The Brown-Vitter bill, which two senators plan to introduce in an effort to dramatically raise bank capital requirements, has caused a range of fairly predictable reactions, and a few strange ones. Here, for instance, is a lobbyist complaining about “raising required capital to comically high levels,” but the comedy is perhaps elusive. But one stylized fact about bank capital that I find a little funny is that it is always the same; after a certain number of drinks this chart is hilarious:

What that says – perhaps a bit unclearly – is that if a bank is going to add some assets, it will do it by taking on debt; and if it’s going to reduce its debt, it will do so by selling assets; and the one thing that it won’t ever do is change the amount of equity it has. Capital ratios change, but capital amounts basically don’t (except to grow verrrrrrry slowly and steadily over time); all the action is in the denominator.

Consider what that chart means for Brown-Vitter: on Friday I calculated that the bill would require adding, in round numbers, $1.2 trillion of capital at the top 6 banks, all at once.1 But that holds bank assets constant, which is not how it generally works. Of course it’s possible that this new law would break the pattern of banks always having the same amount of equity and just adjusting their debt, and cause them to actually increase their equity dramatically; I suspect that’s roughly speaking the intention.

Another possibility is that banks would keep doing what they’ve always done and bring up equity ratios by reducing assets; the amount of equity would remain constant, as it has in the past. On that math, the six big banks would have to reduce assets by $7 trillion. Out of a total of $9.5 trillion currently. So like a 72% reduction in bank lending and investing and what-have-you.2 Eep?

Perhaps there is comedy there though I’m not sure. That chart has been floating around various places but I swiped it just now from this paper,3 by Tobias Adrian of the NY Fed and Hyun Song Shin of Princeton, musing about why it might be. Or, rather, they just assume the constantness of bank equity, and question why amounts of bank debt change. What they come up with is that leverage moves inversely to value-at-risk, which you can sort of see in this chart:

If you squint, it looks a little like “Unit VaR” ($ of value-at-risk / $ of assets) and “leverage” ($ of assets / $ of equity) move inversely, while “Var/E” ($ of value-at-risk / $ of equity) remains relatively constant.4 So the basically fixed amount of bank equity funds a variable amount of bank assets, but a constant amount of bank risk. As they put it:

Value-at-Risk turns out to be informative in explaining the way that firms manage their balance sheets. Consider the so-called Value-at-Risk (VaR) rule, which stipulates that the financial intermediary maintains enough equity E to cover its Value-at-Risk. The VaR rule can be stated equivalently as maintaining enough equity E so that the bank’s probability of failure is kept constant, set to the confidence threshold associated with the VaR measure used by the bank.

And then they go off and derive a plausible mechanism for it, which is basically that a bank’s creditors want to keep a constant probability of default and banks want to be as levered as they can be within that constraint.5

I don’t know if their mechanism is right – there are other plausible contenders,6 and the notion of banks being funded by creditors who monitor the risks of their funding is in some conflict with notions of too-big-to-fail subsidies and information-insensitive bank funding – but it’s kind of a neat stylized description of the world. There is an amount of risk that banks fund, and an amount of equity capital that is interested in funding those risks, and the amount of bank assets (and debt) ratchets up and down to provide the right amount of risk.

If you take that model very seriously what would you predict about Brown-Vitter? Here’s a sketch:

  • the amount of bank equity will remain constant – for the biggest six banks, that’s around $874 billion;7
  • the equity ratio will almost triple, meaning that assets and commitments will be reduced by ~$7 trillion from ~$13 trillion to ~$6 trillion;
  • the riskiness of those assets will increase dramatically, so as to keep a constant Equity/VaR ratio: those $6 trillion of new-bank assets will carry as much risk as $13 trillion of old-bank assets.

And that would … work I guess? Brown-Vitter would do away with Basel risk-based capital regulation, although there’s some language saying “but you can have risk-based capital regulation if you want, as long as it’s not Basel-y.” Bankers would find themselves running balance sheets that are much smaller, but much more exciting on a dollar-for-dollar basis. That might actually turn out to be pretty fun.

Procyclical Leverage and Value-at-Risk [NBER (gated), New York Fed (free)]

1. Er technically over about 5 years. Actually 5 years from when the implementing regulations are issued, so, like, 10 years.

2. Math is here. I’m cheating in that my $7 trillion is based on “assets” in the Brown-Vitter sense, including unfunded lending commitments, while my $9.5 trillion is based on “assets” in the GAAP sense excluding those commitments. Still it’s a lot.

3. Which seems to come from 2008 though there’s a new-ish version on NBER today and it’s new to me so whatever. Quotes are from the (gated) NBER version but the NY Fed one is from February 2013 and free.

4. “But not really” would be a fair objection to all of this? This table of correlation coefficients maybe more compelling:

So quarterly leverage growth is uncorrelated to change in VaR-to-equity ratio, but is negatively correlated to increase in VaR-to-assets ratio (“Unit VaR”), CDS spread, and equity implied volatility. The riskier you get on various markety measures, the more you need to deleverage.

5. Obviously that’s like an ex ante probability; it’s not like bank defaults occur at a constant rate. Intuitively they should occur in bunches when VaR was low (and thus leverage was high) and then all of a sudden gets high, wiping out overlevered banks. That’s pretty much 2007-2008.

Ooh for fun let’s sort of size that probability. JPMorgan’s 12/31/2012 95% one-day VaR was $98 million and its stockholders equity was $204 billion. So losing all of its equity in a day would be a … hmm … like a three-thousand-standard-deviation event, which Google Docs calculates at a 0.00E+00 probability. Black swan! Perfect storm!

Lehman Brothers’ VaR at November 30, 2007 was $124mm, while its stockholders’ equity was $22.5 billion, so losing all of its equity in a day would only be a 300-standard-deviation event. Losing it all in a month is only like 60 sigmas. STILL. 0% probability.

So you gotta take all that probability-of-default stuff with a grain of salt. The ratio of equity to VaR is like a directional indicator of likelihood of default, not, like, a quantitative one.

6. A rather glaring one is that bank capital regulation is risk-based, that is, based on VaR. Loosely speaking, Basel capital regulation requires like [Capital >= 8% x Assets x VaR x A Fixed Multiplier], so of course the ratio of Capital to (Assets x VaR) would remain roughly constant. In a certain light all this paper says is “U.S. banks are subject to capital regulation based on risk-weighted assets,” though that isn’t quite true either, since the institutions they cover – the pre-2007 investment banks and the commercial banks with the biggest trading businesses – were not constantly subject to constant Basel rules throughout their period. Still it seems like the sort of thing you might discuss, especially since one of the authors works for the NY Fed. I dunno.

7. Obviously one possible result would be moving of assets etc. to smaller banks, which have lower equity requirements than the big banks under Brown-Vitter; that would somewhat diminish these effects though those banks too would probably have to have higher equity ratios than they do now. But I use just the big banks because it’s easier to get numbers.

8 comments (hidden to protect delicate sensibilities)
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Comments (8)

  1. Posted by The Denominator | April 8, 2013 at 3:42 PM

    Loving all this action!

  2. Posted by Guest | April 8, 2013 at 4:20 PM

    Matt,

    Former FSA Chairman Adair Turner has made a compelling argument for higher bank capital ratios here (beginning on p 24): http://harr123et.files.wordpress.com/2010/07/futu

  3. Posted by Guest | April 8, 2013 at 4:37 PM

    Oh my charts!

  4. Posted by Guest | April 8, 2013 at 5:25 PM

    None of these bastards breathed a peep about this shit back in 2008 when their balls were in a vise. "Oh please, take over Bear Stearns…Oh please, take over WAMU…Oh Please, take over Nat City…Oh Please, take over HBOS!!!" "We promise not to let anyone say it is anti-competitive!" LMHBB.

  5. Posted by Spanishmoon | April 8, 2013 at 6:19 PM

    Or, they could just take our deposits.

  6. Posted by quant me maybe... | April 9, 2013 at 6:48 AM

    Inner thoughts….
    That chart; It's mesmerizing.
    And it has formulas.
    I'll take one. Don't bother with the prospectus. You had me with the cool little blue bubble data points.
    Boobs.

  7. Posted by Guest | April 9, 2013 at 8:34 AM

    And what would these small-balance-sheet banks o' fun do with the QE debt?

    I await the day when the opponents of the debt-financed society finally realize that Big Government has no other way to exist.

  8. Posted by quant me maybe... | April 9, 2013 at 8:46 AM

    Would that be the Chinese State Run Banks and Sovereign Wealth Fund's little banks?
    I'm pretty sure that they will be all right. Hell, they might even read the prospectus' footnotes.