**After completing this reading, you should be able to: **

- Explain the motivations for introducing the Basel regulations, including key risk exposures addressed, and explain the reasons for revisions to Basel regulations over time.
- Explain the calculation of risk-weighted assets and the capital requirement per the original Basel I guidelines.
- Describe measures introduced in the 1995 and 1996 amendments, including guidelines for netting of credit exposures and methods to calculate market risk capital for assets in the trading book.
- Describe changes to the Basel regulations made as part of Basel II, including the three pillars.
- Compare the standardized IRB approach, the Foundation Internal Ratings-Based (IRB) approach, and the advanced IRB approach for the calculation of credit risk capital under Basel II.
- Compare the basic indicator approach, the standardized approach, and the Advanced Measurement Approach for the calculation of operational risk capital under Basel II.
- Summarize elements of the Solvency II capital framework for insurance companies.

## Motivations for Introducing the Basel Regulations

Financial regulations have developed over the years in response to the stressful periods, which showed the weaknesses of the underlying regulations at that time. For instance, before the intervention of the governments, banks, and insurance companies were created without official approval, and their success or failure depends on their strength of persuading the clients. Moreover, they tried to build a good reputation by ramming support from the prominent people in the community, accumulating a large amount of capital at their conception and constructing prominent buildings.

Financial failures from these institutions were frequent due to not only insolvency but also lack of client confidence. This was clear because when the failures occurred, the clients scrambled to withdraw their funds, which, when the withdrawals caused panic, even the solvent institutions could fail if they could not liquidate their assets on time.

These failures prompted the first regulations, such as financial institutions unifying in case of excess withdrawals. Additionally, early clearinghouses were established, which were partial arrangements for mutual support. The clearing houses had the rights for private inspection and hence monitoring and institutionalizing solvency.

However, the privatization of the clearinghouses’ inspection came with some drawbacks:

- In the case that the panic was too massive, the entities that lacked the power to print money lacked enough resources to support the financial system, and thus the clearinghouses and the private banks were gradually replaced by the central banks as the lenders of the last resort
- The financial crisis proved to have a massive impact on the whole economy by the governments, and thus they made necessary efforts so that the financial institutions are solvent enough to survive the distress times.
- The clients of the failed financial institutions were unhappy when they experienced losses because, apart from fraud, customers complained of biasness and the difficulty of monitoring the safety and soundness of a financial institution.
- Globalization of the financial systems such as international coordination of the regulations accelerated the need for regulation.

International trade emerged in the 1960s and 1970s, which saw growth of the multinational corporation, and thus foreign exchange and capital flow increased. As a result, multinational valued the financial providers in many countries, giving rise to the following problems:

- Huge financial institutions such as international banks linked together such that if one of the entities fails, this would have been amplified to many countries
- The banks and regulators recognized the competitive advantages and disadvantages as a result of the difference in capital requirements in diverse countries.
- There was a need for a sound technical procedure in clearing and settlements, such as delivery times for currencies and inclusion of time zone difference.

Due to these problems, it was evident that official sector cooperation and coordination was the solution. Therefore, the Basel Committee on Banking Supervision (BCBS) was formed in 1974, which was just after the failure of Hertatt Bank.

## The Basel Accord: Basel I

Basel I accord was the specification for capital regulation developed in the late 1980s by the members of BCBS (mostly G10 nations). The accord was published in December 1987, agreed in July 1988, and implemented by the end of 1992. However, by early 2000, it was recognized as a global minimum capital standard.

The Basel I bears no legal ground, but rather countries chose to include Basel I standard through domestic laws and regulations.

### Motivations of Basel 1

The following events triggered the Basel I accord:

- The was rise international financial transactions even after the Herstat Bank fall in the summer of 1974, and there was a common interest among the G10 countries that banks should have enough equity to absorb huge losses.
- There was an imminent competition between the international banks in their respective countries, and since the minimum capital requirements varied substantially across countries, there was a feeling that the banks in countries with minimum capital requirements had a competitive advantage. The BCBS wanted to establish a level playing field.

The key features of the Basel I are the minimum risk-based capital ratio and numerators and denominators of this ratio.

### The Basel-Based Capital Ratios

The BCBS chief goal is to ensure that the financial institutions would have enough assets to remain solvent in time of stress. Therefore, they had to find a way of measuring this sufficiency.

The specification of the minimum capital requirement in terms of the leverage ratio (ration of capital to book value of assets) would undermine the financial institutions with low-risk portfolios and advantage those with high-risk capital ratios. Therefore, BCBS developed a risk-based ratio which is the ratio of capital to risk-weighted assets (RWA) instead.

Additionally, the risk-weighted assets included the assets on the balance sheet according to accounting requirements and off-balance sheet exposures (such as loan commitments) and derivative exposures.

### Ratios and Minimum Values

The requirement of the Basel I was that the consolidated bank should maintain the following ratios:

$$ \cfrac { \text{Tier 1 capital} }{ \text{RWA} } >4\% $$

and

$$ \cfrac { \text{Total capital} }{ \text{RWA} } >8\% $$

Note that the total capital is the sum of Tier 1 and Tier 2 capital, where Tier 2 should not be more than half of the capital.

According to Basel I, Tier 1 capital is defined as common equity and disclosed reserves (retained earnings and types of minority interest in subsidiaries) less goodwill. Some of the later Basel I frameworks include a limited amount of non-cumulative perpetual preferred stock.

The Tier 2 capital consists of:

- Undisclosed reserves and some reevaluation reserves
- Hybrid investments such as subordinated debt
- Loan loss reserves not allocated to particular assets

The proportion of the loan reserves to be included in the capital was lowered from 2% to 1.25% of RWA.

### The Assumptions of BCBS

The assumptions were not put forward by the BCBS committee, but intuitively, the assumptions were:

- Tier I capital was meant to maintain the solvency while Tier 2 capital would avail the resources for the recapitalization of an institution in resolution and lower the impact of failure on depositors
- The general loan reserves were not considered as loss-absorbing capacity to sustain solvency. However, loan reserves were often taken to cover for losses that are already attached to an entity’s portfolio but have not yet occurred.

### The Risk-Weighted Assets (RWA)

The on-balance sheet amount of each asset was made risk-sensitive by multiplying each asset by the percentage weight according to the risk it has. The RWA is defined:

$$ \text{RWA} = \sum _{ \text{i}=1 }^{ \text{N} }{ { \text{W} }_{ \text{i} }{ \text{A} }_{ \text{i} } } $$

Where:

\({\text{W}}_{\text{i}}\) = risk weight

\({\text{A}}_{\text{i}}\) = size of the asset

The weights, according to Basel I accord are as follows:

$$ \begin{array}{c|l} {\textbf{Percentage Weight}} & \textbf{Asset Category} \\ \hline {0\%} & {\text{Cash, claims on OECD governments(such as bonds), and other} \\ \text{instruments with an assured guarantee from and OECD government.}} \\ \hline {20\%} & \text{Claims on OECD banks and OECD public sector institutions.} \\ \hline {50\%} & \text{Uninsured residential mortgages} \\ \hline {100\%} & \text{Other exposures such as Commercial or consumer loans} \\ \end{array} $$

Looking at the risk, weights, the maximum risk weight is 100%. Moreover, the government OECD has 0% weight implying that the government OECD would not default in its obligation.

**Example: Calculating the RWA**

The constituents of an Australian bank are 100 million AUD of American government bonds, 400 million AUD of loans to corporations, 200 million AUD of uninsured residential mortgages, and 150 million AUD of residential mortgages issued by the public sector. What is the value of risk-weighted assets (RWA) based on Basel I accord?

**Solution**

Using the weight ratios under the Basel I accord; the weights are as follows:

- American government bonds = 0%
- Loans to corporations = 100%
- Uninsured residential mortgages = 50%
- Residential mortgages = 20%

So that the RWA is given by:

$$ \text{RWA}=0\%\times100+100\%\times400+50\%\times300+20\%\times250=$600 \text{ million} $$

### Credit Equivalent of Off-Balance Sheet Exposures

As stated earlier, RWA included both the balance sheet exposures and off-balance sheet exposures and other non-traditional on balance sheet exposures such as derivatives. The traditional off-balance sheet exposures were converted to credit-equivalent amount, which is an on-balance sheet equivalent by multiplying one of the credit conversions factors by the risk weights according to Basel I accord (shown by the table above) after multiplying by the conversion factors. The Basel I credit conversion factors are shown below:

$$ \begin{array}{c|l} \hline {\textbf{Credit} \\ \textbf{Conversion} \\ \textbf{Factor}} & \textbf{Off-Balance sheet Asset Category} \\ \hline {100\%} & {\text{Guarantees on loans and bonds, bankers acceptances and equivalents}} \\ \hline {50\%} & {\text{Warrantees and standby letters of credit related to transactions}} \\ \hline {20\%} & {\text{Loan commitments with an original maturity greater than or equal to one year}}\\ \hline {0\%} & {\text{Loan commitments with original maturity less than one year} } \\ \end{array} $$

For example, a $200 million five-year loan commitment converted to 20% ×200=$40 million and then weighted again to get 20%×40=$8 million, which is the credit equivalent amount.

### Credit Equivalent of the Derivative Exposures

In the case of derivative instruments, the Basel I offered two methods of calculating credit equivalent amount: Current Exposure Method and Original Exposure Method. Note that the credit equivalent amount was amended in 1995 to include the maturities of over five years.

**Current Exposure Method,**

The steps of calculating the credit equivalent of the derivative under this method include:

- Compute the current market value of the contract denoted by V. If the value of the contract is negative, then let V=0.
- Add the amount denoted by D, which accounts for the changes in the contract’s future market value:
- Specifically for the interests swaps:
- D = 0 for the maturities of less than one year
- D = 0.5% of the notional value of the swap for the remaining maturities of five years or less
- D = 1.5% of the notional value of the swap for maturities for more than five years.

- For Foreign Exchange Swaps, D was defined by:
- D = 1% of the notional value of maturities less than one year
- D = 5% of the notional value of maturities between one year and five years
- D = 7.5% of the notional value for the maturities greater than five years

- Specifically for the interests swaps:

### Original Exposure Method

The original exposure method was specifically for interest rate and foreign exchange contracts. The steps included:

- The nations could ignore the current market value of the contract and choose whether to utilize the initial or remaining maturity
- For interest-rate contracts, D was defined as:
- 0.5% of the notional value for maturities of less than one year
- 1% for the maturities of between one and two years
- 1% + 1% × INT[M-1] for the maturities greater than two years. Note that INT[X] returns the integer closest to X

- For the foreign exchange contracts
- 2% for the maturities of less than a year
- 5% for maturities between one and two years
- 5% + 3% × INT[M-1] for maturities of more than two years.

- For interest-rate contracts, D was defined as:

It is worth to note that equity and commodity derivatives were not included in Basel I and that the risk weight was determined based on the counterparty while making sure that no weight exceeded 50%

**Example: Calculating the Credit Equivalent of Derivative Equivalent**

The derivative book of American bank consists of $600 million of the notional value of interest swaps with each $200, has a remaining maturity of 0.5, 1.5, and 2.5 years. The market value of swaps is $30 million. Additionally, the book contains $300 million of foreign exchange swaps with each $100 million, has the same maturity description as in interest swaps. We wish to calculate the credit equivalent amount of the Bank’s derivative book?

**Solution**

Using the description given:

$$ \text{V} = 30 $$

And the value for the swaps is

$$ \text{D} = 0\% \times 200 +0.5\% \times 400=$2 \text{ million} $$

And for the foreign exchange swaps is

$$ \text{D} = 1\% \times 200 +5\% \times 400=$22 \text{ million} $$

So that the Credit equivalent is given by:

$$ \text{CE}=30+2+22=$54 \text{ million} $$

### The Market Risk Amendment of 1995 and 1996

The handling of the derivative exposures by the Basel I was basic. However, by 1995 much had changed after the 1987 stock crush, the rising popularity of the VaR, and the existence of developed quantitative market risk management systems. The Market Risk Amendment revolved around the issues of netting and capital for market risks associated with trading activities.

### Netting

In a conventional market, the entities that transact in over-the-counter derivatives which involves the signing of an International Swaps and Derivatives Association. The agreement states that in case one party defaults in its obligations, the transaction of the defaulting party with its counterparty is considered a single transaction. Moreover, depending on the choices in the transactions, the agreement allows bilateral transactions with negative and positive values to offset one another. For instance, if two entities A and B enter an interest swap agreement against interest movements, the overall net exposure in each of the entities is zero if the interest rate does not move, and thus the value of the portfolio is zero.

The initial Basel I did not incorporate the capital credit for netting. Conventionally, the changes in the interest would have an offsetting effect on the market value of two swaps, but Basel I would include an add-on in each swap discouraging the hedging strategy. The reason behind this strategy was that bankruptcy courts had not sufficiently tested the master agreements.

By 1995, the BCBS members were confident that “add-on” agreements would work as required, and thus the 1995 amendment incorporated reductions in the credit equivalent amounts when enforceable bilateral netting agreements were set.

When calculating the equivalent amount, complete netting of the market values of all positions was allowed for each counterparty i, and add-on Dj for the future changes in the value was decreased for each category of derivative j. So, the credit equivalent amount (CEA) is given by:

$$ \text{CEA} = {\text{max}\left( \sum _{ \text{i}=1 }^{ \text{N} }{ { \text{V} }_{ \text{i} },0 } \right) } + \sum _{ \text{j} }^{}{ {\left(0.4\times {\text{D}}_{\text{j}} + 0.6 \times {\text{D}}_{\text{j}} \times \text{NRR}\right) } }$$

Where NRR (Net Replacement Ratio) is defined as:

$$ \text{NRR} =\cfrac { \text{max}\left( \sum _{ \text{i}=1 }^{ \text{N} }{ { \text{V} }_{ \text{i} },0 } \right) }{ \sum _{ \text{i}=1 }^{ \text{N} }{ \text{max}\left( { \text{V} }_{ \text{i} },0 \right) } } $$

Note that \({ \text{max}\left( \sum _{ \text{i}=1 }^{ \text{N} }{ { \text{V} }_{ \text{i} },0 } \right) }\) (numerator) is the market value of positions of type j with netting while \(\sum _{ \text{i}=1 }^{ \text{N} }{ \text{max}\left( { \text{V} }_{ \text{i} },0 \right) } \) (denominator)is the market value with no netting.

NRR is the average across the positions. The add-on and effect of netting differences across different derivatives were ignored.

**Example: Equivalent Credit Amount for Derivatives under the Market Risk Amendment**

Assume that a bank has a portfolio of four derivatives with two counterparties, as shown in the table below:

$$ \begin{array}{c|c|c|c|c|c} \textbf{Counterparty} & {\textbf{Derivative}\\ \textbf{type}} & \bf{\text{Maturity} \\ \text{Period}} & \bf{\text{Notional} \\ \text{Amount}} & \bf{\text{Market } \\ \text{Value}} & {}\\ \hline {1} & \text{Interest rate} & {2} & {100} & {-5} & {0.10\%} \\ \hline {1} & \text{Interest rate} & {2} & {200} & {14} & {0.10\%} \\ \hline {2} & \text{Equity Option} & {3} & {100} & {0} & {10\%} \\ \hline {2} & \text{Wheat Option} & {6} & {200} & {-10} & {5\%} \\ \end{array} $$

What is the value of the credit equivalent of the derivative portfolio?

**Solution**

We know the credit equivalent is given by:

$$ \text{CEA} = {\text{max}\left( \sum _{ \text{i}=1 }^{ \text{N} }{ { \text{V} }_{ \text{i} },0 } \right) } + \sum _{ \text{j} }^{}{ {\left(0.4\times {\text{D}}_{\text{j}} + 0.6 \times {\text{D}}_{\text{j}} \times \text{NRR}\right) } }$$

Now,

$$ {\text{max}\left( \sum _{ \text{i}=1 }^{ \text{N} }{ { \text{V} }_{ \text{i} },0 } \right) } = \text{max} \left(0,9\right) =9 $$

Note that the current exposure portion of the credit equivalent is 9 for counterparty 1 because -5 exposure on the first interest rate is netted against 14 on the second interest rate. Moreover, the current exposure for counterparty 2 is 0 current since exposure cannot be negative (-10).

Now,

$$ \text{NRR}=\cfrac {\text{max}\left( \sum _{ \text{i}=1 }^{ \text{N} }{ { \text{V} }_{ \text{i} },0 } \right) }{ \sum _{ \text{i}=1 }^{ \text{N} }{ \text{max}\left( { \text{V} }_{ \text{i} },0 \right) } } =\cfrac { \text{Current exposure} }{ \text{sum of positive Exposure} } =\cfrac { 9 }{ 14 } =0.6429 $$

The add-on for the potential future exposures is calculated for each derivative

$$ \begin{align*} \text{Interest rate} & =0.10\% \left(100+200 \right)=0.3 \\ \text{Equity Option} & =10\%\times100=10 \\ \text{Wheat Option} & =5\%\times200=10 \end{align*} $$

So,

$$ \begin{align*} & =\sum _{ j }^{ }{ \left( 0.4\times { \text{D} }_{ \text{j} }+0.6\times { \text{D} }_{ \text{j} }\times \text{NRR} \right) } \\ &=\left[0.4\times0.3+0.6\times0.3\times0.6429\right]+\left[0.4\times10+0.6×10\times0.6429\right] \\ & +\left[0.4\times10+0.6×10\times0.6429\right] \\ & =0.2357+7.8574+7.8574=15.95 \end{align*} $$

Therefore:

$$ \text{CEA}=\text{max}\left( \sum _{ \text{i}=1 }^{ \text{N} }{ { \text{V} }_{ \text{i} },0 } \right) +\sum _{ j }^{ }{ \left( 0.4\times { \text{D} }_{ \text{j} }+0.6\times { \text{D} }_{ \text{j} }\times \text{NRR} \right) } =9+15.95=24.95 $$

### Capital for Market Risks Associated with Trading Activities

The initial description of Basel I left out the market risk. Recall that market risk is a potential change in the market value of trading book values.

The Amendment of the Basel I gives two ways to measure market risk: **Standardized Approach and Internal Model Approach.**

The internal-models approach is suitable for the banks with material size trading books because it generated lesser capital requirements because the asset values were assumed to be uncorrelated as they were in a standardized approach.

The standardized explains the following classification of positions separately:

- Foreign exchange
- Commodities
- All types of options
- Equity securities and equity derivatives other than options
- Fixed income securities and interest rate derivatives other than options

Note that the internal models-based method allows the banks to use their own internally developed risk measure as the input to the inputs specified by the regulators where monitoring is incorporated to control the manipulation of internal measures. On the other hand, a standardized approach explained most of the details and was based on observable features of positions such as remaining maturity.

Under both approaches, capital charges were computed distinctively for specific risk (SR) and general market risk (MR) for each of the five categories stated above. The SR and the MR were then summed and then multiplied by 12.5 so that the usual multipliers on RWA could be applied to them. It is worth noting that 12.5 is the inverse of 8% so that the multiplier transforms the capital requirements into an RWA measure with particular attention to total capital. So the capital charges is given by:

$$ \text{Total capital for trading assets}=0.08\times 12.5\sum _{ \text{j}=1 }^{ 5 }{ \left( { \text{MR} }_{ \text{j} }+{ \text{SR} }_{ \text{j} } \right) } $$

For a bank using an internal model-based approach, the bank is obliged to calculate value at risk (VaR) for each asset category. According to the amendment, a 10-day 99% VaR was required on at one year of daily data, usually using the scale √10. Moreover, adjustments for correlations across the asset categories with the based-on direction by national supervisor.

Therefore, the market risk is given by:

$$ \text{MR}=\text{max}\left( { \text{VaR} }_{ \text{t}-1 },\text{m}\times { \text{VaR} }_{ \text{avg} } \right) $$

Where

\({\text{VaR}}_{\text{avg}}\) = average VaR over the past 60 days

m = multiplier that was not less than 3 (but can be more than 3).

Capital for specific risks required for fixed income could be computed using a standardized approach or the bank’s internal models. However, if the internal models were used, the method is similar to that of market risk, but the multiplier is 4 rather than 3, but the bank was required to calculate the capital under the standardized approach.

The 1996 amendment introduced Tier 3 capital, which consisted of unsecured subordinated debt with an initial maturity of at least two years that could be utilized to cover for the market risk requirements. However, only 70% of the market risk capital requirements could be covered by Tier 3 capital.

The 1996 Amendment concentrated on the requirements of the banks using the internal model-based approach. For instance, the amendment required the banks to have sound risk management and independent risk management units.

Moreover, the 1996 amendment required daily backtesting. For each model, the banks were required to compute one-day 99% VaR for each of the significant 250 days and draw a comparison between the actual loss for the day and the VaR. If for a day, the actual loss is larger than VaR, it is termed as an **exception**. For less than five exceptions, the multiplier was taken to be 3. However, more than five exceptions, the supervisor has an option to choose larger multipliers, but for ten or more exceptions, a multiplier of 4 was recommended.

## The Basel Accord: Basel II

By the mid-1990s, some supervisors had become alarmed that the Basel I was not risk-based enough as it claimed to be. For instance, the 100% risk weight included exposures that put the corporations to a variety of risks.

Moreover, the banking crisis in Nordic countries proved that the issues could occur in banks with sound chaptalization. Additionally, there was an advancement in the market and credit risk quantification and management from 1987 onwards, which incentivized for accurate risk weighting and risk management at all hierarchies of banking institutions.

As a result, Basel II was initiated in the late 1990s, and the finalized version was published in 2004 and later revised several.

### Significant Innovations of the Basel II Accord

Basel II accord contained the following significant improvements:

- Basel II required the capital for the operational risk, in addition to credit and market risk
- The risk weights formulas for the credit risk were to be based on modern credit risk management concepts and the bank’s internal risk measures
- The Basel II introduced the three pillars: minimum capital requirements (Pillar 1), Specific requirements for supervision related to capital and risk management (Pillar 2) and required public disclosures (Pillar 3)
- Basel II introduced repeated utilization of Quantitative Impact Studies (QIS) to improve the structure if the accord. In a given QIS, banks were required to provide the crucial data which was to be analyzed by the supervisors.

### The Basel II Pillars

The Basel II pillars formed a basis for common national practices. Pillar 2 mandated the supervisors to have the banks posses more than the minimum amount of capital and the internal capital adequacy process (ICAAP) that considers their risk profile. The supervisors were required to act on the earlier signs that the bank’s capital would go below the minimum by recommending appropriate actions. Moreover, the supervisors were required to motivate the banks to improve the risk management systems and adequately address the deficiencies.

Pillar 3 advocated for ore qualitative and quantitative disclosures with an aim that the market participants’ pressures would assist in improving bank’s practices. It required qualitative disclosures such as corporate structure, the applicability of the Basel accord, and accounting procedures. The quantitative disclosures include the features of the bank’s capital, risk measures, and exposures.

Some banks found the pillars challenging to interpret, and the disclosure practices were not uniform over the years, but Basel emphasized them while providing more clarity.

### Capital for the Credit Risk

Supervisors were wary that the banks could distort the interval risk measure to lower the required capital due to a lack of sufficient supporting data and analysis in Basel II. Therefore, the negotiators introduce three methods of determining minimum capital requirements to cover credit risk:

- The standardized approach: similar to Basel I, it included increased sensitivity of risk weights to credit quality for borrowers with external ratings.
- The Foundation Internal Ratings-Based (IRB) method: In this method, risk weights were responsive to internal measures of default probability with the included regulatory-specified loss given default parameters.
- The Advance IRB approach. In this method. Risk weights were sensitive to internal measures of default probability, exposure to default, and loss given default.

### The Standardized Approach

This method was directed to banks with internal risk measures and risk management systems that were insufficient to support IRB methods. The risk weights were sensitive to fluctuation in risks. Unlike Basel I standardized approach whose risk weights were based on asset type and the country to which obligor belongs, Basel II risk weights depended on the type of the obligor, ratings of some obligors, and asset types of some obligors. The table below shows some examples of some risk weights

$$ \begin{array}{c|c|c|c|c|c|c} \textbf{Obligation of:} & {\textbf{AAA to}\\ \textbf{AA-}} & \textbf{A+ to A-} & \bf{\text{BBB+ to} \\ \text{BBB-}} & \bf{\text{BB+ to} \\ \text{BB-}} & \textbf{B+ to B-} & \textbf{Unrated}\\ \hline \text{Countries} & {0} & {20} & {50} & {100} & {150} & {100} \\ \hline \text{Banks} & {20} & {50} & {50} & {100} & {150} & {50} \\ \hline \text{Corporations} & {20} & {50} & {100} & {100} & {150} & {100} \\ \end{array} $$

As seen above, the weights were somewhat severe for the banks as compared to Basel I, but the severity could be lowered at the national discretion. For instance:

- A supervisor could decide to assign a risk weight of 0 on the holding of the bank on its sovereign debt issued in the home country’s currency. As a result, other nations could follow suit and assign 0 risk weight on that particular sovereign.
- The claims that were issued by banks posed a risk weight of one category less favorable than the sovereign or risk-weighted based on the bank’s ratings. All these risk weights were limited to 100%.

### Adjustment for Collaterals in Basel II Standardized Approach

The Basel II standardized included two methods for adjusting for collateral: a simple approach and a comprehensive approach.

The simple approach is similar to Basel I in that, the risk weight of a counterparty could be substituted with the risk weight of a collateral for the proportion of the exposure covered by the collateral. The minimum weight of a collateral weight was 20%, except the collateral, which was a sovereign debt collateral expressed in the same currency as the exposure.

The comprehensive approach required the variations of the exposures and the collaterals amounts to incorporate the possible deviations of value. That is, the risk weight of a collateral was applied to a minimum collateral, and the counterparty’s risk weight was applied to the remaining exposure. Moreover, netting was applied distinctively to exposures and collaterals, and either of the Basel or approved internal models could be applied to facilitate the adjustments.

### The IRB Approach

The IRB model is based on the Gordy’s (2003) one factor Gaussian copula model. Gordy postulated that, given a well-diversified portfolio, there exists a positive link between the default probability of an obligor and the obligor’s contribution to the capital required to cap the probability of the portfolio losses surpassing the loss distribution percentile.

The Basel committee chose a one-yar time horizon for the credit losses and desired \({99.9}^{\text{th}}\) percentile credit loss distribution. So the formula for the capital required for the credit risk is given by:

$$ \text{Capital}=\sum \left[\text{EAD}_\text{i}\times \text{LGD}_\text{i} \times \text{DR}{99.9}_{\text{i}} \right]-\text{EL} $$

Where

\({\text{EAD}}_{\text{i}}\)= the exposure at default for an asset i , that is, amount expected to be owed by the counterparty on an asset i in case of a default

\({\text{LGD}}_{\text{i}}\)= expected loss given a default for asset i (expected proportion of \({\text{EAD}}_{\text{i}}\) to be lost)

\(\text{DR}{99.9}_{\text{i}}\)= default rate at the \({99.9}^{th}\) percentile for a large portfolio of assets of category i. This quantity is defined as:

$${ \text{DR}99.9 }_{ \text{i} }=\text{N}\left[ { \text{N} }^{ -1 }\left( { \text{PD} }_{ \text{i} } \right) +\cfrac { \sqrt { \rho } { \text{N} }^{ -1 }\left( 0.999 \right) }{ \sqrt { 1-\rho } } \right] $$

EL=expected loss (annual credit loss) on a portfolio and defined as:

$$ \text{EL}=\sum \left[\text{EAD}_\text{i}\times \text{LGD}_\text{i} \times \text{PD}_{\text{i}} \right] $$

Note that the capital was expressed in terms of dollars.

### Types of IRB Model

The Basel Committee did not consider the loan reserves as part of Tier 1 capital, but the loan reserves were taken to be approximately equivalent to the expected loss.

Therefore, the committee decided to make the capital, the function of the unexpected losses (net expected losses), and in the case where the loan reserves were less than expected losses (EL), the capital is reduced for the shortfall.

Consequently, the Basel Committee was able to state the loss percentile and the asset correlation (\(\rho\)) for each asset category, and thus the contribution of each asset to the capital was be based only on the bank’s estimates of EAD, LGD, and PD for a particular asset.

There were two types of IRB model:

- Foundation IRB: the bank provided PD (probability of default) only, and the Basel accord stated the values of EAD and LGD for each given type an asset. This method was suitable for large banks since most large banks had internal rating systems that could be used to determining PD
- Advance IRB: the banks specified all the three variables. Most banks did not use this method due to the limited availability of the EAD and LGD data.

**Bank, Corporate and Sovereign Exposure Under IRB**

Basel II assumes that the correlation (ρ) and the probability of default (PD) depends on Lopez’s (2004) model, which defines the correlation as:

$$ \rho =0.12\left[ \cfrac { 1-{ \text{e} }^{ -50\text{PD} } }{ 1-{ \text{e} }^{ -50 } } \right] +0.24\left[ 1-\cfrac { 1-{ \text{e} }^{ -50\text{PD} } }{ 1-{ \text{e} }^{ -50 } } \right] $$

Lopez’s model gives the relationship between the average asset correlation, the firm’s PD, and the asset size. Looking at the formula above, the average asset correlation decreases as PD increases confirming that the default for high-risk borrowers are usually idiosyncratic while that of middle-class borrowers tend to default when the aggregate economy is in distress. Moreover, the safest borrowers also idiosyncratic, but their default rates (DR) are ignored since they are usually minimal.

### The Maturity Adjustment for Bank, Corporate and Sovereign Exposure Under IRB

The computation of capital for banks, corporate, and sovereign exposures incorporates the maturity adjustment to account for the assets with maturity more than one year of remaining maturity, which usually remains on the balance sheet at the end of the loss-forecasting period and may have lowered in credit quality. The Maturity adjustment is given by:

$$ \text{MA}=1+\cfrac { \text{b}\left( \text{M}-2.5 \right) }{ 1-1.5\text{b} } $$

Where

MA=maturity of the asset

\(\text{b}=\left[0.11852-0.05478 \text{ln(PD)} \right]^{2}\)

Now, recall that the Basel II expressed the required capital in terms of RWA, the RWA for the banks, corporations and the sovereign exposures is given by:

$$ \text{RWA}=12.5\times \text{EAD} \times \text{LGD}\times \text{(DR-PD)} \times \text{MA} $$

### Example: Calculating RWA under Basel II

An American bank’s assists consist of $200 million BB-rated drawn loans. The MA is estimated to be 1.25. The probability of default is estimated to be 0.02, and the LGD is 40%, and DR is estimated to be 0.15. What is the RWA for the bank with regard to the Basel II accord?

**Solution**

We know that:

$$ \begin{align*} \text{RWA} & =12.5\times \text{EAD} \times \text{LGD}\times \text{(DR-PD)} \times \text{MA} \\ & =12.5\times 200\times 0.4\times \left(0.15-0.02\right)\times 1.25 \\ & =$162.5 \text{ million} \end{align*} $$

### Retail Exposures Under IRB

The retail exposures were calculated similarly to that of advanced IRB only that there is no maturity adjustment. Moreover, a set of three correlations are used: ρ=0.15 for the residential mortgages, ρ=0.04 for qualifying assets ( such as credit card balances) and other retail assets, the correlation is defined as:

$$ \rho =0.03\left[ \cfrac { 1-{ \text{e} }^{ -35\text{PD} } }{ 1-{ \text{e} }^{ -35 } } \right] +0.16\left[ 1-\cfrac { 1-{ \text{e} }^{ -35\text{PD} } }{ 1-{ \text{e} }^{ -35 } } \right] $$

Looking at the formula, it evident to see that the correlations are lower for retail than wholesale exposures.

### Example: Calculating RWA for Retail Exposures Under Basel II

An American bank’s assets consist of $200 million BB-rated drawn loans. The probability of default is estimated (PD) to be 0.02, the LGD is 40%, and DR is estimated to be 0.10. What is the RWA for the bank with regard to the Basel II accord?

**Solution**

Recall that retail exposures were calculated similarly to that of advanced IRB only that there is no maturity adjustment. So,

$$ \begin{align*} \text{RWA} & =12.5\times \text{EAD} \times \text{LGD}\times \text{(DR-PD)} \\ & =12.5\times 200\times 0.40\times \left(0.10-0.02\right)=$80 \text{ million} \end{align*} $$

### Credit Mitigants other than Collateral

A credit substitution was utilized for arrangements such as guarantees and the credit default swaps. This method involved substituting the credit rating of the guarantor for that of the obligor in the capital calculations, until the amount covered by the mitigant is reached.

However, this method has low sensitivity to actual loss occurrence because it involves double default (guarantor and the borrower. Nevertheless, Basel II assumes low correlations of the wholesale counterparty defaults, and thus the frequency of the double defaults low.

Alternatively, the Basel Committee amendment in 2005 allowed capital without the mitigant to be multiplied by 0.15+160\({\text{PD}}_{\text{g}}\) where \({\text{PD}}_{\text{g}}\) is defined as the one-year PD of the guarantor

### Capital for the Operational Risk

According to Basel, operational risk is the risk that occurs due to inadequate or failed internal processes, people and systems or from external events. The Basel II implemented two three methods of determining the capital required for the operational risk:

- Basic Indicator Approach: this method computes the capital for the operational risk as the 15% of the bank’s average annual gross income over the past three years while ignoring years that resulted in negative gross income.
- Standardized approach: this method is similar to the basic indicator method, but the multipliers are distinct for each business line, and then the average is calculated.
- Advanced Measurement Approach (AMA): this method involves using the internal models to compute a one-year 99.9% VaR (measure of operational risk losses at the 99.9th percentile). The operational risk capital is the 99.9% VaR less expected operational losses.

### Example: Calculating the Required Capital for Operational Risk under Standardized and Basic Indicator Approach

The sample of four business lines, their corresponding multipliers and gross income (in millions) is given in the table below:

$$ \begin{array}{c|c|ccc} \textbf{Business Line} & \textbf{Multiplier}& \textbf{Annual} & \textbf{Gross} & \textbf{Income} \\ {} & {} & \text{Year 1} & \text{Year 2} & \text{Year 3} \\ \hline {\text{Retail Banking}} & {13\%} & {10} & {20} & {10} \\ \hline {\text{Asset Management}} & {14\%} & {10} & {10} & {20} \\ \hline {\text{Trading and Sales}} & {19\%} & {10} & {-50} & {30} \\ \hline {\text{Corporate Finance}} & {18\%} & {50} & {30} & {60} \\ \end{array} $$

What is the value of the required capital for operational risk under standardized and Basic Indicator approaches?

**Solution**

__Standardized approach__

Under this method, it is similar to the basic indicator method, but the multipliers are distinct for each business line. So,

$$ \begin{array}{l|c|ccc|ccc} \textbf{Business} & \textbf{Multiplier}& \textbf{Annual} & \textbf{Gross} & \textbf{Income} & {} & {\textbf{Capital}} & {} \\ \textbf{Line} & {} & \text{Year 1} & \text{Year 2} & \text{Year 3} & \text{Year 1} & \text{Year 2} & \text{Year 3} \\ \hline {\text{Retail} \\ \text{Banking}} & {13\%} & {10} & {20} & {10} & {1.3} & {2.6} & {1.3} \\ \hline {\text{Asset} \\ \text{Management}} & {14\%} & {10} & {10} & {20} & {1.4} & {1.4} & {2.8} \\ \hline {\text{Trading and} \\ \text{Sales}} & {19\%} & {10} & {-50} & {30} & {1.9} & {-9.5} & {5.7} \\ \hline {\text{Corporate} \\ \text{Finance}} & {18\%} & {50} & {30} & {60} & {9.0} & {5.4} & {10.8} \\ \hline {\textbf{Sum}} & {} & {} & {}& {} & \textbf{13.6} & \textbf{-0.1} & \textbf{20.60} \\ \end{array} $$

Note that the negative capital offsets the positive capital within a given year and thus ignored. So the required capital under the standardized approach is given by:

$$ \cfrac { 1 }{ 2 } \left[ 13.6+20.6 \right] =17.10 million $$

__Basic Indicator Approach__

Under this method, computes the capital for the operational risk as the 15% of the bank’s average annual gross income over the past three years while ignoring years that resulted in negative gross income.

So,

$$ \begin{array}{c|ccc} \hline \textbf{Business Line} & \textbf{Annual} & \textbf{Gross} & \textbf{Income} \\ {} & {\text{Year 1}} & \text{Year 2} & \text{Year 3} \\ \hline {\text{Retail Banking}} & {10} & {20} & {10} \\ \hline {\text{Asset Management}} & {10} & {10} & {20} \\ \hline {\text{Trading and Sales}} & {10} & {-50} & {30} \\ \hline {\text{Corporate Finance}} & {50} & {30} & {60} \\ \hline {\textbf{Sum}} & \textbf{80} & \textbf{10} & \textbf{120} \\ \end{array} $$

Note that the multiplier column has been excluded since we do not need it here. Therefore, the required capital for the operational risk is given by:

$$ 0.15\left[ \cfrac { 80+10+120 }{ 3 } \right] =10.5 \text{ million} $$

**Detailed Information on AMA Approach**

The banks that decide the AMA approach are required to approximate the distribution of the operational risk losses in seven classes that include both the estimates of occurrences and severity of the loss events. The classes of operational losses are Clients, Products and Business Practices; Execution, Delivery and Process Management; External Fraud; Internal Fraud; Damage to Physical Assets; Employee Practices and Workplace Safety; Business Disruption and System Failures.

Although banks use different AMA approaches, the most used methods are:

- Parametric and Monte Carlo Approach: the data is used. Such data are the probability distribution of occurrences (such as Poisson) and severity (such as Weibull). These distributions are used to simulate a large number of loss observations, and then the 99.9th percentile data is obtained.
- Scenario Analysis: a relatively small number of scenarios in which losses are experienced is generated, and operational loss is measured in each scenario for each class of operational loss. Finally, the 99.9th percentile loss is read.

Scenario analyses are merited to creating an informative scenario, and they look into the future (forward-looking). However, the data used is small, and thus generating the 99.9th percentile loss is a challenging task.

### Solvency II

The Minimum capital requirements for insurance companies are present in many countries. However, there are no international standards, but the United States and the European Union have implemented come complex standards.

The US-based National Association of Insurance Commissioners (NAIC) enacted capital requirements that predicted some features of Basel II. Moreover, capital requirements on the risky assets depended on the ratings given by NAIC on each asset, which was additional to capital requirements on the liabilities. The insurance level was at the state level, where most of the states have implemented these requirements.

In the European Union, the European Insurance and Occupational Pensions Authority (EIOPA) regulates the insurance companies. The first capital requirement at the EU level was termed Solvency I, which was later replaced by Solvency II.

The Solvency II is similar to Basel II in many aspects. For instance, the capital requirements are based on the one year, 99.5% VaR and have three pillars: quantitative requirement, internal governance and supervision, and disclosure and transparency. Moreover, when underwriting the risks, market, credit, and operational risk are taken into consideration, which is further segmented into risk originating from life, property and casualty, and health insurance.

Solvency II also borrows some Basel III elements, such as it requires controls on the capital, which above the minimum. In the case, an insurance company breaks the Solvency II minimum capital requirement (MCR), the supervisors are advised to stop the stressed firm from accepting new policies or put into resolution. A firm put in resolution means that it can be sold to a stronger firm or liquidated.

The required buffer above the MCR is provided by the Solvency Capital Requirement (SCR), which is less than MCR. In case the SCR requirement is breached, the concerned insurance company should give a detailed plan of recapitalization, and more requirements might be imposed by the supervisor.

It is worth noting that Solvency II uses both standardized and internal model-based approaches to compute SCR. However, the models used must take into consideration the following factors:

- The data used and methods used should efficient
- The model employed must be utilized in real business decision making
- The risk assessment must be calibrated based on the target criteria set by the regulator.

Solvency II can be made to fruition through a combination of Tier 1, Tier 2, and Tier 3 capital.