 ### Capital Modeling

In this chapter, we provide a comparison of the standardized approach, the alternative standardized approach, and the basic indicator approach for the computation of the operational risk capital charge.

The modeling necessities for a bank to apply the advanced measurement approach ($$AMA$$) will be described. Another description provided will be on modeling operational risk capital using the loss distribution approach.

The process of obtaining the frequency and severity distribution of operational losses will be explained, and this shall include popular distributions and suitability guidelines for probability distributions.

The application of Monte Carlo simulations in the generation of additional data points – most often for the $${ 99 }^{ th }$$ percentile to be established for an operational loss distribution– shall be studied. A further explanation of the application of scenario analysis with an inclusion of the hybrid approach in operational risk capital modeling will be provided.

# Operational Risk Capital

Several approaches can be used by companies for the computation of operational risk capital. The three main ones provided by Basel II include: the basic approach ($$BIA$$), the standardized approach ($$TSA$$), and the advanced measurement approach ($$AMA$$).

In case of an $$AMA$$ application, the underlying elements will be drawn on computation. However, it is not a necessity for the underlying elements to feed the model in the event of a simpler model being in application.

Furthermore, banks are encouraged to adopt cutting-edge approaches as they create their own operational risk management tools. Basel II expects that international banks select either the standardized or advanced measurement approach.

# Basic Indicator Approach (BIA)

The product of a simple computation of the average risk revenue for the past three years and 15% is a way of arriving at the capital calculation under the basic indicator approach ($$BIA$$). When computing the average, figures for any year with a negative or zero annual gross income should not be included in both the numerator and denominator. The following is the expression of the charge:

$${ K }_{ BIA }={ \left[ { GI }_{ 1…n }\times \alpha \right] }/{ n }$$

Where $${ K }_{ BIA }$$ is the capital charge under the $$BIA$$, $$GI$$ is the positive gross income per year over the previous three years, $$n$$ is the number of the past three yearswith positive gross income, and $$\alpha$$ is 15%.

# Standardized Approach (TSA)

This approach has almost all similarities with the basic approach apart from the different multipliers by business lines. It captures the operational risk factors not present in the basic approach using the assumption that different operational risk levels are contained in different business activities.

The total capital charge is computed via the following expression:

$${ K }_{ TSA }={ \left\{ { \Sigma }_{ years 1-3 }max\left[ \Sigma \left( { GI }_{ 1-8 }\times { \beta }_{ 1-8 } \right) ,0 \right] \right\} }/{ 3 }$$

Where $${ K }_{ TSA }$$ is the capital charge based on the standardized approach, $${ GI }_{ 1-8 }$$ is the gross income per annum in a particular year,and $${ \beta }_{ 1-8 }$$ is a ratio set by the committee and is fixed.

A different treatment approach will be taken by the standardized approach, in comparison to the basic indicator approach, in the event of a negative or zero income in one of three previous years.

In the standardized approach, the positive capital charges in other business lines may be offset without limit by negative capital charges in any business line any given year. The input to a particular year’s numerator will be zero if the aggregate capital charges across all business lines in that year are negative.

## Alternative Standardized Approach (ASA)

A bank is permitted to apply an alternative standardized approach ($$ASA$$) by the national regulator on condition that it is able to satisfy its supervisor an improved basis is provided by this alternative approach.

The following formula is applicable for $$ASA$$ operational risk capital charge:

$${ K }_{ RB }={ \beta }_{ RB }\times m\times { LA }_{ RB }$$

Where $${ K }_{ RB }$$ is the retail banking business line’s capital charge,$${ \beta }_{ RB }$$ is the retail banking business line’s beta, $${ LA }_{ RB }$$ constitutes the retail loans and advances outstanding as an average of the past three years given, and $$m$$ is a constant equal to $$0.035$$.

## Future of the Basic Standardized Approaches

The Basel Committee recognizes that inappropriate results might be produced by making allowances for negative income. The Basel Committee recognizes this possibility such that “If negative gross income distorts a bank’s Pillar 1 capital charge, supervisors will consider appropriate supervisory action under Pillar 2.”

Under this approach banks can design their own operational risk capital computation model based on three requirements:

Firstly, the confidence level to hold the capital for a one-year horizon by the model is 99.9%. This implies that the institution should hold an operational risk capital for protection from one in a thousand year fat-tail event.

The $$AMA$$ soundness standard should demonstrate that potentially severe tail loss events can be modeled. Moreover, the soundness standard that should be achieved by banks’ operational risk measures should be similar to that of the internal ratings-based approach for credit risk.

Secondly, internal loss data, external loss data, scenario analysis, and internal control factors of a business environment should be added to the model. Based on the combined use of these four elements, the internal measurement system of an institution must reasonably approximate the unexpected losses.

Finally, the allocation of capital for good behaviors to be incented requires an appropriate method. The economic capital for operational risk across business lines should improve the management of business lines’ operational risks.

The following are crucial necessities:

1. The operational risk framework should be represented as outlined in Basel II;
2. All the expected and unexpected losses must be must be captured in the model with only an exclusion of expected losses under certain strict criteria;
3. To ensure the bank captures fat tail events, the model should provide sufficient detail and granularity;
4. Any correlation assumptions created in the $$AMA$$ models should be defended and all computed cells have to be summed by the bank;
5. The four elements previously stated must be captured in the model to meet the supervisory soundness standard that has been set out; and
6. The bank must appropriately weight these four elements.

## Loss Distribution Approach to Modeling Operational Risk Capital (LDA)

Strictly internal losses are applied as direct inputs into a simple $$LDA$$ model and the remaining three elements should be applied for the purpose of stress testing or allocation.

A bank should have a minimum of three years of loss data, regardless of its design, to put into its $$AMA$$ model. However, regulators feel that all the inclusion of all information is necessary and should surpass the minimum five-year observation period necessity.

Since the data collection period is likely to be short, the fat-tail events that the company should be protected against by the capital computation may not be accurately captured. Secondly, the future is not necessarily reflected by the historical data. These twoelements combined weaken the use of the $$LDA$$ approach.

The following are some standard methods worth studying, despite there being a wide range of $$AMA$$ practices:

## Step 1: Modeling Frequency

A determination of the frequency of events per annum is the first step in the creation of a model of expected operational risk losses. The Poisson distribution, with only parameter $$\left( \lambda \right)$$ representing the average number of events in the year, is the most commonly applied distribution selection for frequency modeling.

The following formula is applicable for creating the Poisson distribution:

$$f\left( n \right) =\frac { { \lambda }^{ N }{ e }^{ -\lambda } }{ n! }$$

Where: $$n=1,2,\dots ,$$ and $$\lambda$$ is the average number of events in the year.

The likelihood of a certain number of events occurring in one year is represented by the Poisson distribution obtained from this approach.

## Step 2: Modeling Severity

Supposing an event has happened, the probable size of the event should be determined. The event’s severity does not necessarily need to be an integer; it establishes the likelihood of the occurrence of an event over a wide range of values.

The logarithmic distribution is the most popular and simplest severity modeling method. However, when selecting the applicable method, the model’s suitability should be assessed through the following criteria: realistic, well specified, flexible, and simple.

## Step 3: Monte Carlo Simulation

For a better estimation of the required capital to ensure 99.9% certainty that the following year’s probable losses are appropriately covered, more data points should be approximated using the established severity and frequency distributions.

These two distributions can be combined through the Monte Carlo distribution to produce more data points with similar features as the observed data points.

Data is first selected from the frequency distribution, giving the number of events predicted to happen in the first year.

Then, each event size is selected from the severity distribution, producing a given number of losses for that year. The total value of losses is then obtained by a summation of the value of those losses.

Afterward, the process is repeated for the subsequent years and those totals placed in order from the largest to the smallest. Selecting a thousandth item from the ordered list gives the 99.9% certainty.

## Correlation

To produce the total capital required, all cells from populated operational risk capital matrices plus an amount of capital computed for each class of risk must be summed together.

The appropriate quantitative and qualitative methods must be applied to validate their correlations assumptions. These assumptions might be applied to the internal $$AMA$$ model of a member company.

## Scenario Analysis Approach to Modeling Operational Risk Capital

Fat-tail events can be identified through scenario analysis data, hence providing rich information for computing the appropriate operational risk capital.

Since the data is captured in a process designed to consider what-if scenarios, it reflects the future. This is an advantage as the $$LDA$$ approach only considers the past.

However, the data is subjective as it has been collected in an interview or workshop approximation activity. Moreover, the fitting of distributions can be challenging since this approach lacks data.

## Hybrid Approach to Modeling Operational Risk Capital

The loss data and scenario analysis output are both applied in the computation of operational risk capital. Some companies use these two approaches combined to stitch together two distributions, whereas other companies develop $$LDA$$ models and apply scenario analysis for the model to be stressed, producing a distribution that is more appropriate.

# Insurance

For evaluating the operational risk applied for minimum capital requirements, the risk mitigating effect of insurance will be recognized by an institution as allowed under the $$AMA$$. The following criteria should be compiled by the institution for it to benefit from such risk mitigation:

1. The least claims-paying ability of the insurance provider should be rated A;
2. An initial term of over one year should be possessed by the insurance policy;
3. The minimum cancellation notice period for the insurance policy should be 90 days;
4. There should be no exclusions or limitations triggered by supervisory actions, in the insurance policy;
5. The manner in which the banks’ insurance coverage is reflected by the risk mitigation computations should be transparent in its relationship, and consistent with the probability and effect of losses applicable in the banks’ overall determination of its operational risk capital;
6. A third-party entity should provide the insurance;
7. A well-reasoned and documented framework for recognizing the insurance should be assembled;
8. A description of the bank’s use of insurance to mitigate operational risk should be disclosed by the bank;

The following elements should be captured by the bank’s methodology for insurance recognition based on the $$AMA$$:

1. The policy’s residual term;
2. The terms of cancellation of the policy; and
3. Mismatches in the coverage of insurance policies and payments uncertainty.

# Disclosure

Pillar III of the Basel II Accord requires the results of capital computations to be disclosed and the applied methodologies to be explained.The stress testing and back testing of models for validity is a must regardless of the approach applied for operational risk capital modeling.

# Future of Capital Requirements

Despite continued refining in the industry, there is a possibility of the Basel Committee issuing new requirements for operational risk capital and guidance on modeling approaches that will be allowed.

# Practice Questions

1) Assume that the following information on revenue has been provided by SunBank(in \$USD million):

$$\begin{array}{|c|c|c|c|} \hline {} & Year \quad 1 & Year \quad two & Year \quad Three \\ \hline Corporate \quad financing & 9.8 & 12.4 & 15.2 \\ \hline Retail \quad banking & -19.2 & 11.3 & -16.3 \\ \hline \end{array}$$

If the beta values for corporate finance and retail banking are 17.99% and 12.11%, respectively, then how much operational risk capital should the bank hold under Basel II by applying the Standardized Approach?

1. USD 2.1799million
2. USD 1.4533 million
3. USD 1.2659million
4. USD 3.7977 million

Recall that:

$${ K }_{ TSA }=\frac { \left\{ { \Sigma }_{ years1-3 }max\left[ E\left( { GI }_{ 1-8 }\times { \beta }_{ 1-8 } \right) ,0 \right] \right\} }{ 3 }$$

Year one: $$9.8\times 0.1799-19.2\times 0.1211=-0.5621$$

Year two: $$12.4\times 0.1799+11.3\times 0.1211=3.5992$$

Year three: $$15.2\times 0.1799-16.3\times 0.1211=0.7606$$

Therefore:

$${ K }_{ TSA }=\frac { \left( 0+3.5992+0.7606 \right) }{ 3 } =1.4533$$

Under this approach, the bank should hold a minimum of USD 284.5 million of operational risk capital.