# Rate of Change Definition, Formula, and Importance

![Rate of Change](https://www.investopedia.com/thmb/bzeC3PUuaozP9JUJIygV41yYQb0=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/rateofchange.asp-final-f20db45c203b4651b8bdcd6d34b3d6d1.png)

*Analysts use rates of change to determine how quickly data changes over time. — Investopedia / Julie Bang*

> **Definition**
>
> A rate of change measures how quickly a measurement or value changes over time.

## What Is Rate of Change (ROC)?

Commonly used in finance and economics, a rate of change (ROC) is the speed at which a metric, ratio, or variable changes over a specified period. ROC is often used by analysts when discussing momentum, and it can generally be expressed as a ratio between a change in one variable relative to a corresponding change in another. Graphically, the rate of change is represented by the slope of a line. ROC is often illustrated by the Greek letter delta (Δ).

### Key Takeaways

*   ROC is the acceleration or deceleration of changes (i.e., the rate) and not the magnitude of individual changes themselves.
*   In finance, the rate of change is used to understand price returns and identify momentum in trends.
*   Moving averages are used by traders to understand the rates of change in asset prices by smoothing them out.
*   The Price Rate of Change indicator is a technical tool that measures the percentage change in price between the current price and the price a certain number of periods ago.

## Understanding the Rate of Change (ROC)

The rate of change is used to mathematically describe the percentage change in value over a defined period of time, representing the momentum of a variable. It is used in a variety of mathematical and scientific situations: For example, when determining a change in distance over time (acceleration).

It can also be used in finance and investing to describe the change in the value of an asset or index over time.

The rate of change is an important financial concept because it allows investors to spot security momentum and other [trends](https://www.investopedia.com/terms/t/trend.asp). For example, a security with high momentum, or one that has a positive ROC, normally outperforms the market in the short term. Conversely, a security is more likely to decline if it has an ROC that falls below its moving average or one that has a low or negative ROC. In this case, the rate of change could be seen as a sell signal to investors.

The rate of change is also a good indicator of market bubbles. Even though momentum is good and traders look for securities with a positive ROC, if a broad-market ETF, index, or mutual fund has a sharp increase in its ROC in the short term, it may be a sign that the market is unsustainable.

> **Important**
>
> If the rate of change of an index or other broad-market security is over 50%, investors should be wary of a bubble.

Traders can also pay close attention to the speed at which one price changes relative to another. For example, options traders study the relationship between the rate of change in the price of an option relative to a small change in the price of the underlying asset, known as an option's delta.

Options traders use rates of change in various risk metrics, known as the "[Greeks](https://www.investopedia.com/terms/g/greeks.asp)." The gamma, for example, is the rate of change of the delta (where the delta is how the option's price changes with movements in the underlying).

## How to Find the Rate of Change

The general formula for the rate of change is ROC = (X1-X2)/(T1-T2), where (X1-X2) is the change in variable being measured and (T1-T2) is the amount of time it took for the change to happen.

This formula can also be written as:

> R = (D2 - D1)/T

Where:

*   R = rate of change
*   D = distance (or some other variable) measured at the beginning and end of the period
*   T = the time it took for that change to occur

In finance, the calculation for ROC can also be computed as a return over time. In this instance, the formula uses the current value of a stock or [index](https://www.investopedia.com/terms/m/marketindex.asp) and divides it by the value from an earlier period. Subtract one and multiply the resulting number by 100 to give it a percentage representation:

$ROC = (\frac{\text{current value}}{\text{previous value}} - 1)*100$

## The Price Rate of Change Indicator

The rate of change is most often used to measure the change in a security's price over time. This is also known as the [price rate of change](https://www.investopedia.com/terms/p/pricerateofchange.asp) (also abbreviated ROC). The price rate of change can be derived by taking the price of a security at time B minus the price of the same security at time A and dividing that result by the price at time A.

$\begin{aligned} &\text{Price ROC} = \frac{B - A}{A} \times 100 \\ &\textbf{where:}\\ &B=\text{price at current time}\\ &A=\text{price at previous time}\\ \end{aligned}$

The indicator is an unbounded momentum indicator used in technical analysis set against a zero-level midpoint. When it is positive, prices are accelerating upward; when it is negative, price acceleration is downward.  

## What Are Other Terms for Rate of Change?

The rate of change may be referred to by other terms, depending on the context. When discussing speed or velocity, for instance, acceleration or deceleration refers to the rate of change. In statistics and regression modeling, the rate of change is defined by the slope of the line of best fit. For populations, the rate of change is called the growth rate. In financial markets, the rate of change is often referred to as momentum.

## How Do You Solve Rate of Change Problems?

Rate of change problems can generally be approached using the formula **R = D/T**, or rate of change equals the distance traveled divided by the time it takes to do so. Depending on the context involved in the problem, the change in distance can be replaced with a different variable, such as the change in value or price.

## How Do Traders Use the Price Rate of Change Indicator?

The price rate of change (ROC) indicator is used in technical analysis to measure momentum. A positive ROC can confirm a bullish trend, while a negative ROC indicates a bearish one. When the price is consolidating, the ROC will hover near zero.

## The Bottom Line

Rate of change (ROC) is an important concept that tells us not just that things are changing, but how fast things are changing. It doesn't measure the magnitude or size of the change; instead, rate of change shows how quickly or slowly that change is happening over time.

The rate of change can be applied in various mathematical and scientific contexts. In finance, it is often used by traders to understand changes in price returns and identify the momentum of market trends. For example, if the rate of change of an index is over 50%, this can indicate that the current trend is a bubble, rather than a sustainable, long-term change.