CFA Level II Exam · Analysis of Active Portfolio Management

Optimal Active Risk and Portfolio Construction

Updated 7 October 2026 · Fact-checked

The optimal level of active risk is the amount that maximizes the Sharpe ratio of benchmark plus active portfolio. It equals the information ratio divided by the benchmark Sharpe ratio, times benchmark risk. The combined Sharpe ratio is √(SR_B² + IR²). Constraints lower IR through the transfer coefficient.

Understand Optimal Active Risk and Portfolio Construction

You hold a benchmark portfolio. You can add an active portfolio on top of it. The active portfolio earns active return (return minus benchmark) and carries active risk (the standard deviation of that active return). The question is how much active risk to take.

The key measure is the information ratio (IR): expected active return divided by active risk. It tells you how much active return you get per unit of active risk. The benchmark has its own Sharpe ratio (SR_B): expected excess return divided by benchmark risk.

When the active portfolio's returns are uncorrelated with the benchmark's (the standard assumption in this topic), the best combined Sharpe ratio satisfies SR_P² = SR_B² + IR². Active management adds to the Sharpe ratio only through IR. A higher IR always raises the maximum Sharpe ratio. To reach that maximum you must scale active risk to the right level: σ_A* = (IR ÷ SR_B) × σ_B. If you take less, you leave Sharpe ratio on the table. If you take more, you also lose it.

In practice, managers face limits such as no short selling, position caps and turnover limits. The transfer coefficient (TC) measures how well the actual active weights reflect the manager's ideal views. It is the correlation between the ideal and the actual active weights, so it lies between 0 and 1. The constrained IR is TC × IC × √BR, where IC is the information coefficient and BR is breadth. A lower TC means a lower IR and so a lower achievable Sharpe ratio.

A core-satellite structure applies this idea. The core is a low-cost passive (or low-active-risk) portfolio. The satellites are active strategies, ideally with low correlation to each other, sized by their IR and risk budget. You can also split money between a passive fund and an active fund. The same logic applies: you fund the active share up to the point where the Sharpe ratio is highest.

Key formulas to remember

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.

How to solve Optimal Active Risk and Portfolio Construction questions

Use this order for any item set question on optimal active risk, combining active and passive portfolios, or implementation constraints.

1Find the data in the vignette: benchmark return, risk-free rate, benchmark risk, active return, active risk, IC, breadth, TC. Note whether they are annual and in the same units.
2Compute SR_B = (E(R_B) − R_F) ÷ σ_B if it is not given.
3Get IR. Use E(R_A) ÷ σ_A, or IC × √BR, or TC × IC × √BR if constraints apply. If the question gives constraints, use the constrained IR.
4Find the optimal active risk: σ_A* = (IR ÷ SR_B) × σ_B.
5Find the active return at that risk: IR × σ_A*. Find total risk with √(σ_B² + σ_A²) if asked.
6Find the maximum Sharpe ratio: √(SR_B² + IR²). Compare it with SR_B to see the value added.
7For a qualitative question (core-satellite, constraints, which manager), reason from IR and TC. Higher IR earns more risk budget. Lower TC cuts IR.
8Check that the answer is sensible: the combined Sharpe ratio must be at least SR_B, and it cannot exceed the unconstrained value.

Quickest way: Ratio shortcut for optimal active risk

When to use it: Use when the vignette gives IR (or IC, BR, TC) and SR_B, and asks for optimal active risk or the best Sharpe ratio.

Write IR and SR_B. Divide IR by SR_B to get the scaling ratio.
Multiply the ratio by σ_B to get optimal active risk.
Multiply IR by that active risk for the active return.
Square SR_B and IR, add them, and take the square root for the maximum Sharpe ratio.
If TC is given, multiply it into IR before every step. Do not multiply it in twice.

Common mistakes in Optimal Active Risk and Portfolio Construction

Using the active return directly instead of the information ratio when finding the best Sharpe ratio.
Students mix up IR with active return or with the Sharpe ratio of the active portfolio.
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.
The formula looks like a simple sum, and under time pressure the square root is forgotten.
Fix: Write SR_B² and IR² as separate numbers, add them, then take the root. The combined ratio is less than SR_B + IR.
Dividing by the wrong ratio in the optimal active risk formula, using SR_B ÷ IR.
The two ratios look alike and the order is easy to swap.
Fix: Think logically: higher IR should mean more active risk. So IR is in the numerator.
Ignoring the transfer coefficient when the vignette describes constraints such as long-only rules.
Students use the unconstrained IC × √BR because it is the familiar formula.
Fix: If the vignette mentions limits on shorting, position size or turnover, check whether TC is given. Use IR = TC × IC × √BR and note that the achievable Sharpe ratio is lower.
Treating a TC of 1 as normal and thinking a lower TC raises risk.
Students misread TC as a risk measure.
Fix: TC is a correlation between ideal and actual active weights. TC below 1 means lost efficiency and a lower IR, not extra risk.
Taking total risk as σ_B + σ_A.
Students forget that risks add in variance terms.
Fix: With uncorrelated active return, σ_P = √(σ_B² + σ_A²). Do not add standard deviations.

Worked examples

Example 1

Vignette: A global equity benchmark has expected excess return of 6% and standard deviation of 15%. A manager's active strategy has an expected information ratio of 0.50. Active return is uncorrelated with the benchmark. Questions: (1) What is the optimal active risk? (2) What is the maximum Sharpe ratio of the combined portfolio? (3) What is the expected active return at the optimal active risk?

Show the solution

Benchmark Sharpe ratio: SR_B = 6% ÷ 15% = 0.40.
Optimal active risk: (IR ÷ SR_B) × σ_B = (0.50 ÷ 0.40) × 15% = 1.25 × 15% = 18.75%.
Maximum Sharpe ratio: √(0.40² + 0.50²) = √(0.16 + 0.25) = √0.41 = 0.6403.
Expected active return: IR × σ_A = 0.50 × 18.75% = 9.375%.
Check: total risk = √(15² + 18.75²) = √576.5625 = 24.01%. Total excess return = 6% + 9.375% = 15.375%. Divide: 15.375 ÷ 24.01 = 0.640, which matches.

Answer: (1) 18.75%. (2) About 0.64. (3) About 9.38%.

Example 2

Vignette: A long-only manager has an information coefficient of 0.08 and makes 100 independent decisions a year. The manager's transfer coefficient is 0.70 because of the long-only rule. The benchmark has a Sharpe ratio of 0.35 and a standard deviation of 12%. Questions: (1) What is the constrained information ratio? (2) What is the optimal active risk? (3) What is the maximum Sharpe ratio, and how much does it fall compared with an unconstrained manager?

Show the solution

Unconstrained IR = IC × √BR = 0.08 × √100 = 0.08 × 10 = 0.80.
Constrained IR = TC × IC × √BR = 0.70 × 0.80 = 0.56.
Optimal active risk = (0.56 ÷ 0.35) × 12% = 1.6 × 12% = 19.2%.
Constrained maximum Sharpe ratio = √(0.35² + 0.56²) = √(0.1225 + 0.3136) = √0.4361 = 0.6604.
Unconstrained maximum Sharpe ratio = √(0.1225 + 0.64) = √0.7625 = 0.8732.
Fall in Sharpe ratio = 0.8732 − 0.6604 = 0.2128, about 0.21.

Answer: (1) 0.56. (2) 19.2%. (3) About 0.66, which is about 0.21 lower than the unconstrained value of about 0.87.

Exam tips

Most questions are two-step: get IR, then plug it into either the optimal risk formula or the Sharpe formula. Set out SR_B and IR first so you do not mix them up.
Check whether the vignette gives a transfer coefficient. If it does, IR must include it. If the vignette only talks about constraints in words, answer qualitatively: lower TC, lower IR, lower Sharpe ratio.
For core-satellite questions, link the answer to IR: a passive core keeps cost and tracking error low, and satellites receive risk in line with their IR and low correlation with each other.
Use the check that SR_P ≥ SR_B. If your answer is below the benchmark Sharpe ratio, you made an arithmetic or formula error.
Keep units consistent: convert percentages to decimals when squaring, then convert back.

Optimal Active Risk and Portfolio Construction in other exams

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

Optimal Active Risk and Portfolio Construction: frequently asked questions

What is the formula for the optimal level of active risk?

It is σ_A* = (IR ÷ SR_B) × σ_B, where IR is the information ratio, SR_B is the benchmark Sharpe ratio and σ_B is benchmark risk. It is the active risk that maximizes the combined Sharpe ratio. It assumes active return is uncorrelated with the benchmark.

How do I combine active and passive portfolios to maximize the Sharpe ratio?

Hold the benchmark (passive) portfolio and add the active portfolio, sized to the optimal active risk. The result has a Sharpe ratio of √(SR_B² + IR²). A higher IR lets you take more active risk and earn a higher Sharpe ratio.

What is a core-satellite portfolio in the CFA curriculum?

The core is a passive or low-cost, low-active-risk holding that provides market exposure. The satellites are active strategies that add active return. You assign each satellite risk according to its information ratio and its correlation with the other satellites.

What does the transfer coefficient do?

The transfer coefficient is the correlation between the manager's ideal active weights and the weights actually held after constraints. It lies between 0 and 1. The constrained information ratio is TC × IC × √BR, so a lower TC lowers IR and the achievable Sharpe ratio.