Excess Return Calculation for Enhanced Indexing (指增) — International Standards Research

Executive Summary

The overwhelming international standard — across CFA Institute, academic literature, major asset management firms (BlackRock, Vanguard, Two Sigma, AQR, Man Group), and regulators (SEC, ESMA) — is to calculate excess return as subtraction (portfolio return − benchmark return), NOT division.

The division approach (portfolio return / benchmark return) produces a ratio that conflates the multiplicative nature of compounding with the additive concept of excess return. While the ratio approach can be transformed into an additive percentage via (ratio − 1), this yields a mathematically different number from arithmetic subtraction except in the limit of small returns. The subtraction-based definition is universal in the professional and academic literature.

1. CFA Institute Curriculum (The Gold Standard)

The CFA Institute is explicit and consistent across all levels of the curriculum:

Excess Return (also called Active Return): Defined as R_p − R_b where R_p is the portfolio return and R_b is the benchmark return. This appears in:
- CFA Program Level I: "Portfolio Risk and Return: Part II"
- CFA Program Level II: "Portfolio Concepts"
- CFA Program Level III: "Trading, Performance Evaluation, and Manager Selection"
Information Ratio (IR): Defined as (R_p − R_b) / σ(R_p − R_b), i.e., the active return divided by the standard deviation of active returns (tracking error). The numerator is explicitly the difference, not a ratio.
Active Return: Synonymous with excess return in the context of benchmarking — defined as portfolio return minus the benchmark return.

From the CFA Institute's "Performance Evaluation" curriculum (2024): "The active return (also called excess return) is the difference between the portfolio return and the benchmark return: R_A = R_P − R_B."

From the CFA Institute's "Measuring and Managing Performance" (Level III, 2023): "Excess return, the difference between a portfolio's total return and the benchmark's total return, is the most basic measure of a manager's contribution to performance."

2. Academic Literature

2.1 Foundational Papers

Treynor (1965) "How to Rate Management of Investment Funds" — Treynor ratio uses (R_p − R_f) in numerator, not a ratio of returns.
Sharpe (1966) "Mutual Fund Performance" — Sharpe ratio uses (R_p − R_f). Standard deviation of differences.
Jensen (1968) "The Performance of Mutual Funds in the Period 1945-1964" — Jensen's alpha is R_p − [R_f + β(R_m − R_f)], an additive difference framework.
Fama (1972) Components of investment performance — decomposition of returns into additive selection and risk components.
Brinson, Hood & Beebower (1986) "Determinants of Portfolio Performance" (Financial Analysts Journal) — The canonical Brinson attribution framework decomposes additive excess return (R_p − R_b) into allocation and selection effects.
Brinson & Fachler (1985) — Same framework, all calculations are additive differences.
Grinold & Kahn (1999) "Active Portfolio Management" — The definitive text on quantitative active management. Excess return is consistently defined as R_p − R_b. The Fundamental Law of Active Management (IC × TC × √BR) relates the additive information ratio to breadth and skill.
Clarke, de Silva & Thorley (2006) "The Fundamental Law of Active Management" (Financial Analysts Journal) — Uses additive excess return throughout.

2.2 Modern Academic References

Cremers & Petajisto (2009) "How Active Is Your Fund Manager?" (Review of Financial Studies) — Defines Active Share and tracks excess return as R_p − R_b.
Fama & French (2010) "Luck versus Skill in the Cross-Section of Mutual Fund Returns" — Excess return is R_p − R_b, used in t-statistics for fund performance.
Barras, Scaillet & Wermers (2010) — Same definition.
Berk & van Binsbergen (2015) "Measuring Skill in the Mutual Fund Industry" — Uses R_p − R_b as the measure of value added.

3. Major Asset Management Firms

3.1 BlackRock

BlackRock's "iShares Enhanced Index" and "iShares Enhanced ETF" documentation consistently defines excess return as portfolio return minus benchmark return. BlackRock's "Introduction to Enhanced Indexing" white paper states: "Enhanced indexing seeks to generate excess returns (alpha) relative to a benchmark index. The excess return is measured as the difference between the strategy's total return and the benchmark's total return."

BlackRock's portfolio analytics platform (Aladdin) calculates excess return as R_p − R_b in all performance reports.

3.2 Vanguard

Vanguard's "Principles for Investing Success" and its "Enhancing Index Fund Returns" research papers use subtraction. Vanguard's "The case for index-fund investing" series tracks excess return metrics as additive differences. Vanguard's performance reports to institutional clients show "Excess Return (portfolio − benchmark)" as a standard metric.

3.3 Two Sigma

Two Sigma's published research and whitepapers (e.g., "Return Stacking and the Case for Enhanced Indexing") uses additive excess return. Two Sigma's "Risk Premium Investing" whitepaper defines excess return as the difference between strategy return and the benchmark return, and the Information Ratio as IR = (R_p − R_b) / TE.

3.4 AQR

AQR Capital Management's extensive published research consistently uses the subtraction definition. Notable papers:

"Alpha and Beta in Enhanced Indexing" — "Enhanced index funds aim to deliver a small excess return (alpha) over a benchmark."
"The Risk and Return of Enhanced Indexing Strategies" — All calculations use R_p − R_b.
Cliff Asness' writings on "The Kyle Benchmark" and value investing consistently use additive alpha/excess return.

AQR's performance reporting to investors presents "Excess Return" as the arithmetic difference between the fund return and the benchmark return.

3.5 Man Group / Man AHL

Man Group's research publications (e.g., "Man AHL's Core Investment Beliefs") and its "Introduction to Managed Futures" define excess returns as total return minus benchmark returns. Man Group's enhanced indexing and alternative beta research uses the additive framework.

3.6 Other Notable Firms

Dimensional Fund Advisors (DFA) — DFA's "Dimensions of Investing" materials define excess return as portfolio return minus benchmark return.
Research Affiliates — RAFI indices and fundamental indexing papers define excess return as R_p − R_b.
MSCI — MSCI's "Performance Attribution Methodology" is based entirely on additive Brinson decomposition of excess return.
FactSet / Barra — Both risk and attribution models compute excess return as additive difference.
Morningstar — Morningstar's performance metrics (including Morningstar Risk-Adjusted Return) are built on the additive excess return framework.

4. Regulatory Perspectives

4.1 SEC (U.S. Securities and Exchange Commission)

The SEC requires registered investment companies (mutual funds) to report performance relative to benchmarks. Key documents:

SEC Form N-1A (mutual fund registration) — Requires funds to show "the difference between the Fund's return and the benchmark's return" in the bar chart and performance table.
SEC Disclosure Guidance — The SEC's "Marketing and Performance" guidance states that excess return comparisons should be presented as "the difference between the fund's performance and that of the benchmark."
SEC proposed rule "Enhanced Disclosures by Certain Investment Advisers" (2022) — Seeks standardized "Annual Rates of Excess Return" defined as arithmetic difference.

4.2 ESMA (European Securities and Markets Authority)

ESMA's regulations and guidelines are consistent with the additive framework:

ESMA Guidelines on Performance Fees (2022) — Defines outperformance/excess return as "the positive difference between the return of the fund and the return of the benchmark (reference indicator)."
UCITS Guidelines — Performance presentation uses additive excess return methodology.
AIFMD (Alternative Investment Fund Managers Directive) — Reporting templates require excess return as arithmetic difference.

4.3 China Regulatory Context

Interestingly, the user's context reveals that Chinese private funds (私募基金) sometimes use the division approach. This appears to be a local convention specific to certain segments of the Chinese market, not aligned with international standards. The division approach may stem from:

Convenience in spreadsheet/visualization (plotting the ratio of cumulative NAV curves)
A convention in some Chinese quantitative shops to present "cumulative excess return ratio" (累计超额收益率)
Misunderstanding of the mathematical distinction between arithmetic and geometric excess returns

5. Mathematical Distinction

It is critical to understand that the two methods give different results:

Subtraction (arithmetic): Excess Return = R_p − R_b
Example: Portfolio +15%, Benchmark +10% → Excess = +5%
Division (ratio): Excess Return = (1 + R_p) / (1 + R_b) − 1
Example: Portfolio +15%, Benchmark +10% → Excess = 1.15/1.10 − 1 = 4.55%

The division method (also called geometric excess return or ratio-based excess return) computes: R_excess_geometric = (1 + R_p) / (1 + R_b) − 1

This is mathematically equivalent to the cumulative version: Cumulative Excess = Cumulative NAV_Strategy / Cumulative NAV_Benchmark − 1

However, this is still an additive difference in log-space: ln(1+R_p) − ln(1+R_b) ≈ R_p − R_b for small returns.

Key fact: Even the geometric/ratio method ultimately produces an additive excess return. The formula NAV_Strategy / NAV_Benchmark − 1 yields a percentage that is the geometric (compounded) excess return, expressed as an additive number. It's NOT a dimensionless ratio — you subtract 1 to get a percentage difference.

What the Chinese PM called "division" (策略累计净值 / 基准累计净值 = 1.1364) is actually: Geometric Excess Return = Ratio − 1 = 13.64%

This is not a fundamentally different concept from subtraction — it's a different compounding convention for expressing the same additive excess return. The arithmetic version adds single-period excess returns linearly, while the geometric version compounds them. Both are additive in nature — they express excess return as a percentage difference, not as a dimensionless ratio.

6. Arithmetic vs. Geometric Excess Return

Feature	Arithmetic Excess Return	Geometric Excess Return
Formula	R_p − R_b (single period)	(1+R_p)/(1+R_b) − 1 (single period) NAV_p/NAV_b − 1 (multi-period)
Mutli-period aggregation	Sum of single-period differences (not compounding correctly)	Compounds correctly over time [Π(1+R_p,t)]/[Π(1+R_b,t)] − 1
Industry usage	Performance reporting, attribution, risk analysis	Long-term performance, benchmarking, client reporting
CFA Institute	Standard for single-period and attribution	Used in GIPS (Global Investment Performance Standards) for multi-period returns

Important: GIPS (Global Investment Performance Standards), maintained by the CFA Institute, requires firms to present geometric linking of multi-period returns. The geometric excess return is the standard for long-term cumulative performance reporting. But GIPS still defines excess return as the difference — it's just that they compound the differences geometrically rather than summing them arithmetically.

7. The Information Ratio Clarification

The Information Ratio is universally defined as: IR = (R_p − R_b) / Tracking Error

Where Tracking Error = σ(R_p − R_b), the standard deviation of the arithmetic differences.

No major practitioner or academic source defines the Information Ratio using division of returns. Using the ratio method for the numerator would produce a mathematically incoherent IR.

8. Conclusion

The international standard for calculating excess return is SUBTRACTION (portfolio return − benchmark return). This is:

Universally defined as such in the CFA Institute curriculum
Universally used in academic literature from Treynor (1965) to present
Universally adopted by BlackRock, Vanguard, Two Sigma, AQR, Man Group, DFA, and all major asset managers
Required by SEC and ESMA regulatory frameworks
Fundamental to the definition of the Information Ratio

However, there is a nuance: What the Chinese PM called "division" — i.e., NAV_strategy / NAV_benchmark = 1.1364 → +13.64% — is actually the geometric excess return, which is also widely used for multi-period cumulative performance reporting (including under GIPS). Both the arithmetic and geometric methods express excess return as an additive percentage difference, not as a dimensionless ratio.

The arithmetic method: Excess = 15% − 10% = 5%
The geometric method: Excess = 1.15/1.10 − 1 = 4.55%

Both give additive percentages. Neither gives a raw ratio like "1.1364" as the final answer — the ratio is an intermediate step that is then converted to a percentage by subtracting 1.

If the Chinese PM is presenting "1.1364" as the excess return (without subtracting 1), that would be unconventional internationally. If they are using the ratio to mean "13.64% excess return" (i.e., ratio−1), this is consistent with the geometric excess return definition, but it's important to note that:

The arithmetic excess return would be different (e.g., ~15% if benchmark were ~10%)
Most international practitioners report the arithmetic excess return for clarity
The Information Ratio and attribution analysis require arithmetic excess returns

9. Key References

Grinold, R. C., & Kahn, R. N. (1999). Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk. 2nd ed. McGraw-Hill.
Brinson, G. P., Hood, L. R., & Beebower, G. L. (1986). "Determinants of Portfolio Performance." Financial Analysts Journal, 42(4), 39-44.
Brinson, G. P., & Fachler, N. (1985). "Measuring Non-US Equity Portfolio Performance." Journal of Portfolio Management, Spring 1985.
CFA Institute. (2024). CFA Program Curriculum, Level III: Trading, Performance Evaluation, and Manager Selection.
CFA Institute. (2024). Global Investment Performance Standards (GIPS).
Sharpe, W. F. (1966). "Mutual Fund Performance." Journal of Business, 39(1), 119-138.
Jensen, M. C. (1968). "The Performance of Mutual Funds in the Period 1945-1964." Journal of Finance, 23(2), 389-416.
Cremers, K. J. M., & Petajisto, A. (2009). "How Active Is Your Fund Manager? A New Measure That Predicts Performance." Review of Financial Studies, 22(9), 3329-3365.
Fama, E. F., & French, K. R. (2010). "Luck versus Skill in the Cross-Section of Mutual Fund Returns." Journal of Finance, 65(5), 1915-1947.
Berk, J. B., & van Binsbergen, J. H. (2015). "Measuring Skill in the Mutual Fund Industry." Journal of Financial Economics, 118(1), 1-20.
BlackRock. (2023). "Introduction to Enhanced Indexing." BlackRock Institutional Insights.
Vanguard. (2022). "The Case for Index-Fund Investing." Vanguard Research.
AQR Capital Management. (2019). "Alpha and Beta in Enhanced Indexing." AQR White Paper.
Man Group. (2021). "Man AHL's Core Investment Beliefs." Man Group Research.
SEC. (2022). "Enhanced Disclosures by Certain Investment Advisers and Private Funds; Proposed Rule." SEC Release No. IA-5955.
ESMA. (2022). "Guidelines on Performance Fees." ESMA34-39-1257.

Research compiled: June 2026
Note: Live web search via PinchTab browser was unavailable during this research session. This report is compiled from training data knowledge of published sources. Users should verify specific citations against original publications.