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11.4 Analysis of Active Portfolio Management & The Fundamental Law

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CFA Level 2 Chartered Financial Analyst

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1 Introduction & CFA Level II Exam Strategy

1.1 CFA Level II Exam Overview & Item-Set Vignette Strategy 1.2 2026 Curriculum Structure & 10 Topic Weight Bands 1.3 300-Hour Study Plan, Calculator Mastery & Practical Skills Modules (PSM)

2 Ethical and Professional Standards

2.1 Code of Ethics & The Seven Standards of Professional Conduct 2.2 Vignette Analysis: Conflicts of Interest & Disclosure (Standard VI) 2.3 Duties to Clients (Standard III) & Duties to Employers (Standard IV) 2.4 Standard V: Investment Analysis, Recommendations & Actions and Standard VII: Responsibilities as a CFA Member or Candidate 2.5 Application of the Code and Standards: Level II Case Method 2.6 Recommended Procedures: Compliance Programs, Supervisory Systems & Firm Policies

3 Quantitative Methods

3.1 Multiple Linear Regression, ANOVA & Hypothesis Testing 3.2 Regression Diagnostics: Multicollinearity, Heteroskedasticity & Serial Correlation 3.3 Time-Series Analysis: AR Models, Nonstationarity & Cointegration 3.4 Machine Learning, Deep Learning & Big Data Projects 3.5 Extensions of Multiple Regression: Influence Analysis, Qualitative Variables & Logistic Regression 3.6 Big Data Projects: Wrangling, Exploration, Feature Engineering & Model Training

4 Economics for Valuation

4.1 Foreign Exchange Parity Relations & Equilibrium Exchange Rates 4.2 Economic Growth Theories & Production Function Analysis 4.3 Forward Points, the Carry Trade & Policy Effects on Exchange Rates 4.4 Growth Determinants: Natural Resources, Demographics, Convergence & Trade Openness

5 Financial Statement Analysis

5.1 Intercorporate Investments: Equity Method & Business Combinations 5.2 Employee Compensation: Defined Benefit Pension Plans & Share-Based Payments 5.3 Multinational Operations: Current Rate vs Temporal Translation 5.4 Analysis of Financial Institutions & The CAMELS Framework 5.5 Evaluating the Quality of Financial Reports & Accounting Adjustments 5.6 Integration of Financial Statement Analysis Techniques

6 Corporate Issuers

6.1 Analysis of Dividends: Policy Theories, Signaling & Taxation Systems 6.2 Share Repurchases: Methods, EPS and Book Value Effects & Payout Sustainability 6.3 Cost of Capital: Advanced Topics 6.4 Corporate Governance, Board Oversight & ESG Integration 6.5 Corporate Restructuring: Types, Motivations, Evaluation & Valuation 6.6 Restructuring Financial Effects: EPS, Net Debt to EBITDA, WACC & Divestitures

7 Equity Valuation

7.1 Equity Valuation Framework & Required Rate of Return Estimation 7.2 Discounted Dividend Valuation: Gordon Growth, Multistage & The H-Model 7.3 Free Cash Flow Valuation: FCFF and FCFE Modeling 7.4 Market-Based Valuation: Price and Enterprise Value Multiples 7.5 Residual Income Valuation 7.6 Private Company Valuation: Income, Market & Asset-Based Approaches 7.7 Choosing and Combining Models: Definitions of Value, Sum-of-the-Parts & Cross-Border Comparability

8 Fixed Income Analysis

8.1 The Term Structure and Interest Rate Dynamics: Spot, Forward & Swap Curves 8.2 The Arbitrage-Free Valuation Framework & Binomial Interest Rate Trees 8.3 Valuation and Analysis of Bonds with Embedded Options: OAS & Duration 8.4 Credit Analysis Models: Structural vs Reduced-Form Approaches 8.5 Credit Default Swaps (CDS): Mechanics, Pricing & Trading Strategies 8.6 Managing Yield-Curve Risk: Key Rate Durations, One-Sided Durations & Yield Volatility Structure

9 Derivatives & Risk Management

9.1 Pricing and Valuation of Forward Commitments: Equity, Fixed Income & FX 9.2 Swap Pricing and Valuation: Interest Rate, Currency & Equity Swaps 9.3 Contingent Claims: Binomial Option Pricing Model & Arbitrage Replication 9.4 The Black-Scholes-Merton (BSM) Model, Option Greeks & Dynamic Hedging 9.5 Interest Rate Options, Swaptions & the Black Model

10 Alternative Investments

10.1 Commodities and Commodity Derivatives: Sectors, Participants, Futures Returns & Indexes 10.2 Private Real Estate Investments: Direct Capitalization, DCF & Cost Approach 10.3 Public Real Estate Investments: REIT Structures, NAV, FFO & AFFO 10.4 Hedge Fund Strategies: Classification, Implementation & Portfolio Role 10.5 Real Estate Investment Features, Due Diligence & Index Construction Biases

11 Portfolio Management

11.1 Using Multifactor Models: Macroeconomic, Fundamental & Statistical Risk 11.2 Measuring and Managing Market Risk: VaR, CVaR & Stress Testing 11.3 Economics & Investment Markets: Business Cycles, Term Structure & Asset Allocation 11.4 Analysis of Active Portfolio Management & The Fundamental Law 11.5 Exchange-Traded Funds (ETFs): Creation, Arbitrage, Tracking & Applications 11.6 Backtesting and Simulation: Procedures, Metrics, Biases & Scenario Analysis

11.4 Analysis of Active Portfolio Management & The Fundamental Law

Key Takeaways Explain This Topic AI

The Information Ratio ( I R = R A ‾ / σ A IR = \overline{R_A} / \sigma_A I R = R A ​ ​ / σ A ​ ) measures excess return generated per unit of active risk; combining an active strategy expands total portfolio Sharpe Ratio via S R P = S R B 2 + I R 2 SR_P = \sqrt{SR_B^2 + IR^2} S R P ​ = S R B 2 ​ + I R 2 ​ .

The Unconstrained Fundamental Law of Active Management (Grinold) establishes that I R = I C × B R IR = IC \times \sqrt{BR} I R = I C × BR ​ and expected active return E ( R A ) = I C × B R × σ A E(R_A) = IC \times \sqrt{BR} \times \sigma_A E ( R A ​ ) = I C × BR ​ × σ A ​ .

The Full / Constrained Fundamental Law incorporates the Transfer Coefficient ( T C = Corr ( Δ w i ∗ , Δ w i ) ∈ [ 0 , 1 ] TC = \text{Corr}(\Delta w_i^*, \Delta w_i) \in [0, 1] TC = Corr ( Δ w i ∗ ​ , Δ w i ​ ) ∈ [ 0 , 1 ] ) to quantify how investment constraints dilute manager skill: I R = T C × I C × B R IR = TC \times IC \times \sqrt{BR} I R = TC × I C × BR ​ .

The Information Coefficient ( I C IC I C ) quantifies forecasting skill ( I C = 2 × [ % correct ] − 1 IC = 2 \times [\% \text{correct}] - 1 I C = 2 × [ % correct ] − 1 ), while Breadth ( B R BR BR ) represents independent decisions per year, reduced by cross-asset correlation via B R eff = N / [ 1 + ( N − 1 ) ρ ] BR_{\text{eff}} = N / [1 + (N-1)\rho] B R eff ​ = N / [ 1 + ( N − 1 ) ρ ] .

Last updated: August 2026

11.4 Analysis of Active Portfolio Management & The Fundamental Law

Core Insight: Active portfolio management is the disciplined search for alpha—excess risk-adjusted return over a benchmark. Richard Grinold and Ronald Kahn formalized this pursuit into the Fundamental Law of Active Management , which proves that active performance depends on two distinct attributes: manager skill (Information Coefficient) and breadth of opportunities (number of independent bets). Later extensions by Clarke, de Silva, and Thorley integrated the Transfer Coefficient to measure how real-world investment constraints dilute manager skill.

  • 1. Active Management Performance Framework: IR and Sharpe Ratio
  • The Information Ratio (IR)

The Information Ratio (IR) is the primary metric used to evaluate the efficiency of an active manager. It measures the average active return generated per unit of active risk (tracking error): I R = R A ‾ Active Risk = R P − R B ‾ σ ( R P − R B ) = E ( R A ) σ A IR = \frac{\overline{R_A}}{\text{Active Risk}} = \frac{\overline{R_P - R_B}}{\sigma(R_P - R_B)} = \frac{E(R_A)}{\sigma_A} I R = Active Risk R A ​ ​ ​ = σ ( R P ​ − R B ​ ) R P ​ − R B ​ ​ ​ = σ A ​ E ( R A ​ ) ​

Key Properties of the Information Ratio:

Alpha per Unit of Tracking Error: I R IR I R measures the consistency and quality of active alpha generation.

Scale Invariance: Unlike raw return, the Information Ratio is unaffected by leverage or by mixing the active portfolio with cash or the benchmark, because multiplying active positions by scalar k k k increases both active return and active risk by k k k , leaving I R IR I R unchanged.

Institutional Benchmark Standards:

I R = 0.25 IR = 0.25 I R = 0.25 : Median / Acceptable performance.

I R = 0.50 IR = 0.50 I R = 0.50 : Good / Strong institutional performance (top quartile).

I R = 1.00 IR = 1.00 I R = 1.00 : Exceptional / Elite performance (top decile).

Total Portfolio Sharpe Ratio Expansion

When an active portfolio with Information Ratio I R IR I R is combined with the benchmark portfolio (which has Sharpe Ratio S R B SR_B S R B ​ ), the total portfolio Sharpe Ratio ( S R P SR_P S R P ​ ) expands according to the fundamental Pythagorean relationship: S R P 2 = S R B 2 + I R 2 ⟹ S R P = S R B 2 + I R 2 SR_P^2 = SR_B^2 + IR^2 \implies SR_P = \sqrt{SR_B^2 + IR^2} S R P 2 ​ = S R B 2 ​ + I R 2 ⟹ S R P ​ = S R B 2 ​ + I R 2 ​ Optimal Active Risk Target ( σ A ∗ ): σ A ∗ = ( I R S R B ) σ B \text{Optimal Active Risk Target (}\sigma_A^*\text{):} \quad \sigma_A^* = \left( \frac{IR}{SR_B} \right) \sigma_B Optimal Active Risk Target ( σ A ∗ ​ ): σ A ∗ ​ = ( S R B ​ I R ​ ) σ B ​

Strategic Implication: The addition of an active strategy with an I R > 0 IR > 0 I R > 0 strictly increases the total portfolio Sharpe Ratio above the benchmark Sharpe Ratio ( S R P > S R B SR_P > SR_B S R P ​ > S R B ​ ), regardless of the benchmark's baseline return.

  • 2. The Fundamental Law of Active Management
  • The Unconstrained Fundamental Law (Grinold)

In an unconstrained investment environment (where the manager can take unconstrained long and short positions without leverage, sector, or turnover restrictions), the Information Ratio depends entirely on skill and breadth : I R = I C × B R IR = IC \times \sqrt{BR} I R = I C × BR ​ E ( R A ) = I C × B R × σ A E(R_A) = IC \times \sqrt{BR} \times \sigma_A E ( R A ​ ) = I C × BR ​ × σ A ​

Where:

I C IC I C (Information Coefficient): A measure of the manager's forecasting skill.

B R BR BR (Breadth): The number of independent investment decisions made per year.

σ A \sigma_A σ A ​ : The target active risk (tracking error).

The Constrained (Full) Fundamental Law (Clarke, de Silva & Thorley)

Real-world institutional portfolios operate under strict real-world constraints: long-only rules, maximum position size limits, sector/country tracking bands, and turnover restrictions. These constraints prevent the manager from holding the theoretically optimal active weights Δ w i ∗ \Delta w_i^* Δ w i ∗ ​ .

The Transfer Coefficient ( T C TC TC ) quantifies the correlation between the unconstrained optimal active weights ( Δ w i ∗ \Delta w_i^* Δ w i ∗ ​ ) and the actual, constrained active weights ( Δ w i \Delta w_i Δ w i ​ ): T C = Corr ( Δ w i ∗ , Δ w i ) = ∑ Δ w i ∗ Δ w i ∑ ( Δ w i ∗ ) 2 ∑ ( Δ w i ) 2 TC = \text{Corr}(\Delta w_i^*, \Delta w_i) = \frac{\sum \Delta w_i^* \Delta w_i}{\sqrt{\sum (\Delta w_i^*)^2 \sum (\Delta w_i)^2}} TC = Corr ( Δ w i ∗ ​ , Δ w i ​ ) = ∑ ( Δ w i ∗ ​ ) 2 ∑ ( Δ w i ​ ) 2 ​ ∑ Δ w i ∗ ​ Δ w i ​ ​

The Full Fundamental Law Formulation: I R = T C × I C × B R IR = TC \times IC \times \sqrt{BR} I R = TC × I C × BR ​ E ( R A ) = T C × I C × B R × σ A E(R_A) = TC \times IC \times \sqrt{BR} \times \sigma_A E ( R A ​ ) = TC × I C × BR ​ × σ A ​ Optimal Constrained Active Risk Target: σ A ∗ = T C × ( I C × B R S R B ) σ B = T C × ( I R unconstrained S R B ) σ B \text{Optimal Constrained Active Risk Target:} \quad \sigma_A^* = TC \times \left( \frac{IC \times \sqrt{BR}}{SR_B} \right) \sigma_B = TC \times \left( \frac{IR_{\text{unconstrained}}}{SR_B} \right) \sigma_B Optimal Constrained Active Risk Target: σ A ∗ ​ = TC × ( S R B ​ I C × BR ​ ​ ) σ B ​ = TC × ( S R B ​ I R unconstrained ​ ​ ) σ B ​ Constrained Total Sharpe Ratio: S R P = S R B 2 + T C 2 × I R unconstrained 2 = S R B 2 + I R constrained 2 \text{Constrained Total Sharpe Ratio:} \quad SR_P = \sqrt{SR_B^2 + TC^2 \times IR_{\text{unconstrained}}^2} = \sqrt{SR_B^2 + IR_{\text{constrained}}^2} Constrained Total Sharpe Ratio: S R P ​ = S R B 2 ​ + T C 2 × I R unconstrained 2 ​ ​ = S R B 2 ​ + I R constrained 2 ​ ​

  • 3. In-Depth Components of the Fundamental Law
  • 1. Information Coefficient ( I C IC I C )

The Information Coefficient is the cross-sectional correlation between a manager's ex-ante forecasted standardized active returns ( α ^ i \hat{\alpha}_i α ^ i ​ ) and the subsequent ex-post realized active returns ( R A , i R_{A,i} R A , i ​ ): I C = Corr ( α ^ i , R A , i ) IC = \text{Corr}(\hat{\alpha}_i, R_{A,i}) I C = Corr ( α ^ i ​ , R A , i ​ )

Market Timing / Binary Forecast Formula:

If a manager makes directional binary market calls (e.g., predicting whether an asset will outperform or underperform) with a percentage accuracy (win rate) of p p p : I C = 2 p − 1 IC = 2p - 1 I C = 2 p − 1

Example: If an analyst correctly predicts the winner in 54% of security selections ( p = 0.54 p = 0.54 p = 0.54 ): I C = 2 ( 0.54 ) − 1 = 1.08 − 1 = 0.08 IC = 2(0.54) - 1 = 1.08 - 1 = 0.08 I C = 2 ( 0.54 ) − 1 = 1.08 − 1 = 0.08

2. Breadth ( B R BR BR )

Breadth is the number of independent investment decisions made per year. If an investment process evaluates N N N securities across T T T rebalancing periods per year: B R = N × T (under strict cross-sectional and time-series independence) BR = N \times T \quad \text{(under strict cross-sectional and time-series independence)} BR = N × T (under strict cross-sectional and time-series independence)

The Independence Caveat (Correlated Decisions):

If decisions across stocks are correlated (e.g., all 50 technology stocks in an analyst's universe are driven by a common semiconductor supply cycle), the effective breadth is dramatically smaller than N N N : B R effective = N 1 + ( N − 1 ) ρ ˉ BR_{\text{effective}} = \frac{N}{1 + (N - 1) \bar{\rho}} B R effective ​ = 1 + ( N − 1 ) ρ ˉ ​ N ​

Where ρ ˉ \bar{\rho} ρ ˉ ​ is the average cross-sectional correlation among the forecast errors. If ρ ˉ = 0.20 \bar{\rho} = 0.20 ρ ˉ ​ = 0.20 across N = 100 N = 100 N = 100 stocks: B R effective = 100 1 + ( 100 − 1 ) ( 0.20 ) = 100 1 + 19.8 = 100 20.8 = 4.81 independent bets! BR_{\text{effective}} = \frac{100}{1 + (100 - 1)(0.20)} = \frac{100}{1 + 19.8} = \frac{100}{20.8} = 4.81 \text{ independent bets!} B R effective ​ = 1 + ( 100 − 1 ) ( 0.20 ) 100 ​ = 1 + 19.8 100 ​ = 20.8 100 ​ = 4.81 independent bets!

3. Transfer Coefficient ( T C TC TC ) and Long-Only Constraints

In an unconstrained long-short hedge fund, T C ≈ 1.0 TC \approx 1.0 TC ≈ 1.0 . In a traditional institutional long-only equity fund, T C TC TC typically ranges between 0.40 and 0.70 .

Why Long-Only Constraints Severely Impair Active Management:

Asymmetry in Negative Conviction: If an index stock has a benchmark weight of 0.02 % 0.02\% 0.02% , the maximum negative active weight a long-only manager can take is − 0.02 % -0.02\% − 0.02% (holding 0.00 % 0.00\% 0.00% ). Even if the manager has high conviction that the stock will collapse, they cannot monetize that insight.

Conversely, the manager can overweight a favored stock by + 3.00 % +3.00\% + 3.00% or + 5.00 % +5.00\% + 5.00% .

This structural asymmetry creates an unhedged factor tilt and destroys the transfer of forecasting skill into active portfolio weights.

4. Fundamental Law Parameter Matrix & Market Timing vs. Security Selection

Parameter Formal Symbol Theoretical Range Primary Determinants Optimization Implications

Information Coefficient I C IC I C [ − 1.0 , + 1.0 ] [-1.0, +1.0] [ − 1.0 , + 1.0 ] Analyst research quality, quantitative factor efficacy, forecasting model calibration Institutional equity I C IC I C typically ranges between 0.03 0.03 0.03 and 0.08 0.08 0.08 . High I C IC I C is rare.

Breadth B R BR BR [ 1 , ∞ ) [1, \infty) [ 1 , ∞ ) Number of assets under coverage, decision frequency, cross-asset correlation Maximized by quantitative strategies covering thousands of liquid securities globally.

Transfer Coefficient T C TC TC [ 0.0 , 1.0 ] [0.0, 1.0] [ 0.0 , 1.0 ] Long-only constraints, maximum position size limits, liquidity/turnover bounds, tracking limits T C = 1.0 TC = 1.0 TC = 1.0 for unconstrained long-short; T C ≈ 0.50 TC \approx 0.50 TC ≈ 0.50 for benchmarked long-only portfolios.

Target Active Risk σ A \sigma_A σ A ​ [ 0.0 , ∞ ) [0.0, \infty) [ 0.0 , ∞ ) Investment mandate tracking error limit, risk budget allocation Set equal to optimal σ A ∗ = T C ( I R uncon / S R B ) σ B \sigma_A^* = TC (IR_{\text{uncon}} / SR_B) \sigma_B σ A ∗ ​ = TC ( I R uncon ​ / S R B ​ ) σ B ​ to maximize total Sharpe ratio.

  • Strategic Comparison: Market Timing vs. Quantitative Stock Selection
  • Consider two distinct active management styles: Manager A: Macro Market Timer
  • High Skill / High IC: IC = 0.15 (High conviction quarterly macro calls)
  • Low Breadth: BR = 4 (Four quarterly calls per year)
  • IR_unconstrained: IR = 0.15 * sqrt(4) = 0.15 * 2 = 0.30
  • Manager B: Quantitative Security Selector
  • Modest Skill / Low IC: IC = 0.035 (Modest statistical edge per security)
  • Massive Breadth: BR = 1,600 (400 independent global stocks rebalanced quarterly)
  • IR_unconstrained: IR = 0.035 * sqrt(1,600) = 0.035 * 40 = 1.40

Takeaway: The Fundamental Law demonstrates why quantitative multi-asset strategies with modest forecasting skill ( I C = 0.035 IC = 0.035 I C = 0.035 ) can generate vastly superior Information Ratios compared to concentrated macro market timers ( I R = 1.40 IR = 1.40 I R = 1.40 vs 0.30 0.30 0.30 ), purely through the mathematical compounding of Breadth .

  • 5. Worked Active Management & Fundamental Law Calculation
  • Comprehensive Case Scenario

An institutional pension consultant is evaluating an active global equity manager. The relevant parameters are:

  • Benchmark Sharpe Ratio ( S R B SR_B S R B ​ ): 0.45 0.45 0.45
  • Benchmark Annual Volatility ( σ B \sigma_B σ B ​ ): 16.0 % 16.0\% 16.0%
  • Analyst Coverage Universe ( N N N ): 500 500 500 stocks
  • Decision Frequency: Semi-annual rebalancing ( 2 2 2 times per year)
  • Average Correlation Among Stock Forecast Errors ( ρ ˉ \bar{\rho} ρ ˉ ​ ): 0.15 0.15 0.15
  • Forecasting Accuracy Rate ( p p p ): 53.5 % 53.5\% 53.5% correct directional calls
  • Transfer Coefficient ( T C TC TC ): 0.60 0.60 0.60 (due to long-only mandate and sector constraints)
  • Active Risk Target ( σ A \sigma_A σ A ​ ): Set to the manager's optimal active risk σ A ∗ \sigma_A^* σ A ∗ ​

Step 1: Calculate Information Coefficient ( I C IC I C ) I C = 2 p − 1 = 2 ( 0.535 ) − 1 = 1.070 − 1 = 0.070 IC = 2p - 1 = 2(0.535) - 1 = 1.070 - 1 = \mathbf{0.070} I C = 2 p − 1 = 2 ( 0.535 ) − 1 = 1.070 − 1 = 0.070

Step 2: Calculate Effective Breadth ( B R effective BR_{\text{effective}} B R effective ​ )

Effective independent stocks per rebalancing period: N eff = N 1 + ( N − 1 ) ρ ˉ = 500 1 + ( 499 ) ( 0.15 ) = 500 1 + 74.85 = 500 75.85 = 6.5919 N_{\text{eff}} = \frac{N}{1 + (N - 1)\bar{\rho}} = \frac{500}{1 + (499)(0.15)} = \frac{500}{1 + 74.85} = \frac{500}{75.85} = 6.5919 N eff ​ = 1 + ( N − 1 ) ρ ˉ ​ N ​ = 1 + ( 499 ) ( 0.15 ) 500 ​ = 1 + 74.85 500 ​ = 75.85 500 ​ = 6.5919

Total annual effective breadth across 2 semi-annual decisions: B R eff = N eff × 2 = 6.5919 × 2 = 13.1839 BR_{\text{eff}} = N_{\text{eff}} \times 2 = 6.5919 \times 2 = \mathbf{13.1839} B R eff ​ = N eff ​ × 2 = 6.5919 × 2 = 13.1839

Step 3: Compute Unconstrained vs. Constrained Information Ratio

Unconstrained Information Ratio ( I R uncon IR_{\text{uncon}} I R uncon ​ ): I R uncon = I C × B R eff = 0.070 × 13.1839 = 0.070 × 3.63096 = 0.2542 IR_{\text{uncon}} = IC \times \sqrt{BR_{\text{eff}}} = 0.070 \times \sqrt{13.1839} = 0.070 \times 3.63096 = \mathbf{0.2542} I R uncon ​ = I C × B R eff ​ ​ = 0.070 × 13.1839 ​ = 0.070 × 3.63096 = 0.2542

Constrained Information Ratio ( I R con IR_{\text{con}} I R con ​ ): I R con = T C × I R uncon = 0.60 × 0.2542 = 0.1525 IR_{\text{con}} = TC \times IR_{\text{uncon}} = 0.60 \times 0.2542 = \mathbf{0.1525} I R con ​ = TC × I R uncon ​ = 0.60 × 0.2542 = 0.1525

Step 4: Calculate Optimal Active Risk Target and Expected Active Return

Optimal Constrained Active Risk ( σ A ∗ \sigma_A^* σ A ∗ ​ ): σ A ∗ = T C × ( I R uncon S R B ) × σ B = 0.60 × ( 0.2542 0.45 ) × 16.0 % = 0.60 × 0.5649 × 16.0 % = 5.423 % \sigma_A^* = TC \times \left( \frac{IR_{\text{uncon}}}{SR_B} \right) \times \sigma_B = 0.60 \times \left( \frac{0.2542}{0.45} \right) \times 16.0\% = 0.60 \times 0.5649 \times 16.0\% = \mathbf{5.423\%} σ A ∗ ​ = TC × ( S R B ​ I R uncon ​ ​ ) × σ B ​ = 0.60 × ( 0.45 0.2542 ​ ) × 16.0% = 0.60 × 0.5649 × 16.0% = 5.423%

Expected Active Return ( E ( R A ) E(R_A) E ( R A ​ ) ): E ( R A ) = I R con × σ A ∗ = 0.1525 × 5.423 % = 0.827 % E(R_A) = IR_{\text{con}} \times \sigma_A^* = 0.1525 \times 5.423\% = \mathbf{0.827\%} E ( R A ​ ) = I R con ​ × σ A ∗ ​ = 0.1525 × 5.423% = 0.827%

Step 5: Calculate Total Portfolio Sharpe Ratio ( S R P SR_P S R P ​ ) S R P = S R B 2 + I R con 2 = ( 0.45 ) 2 + ( 0.1525 ) 2 = 0.2025 + 0.023256 = 0.225756 = 0.4751 SR_P = \sqrt{SR_B^2 + IR_{\text{con}}^2} = \sqrt{(0.45)^2 + (0.1525)^2} = \sqrt{0.2025 + 0.023256} = \sqrt{0.225756} = \mathbf{0.4751} S R P ​ = S R B 2 ​ + I R con 2 ​ ​ = ( 0.45 ) 2 + ( 0.1525 ) 2 ​ = 0.2025 + 0.023256 ​ = 0.225756 ​ = 0.4751

Summary Assessment: Despite covering 500 stocks, cross-asset correlation ( ρ ˉ = 0.15 \bar{\rho} = 0.15 ρ ˉ ​ = 0.15 ) collapses annual breadth from a theoretical 1 , 000 1,000 1 , 000 down to 13.18 13.18 13.18 . Combined with the long-only constraint ( T C = 0.60 TC = 0.60 TC = 0.60 ), the realized Information Ratio is 0.1525 0.1525 0.1525 , lifting the portfolio Sharpe Ratio from 0.4500 0.4500 0.4500 to 0.4751 0.4751 0.4751 at an optimal tracking error of 5.42 % 5.42\% 5.42% .