Just a simple "teaching" post.
There are two different ways of thinking about central bank profits (of doing the accounting). A simple numerical example will illustrate the difference.
Assume 2% inflation, and 1% real GDP growth, so Nominal GDP grows at 3%. Assume the stock of currency is 5% of annual NGDP. Assume the nominal interest rate on bonds is 4%, but currency pays 0% interest. Assume central banks have zero operating costs.
Method 1: Every year the central bank prints currency worth 0.15% of NGDP (that's 5% x 3%) so its profits are 0.15% of NGDP, and it gives those profits to the government which owns the central bank.
Method 2: When the central bank prints currency it buys bonds, so owns bonds worth 5% of NGDP, and earns interest on those bonds equal to 0.2% of NGDP (that's 5% x 4%), and it gives those profits to the government which owns the central bank.
The Bank of Canada uses Method 2, which JKH tells me is called "accrual accounting". But economists like me sometimes prefer to think in terms of Method 1. (What's a good name for Method 1?) [Actually, it's a little more complicated than this, because I have ignored capital gains/losses from fluctuations in interest rates etc.]
Those two methods seem to give different answers -- 0.15% of NGDP vs 0.2% of NGDP (Method 2 gives bigger/smaller profits than Method 1 if the nominal interest rate is greater/less than the growth rate of currency). But both methods give exactly the same answer for the Present Value of profits if we start a new central bank from scratch.
[Remember the formula for the Present Value of X, if X grows at rate g, discounted at an interest rate r, is X/(r-g).]
Method 1: 5% of NGDP in year zero. Plus the Present Value of 0.15% of NGDP, growing at 3%, discounted at 4%, equals 15% of year zero NGDP. Equals 20% of year zero NGDP.
Method 2: The Present Value of 0.2% of NGDP, growing at 3%, discounted at 4%, equals 20% of year zero NGDP.
We get the same Present Value equal to 20% of year zero NGDP whichever way we do the accounting. The only difference is that Method 1 "books" the profits immediately when the currency is printed, while Method 2 only books the profits as the bonds bought by the newly-printed currency earn interest. [But how can I show we get the same answers if the growth rate of NGDP exceeds the interest rate on bonds??]
The intuition is simple: If the central bank prints $100 and buys a bond worth $100, the present value of the payments earned on that bond is also $100.
People who think that currency is a liability of the central bank that created it, just like demand deposits are a liability of the commercial banks that create them, may prefer to think in terms of Method 2.
People who think that irredeemable currency is not a liability of the central bank that created it, just like newly-mined gold is not a liability of the gold miner (Bill Woolsey's analogy), may prefer to think in terms of Method 1.
But we end up with the same answer either way, in Present Value terms.
If central banks were to pay interest to those who hold its currency (and electronic money issued by central banks (aka "reserves") typically does pay interest) then we would need to subtract those interest liabilities from the central bank's profits, whichever method we use. And in this case, Method 2 might be the simpler of the two methods, because we simply multiply the stock of currency by the interest rate differential (bond rate minus currency rate) to calculate the central bank's profits.