Skip to content

CFA Level II Exam · Analysis of Active Portfolio Management

Active Risk and Active Return Decomposition Explained

Updated 7 October 2026 · Fact-checked

Active return is portfolio return minus benchmark return. You split it into asset allocation (over- or underweighting segments) and security selection (picking better or worse assets within segments). Active risk, or tracking error, splits into active factor risk and active specific risk, and the two variances add. Active share measures holdings difference, not return volatility.

Understand Active Risk and Active Return Decomposition

Active management means holding a portfolio that differs from its benchmark. The difference in returns is active return: portfolio return minus benchmark return. The standard deviation of that difference over time is active risk, also called tracking error.

You can explain active return in two ways. The first is by decision. Asset allocation is the return from holding different weights in segments (sectors, countries, asset classes) than the benchmark. Security selection is the return from holding assets inside a segment that did better or worse than the benchmark's assets in that same segment. Under the definitions used on this page (allocation measured at benchmark segment returns, selection measured at portfolio weights), the two parts add up exactly to total active return. Some other conventions, such as a Brinson three-effect model, add a separate interaction term, so check which definitions the question uses.

The second way is by source of risk. Active factor risk comes from the portfolio having different exposures to systematic factors (such as market, size, value, or industry) than the benchmark. Active specific risk comes from weights in individual assets that differ from the benchmark, which exposes you to asset-specific returns not explained by the factors. Variances add: active risk² = active factor variance + active specific variance. Standard deviations do not add.

Active share is a different idea. It measures how much the holdings differ from the benchmark: half the sum of the absolute weight differences. It ranges from 0% (identical to the benchmark) to 100% (no overlap). Tracking error measures how much the return difference fluctuates. A portfolio can have a high active share and low tracking error, for example if it holds different stocks with similar factor exposures. A portfolio can have low active share and still have meaningful tracking error if its few bets are in volatile or highly correlated assets.

In an item set, the vignette usually gives weights and returns in an exhibit, or factor betas and variances. Your job is to pick the right numbers, apply the matching formula, and keep units consistent.

Key formulas to remember

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.

How to solve Active Risk and Active Return Decomposition questions

Use this order for any question on active return or active risk decomposition.

  1. 1Read the question and decide which measure is asked: return decomposition (allocation or selection), risk decomposition (factor or specific), or active share.
  2. 2Find the data in the exhibit. Separate portfolio weights, benchmark weights, portfolio returns and benchmark returns, and note whether they are in percent or decimals.
  3. 3Compute active weights (wP − wB) for each segment or asset. Check that they sum to zero.
  4. 4For return: apply allocation = active weight × benchmark segment return, and selection = portfolio weight × (portfolio return − benchmark return). Sum each across segments.
  5. 5For risk: compute active factor variance from the beta differences and factor variances, and active specific variance from squared active weights times specific variances. Add the variances, then take the square root for tracking error.
  6. 6For active share: sum the absolute active weights and halve it.
  7. 7Check: under these definitions, allocation + selection should equal total active return, and factor + specific variance should equal total active variance. If the question uses a model with an interaction term, include it in the check.
  8. 8Answer in the unit requested. Do not forget the square root when the question asks for tracking error rather than variance.

Quickest way: Active-weight table shortcut

When to use it: Use when the exhibit gives weights and returns by segment and you must find allocation, selection or active share.

  1. Build one table with columns: wP, wB, wP − wB, RP,i, RB,i.
  2. Allocation per row is column 3 × RB,i. Selection per row is wP × (RP,i − RB,i).
  3. Add the columns. Compare the sum of allocation and selection with RP − RB computed independently from the totals.
  4. For active share, add the absolute values in column 3 and halve.
  5. If the question only needs the total active return, skip the split and compute RP − RB directly.

Common mistakes in Active Risk and Active Return Decomposition

  • Adding tracking errors instead of variances when combining factor and specific risk.

    Students treat risk as additive like return.

    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.

    The two formulas look alike and the labels get mixed up.

    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.

  • Forgetting to halve the sum in active share.

    Students stop after summing the absolute differences.

    Fix: Each overweight is matched by an underweight, so the sum double counts the shift. Always multiply by ½.

  • Treating active share and tracking error as the same measure.

    Both are called measures of how active a manager is.

    Fix: Active share is about holdings. Tracking error is about the volatility of the return difference. They can point in different directions.

  • Forgetting to include assets held only by the portfolio or only by the benchmark in active share.

    Students only list the assets in the portfolio.

    Fix: Include every asset in either list. A benchmark asset not held has a portfolio weight of 0 and an active weight equal to minus its benchmark weight.

  • Not squaring the active weight in active specific risk.

    Students copy the pattern of a weighted average.

    Fix: Variance involves weights squared. Use (wP,i − wB,i)² × σ²(εi). A negative active weight still adds risk.

Worked examples

Example 1

A portfolio and its benchmark hold three sectors. Portfolio weights, benchmark weights, portfolio returns and benchmark returns: Sector A: 50%, 40%, 10%, 8%. Sector B: 30%, 40%, 4%, 5%. Sector C: 20%, 20%, 6%, 7%. Q1: What is total active return? Q2: What is the asset allocation effect? Q3: What is the security selection effect?

Show the solution
  1. Portfolio return = 0.50 × 10% + 0.30 × 4% + 0.20 × 6% = 5.0% + 1.2% + 1.2% = 7.4%.
  2. Benchmark return = 0.40 × 8% + 0.40 × 5% + 0.20 × 7% = 3.2% + 2.0% + 1.4% = 6.6%.
  3. Active return = 7.4% − 6.6% = 0.8%.
  4. Active weights: A +10%, B −10%, C 0%. They sum to zero.
  5. Allocation: A = 10% × 8% = 0.8%. B = −10% × 5% = −0.5%. C = 0. Total = 0.3%.
  6. Selection: A = 0.50 × (10% − 8%) = 1.0%. B = 0.30 × (4% − 5%) = −0.3%. C = 0.20 × (6% − 7%) = −0.2%. Total = 0.5%.
  7. Check: 0.3% + 0.5% = 0.8%, which matches total active return under these definitions (allocation at benchmark returns, selection at portfolio weights).

Answer: Q1: 0.8%. Q2: 0.3%. Q3: 0.5%. Most of the outperformance came from security selection, mainly in Sector A.

Example 2

A portfolio and benchmark hold five stocks. Portfolio weights: W 40%, X 30%, Y 20%, Z 10%, V 0%. Benchmark weights: W 25%, X 25%, Y 20%, Z 15%, V 15%. A risk model estimates active factor variance of 9 (in %²) and active specific variance of 16 (in %²). Q1: What is the active share? Q2: What is the tracking error? Q3: What proportion of active variance is due to factor risk?

Show the solution
  1. Active weights: W +15%, X +5%, Y 0%, Z −5%, V −15%. They sum to zero.
  2. Sum of absolute active weights = 15 + 5 + 0 + 5 + 15 = 40%.
  3. Active share = ½ × 40% = 20%.
  4. Active variance = factor variance + specific variance = 9 + 16 = 25 (%²).
  5. Tracking error = √25 = 5%.
  6. Factor proportion = 9 ÷ 25 = 36%.

Answer: Q1: 20%. Q2: 5%. Q3: 36%. Note that the standard deviations are 3% (factor) and 4% (specific), and they do not add to 5%; only the variances add.

Exam tips

  • Write out the table of active weights first. It serves allocation, selection, active share and specific risk questions.
  • Run the check: allocation plus selection must equal RP − RB when the question uses these definitions. This catches most arithmetic and formula-mix errors.
  • Watch units. Variances may be given in %² while returns are in %. Take the square root only at the end.
  • Expect conceptual options such as 'high active share implies high tracking error'. This is not always true. Choose answers that keep holdings and return volatility separate.
  • If a vignette gives factor betas for portfolio and benchmark, the active exposure is the difference in betas, not the portfolio beta alone.

Active Risk and Active Return Decomposition in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Active Risk and Active Return Decomposition: frequently asked questions

What is the difference between tracking error and active share?

Tracking error is the standard deviation of portfolio return minus benchmark return. It measures how much the return gap varies. Active share is half the sum of absolute weight differences and measures how different the holdings are. They answer different questions.

How do I decompose active return into allocation and selection?

Allocation is the sum of (portfolio weight − benchmark weight) × benchmark segment return. Selection is the sum of portfolio weight × (portfolio segment return − benchmark segment return). Under these definitions they add to total active return. Other conventions add a separate interaction term.

What are active factor risk and active specific risk?

Active factor risk is the part of active variance explained by differences in factor exposures between portfolio and benchmark. Active specific risk is the part from asset-specific returns, caused by active weights in individual assets. Add the variances to get total active variance.

Can active share be above 100%?

For a long-only portfolio, no. It runs from 0% for a portfolio identical to the benchmark to 100% when there is no overlap. Use half the sum of absolute active weights over all assets in either portfolio.