# Variation rate for the period (TVP)

The rate of change is the percentage change between two values.

The rate of change, seen from another point of view, is the relative variation compared to the initial value of the variable. That is, when we say that a variable has grown by 20% in the last month, the last three days or the last 3 years, we are saying that the variable is 20% greater than the reference period.

For the case in which the rate of change is negative, the interpretation is exactly the same but in reverse. For example, a variable that was worth 100 yesterday and that today is worth 20, has suffered a variation rate of -80%.

In this article we will see the formula for the rate of change, its interpretation and an example.

## Rate of change formula

To calculate the rate of change, we will need the absolute values of the variables on those dates. Even if we do not have the intermediate data, we can calculate it. The formula for the rate of change is as follows:

- TV = [(Yt - Yt-n) / Yt-n] x 100 = TV (%)

Or alternatively you can also use this other formula:

- TV = {(Yt / Yt-n) -1} x 100 = TV (%)

Where:

TV: Variation rate of the period in percentage (%)

Yt: Last value of the compared period

Yt-n: Previous value in n periods.

Therefore, we will need the last value of the compared period and the reference value.

In the formula we have used a subscript t in reference to time. So t is now and t-n is the period of n periods before. Do not worry if this expression makes you strange, it is actually mathematical expressions, but with an example you will see it very easily.

We must bear in mind that to calculate the rate of change for the period we need two comparable periods. So, although we can mathematically compare the data of a month with the data of a day, we must ensure that the periods are similar. For example, it makes no sense to compare an annual rate of change with a monthly rate of change.

## Variation rate example

Let's imagine that Juan has a company and wants to know how much his sales have increased during certain periods. As you have a lot of work, you decide to hire us to analyze your accounts and ask us for the following:

- Variation rate of the last 3 years.
- Variation rate for the last year.
- The year-to-year rate of change.

Year | Sales (in dollars) |
---|---|

2014 | 13.260 |

2015 | 14.568 |

2016 | 12.569 |

2017 | 19.768 |

2018 | 25.123 |

2019 | 18.674 |

We will first calculate the rate of change for the last three years. That is, the variation between 2016 and 2019. For this we will apply the formula:

TV16-19 = {[(Y2019 - Y2016) / Yt2016] -1} x 100 = TV (%)

We substitute and we have the following:

TV16-19 = [(18,674 - 12,569) / 12,569] x 100 = 48.57%

Sales increased 48.57% between 2016 and 2019.

The second task that Juan gave us was to calculate the rate of change for the last year, for which we will use the second formula that we have indicated, since it is faster and we arrive at the same result.

TV18-19 = {(18,674 / 25,123) -1} x 100 = -25.67%

Last year sales were down 25.67%.

Third and last, we will calculate the rate of change for each year.

TV14-15 = {(14,568 / 13,260) -1} x 100 = 9.86%

TV15-16 = {(12,569 / 14,568) -1} x 100 = -13.72%

TV16-17 = {(19,768 / 12,569) -1} x 100 = 57.28%

TV17-18 = {(25,123 / 19,768) -1} x 100 = 27.09%

TV18-19 = {(18,674 / 25,123) -1} x 100 = -25.67%

As we can see, the first year they grew, the second they decreased, the third and fourth year they grew again, to end up reducing 25.67% last year.

GDP variation rate
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