Interpolation estimates values inside the span of supplied curve points. The result depends on whether the method operates on yields, discount factors, logarithms of discount factors or forwards. Document that choice, preserve compounding conventions and inspect implied discounting behavior. An interpolated value is not a newly observed market quote.
Separate missing observations from missing definitions
A table of maturities and percentages is not yet a complete discounting specification. Before filling a gap, identify whether each percentage is a spot yield, par yield, forward rate or yield on an individual coupon bond. Also identify compounding and the time basis. An interpolation method cannot repair a table that mixes these meanings; it will simply make the inconsistency look continuous and more precise.
Interpolation stays between supplied maturities. Extrapolation extends beyond them and deserves a separate assumption because the endpoints no longer bracket the estimate. A valuation report should mark both kinds of generated points. This is especially important when a long-dated payment relies on the last available observation: a flat extension, a constant forward rate and convergence toward a chosen long-run rate are different economic assumptions.
Distinguish interpolation from fitting a curve
Treasury describes its official par curve as using bootstrapping and monotone convex interpolation of forward rates. The ECB describes estimation using the Svensson methodology. These brief descriptions show why matching a published curve requires knowing its construction, not simply connecting displayed percentages. They do not establish that either approach is the correct choice for every educational or analytical purpose involving a term structure.
An interpolator commonly preserves its supplied nodes, whereas a fitted model may balance errors across many observations. A smooth fitted curve can therefore differ from individual input yields by design. Review the objective being optimized and the residuals being reported. Calling every smooth line an interpolation obscures whether discrepancies reflect implementation problems, deliberate smoothing or differences between underlying observations and the published output series.
Calculate a straight line in spot-yield space
For a purely hypothetical example, specify annual-compounded spot yields of 3% at two years and 5% at four years. No traded securities or current market levels are implied. Linear interpolation of these annual rates places the three-year spot yield halfway between them, at 4%. The midpoint follows because three years lies halfway through the two-year interval, not because three-year cash flows have been observed.
A hypothetical USD 100 payment in three years then has present value 100 / 1.04^3 = USD 88.899636, rounded to six decimals. The denominator is independently 1.124864. This example uses one payment and annual compounding, with no credit adjustment, fees or timing irregularity. Its purpose is to make the transformation from an interpolated rate to a discounted amount visible before comparing another construction.
Repeat the exercise in discount-factor space
Convert the same endpoints into discount factors: D(2) = 1 / 1.03^2 and D(4) = 1 / 1.05^4. Linear interpolation of those factors gives D(3) = (D(2) + D(4)) / 2. The USD 100 payment is now worth USD 88.264919. Equivalently, average the endpoint present values of USD 100 to reproduce that number independently.
The implied three-year annual spot yield is D(3)^(-1/3) - 1, approximately 4.248694%, rather than 4%. Both methods agree at the supplied endpoints but differ between them because converting yields into discount factors is nonlinear. Neither midpoint is an additional fact about a market. The comparison demonstrates model dependence, not an arbitrage opportunity or a reason to prefer whichever valuation happens to be larger.
| Hypothetical construction | Three-year annual spot yield | PV of USD 100 |
|---|---|---|
| Linear spot yield | 4.000000% | USD 88.899636 |
| Linear discount factor | 4.248694% | USD 88.264919 |
Inspect the forwards implied by the curve
A visually smooth spot curve can imply abrupt or implausible changes in forward rates. Forward rates depend on the relationship between neighboring discount factors, so small features in one representation may become prominent in another. Review the implied forwards around knots and across sparse regions. A pleasing chart is not evidence that the interpolation respects the analytical behavior required by the intended valuation exercise.
Positive discount factors are a basic requirement for ordinary discounting. Requiring them always to decline is a stronger condition and can be inappropriate when negative rates are permitted. Likewise, a negative implied forward is not automatically a coding defect. The reviewer must distinguish an admissible rate environment from an accidental oscillation introduced by the numerical method, using the stated assumptions rather than an unexamined positivity rule.
Keep comparisons reproducible
To compare methods fairly, freeze the same inputs, their precision, valuation time and instrument conventions. Then change only the interpolation rule. Record prices of representative hypothetical payments as well as plotted rates, because small-looking curve differences can accumulate across many coupons. A discrepancy attributable to a method change should not be presented as a change in observed rates or an unexplained revision to source data.
Sensitivity calculations must use the same interpolation policy as the base valuation. Rebuilding shocked curves with a different rule changes both the economic scenario and the numerical mapping. Similarly, rounding node values before interpolation can make independent reproduction needlessly difficult. Preserve full working precision and round only the published display, with enough detail that another reviewer can recover the example within the disclosed rounding tolerance.
Limit the conclusion to the supplied information
A sparse curve can support a clearly labeled educational comparison without supporting a precise market valuation. The important distinction is whether a result comes from an observation, an interpolation rule or an extrapolation assumption. Report where the payments sit relative to available nodes. A concentrated exposure inside a wide gap deserves attention even if the curve fits every input point perfectly and produces stable numerical output.
The site's regular-bond present-value calculation uses one annual nominal yield compounded at the coupon frequency. It does not bootstrap a market curve or choose an interpolation policy for individual payment dates. The arithmetic here explains a broader convention separately. Using a hypothetical interpolated yield as a single input still produces a single-yield illustration, not a complete spot-curve valuation or an executable bond quote.
Review checkpoints
- Specify the interpolated quantity before choosing a numerical method.
- Identical endpoint yields can produce different intermediate discount factors.
- Inspect implied forwards and disclose extrapolation separately from interpolation.
Primary references
Continue the calculation
Spot, Par and Forward Rates
Derive three rate descriptions from consistent discount factors and avoid treating a par curve as a set of zero-coupon discount rates.
Curve Bootstrapping: What the Equations Can and Cannot Identify
Solve a small discount-factor example, then examine missing dates, inconsistent inputs and the limits of exact repricing.
Key Rate Duration: Locating Curve Exposure
Read maturity-specific rate sensitivities, define the curve bumps behind them, and distinguish a local approximation from full scenario repricing.
Negative Yields
Understand above-face prices, discount factors greater than one and rate sensitivity when the assumed nominal yield is below zero.
Edition dated 16 October 2026. Constructed examples illustrate a method, not a current market price or a recommendation.