Rate of Change Definition, Formula, and Importance

What Is Rate of Change (ROC)?

The rate of change (ROC) is the speed at which a variable changes over a specific period of time. ROC is often used when speaking about 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. The ROC is often illustrated by the Greek letter delta (Δ).

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, and it represents the momentum of a variable. It is used in a variety of mathematical and scientific situations, for example, finding a change in distance over time.

It can also be used in finance and investing to describe the change in the value of something over time, such as a market index or individual security.

The rate of change is an important financial concept because it allows investors to spot security momentum and other trends. 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 a 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.

Traders also can 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 rate of change in various risk metrics, known as the "Greeks." 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 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:

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

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

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 negative, downward.

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, rather of change shows how quickly or slowly that change is happening over time.

The rate of change can be used in many mathematical or 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.