Banking Example #7: Calculating Capital vs Equity

The purpose of this example is to demonstrate how to calculate bank capital in various ways vs equity. This post was inspired by a comment thread with Joe in Accounting at pragcap. My basic concern was to distinguish capital from equity. Joe also distinguished capital from a "simple accounting perspective" vs capital from a "regulatory perspective." I'll refer to these two broad categories as "accounting capital" and "regulatory capital" respectively. I also made use of Wikipedia articles, such as this one on Tier 2 capital. As a warning, I really need Joe in Accounting or somebody else who knows about these concepts to review what I've done here! But as is the usual case with this blog, I'll just pretend I know what I'm talking about and hope to be corrected later.

Setup: one bank (A) and one person x. Bank A's balance sheet:

Bank A

Assets	Liabilities
$120 reserves	$75 subordinated debt bond sold w/ maturity < 1 year
------------------	$45 bond sold w/ maturity > 1 year
$100 loan to x	$90 deposit for x
Negative Equity	Equity
------------------	$10

Regulatory Capital:

The Tier 1 capital in this case is the same as equity (shareholder's equity or owner's equity: see here for more details), which is simply:

Tier 1 capital = assets - liabilities = $220 - $210 = $10

After calculating Tier 1, it's possible to calculate Tier 2 capital. The subordinated debt in the liabilities column is the only component in this case. Tier 2 capital is limited to be no more than 100% of Tier 1 capital (I'm also assuming it can't be less than 0). Thus we have:

Tier 2 capital = max(0, min(subordinated debt, Tier 1 capital)) = max($0, min($75, $10)) = $10

Thus the combined Tier 1 + Tier 2 capital is $20. Now for purposes of calculating a regulatory capital adequacy ratio (CAR) we have:

CAR = (Tier 1 capital + Tier 2 capital) / (sum of risk weighted assets) = $20 / ($120*0 + $100*1) = 20%

Here I've assumed the loan to person x is high risk and thus weighted by the maximum weight, 1, while the reserves are by definition of no risk, and thus weighted by the lowest weight, 0. Had the loan been a mortgage, for example, it may have had a weight of 0.5 instead, which would have improved (increased) the CAR.

Accounting Capital:

One kind of accounting capital we might wish to calculate is working capital. This represents funds available in the near term with which to acquire other investments. It's defined as:

working capital = current assets - current liabilities

See the Wikipedia article for definitions. We'll assume the loan to x can be liquidated in less than a year for its book value, which means that

current assets = $120 reserves + $100 loan to x = $220

Current liabilities, however, does not include the $45 bond with a maturity greater than 1 year, thus we have:

current liabilities = $75 subordinated debt + $90 deposit = $165

and thus

working capital = $220 - $165 = $55

Observations:

In summary then, we have:

Bank A

Name	Value
Equity	$10
Tier 1 capital	$10
Tier 2 capital	$10
Tier 1 + Tier 2 capital	$20
Working capital	$55

You might find it strange (as I did) that calculating Tier 2 regulatory capital involved ADDING a liability, whereas the usual way of calculating capital or equity is to follow a formula like

capital or equity = set of assets - set of liabilities

with liabilities SUBTRACTED from the other term. But this isn't so strange if you consider that Tier 2 is calculated after Tier 1 and that it's limited in value to 100% of Tier 1, and that the usual purpose in calculating Tier 2 is to add it to Tier 1 to form the numerator of the CAR. Thus in forming this sum we don't want to double count the assets. In other words, the assets on the balance sheet have already been folded into Tier 1, and in fact the liabilities that contribute to Tier 2 have already been subtracted in the Tier 1 calculation. Thus adding some fraction of certain liabilities back in (up to 100%, depending on how large the "raw" calculation of Tier 2 is compared to Tier 1, i.e. the figure we get before limiting it), really just cancels that fraction of the liabilities out of what was already calculated in Tier 1, to form an expanded measure of capital. In other words:

Tier 1 capital = assets - liabilities

Tier 1 + Tier 2 capital = assets - subset of liabilities

I've oversimplified a bit here, and actually "set of" should precede all occurrences of either "assets" or "liabilities" .. only they are different sets on the two lines. For example, Tier 2 can also include other assets (it just doesn't in this case).

This concept can be further clarified by introducing an alternative method of calculating working capital. In this case we will simply add the qualifying liabilities to the equity (assuming all the assets are current assets, which they are in this case). The qualifying liability here is still just the long term $45 bond. Thus we could write:

working capital = equity + long term liabilities = $10 + $45 = $55

The long term liabilities have already been subtracted in the calculation of equity, thus what we are really doing here is simply cancelling them out in the calculation of working capital. In other words, rather than adding them you could imagine that we are simply NOT subtracting them. Here's another simple example: say bank B starts out with a clear balance sheet:

Bank B

Assets	Liabilities
$0	$0

Clearly all equity and capital calculations result in $0. Now suppose it sells a long term bond (maturity > 1 year):

Bank B

Assets	Liabilities
$100 reserves	$100 bond sold w/ maturity > 1 year

The equity is still $0, but if we note that current assets = assets, we have:

working capital = assets - current liabilities = $100 - $0 = $100

Or, alternatively:

working capital = equity + long term liabilities = $0 + $100 = $100

Now if bank B were to spend all its reserves on donuts (while the regulators weren't looking) and pass them out to the public for free in an unsuccessful attempt to attract customers (the working capital in this case was put into a terrible investment!), we'd have:

Bank B

Assets	Liabilities
$0	$100 bond sold w/ maturity > 1 year
Negative Equity	Equity
$100	-------------------------------------

Now we calculate:

working capital = equity + long term liabilities = -$100 + $100 = $0

The point is that we're not getting something for nothing here. If there are no assets at all, then we'd expect the working capital to be no greater than $0 (and our expectations are met). We're adding in qualifying liabilities in the calculation, but only because for the definition of capital at hand, we've already subtracted them in the equity term. So in a sense, rather than adding them (the qualifying liabilities), we're just NOT subtracting them.

Banking Example #1.2: Loan & Deposit Transfer (w/ Colors!)

Example of a loan and deposit transfer. Coloring of cells inspired by commenter Geoff at pragcap.com (an attempt to make it more clear, by matching assets with their related liabilities across balance sheets). Otherwise this is exactly the same as Example #1. Each color represents a financial entity: loan, deposit, etc. Cells with the same color represent two views of the same entity, one from the creditor's viewpoint (left hand column) and one from the debtor's viewpoint (right hand column). Example #10 demonstrates how these different financial entities are interrelated but have a degree of independence from one another.

Setup: one central bank (CB), two commercial banks A and B, and one person x. No reserve requirements or capital requirements and everyone's balance sheet initially clear (empty).

Initial balance sheets (for CB, A, B, and x):

CB, A, B, x

Assets	Liabilities
$0	$0

Balance sheets after x takes a $100 loan from A:

Bank A

Assets	Liabilities
$100 loan to x	$100 deposit for x

Person x

Assets	Liabilities
$100 deposit at A	$100 borrowing from A

Balance sheets after x transfers deposit from Bank A to Bank B:

Central Bank

Assets	Liabilities
$100 reserve overdraft for A	$100 reserve deposit for B

Bank A

Assets	Liabilities
$100 loan to x	$100 overdraft at CB

Bank B

Assets	Liabilities
$100 reserves	$100 deposit for x

Person x

Assets	Liabilities
$100 deposit at B	$100 borrowing from A

Balance sheet after Bank A borrows $100 of reserves from Bank B and repays CB the overdraft amount by the end of the day (note: Bank A could have borrowed from any other bank, the money markets, the Central Bank's discount window or by attracting transfer deposits, but I've chosen to show the case where it borrows from Bank B):

Central Bank

Assets	Liabilities
$0	$0

Bank A

Assets	Liabilities
$100 loan to x	$100 reserve borrowings from B

Bank B

Assets	Liabilities
$100 loan of reserves to A	$100 deposit for x

Person x

Assets	Liabilities
$100 deposit at B	$100 borrowing from A

Note that at the end, the central bank's balance sheet is again clear, yet there are $100 of reserves on loan to A from B. To see a similar case where the deposit transfer is accomplished with a purchase instead, see Example 1.1.

I recommend this writeup on the monetary system by Cullen Roche for more information. Also, I must credit this article. Also, this site animates balance sheets, and has an especially good macro page with consolidated balance sheets.  Look at this to see this same example with reserve requirements, and here to see it with both reserve and capital requirements.

List of Ways Reserves Leave the Banking System

See this post for a simplified list of where reserves can go (ignoring the distinction between inside and outside the banking system).

First of all, we need to define what is meant by reserves in the banking system. In the US this commonly includes reserves in the private banks' central bank (Federal Reserve Bank or just "Fed") reserve accounts (these are the only accounts on which the Fed pays "interest on reserves" (IOR)). It also includes "vault cash" which is physical currency/cash (paper bills and coins) stored at the private banks. It does not include the Federal government's Fed reserve account (the Treasury General Account (TGA)). It also does not include anything the Fed itself holds as an asset: reserves are never a Fed asset; they are always a Fed liability. All Fed reserve accounts are electronic. The Fed is an independent hybrid public/private institution, and thus not strictly part of the government. Of course the Treasury Department (or just "Treasury") is part of the federal government. We also need to consider other government agency Fed accounts (government sponsored enterprises, or GSEs).

Here, then, is the list of ways in which reserves leave the private banking system:

1. When entities pay taxes/tariffs/fees/fines (and those payments are transferred to the TGA or GSE Fed accounts)
2. When Treasury or GSEs auction bonds (and the proceeds are transferred to the TGA or GSE Fed accounts)
3. When private non-banks withdraw paper bills and coins (physical cash or currency) from their private bank deposits
4. When private banks repay Fed reserve loans or overdrafts (principal and interest)
5. When foreign central banks or institutions (e.g. the IMF) receive funds in their Fed reserve accounts
6. When the Fed/Treasury/GSEs sell assets (typically Treas. bonds, but also could include foreign currency, TARP assets, etc.) to private entities, for example during Fed Open Market Sales (OMSs)

First of all notice that "loaning out reserves to private non-banks" is not on the list. That's an incorrect way to think of reserves. Only chartered banks, Treasury, and certain GSEs, foreign central banks, and institutions (IMF, World Bank, etc.) can hold Fed reserve deposit accounts. Individuals, non-bank businesses, and most organizations cannot hold Fed reserve accounts. Reserves can be loaned by individual banks to other banks, and they can be transferred between banks to back/clear purchases or payments between private entities including when the purchasing entities are the banks themselves (e.g. to pay bank employee salaries, or to buy office supplies for the bank), but these events do not cause reserves to leave the banking system as a whole.

I've not made a distinction between excess reserves (ER) and required reserves (RR) in the above, although you can take the list to apply to ERs since RRs are of course required (by regulation), and thus are only absent (below their required levels) for brief periods of time. There are other ways in which, in aggregate (i.e. taking all banks as a whole), excess reserves can be converted to required reserves (and thus in that sense "leave the banking system"), but those are not covered by the above list since it makes no distinction between reserve types. Of course there are other ways reserves can leave, come into, or be converted from excess to required status by individual banks (as opposed to the banks in aggregate).

Of the six entries on the list, the first two and the interest component of the fourth* are typically reversed when the Federal government spends money. The third is reversed when currency is re-deposited in private banks by private entities. The principal component of the fourth is reversed by the Fed loaning out reserves (note that reserve loans are typically made on the inter-bank market, but the Fed stands ready as a lender of last resort, thus the bulk of Fed reserve loans are typically repaid in short order when replacement reserves are obtained from other entities). The fifth is reversed when funds exit these foreign central banks and institutions (note that foreign owned private banks are part of the banking system). The sixth is reversed by Fed Open Market Purchases (OMPs). Together OMSs and OMPs constitute Fed Open Market Operations (OMOs). So in terms of reserves permanently leaving the banking system, this only happens when the Federal government starts accumulating money in the TGA (i.e. running a surplus), when currency is permanently kept, destroyed or lost by the private sector, when Fed reserve loans are repaid, or when OMSs are not eventually reversed by OMPs. Government surpluses have historically been rare and comparatively little currency is kept, destroyed or lost by the private sector. Net "permanent" Fed reserve loans are really only made to the aggregate banking system to support reserve requirements (typically ~10% of bank demand/checking deposit liabilities). This leaves OMSs as the main way in which reserves leave the banking system most of the time. Conversely, OMPs are the main way reserves are injected into the banking system most of the time. OMPs are how "Quantitative Easing" (QE) is accomplished. Treasury deficit spending does NOT inject reserves into the banking system. Instead it injects net financial assets (i.e. Treasury bonds) into private hands, and at the same time takes the proceeds from the bond auctions and spends them back into the private sector. Thus, in a sense (though not literally), the Treasury obtains and spends private bank created money (inside money) when it deficit spends. This is not literally true, since the Treasury spends from it's Fed deposit (the TGA), but this deposit was funded largely through the private sector: either by taxes or bond auctions. Non-banks purchasing Treasury debt or paying taxes use inside money to do so. In the case of private non-banks, funding the TGA is the mirror image of spending from the TGA: funding involves the elimination of a bank deposit and the simultaneous transfer of outside money from the banks (which may have to borrow from the Fed for this purpose) into the TGA. Spending involves the transfer of outside money from the TGA to the banks, and the simultaneous creation of bank deposits for the non-bank payees. Of course what counts as money can be debated. In our system, bank deposits have a very high degree of moneyness.

Assuming that Treasury immediately spends its Fed deposit (i.e. the TGA is immediately emptied: not a terrible assumption with deficit spending), then the sum of non-bank held commercial bank deposits (& cash) is equal to the sum of private bank loans and the Treasury debt held by the Fed & banks. Why is this? (henceforth in this discussion I'll use "bank deposits" to mean "bank deposits & cash" for simplicity, since cash is mostly just withdrawn deposits). For starters, since "loans create deposits" we'd expect to be able to express the bank deposits of non-banks in terms of loans (loans to Treasury being called "Treasury debt" here). But why doesn't this statement include all Treasury debt? To see why consider the following example: assume we start off with everyone's balance sheets clear (zeros for assets and liabilities), and then the Fed & banks acquire $X in Treasuries. This results in $X in bank deposits when Treasury spends. Now these deposits can be traded by the non-banks for the Treasuries and back again.... in any amount up to $X. Thus bank deposits can vary between $0 and $X under this trade. Now assume that bank loans in the amount of $Y are made to non-banks. Now non-bank deposits can vary between $Y and $(X+Y) depending on the amount of Treasuries traded with the Fed & banks (the maximum still being $X). At this point we've accounted for, in terms of debt, all existing non-bank bank deposits. If the non-banks now use any part of these deposits to purchase Treasuries from Treasury, those deposits will be returned to them (in aggregate) once Treasury spends and thus the aggregate bank deposits will not change. Since this process can be repeated an indefinite number of times (assuming Treasury continues to auction bonds and spends all the proceeds each time), the total amount of T-bonds acquired by the non-banks in this manner can total $Z, where Z can be any positive value, without changing the aggregate non-bank bank deposits. If the Fed & banks purchase some of these new Treasuries from the non-banks, the non-bank bank deposits will increase accordingly. Thus the total amount of bank deposits held by the non-banks can vary between $Y and $(X+Y+Z) depending on how many Treasuries are traded with the Fed & banks. Only to the extent that the Fed & banks hold a part of the $(X+Z) in existing Treasuries, are non-bank bank deposits elevated above $Y, and thus the only case in which these bank deposits total the full $(X+Y+Z) is when the Fed & banks purchase all existing Treasury debt.

Note that I'm simplifying a bit here by glossing over the process by which Treasury Tax and Loan (TT&L) accounts are created as an intermediate step before transferring funds to the TGA -- this transfer being necessary before the funds can be spent by Treasury.

These JKH and Ramanan comments at monetaryrealism.com provides further insight

*Note that almost all the interest paid to the Fed for its reserve loans is remitted by the Fed to Treasury, and thus will almost certainly be spent again into the private sector.

Banking Example #3.2: Capital Requirements (Simplified Stock Issuance)

Example loan and deposit transfer with both reserve and capital regulatory requirements. Required capital is partly raised by a stock issuance and partly by retained earnings (from a loan origination fee). Example #3 is a simpler variation on this without the stock sale (i.e. capital requirements are met entirely through retained earnings). Example 3.1 is a more complicated example documenting the balance sheets associated with a stock investor as well.

Setup: one central bank (CB), two commercial banks A and B and one person (x). Reserve requirements are 10% of deposits, capital requirements are 10% combined Tier 1 and  Tier 2. This example was inspired by this John Carney article at CNBC.  

On the bank balance sheets which follow, loans and the capital requirements they induce will be colored green, while deposits and the reserve requirements they induce will be colored red.  Assume starting out that that any preexisting entries on the CB's balance sheet (e.g. having to do with stock investors owning $5 in the first place*) are summarized in the cells "preexisting assets" and "preexisting liabilities." You can assume, of course, that the CB's balance sheet is initially in balance, i.e.: preexisting assets = preexisting liabilities

Note that all balance sheets are shown only on the first and last steps. Only balance sheets which change are shown on the intermediate steps.


Initial balance sheets for CB, A, B and x:

A, B, x

Assets	Liabilities
$0	$0

CB

Assets	Liabilities
preexisting assets	preexisting liabilities

Balance sheet after Bank A sells $5 of stock to unnamed investors:

CB

Assets	Liabilities
preexisting assets	preexisting liabilities - $5
-----------------	$5 reserve deposit for A

Bank A

Assets	Liabilities
$5 reserves at CB	$0
Negative Equity	Equity
------------------	$5

Balance sheets after x takes a $100 loan from bank A (note: I've chosen to show that bank A meets half its capital requirements by charging a $5 origination fee to x for the loan [please read Carney above for other ways to find capital and reserves], which it keeps as retained earnings. This lowers x's deposit by $5, and thus lowers the reserve requirements from $10 to $9.50. I've chosen to show bank A borrowing $4.50 of this reserve requirement from the CB):

CB

Assets	Liabilities
preexisting assets	preexisting liabilities - $5
$4.50 reserve loan to A	$9.50 reserve deposit for A

Bank A

Assets	Liabilities
$100 loan to x	$95 deposit for x
$9.50 required reserves	$4.50 reserve borrowing from CB
Negative Equity	Equity
------------------------	$10 required capital

Person x

Assets	Liabilities
$95 deposit at A	$100 borrowing from A
Negative Equity	Equity
$5	------------------------

Balance sheets after x transfers deposit from Bank A to Bank B:

Central Bank

Assets	Liabilities
preexisting assets	preexisting liabilities - $5
$4.50 reserve loan to A	$95 reserve deposit for B
$85.50 reserve overdraft for A	--------------------------

Bank A

Assets	Liabilities
$100 loan to x	$85.50 reserve overdraft at CB
--------------------	$4.50 reserve borrowing from CB
Negative Equity	Equity
--------------------	$10 required capital

Bank B

Assets	Liabilities
$95 reserves ($9.50 required)	$95 deposit for x

Person x

Assets	Liabilities
$95 deposit at B	$100 borrowing from A
Negative Equity	Equity
$5	------------------------

Balance sheets after Bank A borrows $85.50 of reserves from Bank B and repays CB overdraft by the end of the day (note: Bank A could have borrowed from any other bank, the money markets or the CB's discount window or by attracting new transfer deposits, but I've chosen to show the case where it borrows from Bank B):

CB

Assets	Liabilities
preexisting assets	preexisting liabilities - $5
$4.50 reserve loan to A	$9.50 reserve deposit for B

Bank A

Assets	Liabilities
$100 loan to x	$85.50 reserve borrowing from B
--------------------	$4.50 reserve borrowing from CB
Negative Equity	Equity
--------------------	$10 required capital

Bank B

Assets	Liabilities
$9.50 required reserves	$95 deposit for x
$85.50 loan of reserves to A	-----------------

Person x

Assets	Liabilities
$95 deposit at B	$100 borrowing from A
Negative Equity	Equity
$5	------------------------

Note: as Carney states, these are simplified reserve and capital requirements. As in the Wikipedia article I link to above, what I'm really doing here for the capital requirements is calculating a capital adequacy ratio (CAR) as the ratio of capital to the sum of risk weighted assets. The loan on bank A's balance sheet is risky, thus it's weighted by the maximum weight 1 (lower risk assets get smaller weights, which increases the CAR, all else being equal). I'm also taking equity = capital. Thus the CAR in this case is $10 equity / $100 loan = 0.10 = 10% which just meets the CAR requirement (10% or greater is satisfactory).

Joe in Accounting helped me with this example, but I still need to run it by him to make sure it's correct.

*To see a related example documenting the balance sheet changes associated with a stock investor as well (rather than having unnamed investors) see Example 3.1.

Banking Example #3.1: Capital Requirements (Stock Issuance)

Example loan and deposit transfer with both reserve and capital regulatory requirements. Required capital is partly raised by a stock issuance and partly by retained earnings (from a loan origination fee). Example #3 is a simpler variation on this without the stock sale (i.e. capital requirements are met entirely through retained earnings). Example 3.2 is slightly simplified in that balance sheets associated with unnamed investors are not shown.

Setup: one central bank (CB), three commercial banks A, B and C and two people x and y. Reserve requirements are 10% of deposits, capital requirements are 10% combined Tier 1 and  Tier 2. This example was inspired by this John Carney article at CNBC.  

On the bank balance sheets which follow, loans and the capital requirements they induce will be colored green, while deposits and the reserve requirements they induce will be colored red. Assume starting out that (similar to Example 3) person y has borrowed $10 from Bank C (with a $1 loan origination fee charged by C to meet its capital requirements), and that she (y) has withdrawn the remaining $9 in cash* (i.e. paper money). Assume further than Bank C has in turn borrowed these cash reserves from the CB.

Note that all balance sheets are shown only on the first and last steps. Only balance sheets which change are shown on the intermediate steps.

Initial balance sheets for CB, A, B, C, x and y:

A, B, x

Assets	Liabilities
$0	$0

CB

Assets	Liabilities
$9 reserve loan to C	$9 cash issued

Bank C

Assets	Liabilities
$10 loan to y	$9 borrowing of (cash) reserves from CB
Negative Equity	Equity
------------------	$1 retained earnings (required capital)

Person y

Assets	Liabilities
$9 cash	$10 borrowings from C
Negative Equity	Equity
$1	-----------------------

Balance sheet after Bank A sells $5 of stock to person y (keep in mind that vault cash is a form of bank reserves: assume it's sold back to the CB for electronic reserves as soon as cash is deposited at the bank rather than being held as a vault cash inventory):

CB

Assets	Liabilities
$9 reserve loan to C	$4 cash issued
---------------------	$5 reserve deposit for A

Bank A

Assets	Liabilities
$5 reserves at CB	$0
Negative Equity	Equity
------------------	$5

Person y

Assets	Liabilities
$4 cash	$10 borrowings from C
$5 stock in A	-----------------------
Negative Equity	Equity
$1	-----------------------

Balance sheets after x takes a $100 loan from bank A (note: I've chosen to show that bank A meets half its capital requirements by charging a $5 origination fee to x for the loan [please read Carney above for other ways to find capital and reserves], which it keeps as retained earnings. This lowers x's deposit by $5, and thus lowers the reserve requirements from $10 to $9.50. I've chosen to show bank A borrowing $4.50 of this reserve requirement from the CB):

CB

Assets	Liabilities
$9 reserve loan to C	$4 cash issued
$4.50 reserve loan to A	$9.50 reserve deposit for A

Bank A

Assets	Liabilities
$100 loan to x	$95 deposit for x
$9.50 required reserves	$4.50 reserve borrowing from CB
Negative Equity	Equity
------------------------	$10 required capital

Person x

Assets	Liabilities
$95 deposit at A	$100 borrowing from A
Negative Equity	Equity
$5	------------------------

Balance sheets after x transfers deposit from Bank A to Bank B:

Central Bank

Assets	Liabilities
$9 reserve loan to C	$4 cash issued
$4.50 reserve loan to A	$95 reserve deposit for B
$85.50 reserve overdraft for A	--------------------------

Bank A

Assets	Liabilities
$100 loan to x	$85.50 reserve overdraft at CB
--------------------	$4.50 reserve borrowing from CB
Negative Equity	Equity
--------------------	$10 required capital

Bank B

Assets	Liabilities
$95 reserves ($9.50 required)	$95 deposit for x

Person x

Assets	Liabilities
$95 deposit at B	$100 borrowing from A
Negative Equity	Equity
$5	------------------------

Balance sheets after Bank A borrows $85.50 of reserves from Bank B and repays CB overdraft by the end of the day (note: Bank A could have borrowed from any other bank, the money markets or the CB's discount window or by attracting new transfer deposits, but I've chosen to show the case where it borrows from Bank B):

CB

Assets	Liabilities
$9 reserve loan to C	$4 cash issued
$4.50 reserve loan to A	$9.50 reserve deposit for B

Bank A

Assets	Liabilities
$100 loan to x	$85.50 reserve borrowing from B
--------------------	$4.50 reserve borrowing from CB
Negative Equity	Equity
--------------------	$10 required capital

Bank B

Assets	Liabilities
$9.50 required reserves	$95 deposit for x
$85.50 loan of reserves to A	-----------------

Bank C

Assets	Liabilities
$10 loan to y	$9 borrowing of reserves from CB
Negative Equity	Equity
------------------	$1 retained earnings (required capital)

Person x

Assets	Liabilities
$95 deposit at B	$100 borrowing from A
Negative Equity	Equity
$5	------------------------

Person y

Assets	Liabilities
$4 cash	$10 borrowings from C
$5 stock in A	-----------------------
Negative Equity	Equity
$1	-----------------------

Note: as Carney states, these are simplified reserve and capital requirements. As in the Wikipedia article I link to above, what I'm really doing here for the capital requirements is calculating a capital adequacy ratio (CAR) as the ratio of capital to the sum of risk weighted assets. The loan on bank A's balance sheet is risky, thus it's weighted by the maximum weight 1 (lower risk assets get smaller weights, which increases the CAR, all else being equal). I'm also taking equity = capital. Thus the CAR in this case is $10 equity / $100 loan = 0.10 = 10% which just meets the CAR requirement (10% or greater is satisfactory).

Joe in Accounting helped me with this example, but I still need to run it by him to make sure it's correct.

*The reason I chose to show cash withdrawn by person y was so that I could slightly simplify the example by eliminating reserve requirements for bank C by eliminating y's deposit at bank C. This puts y in the slightly unrealistic position of purchasing stock in Bank A with cash. I still needed to show bank C meeting its capital requirements though. To see a somewhat simplified example using unnamed stock investors (whose associated balance sheets are not shown) see Example 3.2.

Banking Example #6: QE: CB Holds Gov Debt to Maturity

This example looks at what happens when the central bank (CB) purchases Treasury bonds from the public and holds them to maturity. For just the purchase see Example 4.

Setup: one central bank (CB), one government Treasury Department (Treasury), one commercial bank (A), and one person (x). No reserve or capital requirements. All balance sheets are initially clear (empty). This post was inspired by a question posed in a comment here. This post doesn't really answer that question directly, but reflects a thought experiment I did at the time, wherein I wondered what would happen if the CB kept purchasing and holding Treasury bonds to maturity.

Initial balance sheets :

CB, Treasury, A, x

Assets	Liabilities
$0	$0

Balance sheets after x takes a loan from A:

Bank A

Assets	Liabilities
$100 loan to x	$100 deposit for x

Person x

Assets	Liabilities
$100 deposit at A	$100 borrowing from A

Balance sheets after x buys a Treasury bond from Treasury (I'm skipping the step where A experiences an overdraft at the CB when x's deposit is transferred to Treasury to clear the bond purchase, and I'm going straight to bank A borrowing funds to cover the overdraft, in this case from the CB):

Treasury

Assets	Liabilities
$100 reserves	$100 Treasury bond held by x

CB

Assets	Liabilities
$100 loan of reserves to A	$100 deposit for Treasury

Bank A

Assets	Liabilities
$100 loan to x	$100 reserve borrowings from CB

Person x

Assets	Liabilities
$100 Treasury bond	$100 borrowing from A

Balance sheets after the CB purchases the Treasury bond from x:

Treasury

Assets	Liabilities
$100 reserves	$100 Treasury bond held by CB

CB

Assets	Liabilities
$100 loan of reserves to A	$100 deposit for Treasury
$100 Treasury bond	$100 deposit for A

Bank A

Assets	Liabilities
$100 loan to x	$100 reserve borrowings from CB
$100 reserves	$100 deposit for x

Person x

Assets	Liabilities
$100 deposit at A	$100 borrowing from A

Balance sheets after Bank A repays its reserve borrowings from the CB:

Treasury

Assets	Liabilities
$100 reserves	$100 Treasury bond held by CB

CB

Assets	Liabilities
$100 Treasury bond	$100 deposit for Treasury

Bank A

Assets	Liabilities
$100 loan to x	$100 deposit for x

Person x

Assets	Liabilities
$100 deposit at A	$100 borrowing from A

Balance sheets after Treasury pays x for services rendered to the government:

Treasury

Assets	Liabilities
$0	$100 Treasury bond held by CB
Negative Equity	Equity
$100	--------------------------------

CB

Assets	Liabilities
$100 Treasury bond	$100 reserve deposit for A

Bank A

Assets	Liabilities
$100 loan to x	$200 deposit for x
$100 reserves	------------------

Person x

Assets	Liabilities
$200 deposit at A	$100 borrowing from A
Negative Equity	Equity
------------------	$100

Balance sheets after x repays loan to A:

Treasury

Assets	Liabilities
$0	$100 Treasury bond held by CB
Negative Equity	Equity
$100	--------------------------------

CB

Assets	Liabilities
$100 Treasury bond	$100 reserve deposit for A

Bank A

Assets	Liabilities
$100 reserves	$100 deposit for x

Person x

Assets	Liabilities
$100 deposit at A	$0
Negative Equity	Equity
------------------	$100

Balance sheets after Treasury sells another bond to x to cover its upcoming principal payment to the CB:

Treasury

Assets	Liabilities
$100 reserves	$100 Treasury bond held by CB
------------------	$100 Treasury bond held by x
Negative Equity	Equity
$100	--------------------------------

CB

Assets	Liabilities
$100 Treasury bond	$100 reserve deposit for Treasury

Bank A

Assets	Liabilities
$0	$0

Person x

Assets	Liabilities
$100 Treasury bond	$0
Negative Equity	Equity
------------------	$100

Balance sheets after the first Treasury bond matures and Treasury repays the principal to the CB:

Treasury

Assets	Liabilities
$0	$100 Treasury bond held by x
Negative Equity	Equity
$100	------------------------------

CB

Assets	Liabilities
$0	$0

Bank A

Assets	Liabilities
$0	$0

Person x

Assets	Liabilities
$100 Treasury bond	$0
Negative Equity	Equity
------------------	$100

Balance sheet after CB buys Treasury bond from person x:

Treasury

Assets	Liabilities
$0	$100 Treasury bond held by CB
Negative Equity	Equity
$100	------------------------------

CB

Assets	Liabilities
$100 Treasury bond	$100 reserve deposit for A

Bank A

Assets	Liabilities
$100 reserves	$100 deposit for x

Person x

Assets	Liabilities
$100 deposit at A	$0
Negative Equity	Equity
------------------	$100

Now at this point, if the government wants to continue deficit spending, it must now sell another Treasury bond (e.g. to x), and continue this process. As this cycle of events continues, x's equity will continue to climb by $100 each time and Treasury's equity will continue to decrease by $100 each time (with the CB's and Bank A's equity staying at $0). Person x will not need to take a loan the next time around since he has $100 of deposits with which to purchase the next bond. Of course the government can also tax Person x to reduce the amount of deficit spending it must engage in to keep this process going. If the CB were to purchase every Treasury bond in this manner, the effect would look similar to "monetizing the debt." Ultimately CB and A are performing as intermediaries. In reality the CB only purchases a small fraction of the outstanding Treasury bonds under Quantitative Easing (QE) operations, for example. Plus, it's ultimately up to x whether or not to sell the bonds, since the CB does not directly purchase bonds from Treasury.

This example is over simplified and has many shortcomings. Interest, taxes, fees, and changing bond prices are not covered. These would be significant over the course of a bond's issuance to maturity. Also, Bank A would not likely borrow from the CB to repay its reserve overdraft, but instead would most likely borrow from another bank. Also, x would not likely take a loan out to purchase a Treasury bond, since it would probably experience a negative spread on the two interest rates if it did so. However, someone else may take out such a loan to hire x for his services, which would provide x with the funds to purchase a Treasury bond: thus you can think of x and A as stand-ins for the aggregate non-bank and bank private sectors.

Banking Example #5: Bank 'Spends' Excess Reserves

Examples of a bank spending money; in one case by crediting a deposit and in another by transferring reserves.

Setup: Two commercial banks, A and B, and two people, x and y. Reserve requirements are 10% of deposits, but there are no capital requirements. Person x banks at A and person y banks at B.

Initial balance sheets :

B, x, y

Assets	Liabilities
$0	$0

Bank A

Assets	Liabilities
$100 reserves (excess)	$0
Negative Equity	Equity
-----------------------	$100

Balance sheets after person x does $10 worth of work for Bank A and the bank pays him.

Bank A

Assets	Liabilities
$100 reserves ($1 required, $99 excess)	$10 deposit for x
Negative Equity	Equity
-----------------------------------------	$90

Person x

Assets	Liabilities
$10 deposit at A	$0
Negative Equity	Equity
-----------------------	$10

Balance sheets after person y does $10 worth of work for Bank A and the bank pays her.

Bank A

Assets	Liabilities
$90 reserves ($1 required, $89 excess)	$10 deposit for x
Negative Equity	Equity
----------------------------------------	$80

Bank B

Assets	Liabilities
$10 reserves ($1 required, $9 excess)	$10 deposit for y
Negative Equity	Equity
---------------------------------------	$0

Person x

Assets	Liabilities
$10 deposit at A	$0
Negative Equity	Equity
-----------------------	$10

Person y

Assets	Liabilities
$10 deposit at B	$0
Negative Equity	Equity
-----------------------	$10

Note that since person x banks at A (the same bank buying his services) no reserves leave the bank to pay him. However, since person y banks at B, in order for Bank A to pay her, reserves are transferred to Bank B to back her newly credited deposit. In both cases, however, Bank A's equity decreases. Also note that only banks (and the Treasury/gov) have Fed deposit accounts, thus there are limited ways in which reserves leave the banking system (Treasury/gov Fed deposit accounts are not considered part of this "system."). Notice, for example, above that there are still $100 of reserves in aggregate (banks A and B) at the end of this scenario. Of this, $2, in total, was changed from "excess" reserve status to "required" reserve status. Joe in Accounting helped me with this example here and here.

Aggregation:

Now in terms of aggregates, let's look how the aggregated commercial banks purchased work from the aggregated non-bank private sector in this case (assuming each of these aggregated sectors consists of nothing but Banks A & B for the former, and Persons x & y for the latter):

Commercial Banking Sector

Assets	Liabilities
$100 reserves ($2 required, $98 excess)	$20 deposits
Negative Equity	Equity
----------------------------------------	$80

Non-Bank Private Sector

Assets	Liabilities
$20 deposits	$0
Negative Equity	Equity
-----------------------	$20

So here the aggregated commercial banking sector purchased net $20 of services from the aggregated non-bank private sector (the "public") by crediting their bank deposits. Notice that the reserve level of the aggregate banks did not change. In a sense, reserves were not used for these purchases. This same method of payment for purchases applies to anything the aggregated banks purchase from the aggregated non-banks: the banks credit the non-banks' bank deposits. What they buy/pay-for can be anything*: physical cash (e.g. reserve notes and coins), employee time (salaries & bonuses), electricity, office supplies, donuts, contractor services, shareholder dividends, interest to depositors and bank-bond holders, rents, leases, real-estate, Tsy or other securities, or loans (i.e. the banks make new loans to borrowers).

Likewise (though not shown) the aggregate commercial banks sell-to/get-paid-by the non-bank private sector by debiting their bank deposits. This could be for cash, interest, loan principal, points or other service charges on loans & mortgages, account fees, money wiring charges, late fees, certificates of deposit (CDs), savings deposits, Tsy debt or other investment assets, or bank issued stocks (equity) or bonds.

So in summary when banks buy (sell) anything from (to) non-banks, payments are settled by crediting (debiting) bank deposits. The only exception to this is when the banks make purchases or accept payments with cash, but these cases can be seen conceptually as a two step process: first a credit or debit is made to a conceptual bank deposit, and then this conceptual bank deposit is debited or credited by either selling cash to or by buying cash from this same conceptual bank deposit holder: the net change in balance to the conceptual bank deposit is thus $0.

To summarize then, in aggregate:

1. Banks buy stuff from non-banks (including new loans) by crediting bank deposits
2. Banks sell stuff to non-banks (including the principal on loans) by debiting bank deposits

Notes:
*Subject to capital constraints

Banking Example #4: Quantitative Easing

See Example #4.1 for a simplified comparison of the two kinds of QE. The example below focuses on the second kind, which is more prevalent.

Setup: one central bank (CB), one commercial bank (A), and one person x. Reserve requirements are 10% of deposits, but there are no capital requirements. Person x has previously obtained a $100 loan from Bank A*, and Bank A obtained $10 in required reserves by borrowing it from the CB. Person x also has a $200 Treasury bond, which he intends to sell** (in this case to the CB during a CB Quantitative Easing (QE) operation):

Initial balance sheets :

CB

Assets	Liabilities
$10 loan to A	$10 reserve deposit for A

Bank A

Assets	Liabilities
$100 loan to x	$100 deposit for x
$10 reserves (required)	$10 borrowing from CB
Negative Equity	Equity
-----------------------	$0

Person x

Assets	Liabilities
$100 deposit at A	$100 borrowings from A
$200 Treasury bond	-------------------------
Negative Equity	Equity
-----------------------	$200

Balance sheets after CB purchases $200 Treasury bond from Person x during QE (and in the process credits Person x's bank, Bank A, with $200 of reserves), and Bank A repays the $10 of reserves it borrowed from the CB:

CB

Assets	Liabilities
$200 Treasury bond	$200 reserve deposit for A

Bank A

Assets	Liabilities
$100 loan to x	$300 deposit for x
$200 reserves ($30 required, $170 excess)	-------------------
Negative Equity	Equity
-------------------------------------------	$0

Person x

Assets	Liabilities
$300 deposit at A	$100 borrowings from A
Negative Equity	Equity
-----------------------	$200

Take note that QE did not change the equity on anybody's balance sheet, including the CB's. Bank A ends up with $170 of excess reserves (up from $0 initially), but its equity (assets - liabilities) remains at $0. Also keep in mind that I'm not showing the Treasury Department's balance sheet, on which the $200 bond would appear as a liability (I could also show an offsetting $200 in the Treasury Dept.'s Fed deposit account as an asset, supposing that money had not yet been spent). The CB and the Treasury Dept. are different entities.

Another thing to note is that although person x's equity position doesn't change immediately before or after this bond sale, his position may have changed over the course of a QE operation, since the Fed tends to buy enough of the bonds in the market so as to affect the price of Treasury bonds (and some would argue, all financial assets in general). However even the existence of this asset price effect is somewhat controversial. Of course Treasury continues to auction new bonds. The overall effect is beyond the scope of this post, but others have covered it, such as pragcap and FTAlphaVille.

*Obviously this initial loan is not required for the example. I included it to parallel a ZeroHedge article with a chart demonstrating how loans had previously (before QE) tracked deposits, but since the start of QE deposits have grown faster than loans. As Cullen Roche has pointed out on pragcap, the reason for this divergence is not as mysterious as the ZH article implies it is (it's basically for the reason illustrated in this example). Cullen has also stated that about 20% of Treasury debt is held by commercial banks.

**Note also that the bank could also sell Treasury bonds directly to the Fed if it owned any. In that case the bank's balance sheet would not expand. However, I've chosen to show what I believe is the more typical scenario, wherein the Fed purchases the bonds from a private non-bank entity (perhaps using the bank as an intermediary for the sale).

Cullen Roche has cited a NY Fed SOMA desk official he spoke with to support this assertion about who the primary bond investors are (in this case the quote actually refers to who the ultimate buyer is at the Treasury auction):

“The primary way dealers finance their bond purchases is in the repo market. So here is one scenario. Funds are wired from the dealer’s account at its clearing bank to Treasury on issuance day. During the day, the clearing bank provides intraday credit to the dealer, so the dealer is borrowing from the bank. That same day, the dealer enters into a repo, pledging the newly acquired Treasury as collateral. The other side of the repo is likely to be a money market mutual fund or other money market investor. Therefore, by the end of the day, and for the overnight period, the money market investor is effectively funding the dealer’s position. Of course, there are a variety of ways in which positions can be funded, but the repo market is the key one.”

Banking Example #3: Capital Requirements

Example loan and deposit transfer with both reserve and capital regulatory requirements.

Setup: one central bank (CB), two commercial banks A and B, and one person x. Reserve requirements are 10% of deposits, capital requirements are 10% combined Tier 1 and  Tier 2 and everyone's balance sheet is initially clear (empty). This example was inspired by this John Carney article at CNBC. Examples 3.1 and 3.2 are more complicated versions (with 3.1 being more complicated than 3.2) in that they show the required capital raised partially through a stock sale, whereas in this example capital is raised entirely through a loan origination fee.

On the bank balance sheets which follow, loans and the capital requirements they induce will be colored green, while deposits and the reserve requirements they induce will be colored red.

Initial balance sheets (for CB, A, B, and x):

CB, A, B, x

Assets	Liabilities
$0	$0

Balance sheets after x takes a $100 loan from bank A (note: I've chosen to show that bank A meets its capital requirements by charging a $10 origination fee to x for the loan [please read Carney above for other ways to find capital and reserves], which it keeps as retained earnings. This lowers x's deposit by $10, and thus lowers the reserve requirements from $10 to $9. I've chosen to show bank A borrowing this $9 from the CB):

Central Bank

Assets	Liabilities
$9 reserve loan to A	$9 reserve deposit for A

Bank A

Assets	Liabilities
$100 loan to x	$90 deposit for x ($100 loan - $10 origination fee)
$9 required reserves	$9 reserve borrowing from CB
Negative Equity	Equity
--------------------	$10 retained earnings

Person x

Assets	Liabilities
$90 deposit at A	$100 borrowing from A
Negative Equity	Equity
$10	------------------------

Balance sheets after x transfers deposit from Bank A to Bank B:

Central Bank

Assets	Liabilities
$9 reserve loan to A	$90 reserve deposit for B
$81 reserve overdraft for A	---------------------------

Bank A

Assets	Liabilities
$100 loan to x	$81 reserve overdraft at CB
--------------------	$9 reserve borrowing from CB
Negative Equity	Equity
--------------------	$10 retained earnings

Bank B

Assets	Liabilities
$90 reserves ($9 required)	$90 deposit for x

Person x

Assets	Liabilities
$90 deposit at B	$100 borrowing from A
Negative Equity	Equity
$10	------------------------

Balance sheets after Bank A borrows $81 of reserves from Bank B and repays CB overdraft by the end of the day (note: Bank A could have borrowed from any other bank, the money markets or the Central Bank's discount window or by attracting new transfer deposits, but I've chosen to show the case where it borrows from Bank B):

Central Bank

Assets	Liabilities
$9 reserve loan to A	$9 reserve deposit for B

Bank A

Assets	Liabilities
$100 loan to x	$81 reserve borrowing from B
--------------------	$9 reserve borrowing from CB
Negative Equity	Equity
--------------------	$10 retained earnings

Bank B

Assets	Liabilities
$9 required reserves	$90 deposit for x
$81 loan of reserves to A	-----------------

Person x

Assets	Liabilities
$90 deposit at B	$100 borrowing from A
Negative Equity	Equity
$10	------------------------

Note: as Carney states, these are simplified reserve and capital requirements. As in the Wikipedia article I link to above, what I'm really doing here for the capital requirements is calculating a capital adequacy ratio (CAR) as the ratio of capital to the sum of risk weighted assets. The loan on bank A's balance sheet is risky, thus it's weighted by the maximum weight 1 (lower risk assets get smaller weights, which increases the CAR, all else being equal). I'm also taking equity = capital. Thus the CAR in this case is $10 equity / $100 loan = 0.10 = 10% (10% or greater is satisfactory).

For an interesting discussion of what banks with 100% capital requirements might be like, look at this Nick Rowe article. 

Banking Example #1.1: Loan and Purchase

Example of a bank customer using a loan to make a purchase. See Example 1 for an example of a loan and deposit transfer.

Setup: one central bank (CB), two commercial banks A and B, and two people x and y. Person x banks at A and person y banks at B. No reserve requirements or capital requirements and everyone's balance sheet initially clear (empty).

Initial balance sheets (for CB, A, B, x and y):

CB, A, B, x, y

Assets	Liabilities
$0	$0

Balance sheets after x takes a $100 loan from A:

Bank A

Assets	Liabilities
$100 loan to x	$100 deposit for x

Person x

Assets	Liabilities
$100 deposit at A	$100 borrowing from A

Balance sheets after x purchases $100 of services from y:

Central Bank

Assets	Liabilities
$100 reserve overdraft for A	$100 reserve deposit for B

Bank A

Assets	Liabilities
$100 loan to x	$100 overdraft at CB

Bank B

Assets	Liabilities
$100 reserves	$100 deposit for y

Person x

Assets	Liabilities
$0 deposit at A	$100 borrowing from A
Negative Equity	Equity
$100	------------------------

Person y

Assets	Liabilities
$100 deposit at B	$0
Negative Equity	Equity
------------------	$100

Balance sheet after Bank A borrows $100 of reserves from Bank B and repays CB the overdraft amount by the end of the day (note: Bank A could have borrowed from any other bank, the Central Bank's discount window, or the money market or it could have attracted new transfer deposits but I've chosen to show the case where it borrows from Bank B):

Central Bank

Assets	Liabilities
$0	$0

Bank A

Assets	Liabilities
$100 loan to x	$100 reserve borrowings from B

Bank B

Assets	Liabilities
$100 loan of reserves to A	$100 deposit for y

Person x

Assets	Liabilities
$0 deposit at A	$100 borrowing from A
Negative Equity	Equity
$100	------------------------

Person y

Assets	Liabilities
$100 deposit at B	$0
Negative Equity	Equity
------------------	$100

At the conclusion of this example, person y has $100 of "permanent" bank created (inside/endogenous) money unencumbered by any offsetting liability (note, however, that person x is encumbered by this liability without an offsetting asset!), and yet, just as in Example 1, the central bank's balance sheet still ends up clear.

Note: it doesn't have to be services that x purchases from y of course, but this prevents me from having to put some object on y's initial balance sheet. However, assuming it's some kind of object x purchases from y, such as a car, could help in imagining A giving x the loan in the first place since that could serve as collateral. Also, if we were to add in reserve requirements, A then has a collateralized loan to in turn offer the CB as collateral when borrowing the required reserves which the CB keeps on its balance sheet "permanently" at the conclusion of a similar scenario with reserve requirements (see Banking Example #2).