User:Wyklety/sandbox

$$ MP_l = \frac{\partial}{\partial l} \left ( l^{0.5} \cdot g^{0.5} \right ) $$

$$ x = \left (1+y \right )^\frac{1}{n} - 1 $$

$$ x = 1 - \left (1-y \right )^\frac{1}{n} $$

$$

\text{adjustment} = \left ( \text{FX rate} \right ) \times \frac{ \left ( 1 + r_{n,\text{foreign}} \right )^{\frac{n}{12}} } { \left ( 1 + r_{n,\text{local}} + \text{basis}_{\text{FX}} \right )^{\frac{n}{12}} }

$$

$$

\text{spread}_{\text{liability}} = \text{spread}_{\text{credit}} \times \left ( \text{weight in credit} \right ) \times \frac{\text{duration}_{\text{credit}}}{\text{duration}_{\text{liability}}}

$$

$$ (1+y_{13})^{t_{13}} = (1+y_{12})^{t_{12}} \times (1+f_{1,12})^{t_{13} - t_{12}} $$

$$ (1+y_{x})^{t_{x}} = (1+y_{12})^{t_{12}} \times (1+f_{x-12,12})^{t_{x} - t_{12}} $$

$$ f_{12,x-12} = \left( \frac{(1+y_{x})^{t_{x}}}{(1+y_{12})^{t_{12}}} \right)^{\frac{1}{t_{x}-t_{12}}} - 1 $$

$$

\text{CF}_0 = \left( \text{annuity purchase amount} \right) - \left( \text{NPV annuity}_{\text{IFRS}} + \text{solvency margin} + \text{initial expenses} \right) - \left( \text{asset unit} \right) \cdot \left( \text{asset charge}_{\text{bid mid}}  \right )

$$

$$

\text{cash profit} = \left(\text{IFRS profit CF} \right)_0 + \sum_{i = 1}^{120} \frac{ \left(\text{IFRS profit CF} \right)_i }{ \left( 1 + r_{r} + s \right )^{\frac{i}{2}}}

$$

$$

\text{profit pct} = \frac{ \text{cash profit} }{ \text{annuity purchase amount} } $$

$$ \text{leverage} = \frac{\text{MTM}_{\text{receiving}}}{\text{cash} + \text{MTM}_{\text{swap}}} $$

$$ r_{t,\text{detrended}} = \left [ \left ( r_t - \mu_{r} \right ) \times \left ( \frac{\sigma_{\text{historical}}}{\sigma_{\text{simulation}}} \right ) \right ] + \mu_{r} $$

$$

\text{curveRisk}_{t} = \frac{ \sum_{i=1}^{10} \left [ \left ( \text{PV01}_{\text{asset},i,t} - \text{PV01}_{\text{liab},i,t} \right )^2 \times \text{rebalRatio}_t \right ] + \sum_{i=1}^{10} \left [ \left ( \text{IE01}_{\text{liab},i,t} - \text{IE01}_{\text{liab},i,t} \right )^2 \times \left ( 1 - \text{rebalRatio}_t \right ) \right ] }{ \sum_{i=1}^{10}  \left [ \left ( \text{PV01}_{\text{liab},i,t}  \right )^2 \times \text{rebalRatio}_t \right ] + \sum_{i=1}^{10} \left [ \left ( \text{IE01}_{\text{liab},i,t} \right )^2 \times \left ( 1 - \text{rebalRatio}_t \right ) \right ]  } $$

$$ \text{rebalFlag}_{t} = \begin{cases} \text{true}, & \text{if } \text{curveRisk}_{t} > \text{tolerance}_{\text{rebalancing}} \\ \text{false}, & \text{otherwise} \end{cases} $$

$$ \text{useNomFlag}_{t} = \begin{cases} \text{true}, & \text{if } \left ( \sum_{i = 1}^{10} \text{PV01}_{\text{required},i,t} \right ) < \left ( \sum_{i = 1}^{10} \text{IE01}_{\text{required},i,t} \right ) \\ \text{false}, & \text{otherwise} \end{cases} $$

$$ \text{PV01}_{\text{required},i,t} = \begin{cases} 0, & \text{if } \text{rebalFlag}_{t} = \text{false} \\ \left ( \text{PV01}_{\text{liab},i,t} - \text{PV01}_{\text{asset},i} \right ) & \text{if } \text{rebalFlag}_{t} = \text{true} \end{cases} $$

$$ \text{accountEoY}_{t} = \begin{cases} \left ( \text{CFSurplus}_{t-1} \times \text{reInvRate}_{i,t} \right ), & \text{if } \text{accountEoY}_{t-1} \geq 0 \\ \left ( \text{CFSurplus}_{t-1} \times \text{reInvRate}_{i,t} \right ) + \text{CFSurplus}_{t}, & \text{if } \text{accountEoY}_{t-1} < 0 \end{cases} $$

$$ \text{runOffFlag}_{t} = \begin{cases} 1, & \text{if } \text{surplus}_{t} > 0 \\ 0, & \text{if } \text{surplus}_{t} \leq 0 \end{cases} $$

$$ \text{surplus}_{t} = \text{accountEoY} _{100,t} \times \left [ \left ( 1 + \frac{r_{1200,t}}{2} \right )^{-200} \right ] $$

$$\begin{align} \text{MTM}_{\text{LDI}} & = \sum_{i = 1}^{1200} \left ( \text{swapFixedFixLegCF}_{i,t} \times \text{DF}_{i,t} \right ) - \sum_{i = 1}^{1200} \left ( \text{swapFixedFloatLegCF}_{i,t} \times \text{DF}_{i,t} \right ) \\ & + \sum_{i = 1}^{1200} \left ( \text{swapRealFixLegCF}_{i,t} \right ) - \sum_{i = 1}^{1200} \left ( \text{swapRealFloatLegCF}_{i,t} \right )

\end{align} $$

$$ \text{liabPV01}_{j,t} = \text{HR}_{t} \times \left ( \text{liabFixedPV01}_{j,t} + \text{liabRealPV01}_{j,t} \right )$$

$$ \text{reversePV01}_{i,t} = \frac{\text{requiredPV01}_{i,t} }{\text{DF}_{i,t}} \times 10,000 \times \frac{12}{i} $$

$$ \text{requiredPV01}_{i,t} = \begin{cases} 0, & \text{if } \text{rebalFlag}_{t} = \text{false} \\ \text{liabilityPV01}_{i,t} - \text{assetPV01}_{i,t}, & \text{if } \text{rebalFlag} = \text{true} \end{cases} $$

$$ /frac{\text{OutstandingAmount}}{\text{SinkingFactor}} $$

$$ \left (\text{proportion allocated to credit range}\right) = \frac{\text{target excess return}}{\left ( \text{expected return on 100pct credit portfolio} \right ) } $$

$$ \left (\text{proportion allocated to real range}\right) = \left (\text{proportion of real liabilities}\right)$$