Why Forward Curve Interpolation Is More Stable Than Par Rates
Summary
The document explains why yield-curve construction often interpolates discount or forward rates rather than observed par swap rates. Its central point is that forward rates are more differentiated quantities: par rates aggregate information across periods, while forward rates describe the local term structure. Integration smooths a curve, so smooth forwards tend to produce smooth par rates.
The reverse inference is less stable. A curve that looks smooth after interpolation in par-rate space can imply unexpected variation in forward rates. That matters because forward rates are used in pricing and curve construction. The explanation is conceptual rather than a formal arbitrage proof: it does not provide a specific interpolation formula, data example, or quantitative comparison, and it does not claim that every par-rate interpolation method fails. It offers stability of the implied forward curve as the practical reason to work in a more differentiated space.
Key ideas
- Forward rates are more differentiated term-structure quantities than par swap rates.
- Par rates aggregate forward-rate information across periods.
- Integration smooths a forward curve, so smooth forwards generally imply smooth par rates.
- Interpolating par rates directly can produce unstable or unexpected implied forward rates.
- The explanation is conceptual and does not compare specific interpolation formulas or market data.
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Full text
# Why not fit a par rate curve directly on par rates? # Why not fit a par rate curve directly on par rates? I often see that, in yield-curve construction, practitioners build a forward/spot/discount curve and then price instruments by interpolating within that curve. My question is: why can’t we just take the par swap rates we observe at different maturities and interpolate those directly (assuming we only have swap quotes)? I understand that in practice one might use multiple instruments (FRAs, swaps, futures, deposit rates, etc.), which don’t all have par rates, so you can’t just fit a single curve to par swap rates alone. But if our market data were solely swaps, what’s the fundamental reason that we still prefer to construct a discount or forward curve instead of simply interpolating the par swap rates themselves? I’ve heard the argument that par swap rates represent specific “prices” for each swap, while a yield/discount curve represents the “state variables” of the term structure. Is there a formal argument—perhaps related to consistency, arbitrage-free pricing, or more accurate calibration—that explains why yield-curve-based methods are used rather than direct par-rate interpolation? Thank you for your help! ## Answer by dm63 (score 11, accepted) https://quant.stackexchange.com/a/82090 Fundamentally you always want to interpolate in the most differentiated space. For example forward rates are the differential of par rates, and par rates are the integral of forward rates. This is because integration is a smoothing action , so a smooth forward rate curve guarantees a smooth par curve, but the other way round is not as stable. Your choice of interpolation method in par space can cause unexpected instability in the forward rate curve.
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