When it comes to the stocks, the concept of company size is a well understood concept. It literally relates to the market capitalization (market cap) of a company which is obtained by multiplying the company shares outstanding by the share price each day. In terms of mathematical formula, for stock i with market cap = m, price = p, shares outstanding = s, we have the following formula

Size was one of the first explanatory factors used back when the capital asset pricing model (CAPM) was expanded upon in the 1990s. This CAPM model that related a stocks return to some alpha plus beta times the market return was too simplistic. Professors Fama and French expanded upon this model by introducing a value factor and a size factor, which they called *high-minus-low* (hml) and *small-minus-big* (smb).

This model therefore represents a stocks return as some extra return alpha plus various beta weightings multiplied by the returns on the market, the value factor and the size factor. But what is the size factor itself ?

Academic research has shown that consistent returns were possible if investors went long a basket of small-cap stocks and short a basket of large-cap stocks and this is called the *size factor. *This makes intuitive sense as small cap stocks are viewed as inherently more risky (higher volatility and mostly higher beta) than large-cap stocks and therefore capm says that the return should be higher for small-cap stocks to compensate for this extra risk.

The size factor is intuitively one of the easiest factors to understand and some of the common reasons for its use are given by:

- it is linked to price volatility and higher beta, indicating it is riskier
- economic theory is easy and intuitive to understand
- capm and Fama-French models provides theoretical backing

The size factor is a major part of the Fama-French model and so historically was treated as an alpha factor and quantitatively focused investors seeked to harvest this *size premium*. Over time, this size factor alpha has eroded after the factor became very widespread due to its natural understandability. As a result, it has arguably ceased to become a significant alpha factor and is instead used primarily in investors risk models or for filtering and selecting their investable universe.

We do not construct and invest in a size factor explicitly as an alpha, as highlighted above. We do however use it for universe filtering and also during construction of other quantitative investment strategies to augment those signals to account for some of the size effect.

**Illustration of the size effect**

The volatility of the long-short market neutral size factor is very similar to the value factor, with an average rolling 1y volatility of 9%. As we would expect during covid, this volatility spiked especially as the factor is small-cap focused but like value and momentum, this increase has reversed and is expected now to return to its single digit average.

The size factor has really one natural partner in terms of correlation, or negative correlation, to be more precise. Quality companies as we have discussed previously tend to be long established, historically strong companies and on average these are larger cap companies as a result. Therefore it is natural to see a consistently negative correlation between size and quality, which averages around -40% on a rolling 2y basis.

As has become popular in the investing world, size is now treated as a risk factor or used in universe generation more so than used as a proactive alpha generating strategy which consumes some risk budget. The size factor's recent overall historical performance is indicative of this, as it hugs and oscillates around the zero level with a lack of consistent performance, either positive or negative.

The historical yearly path performance of the size factor also shows us quite nicely the tightly coupled nature of those returns over a 13y history. With the exception of 2020 and covid, there was just a single year where size delivered an absolute performance of greater than 10%. Otherwise it is quite evenly split between up years and down years, 1/3 to 2/3, with most yearly returns at the 5% level.

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