CFA Level II Exam · Measuring and Managing Market Risk

VaR Estimation Methods: Parametric, Historical Simulation and Monte Carlo

Updated 7 October 2026 · Fact-checked

Value at Risk (VaR) is the minimum loss incurred with probability α over a given horizon, not an expected loss. Parametric VaR assumes normality: VaR = (z × σ − expected return) × portfolio value. Historical simulation reads the percentile of ranked past returns. Monte Carlo simulates random outcomes from an assumed model.

Understand VaR Estimation Methods

VaR is the minimum loss that would be incurred with probability α over a given horizon. Put another way, it is the loss threshold that is reached or exceeded only α% of the time. A 5% one-day VaR of ₹10,00,000 means that on 5% of days you expect to lose at least ₹10,00,000. It is a threshold, not the worst case and not an average loss.

The three methods differ in how they get the return distribution. Parametric (variance-covariance) assumes returns are normal, so you only need the mean and standard deviation. It is fast and easy to scale across horizons, but it handles fat tails and skew poorly. It is also weak for portfolios with options, because option payoffs are non-linear. A delta-normal version approximates options with their deltas; a delta-gamma version adds convexity.

Historical simulation uses actual past returns. You apply them to today's portfolio, sort the results, and pick the loss at the chosen percentile. It needs no distribution assumption and captures fat tails and skew that appear in the sample. But it assumes the past represents the future, gives equal weight to all observations unless adjusted, and depends on the window length. Few observations in the tail make the estimate noisy.

Monte Carlo simulation generates thousands of random scenarios from a specified model of risk factors, values the portfolio under each, and reads the percentile. It is the most flexible. It handles non-linear instruments, fat tails and path dependence if you model them. It is also the slowest and most costly, and it is only as good as the model and inputs. Wrong assumptions give precise-looking but wrong answers.

In the exam, you are given a vignette with exhibits. You must pick the right method for the portfolio, compute a simple parametric VaR, or judge which limitation applies.

Key formulas to remember

Parametric VaR (return form): VaR% = −[R̄ − z × σ] = z × σ − R̄; z is the one-tailed critical value: 1.645 for 5%, 2.33 for 1%. Use the same time unit for R̄ and σ.
Parametric VaR (currency): VaR = (z × σ − R̄) × Portfolio value; Reported as a positive loss amount. If the mean is ignored, the VaR is z × σ × value.
Scaling volatility over time: σ(T days) = σ(1 day) × √T; Assumes independent returns. Scale the mean by T, not √T.
Annual to daily conversion: σ(daily) = σ(annual) ÷ √(number of trading days); Use the number of trading days stated in the vignette, often 250 or 252.
Historical simulation percentile: VaR at α% = loss at the α percentile of ranked outcomes; With 500 observations, 5% VaR is around the 25th worst loss. When α% of the observations is not a whole number, the position depends on the convention the question states (round up, round down or interpolate).

How to solve VaR Estimation Methods questions

Use this approach for any VaR-method item set question, whether it asks for a number or a comparison.

1Read the question and note the confidence level, horizon and method asked for.
2Find the data in the vignette and exhibits: portfolio value, mean, standard deviation, trading days, and whether the numbers are daily or annual.
3Match the units. Convert annual figures to the horizon with √T for volatility and T for the mean.
4For parametric VaR, pick z from the confidence level and compute (z × σ − mean) × value.
5For historical simulation, rank the outcomes from worst to best and count down to the required percentile position, using the convention the question states.
6For method comparison, check the portfolio: options or non-linear payoffs point away from parametric; fat tails point away from normality; need for speed points away from Monte Carlo.
7Check the answer: VaR should be positive, larger for a higher confidence level and longer horizon.

Quickest way: Unit-check then z × σ

When to use it: Use for numerical parametric VaR questions where time is short.

Convert σ and the mean to the horizon first.
Compute z × σ, then subtract the mean return.
Multiply by portfolio value.
For 'which method' questions, scan for options, fat tails or speed constraints and eliminate options that contradict them.

Common mistakes in VaR Estimation Methods

Using a two-tailed z-value such as 1.96 for a 5% VaR.
Confidence interval habits from hypothesis testing carry over.
Fix: VaR is one-tailed. Use 1.645 for 5% and 2.33 for 1%.
Scaling the mean by √T along with the standard deviation.
Students apply the square-root rule to everything.
Fix: Scale σ by √T and the mean by T.
Treating VaR as the maximum possible loss.
The word 'value at risk' sounds like a worst case.
Fix: VaR is the minimum loss incurred with probability α over the horizon. It is a threshold, not an expected loss. Losses beyond it can be much larger.
Saying historical simulation assumes a normal distribution.
It is confused with the parametric method.
Fix: Historical simulation is non-parametric. Its assumption is that the past distribution repeats.
Saying parametric VaR works well for option portfolios.
Its simplicity is mistaken for generality.
Fix: Options have non-linear payoffs. Parametric VaR fits them poorly. Prefer Monte Carlo or full revaluation.
Mixing daily and annual inputs.
The vignette gives figures in different periods.
Fix: Write the unit next to every input before you compute.

Worked examples

Example 1

A portfolio is worth ₹50,00,000. Its daily return has a mean of 0.04% and a standard deviation of 1.20%. Returns are assumed normal. Q1: What is the 1-day 5% parametric VaR? Q2: Is the 1-day 1% VaR larger or smaller? Q3: Which feature of the method is a weakness if the portfolio holds many options?

Show the solution

Q1: z for 5% one-tailed is 1.645.
z × σ = 1.645 × 1.20% = 1.974%.
Subtract the mean: 1.974% − 0.04% = 1.934%.
VaR = 0.01934 × ₹50,00,000 = ₹96,700.
Q2: At 1%, z = 2.33, so z × σ = 2.796%. Minus 0.04% gives 2.756%, which is larger.
VaR = 0.02756 × ₹50,00,000 = ₹1,37,800.
Q3: Parametric VaR assumes a linear exposure and normal returns, so it misrepresents non-linear option payoffs.

Answer: Q1: ₹96,700. Q2: Larger (₹1,37,800). Q3: The linear, normal assumption fails for non-linear option payoffs.

Example 2

A risk manager has 200 daily portfolio losses from the past 200 trading days, from a current portfolio of ₹20,00,000 revalued on each day's returns. Ranked from worst to best, the 2nd worst loss is ₹41,000, the 4th worst is ₹38,000, and the 10th worst is ₹27,000. Use this convention: the VaR at α% is the kth worst loss, where k = α × N and N is the number of observations. Q1: Using historical simulation, what is the 5% one-day VaR? Q2: What is the 2% one-day VaR? Q3: Name one advantage of this method over parametric VaR.

Show the solution

Q1: k = 5% × 200 = 10. The 5% VaR is the 10th worst loss, which is ₹27,000.
Q2: k = 2% × 200 = 4. The 2% VaR is the 4th worst loss, which is ₹38,000.
Q3: Historical simulation needs no normality assumption and uses real data, so it captures fat tails and skew present in the sample.

Answer: Q1: ₹27,000. Q2: ₹38,000. Q3: No distributional assumption; captures fat tails and skew in the data.

Exam tips

Write the unit of every input (daily or annual) before computing. Many errors are unit errors.
For 'which method' questions, look for options, non-linear payoffs or fat tails in the vignette. These usually decide the answer.
Remember the trade-off set: parametric is fast but assumes normality; historical is simple but limited to past data; Monte Carlo is flexible but slow and model-dependent.
Use one-tailed z-values: 1.645 and 2.33. Do not use 1.96.
There is no penalty for wrong answers, so answer every question even if you must guess between two options.

VaR Estimation Methods in other exams

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

VaR Estimation Methods: frequently asked questions

What is the difference between parametric, historical simulation and Monte Carlo VaR?

Parametric VaR assumes a normal distribution and uses the mean and standard deviation. Historical simulation ranks actual past returns. Monte Carlo generates many random scenarios from an assumed model. They differ in assumptions, speed and handling of non-linear risk.

How do I calculate parametric VaR for CFA Level II?

Convert the mean and standard deviation to the horizon, choose the one-tailed z-value, and compute (z × σ − mean) × portfolio value. Use 1.645 for 5% and 2.33 for 1%.

What are the advantages and disadvantages of Monte Carlo VaR?

It handles non-linear instruments, fat tails and path dependence if the model is built for them. The costs are computing time, complexity and dependence on the model and inputs. Poor assumptions give misleading results.

Why is historical simulation VaR criticised?

It assumes the past period represents the future and is sensitive to the window length. Tail estimates can be noisy because few observations lie in the tail. Standard versions weight all observations equally.