How Leverage and Short Selling Affect Fama–French Betas and Alpha
Summary
The discussion explains why a long-short portfolio can have a higher market beta than a long-only portfolio. Scaling exposure to the market scales beta, while short positions can offset some of that exposure. The example compares a portfolio beta of 0.5 with one of 1.5 and suggests checking gross long and short holdings to understand how the optimizer produced the difference.
The replies also distinguish leverage from alpha. The optimizer may use short positions to exploit estimated return differences between securities, which can contribute to the portfolio’s historical performance. But those estimates may reflect chance, and aggressive optimization can overbet uncertain or temporary opportunities. The discussion offers conceptual guidance rather than a general regression result: beta depends on the portfolio’s exposures, while alpha depends on the returns left unexplained by the chosen factors. Its numerical examples are illustrative, and future persistence of estimated differentials is uncertain.
Key ideas
- Increasing market exposure through leverage can increase a portfolio’s beta.
- Short positions can reduce net market exposure even when gross exposures are large.
- Gross long and short holdings help reveal how an optimizer constructed leverage.
- Long-short positions can exploit estimated return differences, but those differences may not persist.
- Optimizers can overstate position sizes when their inputs are noisy or overly optimistic.
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# What is the effect of leverage (ability to short stocks) upon the Fama-French regression coefficients? # What is the effect of leverage (ability to short stocks) upon the Fama-French regression coefficients? 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! ## Answer by demully (score 1, accepted) https://quant.stackexchange.com/a/49015 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.
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