Why can we see the Current account as variation in NIIP? Let's designate the Net International Investment Position(NIIP:it's the net international wealth of a country), as $B_t$.

Why can we see the current account at time t, as $CA_t=B_{t+1}-B_t$ , when we know that $CA_t=TB_t+i^wB_{t}$?

$TB_t$ is trade balance at time $t$, and $i^w$ is world nominal interest rate.

Intuitively it makes sense, since the variation of a country net international wealth should mimic its international 'consumption/expenditures'. However, I'm looking for a more formal derivation. I think it would probably be related to the Financial and Capital Accounts.

Any help would be appreciated.
In reality the world is much more complicated than those two equations, but the simple theory is that


the current account net flow is the trade balance in that period plus net primary income in that period, and net primary income is the return on investments abroad less the return on foreign investments at home.  The $i^wB_{t}$ term suggests net primary income can be modelled as an implicit rate of return or interest rate on the net investment stock and this implicit interest rate is the same worldwide for all kinds of investment income in both directions (reality is different, in that different types of invest in different places produce different returns)
the change in the International Investment Position net stock over a period should be equal to the net financial flow in that period (reality is again different as there are also revaluations of the value of the components of existing investment positions, for exchange rate changes and other reasons, and these are not reflected in flows)