What it means
An analyst may have reliable measurements at two dates, maturities or quantities but need a value between them, and interpolation supplies a rule for filling that gap. It is useful when direct measurement is unavailable or a model requires a continuous input.
The known values are often called nodes, and with linear interpolation the estimated value changes at a constant rate between adjacent nodes, so a point halfway between two node positions receives a value halfway between their measured values. The distance must be measured on the relevant scale, because halfway between two calendar dates is not necessarily halfway between two quoted trading-day positions.
Units, dates and the definition of each measurement need to agree before calculation. NIST distinguishes interpolation from fitting a function to data: an interpolating function passes through the original points, whereas a fitted function may not, and choosing interpolation does not remove measurement errors already present in those points.
Piecewise linear interpolation uses a straight segment between each adjacent pair in a larger dataset, while curved methods, including polynomial and spline methods, can produce a smoother shape. Smoothness is a modelling choice, not independent evidence of accuracy.
Financial applications include estimating a yield at an unquoted maturity or filling an intermediate value in a sensitivity table. The analyst must decide what is being interpolated: a yield, discount factor, price or other quantity, because interpolating different quantities can give different results.
Interpolation stays within the range of known positions, whereas extrapolation extends beyond that range and needs assumptions about behaviour outside the observations. Neither label means a predicted value will be realised.
For managers, ask which observations anchor the estimate and why the chosen rule is reasonable. Check whether a discontinuity, threshold or change in conditions lies between them.
A straight line can be a poor description of a stepped price schedule or a sudden market change.
In practice
Real-world examples.
Example
A table shows estimated service costs at volumes of 100 and 200 units. A manager interpolates a value at 150, but first checks that no staffing step or quantity discount changes costs within that interval.
Example
An analyst has yields at two maturities and estimates an intermediate yield. The report labels it as interpolated rather than suggesting that a bond was actually traded at that maturity and yield.
Example
A model needs a value beyond the last measured position. The analyst identifies this as extrapolation and does not present it as interpolation simply because the same straight-line equation is used.
Formula
Calculation
For linear interpolation between positions x1 and x2, estimated y = y1 + (y2 - y1) x (x - x1) / (x2 - x1). The positions must differ, and x must lie between them for interpolation. All positions and values must use consistent units.
Suppose fictional quoted yields are 4.0% at two years and 4.6% at five years. At three years, the position is (3 - 2) / (5 - 2) = one-third of the way across the interval, so the estimated yield is 4.0% + 0.6 x (1/3) = 4.2%. At four years the position is two-thirds, giving 4.0% + 0.6 x (2/3) = 4.4%.
The 0.6 difference is in percentage points. This example estimates a yield, not a guaranteed investment return, bond price or discount factor.Case study
Seen in the real world.
This fictional case follows a company comparing financing proposals with different maturities. Its analyst has benchmark yields at two and five years but needs a three-year reference for an internal comparison. The first spreadsheet copies the two-year quote into the three-year cell. A reviewer asks whether this represents an observation, a deliberately conservative assumption, or an accidental shortcut. The analyst replaces the shortcut with a labelled linear estimate.
The supporting note records the two source observations, their dates, the calculation, and the reason for using yields as the interpolated quantity. Treasury also tests whether a different curve method changes the comparison materially. It keeps lender fees, credit spreads, and actual contract terms separate rather than treating the estimated benchmark as a complete borrowing quote. Management can now see where measurement ends and estimation begins. The method helps compare proposals consistently, but the company still requests current executable financing terms before making a decision.
Watch out
Common mistakes.
- Calling an interpolated estimate an observed market quote or a promised future outcome.
- Mixing units, measurement dates, or quantities, such as interpolating prices while interpreting the result as a yield.
- Using a straight line through a threshold or outside the observed range without identifying the extra assumption.
Questions
People also ask.
Is interpolation the same as forecasting?
No. Interpolation fills an intermediate position between known points. A future forecast may require other assumptions, although interpolation can be one component of a forecasting model.
Must the method be linear?
No. Curved methods are possible. The choice should fit the purpose and data, with assumptions and sensitivity made clear.
Does passing through every node prove accuracy?
No. The nodes may contain errors, and the chosen shape between them may be wrong. Exact agreement at known points does not validate all intermediate estimates.
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