Linear interpolation

Linear Interpolation: Definition, Formula, Example, and FAQs

Linear interpolation is a fundamental mathematical method used to estimate a new data point that falls between two known data points within a given range. This technique operates under the assumption that the relationship between these two known points is linear, meaning a straight line can connect them. In the realm of financial modeling and quantitative analysis, linear interpolation helps fill in gaps in data series where precise observations are unavailable, enabling analysts to approximate values for analytical purposes. When analysts encounter missing data in financial datasets, linear interpolation provides a straightforward and computationally efficient way to infer values, offering a continuous representation where only discrete observations exist.

History and Origin

The concept of interpolation, including its linear form, is ancient, with its roots tracing back to early civilizations focused on practical applications like astronomy. Evidence suggests that linear interpolation was utilized by Babylonian astronomers as far back as 300 BC to predict the positions of celestial bodies. Later, the Greek astronomer Hipparchus of Rhodes (circa 150 BC) employed this method in constructing tables for his "chord function," which is analogous to a sinusoidal function, to compute planetary positions. The technique also appeared in ancient Chinese mathematical texts and in Ptolemy's Almagest in the 2nd century AD, highlighting its long-standing role in filling gaps within tabulated data.⁶ Its enduring appeal stems from its intuitive simplicity and ease of application across various fields requiring data approximation.

Key Takeaways

Linear interpolation is a method for estimating unknown values between two known data points by assuming a linear relationship.
It is widely used in data analysis and financial contexts to fill in gaps and create continuous representations from discrete observations.
The method's simplicity makes it computationally efficient, but its accuracy depends on the actual linearity of the underlying data relationship.
It is particularly useful for short intervals where the assumption of linearity is more likely to hold true.

Formula and Calculation

The formula for linear interpolation calculates an unknown value (y) at a specific point (x) based on two known data points ((x_1, y_1)) and ((x_2, y_2)). Here, (x_1) and (x_2) represent the independent variable values, and (y_1) and (y_2) are their corresponding dependent variable values. The point (x) for which the value (y) is being interpolated must lie between (x_1) and (x_2).

The formula is expressed as:

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Where:

(y): The interpolated value (unknown).
(x): The independent variable value for which (y) is being interpolated.
(x_1, y_1): The coordinates of the first known data point.
(x_2, y_2): The coordinates of the second known data point.

This formula essentially calculates the weighted average of (y_1) and (y_2), with the weights determined by the proximity of (x) to (x_1) and (x_2).

Interpreting Linear Interpolation

Interpreting linear interpolation involves understanding that the estimated value is a straight-line approximation between two known points. This means the method inherently assumes a constant rate of change between the given data points. When applying linear interpolation, for instance, to a time series of stock prices, the interpolated price for an unobserved day between two trading days suggests a smooth, consistent movement.

In financial applications, the interpretation hinges on the context. If interpolating between two closely observed interest rates on a bond, the resulting value is taken as a reasonable estimate. However, if the underlying relationship is known to be non-linear (e.g., due to market volatility or complex asset pricing dynamics), the interpolated value should be treated as an approximation rather than a precise figure. The closer the known points are, and the smoother the underlying trend, the more reliable the interpolated value will be.

Hypothetical Example

Imagine a small business conducting a simple valuation exercise. They have recorded sales figures for two consecutive months:

Month 1 (January): Sales of $100,000
Month 3 (March): Sales of $130,000

The business wants to estimate the sales figure for Month 2 (February) using linear interpolation, assuming a steady growth rate.

Here, our known points are:

((x_1, y_1) = (\text{Month 1, $100,000}))
((x_2, y_2) = (\text{Month 3, $130,000}))
The point we want to interpolate is (x = \text{Month 2}).

Let's assign numerical values to the months: Month 1 = 1, Month 2 = 2, Month 3 = 3.

Using the linear interpolation formula:
$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

$y = 100,000 + \frac{(2 - 1)(130,000 - 100,000)}{3 - 1}$

$y = 100,000 + \frac{(1)(30,000)}{2}$

$y = 100,000 + 15,000$

$y = 115,000$

So, based on linear interpolation, the estimated sales for Month 2 (February) would be $115,000. This example demonstrates how linear interpolation can be used for basic forecasting or estimating intermediate values when direct data is unavailable.

Practical Applications

Linear interpolation finds diverse practical applications across various financial and analytical domains due to its simplicity and effectiveness in approximating values.

Yield Curve Construction: One of the most common applications in finance is in constructing a yield curve. Treasury departments and financial institutions use observed yields for specific maturities (e.g., 3-month, 2-year, 10-year) to interpolate yields for maturities where no direct observations exist. The U.S. Department of the Treasury, for example, uses various interpolation methods, including more complex ones like the monotone convex method, to derive its official yield curve, demonstrating the importance of interpolation in capturing the term structure of interest rates.⁵
Bond Pricing: When pricing a bond with an unusual maturity that falls between standard benchmark bonds, linear interpolation can be used to estimate its yield or price.
Missing Data Imputation: In financial data analysis, linear interpolation is often used to fill in gaps in time series data, such as missing stock prices or commodity rates, ensuring continuity for further analysis or financial modeling.
Option and Derivative Pricing Models: Some simplified option pricing models or internal valuation tools may use linear interpolation to determine implied volatilities or discount factors for specific maturities not directly quoted in the market.
Risk Management: Calculating Value at Risk (VaR) or other risk management metrics might involve interpolating values from discrete probability distributions or historical data series to assess risk at specific points.

Limitations and Criticisms

Despite its widespread use and simplicity, linear interpolation has several limitations, especially in complex financial environments. The primary criticism centers on its fundamental assumption: that the relationship between two data points is perfectly linear.

Assumption of Linearity: If the actual underlying relationship between variables is non-linear, linear interpolation will introduce inaccuracies. For instance, in finance, concepts like the time value of money, compound interest rates, and asset prices often exhibit curvilinear rather than strictly linear behavior. An interpolated value for a future value, based on a linear assumption, might deviate from the true value because the growth is compounded, not linear.⁴ This makes linear interpolation less precise in volatile markets or for data with significant non-linear trends.³
Inability to Capture Curvature: Linear interpolation connects points with straight lines, failing to account for any curvature or fluctuations in the actual data series. This can smooth out important features or introduce distortions if the true function is highly curved between the known points.
Sensitivity to Data Spacing: While simple, its accuracy can be compromised if the data points are unevenly spaced or if the interval between them is large.² The further apart the known points, the greater the potential for error, as the linear assumption becomes less reliable over longer spans.
No Extrapolation: Linear interpolation is designed to estimate values within the range of known data points. It is not suitable for Extrapolation, which involves estimating values outside this range. Attempting to use linear interpolation for extrapolation can lead to highly inaccurate and misleading results.¹
Ignores Volatility and Outliers: In financial markets, factors like market volatility and sudden price jumps are not accounted for by a simple linear model. Outliers or noisy data can also disproportionately influence the interpolated value if they happen to be one of the two anchor points.

For scenarios where higher accuracy is required, or where non-linear relationships are prevalent, more sophisticated methods like polynomial interpolation, spline interpolation, or regression analysis may be more appropriate, though they come with increased complexity.

Linear Interpolation vs. Extrapolation

Linear interpolation and Extrapolation are both methods used to estimate unknown data points from known ones, but they differ fundamentally in the range of estimation. Linear interpolation involves estimating a value that lies between two known data points. For example, if you have sales figures for January and March, linear interpolation can estimate the sales for February. It assumes a linear progression within the observed data range.

In contrast, Extrapolation involves estimating a value that lies outside the range of known data points. Using the same sales example, if you wanted to estimate sales for April, based on January and March data, you would be extrapolating. The key difference is that extrapolation carries a significantly higher risk of inaccuracy. While interpolation assumes the continuation of a trend within observed limits, extrapolation presumes that a trend will continue beyond those limits, which is often not the case in dynamic systems like financial markets. For this reason, interpolation is generally considered more reliable than extrapolation, particularly in quantitative analysis.

FAQs

What is the primary purpose of linear interpolation in finance?

The primary purpose of linear interpolation in finance is to estimate missing or unobserved values within a set of known data points. This is crucial for tasks like constructing continuous yield curves from discrete bond maturities or filling in gaps in a time series of financial data for financial modeling purposes.

When should linear interpolation be avoided?

Linear interpolation should be avoided when the underlying relationship between the independent variable and dependent variable is clearly non-linear, as this will lead to inaccurate estimations. It should also be avoided for Extrapolation (estimating values outside the known data range), and when dealing with highly volatile data or very large intervals between known points where the linearity assumption is unlikely to hold.

Is linear interpolation more accurate than other methods?

Linear interpolation is generally the simplest method and is often accurate enough for straightforward, linear relationships or for estimating values over short intervals. However, for complex or non-linear data series, methods such as polynomial interpolation, spline interpolation, or other numerical techniques may provide more accurate results by better capturing the curvature or underlying patterns in the data. Its accuracy is balanced by its ease of implementation.