Monte Carlo simulation
Also written Monte Carlo method · Monte Carlo VaR · Monte Carlo simulation method
A VaR method that generates the portfolio's returns at random from chosen statistical parameters, repeats this about 10,000 times, sorts the outcomes and reads off the required percentile.
In plain language
The other two ways of estimating Value at Risk both work off a fixed set of numbers. The parametric method needs a formula and a bell curve. Historical simulation needs a run of past returns.
Monte Carlo simulation manufactures the numbers instead. The portfolio's value and returns are generated at random from statistical parameters the user supplies — either drawn from history, or set by the user's own assumptions.
Then it is done again. And again, around 10,000 times.
Each run gives one possible outcome for the portfolio. Sort all of them and read the percentile you want, exactly as in historical simulation.
It earns its keep when the portfolio is complicated. Many constituents, many market factors, or a return pattern that is nothing like a bell curve — Monte Carlo can cope with all three.
How it works
The method (section 17.4.2.3). Monte Carlo simulation is achieved by simulating the portfolio's return over a desired period. The portfolio's value and returns are randomly generated with the given statistical parameters — the random number is generated using either the historical parameter or the user's assumptions. It is then repeated for a large number of times, nearly 10,000 times. The outcomes are arranged in ascending or descending order, as in historical simulation, and VaR is finally estimated using the percentile method, again as in historical simulation.
When it is the right tool. The workbook is specific. Monte Carlo works best compared with the other methods when:
- the factors affecting the market are many, and
- the portfolio is complex, with many constituents.
And secondly, it can accommodate any distribution pattern — not only the normal distribution the parametric method requires.
What it shares and does not share with the other two.
| Parametric | Historical simulation | Monte Carlo | |
|---|---|---|---|
| Source of returns | Formula from mean and standard deviation | Observed past returns | Randomly generated |
| Distribution assumed | Normal | Whatever history was | Any |
| Reading the answer | z-score in a formula | Percentile of sorted returns | Percentile of sorted outcomes |
| Runs | One calculation | One per historical period | About 10,000 |
The limits of VaR still apply. Whichever method produces it, the workbook's table of VaR limitations stands: subjectivity is involved, and VaR underestimates both the extreme loss and the extreme gain scenarios. Its advantages are equally listed — simple and easy to understand, universally accepted including by regulators, and helpful in allocating capital across portfolios.
A worked example
Illustrative figures, following the workbook's method. A manager runs a Rs 250 crore multi-asset portfolio — Indian equity, corporate bonds, gold and a currency-hedged overseas sleeve. Four asset classes, several driving factors, and option positions whose pay-offs are not symmetric. Neither a single formula nor one historical window fits it well.
She sets the parameters: expected returns, volatilities and correlations for each sleeve, and the option pay-off rules. Then she runs 10,000 simulated one-month paths.
The 10,000 monthly portfolio returns are sorted worst to best:
| Percentile | Rank in 10,000 | Simulated return | VaR on Rs 250 crore |
|---|---|---|---|
| 1% (99% VaR) | 100th worst | −8.2% | Rs 20,50,00,000 |
| 5% (95% VaR) | 500th worst | −5.4% | Rs 13,50,00,000 |
| 10% (90% VaR) | 1,000th worst | −4.1% | Rs 10,25,00,000 |
She reports a 95% one-month VaR of Rs 13.5 crore: on 95 months out of 100, the portfolio should not lose more than that.
The number is only as good as her inputs. Change the equity-gold correlation assumption from −0.2 to 0 and re-run, and the 95% figure moves — which is the workbook's point about subjectivity in VaR. And on the 500 worst paths, the average loss is far larger than Rs 13.5 crore, which is its point about VaR underestimating extreme losses.
Why NISM asks about it
Chapter 17 (Risk), section 17.4.2.3 (Monte Carlo Simulation), is the third of the three VaR methods. It is short, so the examinable content is precise: the random generation of returns, the nearly 10,000 repetitions, the percentile method of reading the answer, and the two conditions under which it works best — many market factors and a complex portfolio with many constituents — plus its ability to accommodate any distribution.
Expect a question asking which VaR method suits a complex multi-factor portfolio, or which of the three does not require a normal distribution. The advantages-and-limitations table that follows the section is examined alongside it, including sample question 4 on what VaR does not do.
Common exam traps
- Monte Carlo returns are generated, not observed. That is the single difference from historical simulation, which uses actual past returns. Everything after the generation step — sorting, percentile — is the same in both.
- Nearly 10,000 runs is the workbook's own figure. A question offering 100 or 1,000 is testing whether you read it.
- It accommodates any distribution. Only the parametric method requires normality.
- The random parameters can come from history. Using historical parameters inside Monte Carlo does not make it historical simulation — the returns are still generated.
- VaR's limitations do not disappear with a better method. Subjectivity remains, and every VaR underestimates extreme losses and extreme gains.
- VaR does not take liquidity into account — that is what liquidity-adjusted VaR is for, and it is a listed answer in the chapter's sample question on what VaR is not.
Check yourself
1.Which VaR method randomly generates portfolio returns nearly 10,000 times and can accommodate any distribution pattern?
- a)Parametric method
- b)Historical simulation method
- c)Monte Carlo simulation
- d)Bid-ask spread method
Show the answer
Answer: (c) Monte Carlo simulation
Monte Carlo simulation randomly generates values with given parameters, repeats nearly 10,000 times, sorts the outcomes and reads the percentile. It works best for complex portfolios and can accommodate any distribution.
Parametric VaR assumes normal returns. Historical simulation uses actual past returns, not random ones. The bid-ask spread measures liquidity, not VaR.
Where this is taught
Free preparation for NISM Series XXI-BRelated terms
- Value at RiskA statistical estimate of the most a position is likely to lose over a set holding period at a chosen confidence level — the idea behind the initial margin a clearing corporation charges on a derivatives trade.
- Variance-covariance matrixThe grid of every asset's variance and every pair's covariance that is needed to compute the risk of a portfolio holding more than two securities.
- Factor modelA model that explains a security's or portfolio's return through its sensitivity to chosen factors — macroeconomic, fundamental or statistical — rather than through a single market beta alone.
- Historical simulationA way of estimating Value at Risk with no assumption about the shape of returns: apply today's portfolio to several years of past returns, sort the results, and read off the required percentile.
- Liquidity-adjusted VaRRegular Value at Risk plus the cost of unwinding the positions — because ordinary VaR measures the price move but ignores what it costs to actually get out.
- Parametric VaRValue at Risk estimated from just two parameters — expected return and standard deviation — assuming returns are normally distributed.
- Stress testA scenario-based risk technique where a portfolio manager creates an extreme negative event, real or hypothetical, to estimate the potential loss it would cause to the portfolio.
- Safe Withdrawal RateThe largest share of the starting corpus that can be withdrawn in the first year of retirement, then raised each year by inflation, while the portfolio survives the planned period with high probability.