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FRM Part II · FRM Exam Part II

Portfolio Risk: Analytical Methods for FRM Part II

Portfolio risk analytical methods measure and split a portfolio's risk using formulas, mainly the covariance matrix. You start with portfolio VaR = z × σp × V, then break it into marginal, incremental and component VaR, link it to budgets, tracking error and factors, and test it with stress scenarios.

What this chapter covers

This chapter is about measuring risk at portfolio level with closed-form, formula-based tools. You begin with parametric (delta-normal) VaR, where portfolio volatility comes from weights, volatilities and correlations. Then you ask the harder questions. Which position drives the risk? What happens if you add a trade? How should risk be spread across managers or desks?

The middle of the chapter moves from absolute risk to relative risk. Tracking error measures the volatility of active returns against a benchmark. Factor models split risk into systematic and idiosyncratic parts, so you can say how much comes from market, rates or style exposures. Risk budgeting and optimal allocation tie these together: risk is a scarce resource, and you allocate it where expected return per unit of risk is highest.

The chapter connects to the rest of Part II in several ways. Market risk topics reuse VaR and its limits. Risk Management and Investment Management reuses tracking error, active risk and factor exposures. Stress testing links to liquidity, credit and Current Issues questions, where a scenario matters more than a normal-distribution estimate.

Questions here are numerical but short, and the same few formulas repeat in different dress. Once you can compute portfolio volatility, VaR and the component split quickly, you gain reliable marks and save time for long case-style questions elsewhere in the 80-question, 4-hour paper. The concepts also carry into investment management and market risk, so one solid pass through this chapter pays off in several places. Interpretation questions are common too, such as why component VaR sums to total VaR or when analytical VaR fails, so you need the logic as well as the arithmetic.

Portfolio Risk: Analytical Methods: topics in the order to study them

  1. 1Value at Risk (VaR) Fundamentals for PortfoliosEverything else builds on portfolio volatility and the VaR formula, so learn this first.
  2. 2Marginal, Incremental and Component VaRIt decomposes the VaR you just computed, and the sums-to-total property is a frequent test point.
  3. 3Risk Budgeting and Optimal Portfolio AllocationIt uses marginal and component VaR to decide where risk should go, so it needs the previous topic.
  4. 4Tracking Error and Benchmark-Relative RiskIt reuses the portfolio variance logic but applies it to active weights and active returns.
  5. 5Factor Models and Systematic Risk DecompositionIt explains where risk comes from, and it extends beta and tracking error ideas to multiple factors.
  6. 6Stress Testing and Limitations of Analytical VaRStudy it last, because you need to know the model's assumptions before judging where they break.

How to prepare Portfolio Risk: Analytical Methods

Treat this chapter as one formula family with one set of assumptions. Build the calculation skill first, then the interpretation.

  1. Write the portfolio variance formula for two assets from memory: σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. Practise until it takes under a minute.
  2. Compute VaR end to end: find σp, scale to the horizon with √t, multiply by the z-value (1.645 at 95%, 2.326 at 99% one-tailed) and by portfolio value.
  3. Learn the decomposition chain: marginal VaR is the change per unit of position, component VaR is position × marginal VaR, and components sum to total VaR. Incremental VaR is the actual change from a trade.
  4. Do tracking error and factor questions using the same variance logic, with active weights or factor exposures in place of raw weights.
  5. Practise allocation rules: at the optimum, the ratio of excess return to marginal VaR is equal across positions.
  6. For stress testing, list the assumptions of analytical VaR (normality, linearity, stable correlations) and match each to a way it fails.
  7. Finish with mixed timed sets of short questions. Review every error by asking whether it was a formula, units or interpretation slip.

Common mistakes in Portfolio Risk: Analytical Methods

  • Forgetting the square root of time and scaling VaR linearly with days.

    Fix: Under the usual assumption of independent returns, scale by √t. Check the horizon in the question before you calculate.

  • Mixing up marginal, component and incremental VaR.

    Fix: Marginal is per unit and small, component is position × marginal and adds up to total, incremental is the actual change from a discrete trade.

  • Using weights instead of active weights for tracking error.

    Fix: Subtract the benchmark weights first, then apply the covariance logic to the active weights.

  • Using variances where volatilities are needed, or the reverse.

    Fix: Label each input as σ or σ² before you substitute, and take the square root last.

  • Treating analytical VaR as a worst-case loss.

    Fix: State it as a loss that should be exceeded only with the stated probability. It says nothing about how bad the exceedance is.

  • Assuming correlations hold in a crisis.

    Fix: Remember that correlations tend to rise in stress, so diversification benefits shrink. This is a key reason to add stress tests.

Last-day revision: Portfolio Risk: Analytical Methods

  • Portfolio VaR = z × σp × portfolio value, scaled by √t for the horizon (assuming independent returns).
  • z-values for one-tailed normal VaR: 1.645 at 95% and 2.326 at 99%.
  • Two-asset σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2.
  • Lower correlation means lower portfolio VaR; diversification benefit = sum of standalone VaRs − portfolio VaR.
  • Marginal VaR is the change in portfolio VaR for a small change in a position.
  • Component VaR = position × marginal VaR, and the components sum to total VaR.
  • Incremental VaR is the full change in VaR from adding or removing a position, and it can differ from component VaR for large trades.
  • Optimal allocation: excess return ÷ marginal VaR is equal across positions.
  • Tracking error is the standard deviation of active return (portfolio minus benchmark).
  • Information ratio = active return ÷ tracking error.
  • Factor model: total variance = systematic variance + idiosyncratic variance.
  • Analytical VaR assumes normality and linear exposures, so it understates tail risk and option risk. Stress tests cover what it misses.

Portfolio Risk: Analytical Methods practice questions

Portfolio Risk: Analytical Methods in other exams

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

Portfolio Risk: Analytical Methods: frequently asked questions

Do I need to memorise formulas for this chapter?

Yes, a small set. You need portfolio variance, VaR with z-values, component VaR, tracking error and the information ratio. Practise them until the arithmetic is quick, because the exam is short-question and time-bound.

How is component VaR different from incremental VaR?

Component VaR splits current portfolio VaR across positions and the parts add up to the total. Incremental VaR is the actual change in VaR when you add or remove a position. For small changes they are close, but for large trades they can differ.

Is a calculator allowed for the FRM Part II exam?

Check GARP's current exam rules for the approved calculator models before exam day. Practise with the same calculator so that square roots and the z-value arithmetic feel routine.

Why does this chapter include stress testing if VaR already measures risk?

Analytical VaR relies on assumptions such as normality, linear exposures and stable correlations. These break down in tail events. Stress tests apply specific scenarios to show losses that the VaR model may understate.