.02\%
PD=0.02% → setting X as high as
99.98
%
99.98\%
99.98%
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 99.9 99.9 99.9 percentile of the default rate distribution is therefore around 14.89 % 14.89\% 14.89%. The 99.98 99.98 99.98 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.
E L = ( 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%
3.2 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.
(
W
C
D
R
−
P
D
)
×
E
A
D
×
L
G
D
(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,
V
i
(
i
=
1
,
2
,
.
.
.
)
V_i(i = 1, 2,...)
Vi(i=1,2,...). Each
V
i
V_i
Vi can be mapped to a standard normal distribution
U
i
U_i
Ui in the way we have described.
U
i
=
a
i
F
+
1
−
a
i
2
Z
i
Ui = a_iF +\sqrt{1- a_i^2}Z_i
Ui=aiF+1−ai2
- F F F is a factor common to all the U i U_i Ui
- Z i Z_i Zi is the component of U i U_i Ui that is unrelated to the common factor F F F (idiosyncratic). The Z i Z_i Zi corresponding to the different U i U_i Ui are uncorrelated with each other.
- F ∼ N ( 0 , 1 ) , Z i ∼ N ( 0 , 1 ) F\sim N(0,1),\;Z_i\sim N(0,1) F∼N(0,1),Zi∼N(0,1)
- a i a_i ai are parameters with values between − 1 -1 −1 and + 1 +1 +1.
- U i ∼ N ( 0 , 1 ) U_i \sim N(0,1) Ui∼N(0,1)
-
ρ
=
E
(
U
i
U
j
)
−
E
(
U
i
)
E
(
U
j
)
S
D
(
U
i
)
S
D
(
U
j
)
→
ρ
=
a
i
a
j
\rho=\frac{E(U_iU_j)-E(U_i)E(U_j)}{SD(U_i)SD(U_j)} \to \rho=a_ia_j
ρ=SD(Ui)SD(Uj)E(UiUj)−E(Ui)E(Uj)→ρ=aiaj
- S D ( U i ) = 0 , S D ( U j ) = 0 SD(U_i)=0,\;SD(U_j)=0 SD(Ui)=0,SD(Uj)=0
- E ( U i ) = 0 , E ( U j ) = 0 E(U_i)=0,\;E(U_j)=0 E(Ui)=0,E(Uj)=0
Vasicek model - Unconditional default distribution
Assume the probability of default(
P
D
PD
PD ) is the same for all companies in a large portfolio.
The
a
i
a_i
ai are assumed to be the same for all
i
i
i. Setting
a
i
=
a
a_i=a
ai=a
U
i
=
a
F
+
1
−
a
2
Z
i
U_i=aF+\sqrt{1- a^2}Z_i
Ui=aF+1−a2
The binary probability of the default distribution for company
i
i
i for one year is mapped to a standard normal distribution
U
i
U_i
Ui. Company
i
i
i defaults if:
U
i
≤
N
−
1
(
P
D
)
U_i \leq N^{-1}(PD)
Ui≤N−1(PD)
P
D
=
1
%
PD=1\%
PD=1%, company
i
i
i default if
U
i
≤
N
−
1
(
0.01
)
=
−
2.33
U_i \leq N^{-1}(0.01)=-2.33
Ui≤N−1(0.01)=−2.33
Vasicek model - Conditional default distribution
U
i
=
a
F
+
1
−
a
2
Z
i
U_i=aF+\sqrt{1- a^2}Z_i
Ui=aF+1−a2
The factor F F F can be thought of as an index of the recent health of the economy.
- If F F F is high, the economy is doing well and all the U i U_i Ui will tend to be high (making defaults unlikely).
- If
F
F
F is low, all the
U
i
U_i
Ui will tend to be low so that defaults are relatively likely.
U i ∼ N ( a F , 1 − a 2 ) ⟶ ρ = a 2 U i ∼ N ( ρ F , 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
n
n
n the factor
F
F
F:
99.9
%
99.9\%
99.9% percentile worst case default rate(WCDR)
W
C
D
R
=
N
(
N
−
1
(
P
D
)
−
ρ
N
−
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−ρ
Capital requirement = ( W C D R − P D ) × E A D × L G D \text{Capital requirement}=(WCDR-PD)\times EAD \times LGD Capital requirement=(WCDR−PD)×EAD×LGD
A bank has a USD 100 100 100 million portfolio of loans with a PD of 0.75 % 0.75\% 0.75%. Assume a correlation parameter of 0.2 0.2 0.2. The recovery rate in the event of a default is 30 % 30\% 30%. What is the required regulatory capital using 99.9 99.9 99.9 percentile of the default rate given by the Vasicek model?
N − 1 ( 0.001 ) = − 3.0902 a n d N − 1 ( 0.0075 ) = − 2.432 N^{-1}(0.001)=-3.0902\;and\;N^{-1}(0.0075)=-2.432 N−1(0.001)=−3.0902andN−1(0.0075)=−2.432
W
C
D
R
=
N
(
N
−
1
(
P
D
)
−
ρ
N
−
1
(
0.001
)
1
−
ρ
)
=
N
(
−
2.4324
+
0.2
×
3.0902
1
−
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−ρ
Capital requirement = ( W C D R − P D ) × E A D × L G D = ( 12.1 % − 0.75 % ) × 100 × 70 % = 7.9 million \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
3.3 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.
3.4 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
Q
i
=
Δ
F
Δ
X
i
/
X
i
(
Decomposition
)
→
F
=
∑
i
=
1
n
Q
i
(
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)
- Δ X i \Delta X_i ΔXi is a small change in variable i i i, Δ X / X i \Delta X/ X_i ΔX/Xi is a proportional change.
- Δ F \Delta F ΔF is the resultant small change in F F F.
- Q i Q_i Qi is the risk component decomposition.
Suppose that the losses from loans A and B have standard deviations of $ 2 2 2 and $ 6 6 6. The correlation between two loans is 0.5 0.5 0.5. The standard deviations of portfolio is $ 7.2111 7.2111 7.2111. 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% (
Δ
X
/
X
i
=
1
%
\Delta X/ X_i = 1\%
ΔX/Xi=1%). The SD of loan A is
2
×
1.01
=
2.02
2\times1.01=2.02
2×1.01=2.02
Δ
σ
p
=
2.0
2
2
+
6
2
+
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=0.01388
Δσp=2.022+62+2×2.02×6×0.5
Contribution Lona A = Δ F Δ X i / X i = 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% (
Δ
X
/
X
i
=
1
%
\Delta X/ X_i = 1\%
ΔX/Xi=1%). The SD of loan B is
6
×
1.01
=
6.06
6\times1.01=6.06
6×1.01=6.06
Δ
σ
p
=
2
2
+
6.0
6
2
+
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=0.05826
Δσp=22+6.062+2×2×6.06×0.5
Contribution Lona A = Δ F Δ X i / X i = 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
3.5 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.
3.5 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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4.3.2 Measuring Credit Risk
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