Gilgamath



No Parity like a Risk Parity.

Sun 09 June 2019 by Steven E. Pav

Portfolio Selection and Exchangeability

Consider the problem of portfolio selection, where you observe some historical data on \(p\) assets, say \(n\) days worth in an \(n\times p\) matrix, \(X\), and then are required to construct a (dollarwise) portfolio \(w\). You can view this task as a function \(w\left(X\right)\). There are a few different kinds of \(w\) function: Markowitz, equal dollar, Minimum Variance, Equal Risk Contribution ('Risk Parity'), and so on.

How are we to choose among these competing approaches? Their supporters can point to theoretical underpinnings, but these often seem a bit shaky even from a distance. Usually evidence is provided in the form of backtests on the historical returns of some universe of assets. It can be hard to generalize from a single history, and these backtests rarely offer theoretical justification for the differential performance in methods.

One way to consider these different methods of portfolio construction is via the lens of exchangeability. Roughly speaking, how does the function \(w\left(X\right)\) react under certain systematic changes in \(X\) that "shouldn't" matter. For example, suppose that the ticker changed on one stock in your universe. Suppose you order the columns of \(X\) alphabetically, so now you must reorder your \(X\). Assuming no new data has been observed, shouldn't \(w\left(X\right)\) simply reorder its output in the same way?

Put another way, suppose a method \(w\) systematically overweights the first element of the universe (This seems more like a bug than a feature), and you observe backtests over the 2000's on U.S. equities where AAPL happened to be the first stock in the universe. Your \(w\) might seem to outperform other methods for no good reason.

Equivariance to order is a kind of exchangeability condition. The 'right' kind of \(w\) is 'order …

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Twelve Dimensional Chess is Stupid

Tue 16 October 2018 by Steven

Chess and the Curse of Dimensionality

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Another Confidence Limit for the Markowitz Signal Noise ratio

Wed 28 March 2018 by Steven

Another confidence limit on the Signal Noise ratio of the Markowitz portfolio.

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Markowitz Portfolio Covariance, Elliptical Returns

Mon 12 March 2018 by Steven E. Pav

In a previous blog post, I looked at asymptotic confidence intervals for the Signal to Noise ratio of the (sample) Markowitz portfolio, finding them to be deficient. (Perhaps they are useful if one has hundreds of thousands of days of data, but are otherwise awful.) Those confidence intervals came from revision four of my paper on the Asymptotic distribution of the Markowitz Portfolio. In that same update, I also describe, albeit in an obfuscated form, the asymptotic distribution of the sample Markowitz portfolio for elliptical returns. Here I check that finding empirically.

Suppose you observe a \(p\) vector of returns drawn from an elliptical distribution with mean \(\mu\), covariance \(\Sigma\) and 'kurtosis factor', \(\kappa\). Three times the kurtosis factor is the kurtosis of marginals under this assumed model. It takes value \(1\) for a multivariate normal. This model of returns is slightly more realistic than multivariate normal, but does not allow for skewness of asset returns, which seems unrealistic.

Nonetheless, let \(\hat{\nu}\) be the Markowitz portfolio built on a sample of \(n\) days of independent returns:

$$ \hat{\nu} = \hat{\Sigma}^{-1} \hat{\mu}, $$

where \(\hat{\mu}, \hat{\Sigma}\) are the regular 'vanilla' estimates of mean and covariance. The vector \(\hat{\nu}\) is, in a sense, over-corrected, and we need to cancel out a square root of \(\Sigma\) (the population value). So we will consider the distribution of \(Q \Sigma^{\top/2} \hat{\nu}\), where \(\Sigma^{\top/2}\) is the upper triangular Cholesky factor of \(\Sigma\), and where \(Q\) is an orthogonal matrix (\(Q Q^{\top} = I\)), and where \(Q\) rotates \(\Sigma^{-1/2}\mu\) onto \(e_1\), the first basis vector:

$$ Q \Sigma^{-1/2}\mu = \zeta e_1, $$

where \(\zeta\) is the Signal to Noise ratio of the population Markowitz portfolio: \(\zeta = \sqrt{\mu^{\top}\Sigma^{-1}\mu} = \left\Vert …

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