Interpolated Yield Curve (I Curve)

August 24, 2024 Fixed Income

An interpolated yield curve estimates yields between observed maturities, creating a smoother curve for pricing and rate-risk analysis.

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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.

SVG diagram showing observed yield curve points and an interpolated yield between two maturity points.

For simple linear interpolation:

y(t) = y(t_i) + \frac{(y(t_{i+1}) - y(t_i))(t - t_i)}{t_{i+1} - t_i}

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.

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.

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.

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.

FAQs

Is an interpolated yield curve a market quote?

No. It is an estimate based on observed curve inputs and a chosen interpolation method.

Why interpolate a yield curve instead of using the nearest Treasury maturity?

Interpolation can give a closer maturity match for a bond whose maturity falls between quoted curve points.

Can two interpolated curves give different spreads for the same bond?

Yes. Different data sources, curve types, timestamps, and interpolation methods can produce different benchmark yields and therefore different spreads.

Anticipated Holding Period

On-The-Run Treasury Yield Curve