FRM Exam Part II · Portfolio Construction

Fundamental Law of Active Management and Information Ratio

Updated 11 October 2026 · Fact-checked

Active management is judged by alpha (return above the benchmark) per unit of tracking error. That ratio is the information ratio, IR = α ÷ ω. The fundamental law says IR ≈ IC × √BR, scaled by the transfer coefficient TC when constraints stop you from using every forecast fully.

Understand Active Portfolio Management: Alpha, Information Ratio and Fundamental Law

A passive investor holds the benchmark. An active manager deviates from it to earn active return (alpha): portfolio return minus benchmark return. Deviating adds risk. The risk of the active return is the tracking error (TE), the standard deviation of active return over time.

The information ratio (IR) divides active return by tracking error. It tells you how much benchmark-relative return you earn per unit of benchmark-relative risk. The Sharpe ratio is similar but uses excess return over the risk-free rate divided by total volatility. Use Sharpe for total-portfolio performance. Use IR for a manager measured against a benchmark.

The fundamental law of active management explains where IR comes from. Skill is measured by the information coefficient (IC), the correlation between your forecasts and the actual outcomes. Breadth (BR) is the number of independent bets per year. A modest skill applied to many independent bets gives a good IR: IR = IC × √BR. Breadth counts independent decisions, not the number of securities held.

In practice, constraints such as long-only rules, turnover limits and position caps stop the portfolio from reflecting forecasts perfectly. The transfer coefficient (TC) is the correlation between the active weights you want and the active weights you actually hold. It lies between 0 and 1. The expected IR becomes TC × IC × √BR.

The law also gives the optimal active risk. Higher IR allows a given alpha with less tracking error, since expected alpha ≈ IR × TE. Doubling skill doubles IR. Doubling breadth raises IR only by √2.

Key formulas to remember

Active return (alpha): α = Rp − Rb; Portfolio return minus benchmark return. A beta-adjusted alpha subtracts β × (Rm − Rf) instead if the question says so.
Tracking error: TE (ω) = standard deviation of (Rp − Rb); Annualise monthly TE by multiplying by √12, and quarterly by √4.
Information ratio: IR = α ÷ ω = mean active return ÷ tracking error; Use the same time basis for numerator and denominator.
Sharpe ratio: SR = (Rp − Rf) ÷ σp; Uses the risk-free rate and total volatility, not a benchmark.
Fundamental law: IR = IC × √BR; Assumes unconstrained, independent bets and consistent skill.
Law with transfer coefficient: IR = TC × IC × √BR; TC is between 0 and 1. Lower TC means constraints reduce realised IR.
Expected alpha from IR: Expected α = IR × TE; Rearranged IR formula. Use it to size alpha for a given risk budget.

How to solve Active Portfolio Management: Alpha, Information Ratio and Fundamental Law questions

Most questions give you a few inputs and ask for one of alpha, TE, IR, IC, BR or TC. Work through the same sequence each time.

1Identify what is asked: IR, alpha, TE, IC, breadth or TC.
2Check the time basis of every number. Convert monthly or quarterly figures to annual before combining.
3Compute active return as portfolio minus benchmark, or take the average if given a series.
4Compute or read off tracking error. It is the standard deviation of active returns, not of portfolio returns.
5Choose the formula: IR = α ÷ ω for performance data, or IR = TC × IC × √BR for skill and breadth data.
6Rearrange if needed, for example IC = IR ÷ √BR or TE = α ÷ IR.
7Sanity-check: IR should be modest (often below 1), TC between 0 and 1, and breadth is a count of independent bets.
8Interpret in one line: what the answer says about skill, constraints or risk-adjusted value added.

Quickest way: Shortcut with the square root of breadth

When to use it: Use when the question changes breadth or skill and asks how IR changes, or asks you to compare managers.

Write IR = TC × IC × √BR and note which inputs change.
Scale only the changed inputs: IR multiplies by the ratio of the new to old IC and TC, and by √(new BR ÷ old BR).
Four times the breadth doubles IR. Nine times triples it.
Eliminate options with the wrong direction or that scale breadth linearly.
For TE or alpha, use alpha = IR × TE and finish in one step.

Common mistakes in Active Portfolio Management: Alpha, Information Ratio and Fundamental Law

Using total volatility instead of tracking error in the IR denominator.
Sharpe and IR look alike, so the denominators get mixed up.
Fix: IR always uses the standard deviation of active return. Sharpe uses the standard deviation of portfolio return.
Scaling breadth linearly, so doubling breadth doubles IR.
Students forget the square root in IR = IC × √BR.
Fix: Doubling breadth multiplies IR by √2 ≈ 1.41. Doubling IC doubles IR.
Counting breadth as the number of securities held.
It feels natural to equate bets with positions.
Fix: Breadth is the number of independent forecasts per year. Correlated bets add less than their count.
Mixing monthly and annual figures, for example dividing annual alpha by monthly TE.
Data in the question appear in different periods.
Fix: Annualise first: multiply monthly TE by √12 and monthly mean alpha by 12.
Ignoring the transfer coefficient and reporting the unconstrained IR.
Students memorise IR = IC × √BR only.
Fix: When constraints are mentioned, such as long-only or turnover limits, multiply by TC.
Treating a TC above 1 or an IC above 1 as possible.
Both are correlations, but candidates treat them as scale factors.
Fix: IC and TC are correlations bounded by −1 and 1, and in practice TC is between 0 and 1.

Worked examples

Example 1

A fund returned 9.5% in a year. Its benchmark returned 7.0%. Monthly active returns had a standard deviation of 0.50%. Compute the annual tracking error and the information ratio.

Show the solution

Active return α = 9.5% − 7.0% = 2.5%.
Annualise TE: 0.50% × √12 = 0.50% × 3.4641 = 1.732%.
IR = 2.5% ÷ 1.732% = 1.443.

Answer: Tracking error is about 1.73% and the information ratio is about 1.44.

Example 2

A manager has an information coefficient of 0.05 and makes 100 independent bets per year. A long-only constraint gives a transfer coefficient of 0.60. The portfolio tracking error is 4%. Find the expected IR and expected alpha.

Show the solution

Unconstrained IR = IC × √BR = 0.05 × √100 = 0.05 × 10 = 0.50.
Apply TC: IR = 0.60 × 0.50 = 0.30.
Expected alpha = IR × TE = 0.30 × 4% = 1.2%.

Answer: Expected IR is 0.30 and expected alpha is 1.2% a year.

Exam tips

Read the time basis first. Many wrong options come from unannualised tracking error.
When a question says long-only, turnover cap or position limit, think transfer coefficient.
For compare-and-contrast questions, remember: Sharpe uses the risk-free rate and total risk, IR uses the benchmark and active risk.
Questions on changing breadth test the square root. Four times the bets gives double the IR, not four times.
Expect interpretation: a high IR from few bets or a high TC means different things, so say which input drives it.

Practice questions from Portfolio Construction

Active Portfolio Management: Alpha, Information Ratio and Fundamental Law in other exams

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

Active Portfolio Management: Alpha, Information Ratio and Fundamental Law: frequently asked questions

What is the difference between the Sharpe ratio and the information ratio?

The Sharpe ratio is excess return over the risk-free rate divided by total volatility. The information ratio is active return over a benchmark divided by tracking error. Sharpe suits absolute performance and IR suits benchmark-relative managers.

How do I calculate tracking error?

Take the series of portfolio returns minus benchmark returns and compute its standard deviation. If the data are monthly, multiply by √12 to annualise. Tracking error measures active risk, not total risk.

What do IC, breadth and transfer coefficient mean?

IC is the correlation between forecasts and outcomes, a measure of skill. Breadth is the number of independent bets per year. The transfer coefficient is the correlation between desired and actual active weights, showing how much constraints dilute your forecasts.

Why is breadth under a square root?

Independent bets add variance rather than adding linearly to the risk-adjusted signal. Alpha grows in proportion to the number of bets while risk grows with the square root, so IR scales with √BR.