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In this chapter, the motives behind the introduction of Basel regulations will be explained. The chapter will further explain the key risk exposures addressed and the reasons why the Basel regulations have been revised over time. In addition, there will be an explanation of how risk-weighted assets are computed and the capital requirements per original Basel I guidelines.

The major elements of the two available options to compute market risk will be described and this will include a description of the covered risks. There will also be VaR and capital charge computations via the internal models approach, plus an explanation on the guidelines of VaR backtesting.

The major elements of the standardized approach, foundation IRB approach, and advanced IRB approach of credit risk computation will be discussed. Furthermore, the basic indicator approach, the standardized approach, and the advanced measurement approach of operational risk computation will be studied.

The chapter will include a description of the key elements of the three Basel II pillars. Finally, there will be a comparison between the two approaches of SCR computation in Solvency II, namely: the standardized approach and the internal models approach.

Making sure that enough capital is available to cover the risks taken is the main purpose of bank regulations. Also, the likelihood of default should be considered despite governments seeking to lower the default probability of banks to a minimum. A stable economic environment will, therefore, be created and individuals and businesses will have more confidence in the banking systems.

Regulations are critical in increasing bank capital and informing banks of the risks they are exposed to. Deposit insurance programs are provided by governments for the protection of depositors, without which there will be difficulty in attracting deposits that took excessive credit risk relative to their capital base. The deposit insurance will always be accompanied by regulations regarding capital requirements.

The risk of collapse of the entire financial system due to the failure by a large bank, leading to other large banks failing, is a major concern of governments. This risk is called a systemic risk. Systemic risk leads to very difficult decisions by governments: if the institution is allowed by the government to fail, then the entire financial system is at risk. On the other hand, a bailout by the government may send the wrong signals in the market as large institutions may become less vigilant in controlling risks.

The regulators in the years preceding 1988 had a tendency of setting minimum levels for the ratio of capital to total assets. Questions about the adequacy of capital levels were asked due to the huge exposures created by loans from major international banks to less developed countries.

The sophistication in the types of transactions engaged in by banks was becoming more and more prevalent, therefore posing another challenge. There was a rapid growth in OTC derivatives markets for many products, which had for effect to increase the credit risk taken by banks.

The regulators realized that the value of total assets was no longer a good indicator of the total risks being taken and a more complex approach was necessary.

In 1974, the Basel Committee was formed consisting of representatives from various first world countries in Basel, Switzerland, under the patronage of the Bank for International Settlements.

The first attempt of an international risk-based standard to be set for capital adequacy was the 1988 BIS accord. The accord was a huge achievement despite being criticized as being too simple and somewhat arbitrary. The 12 members of the Basel Committee signed the accord, thereby paving the way for significant increments in the resources banks dedicated to measuring, understanding, and managing risks. The Cooke Ratio, or the amount of capital a bank has as a percentage of its total risk-adjusted assets, was a key innovation in the 1988 accord.

Both on-balance-sheet and off-balance-sheet credit risk exposures are taken into consideration in this ratio. The basis of the Cooke Ratio is called the bank’s total risk-weighted assets or the risk-weighted amount. This is an evaluation of the bank’s total credit risk exposure. The following are the three categories of credit risk exposures:

- The ones due to on-balance sheet assets;
- Those arising for off-balance sheet items; and
- Those arising from OTC derivatives.

In the first category, a risk weight is assigned to each on-balance-sheet asset and reflects its credit risk. Virtually, a zero risk and zero risk weight exist for government-issued cash and securities in OECD countries.

The total risk-weighted assets for \(N\) on balance-sheet items is given by:

$$ \sum _{ i=1 }^{ N }{ { w }_{ i }{ L }_{ i } } $$

Where \({ L }_{ i } \) is the principal amount of the ith item and \({ w }_{ i }\) is its risk weight.

In the second category, bankers’ acceptances, loan commitments, and guarantees are included. Using a conversion factor to the instrument’s principal amount is a way to compute a credit equivalent amount. A conversion factor of 100% exists for instruments with a credit perspective similar to loans. Other instruments have lower conversion factors.

In the third category, the credit equivalent amount for an OTC derivative is computed as follows:

$$ max\left( V,0 \right) +aL $$

Where \(V\) is the current value of the derivative to the bank, \(a\) is the add-on factor, and \(L\) is the principal amount.

To compute the risk-weighted assets, the product of the credit equivalent amount from the second or third category of exposures and the risk weight for the counterparty is determined.

A bank with \(N\) on-balance-sheet items and \(M\) off-balance-sheet items has its total risk-weighted assets computed as follows:

$$ \sum _{ i=1 }^{ N }{ { w }_{ i }{ L }_{ i } } +\sum _{ j=1 }^{ M }{ { W }_{ j }^{ \ast } } { C }_{ j } $$

Where \({ L }_{ i }\) is the principal of the \({ i }^{ th }\) on-balance-sheet asset, \({ w }_{ i }\) is the risk weight of the asset, \({ C }_{ j }\) is the credit equivalent amount for the \({ j }^{ th }\) derivative, and \({ W }_{ j }^{ \ast }\) is the risk weight of the counterparty for this \({ j }^{ th }\) item.

Capital should be kept to at least 8% of the risk-weighted assets as required by the accord. The following are the 2 components of the capital:

**Tier 1 capital:**is made up of items like equity and noncumulative perpetual preferred stock.**Tier 2 capital:**is also called the supplementary capital and is made up of instruments like: cumulative perpetual preferred stock and subordinated debt with an original life of more than five years.

Due to its ability to absorb losses, the most critical capital type is equity capital. A bank becomes insolvent if its equity capital is surpassed by losses, otherwise, it can continue as a going concern. Tier 2 capital is relevant in case losses exceed equity capital since it provides a cushion for depositors.

At least 50% of the required capital should be tier 1, and 2% of the risk-weighted assets should be common equity, as a requirement by the accord.

A report containing 20 risk management recommendations was published by a working group from various professions for dealers and end-users of derivatives and another 4 recommendations were distributed for legislators, regulators, and supervisors. The following points summarize the crucial recommendations briefly:

- A clear definition and approval of the firm’s policies on risk management by the senior management at the board of director’s level.
- At least once daily, the derivatives position should be marked-to-market.
- A consistent measure should be applied by derivatives dealers to evaluate the market risk and set limits to the market risks that are taken.
- The potential losses under extreme market conditions should be determined by stress tests carried out by the derivatives dealers.
- An independent risk management function should be set up so that it is independent of the trading position.
- Assessment of the credit exposures due to derivatives trading should be based on the current replacement value of positions that exist and potential future replacement costs.
- The aggregation of the credit exposure to a counterparty should be in a way that reflects enforceable netting agreements.
- There should independence for individuals who set credit limits and those involved in trading.
- Both the expenses and benefits of credit risk mitigation techniques should be carefully assessed by dealers and end-users.
- The responsibility of derivatives trading, supervision of the trading, and so on should be left to individuals with the appropriate skills and experience.
- Adequate systems should be in place to capture, process, and settle data.
- For the consistency of the income recognition treatment between managed instruments and risks, there should be accountability by dealers and end-users for derivatives transactions applicable in risk management.

The ISDA (International Swaps and Derivatives Association) master agreement covering derivatives trades of participants in the OTC derivatives markets have been originally signed by the market participants. Netting is the clause in the master agreement stating that “all transactions are considered as a single transaction in the event of a default.”

Credit risk can be substantially reduced by netting since a default on a transaction covered in the master agreement by one firm implies a default on all transactions covered in the master agreement.

Let a financial institution possess a portfolio with \(N\) derivatives outstanding with a particular counterparty, and the current value of the \(i\)th derivatives be \({ V }_{ i }\). Therefore, in case of a default, the exposure of the financial institution without netting is given as:

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

With netting:

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

This implies that the exposure without netting is the payoff from a portfolio of options and, with netting, the exposure is the payoff from an option on a portfolio.

For a portfolio of derivatives with a counterparty under the Accord, the credit equivalent amount in the 1988 Basel agreement was:

$$ \sum _{ i=1 }^{ N }{ \left[ max\left( { V }_{ i },0 \right) +{ a }_{ i }{ L }_{ i } \right] } $$

The 1988 Accord was later modified allowing banks to minimize their credit equivalent totals when the Bilateral Netting Agreements are enforceable. A computation of the net replacement ratio (NRR) was the first step. The ratio of the current exposure with netting to without netting is:

$$ NRR=\frac { max\left( { \Sigma }_{ i=1 }^{ N }\left( { V }_{ i },0 \right) \right) }{ { \Sigma }_{ i=1 }^{ N }\left( { V }_{ i },0 \right) } $$

The following is the modification of the credit equivalent amount:

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

A consultative proposal was issued in 1995 to amend the 1988 accord, by the Basel Committee. It was implemented in 1998 and sometimes referred to as BIS 98. It ensures that capital is kept for the market risks linked to trading activities.

The practice of daily valuation of assets and liabilities via models calibrated to prevailing market prices is known as marking-to-market and is sometimes referred to as fair value. Fair value accounting should be applied for all assets and liabilities held for trading purposes.

The credit risk capital charge in the 1988 accord continued to apply to all on-balance sheet and off-balance sheet items in the trade book under the 1996 accord. A capital charge for the market risk associated with all items in the trading book was introduced by the capital amendment.

A standardized approach for evaluating the market risk’s capital charge was outlined in the 1996 amendment. The capital was separately assigned by the standardized approach to each of the debt securities, equity securities, foreign exchange risk, commodities risk, and options. An internal model-based approach for settling market risk capital could be used by complex banks with well-established risk management functions. A VaR measure was therefore computed and converted into capital requirement with a formula specified in the 1996 Amendment.

The calculation of the VaR measure applied in the internal model used ten-day time horizon and a 99% confidence level. The capital requirement is given as:

$$ max\left( { VaR }_{ t-1 },{ m }_{ c }\times { VaR }_{ avg } \right) +SRC $$

The multiplicative factor is given as \({ m }_{ c }\), \(SRC\) is a specified risk charge, \({ VaR }_{ t-1 }\) is the previous day’s VaR, and \({ VaR }_{ avg }\) is the average VaR over the past sixty days.

The risks relating to movements in broad market variables are covered by the first term in the above equation. The risks relating to specific companies are covered by the second term, \(SRC\).

Historical simulation is the most popular VaR computation method. In the first instance, a one day 99% VaR will be calculated by banks almost invariably. When the 1996 amendment was formulated, the 10-day 99% VaR could only be computed as the product of \(\sqrt { 10 } \) and the one-day 99% VaR.

Consider the next SRC. A corporate bond is one security that gives rise to an SRC. The interest rate risk, captured by the first term in the equation, and the credit risk of the corporation issuing out the bond, also captured by the SRC, are the two components to the risk of this security.

Capital is computed by regulators by using a multiplicative factor to the 99% VaR which must be involved in the internal model for the SRC. The multiplicative factor must be greater than 4 and the resultant capital not less than 50% of the standard approach-given capital.

After the implementation of the 1996 Amendment, the sum of the credit risk capital equal to 8% of RWA and the market risk capital, was the total capital a bank was required to keep.

$$ Total\quad Capital=0.08\times \left( credit\quad risk\quad RWA=market\quad risk\quad RWA \right) $$

The one day 99% VaR should be computed by a bank to be back-tested over the previous 250 days as a requirement by the BIS Amendment. The bank’s current procedure for VaR estimation for each of the most recent 250 days should be applied.

\({ m }_{ c }\) is normally set to equal 3 in case the number of expectations in the course of the previous 250 days is below 5. The multiplier should be considered but not necessarily applied in the event that changes in the bank’s positions during the day results in expectations.

A risk weight of 100% should be possessed by all loans to a corporation, as required under the 1988 Basel Accord. There was also no model of default correlation in Basel I.

Before the implementation of new rules proposed by the Basel Committee in June 1999, various Quantitative Impact Studies (QIS) were done to test them by computing the required capital amount had the rules been in place.

The application of the Basel II capital requirements was to internationally active banks. The Basel II regulated all banks and securities firms in Europe. The following three pillars are the basis of Basel II:

- Minimum Capital Requirements;
- Supervisory Review; and
- Market Discipline.

Credit risk of counterparties should be mirrored in the new way to calculate the minimum capital requirement in the banking book, under pillar 1. To convert the capital requirement for a particular risk into an RWA-equivalent, it is multiplied by 12.5% in case a particular risk’s capital requirement is directly computed rather than involving RWAs.

Therefore:

$$ Total\quad Capital=0.08\times \left( Credit\quad Risk\quad RWA+Market\quad Risk\quad RWA+Operational\quad Risk\quad RWA \right) $$

The supervisory review processes and both the qualitative and quantitative processes of managing risks are covered under pillar 2. Making sure that the bank has a process in place to ensure the maintenance of capital levels is the duty of the supervisor. Furthermore, the supervisors are required to encourage banks to develop and apply risk management techniques that are more adequate and evaluate them.

In the third pillar, banks are required to disclose more information about capital allocation and risks taken. Added pressure is therefore pressed on banks for sound risk management decisions to be made should shareholders and potential investors have more data on those decisions.

The following three approaches are specified by Basel II, for credit risk:

- The standardized approach;
- The Foundation Internal Ratings Based (IRB) approach; and
- The advanced IRB approach.

However, only the IRB approach is applicable in the U.S.

Apart from the computation of the risk weights, the standardized approach is similar to Basel I. Under Basel II, the OECD status of a bank or country is no longer considered critical. The range of a country’s risk weight exposures is from 0% to 150%, and an exposure’s risk weight to another bank or a corporation ranges from 20% to 150%.

For bank claims, the rules are complex and the base capital requirements can be chosen by the national supervisors of the country in which the bank is incorporated. Should the national supervisors decide to apply some rules, claims can be treated with a maturity of less than three months, hence leading to another complication.

A risk weight of 75% should be used under the standard rule for retail lending. The risk weight should be 35% in the event that claims are secured by residential mortgages.

The VaR computed via a one-year time horizon and a 99.9% confidence interval is the capital requirement basis for regulators. The pricing of a financial institution’s products covers the expected losses. The required capital is thus the difference between the VaR and the Expected Loss.

The one-factor Gaussian copula model of time to default is applied in the VaR computation. Consider a bank having a very large number of obligors with the ith obligor having a one-year likelihood of default equal to \({ PD }_{ i }\). \(\rho \) is the copula correlation between each pair of obligors. Therefore:

$$ WCD{ R }_{ i }=N\left[ \frac { { N }^{ -1 }\left( P{ D }_{ i }+\sqrt { \rho { N }^{ -1 }\left( 0.0999 \right) } \right) }{ \sqrt { 1-\rho } } \right] $$

\(WCD{ R }_{ i }\) is the worst-case default rate defined for the bank to be 99.9% certain it won’t be exceeded over next year for the \(i\)th counterparty. For a large portfolio of instruments with similar \(\rho\), the one-year 99.9% VaR in a one-factor model is approximately:

$$ \sum _{ i }^{ }{ { EAD }_{ i }\times { LGD }_{ i } } \times { WCDR }_{ i } $$

Where \({ EAD }_{ i }\) is the \(i\)th counterparty’s exposure at default and \({ LGD }_{ i }\) is the \(i\)th counterparty’s loss given default.

The expected loss from a default is given by:

$$ \sum _{ i }^{ }{ { EAD }_{ i }\times { LGD }_{ i } } \times { PD }_{ i } $$

Since the capital requirement is in excess of the 99.9% worst-case loss over the expected loss, then we have:

$$ \sum _{ i }^{ }{ { EAD }_{ i }\times { LGD }_{ i } } \times \left( { { WCDR }_{ i }-PD }_{ i } \right) $$

\(PD\) is the default probability of a counterparty within a year, \(EAD\) is the exposure at default, and \(LGD\) is the loss given default or the exposure that is lost in the event of a default.

Basel II assumes a relationship between the correlation parameter, \(\rho \), and the likelihood of default based on empirical research, in the case of corporate, sovereign, and bank exposures.

Therefore:

$$ \rho =0.12\frac { 1-exp\left( -50\times PD \right) }{ 1-exp\left( -50 \right) } +0.24\left[ 1-\frac { 1-exp\left( -50\times PD \right) }{ 1-exp\left( -50 \right) } \right] $$

As a result of \(exp\left( -50 \right)\) being a very small figure:

$$ \rho =0.12\left( 1+{ e }^{ -50\times PD } \right) $$

An increase in \(PD\) leads to a decrease in \(\rho\). The counterparty’s capital requirement formula is given as:

$$ EAD\times LGD\times \left( WCDR-PD \right) \times MA $$

The \(MA\) is the maturity adjustment and is defined as follows:

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

With:

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

\(M\) is the maturity of the exposure. Furthermore:

$$ RWA=12.5\times EAD\times LGD\times \left( WCDR-PD \right) \times MA $$

\(PD\) is supplied by banks and \(LGD\), \(EAD\), and \(M\) are supervisory values set by the Basel Committee, under the Foundation IRB approach.

\(PD\), \(LGD\), \(EAD\), and \(M\) for corporate, sovereign, and bank exposures will be supplied by banks under the advanced IRB approach.

The estimates of \(PD\), \(EAD\), and \(LGD\) are provided by all banks applying the IRB approach, with the foundation IRB being merged with the Advanced IRB. There lacks a maturity adjustment and the capital requirement is therefore given as:

$$ EAD\times LGD\times \left( WCDR-PD \right) $$

And the risk-weighted assets are:

$$ 12.5\times EAD\times LGD\times \left( WCD-PD \right) $$

A relationship between the \(PD\) and \(\rho\) is specified in the computation of \(WCDR\) for all other retail exposures. Hence:

$$ \rho =0.03\frac { 1-exp\left( -35\times PD \right) }{ 1-exp\left( -35 \right) } +0.16\left[ 1-\frac { 1-exp\left( -35\times PD \right) }{ 1-exp\left( -35 \right) } \right] $$

Since \(exp\left( -35 \right) \) is a very small figure, then:

$$ \rho =0.03+{ 0.13 }{ e }^{ -35\times PD } $$

Banks are required to keep operational risk capital under Basel II. The following are the three approaches applicable to the computation of operational risk capital:

- Basic Indicator Approach;
- Standardized Approach; and
- Advanced Measurement Approach.

The basic indicator approach is the simplest approach. It sets operational risk capital to equal the average annual gross income for the bank. The only difference between the standardized approach and the basic indicator approach is that the factor applied to gross income from different business lines is different. Under the advanced measurement approach, the bank’s internal models are applied in the computation of the operational risk loss that we are 99.9% certain will not be exceeded in a year.

The supervisory review process is the concern of the second pillar of the Basel II accord. The following four key supervisory review principles are specified:

- Banks should possess a process of assessing banks’ overall capital adequacy relative their strategy for capital level maintenance and risk profile.
- There should be a review and evaluation of banks’ internal capital adequacy assessments and strategies, by supervisors, and their ability to monitor and ensure compliance with regulatory capital ratios.
- As expected by supervisors, banks should be operating above the minimum regulatory capital and the supervisors should have the ability to require banks to hold capital in excess of the said minimum.
- An intervention at an early stage by supervisors should prevent capital from falling below the necessary minimum levels to support a particular bank’s risk characteristics, and a rapid remedial action should be required in case the capital is not maintained or resorted.

The concern of this pillar is increasing a bank’s disclosure for its assessment procedures and capital adequacy. Accounting disclosures differ from regulatory disclosures in form and need not be made in the annual reports.

The following are some of the items that can be disclosed by banks:

- The entities to the banking group to which Basel II is applicable plus adjustments made for entities to which it is not applied;
- All capital instruments’ main features’ terms and conditions;
- A list of instruments consisting of Tier 1 capital and the amount of capital provided by each bank;
- The total amount of Tier 2 capital;
- Credit, market, and operational risk capital requirements;
- Other general information on the risks to which a bank is exposed and applicable assessment methods for different risk categories by the bank; and
- The operation and structure of the risk management function.

There are many similarities between solvency II and Basel II. The three pillars also exist in solvency II. The computation of capital requirements and the eligible types of capital is the concern of pillar I. In pillar 2, the supervisory review process is the main concern. Under pillar III, the disclosure of risk management information to the market is the concern.

Solvency capital requirement (SCR) and minimum capital requirement (MCR) are specified under pillar I of solvency II. There are two ways in which SCR is computed, namely: the internal models approach and the standardized approach.

To deal with large advance events, there should be adequate capital. Three steps should be satisfied by the internal models:

**The statistical quality test:**which tests the soundness of the data and applied methodology in the VaR computation;**The calibration test:**which tests if the risk assessment is in accordance with a common SCR target criterion; and**A use test:**which tests if the models are genuinely relevant to risk managers.

The three types of capital in solvency II are:

**Tier 1 capital:**consisting of equity capital, retained earnings, plus other equivalent funding sources;**Tier 2 capital:**which is made up of liabilities subordinated to policyholders and satisfy some criteria concerning their ability in scenarios of wind-down; and**Tier 3 capital:**consisting of liabilities subordinated to policyholders and fail to satisfy these criteria.

1) The following table shows a portfolio of three derivatives possessed by Milson Bank with a particular counterparty:

$$ \begin{array}{|l|c|c|c|} \hline Transaction & Principal\quad { L }_{ i } & Current\quad Value\quad { V }_{ i } \\ \hline 10-year\quad interest & 1000 & 63 \\ rate\quad swap & {} & {} \\ \hline 7.5-year\quad foreign & 1000 & -19 \\ exchange\quad forward & {} & {} \\ \hline 3-month\quad option\quad on & 700 & -13 \\ a \quad stock & {} & {} \\ \hline \hline \end{array} $$

Which of the following is closest to the Net Replacement Ratio?

- 0.49
- 2.03
- 1.97
- 0.50

The correct answer is **A**.

Recall that:

$$ NRR=\frac { max\left( { \Sigma }_{ i=1 }^{ N }\left( { V }_{ i },0 \right) \right) }{ { \Sigma }_{ i=1 }^{ N }max\left( { V }_{ i },0 \right) } $$

The current exposure with netting (the numerator) is computed as:

$$ 63-19-13=31 $$

The current exposure without netting (the denominator) is computed as:

$$ 63+0+0=63 $$

Therefore:

$$ NRR=\frac { 31 }{ 63 } $$

$$ = 0.49 $$