If you look at the balance sheet of a central bank, you will see it has liabilities (mostly currency) and assets (normally mostly government bonds/bills). Why do central banks have assets? Do they need them?
The wrong answer is that central banks need assets to "back" the value of the currency, and that paper currency would be worthless otherwise. The right answer is: since the government gets all the profits from a central bank anyway, there's no point in giving the government the assets; that owning assets lets the bank reverse course and reduce the money supply if it ever needs to; and it stops the accountants freaking out.
Let's deal with the wrong answer first. According to the "backing" theory of the value of money, the value of a central bank's currency is equal to and determined by the value of the central bank's assets backing the currency. (This is different from the fiscal theory of the price level, which says that the value of currency plus bonds is equal to and determined by the present value of primary fiscal surpluses.)
The backing theory sounds good. How can intrinsically worthless paper money have value? Because it is backed by valuable assets. It's just like shares in a mutual fund, which have value equal to and determined by the value of the assets in the fund.
Here are three arguments against the backing theory of money:
1. The assets of central banks are normally nearly all nominal assets, denominated in the same currency as the liabilities. Suppose the price level were to double magically overnight, and the real value of currency halved. The real value of the bonds held by the central bank would also halve. So a magical doubling of the price level would not violate the equality between the value of the currency and the value of the assets backing it. The backing theory leaves the price level indeterminate. It could only pin down the price level if the assets were real assets. If (say) 10% of the bank's assets were real (gold reserves, plus the building), then a 1% loss of its real assets (the building burns down) would cause a 10% jump in the price level.
2. Suppose a mutual fund held bonds, but all the interest on the bonds (minus the administrative expenses of running the fund) were handed over to some third party, and not to the owners of shares in the mutual fund. Who would want to own shares in that mutual fund? The net present value of the dividends paid to the shareholders would be zero, so the shares would be worth zero too. But this is exactly what central banks do. Every year central banks earn profits from the interest on the bonds they own, minus administrative expenses, and hand the whole of that profit to the government, not to the holders of currency.
3. We don't need "backing" to explain why money has value. People want to hold a stock of money because money is a medium of exchange, and holding a stock of the medium of exchange makes shopping easier. This creates a (stock) demand for money. Provided the central bank restricts the supply of money, the intersection of demand and supply curves creates a positive equilibrium value of money (a finite price level). Now you could argue that if paper money were worthless it could not function as a medium of exchange, so you need to assume paper money has value in order to explain the value it has, so the demand and supply theory of the value of money begs the question.
There is some truth in this criticism of standard theories of the value of money. There are indeed two equilibria: the normal one, where paper money has value, and a weird one, where it is worthless. But Ludwig von Mises, for example, addressed this problem in 1912 with his Regression Theory of Money. Historically, money needed to be commodity money, or have commodity backing, in order to get started. But once it does get started, as a social institution, the demand for a medium of exchange supplements the industrial demand for the commodity, and the commodity backing can eventually be withdrawn as custom keeps us out of the weird equilibrium. (When Cambodia reintroduced paper money, after the fall of the Kymer Rouge, it could not create paper money ex nihilo, but initially made it convertible into rice, IIRC.)
A Ponzi scheme is a financial institution with liabilities and no assets backing those liabilities. Paper money can operate just like a Ponzi scheme, but with one important difference. Mr Ponzi promised his clients high rates of interest and/or capital gains. They would not have held his liabilities unless they believed him. The Bank of Canada promises zero interest, zero nominal capital gains, and a minus 2% real rate of interest on people who hold its paper money. Mr Ponzi could not deliver on his promise, even if he hadn't spent the assets. The Bank of Canada can deliver on its promise, even if it gave away all its assets, provided the (real) demand for its paper money does not fall over time more quickly than 2% per year. (If the real demand for money were falling at 2% per year, a constant nominal supply of money would yield 2% annual inflation).
The Bank of Canada does not need assets, because the long run growth in the (real) demand for its paper exceeds the real interest rate at which people are willing to hold its paper. If Mr Ponzi could have met the same test, he wouldn't have needed assets either. People are willing to hold paper money, even at very negative real rates of return (Zimbabwe), because doing so makes shopping easier.
The only reason that the value of a central bank's liabilities are roughly equal to the value of its assets is that whenever the difference between them (its net worth) gets too big, the bank hands its profits over to the government. If central banks choose to keep assets equal in value to their liabilities, and only hand over their annual profits to the government, then saying the value of their assets determines the value of their liabilities gets causality reversed. It is the value of their liabilities that determines the value of their assets.
So why do central banks hold any assets at all? Three reasons (funny how 3 is a magic number):
1. It makes no difference to the owners of the central bank (the government) whether the central bank keeps the bonds and hands the interest over to the government each year, or whether the central bank gives the bonds to the government. The government gets the interest either way; it just passes through another pair of hands. It's a wash.
2. Normally the real demand for paper money grows at about 3% per year (roughly the same as GDP growth rate), but sometimes it rises faster than this (like last year), and sometimes it falls (like next year?). If the demand for money falls, the central bank needs to reduce the supply of money to prevent inflation, and it reduces the supply of money by selling assets. If it had given away all its assets it wouldn't be able to do this.
3. Accountants like double-entry bookkeeping and balance sheets and stuff so they can keep track of things. They like to record assets on one side, and liabilities on the other side, to make sure that everything adds up, to check that everything's been properly recorded. So they like to list currency as a liability of central banks (even though it isn't, because there's no promise to redeem it, or pay interest on it), and assets on the other side. An accountant would freak out if he recorded currency as a liability and couldn't find an equivalent value of assets. He would say that the central bank is a Ponzi scheme. Which of course it is. And it's just not worth the hassle of trying to explain to accountants that some Ponzi schemes are sustainable, really.