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:

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

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:

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:

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

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

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:

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:

Assets | Liabilities |
---|---|

$0 | $0 |

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

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:

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.