Derivation of Affine Coefficient Loadings

Felix Geiger

Chapter Appendix D in The Yield Curve and Financial Risk Premia, 2011, pp 279-281 from Springer

Abstract: Abstract The derivation of the difference equations follows the guess-and-verify strategy similar to the method of undetermined coefficients supposed by McCallum (1983). For convenience, the relevant starting equations are $$\begin{array}{rlrlrl} {X}_{t} & = \mu + \phi {X}_{t-1} + \Sigma {\epsilon }_{t} & & \\ {P}_{n,t} & = {E}_{t}^{\mathcal{P} }[{M}_{t+1}{P}_{n-1,t+1}] & & \\ {M}_{t+1} & =\exp (-{i}_{1,t} - 0.5{\lambda }_{t}^{\top }{\lambda }_{ t} - {\lambda }_{t}^{\top }{\epsilon }_{ t+1}) & & \\ {i}_{i,t} & = {\delta }_{0} + {\delta }_{1}{X}_{t} & & \end{array}$$ Duffie and Kan (1996) guess a solution for bond prices as $${P}_{n,t} =\exp ({A}_{n} + {B}_{n}{X}_{t}).$$ For a one-period bond, it can be easily shown that D.1 $${P}_{1,t} = {E}_{t}{M}_{t+1} =\exp (-{i}_{1,t}) =\exp (-{\delta }_{0} - {\delta }_{1}{X}_{t}). D.1 $$ Matching coefficients yields A1 = − δ0 and B1 = − δ1.

Date: 2011
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DOI: 10.1007/978-3-642-21575-9_12

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