Monte Carlo Simulation, Explained with Examples

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Monte Carlo Simulation

Quantitative Analysis · Chapter 3: Simulation and Monte Carlo MethodsSection 3.3 · lesson 11 of 5512 min · free with a free account

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In this lesson

Rolling the dice deliberately - the scope and business applications.

  • Explain the sampling logic that drives Monte Carlo
  • Choose an iteration count and test for convergence
  • Recognize when input correlation distorts risk
  • Read histograms, S-curves and tornado charts

Transcript

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QAN-201 - Quantitative Analysis · Section 3.3

Monte Carlo Simulation

Most business plans are built on numbers quietly pretending to be certain. Sales will be five million dollars next year. This project will cost two million. Single points, stated with confidence, and almost never exactly right. In reality almost every input is uncertain, and that uncertainty compounds. Monte Carlo simulation is the most widely used simulation technique in business. It exists precisely because single numbers cannot carry compounding uncertainty. This section covers the sampling logic that drives it. How many iterations to run, and how to know when to stop. When correlation between inputs will quietly distort your risk estimate. And how to read the outputs that turn a simulation into a decision.

Why Distributions

The Trouble With Single-Point Estimates

Start with how most forecasts arrive on a manager's desk: as a single point estimate, one number, stated without a range. Sales will be five million dollars next year. This project will cost two million. The plan is then built on that number as if it were a fact. But almost every input in that plan is uncertain. And here is the part people underestimate. The uncertainty compounds. A handful of inputs, each only mildly uncertain, can produce a wildly uncertain output. A point estimate has nowhere to put that. There is no room in one number. Monte Carlo replaces each point estimate with a distribution. It samples those distributions thousands of times, and gives you the whole picture of outcomes.

The Core Idea

Sample, Compute, Aggregate

The mechanics are simpler than the name suggests. You run the same model over and over, with different random inputs each time. The logic has three steps. First, sample. Draw one value from each input's probability distribution. Revenue, cost, demand, lead time, and so on. Second, compute. Run the model on that one set of sampled inputs. You now have one outcome, from one plausible version of the future. Third, aggregate. Collect the thousands of resulting outputs together. Not into a single number, but into a full distribution. A histogram showing every plausible outcome, and how likely each one is. That shift, from a point to a distribution, is the whole idea.

How Many Runs

Iterations, and Knowing When to Stop

So how many runs are enough? The answer flows from the Central Limit Theorem. As the number of samples grows, the sample mean tightens around the true mean. And the standard error shrinks, in proportion to one over the square root of n. That square root is why brute force gets expensive fast. Doubling your iterations cuts the error by only about thirty per cent. To cut it by ninety per cent, you must multiply iterations by one hundred. In practice, one thousand to ten thousand iterations give stable estimates. Fewer for rough screening. More when you need tight estimates of rare events. Convergence is the point where more iterations stop meaningfully changing the answer. The standard check is to plot the running mean as iterations accumulate. When it stops bouncing and settles into a narrow band, you have converged. A second check: run it twice with different seeds, and see whether the answers match.

Correlated Inputs

The Independence Assumption, and What It Costs

Now the assumption that quietly breaks more Monte Carlo models than any other. Default implementations treat inputs as independent. Sample revenue, then sample cost separately. But real businesses do not work that way. Inputs move together. When demand is high, prices and costs may both rise. When the economy weakens, both sales volume and price realization may fall. Assuming independence when inputs are correlated systematically misstates risk. Positively correlated inputs amplify variance, and amplify the tail losses with it. Negatively correlated inputs dampen it. The bad draws partly cancel out. For normal inputs, you model this with a covariance matrix and multivariate sampling. For non-normal inputs, the tool is a copula. But the first step is not technical. Just ask: do my inputs move together in real life? If they do, the independence assumption needs justification, or replacement.

Where It Is Used

Five Canonical Business Applications

Monte Carlo appears in nearly every part of corporate decision-making. Five applications are canonical, and each answers a different kind of question. Capital budgeting simulates net present value and internal rate of return, under uncertain cash flows, discount rates, and terminal value. Should we invest, and what is the chance the net present value comes out negative? Project scheduling simulates total duration when individual task times are uncertain. Will we finish on time, and what is the ninety per cent completion date? Inventory and supply chain simulates stockout probability, under variable demand and lead times. What safety stock keeps stockouts below five per cent? Financial risk simulates portfolio returns and Value at Risk under market scenarios. And pricing strategy simulates revenue under different price elasticity assumptions. How much could we lose in a bad month, and which price best balances upside and downside?

Capital Budgeting

From One NPV to a Distribution of NPVs

Take capital budgeting first, because the contrast with tradition is sharpest there. Traditional analysis uses a single estimate for each year's cash flow. One estimate in, one net present value out. Monte Carlo replaces those point estimates with distributions. Revenue might be normal, with a mean of five million dollars and a standard deviation of one million. Costs might be uniform, somewhere between three million and four million. Notice the distributions need not share a shape. Normal for revenue, uniform for cost. Out of that comes a full distribution of net present value outcomes. Now you can ask: what is the probability of a negative net present value? Or: what does the worst five per cent of outcomes look like? Two projects with identical expected value often have very different risk profiles. The point estimate cannot tell them apart. The distribution can.

The Other Four

Scheduling, Inventory, Financial Risk, Pricing

The other four applications follow the same pattern, applied to different uncertainty. Project scheduling first. Fifty tasks with uncertain durations, and no single finish date. What you have instead is a distribution of possible finish dates. Simulate each task's duration thousands of times, and that distribution appears. Now you can answer the ninety per cent confidence finish date, or the odds of missing a deadline. This connects directly to PERT analysis, in Chapter eight. Inventory next. Demand and lead time are both random. Sample a demand value and a lead time each period, and track the inventory balance. Count how often it goes negative. That is your stockout probability. Run it for any reorder point, and tune the policy to hit a target service level. In finance, Monte Carlo is the backbone of modern Value at Risk calculation. Sample asset returns, compute portfolio returns, and take the first or fifth percentile of losses. And pricing strategy assigns a distribution to price elasticity, then simulates revenue at each candidate price.

Reading The Output

Four Numbers That Describe a Distribution

Running the simulation is only half the work. The real value is in extracting the right summary statistics. The output histogram shows the full distribution of possible outcomes. Four numbers describe it. The mean is the expected value. The long-run average, if the project were repeated many times. The median is the middle outcome. Half the scenarios are better, half worse. Standard deviation is the typical spread around the mean, your measure of total risk. And skewness tells you whether the distribution is lopsided. Right-skewed outcomes have a long upside tail. Left-skewed outcomes have a long downside tail, which is far more concerning. When mean and median differ significantly, the distribution is skewed, and the mean alone misleads.

Decision Visuals

S-Curves and Tornado Charts

Two visuals turn a simulation output into a decision. The first is the cumulative probability curve, the S curve. It plots cumulative probability against outcome value. Read it horizontally: what value is exceeded a given share of the time? Read it vertically: what is the probability of exceeding a given profit? S curves are the standard visual when stakeholders compare two options on risk. If one curve lies further right at every probability level, it stochastically dominates. The second visual answers a different question. Which inputs drive the output variance? A tornado chart ranks inputs from highest to lowest impact. Each bar shows the output range when that one input varies, others held at base case. The widest bar sits at the top, and that is the input that matters most. Most simulations have many uncertain inputs, but only a handful move the needle.

Worked Example

New Product Launch - Building the Model

Let's build one. A firm is considering a new product launch. The task is a spreadsheet Monte Carlo model, run for five hundred iterations. Revenue is modelled as normal, with a mean of five hundred and a standard deviation of eighty. Cost is also normal, with a mean of three hundred and eighty, and a standard deviation of sixty. Profit is not sampled at all. It is computed, as revenue minus cost. Revenue uses the normal inverse function, written NORM dot INV. Its first argument is RAND, which returns a uniform random number between zero and one. Then the mean, five hundred, and the standard deviation, eighty. Cost uses the same function, with three hundred and eighty, and sixty. Here is how that works. The uniform random number is treated as a percentile. The function returns the value sitting at that percentile of the specified normal. Copy the formulas down five hundred rows, and recalculate for a completely fresh set. Notice each draw calls RAND separately, so revenue and cost are sampled independently, an assumption worth checking.

Results

What 500 Iterations Reveal

After five hundred iterations, the simulated mean profit is about one hundred and nineteen dollars. That is close to the deterministic answer: five hundred minus three hundred and eighty is one hundred and twenty. So on the headline number, the simulation told us almost nothing new. That is not a failure. It is evidence the model is wired correctly. The value sits in the second number, which no point estimate can produce. Roughly thirteen per cent of the five hundred scenarios produce a loss. Thirteen per cent of five hundred runs is about sixty-five losing scenarios. Whether that is acceptable is not a mathematical question. It depends on risk tolerance, on the money at risk, and on the competing projects. Without simulation, the analysis stops at an expected profit of one hundred and twenty. With it, you answer what the committee actually wants to know. What is the chance we lose money at all? A point estimate hides risk. A distribution makes it visible.

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Quantitative Analysis carries a NCCRS credit recommendation. These universities work with NCCRS; some are UPI Study's signed partners. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

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FAQ

Questions students ask us

About Monte Carlo Simulation

How does Monte Carlo simulation work?
Monte Carlo simulation replaces each uncertain input with a probability distribution and runs the model many times in three steps. Sample one value from each input distribution, compute the outcome for that set of inputs, then aggregate thousands of outcomes into a distribution. The result shows not just an expected value but how likely each outcome is, such as the probability that a project's NPV turns negative.
How many iterations does a Monte Carlo simulation need?
In practice, 1,000 to 10,000 iterations usually give stable estimates, with more needed for rare events. The standard error shrinks in proportion to 1/\sqrt{n}, so doubling iterations cuts error by only about 30%, and cutting it by 90% requires 100 times as many runs. Check convergence by plotting the running mean until it settles, or by running with different seeds and comparing results.
Why does correlation between inputs matter in Monte Carlo simulation?
Treating correlated inputs as independent misstates risk, and the direction depends on how they combine. When inputs that push the outcome the same way rise and fall together, such as sales volume and price both dropping in a downturn, bad draws stack up, so independence understates the spread and tail losses. When bad draws tend to offset each other, the true spread is narrower. Correlated normal inputs use a covariance matrix; non-normal ones use a copula.
What do an S-curve and a tornado chart show in risk analysis?
An S-curve plots cumulative probability against outcome value, so you can read the chance of exceeding a given profit or compare two options; if one curve lies to the right at every probability level, that option stochastically dominates. A tornado chart ranks inputs by how much the output swings when each one varies with the others held at base case, with the most influential input as the widest bar at the top.

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Quantitative Analysis (Math 250) carries a NCCRS credit recommendation for 3 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
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