Book 3. Operational Risk

FRM Part 2

OR 18. Risk Capital Attribution and Risk-Adjusted Performance Measurement

Presented by: Sudhanshu

Module 1. Risk-Adjusted Return on Capital

Module 2. RAROC, Hurdle Rate and Adjusted-RAROC

Module 3. Diversification Benefits and RARO C Best Practices

Module 1. Risk-Adjusted Return on Capital

Topic 1. Risk Capital, Economic Capital and Regulatory Capital

Topic 2. Economic Capital Approaches

Topic 3. Risk-Adjusted Return on Capital (RAROC)

Topic 4. RAROC Similarities with Sharpe Ratio and NPV

Topic 5. Detailed RAROC Equation

Topic 1. Risk Capital, Economic Capital and Regulatory Capital

Risk Capital:
- Protection against unexpected losses.
- Financial buffer to shield a firm from the economic impact of risks.
- Provides assurance to stakeholders that invested funds are safe.
- Often treated synonymously with Economic Capital.
Economic Capital:
- Generally synonymous with Risk Capital.
- Alternative definition: $E co n o mi c C a p i t a l = R i s k C a p i t a l + St r a t e g i c R i s k C a p i t a l$
Differences between Risk Capital and Regulatory Capital:
- Regulatory capital is only relevant only for regulated industries (e.g., banking, insurance).
- Regulatory capital is computed using general industry benchmarks. It results in a minimum required capital adequacy, often lower than a firm's risk capital.
- Regulatory capital may differ from risk capital within various divisions of a firm. The greater of two should be allocated to a certain division.

Capital as a Cushion for Risk:
- Financial institutions can be highly leveraged at low cost (deposits, debt).
- Involvement in derivatives, guarantees, and other commitments requires economic capital allocation.
Creditworthiness Requirement:
- Main customers are also main liability holders.
- Concern about default risk and counterparty risk (OTC derivatives).
- Sufficient economic capital ensures creditworthiness.
Difficulty in External Creditworthiness Assessment:
- Financial institutions' risk profiles constantly evolve (e.g., complex hedging, derivatives).
- Economic capital mitigates this problem and provides financial stability assurance.
Profitability Impacted by Cost of Capital:
- Economic capital is similar to equity capital (no repayment like debt).
- More expensive to hold than debt capital, increasing the cost of capital and reducing profits.

Topic 2. Economic Capital Approaches

Topic 3. Risk-Adjusted Return on Capital (RAROC)

The necessary amount of economic capital is a function of credit risk,market risk, and operational risk.
The RAROC for a project or loan can be defined as risk-adjusted return

divided by risk-adjusted capital. The basic RAROC equation is as follows:
Tradeoff between risk and return per unit of capital with the numerator acting as return and the denominator acting as risk.
A business unit’s RAROC needs to be greater than its cost of equity in order to create shareholder value.
ROE and ROA Limitations: Return on equity (ROE) and return on assets (ROA) are based solely on accounting book values and cannot account for relevant risks in performance evaluation.
RAROC Risk Adjustments: RAROC makes two specific adjustments - deducting expected loss from returns in the numerator and replacing accounting capital with economic capital in the denominator.

R A R O C=\frac{\text { After-tax expected risk-adjusted net income }}{\text { Economic Capital }}

Topic 4. RAROC Similarities with Sharpe Ratio and NPV

Sharpe Ratio: $(E x p ec t e d R e t u r n - R i s k - f ree R a t e) / St an d a r d De v ia t i o n$
Net Present Value (NPV): Discounted value of future expected after-tax cash flows, with a risk-adjusted discount rate.
Key Difference from NPV: RAROC considers both systematic and unsystematic risk in its earnings figure, unlike CAPM-based NPV which only captures systematic risk.

Detailed RAROC Equation:

Components Explained:
- Expected Revenues: Assume no losses.
- Costs: Direct costs.
- Expected Losses (EL): Mainly expected default losses (loan loss reserve), also from market, operational, and counterparty risks. Captured in the numerator.
- Taxes: Computed using the firm's effective tax rate.
- Return on Economic Capital: Return on risk-free investments based on allocated risk capital.
- Transfers: Head office overhead allocations, transactions with Treasury (borrowing, hedging costs).
- Economic Capital: Includes both risk capital and strategic risk capital.
Risk Capital:
- Buffer against unexpected losses.
- Amount of funds held to cover worst-case loss (over expected loss) at a specific confidence level (e.g., 95% or more).
- Similar to annual Value at Risk (VaR).

RAROC=\frac{\begin{aligned}&(\text { Expected revenues }- \text { costs }- \text { expected losses }- \text { taxes }+ \text { return on economic capital } \pm \text { transfers })\end{aligned}}{\text { Economic Capital }}

Topic 5. Detailed RAROC Equation

Strategic Risk Capital, Goodwill and Burned-Out Capital:
- Strategic Risk Capital: Uncertainty surrounding success and profitability of investments, and includes goodwill and burned-out capital.
- Goodwill: Excess of purchase price over fair value of net assets.
- Burned-out Capital: Risk of lost amounts spent during venture start-up if not pursued due to low projected returns.
Risk Limit Allocation: Firms may allocate risk capital to unused risk limits (such as undrawn credit line amounts) since risk capacity could be utilized at any time, requiring adjustment when capacity is used.
Economic Capital Purpose: Economic capital provides a cushion against unexpected losses at a specified confidence level, representing the probability that firms can absorb unexpected losses over a defined period.
Unexpected Loss Calculation: Unexpected loss equals the difference between worst-case loss and expected loss at a given confidence level, serving as the basis for risk capital allocation.
Practical Example: With expected loss of 20 basis points and worst-case loss of 190 bps at 95% confidence level, the unexpected loss (risk capital requirement) equals 170 bps over one year.
Tail Risk Consideration: Even at 95% confidence level, a 5% probability remains that actual losses will exceed the worst-case scenario of 190 basis points.

Topic 5. Detailed RAROC Equation

Practice Questions: Q1

Q1. Which of the following statements regarding the risk-adjusted return on capital (RAROC) methodology is correct?

A. In the context of performance measurement, RAROC uses accounting profits.

B. In the numerator of the RAROC equation, expected loss is added to the return.

C. If a business unit's cost of equity is greater than its RAROC, then the business unit is not adding value to shareholders.

D. RAROC is useful for determining incentive compensation but it lacks the flexibility to consider deferred or contingent compensation.

Practice Questions: Q1 Answer

Explanation: C is correct.

The cost of equity represents the minimum rate of return on equity required by shareholders. Therefore, if RAROC is below the cost of equity, then there is no value being added.
Response A is not correct because RAROC uses economic profits, not accounting profits. Response B is not correct because in the numerator of the RAROC equation, expected loss is deducted from the return. Response D is not correct because RAROC has the flexibility to consider deferred or contingent compensation.

Practice Questions: Q2

Q2. Assume the following information for a commercial loan portfolio:

$1.2 billion principal amount
6% pre-tax expected return on loan portfolio
Direct annual operating costs of $8 million
Loan portfolio funded by $ 1.2 billion of retail deposits; interest rate =4 %
Expected loss on the portfolio is 0.4 % of principal per annum
Unexpected loss of 7 % of the principal amount
Risk-free rate on government securities is 1 %
30 % effective tax rate
Assume no transfer pricing issues

Based on the information provided, which of the following amounts is closest to the RAROC?

A. 9.33 %.

B. 10.03 %.

C. 12.33 %.

D. 14.66 %.

Practice Questions: Q2 Answer

Explanation: B is correct.

Unexpected loss ( ) is equal to the amount of economic capital required. The return on economic capital is then million. Also, expected revenues

=$72 million; interest expense million; expected losses million.

\begin{aligned} \text { RAROC } & =\frac{\binom{\text { expected revenues }- \text { costs }- \text { expected losses }}{- \text { taxes }+ \text { return on economic capital } \pm \text { transfers }}}{\text { economic capital }} \\ \text { RAROC } & =\frac{(72-8-48-4.8+0.84+0) \times(1-0.3)}{84}=10.03 \% \end{aligned}

\$1.2 billion \times 7 \%=\$ 84 million

\$ 84 million \times1 \%=\$ 0.84

=0.06 \times \$ 1.2 billion

=0.04 \times \$ 1.2 billion =\$ 48

=0.004 \times \$ 1.2 billion =\$ 4.8

Module 2. RAROC, Hurdle Rate, Adjusted RAROC

Topic 1. RAROC for Perfomance Management

Topic 2. Hurdle Rate for Capital Budgeting Decisions

Topic 3. Adjusted RAROC

Topic 1. RAROC for Performance Measurement

Time Horizon:
- Typically one-year horizon for business planning and recovery from unexpected loss.
- Multi-period RAROC for longer-term transactions is possible but raises issues (e.g., changing risk in subsequent periods, over/ undercapitalization with averaging).
- Risk capital could be thought of as the firm’s one-year VaR at a specific confiidence level (e.g., 95% or 99%).
- Credit Risk and Operational Risk: No adjustments are required from one-year VaR to compute risk capital.
- Market Risk Capital: Short-term VaR (1-day, 10-day) needs adjustments for 1-year risk capital. "Square root of time" rule (1-year VaR = 1-day VaR * ) needs fine-tuning for core risk level and time to reduce its risk.
- Time to Reduce Factor: Risk capital computation must factor in the time needed to lower current risk levels to core risk levels, known as "time to reduce," based on position liquidity during difficult market conditions.
- Liquidation Timeline Impact: The time required corresponds to relative liquidity of firm's investment positions, with illiquid positions potentially requiring extended periods for reasonable liquidation during stressed conditions.

\sqrt{252}

Topic 1. RAROC for Performance Measurement

Default Probability:
- Point-in-Time (PIT): Used for short-term expected losses and pricing financial instruments with credit risk. Firm's rating more likely to change.
- Through-the-Cycle (TTC): More commonly used for economic capital, profitability, and strategic decisions. Results in lower volatility of economic capital.
- Advisable to compare PIT vs. TTC results at stable and lowest points of the economic cycle.
Confidence Level:
- Chosen confidence level must correspond with the firm's desired credit rating (e.g., AA/AAA requires >99.95%).
- Lower confidence level reduces risk capital and impacts risk-adjusted performance measures.
- Reduction can be dramatic if firm is primarily exposed to operational, credit, and settlement risks (where large losses are rare).

Practice Questions: Q3

Q3. Which of the following statements regarding the computation of economic capital is correct?
I. Selecting a longer time horizon for RAROC calculations is preferable because risk and return data is more reliable with more time.

II. Choosing a lower confidence level will not likely reduce the amount of risk capital required if the firm has little exposure to operational, credit, and settlement risks.

A. I only.

B. II only.

C. Both I and II.

D. Neither I nor II.

Practice Questions: Q3 Answer

Explanation: B is correct.

Choosing a lower confiidence level will not likely reduce the amount of risk capital required if the firm has little exposure to operational, credit, and settlement risks. The reduction would be much more dramatic only if the firm has significant exposure to such risks because large losses would be rare.
In selecting a time horizon for RAROC calculations, risk and return data for periods over one year is likely to be of questionable reliability.

Practice Questions: Q4

Q4. Which of the following statements regarding the choice of default probability approaches in computing economic capital is correct?

A. A through-the-cycle (TTC) approach should be used to price financial instruments with credit risk exposure.

B. A point-in-time (PIT) approach is more commonly used for computations involving profitability and strategic decisions.

C. A TTC approach is more likely to result in a lower volatility of capital compared to the PIT approach.

D. A firm’s rating will not change when analyzed under the PIT approach versus the TTC approach.

Practice Questions: Q4 Answer

Explanation: C is correct.

A fiirm’s rating is more likely to change when analyzed under the point-in-time (PIT) approach compared to the through-the-cycle (TTC) approach. As a result, the TTC approach results in a lower volatility of economic capital compared to the PIT approach.

A PIT approach should be used to price financial instruments with credit risk exposure and to compute short-term expected losses. A TTC approach is more commonly used for computations involving profitability, strategic decisions, and economic capital.

Topic 2. Hurdle Rate for Capital Budgeting Decisions

Definition:
- After-tax weighted average cost of equity capital.
- Compared to RAROC for business decisions, similar to Internal Rate of Return (IRR) analysis.
- Should be revised periodically (e.g., once or twice a year or when it moves by >10%).
Hurdle Rate ( $h_{A T}$ ) Computation:
- $CE$ : Market value of common equity
- $PE$ : Market value of preferred equity
- $R_{CE}$ : Cost of common equity (derived from CAPM)
- $R_{PE}$ : Cost of preferred equity (yield on preferred shares)

h_{A T}=\frac{\left[\left(C E \times R_{C E}\right)+\left(P E \times R_{P E}\right)\right]}{(C E+P E)}

Capital Asset Pricing Model (CAPM) for $R_{CE}$ :
$R_{CE} = R_{F} + β_{CE} (R_{M} - R_{F})$
- $R_{F}$ : Risk-free rate
- $R_{M}$ : Expected return on market portfolio
- $β_{CE}$ : Firm's common equity market beta
Business Decision Rules (with shortcomings):
- If $R A ROC > h u r d l e r a t e$ , accept the project (value creation).
- If $R A ROC < h u r d l e r a t e$ , reject/discontinue the project (value destruction).
- Shortcoming: High RAROC projects may come with high risk, potentially leading to losses. Low RAROC projects may offer steady returns despite low risk.

Topic 2. Hurdle Rate for Capital Budgeting Decisions

Topic 3. Adjusted RAROC

Purpose:
- Adjusts RAROC to consider systematic risk and a consistent hurdle rate.
Adjusted RAROC Equation:
$A d j u s t e d R A ROC = R A ROC - β_{E} (R_{M} - R_{F})$
- $R_{F}$ : Risk-free rate (also serves as the hurdle rate for adjusted RAROC decisions)
- $R_{M}$ : Expected return on market portfolio
- $β_{E}$ : Firm's equity beta
- $(R_{M} - R_{F})$ : Excess return over risk-free rate to account for non-diversifiable systematic risk.
Revised Business Decision Rules:
- If $A d j u s t e d R A ROC > R_{F}$ , then accept the project.
- If $A d j u s t e d R A ROC < R_{F}$ , then reject the project.

Module 3. Diversification Benefits And RAROC Best Practices

Topic 1. Diversification

Topic 2. Risk Allocation Methods

Topic 3. RAROC Best Practices

Topic 1. Diversification

Diversification Benefits:
- Overall firm risk capital should be less than the sum of individual business unit risk capitals.
- This is due to correlations of returns between business units being less than +1.
- Difficult to measure in practice.
- Business units may use lower confidence levels to avoid excessively high aggregate risk capital.
Challenges in Modeling Diversification Benefits:
- Wide range for overall VaR (e.g., sum of individual VaRs vs. square root of sum of squares).
- Subjectivity in allocating diversification benefits back to business units.
- Business units with earnings highly correlated to the firm need more risk capital than negatively correlated ones.
- Countercyclical business lines allow for stable earnings and lower risk capital for a given credit rating.

Assume that:
- Activity A alone requires $50 of risk capital.
- Activity B alone requires $60 of risk capital.
- Activities A and B together require a total of $90 of risk capital.
Stand-alone Capital: Ignores diversification benefits; useful for incentive pay.
- Stand-alone capital for Activities A and B are $50 and $60, respectively. The stand-alone capital for the business unit is $90.
Fully Diversified Capital: Considers diversification benefits; useful for firm solvency and minimum risk capital for an activity.
- Diversification benefit is $20 (sum of stand-alone capital minus total combined capital)
Marginal Capital: Represents extra capital needed for new activities with full diversification benefits considered - for example, Activity A's marginal capital is $30 ($90 total - $60 for Activity B) while Activity B's is $40 ($90 total - $50 for Activity A), with total marginal capital ($70) below full business unit risk capital ($90), making it useful for portfolio management decisions.
Caution: Correlations between risk factors change over time, and can move to extremes (e.g., -1 or +1) during market crises, reducing diversification benefits.

Topic 2. Capital Allocation Methods

Topic 2. RAROC Best Practices

Senior Management Involvement:
- Active involvement in RAROC implementation.
- Promote RAROC as a means of measuring shareholder value creation.
- Emphasis on profits in relation to risks taken, not just maximizing profits.
Communication and Education:
- Clearly explain the RAROC process to all levels of management.
- Ensure sufficient "buy-in" from management.
- Process of allocating economic capital must be fair and transparent to minimize concerns of excessive attribution.
- Foster open dialogue and debate with business unit leaders on economic capital computation.

Ongoing Consultation:
- Committee with members from business units and risk management should periodically review key metrics impacting economic capital.
- Promotes fairness in capital allocation.
- Metrics for Credit Risk: Probability of default, credit migration frequencies, loss given default, credit line usage. Need regular updates.
- Metrics for Market Risk: Volatility and correlation; update at least monthly.
- Metrics for Operational Risk: Less defined, involve significant subjectivity and debate.
- Other key metrics (core risk level, time to reduce) updated annually.
Data Quality Control:
- Information systems collect data for RAROC calculations.
- Centralized data collection process with built-in edit and reasonability checks for accuracy.
- RAROC Team Responsibility: Data collection, computations, reporting.
- Business Units & Accounting Department Responsibility: Controls to ensure data accuracy.

Topic 2. RAROC Best Practices

Complement RAROC with Qualitative Factors:
- Perform a qualitative assessment of each business unit using a four-quadrant analysis.
- Horizontal Axis: Expected RAROC return.
- Vertical Axis: Quality of earnings (importance to firm, growth opportunities, long-run stability, synergies).
- Four Possibilities:
  - Low quality, low quantity: Correct, reduce, or shut down activities.
  - Low quality, high quantity (managed growth): High returns but low strategic importance.
  - High quality, low quantity (investment): Maintain low returns but high strategic value and growth potential.
  - High quality, high quantity: Allocate most resources.
Active Capital Management:
- Business units submit quarterly limit requests (economic capital, leverage, liquidity, RWA) to the RAROC team.
- RAROC team performs analysis and sets limits collaboratively.
- Senior management makes final decisions.
- Treasury group ensures limits align with funding limits.
- Leverage limitations promote optimal use of scarce capital.

Topic 2. RAROC Best Practices

Practice Questions: Q5

Q5. Which of the following statements regarding best practices in implementing a RAROC approach is correct?

A. A successful RAROC approach is focused on maximizing prots earned by the firm.

B. A restriction on the firm’s growth due to leverage limitations may result in higher profits.

C. The data collection process throughout the firm should be decentralized to allow the various business units to ensure the utmost accuracy of data.

D. Metrics involving credit risk, market risk, and operational risk to compute economic capital are generally clearly defined and may be computed objectively.

Practice Questions: Q5 Answer

Explanation: B is correct.

A restriction on the firm's growth due to leverage limitations may result in higher profits because it requires the firm to be "creative" and to optimize a scarce resource (the limited amount of capital available).
Response A is not correct. A successful RAROC approach is focused on the level of profits earned by the firm in relation to the level of risks taken. Response C is not correct. The data collection process should be the responsibility of the RAROC team; the process should be centralized with built-in edit and reasonability checks to increase the accuracy of the data. Response D is not correct. Metrics involving operational risk are not as defined as credit and market risk, therefore, there is often a significant amount of subjectivity involved in the computations.