Monte Carlo Simulation 2

Purpose

Monte Carlo Simulation 2 repeats an expression many times while randomly sampling several uncertain input variables. It summarizes the resulting outcomes as a histogram, a table, and statistics such as the mean, standard deviation, minimum, maximum, and percentiles.

Compared with the original Monte Carlo calculator, this version can propagate uncertainty from multiple inputs at the same time. Variables are sampled independently by default. One pair of normally distributed variables may optionally be sampled with a specified correlation, which is useful when two inputs tend to move together, such as inflation and interest rates.

Background

Multi-variable uncertainty

An ordinary calculation uses one fixed value for every input. A Monte Carlo calculation instead treats uncertain inputs as probability distributions. Each trial draws one value for every variable and evaluates the complete expression.

For example, if the expression is:

x * y + z

one trial might use x = 10.2, y = 4.8, and z = 3.1. The next trial uses a new combination. The output histogram shows how uncertainty in all three inputs combines and passes through the expression.

This is especially useful for nonlinear expressions, products, ratios, totals, and other calculations where the uncertainty in the result is not obvious from the uncertainty in any single input.

Independent and correlated inputs

By default, each variable is sampled independently. This means that a high sample for one variable does not cause another variable's sample to be high or low.

If two variables are related, enter their names in Correlated Variables and enter their relationship in Correlation. A positive correlation makes the two variables tend to move in the same direction; a negative correlation makes them tend to move in opposite directions. A correlation of zero represents no linear relationship in this model.

Inputs

Expression

The expression or calculator call to repeat. It should contain the variable names listed in Variables.

The default expression is:

x + y

Vary By

Select what the sampled values change:

• Parameters (p) replaces named parameter values inside a calculator call.
• Variables (v) replaces standalone variables in an expression, such as x + y.

Use Parameters for calculator or function arguments and Variables for direct expression variables. The default is Parameters.

For a multi-variable calculation, every variable that should be sampled must be represented by its name in the expression. For example:

inflation * principal + interest

Variables

A comma-separated list of one or more unique variable names:

inflation, interest, principal

Names must begin with a letter or underscore and may contain letters, numbers, and underscores. The order of this list establishes the order used by Distributions, Param1s, Param2s, and Param3s.

Distributions

A comma-separated distribution name for each variable. The supported choices are:

• normal — uses a mean and standard deviation.
• uniform — samples equally between a low and high value.
• triangular — uses a low value, high value, and most-likely mode.
• lognormal — samples a positive, right-skewed quantity using the parameters of an underlying normal distribution.

The number and order of distributions must match Variables.

Param1s and Param2s

These are comma-separated numeric lists. Each position corresponds to the same position in Variables and Distributions.

Distribution	Param1	Param2
normal	mean	standard deviation
uniform	low	high
triangular	low	high
lognormal	mu of the underlying normal distribution	sigma of the underlying normal distribution

For example:

Variables:     inflation, interest, demand
Distributions: normal,    normal,    uniform
Param1s:       3,          5,         80
Param2s:       1,          1.5,       120

The number of values in Param1s and Param2s must match the number of variables. Standard deviations and lognormal sigma values cannot be negative. Uniform and triangular low values cannot exceed their corresponding high values.

Param3s

An optional comma-separated list of triangular distribution modes. Its value is used only for variables whose distribution is triangular; it is ignored for the other distributions.

Use an empty entry for a non-triangular variable when positions need to be preserved:

Variables:     price, demand, duration
Distributions: triangular, normal, triangular
Param1s:       8, 100, 2
Param2s:       15, 20, 10
Param3s:       10, , 5

The mode must lie between the low and high values. A single Param3 value may also be supplied; it is applied to every variable position, but it only affects triangular distributions.

Trials

The number of random trials to perform. Each trial samples every variable and evaluates the expression once. More trials generally make the histogram and percentiles more stable, but increase computation time. The configured qCalc range limit applies.

Bin Count

The number of bins used to group calculated outcomes in the histogram. More bins show more detail but can make a small simulation look irregular; fewer bins give a more compressed view.

Round Off

The number of decimal places used to round each sampled input before it is inserted into the expression. Rounding keeps the evaluated trial expressions manageable and makes the sampled values consistent with the displayed precision. It can introduce a small discretization, normally negligible for ordinary Monte Carlo use.

Correlated Variables

An optional comma-separated pair of variable names:

inflation, interest

Both names must already appear in Variables. Leave this field blank when all variables should be sampled independently.

The pair must use the normal distribution. This version supports only one correlated pair; it does not accept three-way correlation or a general correlation matrix.

Correlation

The correlation coefficient for the pair in Correlated Variables. It must be between -1 and 1:

• 1 — perfect movement in the same direction.
• 0 — no linear correlation in the model.
• -1 — perfect movement in opposite directions.

Leave this field blank when Correlated Variables is blank. It is treated as zero and does not affect the independent-variable case.

Result Columns, Result Units, and Histogram Column

These optional filters control which scalar result columns appear in the table and which single result column supplies the histogram. If the expression returns more than one scalar result, specify Histogram Column to identify the result to plot.

Show

Choose whether to display the table, histogram, or both. Summary statistics are computed regardless of this choice.

Chart Title

The title displayed above the histogram.

Results

Histogram

The histogram groups the calculated output values into the requested number of bins. Its vertical axis is Frequency, meaning the number of successful trials in each bin. Failed trials are not included in the histogram.

Summary statistics

The calculator returns:

• trials_used — trials that produced a usable numeric result.
• trials_failed — trials excluded because evaluation failed or did not produce a numeric result.
• mean — arithmetic average of the successful outcomes.
• stdev — sample standard deviation of the successful outcomes.
• min and max — smallest and largest successful outcomes.
• p5 — 5th percentile.
• p50 — 50th percentile, also called the median.
• p95 — 95th percentile.

The interval from p5 to p95 describes the middle 90 percent of the simulated outcomes in the sample. It is a simulation interval, not a guarantee that future observations must fall inside it.

Understanding the Calculation

For every trial, the calculator draws one value from each variable's selected distribution. It combines those values into one complete input set, substitutes that set into the expression, and evaluates the result.

Independent variables are sampled separately. For a correlated normal pair, the calculator generates two normal samples with the requested means and standard deviations while coupling their standardized random components to produce the requested correlation. Other variables remain independently sampled.

Conceptually, the calculation is:

sample every input
    -> substitute the complete input set
    -> evaluate the expression
    -> collect a successful scalar result
    -> repeat for all trials
    -> histogram and summarize the outcomes

Each variable's values are paired by trial number. The first sampled value of each variable belongs to trial 1, the second value of each belongs to trial 2, and so on. This pairing is what makes a correlated pair meaningful.

Example

Suppose a calculation estimates a result affected by inflation, interest, and an independent demand factor:

inflation * 1000 + interest * 500 + demand

Use:

Variables:             inflation, interest, demand
Distributions:         normal, normal, uniform
Param1s:               3, 5, 40
Param2s:               1, 1.5, 80
Param3s:
Trials:                1000
Bin Count:             20
Correlated Variables:  inflation, interest
Correlation:           0.65

The calculator samples inflation and interest as a positively correlated normal pair. Demand is sampled independently and uniformly from 40 to 80.

The resulting histogram represents the simulated distribution of the complete calculation, not the distribution of any one input. Compared with an independent inflation/interest model, positive correlation will generally make jointly high and jointly low input combinations more common, which can change the spread and tail of the result.

Important Assumptions and Interpretation

• Independent sampling is the default. The calculator does not infer relationships from historical data.
• Only one correlated pair can be specified.
• The correlated pair must use normal distributions. Uniform, triangular, and lognormal variables can be used independently, but cannot currently be part of the correlated pair.
• A full correlation matrix is not supported. If several variables have relationships with one another, representing only one pair may understate or distort the uncertainty in the result.
• A positive correlation is not automatically economically or scientifically correct. Choose it from relevant data or treat it explicitly as a scenario assumption.
• The selected distributions and parameters are model assumptions. They do not forecast the future or establish the true probability of an outcome.
• Rounding sampled values before evaluation can create a small discretization, especially with very narrow distributions or very low precision. Use an appropriate Round Off value for the scale of the inputs.
• Failed trials are excluded from the statistics. A high trials_failed count may indicate invalid ranges, an unsuitable expression, missing units, or parameters that make the calculation invalid.
• If all trials fail, the calculator cannot produce summary statistics.
• The output has the units and meaning of the expression's result. Input units must be expressed in a way accepted by the expression and qCalc's quantity system; this calculator does not independently convert or validate a unit model for the sampled variables.
• The histogram's Frequency values are counts of successful trials, not probability densities or percentages.