# Asset Pricing and inferences - Intercept and relationship to returns

Ideally this question is very similar to What's the meaning of the intercept in asset pricing model?

I am regressing a "buys minus sells" portfolio returns to the Carhart factors. The intercept is coming out to be positive and significant.

However this portfolio has made a massive loss - in terms of profit and loss.

I would like to understand how we can interpret any empirical results with this apparent paradox that though the portfolio has lost a lot of money, its alpha is positive.

I understand that there could be missing factors - but lets assume that these are the factors that we have.

Is there a link between intercept alpha and overall profitability of a portfolio?

As stated in my answer here, we have to distinguish between empirically testing asset pricing models (like the CAPM or Fama/French models) and using asset pricing models as a benckmark portfolio to evaluate the outperformance of our investment strategy.

Consider (without loss of generality) the CAPM with it's (theoretical) risk/return relationship:

$$E[r_i - r_f] = \beta_i E[r_m- r_f]$$

However, we do run the following regression, when we are empirically test if the CAPM holds:

$$r_{i,t} − r_{t,f}= \alpha_i + \beta_i (r_{t,m} − r_{t,f}) + \epsilon_{i,t}$$

The null hypothesis in this test is, that $$\alpha_i$$ should be indistinguishable from zero for any asset $$i$$. If we obtain any significant alpha $$a_i$$ we have to reject that the CAPM is an adequate model to describe real financial markets (and that's indeed the case: The CAPM is "dead", see this excellent answer).

What you are referring to is actually using the Fama/French/Carhart model as a bechmark test for your portfolio strategy. As you obtain a positive significant alpha, your strategy generates (abnormal) positive returns, which may not be explained by size-, value- and momentum-effects.

However, your positive alpha (Jensen's alpha) just states that your portfolio outperforms the benchmark model. This does not imply that your strategy avoids huge losses. According to Kenneth French's data library, the US-market return for March 2020 was -13.39% that month. If your portfolio return is e.g. -10% you obtain a huge loss but still outperform the market.

• Thank you for the answer. This is a basic inference on the factors analogy, though it changes when its either for holdings based or transactions based methodology Commented Jun 8, 2020 at 3:59

In your case, there are five elements that aggregate into your portfolio return. Just treat the intercept as a fifth factor for now. We'll come back to what it means in a second. But you should be able to take your regression betas, multiply these by each style's period-return, to give you 5 style return contributions, that should sum to your portfolio return.

So then you can say something like:

• "We were long of the market, which cost us A%". Assuming of course, that your Beta was >0, which is what matters to absolute returns (greater/less than 1 being more relevant for benchmark-adjusted relative assessments).

• And then "our Value, Size, and Momentum exposures made or lost us B, C, and D% respectively". These are your style returns over and above the market. How much did being on the right or wrong side of your factors generate in the period?

[Although note in passing that it is perfectly possible to own a \$100bn stock with a 50x PE that's down 50% ytd and it behave like a "Smallcap Value Winner"; even though it is actually none of them. And this needn't be a random accident. Something in the company's business model might give it characteristics that are shared by the net bias among stocks with the opposite features]

• And finally, there's your intercept worth E% (which is the average of the residuals ex-your factors, to make the actual regression residuals average zero), without which everything would have been much worse.

Lazily, you might call this "Alpha"; but it might be tempting the gods on Olympus actually using the term. Maybe safer to call it "Sector and Stock Effects"; but it covers a few sources of return that you may or not be able to measure in detail (depending on what data you keep about your portfolio). And many of these will turn out to be related to your four Factors, albeit in ways that are not captured in a regression of period averages.

Style Rotation: imagine, hypothetically, that you really could pick the best performing Carhart factor every day with 100% accuracy and shift the portfolio 100% into this every day at zero cost or friction. Your regression would suggest that you had a 25% exposure to each over the period, an average that obviously represents a zero weight 75% of the time and 100% in the remaining quarter. So all the returns you make from a magic ability to pick and time style factors are statistically "independent" of style returns. From a regression perspective, this is indeed true. But obviously a lot of normal people have a rather different sense of the word "independent"; and they would argue that all of your returns can from Style Factors here! So some of your positive intercept could be due to changes in your style exposures during the period measured.

Style Anomalies: this is the opposite to the observation above that a smallcap might behave like a largecap when it is not, which might give you a slightly misleading level of exposure to smallcaps. It's equally possible that the Carhart factors, although intuitive, are not in themselves significant drivers of return. But they can act as imperfect proxies for different "flavours" of Size or Beta etc. that are very significant.

You might have eg run a beta of 1.2x - but you'll have had a very different outcome achieving this exposure through offshore drillers versus through social media plays. Sectors can often delineate winners and losers within your Large/Small, Value/Growth and Low/Beta factors. Indeed sector effects tend to explain more stock performance than do style effects. The difference simply being that Factors are supposed to deliver a risk premium; while Sectors have no such expectations or pretensions.

But it might not be explicit sector exposure but any number of macro effects that correlate to the Carhart factors. One classic example is supposed to be FX and Smallcaps (Large = more multinational, Small is more domestic, strong dollar is weak global competitiveness for US corporates, so good for Small, bad for Large). Apparently...

Or maybe both Smallcap and Value did badly; but Smallcap Value did OK. Someone was overweight this; and conversely underweight Large Value and Small Growth, which tanked to produce the poor overall style averages. combinations of Style can produce effects not captured in the factor aggregates, so again appear "independent". Google "Simpson's Paradox" for more - some statisticians were famously able to prove that a university admission policy was simultaneously discriminating against both men and women at the same time.

So there might be any number of style effects that cause you to own the right and wrong flavours of Beta, Size, Mom, and Value/Growth; that just aren't captured in a period regression. And all this is before we get to genuine "Stock Selection", ie company returns that have genuinely have nothing to do with your factor, or their sector effects etc.

Your positive intercept - label it as you see fit - is simply some combination of the return drivers above. If the portfolio is down and this was a significant positive, it's a measure of "how much worse it would have been" without them. These effects might be related to your Carhart factors; but just not in the static sense that a regression captures. How much of the overall portfolio loss is simply due to the market beta factor versus the incremental return contributions from the other three factors, you'll have to work out. But it's not too difficult; and doesn't require any data that you didn't already have to run your regression.

hope this helps.

• Thanks for the nice answer demully Commented Jun 8, 2020 at 3:59