Podrobný průvodce již brzy
Pracujeme na komplexním vzdělávacím průvodci pro Risk-Adjusted Return (RAROC). Brzy se vraťte pro podrobné vysvětlení, vzorce, příklady z praxe a odborné tipy.
Risk-adjusted return measures investment performance relative to the risk taken to achieve that return, recognizing that higher returns are meaningless if they come with proportionally higher risk. Multiple metrics have been developed to capture different dimensions of risk-adjusted performance, each with specific strengths and appropriate use cases. Together, they allow investors, fund managers, and bankers to compare performance across strategies with different risk profiles on an equal footing. The Sharpe Ratio, developed by Nobel laureate William Sharpe in 1966, is the most widely used risk-adjusted return metric: it measures excess return above the risk-free rate per unit of total portfolio volatility (standard deviation). Sharpe = (R_p − R_f) / σ_p. A Sharpe of 1.0 means one unit of excess return per unit of standard deviation — generally considered good. Sharpe above 2.0 is exceptional; below 0.5 is poor. The Sharpe ratio works best for well-diversified portfolios where volatility is a good proxy for total risk. The Treynor Ratio replaces volatility with beta (systematic risk) in the denominator: Treynor = (R_p − R_f) / β. This is appropriate for measuring contribution to a diversified portfolio, where only systematic (market) risk is relevant. The Treynor ratio is better for comparing managed funds within a broad portfolio context. Jensen's Alpha measures absolute excess return vs. the CAPM-predicted return: Alpha = R_p − [R_f + β × (R_m − R_f)]. Positive alpha indicates the manager added value beyond systematic market exposure. In banking, RAROC (Risk-Adjusted Return on Capital) is the dominant framework: RAROC = (Revenue − Expected Loss − Operating Costs) / Economic Capital. It measures the after-expected-loss, after-cost return on the capital required to support the risk. A business line or loan is attractive if its RAROC exceeds the bank's hurdle rate (cost of equity capital, typically 12–15%). RAROC is the foundation of performance measurement, incentive compensation, and strategic resource allocation in banks. The Information Ratio (IR) measures active management performance: IR = Active Return / Active Risk (tracking error). An IR above 0.5 is considered good active management; above 1.0 is exceptional. The Sortino Ratio improves on Sharpe by using downside deviation (volatility of negative returns only) in the denominator, better capturing investor asymmetric loss aversion. The Calmar Ratio uses maximum drawdown as the risk measure, particularly relevant for trend-following and alternative strategies.
Sharpe Ratio = (R_p − R_f) / σ_p Treynor Ratio = (R_p − R_f) / β Jensen's Alpha = R_p − [R_f + β × (R_m − R_f)] Sortino Ratio = (R_p − R_f) / σ_downside RAROC = (Net Revenue − Expected Loss − Operating Cost) / Economic Capital Information Ratio = (R_p − R_benchmark) / Tracking Error
- 1Gather portfolio return data (R_p), risk-free rate (R_f), benchmark return (R_m), and portfolio risk metrics (σ_p, β) over the measurement period.
- 2Calculate Sharpe Ratio: (R_p − R_f) / σ_p. For annualized Sharpe from monthly data: multiply numerator by 12 and denominator by √12.
- 3Calculate Treynor Ratio: (R_p − R_f) / β. Compare across funds with different beta exposures to evaluate systematic risk efficiency.
- 4Calculate Jensen's Alpha: compare actual return to CAPM expected return. Positive alpha = outperformance; statistical significance requires testing.
- 5Calculate Sortino Ratio: use only negative return periods to compute downside deviation σ_d. Sortino = (R_p − R_f) / σ_d. Higher than Sharpe for positively skewed strategies.
- 6For banking RAROC: Revenue = interest income + fees; Expected Loss = PD × LGD × EAD; Economic Capital = 99.9th percentile loss at the portfolio level.
- 7Compare RAROC to hurdle rate (cost of equity capital). Business lines with RAROC below hurdle destroy shareholder value; those above create it.
Higher absolute return (Fund A) loses on risk-adjusted basis to Fund B
Fund A: Sharpe = (15% − 4%) / 20% = 11% / 20% = 0.55. Fund B: Sharpe = (10% − 4%) / 8% = 6% / 8% = 0.75. Despite Fund A's 5% higher absolute return, Fund B is superior on a risk-adjusted basis because it achieves a higher return per unit of volatility. An investor who leverages Fund B 2× would have 20% volatility (matching Fund A) but 16% expected return — higher than Fund A's 15%. This leveraged-Fund-B strategy has the same risk as Fund A but higher return, confirming Fund B's risk-adjusted superiority.
Manager added 2% above what beta exposure alone would have produced
CAPM expected return = R_f + β × (R_m − R_f) = 4% + 1.2 × (14% − 4%) = 4% + 12% = 16%. Actual return = 18%. Jensen's Alpha = 18% − 16% = +2%. This positive alpha of 2% represents genuine manager skill — the ability to select securities or time markets beyond what systematic market exposure explains. Statistical significance requires testing across multiple years: a single year's 2% alpha is not statistically distinguishable from luck without at least 3–5 years of data.
RAROC above hurdle rate means this lending activity creates shareholder value
Net income before tax = $5M − $0.8M − $1.5M = $2.7M. RAROC = $2.7M / $15M = 18%. The hurdle rate (cost of equity capital) is 14%. Since RAROC (18%) > hurdle (14%), this business line creates shareholder value — for every dollar of economic capital deployed, it generates an 18% risk-adjusted return vs. the 14% minimum required. If RAROC were 10% (below hurdle), capital would be better deployed elsewhere or returned to shareholders, and the business line should be restructured or exited.
When upside volatility is high, Sortino better reflects investor experience
Sharpe = (12% − 4%) / 12% = 0.67. Sortino = (12% − 4%) / 6% = 1.33. The Sortino is exactly 2× the Sharpe because downside deviation is half of total standard deviation — the fund has significant upside volatility (good) relative to downside (bad). For strategies with positive skew (options selling with insurance-like payoffs, managed futures with trend-following), Sortino is a more appropriate measure because it doesn't penalize upside volatility. For symmetric return distributions, Sharpe and Sortino are proportional and lead to the same relative ranking.
Institutional investor manager selection and performance evaluation, where accurate risk adjusted return analysis through the Risk Adjusted Return supports evidence-based decision-making and quantitative rigor in professional workflows across diverse organizational contexts and analytical requirements
Bank business line profitability assessment and capital allocation (RAROC), where accurate risk adjusted return analysis through the Risk Adjusted Return supports evidence-based decision-making and quantitative rigor in professional workflows across diverse organizational contexts and analytical requirements
Hedge fund due diligence and performance attribution, where accurate risk adjusted return analysis through the Risk Adjusted Return supports evidence-based decision-making and quantitative rigor in professional workflows across diverse organizational contexts and analytical requirements
Portfolio construction — optimizing the Sharpe ratio of a multi-asset portfolio, where accurate risk adjusted return analysis through the Risk Adjusted Return supports evidence-based decision-making and quantitative rigor in professional workflows
Regulatory reporting — risk-adjusted metrics for investment suitability
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in risk-adjusted return on capital (raroc) calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in risk-adjusted return on capital (raroc) calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in risk-adjusted return on capital (raroc) calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
| Metric | Formula | Risk Measure | Best Used For | Typical Good Value |
|---|---|---|---|---|
| Sharpe Ratio | (Rp−Rf)/σ | Total volatility | Complete standalone portfolios | > 1.0 |
| Treynor Ratio | (Rp−Rf)/β | Systematic risk (beta) | Component of diversified portfolio | Relative comparison |
| Jensen's Alpha | Rp−[Rf+β(Rm−Rf)] | CAPM residual | Absolute manager skill | > 0% (positive) |
| Information Ratio | Active return/Tracking error | Active risk | Active vs. benchmark managers | > 0.5 |
| Sortino Ratio | Downside deviation only | Asymmetric return strategies | > 1.5 | |
| Calmar Ratio | Annual return/Max drawdown | Maximum drawdown | Trend following, alts | > 1.0 |
| RAROC | (Revenue−EL)/Econ.Cap. | Economic capital | Bank business lines, loans | > Hurdle rate |
What is a good Sharpe Ratio?
As a general guide: Sharpe below 0 means the portfolio underperforms the risk-free rate; 0–0.5 is poor; 0.5–1.0 is acceptable to good; 1.0–2.0 is very good; above 2.0 is exceptional and rare over multi-year periods. The S&P 500 has historically achieved a Sharpe of approximately 0.4–0.6 over long horizons. Top hedge funds may achieve Sharpe of 1.0–2.0. Strategies reporting Sharpe above 3.0 over sustained periods often involve significant hidden risks (illiquidity, leverage, tail risk) or involve short measurement periods where luck is inflating the metric. Sustainable high-Sharpe strategies are the holy grail of investment management.
When should Treynor Ratio be preferred over Sharpe Ratio?
The Treynor Ratio is preferred when evaluating a component of a larger, well-diversified portfolio. For a fund that represents only a portion of total wealth, investors care about the fund's contribution to portfolio risk — which is systematic risk (beta), not total volatility (which includes diversifiable idiosyncratic risk that is already eliminated in the overall portfolio). Using Sharpe for portfolio components penalizes strategies with high idiosyncratic risk even when that risk is diversified away. Treynor is the theoretically correct metric for ranking portfolio components; Sharpe is better for ranking standalone investments or complete portfolios.
What is the difference between RAROC and ROIC?
ROIC (Return on Invested Capital) = (Net Operating Profit After Tax) / (Total Invested Capital). It uses accounting invested capital (book value of debt plus equity) as the denominator. RAROC = (Risk-Adjusted Net Revenue) / Economic Capital. Economic capital is a model-based estimate of the capital required to support risk at a specified confidence level (typically 99.9%), which may differ substantially from accounting capital. RAROC adjusts the numerator for expected losses (credit cost) and uses risk-based economic capital rather than accounting capital. RAROC is therefore more appropriate for comparing business lines with different risk profiles within a financial institution, while ROIC is standard for comparing non-financial companies.
What is the information ratio and why do institutional investors prefer it?
The Information Ratio (IR) = (Portfolio Return − Benchmark Return) / Tracking Error. It measures active return per unit of active risk (deviation from benchmark). IR is preferred by institutional investors who evaluate active managers relative to a benchmark, not against the risk-free rate. A manager with IR > 0.5 over 3 years is considered skilled; IR > 1.0 is exceptional. The IR accounts for style exposures embedded in benchmarks — a small-cap manager is evaluated against a small-cap benchmark, so benchmark-beta exposure is already removed. IR focuses purely on the value added by active security selection and tactical positioning.
Why is Jensen's Alpha often criticized?
Jensen's Alpha is criticized for several reasons: (1) It depends critically on the choice of benchmark (market portfolio) — different benchmarks produce different alphas; (2) It assumes the CAPM is the correct equilibrium model — if other risk factors (size, value, momentum) explain returns, alpha merely reflects exposure to these unlisted factors, not manager skill; (3) Beta is estimated from historical data with significant error, particularly for concentrated funds; (4) Single-period alpha is not statistically significant — distinguishing skill from luck requires 3–5+ years of data. Fama-French and Carhart 4-factor models are better alternatives, controlling for size, value, and momentum factors before measuring alpha.
How does the risk-free rate choice affect Sharpe ratio calculations?
The risk-free rate affects both the numerator (excess return) and the relative ranking of funds. Short-duration strategies should use short-term T-bill rates; long-duration bond strategies might use the 10-year Treasury yield as the risk-free rate. Using the wrong maturity creates a mismatch — a short-term bond fund compared using the 10-year Treasury rate would show a negative excess return even if it performed well versus similar strategies. During periods of zero or negative interest rates, risk-free rates near zero inflate Sharpe ratios for all strategies (by increasing excess returns), making cross-period comparisons misleading.
What is the Calmar Ratio and when is it used?
The Calmar Ratio = Annualized Return / Maximum Drawdown (absolute value). It is particularly useful for evaluating trend-following, managed futures, and alternative strategies where drawdowns are the primary risk concern for investors. A Calmar of 1.0 means annual return equals maximum drawdown — acceptable. A Calmar above 2.0 indicates strong return-to-drawdown efficiency. Unlike Sharpe (which uses variance as risk), Calmar directly measures the return earned per unit of the worst historical loss, which resonates strongly with investors who think in terms of 'I could lose X% at worst — is the return worth that risk?' Trend-following funds often have low Sharpe but high Calmar due to large wins during crises.
Pro Tip
When evaluating investment managers or strategies, always compute the Sharpe ratio using the same risk-free rate and over the same time period for all alternatives being compared. Even small differences in measurement methodology can reverse the ranking of competing strategies.
Did you know?
William Sharpe developed the Sharpe Ratio in 1966 as a tool to evaluate mutual fund performance for his 1966 paper in the Journal of Business. He called it the 'reward-to-variability ratio' — the term 'Sharpe ratio' was coined by others in his honor. Sharpe received the Nobel Prize in Economics in 1990, shared with Harry Markowitz and Merton Miller, for his contributions to the theory of financial economics. The ratio bearing his name is now computed millions of times daily across investment management, risk management, and regulatory applications worldwide.