Interpolated Yield Curve (I Curve)
An interpolated yield curve estimates yields between observed maturities, creating a smoother curve for pricing and rate-risk analysis.
An interpolated yield curve, often called an I curve, estimates yields for maturities that are not directly observed in the market. It fills the gaps between quoted curve points so analysts can price bonds, compare spreads, and measure rate risk at intermediate maturities.
Interpolation is not the same as observing a traded yield. It is an estimate between known maturity points.
Core Idea
Observed curve points may exist at maturities such as 2 years, 5 years, 10 years, and 30 years. A 7-year bond still needs a benchmark yield. An interpolated curve estimates that 7-year point from the neighboring observed points.
For simple linear interpolation:
Where \(t\) is the target maturity between observed maturities \(t_i\) and \(t_{i+1}\).
Why It Matters
Interpolated curves matter because real portfolios rarely line up perfectly with the quoted benchmark points.
They are used for:
- bond pricing between observed Treasury maturities
- spread calculation for bonds with odd maturities
- Yield Curve Risk measurement
- key-rate or bucketed rate-risk systems
- fair-value checks for new issues and secondary-market bonds
- roll-down analysis over an anticipated holding period
- curve construction for swaps, Treasuries, municipals, agencies, or credit curves
The curve source and interpolation method can change the answer, especially when the yield curve is steep, inverted, or kinked.
Common Interpolation Choices
| Method | How it works | Best use | Main caution |
|---|---|---|---|
| Linear interpolation | Connects two observed points with a straight line | Transparent first-pass estimates | Can miss curvature between points |
| Spline interpolation | Uses piecewise curves for smoother results | Smooth pricing and risk systems | Can create model artifacts |
| Parametric curve model | Fits a curve form to market data | Research, central-bank data, and broad curve estimation | Depends on model design |
| Bootstrapped curve | Builds discount or spot rates from instruments | Valuation and derivatives work | More complex and input-sensitive |
An I curve should disclose the curve inputs and method. Without that, two systems can produce different yields for the same maturity.
Practical Example
Suppose a 7-year corporate bond needs a Treasury benchmark yield, but the available curve points are 5-year and 10-year yields.
The analyst can:
- select the curve source, such as Treasury par yields, on-the-run points, off-the-run fitted yields, or a dealer curve
- interpolate the 7-year Treasury yield between the 5-year and 10-year points
- compare the bond yield with that estimated 7-year benchmark
- calculate a spread using the same curve convention as the benchmark or risk system
If another analyst uses a fitted off-the-run curve instead of on-the-run interpolation, the spread may differ even when the bond price is the same.
I Curve vs. Related Curves
| Curve | What it represents | Best use | Main caution |
|---|---|---|---|
| On-The-Run Treasury Yield Curve | Yields of most recently issued benchmark Treasuries | Current trading benchmark context | Points can reflect liquidity premium |
| Interpolated yield curve | Estimated yields between observed points | Odd-maturity pricing and spread comparison | Method-dependent |
| Federal Reserve fitted nominal curve | Smoothed estimates based on Treasury coupon securities | Historical comparison across maturities | It is a model estimate, not a traded quote |
| Yield Curve | General relationship between yield and maturity | Curve-shape interpretation | Must specify source and convention |
| Forward curve | Implied future rates from current curve data | Rate-expectation and derivatives analysis | Not the same as a yield curve quote |
The practical rule is simple: do not mix curve conventions in one spread, hedge, or attribution analysis without saying so.
What To Verify
Before relying on an interpolated yield curve, verify:
- curve source: Treasury par, on-the-run, off-the-run fitted, swap, municipal, issuer, or credit curve
- observed maturity points and whether they are current
- interpolation method and whether it is linear, spline, parametric, or bootstrapped
- yield type: par yield, spot rate, zero rate, forward rate, or yield to maturity
- day-count, compounding, settlement, and calendar conventions
- whether the target bond maturity falls inside the observed range or requires extrapolation
- whether a kinked, inverted, or illiquid curve makes interpolation fragile
- whether the same curve convention is used for pricing, spread, duration, and attribution
The safest documentation names the data source, timestamp, curve type, method, and target maturity.
Public Source Checks
Useful public references include:
- U.S. Treasury Daily Treasury Rates: Official daily Treasury par yield curve rates by maturity point.
- Federal Reserve Nominal Yield Curve: Smoothed nominal Treasury yield estimates and model data across maturities.
- TreasuryDirect Announcements, Data and Results: Official auction announcements, results, and recently auctioned Treasury securities data.
- MSRB Understanding Yield Curves and Indices: Municipal-bond education on yield curves, indices, benchmarks, and curve interpretation.
These sources support the public curve context. A pricing-grade I curve still needs the exact curve inputs, interpolation method, timestamp, and instrument conventions used by the analyst.
When Interpolation Misleads
Interpolation can mislead when:
- the target maturity is outside the observed maturity range
- the curve has a sharp kink between two quoted points
- on-the-run liquidity premiums are treated as pure rate expectations
- a par yield is mixed with spot, zero, or forward rates
- stale curve points are combined with current security prices
- a smooth curve is assumed to be more accurate than traded market evidence
- the same curve is used for Treasury, municipal, and credit bonds without basis adjustments
Use interpolation as a disciplined estimate, not as proof that an unobserved yield is directly traded.
Related Terms
- On-The-Run Treasury Yield Curve: Observed benchmark Treasury inputs often used for curve context.
- Treasury Yield: The government yield benchmark used in many spread comparisons.
- Yield Curve Risk: Risk that nonparallel curve moves hurt a bond or portfolio.
- Key Rate Duration: Maturity-point sensitivity that often depends on curve interpolation.
- Normal Yield Curve: Upward-sloping curve shape.
- Inverted Yield Curve: Downward-sloping curve shape.