# RDP 2018-02: Affine Endeavour: Estimating a Joint Model of the Nominal and Real Term Structures of Interest Rates in Australia Equation (A6)

$P t n = E t Q ( exp( − r t ) P t+1 n−1 ) = E t ( exp( − r t − 1 2 λ t ′ λ t − λ t ′ ε t+1 + A n−1 + B n−1 ′ X t+1 ) ) =exp( − r t − 1 2 λ t ′ λ t + A n−1 )× E t ( exp( − λ t ′ ε t+1 + B n−1 ′ ( μ+θ X t +∑ ε t+1 ) ) ) =exp( − r t − 1 2 λ t ′ λ t + A n−1 + B n−1 ′ ( μ+θ X t ) )× E t ( exp( ( − λ t ′ + B n−1 ′ ∑ ) ε t+1 ) ) =exp( − r t − 1 2 λ t ′ λ t + A n−1 + B n−1 ′ ( μ+θ X t )+ 1 2 ( − λ t ′ + B n−1 ′ ∑ ) ( − λ t ′ + B n−1 ′ ∑ ) ′ ) =exp( − ρ 0 + A n−1 + B n−1 ′ ( μ−∑ λ 0 )+ 1 2 B n−1 ′ ∑ ∑ ′ B n−1 +( − ρ 1 ′ + B n−1 ′ θ− B n−1 ′ ∑ λ 1 ) X t ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGqbWaa0baaSqaaiaadshaaeaacaWGUbaaaOGaeyypa0Zefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFecFrdaqhaaWcbaGaamiDaaqaaiaabgfaaaGcdaqadaqaaiaabwgacaqG4bGaaeiCamaabmaabaGaeyOeI0IaamOCamaaBaaaleaacaWG0baabeaaaOGaayjkaiaawMcaaiaadcfadaqhaaWcbaGaamiDaiabgUcaRiaaigdaaeaacaWGUbGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaaaqaaiabg2da9iab=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@9689@$