CFA Level II Exam · Analysis of Active Portfolio Management

Fundamental Law of Active Management Formula and Use

Updated 7 October 2026 · Fact-checked

The fundamental law of active management links expected active return to skill and opportunity. The basic form is E(RA) = IC × √BR × σA, so IR = IC × √BR. The full form adds the transfer coefficient: E(RA) = TC × IC × √BR × σA. You plug in the inputs from the vignette.

Understand Fundamental Law of Active Management

Active managers try to beat a benchmark. The information ratio (IR) measures how much active return they earn per unit of active risk. IR = E(RA) ÷ σA, where RA is the portfolio return minus the benchmark return and σA is the standard deviation of that difference (active risk).

The fundamental law, from Grinold and Kahn, says IR depends on two things: skill and opportunity. Skill is the information coefficient (IC). It is the correlation between your forecast active returns and the actual active returns. An IC of 0 means no skill. An IC of 1 means perfect forecasts. Real ICs are small, often below 0.15.

Opportunity is breadth (BR). It is the number of independent decisions you make per year. Independent means each decision uses separate information. Holding 100 stocks that all move on one market view is not 100 independent bets. Because breadth enters as a square root, quadrupling breadth only doubles IR.

The basic law assumes you can implement every forecast with no constraints. In practice you face limits such as no short selling, position caps and turnover limits. The transfer coefficient (TC) is the correlation between the active weights you would like to hold and the active weights you actually hold. TC is between 0 and 1. A lower TC cuts the active return you capture. In the full law, IR = TC × IC × √BR.

For the exam, read the vignette for IC, BR, TC and either IR or active risk. Then solve for the missing item.

Key formulas to remember

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.

How to solve Fundamental Law of Active Management questions

Use this method for any fundamental law question in a vignette.

1Identify what is asked: IR, expected active return, active risk, or a changed input such as breadth.
2Pull the inputs from the vignette: IC, BR, TC, σA, IR. Check whether breadth is per year and independent.
3Decide if the TC applies. If the vignette gives a TC or mentions constraints, use the full law. Otherwise use the basic law.
4Compute IR first: IR = TC × IC × √BR (set TC = 1 if not used).
5If asked for active return, multiply IR by σA. If asked for active risk, divide the target active return by IR.
6For a changed input, recompute rather than guess. Remember breadth is under a square root.
7Sanity check: IR should be modest, and active return should be below what the vignette's skill and risk would plausibly allow.

Quickest way: Scale with square roots

When to use it: When a question changes one input and asks for the new IR or active return.

Write the ratio of new to old input.
For breadth, take the square root of that ratio. Doubling breadth multiplies IR by about 1.414.
For IC, TC or σA, the effect is directly proportional.
Multiply the old IR or active return by these factors.
Check that the answer is between the options in a sensible direction.

Common mistakes in Fundamental Law of Active Management

Forgetting the square root on breadth.
Students remember IC × BR as a simple product.
Fix: Always write √BR. Compute the root before multiplying.
Counting the number of stocks held as breadth.
Breadth feels like portfolio size.
Fix: Use the number of independent decisions per year given in the vignette. Correlated bets reduce breadth.
Ignoring the transfer coefficient when constraints are described.
The basic formula is learned first.
Fix: If the vignette mentions a long-only rule, position limits or a TC value, use the full law.
Stopping at IR when asked for active return.
IR and active return get mixed up.
Fix: Active return = IR × σA. Check the units in the question.
Treating IC as a return or percentage gain.
The word 'coefficient' is unfamiliar.
Fix: IC is a correlation between forecast and actual active returns. It is a number between -1 and 1.
Assuming the law gives realised results.
The formula looks exact.
Fix: It gives expected active return, based on assumed skill and independent decisions.

Worked examples

Example 1

A manager forecasts active returns with an IC of 0.06 across 64 independent decisions a year. Active risk is 4.0%. Questions: (1) What is the expected IR using the basic law? (2) What is the expected active return? (3) If breadth rises to 256 with IC unchanged, what is the new IR?

Show the solution

IR = IC × √BR = 0.06 × √64 = 0.06 × 8 = 0.48.
E(RA) = IR × σA = 0.48 × 4.0% = 1.92%.
New IR = 0.06 × √256 = 0.06 × 16 = 0.96. Breadth rose four times, so IR doubled.

Answer: (1) IR = 0.48. (2) Expected active return = 1.92%. (3) New IR = 0.96.

Example 2

A long-only fund has IC = 0.10 and 100 independent decisions a year. Because of constraints, the correlation between desired and actual active weights is 0.70. Active risk is 5.0%. Questions: (1) What is the IR using the full law? (2) What is the expected active return? (3) What active return would be expected with no constraints?

Show the solution

IR = TC × IC × √BR = 0.70 × 0.10 × √100 = 0.70 × 0.10 × 10 = 0.70.
E(RA) = 0.70 × 5.0% = 3.50%.
Unconstrained: IR = 0.10 × 10 = 1.00. E(RA) = 1.00 × 5.0% = 5.00%.

Answer: (1) IR = 0.70. (2) Expected active return = 3.50%. (3) Unconstrained active return = 5.00%, so constraints cost 1.50 percentage points.

Exam tips

Check first whether the vignette gives a TC. If it does, the full law is almost certainly required.
Look for words such as 'independent' or 'correlated' when breadth is described. They signal that the stated breadth may be the intended input.
Answer in the form asked: IR as a plain ratio, active return as a percentage.
Practise quick square roots of 4, 9, 16, 25, 36, 49, 64, 81, 100 and 144 to save time.
If a question asks what happens when constraints are removed, think of TC rising toward 1.

Fundamental Law of Active Management in other exams

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

Fundamental Law of Active Management: frequently asked questions

What is the fundamental law of active management formula?

The basic form is IR = IC × √BR. The full form is IR = TC × IC × √BR. Multiply IR by active risk to get expected active return.

What is the difference between IC and TC?

IC is the correlation between forecast and actual active returns, so it measures skill. TC is the correlation between desired and actual active weights, so it measures how well constraints let you implement your views.

Is breadth the number of stocks in the portfolio?

No. Breadth is the number of independent decisions per year. Many stocks driven by the same view count as fewer independent decisions.

Why does doubling breadth not double the information ratio?

Breadth sits under a square root. To double IR you must quadruple breadth, if IC and TC stay the same.