Principal Components and the Case for Multi-Factor Models

A fixed income trader running a 2s-5s-10s butterfly position is exposed to a very specific risk: the five-year rate can move independently of the endpoints. A one-factor short-rate model says this cannot happen — all rates move in perfect lockstep, driven by a single source of randomness. This is not just a mathematical inconvenience; it means the model produces wrong prices for any instrument whose payoff depends on the relative movement of rates at different tenors. Principal component analysis (PCA) of historical yield changes tells us exactly how wrong: one factor explains 85–90% of rate variance, leaving a slope component (~8%) and a curvature component (~4%) unmodelled. Two or three factors explain virtually all the action. This note makes the empirical case for multi-factor models, shows how PCA structures guide the choice of factor dynamics, introduces the two-factor Vasicek (G2) model, and identifies the resulting market incompleteness — the structural challenge that distinguishes multi-factor rate modelling from equity and FX.