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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fromo 0.2.0

Sun 13 January 2019 by Steven E. Pav

I recently pushed version 0.2.0 of my fromo package to CRAN. This package implements (relatively) fast, numerically robust computation of moments via Rcpp.

The big changes in this release are:

  • Support for weighted moment estimation.
  • Computation of running moments over windows defined by time (or some other increasing index), rather than vector index.
  • Some modest improvements in speed for the 'dangerous' use cases (no checking for NA, no weights, etc.)

The time-based running moments are supported via the t_running_* operations, and we support means, standard deviation, skew, kurtosis, centered and standardized moments and cumulants, z-score, Sharpe, and t-stat. The idea is that your observations are associated with some increasing index, which you can think of as the observation time, and you wish to compute moments over a fixed time window. To bloat the API, the times from which you 'look back' can optionally be something other than the time indices of the input, so the input and output size can be different.

Some example uses might be:

  • Compute the volatility of an asset's returns over the previous 6 months, on every trade day.
  • Compute the total monthly sales of a company at month ends.

Because the API also allows you to use weights as implicit time deltas, you can also do weird and unadvisable things like compute the Sharpe of an asset over the last 1 million shares traded.

Speed improvements come from my random walk through c++ design idioms. I also implemented a 'swap' procedure for the running standard deviation which incorporates a Welford's method addition and removal into a single step. I do not believe that Welford's method is the fastest algorithm for a summarizing moment computation: probably a two pass solution to compute the mean first, then the centered moments is faster. However, for the …

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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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A Lack of Confidence Interval

Thu 15 February 2018 by Steven E. Pav

For some years now I have been playing around with a certain problem in portfolio statistics: suppose you observe \(n\) independent observations of a \(p\) vector of returns, then form the Markowitz portfolio based on those returns. What then is the distribution of what I call the 'signal to noise ratio' of that Markowitz portfolio, defined as the true expected return divided by the true volatility. That is, if \(\nu\) is the Markowitz portfolio, built on a sample, its 'SNR' is \(\nu^{\top}\mu / \sqrt{\nu^{\top}\Sigma \nu}\), where \(\mu\) is the population mean vector, and \(\Sigma\) is the population covariance matrix.

This is an odd problem, somewhat unlike classical statistical inference because the unknown quantity, the SNR, depends on population parameters, but also the sample. It is random and unknown. What you learn in your basic statistics class is inference on fixed unknowns. (Actually, I never really took a basic statistics class, but I think that's right.)

Paulsen and Sohl made some progress on this problem in their 2016 paper on what they call the Sharpe Ratio Information Criterion. They find a sample statistic which is unbiased for the portfolio SNR when returns are (multivariate) Gaussian. In my mad scribblings on the backs of envelopes and scrap paper, I have been trying to find the distribution of the SNR. I have been looking for this love, as they say, in all the wrong places, usually hoping for some clever transformation that will lead to a slick proof. (I was taught from a young age to look for slick proofs.)

Having failed that mission, I pivoted to looking for confidence intervals for the SNR (and maybe even prediction intervals on the out-of-sample Sharpe ratio of the in-sample Markowitz portfolio). I realized that some of the work I had done …

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