CFA Level II Exam · Backtesting and Simulation

Historical and Parametric Simulation in Portfolio Construction

Updated 7 October 2026 · Fact-checked

Historical simulation applies a portfolio's past returns, or randomly resampled past returns, to today's holdings. Parametric simulation (Monte Carlo) draws returns from an assumed distribution with chosen parameters such as mean, volatility and correlation. To answer exam questions, identify the data source, the assumptions it needs, and the limits each brings.

Understand Historical and Parametric Simulation

A simulation asks: how might this portfolio or plan behave across many possible futures? You cannot know the future, so you generate many paths and study the spread of outcomes, such as final wealth, drawdown or the chance of running out of money.

Historical simulation takes returns that actually happened and uses them as the possible futures. In its simplest form you replay the historical sequence. In a bootstrap version, you draw past periods at random with replacement to build many new paths. It assumes no particular distribution. Fat tails, skewness and real correlations between assets are kept because they are in the data.

Parametric simulation (often called Monte Carlo simulation) starts with assumptions. You choose a distribution, usually normal or lognormal, and parameters: expected return, standard deviation and correlations for each asset. A random number generator then draws returns from that model, many thousands of times. You can set parameters to reflect forward-looking views, not just history.

Each has a weakness. Historical simulation can only produce what the sample contained. If the past had no crisis, the results show none, and the past may not repeat. Parametric simulation is only as good as its inputs and distribution. A normal assumption understates tail risk, and wrong correlations give misleading diversification.

Scenario analysis is different. It tests a small number of specified, often narrative, situations (for example, a rate shock plus an equity fall). Simulation produces a full distribution of outcomes from random draws. Scenario analysis is chosen by the analyst. Simulation outcomes come from the data or the model.

Key formulas to remember

Parametric draw for a single asset (normal model): Rₜ = μ + σ × z, where z is a standard normal random draw; μ is the assumed mean return and σ the assumed standard deviation. Used for one period.
Correlated multi-asset draws: Draws must reflect the covariance matrix (variances and covariances) of the assets; Independent draws would ignore correlation and overstate diversification.
Compounding simulated returns to a terminal value: Vₜ₊₁ = Vₜ × (1 + Rₜ); Compounding simulated arithmetic returns period by period gives a terminal value for each path. If returns are modelled as continuously compounded and normally distributed (a lognormal price path), use Vₜ₊₁ = Vₜ × e^(rₜ) instead.
Probability estimate from simulation: P(event) ≈ number of paths where event occurs ÷ total number of paths; For example, probability of shortfall equals paths ending below the goal divided by all paths.

How to solve Historical and Parametric Simulation questions

Use this method on any vignette about historical or parametric simulation.

1Identify the method: past returns used directly or resampled means historical; assumed mean, volatility and correlation means parametric.
2List the inputs given in the exhibit: sample period, distribution, parameters, number of trials, time horizon.
3Check what the method assumes: historical assumes the past sample represents the future; parametric assumes the chosen distribution and parameters are right.
4Run or interpret the mechanics: apply returns to the current portfolio weights, compound over the horizon, and count paths meeting the target.
5Compute the requested statistic, such as the probability of shortfall, a percentile outcome or the average terminal value.
6Test the result against limitations: tail risk, regime change, correlation instability, estimation error in inputs.
7Choose the answer that matches the method's stated strengths and weaknesses, not a generic statement.

Quickest way: Data source test

When to use it: Use when the question asks which method fits, or which limitation applies.

Ask: where do the returns come from? Actual past observations means historical. A formula with chosen parameters means parametric.
If the vignette mentions fat tails or real correlations being captured, it is historical.
If the vignette mentions forward-looking views or a changed regime, it favours parametric.
If only a few named situations are tested, it is scenario analysis, not simulation.
For probability questions, count paths meeting the condition and divide by total paths.

Common mistakes in Historical and Parametric Simulation

Saying historical simulation needs a distribution assumption
Students mix it up with parametric methods.
Fix: Historical simulation is non-parametric. It uses the empirical data as it is.
Assuming parametric simulation captures fat tails
Random draws look realistic.
Fix: If the model is normal, tails are thin. Fat tails appear only if the distribution is chosen to include them.
Treating simulation and scenario analysis as the same
Both test a portfolio under different conditions.
Fix: Scenario analysis uses a few chosen cases. Simulation generates many random paths and a distribution of results.
Ignoring correlations when drawing multi-asset returns
Each asset is easy to simulate on its own.
Fix: Draws must come from a joint distribution. Ignoring correlation misstates portfolio risk.
Believing more trials fixes bad inputs
Many trials feel precise.
Fix: More trials reduce sampling noise only. Poor parameters or an unrepresentative past still give poor answers.

Worked examples

Example 1

A planner simulates a client's portfolio using 1,000 paths. Returns are drawn from a normal distribution with mean 6% and standard deviation 10% per year. The client's goal needs a 10-year ending value of at least ₹1,00,00,000 from ₹60,00,000, which requires about 5.2% a year. The simulation report states that the goal is met in 540 of the 1,000 paths. This 540 is the reported simulation result, not something you calculate from the inputs. Q1: Which method is this? Q2: What is the estimated probability of meeting the goal? Q3: What is the key limitation?

Show the solution

Q1: Returns come from an assumed distribution with chosen mean and standard deviation, so it is parametric (Monte Carlo) simulation.
Q2: Take the reported count as given. Probability = paths meeting the goal ÷ total paths = 540 ÷ 1,000 = 0.54, or 54%.
Q3: The results depend on the normal assumption and the 6% and 10% inputs. A normal distribution understates tail risk, and wrong parameters mislead.

Answer: Q1: Parametric simulation. Q2: 54%, based on the reported 540 successful paths. Q3: Dependence on the assumed distribution and parameters, with thin tails under the normal model.

Example 2

An analyst resamples 240 months of past returns with replacement to build 5,000 ten-year paths for a global equity and bond portfolio. The sample period included no major equity crash. Q1: Which method is this? Q2: Does it need correlation assumptions? Q3: What is the main concern about the output?

Show the solution

Q1: Actual past returns are resampled, so it is historical simulation, specifically a bootstrap.
Q2: If whole months are drawn (equity and bond returns together), the real historical correlation is kept automatically. No separate correlation assumption is needed.
Q3: The sample had no major crash, so the paths cannot show one. Tail risk will be understated and the past may not represent the future.

Answer: Q1: Historical simulation (bootstrap). Q2: No, observed joint returns carry the correlation. Q3: Output is limited to what the sample contained, so tail risk is likely understated.

Exam tips

Match the method to the data source first. Most questions turn on this.
Memorise one strength and one weakness per method; answer options usually test these.
For probability questions, divide paths meeting the condition by total paths.
Separate scenario analysis (few chosen cases) from simulation (many random paths).
Do not choose an option claiming more trials remove input or model risk.

Historical and Parametric Simulation in other exams

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

Historical and Parametric Simulation: frequently asked questions

What is the difference between historical simulation and Monte Carlo simulation?

Historical simulation uses actual past returns, directly or resampled. Monte Carlo draws returns from an assumed distribution with chosen parameters. The first needs no distribution assumption, while the second needs inputs but allows forward-looking views.

Is parametric simulation the same as Monte Carlo?

In this context, yes. Both mean generating random returns from a specified distribution and parameters. The exam may use either term.

How is scenario analysis different from simulation?

Scenario analysis examines a few specified situations chosen by the analyst. Simulation produces many random paths and a distribution of outcomes. Scenarios suit stress thinking, while simulation suits probability estimates.

How do I run a historical simulation for a portfolio?

Collect past returns for each asset, apply them to current weights, and either replay the sequence or resample periods with replacement to build many paths. Then compound returns over your horizon and study the distribution of results.