Interpolated Yield Curve (I Curve) An interpolated yield curve (I curve) estimates Treasury yields for maturities that are not directly observed in the market by using available on-the-run Treasury securities. It fills gaps between issued maturities so analysts and investors can derive continuous term structures for pricing, risk management, and macro forecasting. Key points
* The I curve is built from on-the-run Treasuries (the most recently issued bills, notes, and bonds) and interpolated values for in-between maturities.
* Common interpolation techniques include linear interpolation, regression methods, and bootstrapping.
* The derived curve is used to compute zero-coupon yields, yield spreads, and expectations about inflation, interest rates, and economic growth.
* Limitations include data sparsity, liquidity differences between on- and off-the-run issues, and model risk from the chosen interpolation method.
How it works
* Plot yields (y-axis) against time to maturity (x-axis) for available on-the-run Treasuries.
* Because Treasuries are issued at discrete maturities, the curve between those points must be estimated. The estimated continuous curve is the interpolated yield curve.
* On-the-run issues tend to trade at higher prices and lower yields than similar off-the-run (seasoned) securities, so the I curve reflects a specific subset of the market.
Common interpolation methods
* Linear interpolation — simple straight-line estimates between adjacent maturities. Useful for quick approximations but may ignore curvature.
* Regression and smoothing splines — fit a functional form (e.g., polynomial, splines) to capture overall shape and curvature of the term structure.
* Bootstrapping — an iterative technique that derives a zero-coupon (spot) yield curve from observed prices and yields of coupon-bearing bonds.
Bootstrapping: step-by-step (overview)
1. Start with short-term instruments whose zero rates are known or easily derived.
2. Use observed prices and coupon payments of the next-maturity coupon-bearing bond to solve for the zero rate that makes the bond’s discounted cash flows equal its market price.
3. Repeat sequentially for increasing maturities, using previously derived zero rates to discount earlier cash flows.
4. Where market maturities are missing, interpolate the required rates between known points before applying the bootstrapping equations.
5. The result is a zero-coupon yield curve consistent with observed par yields and market prices.
Uses and interpretation
* Benchmarking: I curve serves as a reference for pricing other fixed-income securities and calculating yield spreads (e.g., corporate bonds, agency CMOs).
* Economic signals: Curve slope and shape inform expectations about future interest rates, inflation, and economic growth (e.g., upward-sloping, flat, or inverted curves).
* Risk management: Continuous term structures enable consistent valuation, duration, and convexity calculations across instruments with nonstandard maturities.
Important considerations and limitations
* Liquidity and representativeness — on-the-run Treasuries are highly liquid but represent only a portion of outstanding supply; off-the-run issues may convey different yields.
* Method choice matters — linear interpolation can underfit curvature; overly flexible models may overfit noise. Select a method appropriate to purpose and data quality.
* Data gaps and short-end rates — when short-maturity data are sparse, money-market rates may be used, introducing additional assumptions.
* Model risk — all interpolations impose structure; regularly validate the curve against market prices and alternative methods.
Conclusion An interpolated yield curve is a practical tool for converting discrete Treasury observations into a continuous term structure used in valuation, spread analysis, and macroeconomic inference. Understanding the interpolation method, bootstrapping mechanics, and the curve’s limitations is essential for reliable interpretation and application. Explore More Resources