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Growth rates are a useful tool for building many different types of models. So, how do they work, and how do you calculate them?

Mack Grenfell

August 16, 2020

Imagine you have an investment which starts off with a value of $100,000, and grows at 5% a year.

After year 1, your investment is worth $105,000, an increase of $5,000. After year 2, it'll be worth $110,250, with an increase of $5,250 on the previous year.

We can extrapolate this model out for as long as we like, continually applying a 5% increase in value year-on-year.

What we're effectively doing in this example is building a simple model that applies a 5% **Growth Rate** to our investment each year.

But what exactly is a growth rate, and how can you use the idea to build your own models?

The term 'growth rate' describes the change in a variable between two points in time.

The idea of a growth rate is a useful concept to have because it lets us describe mathematically how something grows over time.

What we're describing doesn't just have to be financial, like the example above; anything that changes over time can be described in terms of growth rates:

- You might be a researcher looking to understand how quickly a population is changing over time. You could use a growth rate to quantify changes in the population that you're studying.
- You could trying to sell a product through word of mouth. You could use a growth rate to describe how quickly word of mouth spreads.

There's a reason why the two examples above, and the investment example at the beginning, can be modelled well by growth rates; in each of these examples, the variable we're trying to model is dependent on that variable's value at previous points in time.

In the population example for instance, this is equivalent to saying that the population at a point in time depends on the population in previous years. The higher the population in previous years, the more children born, and so the higher the population in future years.

This kind of dependency makes these examples a great fit for growth rate models.

If you have a set of historical data, broken down by time period, it's easy to calculate the growth rate for that dataset.

We're going to use a financial example below, that looks at the growth of an investment over time, but the method can be applied to any other type of growth.

Let's say you have the following data, describing how the value of a portfolio has grown over the last 5 years, and you're asked to calculate the portfolio's growth rate:

One way you might start by doing this is by looking at the difference between the initial value ($10.00) and the final value ($15.14). You could divide this difference by the initial value to get a growth rate of 51.4%:

We have to be careful about this answer though. What we've worked out above is the 4-year growth rate. i.e. we're saying that this investment has grown 51.4% in 4 years.

When we talk about growth rate, we typically define it at a more granular level than that. It might be defined as a **yearly** growth rate, a **monthly** growth rate, or whatever makes the most sense based on the data at hand.

Given that our data was provided to us broken down by year, it makes most sense to calculate a **yearly** growth rate for it. So how do we do that?

In our very first example, where we had an investment which we knew grew at 5% a year, we could forecast its future value fairly easily.

For each year into the future that we wanted to forecast, we simply multiplied the investment value by 1.05 (a 5% gain) that many times. To forecast its value 3 years ahead, we'd multiple the starting value by 1.05 3 times over (1.05 cubed, or 1.158).

To calculate the yearly growth rate from our data above, we simply do the reverse of what we'd do if we knew the growth rate and wanted to forecast the growth in value. To do this, we use the following equation:

We know the initial and final values of our investment, and we know the number of time periods (the number of years between our start and end). Let's plug these values into our equation, and solve to find the growth rate:

Dividing both sides by $10.00, we get:

We can then take the 4th root of each side:

And simplify to get the yearly growth rate:

And there we have it, our investment works out as having an average yearly growth rate of 10.9%.

Fortunately, forecasting growth rates into the future is fairly easy. If you want to do the calculation manually, you just have to use:

So, for example, if you wanted to calculate the value in 10 years of a $100,000 portfolio that grew at 5% a year, you'd work out:

If you're just looking to do a basic growth rate calculation, then you can often get by on a piece of paper. For anything more complicated than that though, you should look at using Causal.

Causal lets you build interactive, shareable growth models in minutes. To get started, you simply:

- Create the variables for your model. These might be
**Initial Value**,**Growth Rate**, and**Future Value**.

- Decide how many time periods you want your model to look at; this can be in terms of years, quarters, or months - whatever makes most sense to your use case.

- Define how these variables are linked to one another. For example, you can define
**Future Value**as your**Initial Value**multiplied by**Growth Rate**once for each time period in your model.

Causal then lets you see how your values change over time:

One of the difficulties when it comes to dealing with growth rates in the real world is that there's often uncertainty in them. In most cases, just because an investment has grown at 5% historically doesn't mean it'll continue to grow at that rate.

Causal handles uncertainty with ease. Instead of defining a growth rate of 5%, we can allow for uncertainty by inputting a growth rate of, say, 4 to 6%. When we do this, Causal helps you to understand how that uncertainty in growth rate affects the model:

In the graph above, you can see the impact of the uncertainty in growth rate play out over time, as the blue shaded area (indicating uncertainty) grows larger and larger.

Interested in giving Causal a go? Have a play with the interactive demo below. Feel free to change the **Initial Value** and **Growth Rate** variables, to see how they affect the graph.

Once you're done, click **Use this template** in the top right to create your own growth models.

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