CFA Level I Exam · Simulation of Financial Asset Prices and Returns

Monte Carlo Simulation Basics for CFA Level I

Updated 7 October 2026 · Fact-checked

Monte Carlo simulation estimates the distribution of an uncertain outcome by repeatedly drawing random values for the inputs from assumed probability distributions, computing the outcome each time, and studying the results. Steps: specify the model, set input distributions, generate random draws, compute outcomes, repeat many times, then summarize.

Understand Monte Carlo Simulation Basics

Many financial outcomes depend on inputs you cannot know in advance: future returns, interest rates, sales growth. A formula that uses a single guess for each input gives a single answer. It tells you nothing about the range of outcomes.

Monte Carlo simulation fixes this. You describe each uncertain input with a probability distribution, such as normal or lognormal. A computer then draws a random value for each input, plugs them into your model, and records the result. That is one trial. You repeat it thousands of times. The recorded results form an empirical distribution of the outcome.

From that distribution you can read the mean, standard deviation, percentiles, and the probability of a bad result, such as a portfolio falling below a target. Analysts use it for option valuation, value at risk, pension funding, and project risk when no neat closed-form answer exists.

The random numbers start as draws from a continuous uniform distribution between 0 and 1. A computer produces them with a pseudo-random number generator, a deterministic algorithm that looks random. These uniform draws are then converted into draws from the distribution you need, for example by using the inverse of that distribution's cumulative function. For a standard normal variable, the draw is the z-value whose cumulative probability equals the uniform number.

Simulation is only as good as its inputs. If the assumed distributions or correlations are wrong, the output is wrong, however many trials you run. It gives statistical estimates, not exact answers, and it does not give causal insight the way an analytical model can.

Key formulas to remember

Simulated price with normal returns (one period): S(t+1) = S(t) × exp(r), where r = μ + σ × Z; Z is a standard normal draw. Using exp(r) means r is a continuously compounded return and the price cannot go below zero.
Standard normal from a uniform draw: Z = N⁻¹(U), where U is uniform on 0 to 1; N⁻¹ is the inverse standard normal cumulative function. U = 0.5 gives Z = 0.
Mean of simulated outcomes: Mean = (Σ outcomes) ÷ number of trials; Use the same average for any summary statistic, such as the probability of a loss = trials with loss ÷ total trials.
Standard error of the simulated mean: SE = s ÷ √N; s is the standard deviation of the trial results and N is the number of trials. To halve the error you need four times as many trials.

How to solve Monte Carlo Simulation Basics questions

Use this order for most Monte Carlo questions, whether they ask for steps, a calculation, or a judgement about limits.

1Identify the outcome the analyst wants to estimate and the model that links inputs to that outcome.
2List the uncertain inputs and the distribution assumed for each, including any correlation between them.
3Note how the random draws are made: uniform numbers first, then converted to the required distribution.
4If a calculation is asked, convert the given random number or z-value into the input value, then run the model once.
5For summaries, count or average over the trials: mean, percentile, or probability of an event.
6Check the conclusion against the limits: results depend on assumptions and are statistical estimates.
7Eliminate options that claim exact results, certainty, or that more trials fix wrong assumptions.

Quickest way: Three-option elimination for Monte Carlo questions

When to use it: Use it for conceptual questions about what simulation does, its steps, or its limits.

Reject any option saying simulation gives an exact or guaranteed answer.
Reject any option saying more trials correct a wrong distribution assumption.
Keep the option that mentions random draws from assumed distributions, repeated many times, then summarized.
For a calculation, do one trial only: draw to input to outcome, then apply the formula.

Common mistakes in Monte Carlo Simulation Basics

Thinking simulation gives the true answer.
Thousands of trials feel precise.
Fix: Remember it gives an estimate. Its accuracy depends on the number of trials and on the assumed inputs.
Believing more trials fix bad assumptions.
Students confuse sampling error with model error.
Fix: More trials reduce sampling error only, in proportion to 1 ÷ √N. Wrong distributions or correlations stay wrong.
Ignoring correlation between inputs.
Each input is drawn separately in simple examples.
Fix: If inputs move together, the draws must reflect that correlation. Otherwise risk is misstated.
Treating the generated random numbers as truly random.
The word random suggests no pattern.
Fix: Computers use pseudo-random generators, which are deterministic algorithms that produce numbers that look random.
Applying a normal return directly to price and allowing negative prices.
Mixing up price level and return.
Fix: Draw the return, then compound it: S × exp(r). Prices then stay positive.

Worked examples

Example 1

A stock is at $50. An analyst simulates a one-year continuously compounded return as r = 0.06 + 0.20 × Z. In one trial, Z = 1.5. What is the simulated end price? Options: A) $50.00 B) $68.00 C) $71.67

Show the solution

Compute the return: r = 0.06 + 0.20 × 1.5 = 0.06 + 0.30 = 0.36.
Compute exp(0.36) = 1.43333.
Price = 50 × 1.43333 = $71.67 (rounded to the nearest cent).
Option B ($68.00) comes from treating r as a simple return: 50 × (1 + 0.36). That ignores continuous compounding.

Answer: C) $71.67. On a BA II Plus: enter 0.36, press 2ND, eˣ to get 1.4333, then × 50 = 71.67.

Example 2

A simulation of a portfolio's annual return runs 1,000 trials. The trial results have a standard deviation of 12%. What is the standard error of the simulated mean return? Options: A) 0.38% B) 1.20% C) 3.79%

Show the solution

Use SE = s ÷ √N.
√1,000 ≈ 31.62.
SE = 12% ÷ 31.62 = 0.379%.
So the mean estimate is precise to about 0.38 percentage points.

Answer: A) 0.38%

Exam tips

Know the sequence of steps and their order: specify the model, choose distributions, draw random numbers, compute outcomes, repeat, summarize.
Remember the standard limitations: it depends on assumptions, gives statistical estimates, and does not give analytical insight.
Items are standalone with three options, so use the 'exact answer' and 'more trials fix assumptions' traps to eliminate fast.
Practise one-trial calculations: turn a z-value into a return, then into a price with exp.
At about 90 seconds per question, do not simulate by hand; only transform one draw.

Practice questions from Simulation of Financial Asset Prices and Returns

Monte Carlo Simulation Basics in other exams

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

Monte Carlo Simulation Basics: frequently asked questions

What are the steps in Monte Carlo simulation?

Specify the model and the outcome of interest, assign probability distributions to the uncertain inputs, generate random draws, compute the outcome for each set of draws, repeat many times, and summarize the results. The summary gives a mean, spread and probabilities.

How are random numbers generated in a simulation?

A pseudo-random number generator produces uniform numbers between 0 and 1. These are converted into draws from the required distribution, often using the inverse cumulative function. The numbers are deterministic but look random.

Why do analysts use Monte Carlo simulation?

It handles problems with many uncertain inputs or complex payoffs where no closed-form solution exists. Examples are option valuation, value at risk, and project or retirement planning. It shows the whole range of outcomes, not one number.

What are the limitations of Monte Carlo simulation?

It relies on assumed distributions and correlations, so poor assumptions give poor results. It produces estimates, not exact answers, and gives less insight into cause and effect than an analytical model.