Pricing and Monte Carlo

roulette-anglaise-monaco-1100x358.jpgFrank Knight, a great Chicago economists, developed the famous distinction between risk and uncertainty.  Risk is when you don’t know exactly what will happen but nonetheless have a sense of the possibilities and their relative likelihood. Uncertainty is when you’re so unsure about the future that you have no way of calculating how likely various outcomes are.

When in comes pricing products, there is a great deal of uncertainty in terms of adoption of our product and future costs.   In the parlance of Frank Knight, how do we convert that uncertainty into risk?  One possible solution:  Monte Carlo Methods.

Monte Carlo Methods provide a mechanism to simulate multiple possible scenarios that might unfold and quantify risk.    Think in terms of running virtual experiments on your computer.

In this post, I will provide some background and guidelines on how one can leverage Monte Carlo methods for product pricing.  Note,  I have attempted to stay away from complicated math and statistics in this article and try to simplify the process with the hope that a business manager can understand the concepts, process and value of this tool.  Please send me feedback on whether I was successful in that objective.


When it comes to our product adoption, we know that certain scenarios might unfold in future, but we are unsure whether they will happen.   For instance, the outcome for adoption might range between been wildly successful to total failure.  And all of these outcomes are “probable”.   Which means we have to move from deterministic view of the world to probable view of the world.  And Monte Carlo will allow us to introduce probability into our analysis.

Monte Carlo methods allow us to simulate multiple scenarios between the range we specify.   From the data collected during simulation, we can visualize the probability distribution of our output and answer following sample questions  –

  • What is the probability that our product will be profitable?
  • Given 95% confidence level, what will be the range of my profits (confidence intervals)?
  • Or, if I want the profits to be within a certain range, what is the probability (or confidence level) of that occurring?

The Mechanics

Monte Carlo methods involves three steps:

  1. For the uncertain variables in your model (for eg., proposed price, adoption estimates, etc.), choose the upper and lower bounds
  2. Choose the number of scenarios and run the simulation
  3. Plot the histogram for the output variable (for eg., profit) and that will give you the distribution of your output.
  4.  Statistical analysis of the output (to answer our questions)

Let’s look at each step in detail using an example.

Step 1:  The input variables


Figure 1: The input variables

In our example, we are trying to figure out what will be the right price to charge for Workgroup and Enterprise edition of our product to meet our profitability goals.  I started by making estimates of adoption of the two editions of my product for two key scenarios – the worst-case and the best-case – over fixed period of time.  Typically, you will use average product lifecycle as the timeframe for your estimates.   And for Enterprise Software,  four years is a good timeframe to choose, since the technology and market forces might make your current product less desirable after four years.

Estimate the best-case and worst-case scenario for your cost elements as well in the model.  Since compute, storage, etc. follow a general trend of getting less expensive with time.  And that leads us to the lower and upper bound for”uncertain” input variables as illustrated in Figure 1.

Step 2:  Monte-Carlo Simulation

Our profits equal our revenues minus the cost.   We want to simulate a lot of different scenarios that might play out in terms of adoption of our product and the cost elements between the bounds that we estimate in step 1 and collect data on profit.

The assumption we make is that the input variables are “normally” distributed between the upper and lower bound.   Note, you can choose a different distribution as well based on your understanding of your domain.  However, it’s a safe bet to choose the normal distribution (to start with) for most variables.

Click here to download the excel that contains the example scenario.  It will be easier to follow rest of the article once you have that excel.

In my example, I have chosen to simulate 5000 different scenarios.   In other words, adoption by workgroup, enterprise users, cost, etc.  will get randomly assigned 5000 different values.    The use of RAND() function in excel will do this magic for you as shown in the snippet in Figure 2.


Figure2: The Monte-Carlo Simulation

Step 3:  The Output Variable

As shown in Figure 2, we calculate Profit (= income – expense) for 5000 different scenarios.   Next step, is to create a histogram on  5000 data points of Profit as shown in Figure 3.


Figure 3: Histogram for Output (Profit)

 The distribution of Profit  follows a normal distribution or popularly known as “Bell Curve”.   

Step 4: Statistical Analysis

This is the step where we get some real insights.   Now, we are ready to answer some questions using statistics.  For instance, few sample questions we can answer.

How much profit can we expect on average? 


Figure 4: Summary Statistics

Figure 4 shows us that we can expect approximately $115 mln in profit (±  $17 mln – the standard deviation).

Another interesting question one might ask, given 90% (or 95%) confidence, what is the range of profit we can expect? 


Figure 5: Confidence Intervals

From Figure 5, we can say with 95% confidence that we will generate profits between $83 mln and $142 mln.    And this is super insightful for a business/product owner.   First of all, you have moved from trying to be deterministic about future to probabilistic way of thinking about future.  Which I believe is the right way.  More importantly, you have the quantitive information to make a decision whether, given the chosen pricing, you will be able to meet your business goals.  Or, do we need to revisit the pricing.  Or, if you are looking at a portfolio of products, we can prioritize our investments.


Monte Carlo Methods transform the uncertainty into risk and eliminate the bias in our decision-making by allowing us to peek into future by simulating multiple scenarios that might unfold.   I believe these methods are a key tool in getting rid of HiPPO ( Highest Paid Person’s Opinion) and empower a data-driven business professional to make a more informed decision.


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