## Arbitrary CAPM

Emanuel Derman passes on an email from a former student, who’s now working at ****:

Will Sharpe came up with his Capital Asset Pricing Model (which we use at **** all the time, in its most simplistic form) and now it is part of the dogma that asset returns are linearly related to market returns. What is the logical basis to assume a linear relationship here? He could have just as easily assumed a cubic relationship (which will almost by definition fit past data better) and done the same work. His math would have probably gotten messier, but it is hard to find any merits to the linear assumption other than the fact that it is simple. I am all for simplicity, and perhaps if I asked Sharpe he’d tell me that was just a reasonable first approximation, but that is certainly not the way in which people use it now (maybe they are not so rational after all?).

The fact is that **** could be pretty much any buy-side or sell-side firm in the world. And so long as they all talk about concepts like “alpha” in more or less the same way, as though it’s a real scientific thing, in a way it doesn’t matter whether it’s based on empirically-justifiable principles or not. Still, this is a useful reminder that when finance types start blinding you with science, it’s not science in the sense of actually reflecting reality. At best it’s a useful fictional construct, at worst it can help cause a global economic meltdown.

Nonsense. Gauge theory is your friend in the financial business.

I had somewhat the same reaction to Black-Scholes after reading the 1997 not-really-a-Nobel economics announcement. It is certainly more complicated than a linear equation, but the model is explicitly based on several not-true-in-reality assumptions. It seemed that the only reason it worked to price options was that everyone had decided to adopt the formula to price their options!

I’m a little puzzled by Derman’s student’s comment: the linearity in the CAPM is just a consequence of efficient markets/no-arbitrage and the ability of large banks and investors to lend or borrow freely at the risk-free rate. Now both of these are certainly not exactly correct, but they are either almost correct or at least a standard economic assumption.

On the broader question of whether what financial economists and analysts do is “science” is the sense of “reflecting reality” — well, it is as long as it works. But I guess I don’t get how that’s all that different from any other kind of science — especially social science.

AlexR–As an MBA grad myself from a quant grad school many years ago, the simplicity of the CAPM was confused by many as “elegance.” And it certainly isn’t science by any metric, then or now. It doesn’t even meet most criteria for being “statistics.”

You may not know it, but I think you’ve been absorbed by the bankonomics borg. Beware!

I’m seconding Alex R – that’s actually quite an ignorant comment and I’m really surprised that Derman passed it on approvingly. The “linear” relationship of the CAPM isn’t imposed on the model – it’s a consequence of the quadratic programming, the assumption of a lognormal distribution (or alternatively the assumption of quadratic utility IIRC). There are all sorts of reasons to dislike the one-period CAPM, and I blog about them intermittently, but someone who thinks you could just randomly substitute in a cubic relationship is not really someone I want on “team anti-CAPM”.

… adding some intuition to the above, if you have a portfolio of securities X, Y and Z, what is the return on the portfolio? It’s the sum of the returns on X, Y and Z, isn’t it? It’s basically a linear problem (unless you start assuming very complicated things indeed) which is why it’s modelled with a linear model.

The misuse of the word simplistic is almost Freudian.

The linearity comes from investors being mean-variance optimizers. there’s no normality or log-normality anywhere in the model. nevertheless, i agree with others that this is silly.

unless I’ve forgotten or misremembered (which is entirely possible), mean-variance optimisation is optimal for a) anyone who has a quadratic utility function whatever the distribution of returns or b) anyone of any sort of utility function if returns are lognormally distributed.