A Principal Component Based Affine Term Structure Model

We present an essentially affine model with pricipal components as state variables. We show that, once no-arbitrage is imposed, this choice of state variables imposes some unexpected constraints on the reversionspeed matrix, whose N2 elements can be uniquely specified by its N eigenvalues. The requirement that some of its elements should be negative gives rise to a potentially complex dynamics, whose implications we discuss at length. We show how the free parameters of the model can be determined by combining cross-sectional information on bond prices with time-series information about excess returns and by enforcing a 'smoothness' requirement. The calibration in the P and Q measures does not require heavy numerical search, and can be carried out almost fully with elementary matrix operations. Once calibrated, the model recovers exactly the (discrete) yield cuirve shape, the yield covariance matrix, its eigenvalues and eigenvectors. The ability to recover yield volatilities well makes it useful for the estimation of convexity and term premia. The model also recovers well quantities to which it has not been calibrated, and offers an estimation of the term premia for yields of different maturities which we discuss in the last section.