I have optimised a set of portfolio and subsequently regressed the returns against the fama-french-Carhart factors. I have two portfolio, one in which short sales are allowed (the portfolio can take on leverage) and the other limits short sales (so that the leverage is zero). Now my question is: How are the alpha and beta of the levered portfolio affected by the ability to assume leverage??

For the levered portfolio I find a beta close to 1.5. The non-levered portfolio displays a beta of 0.5. As beta is the coefficient explaining the variance of the returns, I would think that this difference is due to leverage as it naturally leads to a higher risk in similar proportion to the return. However, I am not completely sure as I read that levering up the portfolio (simply taking larger positions) would not affect the beta as the exposure remains similar.

I expect the alpha not to be affected by the leverage as it involves no specific skill and therefore will not improve the risk adjusted returns.

However could it also be that establishing long short positions allows the the manager to add alpha?? (as it can short stocks it would normally not be able to do so? i.e. short side alpha)

Looking forward to your thoughts!


Yes, it's a function of leverage.

Your unlevered beta of 0.5 sounds a bit on the low side, but is not directionally surprising. Google "betting against beta" or the "low volatility anomaly" for more details.

To get the beta up from 0.5 to 1.5, that is a function of leverage. The market 2x levered will have a beta of 2.0x to itself, precisely because the return outcomes are double the underlying. Meanwhile, short sales will obviously reduce the beta; so if the net is higher, that just means that the gross longs have risen more than the size of the shorts. The optimiser would be 200 long, 50 short; 250 long: 100 short or 300 long; 150 short etc. It should be relatively easy to verify this, by summing the gross longs and the gross shorts of your model's outputs.

To generate these kinds of outcomes, your optimiser would have to believe there was a set of alpha opportunities to exploit. Give the model highly-correlated return differentials, and it will not hesitate to add in a long-short. The "optimal" size of these positions is often in excess of the size of the rest of the vanilla portfolio! If the model thinks that shorting A to buy B has the same risk-reward as buying the market, it will want to have as much A-B risk as market risk in the portfolio. If it needs to lever up A and B by +5x and -5x respectively to achieve the same risk (given diversification), then so be it. The model is now 600% long; 500 Short ;-) Stylised example, but hopefully you get the point. Again, this should be obvious if you sum the long and short positions in the portfolio.

Of course, whether these historical return differentials will be repeatable in the future is moot. Even if a fraction of a past lucky break were to persist, optimising the portfolio for it continue on the same scale as historically would represent sub-optimal "overbetting". This - and the general sensitivity of "optimal" outputs to input assumptions - is the key reservation that market practitioners have with optimisers.

  • $\begingroup$ Thanks, indeed as you outlined before I have computed the average sum of positive and negative positions. In the levered portfolio the absolute sum of position equals 180% equivalently to a short position of -80% and a long position of 100%. I have optimised the portfolio upon a set of characteristics believed to deliver cross-sect. returns on average. Now also as you pointed out it remains odd why the beta is so low. To be more specific I have conducted three regressions. The portfolio to the 1/N portfolio, Value-weighted and the Fama-French-Carhart factors. --> $\endgroup$ – incognito Oct 1 at 11:09
  • $\begingroup$ The beta of the 1/N& VW are close to 0.5, while the loading on the MKT factor is close to 0.9. The sample includes mainly firms from the SP500. What also seems odd is that the levered portfolio achieves NO alpha over the VW portfolio while the non-levered portfolio does so. Both portfolio achieve significant alpha over the Fama-French-Carhart factors. This last finding, I believe, is due to the relevance of characteristics over factors in line with Kent&Titman 1998 link $\endgroup$ – incognito Oct 1 at 11:14

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