Wednesday, March 27, 2013

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.

13 comments:

  1. Tom,

    I just came across your site today. Just wanted to thank you for all of the content. I have been meaning to study MR and I think your examples will help me a lot through my journey.

    ReplyDelete
    Replies
    1. Bullish_Bear, thanks! I wish you the best.

      Delete
  2. Hi Tom! Interesting discussion. I actually have a Bachelor's in accounting-and am close to a Masters.

    I think it's important to be careful when getting into an accounting discussion with it's partiuclar definitions when having an macroeconomic discussion. I think it's a good discussion to have but it's important to make sure we're talking apples to apples.

    "Capital" can certainly have different meanings if we're talking macro.

    ReplyDelete
    Replies
    1. Hi Mike, obviously I'm a laymen with this stuff. This post in particular I'm shaky on, but I find it's best to just dive right in as if I know what I'm doing sometimes! I'm certainly not against changing it if there are problems, so your input is very welcome. I was really trying to make sense of and summarize what I'd learned from my discussion with Joe here:

      http://pragcap.com/towards-a-mostly-cashless-monetary-system/comment-page-1#comment-141344

      Joe is a bank auditor by profession, and has provided me with lots of information over the past year. Unfortunately my communication channel with him is a little uncertain, since I don't have his email... I just wait for him to show up on pragcap and then I pounce!

      If you read the comment thread above (on pragcap) you'll see me wallowing around in confusion. I didn't really start to see a coherent way to think it through until I started this post. Now whether that "coherent" way to think of it is correct or not is another matter!!

      What gives me some confidence (just a bit) is that Joe started off telling me that "capital is equity plus certain liabilities." That "plus certain liabilities" bit is what I found most confusing and caused me the most mental gymnastics... even though I expected he'd say something like that because I'd seen similar statements before. The part that confused my non-accountant brain is squaring that with this simple view that I (continue to) cling to (at least regarding the principal amounts - rates of return are another matter) summarized below:

      asset = good
      liability = bad
      equity = good*
      capital = good

      *I have a satisfying story I tell myself why equity is "good" like an asset, but goes on the right hand side of the BS like a "bad" liability: 1) It needs to go there to balance the sheet, 2) It represents what's owned to the shareholders, and is in that sense a liability 3) It's NOT a regular liability, because you have to sum those first before calculating equity.

      My problem was that I had a hard time seeing why you'd ever ADD in "bad" liabilities to calculate "good" capital.

      I'm happy that I seem to have come full circle and can now fit that explanation of Joe's into my interpretation above w/o some of the gymnastics I'd adopted previously.

      So, like I say, I'd like to think I'm a little closer to the right answer here, but I'm completely open to being corrected!

      Delete
  3. Overall,you've got a great site. I appreciate your help in understanding how the banking system work better. I feel I have some added clarity after reading a few of your articles.

    ReplyDelete
    Replies
    1. Wow, great! That's a big complement. Thanks!

      Delete
  4. My basic concern was to distinguish capital from equity. Joe also distinguished capital from a "simple accounting perspective" vs capital from a "regulatory perspective. Get Some Dosh

    ReplyDelete
  5. 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! online loan at focusfinancialcorp.com

    ReplyDelete
  6. 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. online loan at focusfinancialcorp.com

    ReplyDelete
  7. 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. millionaire

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