스미스-윌슨 방법

Smith-Wilson 방법은 선도 비율을 추정하는 방법입니다.EIOPA는 금리를 추정할 것을 권장합니다.그것은 A에 의해 2000년에 소개되었습니다.스미스와 T.베이컨 & 우드로의 윌슨입니다.

공식화

UFR을 궁극적인 전진 속도로 $u_{i}$ $({$ 를 i'th 만기로 설정합니다.그런 $P(t)$ P $)$ {\ $displaystyle$ P $(t)}$ 는 $P(t)$ 시간 t에서 제로 쿠폰 채권의 가격을 정의합니다.

$P(t)=e^{-UFR\cdot t}+\sum _{j=1}^{N}\xi_{j}\cdot W(t,u_{j})$

$W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ 서 W $=$ - $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ R $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ ( $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ + $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ ) $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ ( $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ ( $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ , $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ ) - $.$ - $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ max ( $,$ ( $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ ⋅ $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ min ( $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ , $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ ) - e - $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ ⋅ $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ ( $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ t, $uj$ ) $W(t,u_{j})=e^{-UFR\cdot (t+u_{j})}\cdot (\alpha \cdot \min(t,u_{j})-0.5e^{-\alpha \cdot \max(t,u_{j})}\cdot (e^{\alpha \cdot \min(t,u_{j})}-e^{-\alpha \cdot \min(t,u_{j})}))$ \ $displaystyled$ W $(t,u_e$ = e FR $\cdot (t)$ \ $cdt_t_t_t_t_t_t_cd_t_cd_t_e\cd_t$ $u_{j}}-e^{-\alpha \cdot \min(t,u_{j}})}}$

대칭적인 W $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ 은 W $=$ ( $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ ( $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ ) $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ 1 $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ , $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ . $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ , $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ : $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ = $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ , $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ . , $W=(W(u_{i},u_{j}))_{i=1,...,N:j=1,...,N}$ {\ $displaystyle$ W = $(W(u_{$ i, $u_{j}))_{i=$ 1,... $,N:j=1,...$ $,N}}$

$p=(P(u_{1}),...,P(u_{N}))^{T}$ p $=$ ( $p=(P(u_{1}),...,P(u_{N}))^{T}$ ( $p=(P(u_{1}),...,P(u_{N}))^{T}$ 1 $p=(P(u_{1}),...,P(u_{N}))^{T}$ ) $p=(P(u_{1}),...,P(u_{N}))^{T}$ , $p=(P(u_{1}),...,P(u_{N}))^{T}$ . $p=(P(u_{1}),...,P(u_{N}))^{T}$ , $p=(P(u_{1}),...,P(u_{N}))^{T}$ P ( $p=(P(u_{1}),...,P(u_{N}))^{T}$ ) ) $p=(P(u_{1}),...,P(u_{N}))^{T}$ {\ $displaystyle$ p=( $P(u_{1}),$ ..., P $(u_{N})^{$ $T$ $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ $=$ ( $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ 1 $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ , $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ . . $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ , $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ - $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ F ⋅ $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ ) $\mu =(e^{-UFR\cdot u_{1}},...,e^{-UFR\cdot u_{N}})^{T}$ ${\$ = ( $e^{-UFR\cdotu_{1}},$ ..., $e^{-UFR\cdotu_{N}}})^{$ $T$ $=$ - $\xi =W^{-1}(p-\mu )$ ( $\xi =W^{-1}(p-\mu )$ - $\xi =W^{-1}(p-\mu )$ ) $\xi =W^{-1}(p-\mu )$ \ $displaystyle \xi$ = $W^{-1}(p-\mu$ $\xi =W^{-1}(p-\mu )$

레퍼런스

노르웨이 금융감독청 Smith-Wilson Method에 관한 기술노트(2010년 7월 1일)
라게로스, 안드레아스 & 린드홀름, 마티아스(2016).Smith-Wilson 방법에 대한 문제.보험:수학과 경제학. 71. 10. 10. 1016/j. insmatheco. 2016. 08. 009
스미스, A.와 윌슨, T.(2000).장기 제약 조건이 있는 수율 곡선 적합.조사 보고서, 베이컨 & 우드로우.
EIOPA의 무위험 금리 기간 구조를 도출하는 방법론의 기술 문서화

Search

스미스-윌슨 방법

네임스페이스

더

공식화

레퍼런스