Predict beta more accurately by using imaginary returns

Updated October 2014, replacing the word “imaginary” with “reconstructed.” – TMA

If you’re managing portfolios without using a risk model, you’re probably relying on regression statistics of asset returns to estimate risk characteristics such as Beta.

For reporting purposes or for the purpose of measuring sensitivity to a broad index whose constituents don’t change drastically very often (eg, the S&P 500), a time series regression Beta (also known as Historical Beta) is probably good enough.

When Historical Beta is not good enough
The data that go into Beta are portfolio (or index) weights, asset volatility, and correlation coefficients between pairs of all the assets in the portfolio (or index). The more different any of these data are from their current measures, the more distorted Historical Beta will be as an estimate of Predicted Beta.

Volatility and correlation are always unknown and may only be observed with hindsight. But weights ARE observable – you know them. In order to get the least distorted Predicted Beta, use current weights, not historical weights – which means DON’T use Historical Beta.

Where getting Beta right matters
Here are a few scenarios where making the effort to Predict Beta without using Historical Beta can be helpful.

Concentrated portfolios
Portfolios whose benchmarks change drastically, such at Russell reconstitution
Long-short portfolios with very different long and short volatility characteristics

Example scenario: Turnaround investor
Let’s say you invest in turnaround situations. Maybe you’re an activist investor, and you typically hold very few positions. You buy stocks that have been laggards and hope or even try to help the companies turn around. Such a scenario introduces all kinds of portfolio risk.

Let’s say you buy a stock. Chances are it’s been a drastic underperformer for a long time, which means its weight in its sector has dropped a lot over this span. However you really don’t have views on the timeliness of its sector, you’re just picking the stock to rebound and beat its sector peers.

The logical thing to do would be to buy the stock and sell short its appropriate sector ETF. If it’s a member of the S&P 500, the Spider Sector ETFs have been around since the late 1990s, so you have plenty of return data to produce a Beta estimate, which you could use to set your hedge ratio.

Let’s say you buy Alcoa, and decide to short XLB, the Spider Materials Sector ETF. (Disclosure: I chose Materials because it’s got the smallest number of holdings among any Spider Sector ETFs, making this example easier for illustration purposes, not because I have a view on Materials, let alone Alcoa, or whether any activist investor owns Alcoa.)

As of the time of this writing, May 2011, Alcoa (AA) is the sector laggard over the past 8 years, having posted an annualized return of -2.3%, while most of the other XLB constituents have gained, some extraordinarily. XLB, the Spider Materials ETF, has returned over 10% annually over the same span.

With no view on Materials, you decide to buy Alcoa and sell short XLB. But how much do you short?

If you regress Alcoa’s returns against the returns of XLB, using monthly excess returns from May 2003 through April 2011, you get a Beta estimate of about 1.56. That suggests for every dollar of AA you buy, to be hedged you would sell short 1.38 of XLB for every dollar of AA you buy – if you sold latex \sigma latex \rho latex \beta latex \sigma latex \Sigma latex \rho latex \sigma latex \Sigmalatex \Sigma latex \rho Latex \sigma latex \sigma $_j]           (1)                                       

Among those three types of data, we know only one with certainty right now – weights. For volatilities and correlation coefficients we have to guess, typically modeling them with historical observations. But asset weights we don’t have to guess or model – we know! So don’t throw away the one data item you know when you’re estimating portfolio betas, covariances, or volatilities. When you use the Historical Beta, you’re replacing the weights data with historical averages of weights.

Good news: a short cut!
If the thought of grinding through a 30×30 covariance matrix to re-estimate your hedge ratio sickens you, there is a 100% accurate shortcut. Generate a pro-forma historical return series, rebalanced each period, as if the sector’s stock weights had always been identical to the present weights. Over the relevant 96 month period, the resulting regression of returns of Alcoa vs. the imaginary reconstructed pro-forma sector returns will match the matrix-derived answer of 1.38 precisely.