How does cubic spline interpolation smooth the forward rate curve, and what are the potential drawbacks of spline-based methods?

Cubic spline interpolation fits a series of third-degree polynomial segments between each pair of adjacent yield curve nodes, subject to smoothness constraints. Unlike linear interpolation, cubic splines produce continuous first and second derivatives, resulting in a much smoother forward rate curve.\n\nHow Cubic Splines Work on Yield Curves:\n\nBetween each pair of knot points (t_i, r_i) and (t_{i+1}, r_{i+1}), the spline fits a cubic polynomial:\n\nS_i(t) = a_i + b_i(t - t_i) + c_i(t - t_i)^2 + d_i(t - t_i)^3\n\nThe constraints that determine the four coefficients per segment are:\n\n1. Interpolation: S_i passes through both endpoints\n2. First derivative continuity: S_i'(t_{i+1}) = S_{i+1}'(t_{i+1})\n3. Second derivative continuity: S_i''(t_{i+1}) = S_{i+1}''(t_{i+1})\n4. Boundary conditions: Natural splines set S''= 0 at endpoints; clamped splines fix the first derivative\n\n`mermaid\ngraph LR\n A[\"Input: Zero rates
at n knot points\"] --> B[\"Build tridiagonal
system of equations\"]\n B --> C[\"Solve for second
derivatives at knots\"]\n C --> D[\"Compute cubic
coefficients a,b,c,d\"]\n D --> E[\"Evaluate at any t
within the range\"]\n E --> F[\"Smooth zero curve +
smooth forward curve\"]\n`\n\nWorked Example:\nHathaway Fixed Income constructs a spline-based EUR curve using knots at 2Y (2.80%), 5Y (3.15%), and 10Y (3.45%). The natural cubic spline produces the following interpolated rates:\n\n| Tenor | Linear Interp | Cubic Spline |\n|---|---|---|\n| 3Y | 2.917% | 2.935% |\n| 4Y | 3.033% | 3.060% |\n| 7Y | 3.270% | 3.295% |\n| 8Y | 3.330% | 3.352% |\n\nThe forward rates derived from the spline are smooth and continuous, whereas linear interpolation produces a step-function forward curve.\n\nPotential Drawbacks:\n\n1. Overshoot and oscillation: Between widely spaced nodes, cubic polynomials can overshoot significantly, producing negative forward rates or implausible humps\n2. Long-end instability: If the 10Y-30Y segment has few constraints, the cubic term can cause wild extrapolation, pushing 25Y or 30Y forwards to unrealistic levels\n3. Global vs. local: Changing one node affects the entire curve (for natural splines), which can produce unexpected sensitivities\n4. Non-economic shapes: The mathematical smoothness does not guarantee economically sensible forward rates\n\nPractical Variations:\n- Monotone-preserving splines: Constrain the spline to avoid oscillation in monotonic segments\n- Tension splines: Add a tension parameter that pulls the curve closer to linear interpolation, reducing overshoot\n- B-splines: Use a basis of localized spline functions for better local control\n\nCubic splines remain the workhorse of production curve engines, but practitioners must inspect forward rate outputs carefully. Explore advanced curve fitting in our FRM materials.