Overview / Key takeaways
– RAROC (risk‑adjusted return on capital) is a profitability metric that adjusts an investment’s return for expected risk. It helps compare investments with different risk profiles by subtracting expected losses and adding an income-from-capital adjustment, then dividing by the capital allocated.
– RAROC is widely used at financial institutions for pricing, capital allocation and evaluating acquisitions. It was developed at Bankers Trust in the late 1970s and became common in the 1980s. (Investopedia, Michela Buttignol)
– The core inputs are: revenue, expenses, expected loss, income from capital, and capital. Expected loss is typically estimated from default probabilities and loss given default (PD × LGD × exposure). (Capital.com)
The RAROC formula
A commonly used algebraic form:
RAROC = (r − e − EL + IFC) / c
Where:
– r = revenue (expected income over a specified period)
– e = expenses (operating costs associated with the activity)
– EL = expected loss (expected average loss over the same period)
– IFC = income from capital (an adjustment that typically equals capital charges × risk‑free rate, or simply capital × risk‑free rate)
– c = capital (capital allocated or economic capital held against the activity)
Note: Variants exist. Some practitioners use regulatory capital or economic capital; some include other adjustments (taxes, overhead allocation). Always document which convention you use.
How RAROC provides insight into investment risk
– RAROC converts raw profitability into a return per unit of capital after accounting for expected losses. That makes returns comparable across lines of business or projects with different risk profiles.
– A higher RAROC implies higher risk‑adjusted profitability. Decision-makers usually compare RAROC to a hurdle rate (required return or cost of capital). If RAROC > hurdle, the investment is attractive on a risk‑adjusted basis.
Origins: Bankers Trust and the development of RAROC
– RAROC was developed at Bankers Trust under Dan Borge in the late 1970s as financial firms shifted into wholesale banking, securities, and derivatives. It was intended to improve on simple return-on-capital metrics by embedding risk considerations; other banks later adopted similar systems under different names. (Investopedia, Michela Buttignol)
RAROC vs. RORAC (and related measures)
– RORAC (return on risk‑adjusted capital) is closely related; the key difference is conceptual emphasis: RORAC typically adjusts the capital for risk and then computes return, whereas RAROC explicitly adjusts return by expected losses and divides by capital. In practice the formulas and uses overlap and firms often use the terms interchangeably.
– Other related metrics: RARORAC (risk‑adjusted return on risk‑adjusted capital), economic capital‑based metrics, and regulatory measures. Choose the metric whose inputs align with your decision context.
How can you determine the expected loss (EL)?
Expected loss is the average loss you expect over the measurement period. In credit contexts it is commonly estimated as:
EL = PD × LGD × EAD
Where:
– PD = probability of default over the period
– LGD = loss given default (percentage of exposure lost if default occurs)
– EAD = exposure at default (dollar exposure when default happens)
Practical ways to estimate these components:
– Historical default rates and internal credit-cycle data
– Rating-model outputs or external ratings mapped to PDs
– Statistical credit models (logistic regression, survival analysis, machine learning)
– Expert judgment and overlays for low‑data segments
– Stress testing and scenario analysis to explore variability beyond EL (unexpected losses)
(See Capital.com for an accessible primer on expected loss concepts.)
Other methods of assessing risk and return
RAROC should be used alongside other measures:
– Sharpe ratio and Sortino ratio: risk/return metrics for marketable securities, measuring returns relative to volatility or downside deviation.
– Value at Risk (VaR) and Conditional VaR (CVaR): tail‑risk measures for loss distributions.
– Return on invested capital (ROIC), return on equity (ROE), and simple ROI: useful for non‑risk‑adjusted comparisons.
– RORAC: focuses on return per unit of risk‑adjusted capital.
– Economic capital / economic profit models: for enterprise risk allocation and pricing.
Are there drawbacks to using RAROC?
Yes—important limitations to keep in mind:
– Data and model intensity: estimating EL, capital charges and the right capital allocation can require detailed inputs and modeling effort.
– Focus on expected loss: EL ignores unexpected (tail) losses. Two investments with identical ELs can have very different tail risks.
– Model risk and assumptions: PDs, LGDs and capital assumptions can be misspecified or unstable across cycles.
– Procyclicality: models calibrated on benign periods under‑price risk in downturns.
– Misuse in isolation: a high RAROC doesn’t guarantee a good investment if risk of catastrophic loss, correlations, or strategic considerations are unfavorable.
– Incentive distortions: if managers are assessed only on RAROC, they may take risks that increase expected return but create concentrated tail risk.
How RAROC differs from simple ROI/ROC
– ROI/ROC measure returns relative to invested capital or cost, without explicit adjustment for expected loss or capital charges.
– RAROC subtracts expected losses and adds an income-from-capital adjustment before dividing by capital, yielding a return net of expected credit/operational loss — making it a risk‑sensitive profitability metric.
– Example (conceptual): A bet that doubles money with a 50% chance of losing everything has a nominal ROI of 100% if it wins, but RAROC will reduce the expected return to reflect the 50% chance of loss — producing a much lower risk‑adjusted return.
Practical, step‑by‑step guide to calculating and using RAROC
1. Define scope and period
• Decide the activity, time horizon (e.g., 1 year), and whether you use economic or regulatory capital.
2. Gather inputs
• Revenue (r): expected cash inflows (interest, fees, capital gains).
• Expenses (e): operating and direct costs attributable to the activity.
• Expected loss (EL): estimate via PD × LGD × EAD or other suitable methodology.
• Capital (c): capital allocated to the activity (economic capital, regulatory capital, or an internal allocation).
• Risk‑free rate / capital charge: choose a consistent basis to compute IFC.
3. Calculate income from capital (IFC)
• IFC = capital × risk‑free rate (or IFC = capital charges × risk‑free rate, depending on firm convention).
4. Compute RAROC
• RAROC = (r − e − EL + IFC) / c
5. Benchmark against a hurdle rate
• Compare RAROC to the organization’s required return/hurdle (e.g., cost of capital, target RAROC). Accept if RAROC > hurdle.
6. Perform sensitivity and stress analysis
• Vary PDs, LGDs, revenue scenarios and capital levels to see how RAROC reacts.
• Evaluate tail risk separately (VaR, CVaR, scenario losses), since RAROC reflects expected outcomes not extremes.
7. Use RAROC for decisions and governance
• Pricing: set prices or spreads to meet target RAROC.
• Capital allocation: allocate capital to maximize enterprise RAROC subject to risk limits.
• Performance measurement: use alongside risk limits and other metrics (ROE, VaR, capital-at-risk).
Example (simplified)
– Loan portfolio item:
• Expected interest revenue (r) = $70
• Expenses (e) = $10
• Expected loss (EL) = $5
• Capital allocated (c) = $50
• Risk‑free rate = 2% → IFC = 50 × 0.02 = $1
• RAROC = (70 − 10 − 5 + 1) / 50 = 56 / 50 = 112%
Decision rule:
– If the firm’s RAROC hurdle is 10% (or the appropriate cost-of-capital), this opportunity looks attractive on a risk‑adjusted basis. However, you must still check for tail risk, concentration, strategic fit and regulatory constraints.
Best practices and practical tips
– Be explicit about definitions: state whether “capital” is economic/regulatory and how EL is estimated.
– Use EL (expected) for RAROC numerator but complement with measures of unexpected loss and tail risk.
– Run scenario and sensitivity analysis to capture cyclicality and model uncertainty.
– Regularly backtest PD/LGD models and update overlays to reflect changing conditions.
– Combine RAROC with limits, qualitative assessment and strategic criteria before making decisions.
The bottom line
RAROC is a powerful, widely used framework to compare profitability after adjusting for expected risk and capital cost. When properly specified and combined with stress testing and tail‑risk metrics, RAROC supports pricing, capital allocation and performance measurement. However, it requires quality inputs, careful modeling, governance and complementary risk measures—relying on RAROC alone can be misleading.
Sources
– Buttignol, Michela. “Risk-Adjusted Return on Capital (RAROC).” Investopedia. (Accessed via user-supplied content)
– Capital.com. “What is expected loss?” (Accessed Jul. 24, 2024)
Continuing from the previous discussion, below are additional sections that deepen practical application of RAROC, provide worked examples, show alternatives and complementarities, and offer implementation guidance and conclusions.
How to determine the expected loss (EL)
– Core formula for credit-type exposures: EL = PD × LGD × EAD
• PD = Probability of Default over the chosen horizon (e.g., 1 year)
• LGD = Loss Given Default (typically expressed as a percentage of exposure)
• EAD = Exposure at Default (the amount at risk when default occurs)
• Example source for the EL concept: Capital.com (What is expected loss?)1
– Estimating components:
• PD: use historical default rates by borrower rating, internal scorecards, or market-implied PDs (from CDS spreads).
• LGD: estimate recovery rates from historical recoveries or collateral liquidation assumptions.
• EAD: for loans, it may be outstanding balance plus expected undrawn commitments; for derivatives, use potential future exposure models.
– For non-credit risks:
• Market risk: translate into an expected loss estimate by using stressed historical losses, expected shortfall, or probability-weighted scenario losses.
• Operational risk: use internal loss databases and scenario analysis to estimate average annual loss.
– Time horizon and confidence level:
• Be explicit whether EL is an annual expected loss or for another horizon; RAROC calculations must use consistent horizons for revenue, expenses and EL.
Practical steps to calculate RAROC (operational checklist)
1. Define the time horizon (e.g., one year).
2. Select the capital base c:
• Decide whether to use economic capital (preferred for internal performance) or regulatory capital (if benchmarking to regulators).
3. Compute expected income and costs for the horizon:
• Revenue r (interest, fees, mark-to-market gains as appropriate).
• Expenses e (direct operating costs allocated to the product or unit).
4. Estimate expected loss el using PD × LGD × EAD or analogous methods for market/operational risk.
5. Compute income from capital ifc:
• Typically = capital charges × risk-free rate (captures what that capital could earn if invested risk-free).
6. Apply the RAROC formula:
• RAROC = (r − e − el + ifc) / c
7. Compare to hurdle rate:
• Set an internal RAROC target (hurdle) that reflects the cost of equity, opportunity cost and strategic requirements.
8. Perform sensitivity and scenario analysis:
• Stress PDs, LGDs, revenues, and capital assumptions; test for tail events.
Worked examples
Example A — Simple loan product
– Assumptions (1-year horizon):
• Revenue r (interest + fees): $10,000,000
• Expenses e: $2,000,000
• Expected loss el = PD × LGD × EAD = $1,000,000
• Capital allocated c (economic capital): $5,000,000
• Risk-free rate: 2%; capital income ifc = c × risk-free = $5,000,000 × 0.02 = $100,000
– Computation:
• Numerator = r − e − el + ifc = 10,000,000 − 2,000,000 − 1,000,000 + 100,000 = 7,100,000
• RAROC = 7,100,000 / 5,000,000 = 1.42 = 142%
– Interpretation:
• A very high RAROC suggests the product is highly profitable after expected losses. Check assumptions (maybe revenue or capital allocation is atypical) and stress-test.
Example B — Coin-flip investment (illustrative risk adjustment)
– Setup:
• You can double $100 to $200 with probability 50%; otherwise you lose the $100 (single-period bet).
• If you define revenue r only on success, the “gross return on success” is 100% (profit $100 on $100).
– Expected loss:
• Probability of failure = 50% → EL = 0.5 × $100 loss = $50
– If using c = original $100 capital, no operating expenses, risk-free rate = 0:
• RAROC = (r − e − el + ifc) / c = (100 − 0 − 50 + 0)/100 = 0.50 = 50%
– Conclusion:
• Although a successful flip yields 100% gross ROI, the expected (risk-adjusted) return is 50% once expected loss is accounted for. This demonstrates the intuition behind RAROC.
Comparing RAROC, RORAC, and related metrics
– RAROC (Risk-Adjusted Return on Capital): adjusts returns by expected losses and includes income from capital (as shown above).
– RORAC (Return on Risk-Adjusted Capital): usually refers to return / risk-adjusted capital, where the capital denominator is risk-adjusted; emphasis is on adjusting the capital base rather than subtracting EL. Practitioners sometimes use the terms interchangeably; clarify definitions inside an institution.
– RARORAC and variations: some firms use more complex denominators (e.g., capital after regulatory risk buffers) or use economic capital under stressed scenarios.
– Complementary measures:
• Sharpe ratio, Sortino ratio — useful for market portfolios to assess returns per unit of volatility or downside risk.
• VaR / Expected Shortfall — provide tail-loss measures, useful alongside EL which captures average loss but may understate tail risk.
• EVA (Economic Value Added) — focuses on profit above cost of capital, different emphasis but complementary.
Strengths of RAROC
– Brings risk explicitly into profitability measurement — enables apples-to-apples comparisons across lines of business with different risk profiles.
– Aligns pricing and capital allocation decisions with risk.
– Supports incentive design, limits setting and capital budgeting.
– Flexible — can be adapted to different capital definitions (economic vs regulatory), time horizons and risk types.
Drawbacks and practical limitations
– Data and model intensity: accurate EL estimates require quality data (PD, LGD, EAD) and robust models; small-data settings increase estimation error.
– Underestimates tail risk: EL is an average; it may miss low-probability, high-severity events unless complemented by tail-risk measures (VaR, ES).
– Capital allocation disputes: how capital is attributed across products/units affects RAROC outcomes; allocation methodologies can materially change rankings.
– Time horizon mismatches: if revenues and expected losses are measured over different horizons, results can be misleading.
– Procyclicality: EL and capital needs often decline in good times and spike in downturns; relying purely on point-in-time RAROC can lead to mispricing cyclicality.
– Overreliance risk: a high RAROC does not guarantee low absolute risk — an investment can have a high RAROC yet still expose the firm to catastrophic tail loss.
Best practices and governance when using RAROC
– Use economic capital as the preferred c for internal performance measurement; track regulatory capital separately for compliance.
– Combine RAROC with tail-risk metrics (VaR, Expected Shortfall) and stress-testing results.
– Standardize definitions across the firm (horizon, treatment of fees, tax treatments, capital income).
– Backtest EL and capital allocation assumptions periodically; recalibrate PD/LGD models.
– Use scenario analysis and reverse stress testing to understand cases where RAROC remains attractive but leads to unacceptable outcomes.
– Integrate RAROC into capital planning cycles and limit frameworks; have clear hurdle rates tied to cost of capital and strategic objectives.
Implementation roadmap (step-by-step for a finance team)
1. Define purpose (pricing, performance measurement, portfolio selection).
2. Agree definitions (economic vs regulatory capital, horizon, EL approach).
3. Build or refine EL models (PD, LGD, EAD) and collect historical loss data.
4. Determine capital allocation methodology (standalone economic capital, marginal capital, or allocated capital).
5. Set RAROC hurdle(s) that reflect cost of equity and strategic return targets.
6. Run baseline RAROC calculations and rank products/units.
7. Conduct stress tests and scenario analyses to supplement RAROC findings.
8. Implement governance (reporting frequency, model oversight, escalation paths).
9. Monitor and recalibrate periodically.
Other methods of assessing risk and return (when to use them)
– Sharpe ratio: for portfolio-level tradeoffs of return vs volatility (useful for marketable securities).
– Sortino ratio: focuses on downside deviation (better when upside volatility is desirable).
– VaR and Expected Shortfall: quantify potential tail losses at given confidence levels.
– RWA (Risk-Weighted Assets): regulatory construct to measure capital needs under Basel frameworks.
– Marginal contribution to risk: to allocate portfolio-level risk to individual positions.
Use these together: RAROC for capital-oriented profitability, Sharpe/Sortino for volatility-adjusted performance, VaR/ES for tail risk.
Regulatory context and capital definitions
– Basel frameworks and regulators focus on risk-weighted assets (RWAs) and regulatory capital ratios; RAROC typically uses economic capital for internal C-suite and business decisions.
– Firms should reconcile RAROC outputs with regulatory capital metrics to ensure alignment between profitability targets and prudential constraints.
Additional example — RAROC with market risk and VaR
– Suppose a trading desk expects annual mark-to-market gains r = $4M, expenses e = $0.5M, no credit EL but market expected loss over the year estimated as $0.3M from scenario analysis; allocated economic capital c = $10M; risk-free rate 1% so ifc = $100k.
• RAROC = (4,000,000 − 500,000 − 300,000 + 100,000) / 10,000,000 = 3,300,000 / 10,000,000 = 33%
– Augment this with VaR(99%) = $20M and an expected shortfall to see that although RAROC looks attractive, tail risk could be large — a signal to run stress tests and consider capital buffers.
Concluding summary
– RAROC provides a structured, risk-adjusted way to measure return on capital by subtracting expected losses and adding income forgone on capital before dividing by capital. It helps align pricing, capital allocation and strategic decisions with risk.
– Correct calculation hinges on robust estimates of expected loss, a clear choice of capital base (economic vs regulatory), and consistent time horizons.
– RAROC is powerful but not sufficient alone: pair it with tail-risk measures (VaR, Expected Shortfall), stress testing, and careful governance to avoid misleading conclusions driven by estimation error or ignored extreme events.
– In practice, use RAROC to rank and price activities, but always complement it with scenario analysis and capital planning. When implemented carefully, RAROC drives better risk-aware decisions; when misused or used in isolation, it can create blind spots.
References
– Investopedia. “Risk-Adjusted Return on Capital (RAROC).” Michela Buttignol. Accessed [source user provided].
– Capital.com. “What is expected loss?” Accessed Jul. 24, 2024.
Editor’s note: The following topics are reserved for upcoming updates and will be expanded with detailed examples and datasets.