Analysis of Active Portfolio Management: formula sheet

Key formulas

Active return

RA = RP − RB

RP is portfolio return and RB is benchmark return for the same period. Positive means value added.

Active weight

Δwi = wP,i − wB,i

Portfolio weight minus benchmark weight for asset or sector i. Overweight is positive.

Active return from active weights

RA = Σ Δwi × Ri = Σ Δwi × (Ri − RB)

The second form works because active weights sum to zero. Use it when the portfolio and benchmark are both fully invested.

Sum of active weights

Σ Δwi = 0

Holds when portfolio and benchmark weights each sum to 100%. A cash or leverage difference breaks this.

Sharpe ratio

SR_P = (R_P − R_F) ÷ σ_P

Uses total risk. R_F is the risk-free rate. Use the benchmark's own return and standard deviation for SR_B.

Active return

R_A = R_P − R_B

Portfolio return minus benchmark return. Can be negative.

Active risk (tracking risk)

σ_A = standard deviation of (R_P − R_B)

Not the difference between the two standard deviations. It is the standard deviation of the return differences.

Information ratio

IR = (R_P − R_B) ÷ σ_A

For expected values, IR = E(R_A) ÷ σ_A.

Expected active return

E(R_A) = IR × σ_A

Rearranged IR. Useful once you have the optimal active risk.

Optimal active risk

σ_A* = (IR ÷ SR_B) × σ_B

Gives the active risk that maximises the Sharpe ratio, assuming active return is uncorrelated with the benchmark.

Sharpe ratio of optimal portfolio

SR_P* = √(SR_B² + IR²)

Square both ratios, add, then take the square root.

Fundamental law link

IR = TC × IC × √BR

TC is the transfer coefficient, IC the information coefficient, BR the breadth. With full transfer (TC = 1), IR = IC × √BR.

Information ratio

IR = E(RA) ÷ σA

RA is portfolio return minus benchmark return. σA is active risk (tracking risk).

Basic fundamental law

IR = IC × √BR

Assumes no constraints on implementing forecasts.

Expected active return (basic)

E(RA) = IC × √BR × σA

Multiply the basic IR by active risk.

Full fundamental law

IR = TC × IC × √BR

TC is the correlation between desired and actual active weights, from 0 to 1.

Expected active return (full)

E(RA) = TC × IC × √BR × σA

Use when a transfer coefficient is given or constraints are described.

Breadth

BR = number of independent investment decisions per year

Count only independent decisions, not the number of securities held.

Active return

Active return = RP − RB

Portfolio return minus benchmark return, over the same period.

Asset allocation effect

Allocation = Σ (wP,i − wB,i) × RB,i

Sum over segments i. Uses the benchmark segment return. You can also write allocation = Σ(wP,i − wB,i)(RB,i − RB). The total is identical to the version using RB,i because the active weights sum to zero, but the individual segment values differ.

Security selection effect

Selection = Σ wP,i × (RP,i − RB,i)

Uses the portfolio's weights. With the allocation formula above (allocation at benchmark returns, selection at portfolio weights), allocation + selection = RP − RB exactly. Other conventions use a separate interaction term, so the two parts would not sum on their own.

Active risk (tracking error)

Active risk = standard deviation of (RP − RB)

Active risk² = active factor variance + active specific variance.

Active factor risk (variance)

Active factor variance = Σk Σj (βP,k − βB,k)(βP,j − βB,j) Cov(Fk, Fj)

With one factor this reduces to (βP − βB)² × σF².

Active specific risk (variance)

Active specific variance = Σ (wP,i − wB,i)² × σ²(εi)

Assumes asset-specific returns are uncorrelated across assets. Active weights are squared, so signs do not matter.

Active share

Active share = ½ × Σ |wP,i − wB,i|

Sum over all assets, including those held by only one of the two. Result is between 0% and 100% for a long-only portfolio.

Information ratio

IR = E(R_A) ÷ σ_A

E(R_A) is expected active return and σ_A is active risk (standard deviation of active return).

Benchmark Sharpe ratio

SR_B = [E(R_B) − R_F] ÷ σ_B

Uses excess return over the risk-free rate.

Maximum Sharpe ratio of the combination

SR_P = √(SR_B² + IR²)

Holds at the optimal active risk, when active return is uncorrelated with the benchmark. Square first, add, then take the root.

Optimal active risk

σ_A* = (IR ÷ SR_B) × σ_B

Gives the Sharpe-maximizing active risk. Higher IR means more active risk. Higher SR_B means less.

Expected active return at a given risk

E(R_A) = IR × σ_A

Rearranged IR. Use it to find the active return at the optimal risk.

Total risk of benchmark plus active

σ_P = √(σ_B² + σ_A²)

Valid when active return is uncorrelated with the benchmark.

Fundamental law (unconstrained)

IR = IC × √BR

IC is skill, BR is the number of independent decisions per year.

Fundamental law with constraints

IR = TC × IC × √BR

TC is between 0 and 1. Constraints such as long-only rules reduce TC and so reduce IR.

Quick revision

• Active return = portfolio return − benchmark return.
• Active weight = portfolio weight − benchmark weight.
• Sharpe ratio = (Rp − Rf) ÷ σp, which uses total risk.
• Information ratio = active return ÷ active risk, where active risk is the standard deviation of active returns (tracking error).
• Basic fundamental law: expected IR ≈ IC × √BR, where IC is the information coefficient and BR is breadth.
• Expanded form: expected IR ≈ TC × IC × √BR, where TC is the transfer coefficient.
• A transfer coefficient below 1 means constraints stop the portfolio from using all of the manager's insights.
• Breadth means the number of independent decisions per year, not just the number of securities held.
• Active risk squared = factor risk squared + security selection risk squared, under the curriculum's decomposition.
• Optimal active risk rises with a higher information ratio and falls with greater risk aversion.
• Use the Sharpe ratio to compare total-risk performance and the information ratio to judge active skill against a benchmark.
• Check the units: use consistent periods (annual or monthly) for returns and risk before dividing.

Common mistakes

• Subtracting in the wrong order, benchmark minus portfolio. Fix: Always write portfolio minus benchmark. A manager who beats the benchmark has a positive active return.
• Using portfolio weights instead of active weights when computing value added from allocation. Fix: Subtract benchmark weights first. Only then multiply by returns.
• Using portfolio standard deviation in the information ratio denominator. Fix: IR always divides by active risk, the standard deviation of R_P − R_B. Sharpe divides by total standard deviation.
• Subtracting the benchmark's standard deviation from the portfolio's to get active risk. Fix: Active risk is the standard deviation of the return differences. Use the tracking error given, or compute it from the difference series.
• Forgetting the square root on breadth. Fix: Always write √BR. Compute the root before multiplying.
• Counting the number of stocks held as breadth. Fix: Use the number of independent decisions per year given in the vignette. Correlated bets reduce breadth.
• Adding tracking errors instead of variances when combining factor and specific risk. Fix: Square each component to get variance, add, then take the square root. Active risk² = factor variance + specific variance.
• Using the portfolio weight in the allocation formula or the benchmark weight in the selection formula. Fix: Allocation uses the weight difference (wP − wB) times the benchmark segment return. Selection uses the portfolio weight times the return difference. Then confirm the sum matches RP − RB.
• Using the active return directly instead of the information ratio when finding the best Sharpe ratio. Fix: Always compute IR = active return ÷ active risk first. Then use SR_P = √(SR_B² + IR²).
• Adding IR and SR_B instead of adding their squares. Fix: Write SR_B² and IR² as separate numbers, add them, then take the root. The combined ratio is less than SR_B + IR.

Exam tips

• Read which benchmark the mandate uses before touching numbers. Vignettes often list more than one index.
• Always compute active weights first. They also show you where the manager's bets are, which many questions ask about.
• Check that active weights sum to zero. It catches errors fast and costs seconds.
• Watch the units. Options may differ by 10 times or 100 times if you confuse percent and basis points.
• No penalty for wrong answers, so answer every question even if you must estimate the sign and size.
• Read what the standard deviation in the exhibit is labelled. Tracking error, active risk and tracking risk all mean the IR denominator.
• Expect a vignette to give several Sharpe ratios. Pick the benchmark's for σ_A* and the combination formula.
• Do the numerator carefully with signs. A manager can have a positive Sharpe ratio and a negative IR.