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It seems to me that the non-reversible transactions included in the current account, and the reversible transactions included in the capital and financial account, are unrelated, and should therefore be able to fluctuate independent of one another.
Imagine an economy where each person bought a huge number of items online from overseas. The balance on current account would be negative... but the capital and financial account unaffected, meaning they no longer add to zero.
What am I missing?
It was also really strange for me at first. I asked from many people who might happen to know the answer. However, they were all not 100% satisfactory. I think, Kenny LJ's answer is 99.5% correct but i feel that it's still missing something. My explanation is to add to his/her answer something i think very very subtle, but hopefully important, one.
Think you have an organization and record the purchase in your balance sheet. You will purchase the good as an asset with an debit entry. Also, you will credit your cash with the same amount.
When these two entries come to the BOP, the asset will be in CA and the cash will be in FA. Why the money should be recorded in FA? When a country starts to hold a foreign currency /it was well before and independently of your purchase/, it's recorded in BOP as an investment to the foreign country. Now when you pay with the money of the foreign, you stop your investment and make it a physical good. So your stake with the other country doesn't change, but its form. So the balance is not affected in the end.
If you work with other examples, it'll become clearer and clearer. Let me give some.
Imagine you export something to a foreign country. How will it be 0 in BOP? An export debits your goods and increases the CA. If you get a foreign currency from your foreign partner for the payment, it'll be credited as investment to the other country as you hold its currency which will credit or decrease the FA.
Imagine you borrow from a foreign country, it'll be an increase in FA as it's a foreign investment to your country. But in which currency will you hold your investment proceedings? You may start holding that country's currency. This will be recorded as your country's other investment to the other country which will be decrease in FA.
If you sell above investment money into your country's central bank and your central bank holds it as cash, your country's other investment's increase will be undone which will result in positive FA from the investment /it'll look like the foreign country invested positive amount in net/. But the official reserve account of the country will increase with the same amount which will also make the BOP 0.
I hope it helps.
The Balance of Payments (BOP) records, in summary form, every single international transaction.*
Also, the BOP uses a double-entry accounting system, where every entry MUST be balanced (hence the name) by an entry (or entries) that is (or are together) equal and opposite.
In your example, say an American consumer buys \$1,000 of sashimi from a Japanese fisherman. The American consumer pays with cash (could be in USD or JPY, doesn't matter too much here).
Then under the American BOP, the sashimi import is recorded as a debit (-\$1,000) under the Current Account (CA). And the cash is recorded as a credit (+\$1,000) under the Financial Account (in particular the "Other Investments" sub-portion of the FA).
To repeat, the BOP MUST add up to 0. Whenever you enter one entry in the BOP (say in the CA), there must be an equal and opposite entry somewhere else (often this will be in the FA, but it can be anywhere, including even in the CA again).
Sometimes just to make sure that the BOP adds up to 0, the balancing entry can be a bit contrived. But the above is not at all a contrived example; it is a very standard and clear-cut one.
Footnote *: Of course, in practice, these are errors and omissions. Which explains why there is an "Errors and Omissions" component added in the BOP.