How do countries repay debts worth millions, and is the money moved electronically or physically?

It is a little difficult to explain to a first time user, but trust me, it is no rocket science. It is just a concept, if you understand it, you will get it.

Think about money (currency) that each country produces. They create money in physical form (notes & coins) and much of it electronically.

So every country has a very accurate ledger count of how much money is in circulation electronically and in physical form. Because you, the country, are responsible for the creation of your own currency, no one else can make your money (if they do so, your ledger would not have that entry and hence you know someone else is trying to copy your currency).

So now when money is circulated from one country to another, lets take an example between US and UK, it might look like following:

US: Total Money created: US$ 1 Billion
US$ 900 Million Electronic and US$ 100 Million in Currency & Notes.

UK: Total Money created: GBP 1 Billion
GBP 800 Million Electronic and GBP 200 Million in Currency & Notes.

Now if the US loans the UK government say US$ 100 Million, this is how the ledger of the US Treasure will look like (in a crude sense):

US Government Treasury Department
Total Currency Ever Created: US$ 1 Billion.

- Currency & Notes: US$ 100 Million
- Electronic Within The US: US$ 800 Million
- Bank of England: US$ 100 Million.

UK Treasure Department
Total Currency Ever Created: GBP 1 Billion

- Currency & Notes: GBP 200 Million
- Electronic Within The UK: GBP 800 Million
- Received US Dollars from US: US$ 100 Million.

For this transaction to happen, BOTH the US and UK treasury simply agreed to handle their ledgers, by agreeing on a Debit & Credit.

Now are more and more parties (banks) get involved, the ledger becomes more and more complex. So smaller banks report to larger banks, larger banks, compare ledger with central banks, and central banks compare ledgers with each other. There is always a check and balance happening.

Now if the UK Government suddenly say it has US$ 300 Million in its ledger, you know that is incorrect, because the same ledger entry would not be reflected on the US side.

Hence, cheating of ledger entries is not possible. Sure, the UK government can simply increase their ledger entry and say, we now have GBP 5 Billion instead of GBP 1 Billion, which is fine, they can do that, but ONLY they (The UK Government) can do that. No one else can unilaterally just increase the money in circulation of another country.

So to answer your question, the money is electronically sitting in Greece's ledger. It will simply push this out to the IMF (via IMF's designated Bank or Banks) and push the US$ 1.3 Billion out to them and the Central Bank of Greece and the other banks will adjust their ledgers accordingly.

It is all about double entry and two-party (minimum) consensus.

For Further Reading:

The following answers will help you understand how money and money transfer works:

• Faisal Khan's answer to How does a remittance money transfer work? What are all the steps and who is involved?
• Faisal Khan's answer to What exactly happens when money is sent via Western Union versus a company like XOOM? What is the difference?
  
• Faisal Khan's answer to How does money transfer between banks and different countries work?
  
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• Faisal Khan's answer to What exactly happens when a wire transfer is made? How does the money travel from the sender's account to the recipient's account? How much time does each step take? What regulatory measures are in place, if any?
  
• Faisal Khan's answer to How does the settlement of payments work in banks? Specifically, how do payment systems that are connected to multiple banks actually settle the amount between two banks?