How Hagan–West Monotone-Convex Yield-Curve Algorithms Interpolate Between Market Knots: Discrete Forwards, Instantaneous Forwards, Positivity, Locality and Curve-Failure Tests

Reader question: Once a bank or pricing system has bootstrapped a yield curve at a finite set of maturities, how can it fill the gaps without inventing economically absurd forward rates?

Page role: This is a numerical-methods article about curve interpolation. It does not replace our broader guide to yield-curve bootstrapping, nor the parametric Nelson–Siegel–Svensson approach. Its job is narrower: explain the Hagan–West monotone-convex interpolation mechanism, the mathematical invariants it tries to preserve, and the diagnostics that tell us when the interpolation itself is the weak link.

Educational mathematics only. This is not investment, trading or financial-product advice.

The problem begins after bootstrapping, not before it

A yield curve is not observed at every instant of maturity. Markets give us prices or rates at selected instruments and dates: overnight, one month, three months, six months, one year, two years, five years and so on. A bootstrap can solve discount factors or zero rates at those calibration knots. But a valuation engine may need a discount factor at 4.37 years, a forward rate between 7.1 and 7.35 years, or a sensitivity to a tiny movement at a maturity where no calibration instrument exists.

Interpolation therefore becomes part of the model. The dangerous misconception is to think that interpolation is merely graphical smoothing. In fixed income, the interpolated object determines discounting, forward rates, sensitivities and hedges. Two curves can pass through exactly the same market knots while implying very different forward-rate shapes between them.

The objects: discount factors, zero yields and forward rates

Let P(0,t) be the discount factor from time t to today. Under continuously compounded notation, the zero yield r(t) satisfies P(0,t)=exp(-r(t)t). The instantaneous forward rate is related by f(t)=-d ln P(0,t)/dt, so equivalently r(t)t = ∫[0,t] f(s) ds.

At neighbouring knot times t_i and t_{i+1}, Hagan–West works naturally with the discrete forward rate

f_i^d = [r(t_{i+1})t_{i+1} - r(t_i)t_i] / (t_{i+1}-t_i).

This number has a useful interpretation: it is the average instantaneous forward rate over that interval. The interpolation problem can therefore be reframed. Instead of drawing a smooth curve through zero yields, construct a continuous instantaneous-forward curve whose integral over each interval exactly reproduces the already-known discrete forward.

Why directly splining zero yields can go wrong

A cubic spline through apparently well-behaved zero rates can produce oscillating derivatives. Because the forward rate depends on a derivative of the log discount factor, small visual wiggles in the yield curve can become large and economically strange movements in the forward curve. Hagan and West’s 2006 study was motivated in part by this mismatch between a curve that looks smooth and a curve that behaves sensibly when used for pricing and hedging.

The distinction matters mathematically: continuity of r(t) does not guarantee good behaviour of f(t). A smooth picture is therefore not a sufficient diagnostic.

The Hagan–West construction in five algorithmic stages

• 1. Start from ordered market knots. Validate strictly increasing times and consistent units. Convert zero yields to discrete forwards if necessary.
• 2. Estimate instantaneous forwards at knot boundaries. The method combines neighbouring interval-average forwards to obtain boundary values while attempting to preserve the geometry of the observed curve.
• 3. Apply monotonicity/positivity controls. Where the unconstrained boundary estimate would create an overshoot or an implausible sign reversal, the algorithm modifies it according to region-specific rules.
• 4. Build a local piecewise function inside each interval. The interpolating segment is chosen so that its integral equals the known discrete forward for that interval.
• 5. Integrate the instantaneous-forward curve back to yields or discount factors. This closes the accounting identity: the interpolated curve must still reproduce the knot information from which it started.

The phrase monotone convex does not mean that every possible yield curve is globally monotone or globally convex. It refers to the shape-preserving logic used to interpolate the interval data without the unconstrained overshoots typical of ordinary high-order interpolation.

A worked miniature example

Suppose the continuously compounded zero yields are 4.00% at one year, 4.30% at two years and 4.20% at three years. The one-to-two-year discrete forward is approximately (0.043×2 - 0.040×1)/(2-1)=4.60%. The two-to-three-year discrete forward is approximately (0.042×3 - 0.043×2)/(3-2)=4.00%.

An interpolation engine now has to connect an interval averaging 4.60% to the next interval averaging 4.00%. A naive high-order spline might overshoot above both or dip sharply below both. A Hagan–West construction instead chooses knot forward values and interval shapes designed to make the transition controlled while preserving each interval’s required average.

The important verification is not whether the interpolated line is attractive. It is whether integrating the first segment returns 4.60% as its interval average and integrating the second returns 4.00%. If not, the curve has broken its own inputs.

Inputs and outputs

• Inputs: ordered maturity times; zero yields or discrete forward rates; a choice about positivity constraints and any optional smoothing/amelioration; numerical precision rules.
• Intermediate state: interval-average forwards, knot instantaneous forwards, region classification, piecewise interpolation coefficients.
• Outputs: instantaneous forward rates at arbitrary times; integrated yields; discount factors; and therefore values and sensitivities for instruments that depend on the curve.

What positivity really means

In older positive-rate environments, one attractive property was to avoid turning positive discrete forwards into negative instantaneous forwards by interpolation artefact. But positivity is not a universal law of modern interest-rate markets: negative rates have existed. Therefore the correct invariant is not “forwards must always be positive.” It is “the interpolation must not create an economically unintended sign change merely because of its numerical shape.”

This is an important falsifier. If the input market legitimately contains negative forwards, a hard positivity clamp can itself be wrong. The implementation must know whether positivity is a desired constraint, a historical convention, or an invalid assumption for the current dataset.

Locality: the hidden hedging test

Suppose one five-year calibration quote moves by one basis point. A local interpolation scheme should change mostly the nearby part of the curve. If a tiny five-year quote bump materially reshapes the 20-year curve, the interpolation is spreading risk far away from the information that changed.

That matters because curve sensitivities become hedge instructions. Poor locality can generate artificial exposure in distant buckets. Hagan–West therefore treats locality as a practical criterion, not just a mathematical aesthetic. Optional smoothing can improve visual regularity but may reduce locality by allowing a point to depend on more neighbouring intervals. There is no free lunch: smoothness and local risk response can conflict.

Diagnostics before trusting the curve

• Knot repricing test: reprice every calibration instrument. Interpolation must not destroy the bootstrap fit.
• Integral consistency test: integrate the instantaneous forward over every knot interval and recover the original discrete forward within numerical tolerance.
• Discount-factor sanity: inspect for impossible or unintended jumps, sign errors, or non-finite values.
• Forward-shape test: plot instantaneous forwards, not only zero yields. Look for spikes or oscillations invisible in the zero curve.
• One-knot bump test: perturb one market quote and inspect where the curve and PV01/DV01 move. This diagnoses locality.
• Mesh test: evaluate on a much finer time grid. A curve can look acceptable only because a coarse plot missed a narrow spike.
• Boundary test: inspect behaviour near the first and last knot, where interpolation becomes extrapolation or special endpoint logic.
• Precision test: rerun in higher precision or tighter root tolerances. Material changes indicate numerical fragility.

Failure modes and counterexamples

Failure 1: bad inputs disguised as an interpolation problem. If calibration instruments are stale, inconsistent or mis-specified, no interpolation method can make the curve economically correct. Exact repricing can coexist with bad market data.

Failure 2: extreme alternating forwards. A violently zig-zagging set of discrete forwards may force any shape-preserving interpolator into sharp local behaviour. The method can reduce overshoot but cannot manufacture information that is not present.

Failure 3: inappropriate positivity assumptions. A positivity collar is a modelling choice, not a timeless theorem.

Failure 4: over-smoothing. Smoothing may hide genuine local information and distribute a quote’s sensitivity across too much of the curve.

Failure 5: extrapolation mistaken for interpolation. Hagan–West controls behaviour between known knots. Beyond the final liquid point, a different extrapolation assumption takes ownership. Our Smith–Wilson article shows why the long end is a separate problem.

Alternatives and when they may be better

• Piecewise-constant forwards: simple, transparent and local, but discontinuous.
• Linear interpolation in zero rates or log discount factors: easy to audit; may be sufficient when smooth forward derivatives are not required.
• Cubic splines: very smooth, but can overshoot and create unstable forward shapes.
• Nelson–Siegel–Svensson: compact parametric representation, useful for sparse markets and macro interpretation, but not an exact local interpolator.
• Smith–Wilson: specifically useful where extrapolation toward a long-run forward assumption is the job.

What would falsify the claim that this interpolation is suitable?

A useful algorithm must survive attempted disproof. For a particular curve, evidence against Hagan–West suitability would include: persistent repricing error, unintended sign reversals, unstable sensitivities under tiny input bumps, large distant-curve movement from a local quote change, or materially different values under an equally defensible interpolation method. If those effects are large relative to the financial question being answered, interpolation uncertainty is no longer negligible.

Connections to the surrounding mathematics

This topic joins several mathematical ideas already developed in this finance-algorithms lane. The bootstrap supplies the knots. Yield-curve PCA asks how observed curves move statistically. Kalman-filter curve models ask how latent factors evolve through time. HJM imposes dynamic no-arbitrage restrictions on forward-rate evolution. Hagan–West occupies a different layer: given today’s calibrated knot information, construct the between-knot numerical object responsibly.

Verification checklist

• Can every input time and quote be traced to a defined convention?
• Do calibrated instruments reprice?
• Do interval integrals recover the discrete forwards?
• Are discount factors and forwards finite across a dense grid?
• Are any sign restrictions economically intended?
• Does a one-knot bump remain reasonably local?
• Does the answer remain stable against at least one alternative interpolation?
• Is extrapolation handled by an explicit separate rule?

Research anchors

• Patrick S. Hagan and Graeme West, “Interpolation Methods for Curve Construction,” Applied Mathematical Finance 13(2), 2006, pp. 89–129, DOI 10.1080/13504860500396032: Taylor & Francis.
• Google’s TensorFlow Quant Finance implementation and documentation: Hagan–West monotone-convex yield interpolation.
• Society of Actuaries Research Institute, Calibrating Interest Rate Models, including an explicit example where cubic-spline yield interpolation produced an oscillating forward curve.

Final idea

The best interpolation algorithm is not the one that draws the prettiest curve. It is the one whose mathematical consequences remain visible: it reproduces the known market information, behaves sensibly between knots, keeps risk where the information lives, survives perturbation tests, and makes its assumptions easy to challenge.