ハンセン–ジャガナサン境界
m t + 1 ∗ = E t [ m t + 1 ] + C o v t ( m t + 1 , R t + 1 ) ( V a r t ( R t + 1 ) ) − 1 ( R t + 1 − E t [ R t + 1 ] ) {\displaystyle m_{t+1}^{*}=E_{t}[m_{t+1}]+\mathrm {Cov} _{t}(m_{t+1},R_{t+1}){\Big (}\mathrm {Var} _{t}(R_{t+1}){\Big )}^{-1}{\Big (}R_{t+1}-E_{t}[R_{t+1}]{\Big )}}
とすれば
V a r t ( m t + 1 ) = V a r t ( m t + 1 ∗ ) + V a r t ( ε t + 1 ) ≥ V a r t ( m t + 1 ∗ ) {\displaystyle \mathrm {Var} _{t}(m_{t+1})=\mathrm {Var} _{t}(m_{t+1}^{*})+\mathrm {Var} _{t}(\epsilon _{t+1})\geq \mathrm {Var} _{t}(m_{t+1}^{*})}
である。さらに C o v t ( m t + 1 , R t + 1 ) = E t [ m t + 1 R t + 1 ] − E t [ m t + 1 ] E t [ R t + 1 ] = 1 − E t [ m t + 1 ] E t [ R t + 1 ] {\displaystyle \mathrm {Cov} _{t}(m_{t+1},R_{t+1})=E_{t}[m_{t+1}R_{t+1}]-E_{t}[m_{t+1}]E_{t}[R_{t+1}]=\mathbf {1} -E_{t}[m_{t+1}]E_{t}[R_{t+1}]} なので結局