The Capital Asset Pricing Model (CAPM) by Sharpe (1964), Lintner (1965) and Mossin (1966) was long regarded as a comprehensive model for explaining stock returns. However, Fama and MacBeth (1973) found that the CAPM was not able to fully explain stock returns. As a result, further factors were identified that contribute to the understanding of stock returns. The first of these market anomalies is the size premium, which was first discovered by Banz (1981). The size premium states that small companies tend to have higher returns compared to large companies.
Written by
Published on
Category
In modern asset pricing research, the size premium has become indispensable as a control variable. However, the size premium is also increasingly being used in practice to determine the cost of capital as accurately as possible. The theoretical justification for the size premium is that smaller companies have a higher risk and investors demand a higher returnas compensation.
This correlation can be shown in a linear regression. The market capitalisation is used as an independent variable. As the market capitalisation of a stock grows ex-ponentially with constant returns due to compound interest, logarithmic market capitalisation is often used in research. This makes the relationship linear and easier to analyse. The regression equation is as follows:
ri-rf=α+β1*LogMarketCap
Where ri-rf is the excess return of stock i over the risk-free rate. β1 describes the size factor as the percentage additional return perunit of LogMarketCap, which represents the logarithmised market capitalisation. Figure 1 shows the data points for the excess return and the LogMarketCap.
In order to quantify an exact value of how high the size premium is in the data set used, a further regression is carried out using the market capitalisation at a scale of one billion USD. The regression equation then reads:
ri-rf=α+β1*LogMarketCap/1.000.000.000
This results in a value of -0.00016 for the coefficient β1 (t-value: -2,84).Specifically, this means that for an increase in market capitalisation of one billion USD, the average excess return falls by 0.016%. However, these results should be interpreted with caution, as they depend significantly on the underlying data set and the methods used to adjust the data.
In practice, it is much easier to calculate the differences in returns between small and large stocks. The data is sorted by size and grouped into portfolios. The number of portfolios can vary, but in the following analysis, it is organized into ten portfolios. Figure 2 shows the average return for each of the portfolios, where one is the portfolio with the smallest stocks and ten is the portfolio with the largest stocks.
There is a yield difference of 20.91% per year between portfolio one and portfolio ten, with the majority of the difference arising between portfolios one and four. Table 1 shows the average annual excess returns and the market capitalisation upper lim-its for each portfolio.
Table 1 shows that the size premium is particularly relevant for small companies. If this is taken into account in the regression by only using companies with a market capitalisation of less than one billion USD and scaling the market capitalisation to one million USD, this results in a coefficient of -0.00019 (t-value: -12.12). This means that with an increase in market capitalisation of one hundred million USD, the annual excess return falls by an average of 1.96%.
In asset pricing research, the size premium is considered as a control variable by default. In practice, however, the size premium is still rarely used. Taking the size premium into account can make an additional contribution to the calculation of capi-tal costs by increasing the accuracy and informative value of the results. However, the calculation of the size premium depends heavily on the data set used (see breakdown by company size) and therefore requires a careful methodological ap-proach and a sound data basis.
Bibliography
Banz, Rolf W. (1981). The relationship between return and market value of common stocks. Journal of financial economics, 9(1), 3-18.
Fama, Eugene F., and James D. MacBeth. "Risk, return, and equilibrium: Empirical tests." Journal of political economy 81.3 (1973): 607-636.
Lintner, John (1965): The Valuation of Risk Assets and the Selection of Risky In-vestments in Stock Portfolios and Capital Budgets. In: The Review of Economics and Statistic, 47(1): 13–37.
Sharpe, William (1964): Capital asset prices: A theory of market equilibrium under conditions of risk. In: The Journal of Finance, 19(3): 425–442.
Mossin, Jan (1966): Equilibrium in a Capital Asset Market. In: Econometrica, 34: 768–783.
We support you in researching the data — e.g. putting together the peer group — with a short training session on how to use the platform. We are happy to do this based on your specific project.
