How does linear interpolation work on a bootstrapped yield curve, and what artifacts does it introduce?

Linear interpolation is the simplest method for filling gaps between observed yield curve nodes. It connects adjacent known discount factors (or zero rates) with straight-line segments, producing a continuous but piecewise-linear curve.\n\nMechanics of Linear Interpolation:\n\nGiven two observed zero rates r(t_1) and r(t_2) at maturities t_1 and t_2, the interpolated rate at any intermediate maturity t is:\n\nr(t) = r(t_1) + [(t - t_1) / (t_2 - t_1)] x [r(t_2) - r(t_1)]\n\n`mermaid\ngraph TD\n A[\"Known Swap Quotes\"] --> B[\"Bootstrap Zero Rates
at quoted tenors\"]\n B --> C[\"Apply Linear Interpolation
between adjacent nodes\"]\n C --> D[\"Continuous Zero Curve\"]\n D --> E[\"Derive Forward Rates
f(t) = d[r(t) x t] / dt\"]\n E --> F[\"Forward Curve Shows
Jagged Step Pattern\"]\n F --> G[\"Problem: Discontinuous
instantaneous forwards\"]\n`\n\nWorked Example:\nBriarwood Capital is constructing a USD swap curve. They observe:\n\n| Tenor | Zero Rate |\n|---|---|\n| 3Y | 4.15% |\n| 5Y | 4.52% |\n\nTo find the 4Y zero rate:\n\nr(4) = 4.15% + [(4 - 3) / (5 - 3)] x (4.52% - 4.15%)\nr(4) = 4.15% + 0.5 x 0.37% = 4.335%\n\nThe 3.5Y rate would be:\nr(3.5) = 4.15% + [(3.5 - 3) / (5 - 3)] x 0.37% = 4.15% + 0.25 x 0.37% = 4.2425%\n\nThe Forward Rate Problem:\n\nWhile linear interpolation on zero rates produces a smooth-looking zero curve, the implied forward rates are piecewise constant (flat between nodes) with sudden jumps at each node. The instantaneous forward rate between t_1 and t_2 is constant:\n\nf(t) = r(t_1) + [r(t_2) - r(t_1)] x (2t - t_1) / (t_2 - t_1)\n\nBut at each node, the slope changes abruptly, creating a sawtooth pattern in forward rates. This is problematic because:\n\n- Forward rates should reflect gradual changes in economic expectations\n- Hedging ratios derived from jagged forward curves are unstable\n- Pricing path-dependent products off these curves introduces spurious features\n\nWhen Linear Interpolation Is Acceptable:\n- Quick indicative pricing where forward rate smoothness is irrelevant\n- Short-dated curves with closely spaced nodes (1M, 2M, 3M, 6M)\n- Preliminary bootstrapping before applying a smoothing overlay\n- Risk systems that only need spot rates at specific tenors\n\nWhen to Avoid It:\n- Pricing forward-starting swaps, swaptions, or CMS products\n- Any product sensitive to the shape of the forward curve\n- Curves with wide gaps between nodes (the 5Y-10Y segment is particularly problematic)\n\nLinear interpolation remains a useful starting point, but production-quality curves require smoother methods like cubic splines or parametric models. Explore curve construction methods in our FRM course.