시간의 일관성(재무)

금융의 맥락에서 시간 일관성은 서로 모순되는 다른 시점에서의 위험 평가를 하지 않는 특성이다.이 부동산은 투자 A가 미래의 어느 시점에 B보다 위험하다고 판단된다면, A도 이전 어느 시점에 B보다 위험하다고 간주될 것임을 의미한다.

시간의 일관성과 재무상의 리스크

시간 일관성은 동적 위험 측정과 관련된 재무 위험의 특성이다.일관된 부동산의 목적은 포트폴리오(A)가 미래에 포트폴리오(B)보다 더 위험하다면 그 이전 어느 시점에서도 더 위험하다는 조건을 충족하는 위험측정을 분류하는 것이다.이것은 중요한 속성입니다.시간 $t+1$ t + $t+1$ (시간 $t+1$ + 1 ({ $display style$ t+ $1$ 에서 A가 B보다 위험하지만, $시간$ t { $display style$ t $}$ 에서는 $t$ B가 A보다 위험하기 때문입니다.이름에서 알 수 있듯이, 시간 불일치 위험 측정은 재무 위험 관리에서 일관되지 않은 행동을 초래할 수 있습니다.

수학적 정의

동적 리스크 측정 $\left(\rho _{t}\right)_{t=0}^{T}$ δ $\left(\rho _{t}\right)_{t=0}^{T}$ ) $\left(\rho _{t}\right)_{t=0}^{T}$ $\left(\rho _{t}\right)_{t=0}^{T}$ $\left(\rho _{t}\right)_{t=0}^{T}$ T $\left(\rho _{t}\right)_{t=0}^{T}$ (\ $displaystyle \left (\rho$ _ ${t}\right)_{t=0}^{$ $L^{0}({\mathcal {F}}_{T})$ 0 ( F $L^{0}({\mathcal {F}}_{T})$ ) $の$ T $\left(\rho _{t}\right)_{t=0}^{T}$ （ { \ $displaystyle$ L $^$ { $0 }($ { \ $mathcal$ $\forall X,Y\in L^{0}({\mathcal {F}}_{T})$ { $F$ } } $\forall X,Y\in L^{0}({\mathcal {F}}_{T})$ { $}$ ) $x$ X $\forall X,Y\in L^{0}({\mathcal {F}}_{T})$ , Y $\forall X,Y\in L^{0}({\mathcal {F}}_{T})$ L $\forall X,Y\in L^{0}({\mathcal {F}}_{T})$ 0 ( F T ) \ displaystyle \ $forall$ X , $Y$ \ $in$ L^ 0 $（$ { \ \ $mathcal$ { F $\forall X,Y\in L^{0}({\mathcal {F}}_{T})$ } $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)\geq \rho _{t+1}(Y)$ $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)\geq \rho _{t+1}(Y)$ $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)\geq \rho _{t+1}(Y)$ _ { T $\forall X,Y\in L^{0}({\mathcal {F}}_{T})$ } } } } } $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)\geq \rho _{t+1}(Y)$ } } } } ∈ ∈ consist consist consist consist T $\forall X,Y\in L^{0}({\mathcal {F}}_{T})$ ∈ ∈ ∈ ∈ ∈ ∈ ） ∈ ∈ ∈ ∈ ∈ consist ( $Y)}$ 은 $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)\geq \rho _{t+1}(Y)$ $($ 는) $\rho _{t}(X)\geq \rho _{t}(Y)$ ( $\rho _{t}(X)\geq \rho _{t}(Y)$ ( $)\displaystyle \rho$ _ ${t}(X)\geq \rho$ _ ${t}(Y)}$ 를 의미합니다 $\rho _{t}(X)\geq \rho _{t}(Y)$ ^[1]

동등한 정의

평등: $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ t $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ { $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ , $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ , $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ . . $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ } : $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ + $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ ( $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ ) $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ t ( $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ ) $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ t t $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ ( $t\in \{0,1,...,T-1\}:\rho _{t+1}(X)=\rho _{t+1}(Y)\Rightarrow \rho _{t}(X)=\rho _{t}(Y)$ ) $= t$ t ( \ $display style$ t \ in \ $0$ , $1$ . ... $T-1$ \ : \ $rho { t$ + $1$ ( X ) $_$ 1 ( X )

재귀적: $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ t $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ { $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ , $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ , $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ . $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ . . $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ , $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ - 1 $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ : $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ ( X ) $=$ $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ t ( - $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ t $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ + $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ ( $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ ) \ $displaystyle t$ \ $in$ \ { 0 , $... , T-1$ \ : \ $rho$ _ t $}$ ( X ) : \ $rho$ _ { $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ t $t\in \{0,1,...,T-1\}:\rho _{t}(X)=\rho _{t}(-\rho _{t+1}(X))$ } ( X ) _ 0 , 0 . 0 . 0 . 0 . 0 )

수용 세트: $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ t $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ { 0 , $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ , $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ . , $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ T - $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ : $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ t , $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ + $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ + $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ t + $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ { $displaystyle$ t \ $in$ \ { $0$ , 1 , $...$ T - 1 \ $t\in \{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}$ } : $A_{t}=A_{t,t+1}+A_$ $A_{t}$ 서 A $A_{t}$ t { $displaystyle A_{t}$ 는 $A_{t}$ $시간$ t { $displaystyle$ t $}$ 수용 $t$ $A_{t,t+1}=A_{t}\cap L^{p}({\mathcal {F}}_{t+1})$ 이며 $A_{t,t+1}=A_{t}\cap L^{p}({\mathcal {F}}_{t+1})$ $A_{t,t+1}=A_{t}\cap L^{p}({\mathcal {F}}_{t+1})$ t , $A_{t,t+1}=A_{t}\cap L^{p}({\mathcal {F}}_{t+1})$ + 1 $=$ $A_{t,t+1}=A_{t}\cap L^{p}({\mathcal {F}}_{t+1})$ t $A_{t,t+1}=A_{t}\cap L^{p}({\mathcal {F}}_{t+1})$ L $A_{t,t+1}=A_{t}\cap L^{p}({\mathcal {F}}_{t+1})$ p ( $F$ t + $A_{t,t+1}=A_{t}\cap L^{p}({\mathcal {F}}_{t+1})$ ) { $displaystyle$ A _ { $t$ 、 $t$ + $1$ } $LCAP$

코사이클 조건(볼록 위험 측정의 경우): $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ tθ $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ { $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ , $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ , $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ . $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ . . $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ : $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ t $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ ( Q ) $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ t $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ , $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ + $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ ( $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ ) + $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ Q [ $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ + 1 $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ ( $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ ) $t\in \{0,1,...,T-1\}:\alpha _{t}(Q)=\alpha _{t,t+1}(Q)+\mathbb {E} ^{Q}[\alpha _{t+1}(Q)\mid {\mathcal {F}}_{t}]$ F t $]$ \ $display style t$ \ $in$ \ { 0 , $1$ , ... $T-1$ \ : \ $alpha _$ t $=$ { $1 } _ alpha$ } $]$ {\ $displaystyle \alpha _$ {t}( $Q)=\operatorname {*}$ { $esssup}_{$ X\in $A_{t}}\mathbb$ { $E}^{Q}[-X\$ mid { $mathcal {F}_{$ $}}}$ 는 $\alpha _{t}(Q)=\operatorname {*} {esssup}_{X\in A_{t}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ 최소 패널티 함수이며, $\operatorname {*} {esssup}$ * $displaystyle A_{t}$ 는 허용됩니다. $splaystyle$ t $}$ $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ + $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ 1 ( $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ ) $=$ * $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ s $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ X $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ A $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ , $t$ + $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ [ - $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ $\alpha _{t,t+1}(Q)=\operatorname {*} {esssup}_{X\in A_{t,t+1}}\mathbb {E} ^{Q}[-X\mid {\mathcal {F}}_{t}]$ ]{ $displaystyle \alpha$ _ ${t , t+1}(Q$ )=\ $operatorname {*}$ _{ $esssup}_{X\in a_t},$

건설

재귀적 속성으로 인해 시간 일관적인 위험 척도를 쉽게 구성할 수 있습니다.이는 시간 경과에 따른 1주기 측도를 구성함으로써 수행됩니다.즉, 다음과 같습니다.

$\rho _{T-1}^{com:=\rho _{T-1$
${\displaystyle\forall t<T-1:\rho_{t}^{com}:=\rho _{t}(-\rho _{t+1}^{com})$ ^[1]

예

리스크에 따른 가치와 리스크에 따른 평균 가치

위험 시 동적 값과 위험 시 동적 평균 값 모두 시간 일관적인 위험 척도가 아니다.

시간 일관적인 대안

시간 t에서 $\alpha _{t}$ $\alpha _{t}$ t {\ $displaystyle \alpha$ _ ${t}$ 의 $\alpha_t$ 동적 평균 위험에 대한 시간 일치 값은 다음과 같이 정의된다.

\displaystyle \rho _{t}(X)=sup {ess}\sup _{Q\in {mathcal {Q}}E^{Q}[-X {mathcal {F}}_{t}}}}

${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ { ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ : ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ [ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ Q ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ F ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ] ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ - ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ E [ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ - 1 ] ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ 1 ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ , . , ${\mathcal {Q}}=\left\{Q\in {\mathcal {M}}_{1}:E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j}\right]\leq \alpha _{j-1}E\left[{\frac {dQ}{dP}}|{\mathcal {F}}_{j-1}\right]\forall j=1,...,T\right\}$ } { $displaystyle$ { Q } $=$ left \ { $Q$ \ $in$ { $mathcal { M$ } } : { $1$ } : $E\left [ { \ frac { dQ } { dP } \ mathcal { F } _ { j } \ right ]\leq \ alpha _ { j - 1 } E \ left [ { \ frac { dQ } { dP } { \ mathcal { F } _ { j - 1$ $,T\right$ ^[4]

동적 중첩 가격

동적 중첩 가격은 시간 일관적인 위험 ^[5]척도이다.

동적 엔트로피 리스크

동적 엔트로픽 위험 측정은 위험 회피 매개변수가 ^[5]일정할 경우 시간 일관 위험 측정값이다.

연속 시간

연속적인 시간 동안 다음과 같은 방법으로 일관성 있는 위험 측정을 제공할 수 있습니다.

\displaystyle \rho _{g}(X):=\mathbb {E} ^{g}[-X]}

$g$ 의 준선형 선택 ${\$ $displaystyle$ g $}$ 의 $g$ 경우, $\mathbb {E} ^{g}$ 서 E $g$ {\ $displaystyle \mathbb$ {E} $\mathbb {E} ^{g}$ ^{g}는 $\mathbb {E} ^{g}$ g-기대를 나타냅니다. $함수$ g {\ $displaystyle$ g $g$ }가 볼록한 경우 해당 위험 척도는 ^[6]볼록한 것입니다.

레퍼런스

^ ^a ^b Cheridito, Patrick; Stadje, Mitja (October 2008). "Time-inconsistency of VaR and time-consistent alternatives" (PDF). Archived from the original (PDF) on October 19, 2012. Retrieved November 29, 2010. {{cite journal}}:Cite 저널 요구 사항 journal=(도움말)
^ Acciaio, Beatrice; Penner, Irina (February 22, 2010). "Dynamic risk measures" (PDF). Archived from the original (PDF) on September 2, 2011. Retrieved July 22, 2010. {{cite journal}}:Cite 저널 요구 사항 journal=(도움말)
^ Föllmer, Hans; Penner, Irina (2006). "Convex risk measures and the dynamics of their penalty functions" (PDF). Statistics and Decisions. 24 (1): 61–96. Retrieved June 17, 2012.^{[영구 데드링크]}
^ Cheridito, Patrick; Kupper, Michael (May 2010). "Composition of time-consistent dynamic monetary risk measures in discrete time" (PDF). International Journal of Theoretical and Applied Finance. Archived from the original (PDF) on July 19, 2011. Retrieved February 4, 2011.
^ ^a ^b Penner, Irina (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability" (PDF). Archived from the original (PDF) on July 19, 2011. Retrieved February 3, 2011. {{cite journal}}:Cite 저널 요구 사항 journal=(도움말)
^ Rosazza Gianin, E. (2006). "Risk measures via g-expectations". Insurance: Mathematics and Economics. 39: 19–65. doi:10.1016/j.insmatheco.2006.01.002.

[composition-1] Cheridito, Patrick; Stadje, Mitja (October 2008). "Time-inconsistency of VaR and time-consistent alternatives" (PDF). Archived from the original (PDF) on October 19, 2012. Retrieved November 29, 2010. {{cite journal}}:Cite 저널 요구 사항 journal=(도움말)

[2] Acciaio, Beatrice; Penner, Irina (February 22, 2010). "Dynamic risk measures" (PDF). Archived from the original (PDF) on September 2, 2011. Retrieved July 22, 2010. {{cite journal}}:Cite 저널 요구 사항 journal=(도움말)

[3] Föllmer, Hans; Penner, Irina (2006). "Convex risk measures and the dynamics of their penalty functions" (PDF). Statistics and Decisions. 24 (1): 61–96. Retrieved June 17, 2012.^{[영구 데드링크]}

[4] Cheridito, Patrick; Kupper, Michael (May 2010). "Composition of time-consistent dynamic monetary risk measures in discrete time" (PDF). International Journal of Theoretical and Applied Finance. Archived from the original (PDF) on July 19, 2011. Retrieved February 4, 2011.

[penner_thesis-5] Penner, Irina (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability" (PDF). Archived from the original (PDF) on July 19, 2011. Retrieved February 3, 2011. {{cite journal}}:Cite 저널 요구 사항 journal=(도움말)

[6] Rosazza Gianin, E. (2006). "Risk measures via g-expectations". Insurance: Mathematics and Economics. 39: 19–65. doi:10.1016/j.insmatheco.2006.01.002.

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