Another one from Camm’s Business Analytics (excellent book):

Wearable Electronic Product Launch. The management of Madeira Computing is considering the introduction of a wearable electronic device with the functionality of a laptop computer and phone. The fixed cost to launch this new product is 300,000 USD. The variable cost for the product is expected to be between 160 USD and 240 USD, with a most likely value of 200 USD per unit. The product will sell for 300 USD per unit. Demand for the product is expected to range from 0 to approximately 20,000 units, with 4,000 units the most likely.

Develop a what-if spreadsheet model computing profit for this product in the base-case, worst-case, and best-case scenarios.
Model the variable cost as a uniform random variable with a minimum of 160 USD and a maximum of 240 USD. Model the product demand as 1,000 times the value of a gamma random variable with an alpha parameter of 3 and a beta parameter of 2. Construct a simulation model to estimate the average profit and the probability that the project will result in a loss.
What is your recommendation regarding whether to launch the product?

1 Introduction

We have three distributions at play here: * Triangular distribution: in all cases where we are talking about values between X and Y, most likely value being Z. This one is well known in the valuation. * Uniform distribution: mostly used when the historical data is not readily available, but we can infer the plausible range of values. * Gamma distribution: from my experience I can say that I did not use this in valuation context, but this distribution is well known and essential in the context of Bayesian inference.

Let’s find the solutions.

2 Solution to 1. problem

The description of the problem is self-explanatory, but for clarity purposes, let’s model demand for the product, so that we know what can we expect:

2.1 Variable cost

We can see here that the triangular distribution follows the parameters that were defined in problem definition: we can expect values between 160 and 240 USD, with the most likely value being 200 USD.

2.2 Demand

Same story, but this triangular distribution has different parameters, and it is not symmetrical around the most likely value.

2.3 Distribution of Profits

Unfortunately, the problem did not define what it means by base-case, worst-case, and best-case scenarios. But, for our purposes, let’s define these terms as following: * Worst case: 1st percentile * Base case: 50th percentile (median). * Best case: 99th percentile.

In the above graph, we can see that the probability of loss (based upon the given parameters) is 12.18%, with the worst case yielding profit of -0.2 M USD. Base case gives us around 0.4 M USD, and the best case gives us around 1.7 M. Since the probability of loss is 12.18%, and if the analyst did his job well in defining the parameters (which is another can of worms entirely), this project could receive the green light. But, the organization/management should find some ways to mitigate the risk of negative profit.

3 Solution to 2. problem

Again, let’s define the parameters:

3.1 Variable cost

This time we have uniform distribution, where the occurrence of each value in defined range is equally likely.

3.2 Demand

Demand now follows gamma distribution with the parameters of \( = 3, = 2 \). IMO, this is fairly unusual modelling of demand, but for the purposes of this blog post it will suffice. Do note that we will have to multiply random value by 1000 to get the demand in units.

3.3 Distribution of Profits

This distribution is not a surprise. The main reason is that the distribution of gamma has the majority of density in the range of 0 to 4, so our demand is in most cases between 0 and 4,000 units. Compare that with the distribution of demand in 1. solution, where 4.000 units is the most likely value. Triangular vs. uniform distribution of variable costs does not play major effect in my opinion, since the mean for both distributions is same: 200.