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FRM Exam Part I · Calculating and Applying VaR

Monte Carlo Simulation VaR: Steps, Accuracy and Cost

Updated 11 October 2026 · Fact-checked

Monte Carlo VaR generates many random scenarios for risk factors from an assumed model, revalues the portfolio in each scenario, sorts the resulting profits and losses, and reads off the loss at the chosen confidence level. It is flexible for nonlinear positions but slow, and it depends on the model you assume.

Understand Monte Carlo Simulation VaR

Monte Carlo VaR estimates a portfolio's loss distribution by simulation. You do not wait for history to supply scenarios. You build a model for how risk factors move, then draw random numbers to create thousands of possible future outcomes.

The model comes first. You choose distributions for the risk factors (for example, joint normal returns with a given volatility and correlation matrix, or a model with fat tails). You also choose the horizon, such as one day or ten days. Correlated factors are generated by transforming independent random draws so that they carry the assumed covariance structure, usually with a Cholesky decomposition.

For each simulated scenario, you revalue the whole portfolio. This is full revaluation, so options and other nonlinear positions are priced again rather than approximated with delta or gamma. You get one simulated profit or loss per scenario. Sort all of them from worst to best. The VaR at confidence level c is the loss at the (1 − c) quantile of the simulated distribution.

Compare this with the other methods. Historical simulation uses actual past changes and makes no distribution assumption, but it is limited by the data window. Parametric (delta-normal) VaR is fast, but it assumes normality and linear exposures. Monte Carlo handles nonlinearity and any distribution you can specify. The costs are computing time and model risk: if the assumed model is wrong, the answer is precise but wrong.

Monte Carlo also has sampling error. Each run uses a finite number of draws, so the estimate varies from run to run. The standard error of the estimate falls in proportion to 1 ÷ √N, where N is the number of simulations. To halve the error you need four times as many runs. Quantiles far in the tail, such as 99.9%, are noisier than central ones because few draws land there.

Key formulas to remember

VaR from simulated quantile
VaR(c) = − (the (1 − c) quantile of simulated P&L)
With N scenarios sorted worst first, the loss is roughly the k-th worst, where k = N × (1 − c). Quote VaR as a positive loss.
Sampling error scaling
Standard error ∝ 1 ÷ √N
Quadrupling N halves the standard error. This holds for the simulation error only, not for model error.
Correlated normal draws
x = L × z, where Σ = L × Lᵀ and z is a vector of independent standard normals
L is the Cholesky factor of the covariance matrix Σ. This gives the draws the assumed correlations.
Scaling a normal draw to returns
R = μ + σ × z
z is a standard normal draw. For short horizons μ is often set to zero.
Lognormal price step (one step)
S(t+Δt) = S(t) × exp[(μ − σ² ÷ 2) × Δt + σ × √Δt × z]
Used to simulate stock prices under geometric Brownian motion. Prices stay positive.

How to solve Monte Carlo Simulation VaR questions

Use this sequence for any Monte Carlo VaR question, whether it asks you to describe the process, find a number, or compare methods.

  1. 1Identify the risk factors, the horizon and the confidence level. Note whether the answer should be a loss in currency or a percentage.
  2. 2State the model for the risk factors: distribution, volatility, correlation, and drift (often zero for short horizons).
  3. 3Generate random draws. For several factors, make them correlated using the Cholesky factor of the covariance matrix.
  4. 4Revalue the full portfolio under each scenario and compute the profit or loss for each one.
  5. 5Sort the P&L values from worst to best. Find the position k = N × (1 − c).
  6. 6Read the loss at that position. That loss is the VaR. Report it as a positive number.
  7. 7If asked about accuracy, link the error to 1 ÷ √N and to the tail quantile. If asked about method choice, weigh flexibility against cost and model risk.

Quickest way: Rank-and-read shortcut for simulated VaR

When to use it: Use it when the question gives you a list of simulated outcomes, or tells you N and asks which outcome is the VaR.

  1. Compute k = N × (1 − c). For N = 1,000 and 99%, k = 10.
  2. Count to the k-th worst loss in the sorted list. Some questions use the (k+1)-th, so check the wording.
  3. For sampling error questions, scale: new error = old error × √(old N ÷ new N).
  4. For method questions, use the keywords: nonlinear and flexible means Monte Carlo; actual history and no distribution means historical; fast and normal means delta-normal.

Common mistakes in Monte Carlo Simulation VaR

  • Saying Monte Carlo makes no distributional assumption.

    It is confused with historical simulation, because both rely on scenarios.

    Fix: Remember that Monte Carlo scenarios come from an assumed model. Historical simulation draws from actual past data.

  • Believing more simulations remove model risk.

    More runs make results smoother, which looks like higher accuracy.

    Fix: More runs reduce only sampling error. A wrong model or wrong correlation stays wrong however large N is.

  • Thinking quadrupling N cuts error to a quarter.

    Students forget the square root in 1 ÷ √N.

    Fix: Quadrupling N halves the error. To cut error to a tenth you need 100 times the runs.

  • Ignoring correlation when simulating several factors.

    Drawing each factor independently is simpler.

    Fix: Use the covariance matrix and a Cholesky decomposition so the draws carry the correct correlations.

  • Reporting VaR as a negative number or picking the wrong tail.

    The 99% quantile sounds like the top of the distribution.

    Fix: For 99% VaR, take the 1% worst P&L outcome and quote it as a positive loss.

  • Using delta approximation inside a Monte Carlo and calling it full revaluation.

    It saves time, so it feels like an acceptable shortcut.

    Fix: Full revaluation reprices options in each scenario. Delta-based shortcuts lose accuracy for large moves.

Worked examples

Example 1

A bank runs a Monte Carlo simulation with 2,000 scenarios of one-day portfolio P&L. It wants 99% VaR. The sorted losses show that the 20th worst loss is $4.8 million and the 21st worst is $4.6 million. Using the 20th worst loss as the 99% quantile, what is the VaR?

Show the solution
  1. Compute k = N × (1 − c) = 2,000 × (1 − 0.99) = 2,000 × 0.01 = 20.
  2. The 20th worst loss marks the 1% tail boundary.
  3. The question tells you to use the 20th worst loss, which is $4.8 million.
  4. Quote VaR as a positive loss.

Answer: 99% one-day VaR = $4.8 million.

Example 2

A Monte Carlo VaR run with 10,000 simulations gives a standard error on the VaR estimate of $0.40 million. The risk team wants the standard error to fall to $0.10 million. Assuming error is proportional to 1 ÷ √N, how many simulations are needed?

Show the solution
  1. Required error ratio = 0.10 ÷ 0.40 = 0.25.
  2. Error ∝ 1 ÷ √N, so new N = old N × (old error ÷ new error)².
  3. Old error ÷ new error = 0.40 ÷ 0.10 = 4.
  4. Squared, that is 16.
  5. New N = 10,000 × 16 = 160,000.

Answer: 160,000 simulations are needed, 16 times the original number.

Exam tips

  • Expect conceptual questions that ask which method suits a portfolio with options. Monte Carlo with full revaluation is the usual answer.
  • Memorise the 1 ÷ √N rule and practise the scaling arithmetic. It is the most common numerical test.
  • Know that Monte Carlo reduces sampling error with more runs but not model error. Examiners often build a wrong option around this.
  • When comparing methods, think in terms of three factors: accuracy for nonlinear risk, computing cost, and dependence on assumptions.
  • Check whether the question asks for the k-th worst loss or for a quantile, then convert k = N × (1 − c) carefully.

Practice questions from Calculating and Applying VaR

Monte Carlo Simulation VaR in other exams

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

Monte Carlo Simulation VaR: frequently asked questions

What are the steps in Monte Carlo simulation VaR?

Choose the risk factors and a model for them, then generate many random scenarios with the right correlations. Revalue the portfolio in each scenario, sort the P&L, and read the loss at the (1 − c) quantile. That loss is the VaR.

What is the difference between Monte Carlo and historical VaR?

Historical VaR uses actual past changes in risk factors as its scenarios. Monte Carlo VaR uses random scenarios from an assumed model. Monte Carlo can create outcomes never seen in the data, but it relies on the model being sensible.

What are the advantages and disadvantages of Monte Carlo VaR?

It handles nonlinear instruments, fat tails and complex dependence if you can model them. It is computationally heavy and exposed to model risk. Results also vary between runs because of sampling error.

How does sampling error work in Monte Carlo VaR?

Each run uses a finite number of random draws, so the VaR estimate is itself uncertain. The standard error falls in proportion to 1 ÷ √N. Tail quantiles at high confidence levels are noisier than central ones.