KTB Convexity Formula

핵심 요약

국고채(KTB) 가격 공식에서 출발해 볼록성(Convexity) 식을 직접 미분으로 유도하는 글이다. Bond Price, Duration, Convexity 글에서 명기한 가격 공식을 치환 변수로 다시 쓰고, 두 가지 미분 규칙과 네 가지 가정을 둔 뒤 1차 미분을 정리해 최종 볼록성 식을 얻는 순서로 전개된다. 본문 수식은 2022년 12월 25일 워드프레스 데이터베이스 일부 망실 뒤 Wayback Machine으로 복구한 것이다. 본인은 볼록성 식을 그대로 가져다 쓰기보다 이 유도 순서를 한 번 따라가 보는 편이 듀레이션 근사가 어디서 깨지는지 이해하는 데 도움이 된다고 본다.

• 출발점은 Bond Price, Duration, Convexity 글에 명기한 KTB 가격 공식
• 유도 순서는 변수 치환, 미분 규칙 2개·가정 4개 적용, 1차 미분 정리, 최종 볼록성 식 도출
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먼저 대한민국 국고채인 KTB의 가격 구하는 공식은 Bond Price, Duration, Convexity 글에서 명기한 아래 공식 \( (가) \) ¹와 같으며 이를 토대로 \( ; Convexity=displaystyle frac { d^2P }{ dy^2 } cdot displaystyle frac { 1 }{ P } ; \) 식을 도출해보고자 한다. 먼저

\( P=frac { displaystyle 1 }{ left ( displaystyle 1+ frac { displaystyle d_1 }{ displaystyle t_1 } times frac { displaystyle y }{ displaystyle f } right ) } times left [ displaystyle sum_{ i=0 }^{ n-1 } frac { frac{ displaystyle CF_i }{ displaystyle f } }{ left( 1 + frac { displaystyle y }{ displaystyle f } right )^{ i }} + frac { displaystyle A }{ left( 1 + frac { displaystyle y }{ displaystyle f } right)^{ n-1 }} right ] quad quad (가) \)

 

\( (가) \)에서 \( frac { displaystyle d_1 }{ displaystyle t_1 } = a \), \( frac { displaystyle 1 }{ displaystyle f } = b \), \( frac { displaystyle CF_i }{ displaystyle f } = c \) 라고 할 경우 아래와 같이 쓸 수 있고

\( P=frac { displaystyle 1 }{ left ( displaystyle 1+ displaystyle aby right ) } times left [ displaystyle sum_{ i=0 }^{ n-1 } frac { displaystyle c }{ left ( 1 + displaystyle by right )^{ i }} + frac { displaystyle A }{ left ( 1 + by right )^{ n-1 }} right ] \)

 

이를 다시 표현하면, 아래와 같다.

\( P=left ( displaystyle 1+ displaystyle aby right )^{ -1 } times left [ displaystyle sum_{ i=0 }^{ n-1 } displaystyle c, left ( 1 + displaystyle by right )^{ -i } + displaystyle A, left ( 1 + by right )^{ -n+1 } right ] qquad qquad (1) \)

 

상기 \( (1); \)에 대해 아래의 2 가지 미분 규칙을 염두에 두고

\( begin{eqnarray}
① ; y &=& left { f left ( x right ) right }^n &&&&&&& y’ &=& n ,cdot left { f left ( x right ) right }^{ n-1 } , cdot f’ left ( x right ) \
② ; y &=& left { f left ( x right ) cdot g left ( x right ) right } &&&&&&& y’ &=& f’ left ( x right ) cdot g left ( x right ) + f left ( x right ) cdot g’ left ( x right ) \ end{eqnarray} \)

 

아래와 같이 4 가지를 가정하면

\( begin{eqnarray}
① ; h left ( y right ) &=& left ( 1+ aby right )^{ -1 } \ && \ h’ left( y right ) &=& -a cdot b cdot left ( 1+ aby right )^{ -2 } &=& -a cdot b cdot left { h left ( y right ) right }^{ 2 } \
\\
② ; n left ( y right ) &=& left ( 1+ by right )^{ -1 } \
&& \
n’ left( y right ) &=& – b cdot left ( 1+ by right )^{ -2 } &=& – b cdot left { n left ( y right ) right }^{ 2 } \
\\
③ ; k left ( y right ) &=& sum_{ i=0 }^{ n-1 } left ( 1 + by right )^{ -i } \
&& \
k’ left ( y right ) &=& – b cdot i cdot sum_{ i=0 }^{ n-1 } left ( 1 + by right )^{ -i-1 } &=& – b cdot i cdot left ( 1 + by right )^{ -1 } cdot k left ( y right ) \
& & &=& – b cdot i cdot n left ( y right ) cdot k left ( y right ) \
\\
④ ; m left ( y right ) &=& left ( 1 + by right )^{ -n+1 } \
&& \
m’ left ( y right ) &=& – b cdot left ( n-1 right ) cdot left ( 1 + by right )^{ -n } &=& – b cdot left ( n-1 right ) cdot left ( 1 + by right )^{ -1 } cdot m left ( y right ) \
& & &=& – b cdot left ( n-1 right ) cdot n left ( y right ) cdot m left ( y right ) \
end{eqnarray} \)

 

\( (1); \)은 아래와 같다.

\( P = h left ( y right ) cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] qquad qquad qquad qquad qquad qquad qquad (2) \)

 

\( (2); \)에 대해 1차 미분을 구하면

\( begin{eqnarray}
P &=& h left ( y right ) cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
\
frac { dP }{ dy } &=& h’ left ( y right ) cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] + h left ( y right ) cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ]’ \
&=& -a cdot b cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
& & + h left ( y right ) cdot left [ c cdot k’ left ( y right ) + A cdot m’ left ( y right ) right ] \
\
&=& – a cdot b cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
& & + h left ( y right ) cdot left [ c cdot -b cdot i cdot left ( 1 + by right )^{ -1 } cdot k left ( y right ) right . \
& & quad quad quad + left . A cdot – b cdot left ( n-1 right ) cdot left ( 1 + by right )^{ -1 } cdot m left ( y right ) right ] \
\
&=& – a cdot b cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
& & – b cdot left ( 1 + by right )^{ -1 } cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
\
&=& – a cdot b cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
& & – b cdot n left ( y right ) cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
end{eqnarray} \)

 

와 같다. 여기서

\( X left ( y right ) = – a cdot b cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
Y left ( y right ) = -b cdot n left ( y right ) cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \)

 

라고 가정하면

\( begin{eqnarray}
displaystyle frac { dP }{ dy } &=& X left ( y right ) + Y left ( y right ) \
frac { d^{ 2 } P }{ dy^{ 2 } } &=& X’ left ( y right ) + Y’ left ( y right ) \
X’ left ( y right ) &=& -2 cdot a cdot b cdot h left ( y right) cdot h’ left ( y right) cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
& & – a cdot b cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot k’ left ( y right ) + A cdot m’ left ( y right ) right ] \
\
&=& 2 cdot a^{ 2 } cdot b^{ 2 } cdot left { h left ( y right ) right }^{ 3 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
& & – ab cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot -bi cdot n left ( y right ) cdot k left ( y right ) right . \
& & quad quad quad quad + left . A cdot – b, left ( n-1 right ) cdot n left ( y right ) cdot m left ( y right ) right ] \
\
&=& 2 cdot a^{ 2 } cdot b^{ 2 } cdot left { h left ( y right ) right }^{ 3 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
& & + a cdot b^{ 2 } cdot n left ( y right ) cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
\
Y’ left ( y right ) &=& – b cdot n’ left ( y right ) cdot h cdot left ( y right ) left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & – b cdot n left ( y right ) cdot h’ left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & – b cdot n left ( y right ) cdot h left ( y right ) cdot left [ c cdot i cdot k’ left ( y right ) + A cdot left ( n-1 right ) cdot m’ left ( y right ) right ] \
\
&=& b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & + a cdot b^{ 2 } cdot n left ( y right ) cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & – b cdot n left ( y right ) cdot h left ( y right ) cdot left [ c cdot i cdot -bi cdot n left ( y right ) cdot k left ( y right ) right . \
& & quad quad quad quad + left . A cdot left ( n-1 right ) cdot – b, left ( n-1 right ) cdot n left ( y right ) cdot m left ( y right ) right ] \
\
&=& b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & + a cdot b^{ 2 } cdot n left ( y right ) cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & + b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i^{ 2 } cdot k left ( y right ) + A cdot left ( n-1 right )^{ 2 } cdot m left ( y right ) right ] \
end{eqnarray} \)

 

와 같이 표현이 된다. 이를 정리하면

\( begin{eqnarray}
displaystyle frac { d^{ 2 } P }{ dy^{ 2 } }
&=& Bigg ( 2 cdot a^{ 2 } cdot b^{ 2 } cdot left { h left ( y right ) right }^{ 3 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
& & + a cdot b^{ 2 } cdot n left ( y right ) cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] Bigg ) \

& + \

& & Bigg ( b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & + a cdot b^{ 2 } cdot n left ( y right ) cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & + b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i^{ 2 } cdot k left ( y right ) + A cdot left ( n-1 right )^{ 2 } cdot m left ( y right ) right ] Bigg ) \
\\
displaystyle frac { d^{ 2 } P }{ dy^{ 2 } }
&=& 2 cdot a^{ 2 } cdot b^{ 2 } cdot left { h left ( y right ) right }^{ 3 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] \
& & + 2 cdot a cdot b^{ 2 } cdot n left ( y right ) cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & + b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] \
& & + b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i^{ 2 } cdot k left ( y right ) + A cdot left ( n-1 right )^{ 2 } cdot m left ( y right ) right ] \
end{eqnarray} \)

와 같다.

이를 토대로 \( Convexity ; \)는

\( begin{eqnarray}
Convexity
&=& frac { d^{ 2 } P }{ dy^{ 2 } } cdot frac { 1 }{ P } \
\
&=& { small frac { 2 cdot a^{ 2 } cdot b^{ 2 } cdot left { h left ( y right ) right }^{ 3 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] }{ P } } \
&+& { small frac { 2 cdot a cdot b^{ 2 } cdot n left ( y right ) cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] }{ P } } \
&+& { small frac { b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] }{ P } } \
&+& { small frac { b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i^{ 2 } cdot k left ( y right ) + A cdot left ( n-1 right )^{ 2 } cdot m left ( y right ) right ] }{ P } } \
\
Convexity
&=& { small frac { 2 cdot a^{ 2 } cdot b^{ 2 } cdot left { h left ( y right ) right }^{ 3 } cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] }{ h left ( y right ) cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] } } \
&+& { small frac { 2 cdot a cdot b^{ 2 } cdot n left ( y right ) cdot left { h left ( y right ) right }^{ 2 } cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] }{ h left ( y right ) cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] } } \
&+& { small frac { b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] }{ h left ( y right ) cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] } } \
&+& { small frac { b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot h left ( y right ) cdot left [ c cdot i^{ 2 } cdot k left ( y right ) + A cdot left ( n-1 right )^{ 2 } cdot m left ( y right ) right ] }{ h left ( y right ) cdot left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] } } \
\
Convexity
&=& { normalsize 2 cdot a^{ 2 } cdot b^{ 2 } cdot left { h left ( y right ) right }^{ 2 } } \
&+& { small frac { 2 cdot a cdot b^{ 2 } cdot n left ( y right ) cdot h left ( y right ) cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] }{ left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] } } \
&+& { small frac { b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot left [ c cdot i cdot k left ( y right ) + A cdot left ( n-1 right ) cdot m left ( y right ) right ] }{ left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] } } \
&+& { small frac { b^{ 2 } cdot left { n left ( y right ) right }^{ 2 } cdot left [ c cdot i^{ 2 } cdot k left ( y right ) + A cdot left ( n-1 right )^{ 2 } cdot m left ( y right ) right ] }{ left [ c cdot k left ( y right ) + A cdot m left ( y right ) right ] } } \
end{eqnarray} \)

 

로 정리된다. 이를 원래의 식으로 나타내면, 최종 \( Convexity ; \)는 다음과 같다.

\( begin{eqnarray}
Convexity
&=& frac { 2 cdot d_1^{ 2 } }{ t_1^{ 2 } cdot f^{ 2 } cdot left ( 1 + frac { d_1 }{ t_1 } cdot frac { 1 }{ f } cdot y right )^{ 2 } } \
&+& { scriptsize frac { 2 cdot d_1 cdot left [ displaystyle sum_{ i=0 }^{ n-1 } frac { frac { CF_i }{ f } cdot i }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ i } } + frac { A cdot left ( n-1 right ) }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ n-1 } } right ] }{ t_1 cdot f^{ 2 } cdot left ( 1 + frac { 1 }{ f } cdot y right ) cdot left ( 1 + frac { d_1 }{ t_1 } cdot frac { 1 }{ f } cdot y right ) cdot left [ displaystyle sum_{ i=0 }^{ n-1 } frac { frac{ CF_i }{ f } }{ left( 1 + frac { 1 }{ f } cdot y right )^{ i }} + frac { A }{ left( 1 + frac { 1 }{ f } cdot y right)^{ n-1 }} right ] } } \
&+& { small frac { left [ displaystyle sum_{ i=0 }^{ n-1 } frac { frac { CF_i }{ f } cdot i }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ i } } + frac { A cdot left ( n-1 right ) }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ n-1 } } right ] }{ f^{2} cdot left ( 1 + frac { 1 }{ f } cdot y right )^{ 2 } cdot left [ displaystyle sum_{ i=0 }^{ n-1 } frac { frac{ CF_i }{ f } }{ left( 1 + frac { 1 }{ f } cdot y right )^{ i }} + frac { A }{ left( 1 + frac { 1 }{ f } cdot y right)^{ n-1 }} right ] } } \
&+& { small frac { left [ displaystyle sum_{ i=0 }^{ n-1 } frac { frac { CF_i }{ f } cdot i^{ 2 } }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ i } } + frac { A cdot left ( n-1 right )^{ 2 } }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ n-1 } } right ] }{ f^{2} cdot left ( 1 + frac { 1 }{ f } cdot y right )^{ 2 } cdot left [ displaystyle sum_{ i=0 }^{ n-1 } frac { frac{ CF_i }{ f } }{ left( 1 + frac { 1 }{ f } cdot y right )^{ i }} + frac { A }{ left( 1 + frac { 1 }{ f } cdot y right)^{ n-1 }} right ] } } \
\
Convexity
&=& frac { 2 cdot d_1^{ 2 } }{ t_1^{ 2 } cdot f^{ 2 } cdot left ( 1 + frac { d_1 }{ t_1 } cdot frac { 1 }{ f } cdot y right )^{ 2 } } \
&+& { small frac { 1 }{ f^{ 2 } cdot left ( 1 + frac { 1 }{ f } cdot y right )^{ 2 } cdot left [ displaystyle sum_{ i=0 }^{ n-1 } frac { frac{ CF_i }{ f } }{ left( 1 + frac { 1 }{ f } cdot y right )^{ i }} + frac { A }{ left( 1 + frac { 1 }{ f } cdot y right)^{ n-1 }} right ] } } \
&times& { small left [ frac { 2 cdot d_1 cdot left ( 1 + frac { 1 }{ f } cdot y right ) }{ t_1 cdot left ( 1 + frac { d_1 }{ t_1 } cdot frac { 1 }{ f } cdot y right ) } cdot left { displaystyle sum_{ i=0 }^{ n-1 } frac { frac { CF_i }{ f } cdot i }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ i } } + frac { A cdot left ( n-1 right ) }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ n-1 } } right } right . } \
& & quad + { small left . left { displaystyle sum_{ i=0 }^{ n-1 } frac { frac { CF_i }{ f } cdot i }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ i } } + frac { A cdot left ( n-1 right ) }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ n-1 } } right } right . } \
& & quad + { small left . left { displaystyle sum_{ i=0 }^{ n-1 } frac { frac { CF_i }{ f } cdot i^{ 2 } }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ i } } + frac { A cdot left ( n-1 right )^{ 2 } }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ n-1 } } right } right ] } \
\
Convexity
&=& frac { 2 cdot d_1^{ 2 } }{ t_1^{ 2 } cdot f^{ 2 } cdot left ( 1 + frac { d_1 }{ t_1 } cdot frac { 1 }{ f } cdot y right )^{ 2 } } \
&+& { normalsize frac { 1 }{ P } cdot frac { 1 }{ f^{ 2 } cdot left ( 1 + frac { 1 }{ f } cdot y right )^{ 2 } cdot left ( 1 + frac { d_1 }{ t_1 } cdot frac { 1 }{ f } cdot y right ) } } \
&times& { small left [ frac { 2 cdot d_1 cdot left ( 1 + frac { 1 }{ f } cdot y right ) }{ t_1 cdot left ( 1 + frac { d_1 }{ t_1 } cdot frac { 1 }{ f } cdot y right ) } cdot left { displaystyle sum_{ i=0 }^{ n-1 } frac { frac { CF_i }{ f } cdot i }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ i } } + frac { A cdot left ( n-1 right ) }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ n-1 } } right } right . } \
& & quad + { small left . left { displaystyle sum_{ i=0 }^{ n-1 } frac { frac { CF_i }{ f } cdot i }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ i } } + frac { A cdot left ( n-1 right ) }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ n-1 } } right } right . } \
& & quad + { small left . left { displaystyle sum_{ i=0 }^{ n-1 } frac { frac { CF_i }{ f } cdot i^{ 2 } }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ i } } + frac { A cdot left ( n-1 right )^{ 2 } }{ left ( 1 + frac { 1 }{ f } cdot y right )^{ n-1 } } right } right ] } \
end{eqnarray} \)

조회수: 11

1. 해당 글 ‘2번 공식‘에 해당하나 다만, \( r_1 \) 대신 \( y \), \( N \) 대신 \( A \)라고 표기

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