LO 58.8: Define in the context of Basel II and calculate the worst-case default rate

LO 58.8: Define in the context of Basel II and calculate the worst-case default rate (WCDR).
Basel II specifies three approaches that banks can use to measure credit risk: 1. Standardized approach.
2. Foundation internal ratings based (IRB) approach.
3. Advanced IRB approach.
The Standardized Approach
The standardized approach is used by banks with less sophisticated risk management functions. The risk-weighting approach is similar to Basel I, although some risk weights were changed. Significant changes include: OECD status is no longer considered important under Basel II. The credit ratings of countries, banks, and corporations are relevant under Basel II.
For example, sovereign (country) risk weights range from 0% to 150%, and bank and corporate risk weights range from 20% to 150%.
Bank supervisors may apply lower risk weights when the exposure is to the country in
which the bank is incorporated.
Bank supervisors may choose to base risk weights on the credit ratings of the countries in which a bank is incorporated rather than on the banks credit rating. For example, if a sovereign rating is AAA to AA, the risk weight assigned to a bank is 20%. The risk weight increases to 150% if the country is rated below B and is 100% if the countrys bonds are unrated.
Risk weights are lower for unrated countries, banks, and companies than for poorly rated
countries, banks, and companies.
Bank supervisors who elect to use the risk weights in Figure 3 are allowed to lower the
risk weights for claims with maturities less than three months. For example, the risk weights for short-maturity assets may range from 20% if the rating is between AAA to BBBor unrated, to 150% if the rating is below B.
A 75% risk weight is applied to retail loans, compared to 100% under Basel I. A 100%
risk weight is applied to commercial real estate loans. The uninsured residential mortgage risk weights are 35% under Basel II, down from 50% under Basel I.
A sample of risk weights under the standardized approach is presented in Figure 3.
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Figure 3: Risk Weights (as a Percent) Under Basel Us Standardized Approach
AAA to AA- 0 20 20
A+ to A – 20 50 50
BBB+ to BBB-
50 50 100
BB+ to B B – 100 100 100
B+ to B – 100 100 150
Below B Unrated
150 150 150
100 50 100
Country
Bank
Corporation
Collateral Adjustments
Banks adjust risk weights for collateral using the simple approach, similar to Basel I, or the comprehensive approach, used by most banks. Under the simple approach, the risk weight of the collateral replaces the risk weight of the counterparty. The counterpartys risk weight is used for exposure not covered by collateral. Collateral must be revalued at least every six months. A minimum risk weight of 20% is applied to collateral. Using the comprehensive approach, banks adjust the size of the exposure upward and the value of the collateral downward, depending on the volatility of the exposure and of the collateral value.
Example: Adjusting for collateral using the simple approach
Blue Star Bank has a $100 million exposure to Monarch, Inc. The exposure is secured by $80 million of collateral consisting of AAA-rated bonds. Monarch has a credit rating of B. The collateral risk weight is 20% and the counterparty risk weight is 150%. Using the simple approach, calculate the risk-weighted assets.
Answer:
(0.2 x 80) + (1.5 x 20) = $46 million risk-weighted assets
Example: Adjusting exposure and collateral using the comprehensive approach
Blue Star Bank assumes an adjustment to the exposure in the previous example of + 15% to allow for possible increases in the exposures. The bank also allows for a 20% change in the value of the collateral. Calculate the new exposure using the comprehensive approach.
Answer:
(1.15 x 100) – (0.8 x 80) = $51 million exposure
Applying a risk weight of 150% to the exposure:
1.5 x 51 = $76.5 million risk-weighted assets
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The Internal Ratings Based (IRB) Approach
United States regulators applied Basel II to large banks only. As such, regulatory authorities decided that the IRB approach must be used by U.S. banks. Under the IRB approach, the capital requirement is based on a VaR calculated over a one-year time horizon and a 99.9% confidence level. The model underlying this approach is shown in Figure 4.
Figure 4: Capital Requirement
VaR = loss at a very high confidence level (99.9%)
expected loss
The goal of the IRB approach is to capture unexpected losses (UL). Expected losses (EL) should be covered by the banks pricing (e.g., charging higher interest rates on riskier loans to cover EL). The capital required by the bank is thus VaR minus the banks EL. The VaR can be calculated using a Gaussian copula model of time to default. That is:
WCDR; =N
N -1(PDi) + VpN – (0.999)
f – p
In this equation, WCDRj is the worst case probability of default. The bank can be 99.9% certain that the loss from the zth counterparty will not exceed this amount in the coming year. PD is the one-year probability of default of the zth obligor given a large number of obligors, and p is the copula correlation between each pair of obligors.
Professor’s Note: WCDR is called the worst case probability o f default in the assigned reading. It is also called the worst case default rate, hence the acronym WCDR.
Assuming the bank has a large portfolio of instruments such as loans and derivatives with the same correlation, the one-year, 99.9% VaR is approximately:
VaR9 9 9 o/0 iyear ~ T^EAD; x LGDj xWCDRi
1
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EADj is the exposure at default of the zth counterparty or the dollar amount the zth counterparty is expected to owe if it defaults. For example, if the counterparty has a loan outstanding, EAD would likely be the principal amount outstanding on the loan at the time of default. LGDj is the loss given default for the zth counterparty or the proportion of the EADi that is expected to be lost in the event of default. For example, if the bank expected to collect (i.e., recover) 40% in the event of default, the LGDj would be 60% (i.e., 1 0.4 = 0.6).
Recall from Book 2 that the expected loss (EL) from default is computed as:
EL =
1
EAD; x LGD; x PD;
The capital the bank is required to maintain is the excess of the worst-case loss over the banks expected loss defined as follows:
required capital = y^EAD; x LGD| x (WCDRj PD^)
1
Note that WCDR, PD, and LGD are expressed as decimals while EAD is expressed in dollars.
Figure 5 shows the dependence of the one-year WCDR on PD and correlation, p.
Figure 5: Dependence of One-Year, 99.9% WCDR on PD and p = 0.0= 0.4= 0.8 p = 0.0 p = 0.2 p = 0.4 p = 0.6 p = 0.8
PD = 0.1%
0.1% 2.8% 7.1% 13.5% 23.3%
PD = 0.5%
0.5% 9.1% 21.1% 38.7% 66.3%
PD = 1% 1.0% 14.6% 31.6% 54.2% 83.6%
PD = 1.5%
1.5% 18.9% 39.0% 63.8% 90.8%
PD = 2.0%
2.0% 22.6% 44.9% 70.5% 94.4%
It is clear from Figure 5 that WCDR increases as the correlation between each pair of obligors increases and as the probability of default increases. If the correlation is 0, then WCDR is equal to PD.
Basel II assumes a relationship between the PD and the correlation based on empirical research. The formula for correlation is:
p = 0.12 x (1 + e~^ x PD)
Note that there is an inverse relationship between the correlation parameter and the PD. As creditworthiness declines, the PD increases. At the same time, the PD becomes more idiosyncratic and less affected by the overall market, thus the inverse relationship.
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The relationship between WCDR and PD, as shown in Figure 6, is obtained by combining the previous equation with the calculation of WCDR. The WCDR increases as the PD increases, but not as fast as it would if the correlation were assumed to be independent of PD.
Figure 6: Relationship Between WCDR and PD for Firm, Sovereign, and Bank Exposures
PD WCDR
2.0% 0.1% 0.5% 1.0% 3.4% 9.8% 14.0% 16.9% 19.0%
1.5%
>From a counterpartys perspective, the capital required for the counterparty incorporates a maturity adjustment as follows:
required capital = EAD x LGD x (WCDR PD) x MA
where: MA = maturity adjustment = (1 + (M 2.5) x b)l{ 1 – 1.5 x ^) M = maturity of the exposure b [0.11852-0.05478 x In (PD)]2 = [0.11852-0.05478 x In (PD)]2
The maturity adjustment, MA, allows for the possibility of declining creditworthiness and/ or the possible default of the counterparty for longer term exposures (i.e., longer than one year). If M = 1.0, then MA =1.0 and the maturity adjustment has no impact. The risk- weighted assets are calculated as 12.5 times capital required:
RWA = 12.5 x [EAD x LGD x (WCDR – PD) x MA] The capital required is 8% of RWA. The capital required should be sufficient to cover unexpected losses over a one-year period with 99.9% certainty (i.e., the bank is 99.9% certain the unexpected loss will not be exceeded). Expected losses should be covered by the banks product pricing. Theoretically, the WCDR is the probability of default that happens once every 1,000 years. If the Basel Committee finds the capital requirements too high or too low, it reserves the right to apply a scaling factor (e.g., 1.06 or 0.98) to increase or decrease the required capital.
Professors Note: On the exam, i f you begin with RWA, multiply by 0.08 to get the capital requirement. I f instead you begin with the capital requirement, multiply by 12.5 (or divide by 0.08) to get RWA. In other words, these percentages are simply reciprocals (i.e., 1/0.08 = 12.5).
Foundation IRB Approach vs. Advanced IRB Approach
The foundation IRB approach and the advanced IRB approach are similar with the exception of who provides the estimates of LGD, EAD, and M. The key differences between the two approaches are outlined by the following.
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Foundation IRB Approach The bank supplies the PD estimate. For bank and corporate exposures, there is a 0.03%
floor set for PD.
The LGD, EAD, and M are supervisory values set by the Basel Committee. The Basel
Committee set LGD at 45% for senior claims and 75% for subordinated claims. If there is collateral, the LGD is reduced using the comprehensive approach described earlier.
The EAD is calculated similar to the credit equivalent amount required under Basel I. It
includes the impact of netting.
M is usually set to 2.5.
Advanced IRB Approach Banks supply their own estimates of PD, LGD, EAD, and M. PD can be reduced by credit mitigants such as credit triggers subject to a floor of 0.03%
for bank and corporate exposures.
LGD is primarily influenced by the collateral and the seniority of the debt. With supervisory approval, banks can use their own estimates of credit conversion factors
when calculating EAD.
Foundations IRB Approach and Advanced IRB Approach for Retail Exposures The two methods are merged for retail exposures. Banks provide their own estimates of
PD, EAD, and LGD.
There is no maturity adjustment (MA) for retail exposures. The capital requirement is EAD x LGD x (W CD R- PD). Risk-weighted assets are 12.5 x EAD x LGD x (WCDR PD). Correlations are assumed to be much lower for retail exposures than for corporate
exposures.
Example: RWA under the IRB approach
Assume Blue Star Bank has a $150 million loan to an A-rated corporation. The PD is 0.1% and the LGD is 50%. Based on Figure 6, the WCDR is 3.4%. The average maturity of the loan is 2.5 years. Calculate the RWA using the IRB approach and compare it to the RWA under Basel I.
Answer:
b = [0.11852 – 0.05478 x ln(O.OOl)]2 = 0.247
MA = 1/ (1 – (1.5 x 0.247)) = 1.59
risk-weighted assets = 12.5 x 150 x 0.5 x (0.034 0.001) x 1.59 = $49.19 million
Under Basel I, the RWA for corporate loans was 100% or $150 million in this case. Thus, the IRB approach lowers the RWA for higher rated corporate loans, in this case from $150 million to $49.19 million.
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O p e r a t i o n a l R i s k C a p i t a l R e q u i r e m e n t s

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