Beta Quantopian Calculation

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edited Jun 16, 2017

Could someone from Quantopian finally explain to me how you calculate Beta? I've asked multiple times and never got an answer.
Could you at least confirm or decline the following:
1. Are you calculating Beta of the performance of the whole portfolio by taking change in portfolio_value and comparing it against the change of spx? or
2. Are you calculating Beta by summing/subtracting Beta values of each portfolio position during entering/exiting the trade multiplied by each position weight?

If 2: that would be proper in my opinion, as we want full hedge (Beta=0) on entering the trade and max/min Beta on exiting it.
However, I do feel Quantopian is using 1.

I did find the code on GitHub for Alpha, Beta: in statistical.py

def regress(y, x):  
            regr_results = linregress(y=y, x=x)  
            # `linregress` returns its results in the following order:  
            # slope, intercept, r-value, p-value, stderr  
            alpha[i] = regr_results[1]  
            beta[i] = regr_results[0]  
            ...

Yes, this is correct assuming position and hedge are using and will be using the same Beta. Like in case we invested in SPX long and short in the same amount. Then when SPX moves 10% up our long leg increases by 10% and short drops by 10%, however weights are equally increases and decreases proportionally to 10% therefore, as the result Beta of the whole portfolio stays the same at 0.

However, look at the following case: we have an algorithm that generated 1 week event to invest in fully hedged portfolio as there is a signal that SPX will strongly move up while TIP will stay the same during that 1 week. As SPX (Beta=1) and hedge TIP with Beta=-0.128, we created fully hedged portfolio with weights of SPX=1/1=1. and weight of TIP=1/0.128=-7.81. So overall Beta of portfoilo=1*1-7.81*0.128=0
Lets assume that everything has happened as algorithm predicted and in a week SPX moved 10% up, while TIP didn't move at all.
So, portfolio weight of SPX now is 1.1 while weight of TIP is still 7.81. As Betas of SPX is still 1 and TIPs 7.81 we now have full portfolio Beta at: 1*1.1-7.81*0.128=1.1-1=0.1

As you can see: Beta obviously moved and the higher it will move the more Alpha will be generated.
To fight that move would be to rebalance the portfolio more frequently, but who wants that?

4 responses

Dan Dunn

Jun 16, 2017

When talking about beta, it’s important to be clear about beta of what and to what. I’m guilty as anyone else of sometimes being a little too sloppy about that.

On the Quantopian website, the most common use of “beta” is to refer to a specific portfolio’s trailing beta to the performance of SPY. This might also be referred to as "realized beta," and is the historical coefficient of the algorithm's returns with SPY's returns. The code that does the calculation is in empyrical’s stats.py. This realized beta-to-SPY is the calculation that is displayed in most backtest returns, is used in the contest, and is used in pyfolio’s rolling beta calculation. This is the use of beta for “performance attribution”.

The use of beta in performance attribution is backwards looking and we don’t know this beta until after the fact. If, from today, you want to produce a return stream in the future which will have low attribution to the market you need to model and predict security-level beta. A very common way to do this is to calculate how an individual stock has historically moved relative to the market. In that case, we’d calculate the coefficient in a linear regression between the returns of a stock and SPY as the stock's beta. The assumption in this model is that the beta calculated in the historical regression is stable and likely to persist into the future. You need to make modeling choices here: how far is the lookback window, do you use any prior or regularization, how often do you recalculate this value, etc. Taking that concept of a single stock’s beta further: We can calculate the "beta exposure" of a portfolio at a particular point in time, by which we mean "the weighted sum of the algorithm's positions, weighted by the realized beta of each individual position." I think that is what you are referring to in your TIP/SPX example. The Optimize API provides constraints like WeightedExposure which can be used to control your algorithm’s beta exposure. Beta exposure is not yet available in pyfolio tearsheets or in the backtester.

Getting to your final question: you’re correct that your beta exposure can changes as prices change, and excessive rebalancing will eat away at your alpha. Regression estimated betas are modeled values and, as such, are noisy. As an algo designer you need to make good modeling choices to ensure that when your calculated betas change, the change is warranted. One good thing to note is that Quantopian has an experienced trading team, a state-of-the-art execution platform, and a prime broker integration. You can be much more aggressive with your rebalancing than if you were doing it through a retail broker. That said, finding a balance between rebalancing and drifting beta exposure is a real challenge.

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Igor Tebelev

Jun 16, 2017

Dann,
Thank you for an attempt to answer, but looks like you are still "a little too sloppy about that". I understand, as I'm as a conservative investor, would also like to think "... that the beta calculated in the historical regression is stable and likely to persist into the future". However, if Quantopian would always follow that, then any kind of reversal and/or market timing algorithm is not appropriate for Quantopian contest and/or hedge fund. Why not then state that?
Then you just say: As we are limiting Beta to the abs(0.3) range then we are allowing max 30% not hedged portfolio in algorithms, that either momentum following(but then it will not continue forever) and/or algos that exploit market inefficiencies (that eventually after the fact of knowledge would be market corrected) and/or CD like strategies (returning a little bit more then inflation rate).
Aren't you limiting universe of possible algorithms by excluding any reversal or market timing types? I understand you are targeting pretty conservative investors as a majority, but aren't there enough of "gamblers" investor types? Shouldn't any, even "conservative" portfolio needs to have some highly speculative component?
Anyway, just my humble, but obviously not business opinion...

Dan Dunn

Jun 16, 2017

You are correct, reversal and market timing algorithms are not appropriate for the contest or for getting an allocation.

I agree, those strategies are likely appealing to some investors. They just aren't the type of strategy that we're supporting through our allocations at this time. Our allocation decisions are driven by the financial product that we're trying to create for our investors.

Of course, some day in the future we may have reason to use different criteria and make allocations to a broader set of strategies.

Disclaimer

Stephen Hanly

Jun 16, 2017

What time window is used in the beta calculation? Do you look at the correlation of daily returns, hourly, minute-by-minute, etc.?

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