FRM Exam Part II · Portfolio Risk: Analytical Methods

Risk Budgeting and Optimal Portfolio Allocation Explained

Updated 11 October 2026 · Fact-checked

Risk budgeting splits a total risk limit, such as VaR, across positions or managers. A portfolio is optimal when excess return per unit of marginal VaR is equal for every position. If one ratio is higher, shift risk toward it until the ratios match.

Understand Risk Budgeting and Optimal Portfolio Allocation

A portfolio has one total risk limit. Risk budgeting decides how much of that limit each position, asset class or manager may use. Capital is not the scarce resource. Risk is.

To split risk fairly you need each position's contribution to total VaR. That is component VaR: position weight × marginal VaR. Marginal VaR is the change in portfolio VaR from a small increase in that position. Component VaRs add up to total portfolio VaR. Standalone VaRs do not, because of diversification.

Now ask what each unit of risk earns. For position i, divide its expected excess return over the risk-free rate by its marginal VaR. This is the reward per unit of marginal risk. If position A earns more per unit than position B, you gain by moving risk from B to A.

You stop when the ratios are equal across all positions. That is the optimal allocation condition. In a normal-distribution setting, marginal VaR is proportional to beta against the portfolio, so the condition says every position has the same excess return ÷ beta, which is the same Treynor-type ratio. It also means every position sits on the same line as the portfolio.

Risk budgeting is the general idea of assigning risk limits. Risk parity is one specific budget: each position contributes equal risk. Risk budgeting can use any target shares, such as 60% of risk to equities. The optimum above is the budget that maximises return for the risk used.

Key formulas to remember

Marginal VaR (normal): Marginal VaR_i = z × Cov(R_i, R_p) ÷ σ_p = z × β_i × σ_p; β_i is the beta of position i against the portfolio. z is the confidence multiplier, for example 1.645 at 95%.
Component VaR: Component VaR_i = w_i × Marginal VaR_i; Here w_i is the position's weight, or its dollar amount if marginal VaR is per dollar. Sum of components = portfolio VaR.
Percent contribution to risk: % contribution_i = Component VaR_i ÷ Portfolio VaR = w_i × β_i; Contributions sum to 100% with betas measured against the portfolio.
Optimality condition: (E[R_i] − R_f) ÷ Marginal VaR_1 = (E[R_i] − R_f) ÷ Marginal VaR_2 = … for all i; Excess return per unit of marginal VaR is equal across all positions at the optimum. Equivalent to equal (E[R_i] − R_f) ÷ β_i under normality.
Risk parity rule: Component VaR_i = Portfolio VaR ÷ N for all N positions; A special risk budget with equal contributions. It is not the same as the optimality condition.

How to solve Risk Budgeting and Optimal Portfolio Allocation questions

Use this order for any risk budgeting or optimal allocation question.

1Identify what is given: weights, VaRs, betas or marginal VaRs, and expected excess returns.
2Compute or read the marginal VaR of each position. Under normality use z × β × σ_p.
3Compute component VaR as weight × marginal VaR. Check that components sum to portfolio VaR.
4Compute excess return ÷ marginal VaR for each position.
5Compare the ratios. Equal ratios mean the portfolio is optimal.
6If ratios differ, increase risk in the position with the higher ratio and cut the one with the lower ratio.
7Answer in the form asked: a percentage contribution, a direction of trade, or a yes/no on optimality.

Quickest way: Ratio comparison shortcut

When to use it: When the question asks if a portfolio is optimal or which position to increase.

Skip full VaR. Use betas against the portfolio if given, since marginal VaR is proportional to beta.
Compute (expected return − risk-free rate) ÷ beta for each position.
The highest ratio gets more risk. The lowest gets less.
For percent contributions use weight × beta, then check they sum to 100%.

Common mistakes in Risk Budgeting and Optimal Portfolio Allocation

Dividing total expected return, not excess return, by marginal VaR.
The risk-free rate is given in the stem and gets forgotten.
Fix: Always subtract the risk-free rate first.
Using standalone VaR instead of marginal VaR in the ratio.
Standalone VaR is easier to compute and looks similar.
Fix: Standalone VaR ignores diversification. Use marginal or component VaR, which reflect correlation with the portfolio.
Assuming component VaRs should be equal at the optimum.
Confusing the optimum with risk parity.
Fix: The optimum equalises return per unit of marginal VaR. Component VaRs equal only in special cases.
Treating component VaRs as not adding to total VaR.
Mixing them up with incremental VaR, which does not add up exactly.
Fix: Component VaRs sum to portfolio VaR. Incremental VaRs generally do not.
Computing beta against the market or benchmark instead of the portfolio.
Beta usually means CAPM beta.
Fix: For marginal VaR, beta is the position's beta against the portfolio itself.
Moving risk the wrong way when ratios differ.
Intuition about which ratio is bad gets reversed.
Fix: A higher ratio means more reward per risk, so add risk there and cut where the ratio is lower.

Worked examples

Example 1

A portfolio has two managers. Manager A has expected excess return 6% and marginal VaR 0.04 per unit. Manager B has expected excess return 4% and marginal VaR 0.05. Is the allocation optimal? If not, which manager should get more risk budget?

Show the solution

Ratio for A = 6% ÷ 0.04 = 1.5.
Ratio for B = 4% ÷ 0.05 = 0.8.
The ratios are not equal, so the allocation is not optimal.
A earns more per unit of marginal VaR, so shift risk from B to A.

Answer: Not optimal. Increase A's risk budget and reduce B's until the ratios are equal.

Example 2

A portfolio has total VaR of $10 million. Position X has weight 40% and beta 1.25 against the portfolio. Position Y has weight 60% and beta 0.83333 against the portfolio. Find X's component VaR and its percent contribution.

Show the solution

Percent contribution of X = w × β = 0.40 × 1.25 = 0.50.
Check Y: 0.60 × 0.83333 = 0.50. Total = 100%, so the betas are consistent.
Component VaR of X = 50% × $10 million = $5 million.

Answer: X contributes 50% of risk, a component VaR of $5 million.

Exam tips

The questions often hand you marginal VaRs or betas. Go straight to excess return ÷ marginal VaR.
Read whether the stem asks for a trade direction or a yes/no on optimality.
Watch for distractors that use standalone VaR or total return. Both are wrong.
Remember risk parity is one special budget, not the optimum condition.
Check that percent contributions sum to 100% before choosing an answer.

Practice questions from Portfolio Risk: Analytical Methods

Risk Budgeting and Optimal Portfolio Allocation in other exams

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

Risk Budgeting and Optimal Portfolio Allocation: frequently asked questions

What is the optimal portfolio condition with marginal VaR?

Excess return divided by marginal VaR must be equal for every position. If it is not, shifting risk to the position with the higher ratio raises return for the same total VaR.

What is the difference between risk parity and risk budgeting?

Risk budgeting is the general process of setting how much risk each position or manager may use. Risk parity is one budget in which every position contributes the same risk. Other budgets assign unequal shares.

Why use marginal VaR and not standalone VaR?

Marginal VaR measures how a position changes portfolio VaR, so it captures correlation and diversification. Standalone VaR treats the position in isolation and overstates its contribution when it diversifies the portfolio.

Do component VaRs add up to portfolio VaR?

Yes. Component VaR is weight × marginal VaR, and these sum to total portfolio VaR. Incremental VaRs generally do not sum exactly.