## What Is Interpolation?

Interpolation is a statistical method by which related known values are used to estimate an unknown value or set of values. In investing, interpolation is used to estimate prices or the potential yield of a security. Interpolation is achieved by using other established values that are located in sequence with the unknown value.

If there is a generally consistent trend across a set of data points, one can reasonably estimate the value of the set at points that haven't been explicitly calculated. Investors and stock analysts frequently create a line chart with interpolated data points. These charts help them visualize the changes in the price of securities and are an important part of technical analysis.

Interpolation can be compared with extrapolation, which estimates unknown values that extend beyond the known data, rather than values that fall in between known data points.

### Key Takeaways

- Interpolation is a simple mathematical method investors use to estimate an unknown price or potential yield of a security or asset by using related known values.
- By using a consistent trend across a set of data points, investors can estimate unknown values and plot these values on charts representing a stock's price movement over time.
- One of the criticisms of using interpolation in investment analysis is that it lacks precision and does not always accurately reflect the volatility of publicly traded stocks.

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## Understanding Interpolation

Investors use interpolation to create new estimated data points between known data points on a chart. Charts representing a security's price action and volume are examples where interpolation might be used. While computer algorithms commonly generate these data points today, the concept of interpolation is not a new one. Interpolation has been used by human civilizations since antiquity, particularly by early astronomers in Mesopotamia and Asia Minor attempting to fill in gaps in their observations of the movements of the planets.

There are several formal kinds of interpolation, including linear interpolation, polynomial interpolation, and piecewise constant interpolation. Financial analysts use an interpolated yield curve to plot a graph representing the yields of recently issued U.S. Treasury bonds or notes of a specific maturity. This type of interpolation helps analysts gain insight into where the bond markets and the economy might be headed in the future.

Interpolation should not be confused with extrapolation, which refers to the estimation of a data point outside of the observable range of data. Extrapolation has a higher risk of producing inaccurate results compared to interpolation.

## Example of Interpolation

The easiest and most prevalent kind of interpolation is a linear interpolation. This type of interpolation is useful if one is trying to estimate the value of a security or interest rate for a point at which there is no data.

Let's assume, for example, we're tracking a security price over a period of time. We'll call the line on which the value of the security is tracked the function f(x). We would plot the current price of the stock over a series of points representing moments in time. So if we record f(x) for August, October, and December, those points would be mathematically represented as x_{Aug, }x_{Oct, }and x_{Dec, }or x_{1, }x_{3 }and x_{5. }

For a number of reasons, we might want to know the value of the security during September, a month for which we don't have any data. We could use a linear interpolation algorithm to estimate the value of f(x) at plot point x_{Sep}, or x_{2} that appears within the existing data range.

## Criticism of Interpolation

One of the biggest criticisms of interpolation is that although it's a fairly simple methodology that's been around for eons, it lacks precision. Interpolation in ancient Greece and Babylon was primarily about making astronomical predictions that would help farmers time their planting strategies to improve crop yields.

While the movement of planetary bodies is subject to many factors, they are still better suited to the imprecision of interpolation than the wildly variant, unpredictable volatility of publicly-traded stocks. Nevertheless, with the overwhelming mass of data involved in securities analysis, large interpolations of price movements are fairly unavoidable.

Most charts representing a stock's history are in fact widely interpolated. Linear regression is used to make the curves which approximately represent the price variations of a security. Even if a chart measuring a stock over a year included data points for every day of the year, one could never say with complete confidence where a stock will have been valued at a specific moment in time.

## What Type of Interpolation Is Used in Technical Analysis?

In technical analysis, there are two main types of interpolation: linear interpolation and exponential interpolation. Linear interpolation calculates the average of two adjacent data points by drawing a straight line of best fit. Exponential interpolation instead calculates the weighted average of the adjacent data points, which can adjust for trading volume or other criteria.

## How Is Interpolation Used in Trading?

Traders may employ a particular type of interpolation (also called smoothing) to represent the high-low range of price movement between a series of closing price prints. This is done by creating a linear regression line through the highs and lows of a two-day chart as shown above. Then, the slope of the regression line corresponds to (roughly) the shape of the price movement over those consecutive days. This slope can then be used as an approximation for the moving average (MA) of the high-low range. If prices are trading above the regression line (of the moving average), then traders can assume the low-range will support higher prices. On the other hand, if prices fall below the moving average, the low-range is deemed to support lower prices.

## What Is Interpolation vs. Extrapolation?

Interpolation estimates unknown values that fall between two or more known data points, filling in the blanks. Extrapolation instead extends known data points outward.

## The Bottom Line

Interpolation is a mathematical technique to estimate the values of unknown data points that fall in between existing, known data points. This process helps fill in the blanks. Technical traders use interpolation to understand how prices have behaved in the past, even when they do not have full information. Doing so can thus help predict future trends based on a more complete picture of past price action.