ハンセン–ジャガナサン境界 （ハンセン–ジャガナサンきょうかい、英 : Hansen–Jagannathan bound ）とは、金融経済学 とマクロ経済学 において金融資産 の資産価格モデルにおける確率的割引ファクター （英 : stochastic discount factor ）の分散 の下限を決定する理論である。1991年 にラース・ハンセン とラビ・ジャガナサン （英語版 ） [1]
金融資産
i
{\displaystyle i}
t
{\displaystyle t}
p
i
,
t
{\displaystyle p_{i,t}}
p
i
,
t
=
E
t
[
m
t
+
1
(
p
i
,
t
+
1
+
d
i
,
t
+
1
)
]
{\displaystyle p_{i,t}=E_{t}[m_{t+1}(p_{i,t+1}+d_{i,t+1})]}
ただし、
d
i
,
t
+
1
{\displaystyle d_{i,t+1}}
t
+
1
{\displaystyle t+1}
i
{\displaystyle i}
インカム・ゲイン のことで、例えば株式 なら配当 、債券 ならクーポンなど）で、
E
t
{\displaystyle E_{t}}
t
{\displaystyle t}
期待値 である。
m
t
+
1
{\displaystyle m_{t+1}}
t
+
1
{\displaystyle t+1}
確率的割引ファクター である。
ここで、金融市場 に存在する全てのリスクのある金融資産のグロスのトータルリターン
p
i
,
t
+
1
+
d
i
,
t
+
1
p
i
,
t
{\displaystyle {\frac {p_{i,t+1}+d_{i,t+1}}{p_{i,t}}}}
R
t
+
1
{\displaystyle R_{t+1}}
V
a
r
t
(
m
t
+
1
)
≥
(
1
−
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
)
′
(
V
a
r
t
(
R
t
+
1
)
)
−
1
(
1
−
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
)
{\displaystyle \mathrm {Var} _{t}(m_{t+1})\geq {\Big (}\mathbf {1} -E_{t}[m_{t+1}]E_{t}[R_{t+1}]{\Big )}^{\prime }{\Big (}\mathrm {Var} _{t}(R_{t+1}){\Big )}^{-1}{\Big (}\mathbf {1} -E_{t}[m_{t+1}]E_{t}[R_{t+1}]{\Big )}}
ここで、
V
a
r
t
(
m
t
+
1
)
{\displaystyle \mathrm {Var} _{t}(m_{t+1})}
m
t
+
1
{\displaystyle m_{t+1}}
分散 、
V
a
r
t
(
R
t
+
1
)
{\displaystyle \mathrm {Var} _{t}(R_{t+1})}
1
{\displaystyle \mathbf {1} }
′
{\displaystyle \prime }
ハンセン–ジャガナサン境界 と呼ぶ[1] [2]
ここで
E
t
[
m
t
+
1
]
{\displaystyle E_{t}[m_{t+1}]}
利子率 を
R
f
,
t
+
1
{\displaystyle R_{\mathrm {f} ,t+1}}
R
f
,
t
+
1
=
1
E
t
[
m
t
+
1
]
{\displaystyle R_{\mathrm {f} ,t+1}={\frac {1}{E_{t}[m_{t+1}]}}}
であるので、ハンセン–ジャガナサン境界の両辺を
(
E
t
[
m
t
+
1
]
)
2
{\displaystyle {\Big (}E_{t}[m_{t+1}]{\Big )}^{2}}
V
a
r
t
(
m
t
+
1
)
(
E
t
[
m
t
+
1
]
)
2
≥
(
E
t
[
R
t
+
1
−
R
f
,
t
+
1
1
]
)
′
(
V
a
r
t
(
R
t
+
1
)
)
−
1
(
E
t
[
R
t
+
1
−
R
f
,
t
+
1
1
]
)
{\displaystyle {\frac {\mathrm {Var} _{t}(m_{t+1})}{{\Big (}E_{t}[m_{t+1}]{\Big )}^{2}}}\geq {\Big (}E_{t}[R_{t+1}-R_{\mathrm {f} ,t+1}\mathbf {1} ]{\Big )}^{\prime }{\Big (}\mathrm {Var} _{t}(R_{t+1}){\Big )}^{-1}{\Big (}E_{t}[R_{t+1}-R_{\mathrm {f} ,t+1}\mathbf {1} ]{\Big )}}
金融資産の超過リターンベクトルを
r
t
+
1
e
{\displaystyle r_{t+1}^{e}}
r
t
+
1
e
=
R
t
+
1
−
R
f
,
t
+
1
1
{\displaystyle r_{t+1}^{e}=R_{t+1}-R_{\mathrm {f} ,t+1}\mathbf {1} }
V
a
r
t
(
r
t
+
1
e
)
=
V
a
r
t
(
R
t
+
1
)
{\displaystyle \mathrm {Var} _{t}(r_{t+1}^{e})=\mathrm {Var} _{t}(R_{t+1})}
V
a
r
t
(
m
t
+
1
)
(
E
t
[
m
t
+
1
]
)
2
≥
(
E
t
[
r
t
+
1
e
]
)
′
(
V
a
r
t
(
r
t
+
1
e
)
)
−
1
(
E
t
[
r
t
+
1
e
]
)
{\displaystyle {\frac {\mathrm {Var} _{t}(m_{t+1})}{{\Big (}E_{t}[m_{t+1}]{\Big )}^{2}}}\geq {\Big (}E_{t}[r_{t+1}^{e}]{\Big )}^{\prime }{\Big (}\mathrm {Var} _{t}(r_{t+1}^{e}){\Big )}^{-1}{\Big (}E_{t}[r_{t+1}^{e}]{\Big )}}
という表現も可能になる。
またハンセン–ジャガナサン境界は無条件の期待値と分散についても成立する。ここで、
(
E
[
r
t
+
1
e
]
)
′
(
V
a
r
(
r
t
+
1
e
)
)
−
1
(
E
[
r
t
+
1
e
]
)
{\displaystyle {\Big (}E[r_{t+1}^{e}]{\Big )}^{\prime }{\Big (}\mathrm {Var} (r_{t+1}^{e}){\Big )}^{-1}{\Big (}E[r_{t+1}^{e}]{\Big )}}
接点ポートフォリオ のシャープ・レシオ の2乗であり、接点ポートフォリオはシャープ・レシオを最大化するポートフォリオでもあるので、
V
a
r
(
m
t
+
1
)
(
E
[
m
t
+
1
]
)
2
≥
max
p
S
p
2
{\displaystyle {\frac {\mathrm {Var} (m_{t+1})}{{\Big (}E[m_{t+1}]{\Big )}^{2}}}\geq \max _{p}S_{p}^{2}}
とも書ける。ただし、
S
p
{\displaystyle S_{p}}
p
{\displaystyle p}
CAPM などがそれにあたる。
ハンセン–ジャガナサン距離 [ 編集 ] ハンセン–ジャガナサン距離 （英 : Hansen–Jagannathan distance ）とは、確率的割引ファクターの特定化の誤りの程度を表す一つの指標である[3]
m
t
+
1
∗
=
1
′
(
E
t
[
R
t
+
1
R
t
+
1
′
]
)
−
1
R
t
+
1
{\displaystyle m_{t+1}^{*}={\mathbf {1} }^{\prime }{\Big (}E_{t}[R_{t+1}R_{t+1}^{\prime }]{\Big )}^{-1}R_{t+1}}
また、想定しているモデルの確率的割引ファクターをパラメーター
θ
{\displaystyle \theta }
m
t
+
1
θ
{\displaystyle m_{t+1}^{\theta }}
m
^
t
+
1
θ
=
(
E
t
[
m
t
+
1
θ
R
t
+
1
]
)
′
(
E
t
[
R
t
+
1
R
t
+
1
′
]
)
−
1
R
t
+
1
{\displaystyle {\widehat {m}}_{t+1}^{\theta }={\Big (}E_{t}[m_{t+1}^{\theta }R_{t+1}]{\Big )}^{\prime }{\Big (}E_{t}[R_{t+1}R_{t+1}^{\prime }]{\Big )}^{-1}R_{t+1}}
この時、ハンセン–ジャガナサン距離
H
J
D
{\displaystyle HJD}
[4]
H
J
D
=
E
t
[
(
m
t
+
1
∗
−
m
^
t
+
1
θ
)
2
]
=
(
1
−
E
t
[
m
t
+
1
θ
R
t
+
1
]
)
′
(
E
t
[
R
t
+
1
R
t
+
1
′
]
)
−
1
(
1
−
E
t
[
m
t
+
1
θ
R
t
+
1
]
)
{\displaystyle HJD={\sqrt {E_{t}[(m_{t+1}^{*}-{\widehat {m}}_{t+1}^{\theta })^{2}]}}={\sqrt {{\Big (}\mathbf {1} -E_{t}[m_{t+1}^{\theta }R_{t+1}]{\Big )}^{\prime }{\Big (}E_{t}[R_{t+1}R_{t+1}^{\prime }]{\Big )}^{-1}{\Big (}\mathbf {1} -E_{t}[m_{t+1}^{\theta }R_{t+1}]{\Big )}}}}
もし、あるパラメーター
θ
0
{\displaystyle \theta _{0}}
1
=
E
t
[
m
t
+
1
θ
0
R
t
+
1
]
{\displaystyle \mathbf {1} =E_{t}[m_{t+1}^{\theta _{0}}R_{t+1}]}
m
t
+
1
θ
0
{\displaystyle m_{t+1}^{\theta _{0}}}
m
t
+
1
∗
−
m
^
t
+
1
θ
0
=
0
{\displaystyle m_{t+1}^{*}-{\widehat {m}}_{t+1}^{\theta _{0}}=0}
H
J
D
=
0
{\displaystyle HJD=0}
ハンセン–ジャガナサン距離は
H
J
D
=
min
m
t
+
1
E
t
[
(
m
t
+
1
θ
−
m
t
+
1
)
2
]
{\displaystyle HJD=\min _{m_{t+1}}{\sqrt {E_{t}[(m_{t+1}^{\theta }-m_{t+1})^{2}]}}}
subject to
E
t
[
m
t
+
1
R
t
+
1
]
=
1
{\displaystyle {\mbox{subject to }}E_{t}[m_{t+1}R_{t+1}]=\mathbf {1} }
という形で表現でき、
E
t
[
m
t
+
1
R
t
+
1
]
=
1
{\displaystyle E_{t}[m_{t+1}R_{t+1}]=\mathbf {1} }
m
t
+
1
{\displaystyle m_{t+1}}
m
t
+
1
θ
{\displaystyle m_{t+1}^{\theta }}
m
t
+
1
θ
{\displaystyle m_{t+1}^{\theta }}
[3]
歴史と応用 [ 編集 ] ハンセン–ジャガナサン境界の原型となる不等式はロバート・シラー によって1982年 にもたらされている[5] 1991年 にハンセン–ジャガナサン境界を提示した[1] 相対的リスク回避度一定(CRRA)型効用関数 を想定した場合の確率的割引ファクターはハンセン–ジャガナサン境界の不等式を満たしていないことを実際のデータを使って実証した。この実証結果はエクイティプレミアムパズル の結果[6] [1]
ハンセン–ジャガナサン境界の導出 [ 編集 ] リスク資産が一つである場合 [ 編集 ] リスク資産が一つであるならば、そのグロスのトータルリターンを
R
t
+
1
{\displaystyle R_{t+1}}
1
=
E
t
[
m
t
+
1
R
t
+
1
]
=
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
+
C
o
v
t
(
m
t
+
1
,
R
t
+
1
)
{\displaystyle 1=E_{t}[m_{t+1}R_{t+1}]=E_{t}\left[m_{t+1}\right]E_{t}\left[R_{t+1}\right]+\mathrm {Cov} _{t}(m_{t+1},R_{t+1})}
=
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
+
V
a
r
t
(
m
t
+
1
)
V
a
r
t
(
R
t
+
1
)
C
o
r
r
t
(
m
t
+
1
,
R
t
+
1
)
{\displaystyle =E_{t}\left[m_{t+1}\right]E_{t}\left[R_{t+1}\right]+{\sqrt {\mathrm {Var} _{t}(m_{t+1})}}{\sqrt {\mathrm {Var} _{t}(R_{t+1})}}\mathrm {Corr} _{t}(m_{t+1},R_{t+1})}
となる。よって相関係数
C
o
r
r
t
(
m
t
+
1
,
R
t
+
1
)
{\displaystyle \mathrm {Corr} _{t}(m_{t+1},R_{t+1})}
V
a
r
t
(
m
t
+
1
)
=
1
(
C
o
r
r
t
(
m
t
+
1
,
R
t
+
1
)
)
2
(
1
−
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
)
2
V
a
r
t
(
R
t
+
1
)
≥
(
1
−
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
)
2
V
a
r
t
(
R
t
+
1
)
{\displaystyle \mathrm {Var} _{t}(m_{t+1})={\frac {1}{{\Big (}\mathrm {Corr} _{t}(m_{t+1},R_{t+1}){\Big )}^{2}}}{\frac {{\Big (}1-E_{t}\left[m_{t+1}\right]E_{t}\left[R_{t+1}\right]{\Big )}^{2}}{\mathrm {Var} _{t}(R_{t+1})}}\geq {\frac {{\Big (}1-E_{t}\left[m_{t+1}\right]E_{t}\left[R_{t+1}\right]{\Big )}^{2}}{\mathrm {Var} _{t}(R_{t+1})}}}
となる。特に両辺を
E
t
[
m
t
+
1
]
{\displaystyle E_{t}\left[m_{t+1}\right]}
平方根 を取れば、
m
t
+
1
{\displaystyle m_{t+1}}
V
a
r
t
(
m
t
+
1
)
E
t
[
m
t
+
1
]
≥
|
E
t
[
R
t
+
1
−
R
f
,
t
+
1
]
V
a
r
t
(
R
t
+
1
−
R
f
,
t
+
1
)
|
{\displaystyle {\frac {\sqrt {\mathrm {Var} _{t}(m_{t+1})}}{E_{t}\left[m_{t+1}\right]}}\geq {\Big |}{\frac {E_{t}\left[R_{t+1}-R_{f,t+1}\right]}{\sqrt {\mathrm {Var} _{t}(R_{t+1}-R_{f,t+1})}}}{\Big |}}
となる。右辺はシャープ・レシオの絶対値である[7]
リスク資産が複数である場合 [ 編集 ] 確率的割引ファクター
m
t
+
1
{\displaystyle m_{t+1}}
R
t
+
1
{\displaystyle R_{t+1}}
直交射影 を考えると、
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
]
)
+
ϵ
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 )}+\epsilon _{t+1}}
が成立する。ただし
E
t
[
ϵ
t
+
1
]
=
0
{\displaystyle E_{t}[\epsilon _{t+1}]=0}
E
t
[
R
t
+
1
ϵ
t
+
1
]
=
0
{\displaystyle E_{t}[R_{t+1}\epsilon _{t+1}]=0}
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}]}
V
a
r
t
(
m
t
+
1
∗
)
=
(
1
−
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
)
′
(
V
a
r
t
(
R
t
+
1
)
)
−
1
(
1
−
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
)
{\displaystyle \mathrm {Var} _{t}(m_{t+1}^{*})={\Big (}\mathbf {1} -E_{t}[m_{t+1}]E_{t}[R_{t+1}]{\Big )}^{\prime }{\Big (}\mathrm {Var} _{t}(R_{t+1}){\Big )}^{-1}{\Big (}\mathbf {1} -E_{t}[m_{t+1}]E_{t}[R_{t+1}]{\Big )}}
となる。よって
V
a
r
t
(
m
t
+
1
)
≥
(
1
−
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
)
′
(
V
a
r
t
(
R
t
+
1
)
)
−
1
(
1
−
E
t
[
m
t
+
1
]
E
t
[
R
t
+
1
]
)
{\displaystyle \mathrm {Var} _{t}(m_{t+1})\geq {\Big (}\mathbf {1} -E_{t}[m_{t+1}]E_{t}[R_{t+1}]{\Big )}^{\prime }{\Big (}\mathrm {Var} _{t}(R_{t+1}){\Big )}^{-1}{\Big (}\mathbf {1} -E_{t}[m_{t+1}]E_{t}[R_{t+1}]{\Big )}}
が得られる[2]
^ a b c d Hansen and Jagannathan & (1991)
^ a b Ferson & (2003) , p.769
^ a b Hansen and Jagannathan & (1997)
^ Ferson & (2003) , p.773^ Shiller & (1982) ^ Mehra and Prescott & (1985) ^ Cochrane & (2005) , p.93
参考文献 [ 編集 ] Cochrane, John H. (2005), Asset Pricing (2 ed.), Princeton, NJ: Princeton University Press, ISBN 9780691121376 Ferson, Wayne E. (2003), “Tests of Multifactor Pricing Models, Volatility Bounds and Portfolio Performance”, in Constantinides, George M.; Harris, Milton; Stulz, René M., Handbook of the Economics of Finance 1 , Elsevier, pp. 743-802, doi :10.1016/S1574-0102(03)01021-5 , ISBN 9780444513632 Hansen, Lars P. ; Jagannathan, Ravi (1991), “Implications of Security Market Data for Models of Dynamic Economies” , Journal of Political Economy 99 (2): 225-262, doi :10.1086/261749 , JSTOR 2937680 , https://jstor.org/stable/2937680 Hansen, Lars P. ; Jagannathan, Ravi (1997), “Assessing Specification Errors in Stochastic Discount Factor Models” , The Journal of Finance 52 (2): 557–590, doi :10.1111/j.1540-6261.1997.tb04813.x , JSTOR 2329490 , https://jstor.org/stable/2329490 Mehra, Rajnish; Prescott, Edward C. (1985), “The Equity Premium: A Puzzle”, Journal of Monetory Economics 15 (2): 145-161, doi :10.1016/0304-3932(85)90061-3 Shiller, Robert J. (1982), “Consumption, Asset Markets and Macroeconomic Fluctuations”, Carnegie-Rochester Conference Series on Public Policy 17 : 203-238, doi :10.1016/0167-2231(82)90046-X 関連項目 [ 編集 ] 
![{\displaystyle p_{i,t}=E_{t}[m_{t+1}(p_{i,t+1}+d_{i,t+1})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a49997ca02e9501b0b569a1e2bdb73ab46292e50)
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![{\displaystyle {\Big (}E_{t}[m_{t+1}]{\Big )}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93ec0240f58fdc6fda85315bb5925610abf5b9fe)
![{\displaystyle {\frac {\mathrm {Var} _{t}(m_{t+1})}{{\Big (}E_{t}[m_{t+1}]{\Big )}^{2}}}\geq {\Big (}E_{t}[R_{t+1}-R_{\mathrm {f} ,t+1}\mathbf {1} ]{\Big )}^{\prime }{\Big (}\mathrm {Var} _{t}(R_{t+1}){\Big )}^{-1}{\Big (}E_{t}[R_{t+1}-R_{\mathrm {f} ,t+1}\mathbf {1} ]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a72f0d65d0231b1d992e54260c501b3e34cb3ce)
![{\displaystyle {\frac {\mathrm {Var} _{t}(m_{t+1})}{{\Big (}E_{t}[m_{t+1}]{\Big )}^{2}}}\geq {\Big (}E_{t}[r_{t+1}^{e}]{\Big )}^{\prime }{\Big (}\mathrm {Var} _{t}(r_{t+1}^{e}){\Big )}^{-1}{\Big (}E_{t}[r_{t+1}^{e}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/044e23c77acdfc53112dad599794d77fa45c3cf1)
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![{\displaystyle {\frac {\mathrm {Var} (m_{t+1})}{{\Big (}E[m_{t+1}]{\Big )}^{2}}}\geq \max _{p}S_{p}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81492738d008bfe8cbe9a63fbc6424ea8b4b07fc)
![{\displaystyle m_{t+1}^{*}={\mathbf {1} }^{\prime }{\Big (}E_{t}[R_{t+1}R_{t+1}^{\prime }]{\Big )}^{-1}R_{t+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39e0e4f8fb94e2d57d2a080beda142e48a5424b2)
![{\displaystyle {\widehat {m}}_{t+1}^{\theta }={\Big (}E_{t}[m_{t+1}^{\theta }R_{t+1}]{\Big )}^{\prime }{\Big (}E_{t}[R_{t+1}R_{t+1}^{\prime }]{\Big )}^{-1}R_{t+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f92463f23075906924482decec7589bc5ceed166)
![{\displaystyle HJD={\sqrt {E_{t}[(m_{t+1}^{*}-{\widehat {m}}_{t+1}^{\theta })^{2}]}}={\sqrt {{\Big (}\mathbf {1} -E_{t}[m_{t+1}^{\theta }R_{t+1}]{\Big )}^{\prime }{\Big (}E_{t}[R_{t+1}R_{t+1}^{\prime }]{\Big )}^{-1}{\Big (}\mathbf {1} -E_{t}[m_{t+1}^{\theta }R_{t+1}]{\Big )}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbda4a56ea0cd2645b85b507b0af9270657aa2b5)
![{\displaystyle \mathbf {1} =E_{t}[m_{t+1}^{\theta _{0}}R_{t+1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7aea1ff3594ab6b99faa112e78e9505365cf64)
![{\displaystyle HJD=\min _{m_{t+1}}{\sqrt {E_{t}[(m_{t+1}^{\theta }-m_{t+1})^{2}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db6c6943f3cb0c7b2897aac777f13a436664348a)
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![{\displaystyle E_{t}[m_{t+1}R_{t+1}]=\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a90a27c2b8a359575dc05fd49cd2132c2591cea)
![{\displaystyle 1=E_{t}[m_{t+1}R_{t+1}]=E_{t}\left[m_{t+1}\right]E_{t}\left[R_{t+1}\right]+\mathrm {Cov} _{t}(m_{t+1},R_{t+1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b101e5b8c22fb031c43141679c940f938932dad4)
![{\displaystyle =E_{t}\left[m_{t+1}\right]E_{t}\left[R_{t+1}\right]+{\sqrt {\mathrm {Var} _{t}(m_{t+1})}}{\sqrt {\mathrm {Var} _{t}(R_{t+1})}}\mathrm {Corr} _{t}(m_{t+1},R_{t+1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b261a24f2727620471c7c6736ed2838cb44cb83a)
![{\displaystyle \mathrm {Var} _{t}(m_{t+1})={\frac {1}{{\Big (}\mathrm {Corr} _{t}(m_{t+1},R_{t+1}){\Big )}^{2}}}{\frac {{\Big (}1-E_{t}\left[m_{t+1}\right]E_{t}\left[R_{t+1}\right]{\Big )}^{2}}{\mathrm {Var} _{t}(R_{t+1})}}\geq {\frac {{\Big (}1-E_{t}\left[m_{t+1}\right]E_{t}\left[R_{t+1}\right]{\Big )}^{2}}{\mathrm {Var} _{t}(R_{t+1})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/455721e2c82ecd8d1a52162c461432f9e4bf5a2f)
![{\displaystyle E_{t}\left[m_{t+1}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb60813feec7412f364141a413f8cdd749f211c)
![{\displaystyle {\frac {\sqrt {\mathrm {Var} _{t}(m_{t+1})}}{E_{t}\left[m_{t+1}\right]}}\geq {\Big |}{\frac {E_{t}\left[R_{t+1}-R_{f,t+1}\right]}{\sqrt {\mathrm {Var} _{t}(R_{t+1}-R_{f,t+1})}}}{\Big |}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2f7cf22a8dd14e8f82185826c95983776cce1c)
![{\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 )}+\epsilon _{t+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f359f25d68ba5a4245fafb2800b105cd7794ab)
![{\displaystyle E_{t}[\epsilon _{t+1}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d3cb88deab8b5bdd1432f26bc9f09ffaba9388f)
![{\displaystyle E_{t}[R_{t+1}\epsilon _{t+1}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f3d0305746b9ce4b9cedb0a1dc6e27d92d38bc8)
![{\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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3248d229a2572240d6a436932b00f8407c3e2c)
![{\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}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/470d204f4f96326de0976a5bb332a99ba2b3aca3)
![{\displaystyle \mathrm {Var} _{t}(m_{t+1}^{*})={\Big (}\mathbf {1} -E_{t}[m_{t+1}]E_{t}[R_{t+1}]{\Big )}^{\prime }{\Big (}\mathrm {Var} _{t}(R_{t+1}){\Big )}^{-1}{\Big (}\mathbf {1} -E_{t}[m_{t+1}]E_{t}[R_{t+1}]{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7b4dcc4541c2aba28b8087cd68a55aa6f7bb7b)