FRM Exam Part II · Non-parametric Approaches

Historical Simulation Approach to VaR Explained

Updated 11 October 2026 · Fact-checked

Historical simulation VaR ranks past returns or P&L over a lookback window and reads the loss at the chosen percentile. For 99% VaR with 500 observations, the tail holds 5 outcomes (1% of 500). The convention varies: the 5th worst, the 6th worst, or an interpolation. Follow the question's convention. It assumes the past window represents the future.

Understand Historical Simulation Approach to VaR

Historical simulation (HS) is a non-parametric method. You do not assume returns are normal. You let the data speak. You collect past returns (or P&L) over a lookback window, apply them to today's portfolio, and rank the results.

The idea is simple. If the last 500 days are a fair guide to tomorrow, then tomorrow's loss distribution looks like those 500 outcomes. The VaR at confidence level c is the loss that was exceeded in only (1 − c) of the observations. For 95% VaR, that is the 5th percentile of the P&L distribution, with losses read as positive numbers.

The key assumption is that returns are independent and identically distributed (i.i.d.), so the past sample is a good picture of the future distribution. This is also the weakness. If volatility has just risen, a long window reacts slowly. If the window holds a calm period, VaR will be too low. Every observation gets equal weight, so a crisis day drops out suddenly when it leaves the window, and VaR jumps.

HS handles fat tails, skewness and non-linear positions (such as options) naturally, because you revalue the full portfolio under each historical scenario. It also captures actual correlations without modelling them. But it cannot show losses worse than the worst observation, and it is limited by data length. Choice of window length is a trade-off: longer gives more tail data but is slower to adapt; shorter adapts quickly but gives noisy tail estimates.

Key formulas to remember

Historical return (simple): R_t = (P_t − P_(t−1)) ÷ P_(t−1); Use the same return definition consistently. Log returns are also common: ln(P_t ÷ P_(t−1)).
Scenario P&L: P&L_t = V × R_t; Apply each historical return to today's portfolio value V (or revalue the portfolio under each scenario's risk-factor changes).
Tail count: k = n × (1 − c); Number of observations in the tail. For n = 500 and c = 99%, k = 5. Under the convention used on this page, VaR is the (k+1)-th worst outcome, so k outcomes are worse than the VaR loss. Other conventions interpolate between the k-th and (k+1)-th worst outcomes; follow the question's convention.
HS VaR: VaR(c) = −(percentile of P&L at level 1 − c); Report as a positive loss. Equivalently, the loss ranked just beyond the tail count in the sorted list.
Horizon scaling (rule of thumb): VaR(T days) ≈ VaR(1 day) × √T; Valid only under i.i.d. returns with zero mean; it is an approximation, not an HS output.

How to solve Historical Simulation Approach to VaR questions

Use this sequence for any HS VaR question, whether it gives a short list of returns or a large sample.

1Note the confidence level c, the window length n and the portfolio value or position sizes.
2Compute or read the historical returns or P&L for each day. For a portfolio, apply the returns to today's holdings.
3Sort the P&L from worst loss to best gain.
4Compute the tail count k = n × (1 − c).
5Pick the loss at the tail position. If k is a whole number, the VaR is usually the (k+1)-th worst loss or the k-th, depending on the stated convention; use the one the question states, otherwise interpolate or use the quantile given.
6Report VaR as a positive number in currency terms, with the horizon and confidence level.
7If the horizon is longer than the data frequency, scale by √T only if the question allows, and state the i.i.d. assumption.
8Add the interpretation: the loss is exceeded on about (1 − c) of days under the assumption that the window represents the future.

Quickest way: Count from the bad end

When to use it: When the question gives a table of sorted or easy-to-rank outcomes and asks for VaR at a stated confidence level.

Compute k = n × (1 − c) in your head.
Mark the k worst losses from the bad end of the list.
The VaR is the next loss after those k (the (k+1)-th worst) unless the question says otherwise.
If answer options are close, check which convention the numbers fit, and whether sign and currency match.

Common mistakes in Historical Simulation Approach to VaR

Counting from the wrong end of the ranking
Students sort best to worst and take the 99th percentile from the top, picking a gain instead of a loss.
Fix: Sort losses from worst to best and count k worst outcomes from the loss end.
Using 1 − c as the tail on the wrong side, e.g. 5% for a 99% VaR
Confusing the confidence level with the tail probability.
Fix: Tail probability = 1 − c. For 99%, the tail is 1%.
Reporting VaR as a negative number or as a return when a currency amount is asked
The percentile of P&L is naturally negative, and the percentage return is not converted.
Fix: Multiply the return by portfolio value and quote VaR as a positive loss.
Claiming HS assumes normality
Mixing it up with parametric VaR.
Fix: HS assumes no distribution, only that past returns are i.i.d. and representative of the future.
Applying historical returns to historical prices rather than today's portfolio
Forgetting that scenarios should reflect current positions.
Fix: Apply each historical risk-factor change to today's holdings and revalue.
Saying a longer window is always better
More data feels safer.
Fix: A longer window gives more tail observations but responds slowly to changes in volatility. State the trade-off.

Worked examples

Example 1

A portfolio is worth $10 million. Using 500 daily returns, the 5 worst are −3.2%, −2.9%, −2.6%, −2.4% and −2.2%, and the 6th worst is −2.0%. Estimate 99% one-day HS VaR, taking VaR as the (k+1)-th worst loss where k = n × (1 − c).

Show the solution

n = 500 and c = 99%, so k = 500 × 0.01 = 5.
k is a whole number, so the (k+1)-th worst loss is the 6th worst.
The 6th worst return is −2.0%.
Loss = 2.0% × $10,000,000 = $200,000.
Check: by the stated convention, the VaR is the 6th worst return, −2.0%. Five observations (1% of 500) are worse than it: −3.2%, −2.9%, −2.6%, −2.4% and −2.2%. Under the k-th worst convention, you would use the 5th worst return, −2.2%, giving $220,000.

Answer: 99% one-day HS VaR = $200,000 under the (k+1)-th worst convention stated in the question (the 6th worst outcome, with 5 worse outcomes beyond it).

Example 2

A bank uses 500 days of P&L. At 95% confidence, the 25th worst loss is €1.8 million, the 26th worst is €1.7 million. Take VaR as the (k+1)-th worst loss. What is the VaR, and what does it mean?

Show the solution

k = 500 × (1 − 0.95) = 25.
The VaR is the (k+1)-th worst loss, which is the 26th worst.
The 26th worst loss is €1.7 million.
Interpretation: under this convention, losses larger than €1.7 million occurred on 25 of 500 days, which is 5%.
Convention note: under the k-th worst convention, the VaR would be the 25th worst loss, €1.8 million.

Answer: 95% one-day HS VaR = €1.7 million under the (k+1)-th worst convention (€1.8 million under the k-th worst convention). Under the assumption that the window represents the future, losses should exceed the VaR on about 5% of days.

Exam tips

Read the convention in the question for the tail position. Different texts use the k-th or (k+1)-th worst loss, and options may be built to catch this.
Check that the answer is a positive currency amount at the right confidence level and horizon.
Questions often ask for advantages and disadvantages. Advantages: no distribution assumption, captures fat tails and non-linearity, simple. Disadvantages: equal weights, slow reaction, limited by window data, cannot exceed the worst observation.
When volatility has just risen, expect HS VaR to understate risk; when a crisis day leaves the window, expect VaR to drop suddenly.
Link to related fixes: weighted and bootstrap HS are the standard answers to HS weaknesses.

Practice questions from Non-parametric Approaches

Historical Simulation Approach to VaR in other exams

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

Historical Simulation Approach to VaR: frequently asked questions

How do you calculate VaR using historical simulation?

Collect returns over a lookback window, apply them to today's portfolio to get scenario P&L, and sort from worst to best. Find the tail count n × (1 − c) and read the loss at that position. Quote it as a positive amount.

What are the main assumptions of historical simulation VaR?

It assumes past returns are independent and identically distributed, and that the lookback window represents the future. It does not assume any specific distribution such as the normal.

What are the advantages and disadvantages of historical simulation VaR?

It is simple, needs no distribution assumption, and handles fat tails and options. It weights all observations equally, reacts slowly to volatility changes, depends on the window chosen, and cannot show losses beyond the worst observation.

How does window length affect HS VaR?

A longer window gives more tail data and a more stable estimate but adapts slowly to new conditions. A shorter window adapts faster but has few tail observations and is noisier.