This is a regulatory way to calculate a reserve for credit loss. I want to interpret this formula.
$$Lifetime\ Reserve=\frac{PD\times LGD\times EAD}{(1+i)}\left[\frac{1-(1-PD)^n}{PD}\right]-$$ $$\frac{PD\times LGD\times Payment}{i(1+i)}\left[\frac{1-(1-PD)^n}{PD}\right]+\frac{PD\times LGD\times Payment}{i(i+PD)}\left[1-\left(\frac{1-PD}{1+i}\right)^n\right]$$
where
$PD=Probability\ of\ Default$
$LGD=Loss\ Given\ Default$
$EAD=Exposure\ at\ Default$
$i=rate\ of\ interest$
$n=remaining\ term\ of\ a\ credit$
$EAD=Payment\frac{1-(1+r)^{-n}}{r}$
After doing a bit (a lot) of algebra, I get to this expression
$$\frac{PD\times LGD\times Payment}{(1+i)}\left[\sum_{j=0}^{n-1}\left(\frac{1-PD}{1+i}\right)^j\frac{1-(1+r)^{-(n-j)}}{r}\right]$$
Which seems a little bit simplier to understand but still, I cant find a way to interpret it. I tried putting the payments of the annuity in a timeline, but again, still dont understand. I know it has something to do with a annuity with contingent claims. Could you help me understand the formula? Thanks in advance.