User:Jaffa3027/Sandbox

Our goal is to calculate a general measure for an finance account, it is similar for a fund price NAV (Net Asset Value) Balance equation $A=L+C-D+I-E$

where A = Assets L = Liability C = Capital D = Drawing I = Revenue E = Expense

and we defined $D=A-L$ (Equity), $U=C-D$ (Net drawing) and $R=I-U$ (Result)

$D=U+R$

Differance

$d=u+r$

In countinues case we have

$z=xy=d$

$|z|=xy$

$z'=x'y+xy'=u+r$

${\frac {z'}{|z|}}={\frac {x'y}{|z|}}+{\frac {xy'}{|z|}}={\frac {u}{|z|}}+{\frac {r}{|z|}}$

$x'y=r$ → ${\frac {x'}{x}}={\frac {r}{z}}$→ $x=x_{0}e^{\sum {\frac {r}{z}}}$

$xy'=u$ ,→ ${\frac {y'}{y}}={\frac {u}{z}}$ → $y=y_{0}e^{\sum {\frac {u}{z}}}$

$x_{0}=100$ → $y_{0}={\frac {z_{0}}{100}}$

Now

$z=z_{0}e^{\sum {\frac {r}{z}}}e^{\sum {\frac {u}{z}}}$

To handle sampled data we have work with difference

$z_{i}=x_{i}y_{i}$

and

$|z_{i}|=x_{i}y_{i}$

$\Delta z_{i}=\Delta (x_{i}y_{i})=x_{i}y_{i}-x_{i-1}y_{i-1}=z_{i}-z_{i-1}$

Now

(A) $z_{i}-z_{i-1}=x_{i}y_{i}+x_{i}y_{i-1}-x_{i}y_{i-1}-x_{i-1}y_{i-1}=x_{i}(y_{i}+y_{i-1})-(x_{i}-x_{i-1})y_{i-1}$

or

(B)$z_{i}-z_{i-1}=x_{i}y_{i}+x_{i-1}y_{i}-x_{i-1}y_{i}-x_{i-1}y_{i-1}=(x_{i}+x_{i-1})y_{i}-x_{i-1}(y_{i}-y_{i-1})$,

We have

$x_{i-1}(y_{i}-y_{i-1})=u_{i}$

${\frac {x_{i-1}(y_{i}-y_{i-1})}{|z_{i-1}|}}={\frac {u_{i}}{|z_{i-1}|}}$, ${\frac {y_{i}-y_{i-1}}{y_{i-1}}}={\frac {u_{i}}{|z_{i-1}|}}$

and we get

(1) $y_{i}=y_{i-1}(1+{\frac {u_{i}}{|z_{i-1}|}})$

and start with $y_{0}={\frac {z_{0}}{100}}$

$x_{i}={\frac {z_{i}}{y_{i}}}$

$(x_{i}+x_{i-1})y_{i}=r_{i}$

${\frac {(x_{i}+x_{i-1})y_{i}}{|z_{i}|}}={\frac {r_{i}}{|z_{i}|}}$,${\frac {(x_{i}+x_{i-1})}{x_{i}}}={\frac {r_{i}}{|z_{i}|}}$

(2) $x_{i}=x_{i-1}{\frac {1}{1-{\frac {r_{i}}{|z_{i}|}}}}=x_{i-1}(1+{\frac {\frac {r_{i}}{|z_{i}|}}{1-{\frac {r_{i}}{|z_{i}|}}}})$

and

$y_{i}={\frac {z_{i}}{x_{i}}}$

So if $u_{i}>=0$ the we use (1) else (2)

If $x<=0$ and $y<=0$ the we set $x_{i}=100$

$x_{i}(y_{i}-y_{i-1})=u_{i}$

${\frac {x_{i}(y_{i}-y_{i-1})}{|z_{i}|}}={\frac {u_{i}}{|z_{i}|}}$, ${\frac {y_{i}-y_{i-1}}{y_{i}}}={\frac {u_{i}}{|z_{i}|}}$

$y_{i}=y_{i-1}{\frac {1}{1-{\frac {u_{i}}{|z_{i}|}}}}=y_{i-1}(1+{\frac {\frac {u_{i}}{|z_{i}|}}{1-{\frac {u_{i}}{|z_{i}|}}}})\approx y_{i-1}(1+{\frac {u_{i}}{|z_{i}|}})$