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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} \)