Title: Understanding and Using Excess Returns — A Practical Guide
Key takeaways
– Excess return = investment return minus the return of a chosen comparator (benchmark or risk-free rate).
– Excess returns can be raw (vs. a risk-free rate) or risk-adjusted (alpha vs. a like-for-like benchmark).
– Interpreting excess return requires consideration of risk (beta), implementation costs, time horizon, and statistical significance.
– Use complementary metrics (Jensen’s alpha, Sharpe ratio, information ratio, tracking error) to determine whether excess performance is meaningful.
Introduction
Excess return measures how much an investment outperformed (or underperformed) another reference return. Investors use excess returns to judge whether a strategy, fund manager, or individual security produced value beyond what could have been obtained from a chosen alternative — typically a risk-free instrument or a market benchmark.
Definitions and core formulas
– Excess return (simple): Excess = R_investment − R_benchmark
Example: One‑year Treasury = 2.0%, Tech stock = 15.0% → Excess = 15.0% − 2.0% = 13.0%.
– Capital Asset Pricing Model (CAPM) (expected return):
R_expected = R_rf + β × (R_m − R_rf)
where R_rf = risk-free rate, R_m = market return, β = security beta.
– Jensen’s alpha (CAPM alpha):
α_Jensen = R_portfolio − [R_rf + β_portfolio × (R_m − R_rf)]
This measures the portfolio return above CAPM-predicted return.
– Sharpe ratio (risk-adjusted return vs. risk-free):
Sharpe = (R_portfolio − R_rf) / σ_portfolio
where σ_portfolio is the standard deviation of portfolio returns.
– Information ratio:
Information ratio = Active return / Tracking error
Active return = R_portfolio − R_benchmark
Tracking error = standard deviation of (R_portfolio − R_benchmark)
Practical steps to calculate and evaluate excess returns
1. Choose the comparator
– For absolute “extra” return, use a risk-free rate (e.g., matching-maturity U.S. Treasury yield).
– For like‑for‑like performance, use a closely comparable benchmark (e.g., S&P 500 for large-cap U.S. equities, Nasdaq 100 for large-cap tech).
– Document the choice and the rationale.
2. Match measurement periods and return type
– Use the same time window and return frequency (daily, monthly, annual) for both series.
– Decide arithmetic vs. geometric returns; for multi-period performance prefer geometric (compound) returns when annualizing.
3. Calculate raw excess return
– Period excess = R_investment_period − R_benchmark_period.
– Cumulative/annualized excess: compute annualized returns for each and subtract.
4. Adjust for implementation costs
– Subtract fees, transaction costs, taxes, and any model replication costs that would affect an investor’s realized return.
– Note: benchmark indices do not include trading costs or some real-world frictions.
5. Assess risk and risk-adjusted excess
– Compute beta (regress investment excess returns on market excess returns to estimate β).
– Compute Jensen’s alpha to see whether the investment beat its CAPM-predicted return.
– Compute Sharpe ratio to see return per unit of volatility (using risk-free rate).
– Compute Information ratio to assess consistency of active outperformance vs. benchmark.
6. Test statistical significance and persistence
– Run t-tests or regression significance tests on alpha to evaluate whether excess is likely due to skill or noise.
– Use rolling windows or out-of-sample tests to check if excess returns persist over time.
Worked examples
Example A — Simple excess vs. risk-free:
– R_Treasury (1‑yr) = 2.0%, R_stock = 15.0%
– Excess = 15.0% − 2.0% = 13.0%.
Example B — Fund alpha (illustrative):
– Fund return = 12.0%, S&P 500 = 7.0%, Rf = 2.0%, fund β = 1.0.
– Jensen’s alpha = 12.0% − [2.0% + 1.0×(7.0% − 2.0%)] = 12.0% − (2.0% + 5.0%) = 5.0%.
Example C — Sharpe and Information ratios:
– Fund return = 12.0%, Rf = 2.0%, σ_fund = 10.0% → Sharpe = (12% − 2%)/10% = 1.0.
– Active return vs benchmark = 5.0%, tracking error = 4.0% → Information ratio = 5%/4% = 1.25.
Interpreting results
– Positive raw excess means the investment outperformed the comparator; negative means underperformance.
– A positive alpha (Jensen’s) indicates performance above CAPM expectation after accounting for systematic risk.
– Sharpe > 1.0 is generally considered good (varies by asset class/time); Information ratio > 0.5 is often viewed as solid for active managers.
– Consider magnitude, consistency, and statistical significance. A large one‑off excess may be luck; persistent, significant excess is more evidence of skill.
Practical checklist for investors and analysts
– Select the right benchmark and risk-free rate matched to the investment horizon and style.
– Use consistent data frequency and compound appropriately when annualizing.
– Net out fees, slippage, and taxes from realized returns.
– Compute both raw and risk‑adjusted excess (alpha, Sharpe, info ratio).
– Check statistical significance (confidence intervals, p-values).
– Perform out-of-sample or rolling-window tests to evaluate persistence.
– Consider transaction costs and whether the benchmark was truly investable for an ordinary investor.
– Be wary of data biases (survivorship bias, look-ahead bias).
Special considerations and common pitfalls
– Benchmark mismatch: Choosing a broad index for a niche strategy can misstate excess and risk.
– Fees and implementation: Index returns ignore trading costs; a manager’s required trades, liquidity, or fees reduce realized excess.
– Leverage and higher beta: Higher raw excess that comes from higher systematic risk (β > 1) is different from manager skill; use risk‑adjusted metrics.
– Multi-factor risks: A single‑factor CAPM alpha may miss exposure to value, size, momentum, or other systematic factors — consider multi-factor models for deeper attribution.
– Short windows and small samples: Small-sample excess returns may be noise; longer horizons and more observations improve reliability.
– Survivorship bias: Historical queries that exclude failed funds or delisted stocks overstate average excess.
Tools and simple formulas (Excel / quick code)
– Excel: Excess = B2 − C2 (where B2 = investment return, C2 = benchmark return).
– Excel annualize monthly return r_month: (1 + r_month)^12 − 1.
– Beta and alpha: Run linear regression in Excel (LINEST) or use slope/intercept of regressing (R_invest − Rf) on (R_benchmark − Rf).
– Python (pandas/statsmodels): compute returns, then OLS to get intercept (alpha) and slope (beta).
Conclusion
Excess return is a foundational concept for evaluating whether an investment delivered value beyond a chosen alternative. But raw excess is only a starting point — adjusting for risk, costs, investment constraints, and statistical significance is essential to determine whether the excess is meaningful and repeatable. Combining simple excess return calculations with risk‑adjusted metrics (Jensen’s alpha, Sharpe, information ratio) and a disciplined analytic process helps investors decide whether outperformance reflects skill, compensation for extra risk, or merely chance.
Source
– Investopedia — “Excess Return” (https://www.investopedia.com/terms/e/excessreturn.asp)
Jensen’s Alpha and How It’s Used
Jensen’s alpha (often just “Jensen’s α”) is a risk-adjusted performance measure that compares a portfolio’s actual return to the return predicted by the Capital Asset Pricing Model (CAPM). It isolates the part of a portfolio’s return that cannot be explained by market exposure (beta). A positive Jensen’s alpha suggests that a manager generated returns beyond what would be expected for the portfolio’s systematic risk; a negative alpha suggests underperformance.
Formula
Jensen’s alpha = R_p − [R_f + β_p × (R_m − R_f)]
Where:
– R_p = portfolio (or fund) return
– R_f = risk-free rate
– β_p = portfolio beta relative to the market
– R_m = market return
Numeric example
– Portfolio return (R_p) = 12%
– Risk-free rate (R_f) = 2%
– Market return (R_m) = 8%
– Portfolio beta (β_p) = 1.1
Expected return by CAPM = 2% + 1.1 × (8% − 2%) = 2% + 6.6% = 8.6%
Jensen’s alpha = 12% − 8.6% = +3.4%
Interpretation: This portfolio generated 3.4 percentage points of excess return above what CAPM predicts for its level of market risk.
Sharpe Ratio and Other Risk-Adjusted Metrics
Sharpe ratio
The Sharpe ratio measures return in excess of a risk-free rate per unit of total risk (standard deviation). It’s commonly used to compare risk-adjusted performance across portfolios.
Sharpe ratio = (R_p − R_f) / σ_p
Where σ_p is the standard deviation of portfolio returns.
Example
Using the same portfolio return of 12%, risk-free rate 2%, and portfolio volatility σ_p = 10%:
Sharpe = (12% − 2%) / 10% = 1.0
Information ratio
The information ratio compares active excess return relative to a benchmark divided by the tracking error (volatility of active returns).
Information ratio = (R_p − R_b) / σ_active
Where R_b is benchmark return and σ_active is the standard deviation of (R_p − R_b).
Example: If a fund outperforms its benchmark by 3% with a tracking error of 4%, the information ratio = 0.75.
Practical Steps to Calculate and Evaluate Excess Returns
1. Decide your comparison benchmark(s)
– Risk-free rate (e.g., short-term U.S. Treasury yields) for absolute excess returns.
– A relevant market index (S&P 500, Nasdaq 100, sector index) for alpha or active excess returns.
– A style-consistent benchmark for funds that target a specific market segment.
2. Choose the return measure and time horizon
– Use total return (price appreciation + dividends/interest) over an appropriate horizon (monthly, annual, multi-year).
– Match benchmark and portfolio return frequency (daily, monthly, annually).
3. Calculate simple excess return
– Excess = R_asset − R_benchmark.
4. Adjust for risk (calculate beta, Jensen’s alpha, Sharpe ratio)
– Regress portfolio excess returns vs market returns to estimate beta.
– Compute Jensen’s alpha using CAPM if you want to control for systematic risk.
– Use Sharpe to evaluate return per unit of total volatility.
5. Adjust for costs and taxes
– Subtract fees, transaction costs, and estimate tax impacts to find net excess return to the investor.
– Consider whether the benchmark’s replication costs are reflected.
6. Test statistical significance and persistence
– Run a t-test on alpha to see if excess return is statistically different from zero.
– Analyze rolling windows and multiple periods to look for persistent skill rather than luck.
7. Consider capacity, liquidity, and implementation
– Some strategies lose effectiveness as assets under management grow.
– Illiquid securities can produce high gross excess return but be costly to implement.
Worked examples (step-by-step)
Example A — Excess vs. Risk-Free (simple)
– Asset: Tech stock return = 15% for year.
– One-year Treasury (R_f) = 2%.
– Excess vs. risk-free = 15% − 2% = 13%.
Example B — Fund alpha vs. stated benchmark
– Fund return = 12%, benchmark (S&P 500) = 7% for the year.
– Simple alpha (active excess) = 12% − 7% = 5%.
Example C — Jensen’s alpha (risk-adjusted)
– Fund return = 12%
– Risk-free = 2%
– Market return = 8%
– Fund beta = 1.1
– Jensen’s alpha = 12% − [2% + 1.1×(8%−2%)] = 12% − 8.6% = 3.4%
Example D — Sharpe and Information ratio
– Fund return = 12%, risk-free 2%, volatility = 10% → Sharpe = (12−2)/10 = 1.0
– Fund outperforms benchmark by 3% with tracking error 4% → Information ratio = 3/4 = 0.75
Special Considerations and Common Pitfalls
– Fees and transaction costs: Gross excess returns can look attractive but may disappear after management fees, performance fees, and trading costs. Always analyze net-of-fee returns.
– Benchmark choice: Using an inappropriate benchmark can misstate performance. Compare like-with-like (style, capitalization, region, sector).
– Survivorship bias: Published performance may exclude failed funds. Use databases that include delisted funds to avoid bias.
– Look-ahead/data-snooping bias: Be wary of backtests that implicitly use future information or overfit parameters.
– Short sample size: Small samples increase the chance of mistaking luck for skill.
– Risk adjustment: Raw excess returns are insufficient. Use measures like Jensen’s alpha, Sharpe, Information ratio, and multi-factor alphas (Fama–French) when appropriate.
– Leverage and derivatives: These can inflate gross returns but also raise risk and costs.
– Taxes: Different investors face different after-tax returns. Compare after-tax excess returns if relevant.
How to Test Whether Excess Return Is Skill or Luck
1. Use longer sample periods and out-of-sample testing.
2. Apply bootstrapping or Monte Carlo simulations to assess how often a given alpha could arise by chance.
3. Check consistency across market regimes (bull vs bear markets).
4. Analyze turnover and trade-level data — high turnover with little net excess after costs suggests luck or noise.
5. Evaluate manager process and repeatability — documented, repeatable processes increase the plausibility of skill.
When Excess Return Is Worth Pursuing
– Excess return that remains positive after fees, taxes, and realistic implementation costs.
– Risk-adjusted measures (Sharpe, Jensen’s alpha) show improvement versus passive alternatives.
– The excess is persistent over multiple, varied market conditions.
– The strategy aligns with investor goals, risk tolerance, liquidity needs, and capacity constraints.
Applying Excess Return Analysis — A Practical Checklist for Investors
1. Define your objective: beat a risk-free rate, outperform a benchmark, or achieve a target Sharpe.
2. Select appropriate benchmark(s) and time horizon.
3. Compute gross and net excess returns.
4. Adjust for risk (Jensen’s alpha, Sharpe, Information ratio).
5. Test statistical significance and persistence.
6. Consider fees, taxes, liquidity, and replication costs.
7. Evaluate manager/process quality and capacity.
8. Reassess periodically using rolling windows and scenario analyses.
Concluding Summary
Excess return is a fundamental concept for measuring how much an investment outperforms a chosen comparator—whether a risk-free rate, a broad market index, or a carefully chosen benchmark. Simple excess-return calculations are easy to compute, but a meaningful assessment requires careful benchmarking, risk adjustment, and attention to implementation costs and statistical robustness. Jensen’s alpha, the Sharpe ratio, and the information ratio each provide complementary perspectives: Jensen’s alpha isolates performance after accounting for market risk (beta), the Sharpe ratio evaluates returns per unit of total volatility, and the information ratio measures active performance relative to tracking error. For investors, the central questions are (1) is the excess return positive after realistic costs and taxes; (2) is it risk‑adjusted and statistically significant; and (3) is it persistent and implementable at scale? Following the practical steps and checks above will help determine whether excess returns represent genuine skill, acceptable compensation for risk, or merely luck.
References and Further Reading
– Investopedia — “Excess Return” (Matthew Collins). https://www.investopedia.com/terms/e/excessreturn.asp
– Sharpe, William F., “Mutual Fund Performance,” Journal of Business, Vol. 39, No. 1 (1966): 119–138.
– Jensen, Michael C., “The Performance of Mutual Funds in the Period 1945–1964,” Journal of Finance (1968).
– Fama, Eugene F., and Kenneth R. French, “Common risk factors in the returns on stocks and bonds,” Journal of Financial Economics (1993).
[[END]]