Setting the right price for your product keeps your business humming; the wrong price can lead to a swift and painful demise. What then makes the price right? And how do you determine it? That's what we're going to explore via simulation.

The subscription business we're going to start is a simple one. You begin at the start of year 1 with an initial number of subscribers. You've doubtless worked hard to acquire them, so what do you have to do to keep them (hopefully) and add even more subscribers?

Well, you have two things you can try to control - the first is your renewal rate. By providing good value you hope your subscribers will renew their subscription. And you can go out and acquire new subscribers. Together, they determine how you grow your customers.

By simulating changes in renewal and acquisition rates, we can see how customer growth changes. Here are three scenarios. In figure 1 the sum of the renewal and acquisition rates are less than 100%; in figure 2 the sum is exactly 100% and in figure 3 the sum is greater than 100%.

In figures 1, 2, and 3 the renewal rate is fixed at 0.7 or 70%. What's different is the rate of acquisition which varies from 0.29 to 0.30 to 0.31. When the sum of the renewal and acquisition rates reaches exactly 1, customer growth is zero; when it's greater than 1, customers increase year on year.

You can play with these scenarios if you download the Mathematica notebook and can run it on your local system.

The renewal and acquisition rates will vary from year to year. They are also dependent on the subscriber base. It might be realistic to acquire new customers at the rate of 30% a year when you don't have many subscribers to begin with, but this rate of growth is probably unrealistic when you already have a thousand subscribers. You can change the acquisition rate in the simulation above to see how this affects subscriber growth.

As your subscribers grow, it becomes hard to keep acquisition rates as high as they used to be when you had fewer subscribers. An exception to this rule happens when your product has "network effects" -- the more people who subscribe, the more value it provides all subscribers. What are the key attributes of such a product? It's worth thinking about this but let's move on for now.

Your subscribers need care and feeding. And that costs money. So does acquiring new subscribers. And so does renewing your existing subscribers. Add these costs up and you'll know how much you need to earn from your subscriptions to break even. Go ahead and set your costs, your renewal and acquisition rates. Then vary your subscription price to see what it does to your revenues and your profits.

Figure 4 shows the model with all of the relevant inputs. As you change the inputs you'll notice the following.

1. The sum of the renewal and acquisition rate needs to be well above 1. When it hovers around 1 you can make a profit, but you have to keep costs really low and prices relatively high. That's no fun at all. So you'd better invest in some good people who achieve the renewal and acquisition rates that are required.

2. The number of subscribers you begin with doesn't matter to your profit trajectory. If you keep everything else the same and change only the number of subscribers you begin with, you won't change the trajectory of your earnings. Of course the curves will shift up or down, but their trajectories (and hence their relative trajectories) will not change.

3. It's good to hold maintenance costs low -- profits are sensitive to that cost. The cost of acquisition and the acquisition rate are connected. You can slack off on one if you perk up on the other.

4. Try this out. Set the renewal and acquisition rates at 1 -- so you're doubling your customer base every year! You've hit the big time! Or have you? What do you observe about the relationship between the three costs and the price? (Compare figures 5 and 6.)

When renewal and acquisition rates are at their maximum values of 100%, costs can be very high if the subscription fee is also very high (figure 5 shows a \$6B profit in year 10). And conversely, if costs are low then even with the relatively low subscription price of \$20,000 a year, profits still rise to a healthy \$1.25B in year 10 (figure 6).

But, as shown in figure 7, if costs are high and you don't have pricing power, you can have the maximum renewal and acquisition rates of 100% and have higher and higher losses as the years go by.

Simulations like these can help you see the effect of the interplay between many different factors as they change simultaneously. They give you a way to get an intuitive feel for questions like the following: If I can only renew at 62% and acquire new subscribers at 40% and my customers won't pay more than \$25,000 for a subscription, what combination of costs do I have to stay under to make a profit of at least \$X in year 5? What is the maximum profit possible?

When there are a number of variables in play, even simple problems can get quickly complicated. In these situations simulations are invaluable. In some cases they may be the only way to make progress on the problem.

Real life is more complicated. Renewal and acquisition rates (and costs too) vary from year to year due to many factors that are not within your control. Simulating these variations from year to year will give you a more certain sense of how things will evolve.

Let's make some assumptions.

Assumption 1) Renewal rates are normally distributed with a mean of 70% and a standard deviation of 20% (accounting for some large fluctuations in annual renewal rates).

Figure 8 above will give you a feel for how renewal rates (values along the x axis) are distributed (values along the y axis) compared with some other potential distributions. Knowing that probabilities can never be greater than 1, you might be surprised to see a peak that goes up to 2 on the y axis. There is no mistake here -- keep in mind these are probability density curves, not raw probabilities. To get the raw probability for a range of renewal rates, you'd have to integrate over that range.

Assumption 2) Acquisition rates (figure 9) are normally distributed with a mean of 20% and a standard deviation of 5% (that's aggressive growth and you'll need a good sales team and a good sales process engine to make this happen). This should look familiar to the sharply-peaked curve in the probability density figure above.

Assumption 3) The cost of renewal ranges from \$500 to \$5,000 and is distributed as a Gamma distribution of a particular kind with a peak of \$1,000. This is an appropriate distribution to choose because it is right-skewed -- most subscribers can be renewed around a cost of \$1,000 (and a few at less than that), but it's much more expensive to renew a handful of subscribers. Figure 10 shows how the Gamma distribution behaves so you can get a feel for it.

Assumption 4) Similarly, the cost of acquiring subscribers ranges from \$5,000 to \$30,000 (it's expensive to fly to Australia!) and is also distributed as a Gamma function for the same reason as above: most subscribers can be acquired for around \$8,000 but some will cost a lot more. Figure 11 shows the distribution for $\alpha = 2$.

Assumption 5) Maintaining subscribers during their subscription period is also distributed as a Gamma function. Like the cost of acquiring subscribers, it too ranges from \$5,000 to \$30,000. Figure 12 shows the distribution for $\alpha = 2$.

Assumption 6) Finally, not all subscribers will pay the same price for the subscription. The range here is from \$20,000 to \$120,000 with most subscribers paying around \$25,000. Again, no surprise, the values are modeled by a Gamma distribution (here shown with $\alpha = 2$ in figure 13).

For the renewal rate (figure 10), the cost of acquisition (figure 11), and the cost of maintenance (figure 12) distributions, we'll set up the simulation so you can vary the value of $\alpha$ from 2 (low level of variation) to 6 (high level of variation).

Let's set up the initial conditions and run some simulations where the costs and the subscription price vary according to a Gamma distribution where $\alpha$ varies from 2 to 6 much like in figure 10 above.

As $\alpha$ varies from 2 to 6 the level of variation in the costs to acquire, maintain, and renew subscribers increases (the Gamma distributions have "fatter tails" as the value of $\alpha$ increases). The vagaries of the subscription business can leave you with many fewer subscribers than you started with; and profits can tank too as the reality of the marketplace takes its toll on your business.

Even when it's just a few things that vary, the results of simulations like these are essential to fully understand what can occur. Based on these simulations we can start to understand which variations drive customer churn, the impact of pricing power, and much more. Based on this understanding we can prioritize actions that try to better control the variations that have the most impact on our business goals.

You can play with these scenarios if you download the Mathematica notebook and can run it on your local system.

### The Business

The subscription business we're going to start is a simple one. You begin at the start of year 1 with an initial number of subscribers. You've doubtless worked hard to acquire them, so what do you have to do to keep them (hopefully) and add even more subscribers?

Well, you have two things you can try to control - the first is your renewal rate. By providing good value you hope your subscribers will renew their subscription. And you can go out and acquire new subscribers. Together, they determine how you grow your customers.

### Customer Growth

By simulating changes in renewal and acquisition rates, we can see how customer growth changes. Here are three scenarios. In figure 1 the sum of the renewal and acquisition rates are less than 100%; in figure 2 the sum is exactly 100% and in figure 3 the sum is greater than 100%.

Figure 1: Customer growth when sum of renewal and acquisition rate is < 1 |

Figure 2: Customer growth when sum of renewal and acquisition rate is equal to 1 |

Figure 3: Customer growth when sum of renewal and acquisition rate is > 1 |

In figures 1, 2, and 3 the renewal rate is fixed at 0.7 or 70%. What's different is the rate of acquisition which varies from 0.29 to 0.30 to 0.31. When the sum of the renewal and acquisition rates reaches exactly 1, customer growth is zero; when it's greater than 1, customers increase year on year.

You can play with these scenarios if you download the Mathematica notebook and can run it on your local system.

The renewal and acquisition rates will vary from year to year. They are also dependent on the subscriber base. It might be realistic to acquire new customers at the rate of 30% a year when you don't have many subscribers to begin with, but this rate of growth is probably unrealistic when you already have a thousand subscribers. You can change the acquisition rate in the simulation above to see how this affects subscriber growth.

As your subscribers grow, it becomes hard to keep acquisition rates as high as they used to be when you had fewer subscribers. An exception to this rule happens when your product has "network effects" -- the more people who subscribe, the more value it provides all subscribers. What are the key attributes of such a product? It's worth thinking about this but let's move on for now.

### The Cost of Staying Alive

Your subscribers need care and feeding. And that costs money. So does acquiring new subscribers. And so does renewing your existing subscribers. Add these costs up and you'll know how much you need to earn from your subscriptions to break even. Go ahead and set your costs, your renewal and acquisition rates. Then vary your subscription price to see what it does to your revenues and your profits.

Figure 4: Subscribers, costs, and revenues for a particular set of inputs |

Figure 4 shows the model with all of the relevant inputs. As you change the inputs you'll notice the following.

1. The sum of the renewal and acquisition rate needs to be well above 1. When it hovers around 1 you can make a profit, but you have to keep costs really low and prices relatively high. That's no fun at all. So you'd better invest in some good people who achieve the renewal and acquisition rates that are required.

2. The number of subscribers you begin with doesn't matter to your profit trajectory. If you keep everything else the same and change only the number of subscribers you begin with, you won't change the trajectory of your earnings. Of course the curves will shift up or down, but their trajectories (and hence their relative trajectories) will not change.

3. It's good to hold maintenance costs low -- profits are sensitive to that cost. The cost of acquisition and the acquisition rate are connected. You can slack off on one if you perk up on the other.

4. Try this out. Set the renewal and acquisition rates at 1 -- so you're doubling your customer base every year! You've hit the big time! Or have you? What do you observe about the relationship between the three costs and the price? (Compare figures 5 and 6.)

Figure 5: Profit when inputs are all at extremely high values |

Figure 6: Profit when renewal and acquisition rates are maximized while costs are minimized |

When renewal and acquisition rates are at their maximum values of 100%, costs can be very high if the subscription fee is also very high (figure 5 shows a \$6B profit in year 10). And conversely, if costs are low then even with the relatively low subscription price of \$20,000 a year, profits still rise to a healthy \$1.25B in year 10 (figure 6).

But, as shown in figure 7, if costs are high and you don't have pricing power, you can have the maximum renewal and acquisition rates of 100% and have higher and higher losses as the years go by.

Figure 7: Profit when renewal and acquisition rates are maximized and costs are maximized |

Simulations like these can help you see the effect of the interplay between many different factors as they change simultaneously. They give you a way to get an intuitive feel for questions like the following: If I can only renew at 62% and acquire new subscribers at 40% and my customers won't pay more than \$25,000 for a subscription, what combination of costs do I have to stay under to make a profit of at least \$X in year 5? What is the maximum profit possible?

When there are a number of variables in play, even simple problems can get quickly complicated. In these situations simulations are invaluable. In some cases they may be the only way to make progress on the problem.

### The Vagaries of the World

Real life is more complicated. Renewal and acquisition rates (and costs too) vary from year to year due to many factors that are not within your control. Simulating these variations from year to year will give you a more certain sense of how things will evolve.

Let's make some assumptions.

Assumption 1) Renewal rates are normally distributed with a mean of 70% and a standard deviation of 20% (accounting for some large fluctuations in annual renewal rates).

Figure 8: Three renewal rate scenarios |

Figure 9: How acquisition rates vary |

Figure 10: Cost of renewal scenarios |

Assumption 4) Similarly, the cost of acquiring subscribers ranges from \$5,000 to \$30,000 (it's expensive to fly to Australia!) and is also distributed as a Gamma function for the same reason as above: most subscribers can be acquired for around \$8,000 but some will cost a lot more. Figure 11 shows the distribution for $\alpha = 2$.

Figure 11: How cost of acquisition varies |

Assumption 5) Maintaining subscribers during their subscription period is also distributed as a Gamma function. Like the cost of acquiring subscribers, it too ranges from \$5,000 to \$30,000. Figure 12 shows the distribution for $\alpha = 2$.

Figure 12: How maintenance cost varies |

Assumption 6) Finally, not all subscribers will pay the same price for the subscription. The range here is from \$20,000 to \$120,000 with most subscribers paying around \$25,000. Again, no surprise, the values are modeled by a Gamma distribution (here shown with $\alpha = 2$ in figure 13).

Figure 13: How subscription price varies |

For the renewal rate (figure 10), the cost of acquisition (figure 11), and the cost of maintenance (figure 12) distributions, we'll set up the simulation so you can vary the value of $\alpha$ from 2 (low level of variation) to 6 (high level of variation).

Let's set up the initial conditions and run some simulations where the costs and the subscription price vary according to a Gamma distribution where $\alpha$ varies from 2 to 6 much like in figure 10 above.

### Initial Conditions

- Number of Initial Subscribers = 500
- Renewal Rate = Normal distribution with average 0.7 and standard deviation 0.2
- Client Acquisition Rate = Normal distribution with average with 0.35 and standard deviation 0.05
- Most Likely Renewal Cost = \$4,000 (Gamma distribution with $\alpha$ varying from 2 to 6)
- Most Likely Acquisition Cost = \$5,000 (Gamma distribution with $\alpha$ varying from 2 to 6)
- Most Likely Maintenance Cost = \$4,000 (Gamma distribution with $\alpha$ varying from 2 to 6)
- Most Likely Subscription Price = \$25,000 (Gamma distribution with $\alpha$ varying from 2 to 6)

### Results from Simulations

Figure 14: Distribution of total subscribers after 10 years (based on 1000 simulations) |

Figure 15: Distribution of profits after 10 years (based on 1000 simulations) |

As $\alpha$ varies from 2 to 6 the level of variation in the costs to acquire, maintain, and renew subscribers increases (the Gamma distributions have "fatter tails" as the value of $\alpha$ increases). The vagaries of the subscription business can leave you with many fewer subscribers than you started with; and profits can tank too as the reality of the marketplace takes its toll on your business.

Even when it's just a few things that vary, the results of simulations like these are essential to fully understand what can occur. Based on these simulations we can start to understand which variations drive customer churn, the impact of pricing power, and much more. Based on this understanding we can prioritize actions that try to better control the variations that have the most impact on our business goals.

You can play with these scenarios if you download the Mathematica notebook and can run it on your local system.