Accounting for central bank profits

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.

Comments

[Roger Sparks]

Both methods ignore that government is borrowing from itself . This over-riding fact turns both "methods" into an illusion.

There can be no "profits" when bonds issued to yourself (to collateralize currency you print and spend yourself) pay "interest". There is only dilution of previously existing assets.

[Mike Sproul]

The trouble is that profits attract rival money producers. For example, Citibank could issue paper eurodollars from a base in the Cayman islands, and if those Eurodollars were a little more convenient than Central Bank notes, then those eurodollars would displace the central bank's dollars and Citibank could grab all the central bank's profits. Of course, we don't see this happening in practice, not even in the smallest and most pathetic countries. That's either because people stubbornly insist on using central bank dollars no matter what, or because the costs of printing, handling, chasing counterfeiters, etc, were enough to eat up the central bank's profits.

[Nick Rowe]

Roger: Assume the government issues no bonds, so the central bank buys commercial bonds. It's the same.

Mike: yep, another way of looking at central bank profits is as (de facto or de jure) monopoly profits. And you could say that government creates those monopoly profits by punishing those who counterfeit its own currency more severely (or with greater probability) than those who counterfeit competitors' currencies.

[JKH]

Nick

Regarding your comment from the previous post:

" JKH: "...because the buyer/assumer would be liable to repay the book value of that deposit if/when the depositor wants his money back - and the buyer/assumer of the deposit liability won't have the cash to do that because he hasn't been paid to assume the liability..."

But if central bank money is irredeemable (unlike under the gold standard), it's not like commercial bank money. And the whole point of helicopter money is to create an excess supply of money (at the existing level of NGDP) so that people go out and spend it and increase NGDP, rather than redeeming it at the central bank.

(But I'm not sure whether this discussion of the two methods of doing the accounts really matters, since you get the same Present Value of central bank profits either way you do it. But it's still an interesting (to me) discussion.)) "

------->

I think redeemability is secondary to the point of the analogy.

Think of it this way - the CB has an option to force redeemability contingent on bond sales that are designed to drain reserves.

The point being that the optionality tweak is secondary to the fact that the CB under certain circumstances may want to pay for a reserve "outflow" from its balance sheet (i.e. reserve extingishment) - just as a commercial bank wants to pay for a deposit outflow.

And the accounting analogy holds from there - and there is no accounting method under which a commercial bank would calculate the PV of a future interest margin differential as profit.

And I understand the general objective of HM as monetary base expanding - but permanence is a shaky idea - and the accounting shouldn't preclude reasonable contingencies.

[JKH]

I'd be more comfortable if you said that there are 2 ways that economists think of central bank profits (maybe that's implicit). From there I would say that there is no private sector correspondent to the PV approach for CB'S. And that should be pointed out, because there is definitely a private sector analogy in the case of a commercial bank creating a deposit from a loan. And the accounting comparison should really not be affected by who has the option to extinguish reserves or bank money respectively.

[Roger Sparks]

When we discuss "money", I always orient myself by asking how the owner-of-the-money got it.

Did the owner exchange labor or property for money he is now about to spend?

Did the owner borrow money he is now about to spend?

Did the owner borrow from-himself money he is now about to spend?

In the case of a counterfeiter, did the owner print-himself the money he is now about to spend?

I also treat "spending" as a broad description of releasing control of money. Lending money, or buying labor or resources, are both "spending" in the sense that control of money has been relinquished.

[Nick Rowe]

JKH: "Think of it this way - the CB has an option to force redeemability contingent on bond sales that are designed to drain reserves."

I think that's a useful point to make. In a world of certainty (like my toy example) that option is never exercised. But an inflation-targeting (for example) central bank might need to "redeem" currency (for bonds) if the demand for currency fell and inflation would rise above target if the central bank did not redeem it.

"I'd be more comfortable if you said that there are 2 ways that economists think of central bank profits (maybe that's implicit)."

Yep. It's implicit. ;-) I find myself flipping back and forth between those two ways. The second is more useful for normal times; the first is more useful when we want to do a one-time big increase in money (like Helicopter Money).

[Market Fiscalist]

Going back to yesterdays post, if you use method 2 for calculating CB profits its r x 0.25% of NGDP in that scenario.

Does that mean that the statement 'So helicopter money finances a deficit of 0.25% of NGDP for one year' would not holds if you use that method?

[Henry]

"The Bank of Canada uses Method 2, which JKH tells me is called "accrual accounting"."

In accrual accounting, an entry in the accounts recording a transaction is made when the transaction notionally occurs, i.e. cash has not changed hands.

In cash accounting, an entry in the accounts recording a transaction is made when the transaction is effected in cash , that is when cash changes hands.

In book value accounting, entries in the accounts are made at the value of the transaction as actually incurred.

In market value accounting, entries in the accounts are made adjusting for changes in the value of the transaction because of subsequent change in market prices.

The first distinction is between accrual and cash and and the second distinction is between book and market value - not between accrual and market, as appeared to be suggested in Nicks's prior blog.

Whether the transactions in "Method 1" and "Method 2" are reported in accrual/cash terms, I don't think, can be said with the information given.

Apart from the mundane business immediately above, I don't get what all the fuss is about. I guess I am just plain stoopid.

What's accounting and CB profits got to do with how HM is implemented and whether it's effective or not in its goals (presumably increasing output/employment/inflation)?

[JKH]

http://worthwhile.typepad.com/worthwhile_canadian_initi/2009/02/guest-post-mark-to-market.html

[Henry]

Interesting post JKH.

However, I will still argue (well it's terminological nitpicking really), that the distinction you ought to be making is between book value accounting and market value accounting. There is no distinction between accrual accounting and market value accounting.

[Too Much Fed]

Roger said: "Did the owner borrow from-himself money he is now about to spend?"

What is that? It sounds like savings.

Roger said: "In the case of a counterfeiter, did the owner print-himself the money he is now about to spend?"

Nick said something similar to: The central banks prints $100 and does not buy assets with it.

If the central bank did this, I do not think that should be called "profits". That is probably going to confuse most people.

[John Handley]

Nick,

Your post confused me, so I'm going to try to work out the math, tell me if I get what you're talking about:

So first there's the CGBC:
M_t + B_t + T_t = R_t-1 B_t-1 + M_t-1

Divide by the price level to get everything in real terms (and assume constant real interest rate = 1/B - 1, constant inflation = pi*):

m_t + b_t + t_t = (1/B)b_t-1 + m_t-1(1-pi*)

real money demand = y, which is constant, so

m(pi*) = (1/b)b_t-1 - b_t - t_t

Seingiorage revenue (CB profits) should be equal to the LHS of the previous equation.

Seigniorage can also be seen as interest not paid on government debt, so if you rewrite the CGBC as

(M_t + B_t) + T_t = R_t-1(B_t-1 + M_t-1) + (1-R_t-1)M_t-1

in real terms this is

(m + b_t) + t_t = (1/B)(b_t-1 + m) + (1-1/B-pi*)m

solving for seigniorage yields the exact same thing as before:

m(pi*) = (1/B)(b_t-1) - b_t - t_t

So how do you get a difference between using interest differential in your post? Our assumptions are superficially equivalent: I assume constant NGDP growth equal to pi* and a constant M to NGDP ratio of 1, which should give me the same results as you (i.e., seigniorage is different when defined differently). As far as I knew until now the two definitions are literally exactly the same, but maybe I'm wrong, unless of course central bank profits are somehow different from seigniorage...

[JKH]

Henry

You may have a point and perhaps a good one.

I can tell you for sure that this was the distinction we definitely used to use in banking - although to your point it may have been because there is an element of short hand convenience to it. But it was definitely used.

The intuition in part is that booked premiums or discounts on bonds or mortgages for example would be amortized or accreted without marking to market the principal otherwise. That's one way to think about it as a distinction.

But as you say, there are elements of asymmetry to it. For example, the accrual of interest on a mortgage or bond is in effect a component of the marked to market value.

The issue of cash flow accounting is broader again I think. But safe to say that cash flow accounting is not a formally legitimate way to calculate either corporate profit or macroeconomic income. But it is useful in the broader sources and uses of funds or macroeconomic flow of funds sense, which is properly viewed as a suplement to income statement and balance sheet accounting.

(interesting macro observation - David Glasner uses cash flow accounting to assert that investment is not identically equal to investment)

Anyway, you have a point. I've never drilled down on the entire taxonomy to my complete satisfaction. But no question that the distinction used to be used in banking and probably still is - except that fair value terminology often replaces marked to market terminology these days.

[JKH]

meant above:

"to assert that investment is not identically equal to saving"

[Ralph Musgrave]

Mike Sproul,

Re your point about private bank money displacing publicly issued money (base money), I don’t agree with your claim that “we don't see this happening in practice..” George Selgin wrote an article claiming that in an economy which initially had just base money, and where private banks were allowed to start issuing their own money, base money would be driven to near extinction. See link below and start at para starting “Perhaps the simplest way…”.

http://capitalismmagazine.com/2012/06/is-fractional-reserve-banking-inflationary/

If private money doesn’t displace public money, why does private money make up a good 95% of the money supply normally? (I say “normally” because the percentage is down a bit right now – to about 90% - thanks to QE).

The actual process via which private money largely displaces public money, far as I can see, is thus. In a public money only system, interest rates would presumably settle down to some sort of optimum free market rate. Then when private banks are allowed to issue money, they can print and lend out money at BELOW the going rate because they do not have to carry the cost that most lenders carry: i.e. do some work, abstain from consumption, save up money and lend it out. That is, private banks can just “print and lend”. Not all that different to what counterfeiters do, i.e. “print and spend”.

[Henry]

JKH,

The way I see it, rightly or wrongly, is that accrual accounting and market value are not mutually exclusive. Most accounting for business etc., if not all accounting, uses the accrual method over the cash method as a matter of practice. Whether market value accounting is used is matter of accounting standards. As you have pointed out, banking uses market value accounting in its various forms.

[Nick Rowe]

John: I'm afraid I'm not following you.

Take a *really* simple example. 0% inflation target, 0% growth, so 0% NGDP growth, constant velocity, so money growth is 0%, so the central bank earns zero profits by method 1. But (nominal and real) interest rate is r, so central bank profits are rM/P by method 2.

MF: Let's put it this way: when we ask the question "How big a deficit can be money-financed (for a given growth in NGDP)?" we get two different answers. Method 1 assumes the newly-printed money is spent immediately (which is presumably what advocates of HM normally mean). Method 2 assumes the central bank always has assets equal to its "liabilities" (which is how central bank profits are normally accounted for).

[JKH]

Henry

Let's not grind teeth on labels over substance. There's too much dysfunction out there with respect to the relationship between accounting and economics to let that get in the way.

I'm just giving you the facts about how practioners used the terminology.

And it's not so illogical.

First, cash accounting is a non issue to the extent that it is considered universally illegitimate for both corporate and macroeconomic income reporting. So the accrual/cash distinction is moot for that purpose, which is what we're talking about here.

Second, accrual accounting is a book value concept. That is already inherent in the stand alone use of the term. Accruals are accruals of book value.

Third, accrual accounting in the absence of market value accounting is book value accounting. That is the context for how practioners used the term. You prefer to call that book value accounting. That may be better. I'm just telling you that practioners referred to it as accrual accounting - to emphasize the relative importance of unadjusted book value accruals in the determination. That's a fact as to how the terminology was used.

Accrual accounting in the presence of market value accounting is effectively a subset of market value accounting. But that situation is referred to as market value accounting.

You can refer to it as book value accounting. That would not be incorrect. I'm just giving you the facts that practioners referred to it as accrual accounting for the reasons I've given. You can argue the wisdom of the usage, but you can't argue the facts.

[Market Fiscalist]

Imagine a simple economy with following attributes:

- The CB bootstrapped the money supply by buying lots of interest bearing assets
- There is no growth
- The CB targets a positive inflation rate.

If the CB tries to hit the IT just by buying more private assets each year it will eventually own all private assets. The lower the velocity and the higher inflation the sooner this will happen.

If the CB just gives new money to the CB to spend - that would generate inflation in a sustainable way.

If the govt sells some bonds to the public that the CB buys back and gives the proceeds to the govt then this is the same as just giving the money directly and should count as HM too.

I think some confusion (at least for me) arises because 1) the CB may also buy govt assets to address variations in velocity 2) In a steady state the mechanism where the CB buys govt assets to accomplish HM is pretty much hidden 3) at the ZLB everyone says weird things abt HM.

[Henry]

JKH,

"Let's not grind teeth on labels over substance."

I agree. The point I was making is a very minor one in the scheme of things.

"I'm just giving you the facts that practioners referred to it as accrual accounting for the reasons I've given."

Thanks for making that more than clear.

[Neil]

This doesn't seem right.

In your method one, how is the cash distributed? Are we handing over all newly created money directly to the government to spend as it sees fit? If the behaviour is the same as in method 2, and bonds are purchased with new money, then you would record the new asset (the bond) from when the money is created, and then book a profit in the amount of the purchase rather than recording an offsetting liability. But it would just be a paper profit, since there'd be no cash available to pay the government a cash dividend.

The trouble with this is that (some) of that outstanding money really is a liability. If inflation is above target (or the NGDP path or whatever), and the bank needs to shrink the money supply, method one would require the bank to book a loss as it shrinks base money. This loss would then directly show up as an increase in the government deficit/reduction of government surplus.

[Eric Lonergan]

Here's a thought experiment: Correctly measured, the present value of an asset should equal its market value - how much we would pay to buy it. So imagine a society without a central bank, how much (or how many year's worth) of current GDP would it forgo to have a central bank? A lot more than the 'PV of seigniorage' ... ? In which case, it is not the PV, at all.

Here's another thought experiment: Given that the CB's value fluctuates with the demand for base money - the higher the demand, the more it can issue at zero cost - and base money demand rises in recessions, a central bank should buy equities and not government bonds (or more accurately, it should buy cyclical 'risk' assets and not counter-cyclical 'safe' assets). Its balance sheet 'risk preferences' are the opposite of the private sector's, so it should earn a large premium on its assets. But what is the correct discount rate? The discount rate of any entity should be its cost of capital, which in the case of many CB's appears to be zero, or negative. What would the PV of a private sector entity be if it's cost of capital was zero and its assets generate an equity-like return?

Ironically, I suspect the PVs in both these cases may be similar.

[JKH]

OT

NR

twitter:

sure sounds like Paul Demarais

but can't be

you would have remembered that name

[JKH]

After having graduated from the University of Ottawa, Desmarais went to Osgoode Law School[13] until he began working at a railroad and bus line, Sudbury Bus Lines, established by his grandfather. The company was sold to him for a symbolic 1 CAD, because it was almost bankrupt. He rescued the company and acquired additional bus lines in the Ottawa area and Quebec City (including Quebec Autobus, Provincial Transport and Regional Transport).[14] By 1968 the holding company which Desmarais had acquired three years earlier, Trans-Canada Corporation Fund (TCCF), owned the bus line Provincial Transport, an interest in Toronto-based Imperial Life Assurance and Gesca Ltée, (which had an interest in the Montreal paper La Presse). That year TCCF made a share-exchange offer with the Power Corporation of Canada, headquartered in Montreal, Quebec, whereby Paul Desmarais became Chairman and Chief Executive Officer and controlling shareholder.