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

Risk Budgeting Across Managers and Decentralised Risk Management

Q: How do pension funds allocate risk budget to managers?

They set a total risk limit, such as VaR or tracking error. They then split it across managers using expected alpha, information ratios and correlations. The aim is to put more risk where return per unit of marginal risk is highest.

Q: Why is fund VaR less than the sum of manager VaRs?

Manager returns are not perfectly correlated, so losses rarely peak together. The gap is the diversification benefit. It disappears if all correlations are 1.

Q: What is the difference between marginal and component VaR here?

Marginal VaR is the change in total VaR from a small increase in one manager's allocation. Component VaR is the marginal VaR per dollar times the dollar position in manager i, which equals wᵢ × βᵢ × VaR_p. The components sum to total VaR.

Q: How does a sponsor monitor VaR with multiple managers?

It collects positions or risk exposures from each manager and computes VaR at fund level. It compares manager and total figures with limits and watches for style drift and rising correlations.

Updated 11 October 2026 · Fact-checked

Risk budgeting across managers means a sponsor splits its total risk allowance (usually a VaR or tracking-error limit) among external managers, sets each manager's limit, and monitors VaR centrally. Total risk is below the sum of manager VaRs because correlations are below 1. You compute it with the portfolio VaR formula and component VaR.

Understand Risk Budgeting Across Managers and Decentralised Risk Management

A pension fund or other plan sponsor rarely runs money itself. It hires many external managers, each with a mandate. The sponsor still carries the total risk. Decentralised investment management means each manager makes decisions on their own, inside limits the sponsor sets. The sponsor does not approve each trade.

The sponsor starts with a total risk budget. This is often a limit on total VaR, or on tracking error against a policy benchmark. The sponsor then allocates part of it to each manager. The allocation depends on expected alpha, the manager's risk-adjusted skill (such as the information ratio), and how the manager's returns correlate with the others.

The key point is diversification. Manager VaRs do not add up. Total VaR = √(VaR₁² + VaR₂² + 2ρ·VaR₁·VaR₂) for two managers. If ρ < 1, total VaR is below the simple sum. The gap is the diversification benefit. If managers all hold similar positions, correlation is high and the benefit disappears. Managers may also take offsetting bets that cancel out and still earn fees.

To monitor, the sponsor collects positions or exposures from each manager and computes VaR at the total fund level. It also computes each manager's component VaR, which shows how much of the total each manager contributes. Components sum to total VaR. A manager with a small stand-alone VaR can still have a large component if they are highly correlated with the fund.

Limits are usually set on each manager's tracking error or VaR relative to their benchmark, with zones. A manager in a warning zone is reviewed. A manager over the hard limit must cut risk. The sponsor also checks that managers stay in style, since drift changes correlations and so total risk.

Key formulas to remember

Two-manager total VaR: VaR_p = √(VaR₁² + VaR₂² + 2ρ·VaR₁·VaR₂); Assumes normal, elliptical-type behaviour and the same confidence level and horizon. If ρ = 1, VaR_p = VaR₁ + VaR₂.
Diversification benefit: Benefit = ΣVaRᵢ − VaR_p; Undiversified sum minus portfolio VaR. It is zero only when all pairwise correlations equal 1.
Component VaR: CVaRᵢ = wᵢ × βᵢ × VaR_p, where wᵢ = manager i's fraction of fund value and βᵢ = Cov(Rᵢ, R_p) ÷ σ_p²; Because Σwᵢβᵢ = 1, the component VaRs sum exactly to VaR_p. They can be negative for a hedging manager.
Marginal VaR (parametric): Marginal VaRᵢ = z × Cov(Rᵢ, R_p) ÷ σ_p; Change in VaR for a small change in exposure to manager i.
Information ratio: IR = α ÷ tracking error; Used to compare risk budget efficiency. Optimal allocation sets alpha per unit of marginal risk equal across managers.
VaR scaling: VaR(T days) = VaR(1 day) × √T; Valid only when returns are independent with constant volatility.

How to solve Risk Budgeting Across Managers and Decentralised Risk Management questions

Use this order for any question on risk budgets, manager limits or decentralised VaR.

1Identify the risk measure: absolute VaR, or tracking-error (relative) VaR against a benchmark. Check confidence level and horizon.
2Write down each manager's stand-alone VaR or volatility, and the correlations between managers.
3Convert all numbers to the same confidence level and horizon before combining.
4Compute total VaR with the portfolio formula. Do not add VaRs unless ρ = 1.
5Find the diversification benefit as the sum of stand-alone VaRs minus total VaR.
6If asked about contributions, compute component or marginal VaR and check that components sum to total.
7Compare with limits and state the action: within limit, warning zone, or breach, and what the sponsor should do.
8Interpret in words: high correlation, style drift or offsetting bets reduce the real diversification.

Quickest way: Square, add, root, then compare

When to use it: Use for two- or three-manager numerical questions where the answer options are far apart.

Check the limits first: if ρ = 1 the answer is the plain sum, if ρ = 0 it is √(a² + b²).
For ρ = 0 and 3-4-5 style numbers, spot the triple (for example 30 and 40 give 50).
For 0 ≤ ρ ≤ 1, total VaR lies between the larger single VaR and the sum. Remove options outside that range. For ρ < 0, total VaR can fall below the larger single VaR, though not always.
For component VaR, check the shares add to 100 percent of total VaR.
Verify the same confidence level was used for every figure.

Common mistakes in Risk Budgeting Across Managers and Decentralised Risk Management

Adding manager VaRs to get fund VaR
VaR looks like a loss amount, so adding feels natural.
Fix: Use the correlation formula. Add only when ρ = 1. That is the upper bound under the normal assumption; VaR is not subadditive in general.
Assuming that low manager correlation is stable
Historical correlations look comfortable.
Fix: Remember correlations rise in stress and style drift changes them. Stress test correlations toward 1.
Treating stand-alone VaR as a manager's risk contribution
Stand-alone VaR is easy to read from the manager report.
Fix: Use component VaR. A manager with a small VaR but high correlation to the fund can contribute more than a larger, diversifying one.
Mixing confidence levels or horizons
Managers report on different bases, such as 95% daily and 99% monthly.
Fix: Rescale first. Multiply by the ratio of z-scores (for example 2.326 ÷ 1.645 to move from 95% to 99%) and by √(T₂ ÷ T₁) when returns are i.i.d. normal.
Ignoring offsetting positions between managers
Each manager looks fine alone.
Fix: Aggregate positions at the fund level. Offsets lower risk but cost fees twice, and netting hides gross exposure.
Setting limits only on total risk, not on each manager
Total risk is what the sponsor cares about.
Fix: Set manager-level limits as well, with warning zones, so a single manager cannot dominate and breaches are caught early.

Worked examples

Example 1

A sponsor has two managers. Manager A has a monthly 95% VaR of $20 million and Manager B has $30 million. The correlation between their returns is 0.5. Find total VaR and the diversification benefit.

Show the solution

Use VaR_p = √(VaR₁² + VaR₂² + 2ρ·VaR₁·VaR₂).
VaR₁² = 400 and VaR₂² = 900.
2 × 0.5 × 20 × 30 = 600.
Sum = 400 + 900 + 600 = 1,900.
VaR_p = √1,900 ≈ 43.59.
Undiversified sum = 20 + 30 = 50.
Benefit = 50 − 43.59 = 6.41.

Answer: Total VaR is about $43.6 million. The diversification benefit is about $6.4 million.

Example 2

Two managers have tracking-error VaRs of $40 million and $30 million at the same confidence level and horizon, with zero correlation. The sponsor's limit on total tracking-error VaR is $60 million. Is the limit met, and what happens if the correlation rises to 1?

Show the solution

With ρ = 0, VaR_p = √(40² + 30²).
40² = 1,600 and 30² = 900, sum = 2,500.
√2,500 = 50.
50 is below the $60 million limit, so the limit is met with $10 million of room.
With ρ = 1, VaR_p = 40 + 30 = 70.
70 exceeds the limit of 60 by $10 million.

Answer: At zero correlation total VaR is $50 million, within the limit. If correlation rises to 1, total VaR is $70 million, which breaches the limit, so the sponsor should cut risk or reduce manager limits.

Exam tips

Always check confidence level and horizon before combining numbers. Many wrong options come from mixed bases.
Expect questions on why the sum of manager VaRs overstates risk, and when it does not (ρ = 1).
Know that component VaRs add to total VaR, and marginal VaR shows the effect of a small change.
For conceptual items, link decentralisation to central monitoring: managers act independently, but the sponsor aggregates positions and checks limits and style drift.
Read the option range: under the normal (elliptical) assumption and ρ ≤ 1, total VaR does not exceed the sum of the manager VaRs, which removes options fast.

Practice questions from VaR and Risk Budgeting in Investment Management