Economics and similar, for the sleep-deprived

Monday, June 26, 2006

 
The Cave-in theory of economics

My four year old boy came up with this piece of philosophical poetry at the weekend. I think it has echoes of Ezra Pound.

Rock, Rock, Rock the boat
Gently down the stream
Merrily, merrily merrily merrily
Life is but a TUNNEL

Truly he is his father's son.

Hmmm, this blog appears to have degenerated into a parish magazine of the 1970s, and I suspect that the frequent promises to do something about the Stephen Levitt review might be losing their ability to retain readers, so here's a little economic thought (btw, a contribution to financial theory which might have been better put on D^2D is here on the Guardian blog). I've been reading Perry Mehrling's biography of Fischer Black (capsule review: disappointing) and thinking once more about the Capital Asset Pricing Model. Specifically about one little quasi-inconsistency (as in, I don't think it's serious enough to bring down the mathematical structure of the theory, but it makes the economic interpretation a lot more difficult to fathom) in the concept of beta as a measure of covariance risk.

The idea is this; consider a stock with negative beta (this makes it clearest but there is no loss of generality here). In other words, it's a stock which you own because it tends to move in the opposite direction from the market. That's an attractive quality for which you are prepared to pay a premium (in the sense of accepting a lower return from negative beta stocks), because it reduces the risk of your portfolio. I'll set out the CAPM equation below:

(1) Expected return on your stock = Risk free rate of return + beta (Expected return on market - risk free rate)

(where beta is, in this case, a negative number).

Now this equation is set out in terms of "expected" values ex ante, which means that it is basically unfalsifiable in the Popperian sense. In order to actually use it for practical purposes (and, biographers of Fischer Black, spare me any claims that a full intertemporal CAPM is what people really use for practical purposes; I went to business school and you can't fool me), you have to assume that "expected" refers to the large-sample properties of the underlying data generating process, rather than to a metaphysical "true" expected value. This means that you have to be prepared to make statements like:

Conditional on a market return of X, the expected return on your stock is equal to:

(2) Expectation of return on your stock : X = Risk-free rate of return + beta (X - risk free rate).

A lot of people will tell you that this is basically the definition of beta; a number which tells you what the "sensitivity" of your stock to the market return, with "sensitivity" defined in terms of conditional expectations. However, a bit of thought about the equation above shows that this can't be literally true. For example, let's think about the case when the market return is minus 100%.

The misleading equation (2) above would suggest that our stock would go up quite a lot; if the beta was close to -1 we'd expect it to nearly double. But of course, this can't possibly be the right answer. Our stock is part of the market, so if the market has had a return of -100% (ie, it has gone to zero), then our stock must have gone to zero as well.

Is this a silly technical point of no empirical relevance? Not at all. Stock markets do go to zero, more often than you'd think (the Russian stock exchange did so pretty emphatically in 1917 for example). I don't have the data to hand but I would not be surprised to find out that as much as a third to a half of all stock exchanges that have ever existed have gone to zero. The point here is that the simple linear interpretation of beta as a factor that lets you read off your stock's expected return from the market return is wrong. Beta is an empirical measure of the normalised covariance of a stock with the market, which is consistent with all sorts of underlying distributions. As the example of considering what happens in a market collapse shows, the copula[1] which links the marginal distributions of your stock and the market has to give very strong interdependence at extreme values in the lower tail of the distribution; empirically, one thing we know about stock returns is that dependence increases markedly even in crashes much less serious than the one I talk about above.

But the problem actually goes much deeper; the previous paragraph suggests that this is something that can be dealt with by taking a slightly more nuanced approach to one's estimation of CAPM type models and I don't think that's true. I think that the real problem here is the one that's missed by all equilibrium models; that there is an implicit assumption that the ground rules of the optimisation problem are in some way stable and this assumption is empirically wrong. To put this in the bluntest terms possible, you are not entitled as an economist modelling financial risks, to ignore the possibility of something like the Second World War or the Russian Revolution. The whole of the Fischer Black biography (and indeed, Black's own books; Mehrling gives a very accurate account of Black's economics although not nearly a critical enough assessment) is shot through with this fundamental error; Black's oddball economic arguments almost always rely at a crucial stage on being able to assume away idiosyncratic risks by diversification, which is tantamount to ignoring the very low probability events that can't be insured against or made smaller by sharing them precisely because when they happen, they sweep everything away in one fell swoop.

In Martin Weitzman's Bayesian theory of the equity risk premium, very small probabilities of very bad events are a big driver of the economic behaviour of the whole system, leading to empirically observed behaviour that doesn't seem to make sense when you only observe the system in equilibrium. I think that one of the central insights of (Post-)Keynesian economics is that this is a very general point about economics; that an important motivating factor is the fear of very bad events that are not part of the model. This is also a big reason why there is a liquidity preference and I think it's Paul Davidson's main theoretical insight; that the reason why central bank money is special is that it is the asset that can be converted into consumption at a moment's notice even if everything goes to hell. As a wise man once said to me, life is but a tunnel, and the thing about tunnels is that although most of your time spent in them is a slow incremental progress forward toward the light, it does well to consider once in a while that there is a risk of the damn thing caving in.

[1] In talking about copulas here, I am bullshitting a bit; this is the way I think about modelling the kind of thing I am talking about here, but if I was a better mathematician I would be talking about the bivariate distribution (stock, market) as a representation of the entire multivariate distribution of asset prices. Here's a rather good diatribe against copulas; the general theme is "it's much more complicated than that", which was also the battle cry of the copula posse when the enemy was simple correlation coefficients, and in such a manner does science progress.
0 comments this item posted by the management 6/26/2006 04:14:00 AM