### Risk Capital Attribution and Risk-Adjusted Performance Measurement

The objective of this chapter is to define, compare and contrast risk capital, economic capital, and regulatory capital and to give an explanation of the models and motivation allocating risk capital through the use of economic capital models. The Risk-Adjusted Return on Capital (RAROC) methodology will also be studied and its application in capital budgeting.

By the end of the chapter, the leaner should be able to do a calculation and an interpretation of the RAROC for a project, loan, or loan portfolio and apply RAROC to compare the performance of different units of the business. When RAROC is applied to evaluate performance, the problems arising, ranging from the choice of a time horizon, and measuring the likelihood of default to the choice of a confidence level, will be explained in details.

Furthermore, the hurdle rate will be computed and used as the basis of business decisions. We will also calculate a project’s RAROC for the determination of its viability.

The challenges in diversifying modeling benefits will be studied and the best practices in the implementation of a methodology that applies RAROC in economic capital allocation will be explained.

In general, the focus of the chapter is on the role of risk capital and its attribution to lines of business as part of a Risk-Adjusted Performance Measurement (RAPM).

# The Role of Risk Capital

Risk capital is used as protection against the prevalence of various risks in a corporation’s business for a company to maintain its financial integrity and remain in business even in the worst case scenario. Therefore, the stakeholders in the corporation will be essentially confident due to the risk capital.

The intention of risk capital concept is to capture the economic realities of the risks ran by a company, and should not be confused with regulatory capital. Regulatory capital is used only for a few regulated industries, e.g., insurance and banking. This because the interests of small depositors or policyholders need to be protected by regulators. Moreover, a set of rules and formulas are used in the computation of regulatory capital and only a minimum level of required capital adequacy is set. Also, there may be no similarity in the risk and regulatory capital number for each constituent business line despite them having a similar number.

Regulatory capital for activities like securitization may end up much higher than economic capital due to the new regulatory capital requirements that were imposed by Basel III. The computation of economic capital is still crucial for senior management as a benchmark for a financial institution’s economic viability assessment.

The same concepts applied by methodologies of VaR computation are used as the basis of evaluating risk capital. Internal VaR models that are complex often support the figures used in risk capital.

The computation of risk capital should be in such a way that unexpected losses can be absorbed by the institution to a confidence level that is aligned with the various stakeholders’ requirements. Some target credit ratings from a given rating agency are used to choose the exact confidence level. This is because the ratings are linked explicitly to a default likelihood.

# Emerging Uses of Risk Capital Numbers

Originally, risk capital was applied in the evaluation of the necessary capital required by an institution to remain solvent given the risky activities they engage in. However, the additional applications of risk capital that have been included – particularly by banks and other financial institutions – are:

1. The evaluation of performance and incentive compensation at the business unit and individual levels: This can be achieved by including risk capital into risk-based capital attribution systems under the RAPM or RAROC. Economic profitability can also be compared through the measure, rather than the accounting profitability of different activities.
2. Active management of portfolio for entry/exit decisions: A business’ RAPM and its risk diversification effect should be the basis for entering or exiting a specified business.
3. Pricing transactions: Individual transactions’ risk-based pricing can be computed using figures of risk capital. The attraction of risk-based pricing can be attributed to the fact that compensation to a company due to economic risk generated by a transaction is ensured.

However, the above-described roles cannot be accommodated by a single measure of risk capital.

# RAROC: Risk-Adjusted Return on Capital

To evaluate the economic performance of a business unit and individual transactions, the RAROC approach is applied to allocate risk capital. Across all activities of a business unit, the risk-adjusted performance can be evaluated in a uniform and comparable way since the trade-off between risk and reward of a unit of capital is made clear by the RAROC.

This evaluation can be applied to measure the performance for the purposes of capital budgeting by top management and also as an input to compensate business unit managers. The generic RAROC equation reads:

$$RAROC={ \left( After\quad tax\quad expected\quad risk-Adjusted\quad net\quad income \right) }/{ \left( Economic\quad capital \right) }$$

# RAROC for Capital Budgeting

After the generic after-tax RAROC equation is implemented, the meaning interpreted by industry practices is as:

$$RAROC=\frac { Expected\quad revenues-Costs-Expected\quad losses-Taxes+Return\quad on\quad risk\quad capital\pm Transfers }{ Economic\quad capital }$$

Where:

1. Expected revenues are the revenues to be generated by the activity;
2. Costs are the direct expenses incurred in running the activity;
3. Expected losses refer to the losses from default;
4. Taxes are the expected taxation amounts added to the activity via the company’s effective tax rate;
5. Return on risk capital refers to the activity’s return on risk capital;
6. Transfers refer to the corresponding transfer mechanisms of pricing between the business unit and the treasury groups; and
7. Economic capital refers to the total of risk capital and strategic capital where:

$$Strategic \quad risk \quad capital = Goodwill + Burned \quad out \quad capital$$

To cover worst-case losses from market, credit, operational, and other risks at the necessary confidence threshold, some capital cushion must be set aside by banks, usually referred to as risk capital. Risk capital is directly related to the computation of VaR at a specified confidence and a time horizon of one year.

Moreover, the risk of significant investments whose success and profitability are highly uncertain is normally referred to as strategic risk capital.

Currently, strategic risk capital is measured in terms of the totals of burned-out capital and goodwill. Burned-out capital is the capital allocated to account for the risk of strategic failure of recent acquisitions or other strategic initiatives that are organically built.

Goodwill is an intangible asset that arises when one company purchases another for a premium value. The value of a company’s brand name, solid customer base, good employee relations, etc. represents goodwill.

Finally, the loss corresponding to the risk capital allocated to an activity is called unexpected loss and is the difference between the total loss and the expected loss over a one-year horizon.

# RAROC for Performance Measurement

Originally, RAROC was suggested to be a tool to allocate capital on an anticipatory or ex-ante basis. The implication is, therefore, that the numerator of the RAROC equation should contain the expected revenues and losses in order to budget capital.

In the event that RAROC is applied for ex-post, realized losses and revenues can be applied in our computation instead of expected revenues and losses.

## RAROC Horizon

The computation of all the quantities relevant to the RAROC equation should be based on a particular time horizon (or over the deal’s lifetime). The most reasonable time horizon applied by practitioners is usually one year because it corresponds to the period taken for the firm to be recapitalized in the event of a major unexpected loss.

RAROC’s risk horizon is mostly chosen somewhat arbitrarily. Capital is not necessarily increased by computing capital over longer time horizons due to the reduction of the confidence level in the solvency of any company.

## Default Probabilities: Point-in-Time (PIT) versus Through-the-Cycle (TTC)

When pricing financial instruments subject to credit risk and computing near-term expected losses, the most reasonable approach to apply is the point-in-time (PIT) probability of default (PD).

On the other hand, economic capital, current profitability, and strategic decisions on products, geographies, and new business ventures are reasonably computed using the through-the-cycle (TTC) probability of default.

The likelihood of a company maintaining its rating when assessed using a PIT approach is minimal as compared to when the TTC approach is applied in the assessment. Both the normal part and the worst part of the economic cycle should periodically be compared using PIT PD versus TTC PD in RAROC computations.

## Confidence Level

There should be consistency in the confidence level for computing economic capital and the target credit rating of the company.

The amount of risk capital allocated to an activity may significantly decline when lower confidence levels are set, particularly in the event of domination by operational, credit, and settlement risks in the institution’s risk profile.

## Hurdle Rate and Capital Budgeting Decision Rule

For most companies, the single hurdle rate for their business activities is the after-tax weighted-average cost of equity capital.

The hurdle rate,$${ h }_{ AT }$$,is given by:

$${ h }_{ AT }=\frac { CE\times { r }_{ CE }+PE\times { r }_{ PE } }{ CE+PE }$$

Where:

• $$CE$$ is the market value for common equity and $$PE$$ is the market value for preferred equity; and
• $${ r }_{ CE }$$ is the cost of common equity and $${ r }_{ PE }$$ the cost of preferred equity:

$${ r }_{ CE }={ r }_{ f }+{ \beta }_{ CE }\left( { \bar { R } }_{ M }-{ r }_{ f } \right)$$

Where $${ r }_{ f }$$ is the risk-free rate,$${ \bar { R } }_{ M }$$ is the expected return on the market portfolio,$${ \beta }_{ CE }$$ is the firm’s common equity market beta. There should be a periodic resetting of the hurdle rate.

Supposing a company considers investing in a business or closing down an activity, the business after-tax RAROC is calculated and compared with the company’s hurdle rate, then the following simple decision is applied:

1. The activity only adds value to the company if the RAROC ratio is greater than the rate;
2. Otherwise, the activity only destroys the company’s value and should be rejected or closed down.

This simple rule can sometimes lead a company to accept high-risk projects that eventually lower its value and reject low-risk projects that can lead to an increase in the company’s value.

## Diversification and Risk Capital

Viewing the business on a stand-alone basis and applying the hurdle rate is a way to evaluate a particular business unit’s risk capital within a larger firm. But the total of the stand-alone risk capitals of the individual business units should significantly exceed the company’s risk capital since the profits generated by the businesses are not likely to be perfectly correlated.

An unnecessarily large amount of overall risk capital can be produced when business-specific models are ran at the confidence levels targeted at the top of the house since the effects of diversification are neglected.

Some boundaries can also be applied around the problem if the above approach is unsatisfactory. The VaR figure should, therefore, fall between the extreme cases of perfect and zero correlation between the types of risks across businesses. However, the boundaries are quite wide and do not solve the problem.

More risk capital should be allocated to a business with operating cash flows strongly correlated with the earnings of other activities as compared to a business with a similar volatility but whose earnings move in a countercyclical fashion.

Matters can also be complicated due to the impacts of diversification within business units. Suppose a business unit undertakes two activities – $$A$$ and $$B$$. Assume further that all the diversification effects due to a combination of activities $$A$$ and $$B$$ are taken into account by risk analysts in the computation of the business unit’s risk capital which is \$100. As an effort to allocate risk capital at the activity level of the business unit, the following are the three different measures of risk capital:

1. Stand-alone capital: is the amount of capital an activity uses and is considered independently of other activities in the same business unit.
2. Fully diversified capital: is the capital attributed to each activity, accounting for all benefits of diversification from combining them under the same leadership.
3. Marginal capital: is the extra capital necessary in an incremental activity.

This implies that the desired objective affects the choice of capital measure. The company’s solvency and minimum risk should be assessed by fully diversified measures.

On the other hand, marginal risk capital should be used as the basis to actively manage a portfolio or business mix decisions, while still accounting for the benefits of full diversification.

Finally, stand-alone risk capital for incentive compensation and fully diversified risk capital to assess the additional performance generated by the effect of diversification should both be involved in performance measurement.

# RAROC in Practice

The risk management function is often challenged by business units concerning the fairness of the economic capital amount attributed to them in companies where the RAROC has been implemented. The RAROC unit should be transparent about the applied methodology in risk assessment and institute forums where issues related to the determination of economic capital can be analyzed and debated.

The following are the various recommendations for the implementation of a RAROC system:

1. Commitment by the senior management: The marching orders should originate from the top management of the company due to the strategic nature of the decisions steered by the RAROC system.
2. Communication and education: There should be transparency in the RAROC group and an explanation of the RAROC methodology should be given not only to the business heads, but also to the business line managers and the CFO for the methodology to be generally accepted in the company.
3. Ongoing consultation: A forum to review the critical parameters driving risk and economic capital should be instituted by the company.
4. Maintaining the integrity of the process: The quality of the data about risk exposures and collected positions from the system of management are crucial for the validity of the RAROC figures.
5. Combining RAROC with qualitative factors: In addition to the requirement that only projects where RAROC is greater than the hurdle rate should be accepted, other qualitative factors should also be considered.
6. Active capital management processes: Every quarter, there should be channeling of balance sheet requests from business units like leverage ratio, liquidity ratios, and so on to the RAROC group.

# Practice Questions

1) We have been given the following information for a project:

• Expected revenue: USD 75 million
• Return on risk capital: USD 21 million
• Economic capital: USD 73 million
• Tax expense: USD 22 million
• Operating cost: USD 8 million
• Expected loss: USD 5 million

What is the RAROC for the project?

1. 0.2627
2. 0.8356
3. 0.9726
4. 1.0548

The correct answer is B.

Recall that:

$$RAROC=\frac { Expected\quad revenues-Costs-Expected\quad losses-Taxes+Return\quad on\quad risk\quad capital\pm Transfers }{ Economic\quad capital }$$

Therefore:

$$RAROC=\frac { 75-8-22-5+21 }{ 73 }$$

$$=0.8356$$