# Can alpha be positive if cumulative returns underperform the benchmark?

According to my portfolio analysis program (pyfolio), the alpha of the following strategy is .17 (I am assuming 17%). [Based on pyfolio documentation, alpha here is the "annualized alpha".]

However, the cumulative returns of the benchmark are about 10-fold higher than that of the strategy (~12% vs ~125%), see graph.

To me this is non-intuitive, and I was wondering if someone had a good explanation.

Does beta (negative in this case) have something to do with it?

Of course, it is also possible that I am not using the software correctly, or misinterpreting its output. I will go into its source code to try and figure out the calculation it performs. But I hoped someone knows what might be going on here. Thanks!

• That's possible yes. Can you share a spreadsheet with both time-series of returns? It would make it easier to shed some light on that. Commented Jul 29, 2020 at 23:05
• @phdstudent Here we go: drive.google.com/file/d/1m04SfUPzYdB9fPPSLMHbNq5iM0LWf3fc/…. Thanks! Commented Jul 29, 2020 at 23:28
• This strategy is statistically a decent hedge to the benchmark. According to a CAPM style model, zero alpha would mean a return clearly below the risk free rate. If it yields the risk free rate, it already has a clearly positive alpha.
– fes
Commented Jul 30, 2020 at 7:16
• @fesman I think I see what you are saying. By the way, I believe the software I am relying on uses a 0 risk-free rate by default. Not sure, if I should start incorporating Rf into my analyses. How critical do you think the use of Rf is in portfolio analytics? I am only working with data from the last two decades. Commented Jul 30, 2020 at 8:31
• Typically it is good to have them, they can affect alphas quite a lot. Anyway, the economic intuition in your example is that you can think of your strategy as insurance. Such insurance is valuable even when it provides low returns.
– fes
Commented Jul 30, 2020 at 8:41

Yes, this is absolutely possible. Here is a simple thought experiment to show how.

We want to benchmark to the S&P 500. We allocate 90% of our capital to an index tracking strategy and 10% to some new portfolio manager with a good track record. (We'll call the new PM "Rumplestiltskin," for ease of reference.)

Unfortunately, there is a bug in our index tracking strategy and it ends up being mostly mean-zero noise with an S&P 500 beta of 0.1. (This may sound implausible. Check the returns of some hedge funds and you will see it is very plausible.)

On the other hand, Rumplestiltskin turns everything to gold: he makes consistent gains uncorrelated with the S&P 500.

If Rumplestiltskin's gains are small, we will likely underperform the S&P 500 when it rises; however, we will have alpha that will be shown to be significant with enough data.

• Thanks! To be truthful, I am a little fuzzy on what the index tracking strategy illustrates in this example. Since it's broken and amounts to zero, as you point out. Otherwise, am I correct in understanding you're saying if strategy's absolute returns are small but the strategy is making money, when the benchmark is losing, then alpha can be positive? That also doesn't quite make sense to me, since the difference between cumulative returns is so drastic in my case. Could you please elaborate? Commented Jul 30, 2020 at 5:01
• The index tracking strategy does not amount to 0. Rather, it earns 10% of the excess return of the S&P 500 -- times 90% for a complete portfolio beta of 0.09. In your case, the beta is negative. The two are not very different; your case just needs a higher alpha to get a positive portfolio return. A strategy could even lose money but have positive alpha. That would even be attractive if most of the losses were due to exposures to the S&P 500, other indices, or other risk factors. In that case, we would hedge those risk factors and be left with positive alpha -- which is very desirable. Commented Jul 30, 2020 at 5:44
• I think I get it more or less. Thanks! Perhaps another way to think of it is that in the end alpha and beta are just regression coefficients, so like you say it is possible for the alpha to be positive and yield negative return (provided beta is sufficiently negative). So it does seem to be very beta dependent. Right? Also, I am still confused by the index tracking in your example. If you find a better way to word your answer to be more clear (even for dummies like me), please update it (for posterity). Commented Jul 30, 2020 at 6:00
• Close, yes. Alpha is always in light of a factor model (e.g. the CAPM). So alpha plus beta times the market excess return may be less than the market return may be lower than the market excess return; however, if the alpha is consistent then it will likely be significant. So this happening is beta-dependent and dependent on the return of "the market" (the S&P 500 or whatever index is used). What is confusing about the index tracking setup? Would be glad to make the answer more clear. Commented Jul 30, 2020 at 8:04
• Thanks! The part that's confusing is that we are talking about the returns of the strategy in relation to the benchmark's returns. So, I'm not sure what introducing the third series of returns, the "broken" (beta=.1) benchmark, accomplishes. Commented Jul 30, 2020 at 8:25

The best example of an underperforming strategy with big alpha, is insurance.

Every year you pay a premium to insure your house. That strategy has negative expected return, negative beta, but super high alpha (as it is uncorrelated with the market and diversifies well your portfolio).