FRM Exam Part II · VaR and Risk Budgeting in Investment Management

Fundamental Law of Active Management Explained

Updated 11 October 2026 · Fact-checked

The fundamental law of active management says the information ratio equals the information coefficient times the square root of breadth: IR = IC × √BR. IC measures forecasting skill. Breadth is the number of independent bets per year. To solve questions, identify IC and BR, then compute IR or solve for the missing input.

Understand Fundamental Law of Active Management

Active managers try to beat a benchmark. The information ratio (IR) measures how well they do it. It is active return divided by active risk, where active risk is tracking error.

The fundamental law explains where a high IR comes from. It has two sources. The first is skill, measured by the information coefficient (IC). IC is the correlation between your forecasts and the returns that actually occur. An IC of 0 means your forecasts are no better than random. An IC of 1 means perfect foresight. Real values are small, often a few hundredths.

The second source is breadth (BR). This is the number of independent, uncorrelated bets you make in a year. More independent bets let skill show up repeatedly and diversify away luck. Because independent risks add up in variance terms, IR grows with the square root of breadth, not linearly.

The law gives a trade-off. A manager with modest skill who makes many independent bets can match a manager with high skill who makes few. Doubling breadth raises IR only by a factor of √2, about 1.41. To double IR through breadth alone you need four times as many independent bets.

The basic law assumes forecasts are independent, the manager uses all information efficiently, and there are no constraints. Constraints such as long-only limits reduce the result. The extended form adds a transfer coefficient for this.

Key formulas to remember

Fundamental law (basic): IR = IC × √BR; BR is independent bets per year. IR is annualised if BR is per year.
Fundamental law with transfer coefficient: IR = TC × IC × √BR; TC is between 0 and 1. It measures how well forecasts convert into portfolio weights after constraints.
Information ratio: IR = α ÷ ω = active return ÷ tracking error; α is expected active return. ω is tracking error, the standard deviation of active return.
Information coefficient: IC = correlation(forecast return, realised return); Ranges from -1 to +1. Positive and above zero shows skill.
Implied active return: α = IR × ω; Use this to get expected active return from a target tracking error.
Breadth scaling: BR needed = (IR ÷ IC)²; Rearranged law to solve for the number of independent bets.

How to solve Fundamental Law of Active Management questions

Use this method for any question on the law. Most questions give two of IC, BR and IR and ask for the third, or ask you to link IR to active return.

1Write down what is given: IC, BR, IR, tracking error, active return, transfer coefficient.
2Check that breadth means independent bets per year. If bets are correlated, use only the independent number.
3Apply IR = IC × √BR, adding TC as a multiplier if the question mentions constraints.
4If solving for IC, use IC = IR ÷ √BR. If solving for BR, use BR = (IR ÷ IC)².
5If the question asks for active return, multiply IR by tracking error.
6Check the result for sense: IC should be between 0 and 1, and IR should be modest.
7State the interpretation: skill or breadth drives the result, and note the √ effect.

Quickest way: Square-root shortcut

When to use it: Use when comparing two managers or asking what happens when skill or breadth changes.

Scale IR directly with IC: doubling IC doubles IR.
Scale IR with √BR: quadrupling BR doubles IR, and doubling BR multiplies IR by about 1.41.
For a new IR, multiply the old IR by the IC ratio and the √BR ratio.
Eliminate options that treat breadth as linear.

Common mistakes in Fundamental Law of Active Management

Using BR instead of √BR
The formula looks like a simple product, so students forget the square root.
Fix: Always take √BR first. Write it as a separate line before multiplying.
Counting correlated bets as breadth
Students count every position in the portfolio.
Fix: Breadth counts only independent bets. Correlated positions give less breadth than their count.
Treating IC as a return or probability of being right
The word coefficient is not obvious.
Fix: IC is a correlation between forecasts and outcomes, from -1 to +1.
Ignoring the transfer coefficient when constraints are stated
Students memorise only the basic law.
Fix: If the question mentions long-only or other constraints, multiply by TC. The realised IR is lower than the unconstrained IR.
Assuming doubling breadth doubles IR
Linear thinking about diversification.
Fix: IR rises by √2 when breadth doubles. You need four times the breadth to double IR.

Worked examples

Example 1

A manager has an information coefficient of 0.05 and makes 100 independent bets per year. The tracking error target is 4%. What is the IR and the expected active return?

Show the solution

Take √BR = √100 = 10.
IR = IC × √BR = 0.05 × 10 = 0.50.
Active return = IR × tracking error = 0.50 × 4% = 2.0%.

Answer: IR = 0.50 and expected active return = 2.0% a year.

Example 2

A manager has an IC of 0.04 and 64 independent bets per year. The manager wants an IR of 0.60. By how much must breadth rise, holding IC constant?

Show the solution

Current IR = 0.04 × √64 = 0.04 × 8 = 0.32.
Required breadth = (IR ÷ IC)² = (0.60 ÷ 0.04)² = 15² = 225.
Increase = 225 − 64 = 161 additional independent bets.

Answer: Breadth must rise from 64 to 225 independent bets, an increase of 161.

Exam tips

Expect scenario questions that change one input and ask for the new IR. Use ratio scaling to save time.
Read whether bets are independent. Questions often add correlated positions to test breadth.
If constraints appear, look for the transfer coefficient and apply it as a multiplier.
Interpret the answer: say whether skill or breadth is the driver and note the square-root effect.

Practice questions from VaR and Risk Budgeting in Investment Management

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 does IR = IC × √BR mean?

It says the information ratio comes from forecasting skill and the number of independent bets. Skill scales IR linearly. Breadth scales it by the square root.

What is a good information coefficient?

IC values are usually small because markets are competitive. Even a few hundredths of positive IC can produce a useful IR if breadth is large.

How do I count breadth?

Count independent bets per year, not positions. If forecasts or returns are correlated, the effective breadth is lower than the raw count.

What does the transfer coefficient add?

It captures how well forecasts translate into actual portfolio weights. Constraints such as no short selling lower it, so realised IR falls below the unconstrained value.