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FRM Exam Part II · Estimating Market Risk Measures: An Introduction and Overview

Historical Simulation VaR: Step-by-Step Method

Updated 11 October 2026 · Fact-checked

Historical simulation VaR is a nonparametric method. You collect past returns or P&L, apply them to today's portfolio, sort them from worst to best, and read off the loss at the chosen quantile. At 95% confidence with 100 observations, VaR is the loss near the 5th worst outcome.

Understand Historical Simulation VaR

Historical simulation (HS) assumes the recent past is a fair sample of the future. You do not assume a normal or any other distribution. The data are the distribution.

The idea is simple. Take a window of past observations, for example the last 500 daily returns. Each one is a possible outcome for tomorrow. Apply each return to your current portfolio value to get a loss or profit for each scenario. Then sort the results.

VaR is a quantile of that sorted list. At 95% confidence, you want the loss that is exceeded only 5% of the time. With 500 observations, 5% is 25 observations. So VaR sits around the 25th worst outcome. Exact conventions differ, and an exam question will tell you which to use, such as taking the 5th worst of 100 or interpolating between two points.

Strengths: it is easy to explain, needs no distribution assumption, captures fat tails, skewness and non-normal dependence that are present in the sample, and handles nonlinear positions if you reprice them fully.

Limits: results depend on the window length. Every observation gets equal weight, so old data matter as much as recent data, and the VaR reacts slowly to a change in volatility. It cannot show losses worse than anything in the sample. Tail estimates at high confidence are noisy because few observations sit beyond the quantile. It also gives a single estimate with no built-in standard error, and extreme past events can drop out of the window and suddenly change VaR.

Key formulas to remember

Scenario P&L
P&L(i) = Portfolio value today × return(i)
For a linear position. For options or nonlinear positions, reprice the portfolio under each historical scenario instead.
Number of tail observations
k = n × (1 − confidence level)
With n = 500 at 99%, k = 5. VaR is read near the 5th worst loss. Follow the convention given in the question.
HS VaR
VaR = −(P&L at the (1 − c) quantile of the sorted scenario P&Ls)
Reported as a positive loss number.
Scaling to longer horizons (rule of thumb)
VaR(h days) ≈ VaR(1 day) × √h
Valid only if returns are independent and identically distributed with zero mean. It is an approximation, not a property of HS.

How to solve Historical Simulation VaR questions

Use this order for any historical simulation VaR question.

  1. 1Note the portfolio value, the confidence level, the horizon and the number of observations n.
  2. 2Convert each historical return into a P&L for today's portfolio (or reprice for nonlinear positions).
  3. 3Sort the P&Ls from worst loss to best gain.
  4. 4Compute the tail count k = n × (1 − confidence level).
  5. 5Read the loss at that position, using the convention in the question. If k is not a whole number, interpolate or use the stated rule.
  6. 6Report VaR as a positive loss in the currency asked, and scale the horizon only if told to and if the assumptions hold.
  7. 7Add the interpretation: with the stated confidence, the loss should not exceed VaR over the horizon, based on the sample.

Quickest way: Count from the bad end

When to use it: When the exam lists sorted or partly sorted losses and asks for VaR at a given confidence.

  1. Compute k = n × (1 − c).
  2. Count k losses from the worst end of the list.
  3. Read the loss there, unless the question states a different convention.
  4. Multiply return by portfolio value if the list is in returns.
  5. Check the sign: VaR is a positive loss.

Common mistakes in Historical Simulation VaR

  • Counting from the best outcome instead of the worst

    Lists are sometimes sorted ascending by profit and students read the wrong end.

    Fix: Always locate the worst losses first, then count k observations in from that end.

  • Using the confidence level instead of the tail probability to find the position

    Students take 95% of n rather than 5% of n.

    Fix: Use k = n × (1 − c). Then read the kth worst loss.

  • Forgetting to apply returns to today's portfolio value

    Students read return numbers as if they were already in currency.

    Fix: Multiply each return by the current position value, or reprice the portfolio under each scenario.

  • Claiming HS assumes normality or needs a volatility estimate

    Confusion with parametric VaR.

    Fix: HS is nonparametric. The empirical quantile replaces the normal quantile times sigma.

  • Saying HS reacts quickly to new volatility or can predict losses beyond the sample

    Students assume more data means a more current estimate.

    Fix: Equal weighting makes HS slow to adjust, and it cannot show losses worse than the worst observation. Weighted schemes address the first problem.

  • Applying √h scaling as a property of HS

    The rule is common in parametric VaR.

    Fix: Treat it as an approximation that needs i.i.d. returns and zero mean, or use multi-day historical returns.

Worked examples

Example 1

A portfolio is worth $10 million. You hold 250 daily returns. Find the 1-day 99% historical simulation VaR if the worst three returns are −4.2%, −3.1% and −2.8%, the next ones are milder, and you take VaR as the 3rd worst outcome.

Show the solution
  1. Tail probability is 1% of 250 = 2.5 observations.
  2. The question states to use the 3rd worst outcome.
  3. The 3rd worst return is −2.8%.
  4. Loss = 2.8% × $10,000,000 = $280,000.

Answer: 1-day 99% VaR = $280,000.

Example 2

A fund holds ₹50,00,00,000 in equities. From 500 daily returns, the 25th worst return is −1.9% and the 26th worst is −1.8%. Estimate 1-day 95% HS VaR using the 25th worst return.

Show the solution
  1. Tail count k = 500 × 5% = 25.
  2. The 25th worst return is −1.9%.
  3. Loss = 1.9% × ₹50,00,00,000 = ₹95,00,000.

Answer: 1-day 95% VaR = ₹95,00,000. Over one day, losses should exceed this about 5% of the time if the past is representative.

Exam tips

  • Check which quantile convention the question uses before you count.
  • Expect conceptual MCQs on strengths and limits. Link each limit to equal weighting, window length or sample tail data.
  • If the position has options, the right answer usually involves full revaluation under each scenario.
  • Be ready to say why HS VaR at 99% is noisy: very few observations lie in the tail.
  • Read the sign: VaR is quoted as a positive loss.

Practice questions from Estimating Market Risk Measures: An Introduction and Overview

Historical 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.

Historical Simulation VaR: frequently asked questions

How do you calculate historical simulation VaR step by step?

Convert past returns into P&L on today's portfolio, sort from worst to best, and find k = n × (1 − confidence). VaR is the loss at that position. Use the convention the question states.

How do you do historical simulation VaR in Excel?

Put the scenario P&Ls in a column. Use PERCENTILE.INC with the tail probability, for example 0.05 for 95% confidence, and flip the sign to report a positive loss. Excel interpolates between observations, so the result can differ from a simple counting rule.

What are the advantages and disadvantages of historical simulation VaR?

It is simple, makes no distribution assumption and keeps fat tails and skewness present in the data. It weights all observations equally, reacts slowly to volatility changes, depends on the window and cannot show losses beyond the sample.

Does historical simulation need a normal distribution?

No. It is nonparametric, so the empirical quantile of past outcomes replaces any distribution assumption.