Key takeaways
– An interpolated yield curve (I curve) is a yield curve constructed from the yields of on‑the‑run Treasuries with missing maturities filled in by interpolation.
– Interpolation methods (linear, spline, polynomial, regression) and bootstrapping are commonly combined to convert par/coupon yields into a zero (spot) curve.
– The I curve is a widely used benchmark for pricing, spread analysis, risk management, and mortgage/CMO valuation, but it has limitations (liquidity effects, on‑ vs off‑the‑run biases, extrapolation risk).
– Practical construction requires careful choices about market data, conventions (day count, compounding), interpolation domain (par yields vs zero rates vs log yields), and validation.
What is an interpolated yield curve (I curve)?
– The yield curve maps bond yield (vertical axis) against time to maturity (horizontal axis).
– On‑the‑run Treasuries are the most recently issued U.S. Treasury securities for particular standard maturities (e.g., 2‑year, 5‑year, 10‑year). Only those issues have readily quoted yields at each standard maturity.
– Market participants often need yields (or spot rates) at intermediate maturities that are not directly quoted (for example, 3.5 years or the weighted average life of a CMO). The I curve is produced by interpolating between the on‑the‑run points to estimate yields at those intermediate maturities.
Why derive an I curve?
– Benchmarks and spreads: Many securities trade at a spread to Treasury yields at nonstandard maturities; an interpolated curve supplies the benchmark yield.
– Pricing and valuation: Valuing coupon bonds, swaps, CMOs, and derivatives often requires a continuous term structure or spot rates at arbitrary dates.
– Risk and macro inference: The shape of the curve is used to infer market expectations about growth, inflation, and policy.
Interpolation vs. bootstrapping — how they fit together
– Interpolation: An estimation technique used to fill in missing yields between known data points. Methods include linear interpolation, cubic spline, monotone splines, polynomial fits, or regression models. The choice affects smoothness and economic plausibility of the curve.
– Bootstrapping: An iterative technique that uses the prices (or quoted yields) of coupon‑bearing bonds to extract zero‑coupon (spot) rates for each maturity. Bootstrapping often requires a complete set of term structure rates; where those rates are missing, interpolation is used to estimate them before or during the bootstrap. In practice one might interpolate a par curve and then bootstrap to a zero curve, or bootstrap while applying interpolation at intermediate steps.
Practical steps to construct an I curve — end‑to‑end
Below is a stepwise practical procedure that you can implement in Excel or code (Python/R)
1) Data collection and preparation
• Gather on‑the‑run Treasury yields (or prices) for the full set of standard maturities (e.g., 1M, 3M, 6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 20Y, 30Y). Use reliable sources (Treasury, Bloomberg, Reuters).
• Record market conventions: coupon frequency (semiannual for US Treasuries), day‑count convention (ACT/ACT for U.S. Treasuries), and quoting convention (par yields or yield to maturity).
• If you work from prices, convert to yields consistently; if you use par yields, note that bootstrapping to spot rates uses par prices.
2) Decide the target curve and interpolation domain
• Target curve options: par yield curve, zero (spot) curve, or forward curve. Many applications require the zero curve (spot rates).
• Interpolate in an appropriate variable: common choices are yields (linear in yield), zero rates, continuously compounded zero rates, or log discount factors. The choice affects economics (e.g., linear interpolation in log discount factors preserves monotonicity of discount factors).
3) Choose an interpolation method
• Simple/fast: linear interpolation between adjacent maturities (works well for many automation needs but can be kinked).
• Smoother: cubic spline, monotone spline, or Nelson‑Siegel / Svensson parametric fits (these produce smooth, economically plausible shapes).
• For certain quant workflows, perform local interpolation (piecewise) rather than a global polynomial to limit oscillation.
4) Interpolate yields at required intermediate maturities
• Use your chosen method to compute yields (or discount factors/zero rates) at all required dates (e.g., every month up to 30 years, or specific bespoke dates).
5) Bootstrapping to derive zero (spot) rates (if starting from coupon/par yields)
• If you have coupon‑bearing instruments (par yields or prices), bootstrap iteratively:
a. Start with the earliest maturity for which cash flows are simple (e.g., a short zero or a 6M bill). The spot rate for that maturity is directly observed.
b. For the 2nd instrument (e.g., a 1‑year coupon bond), discount known earlier cash flows using previously obtained spot rates to solve for the spot rate that makes present value equal to observed market price.
c. Repeat sequentially for longer maturities. Where cash flows fall between on‑the‑run maturities, use interpolated rates/discount factors as needed during the iteration.
• Keep compounding conventions consistent (convert periodic to continuous if your chosen output uses continuous compounding).
6) Validation and smoothing
• Check for arbitrage violations (e.g., negative discount factors, nonmonotonic discount curves).
• Compare interpolated yields to off‑the‑run or liquid swap rates, if available, to test plausibility.
• Apply smoothing or regularization if the curve is noisy.
7) Produce outputs for use
• Export discount factors, zero rates, par yields, forward rates, and a continuous curve object.
• Use curve to price instruments, compute spread to Treasury, or feed into risk models.
Numerical examples (short)
1) Linear interpolation example (simple)
• Given on‑the‑run par yields: 2‑year = 1.50%, 5‑year = 2.00%. Estimate 3‑year yield by linear interpolation:
slope = (2.00% − 1.50%) / (5 − 2) = 0.1667% per year.
3‑year yield = 1.50% + 0.1667% × (3 − 2) = 1.6667%.
2) Bootstrapping example (solve for 2‑year spot)
• Suppose you know the 1‑year spot s1 = 1.00% (annual compounding). A 2‑year coupon bond pays 4% annually, price = 99 (par = 100).
• Present value equation: 99 = 4/(1+s1) + 104/(1+s2)^2.
• Plug s1 = 1%: 4/1.01 = 3.9604. Remaining PV = 99 − 3.9604 = 95.0396.
• Solve for s2: (1+s2)^2 = 104 / 95.0396 = 1.0939 → 1+s2 = sqrt(1.0939) = 1.045 → s2 = 4.5%.
Special considerations and caveats
– On‑the‑run vs off‑the‑run: On‑the‑run Treasuries are more liquid and typically trade at slightly lower yields (higher prices) than off‑the‑run issues. The I curve reflects on‑the‑run pricing and can therefore be a biased benchmark for less liquid instruments.
– Interpolation domain matters: Interpolating par yields and then bootstrapping may produce a different zero curve than bootstrapping first and interpolating discount factors or zero rates. Choose the approach that matches your valuation needs.
– Day count and compounding: Mistakes here produce material valuation errors — be consistent and explicit about conventions.
– Extrapolation risk: Estimating yields beyond the longest quoted maturity (e.g., beyond the 30‑year) requires extrapolation and is especially unreliable.
– Liquidity and market dislocations: During times of stress, on‑the‑run quotes can move irregularly; smoothing and cross‑checks with other instruments (swaps, bills, repo) can improve robustness.
– Model risk: Parametric fits (Nelson‑Siegel, Svensson) impose a functional shape — they are powerful but can hide short‑term anomalies.
Common interpolation choices and tradeoffs
– Linear interpolation in yields: simple, fast, but not smooth and may produce kinks in derived forward rates.
– Cubic spline on zero rates or discount factors: smooth curve and smoother forward rates, but can overshoot between points unless monotone splines are used.
– Nelson‑Siegel / Svensson: parametric, low‑dimensional, good for macro analysis, but may not fit idiosyncratic short‑term features.
– Interpolate discount factors or log discount factors to preserve arbitrage‑free monotonicity.
Implementation tips and tooling
– Excel: Good for small examples; use RATE, PRICE, and solver routines; be explicit with day‑count conventions.
– Python: Use pandas, numpy, scipy.interpolate (UnivariateSpline, PCHIP for monotone), and quant libraries (QuantLib) to bootstrap and fit curves.
– R: Packages like termstrc or YieldCurve offer fitting routines.
– Automate validations: ensure discount factors are in (0,1) and monotonically decreasing with maturity.
Use cases
– Spread measurement: ABS, agency CMOs, corporate bonds, and municipal securities trade at spreads to Treasuries; the I curve provides the reference yield at the instrument’s effective maturity.
– Pricing and risk: Derivatives, swap valuation, and portfolio risk metrics rely on accurate spot/forward curves.
– Arbitrage detection and relative‑value trading: Traders look for mispricings versus an interpolated benchmark.
References and further reading
– U.S. Department of the Treasury. “Treasury Yield Curve Methodology.” (Treasury provides guidance and primary data used to construct Treasury yield curves.)
– Investopedia. “Interpolated Yield Curve (I Curve).” (Introductory overview and context for interpolation and bootstrapping.)
(Access these sources for official definitions and methodology notes.)
– Produce an Excel template that builds an interpolated par curve and bootstraps a zero curve from on‑the‑run yields; or
– Provide Python code (pandas + scipy or QuantLib) implementing linear and spline interpolation plus bootstrapping; or
– Walk through a worked example with real market quotes for a given date. Which would you prefer?