A stock market model based on CAPM and market size

Abstract

We introduce a new system of stochastic differential equations which models dependence of market beta and unsystematic risk upon size, measured by market capitalization. We fit our model using size deciles data from Kenneth French’s data library. This model is somewhat similar to generalized volatility-stabilized models. The novelty of our work is twofold. First, we take into account the difference between price and total returns (in other words, between market size and wealth processes). Second, we work with actual market data. We study the long-term properties of this system of equations, and reproduce observed linearity of the capital distribution curve. In the “Appendix”, we analyze size-based real-world index funds.

Appendix: Statistical analysis of size-based index funds

These size deciles of the CRSP universe are not directly investable. But there exist size-based funds available for individual investors. Among many of them, let us take JKJ, JKG, JKD: iShares Morningstar Small-Cap, Mid-Cap, and Large-Cap exchange-traded funds. These are based on largest 70%, next 20%, and next 7% of the total universe of stocks. In other words, Large-Cap corresponds to Deciles 1–7 weighted by their market capitalizations, Mid-Cap corresponds to Deciles 8–9 weighted by their market capitalizations, and Small-Cap corresponds to the top 7% of the bottom Decile 10.

Monthly total arithmetic returns for these funds are taken from BlackRock web site, July 2004 – August 2020. For risk-free returns, we take 1-month Treasury Constant Maturity Rate from Federal Reserve Economic Data web site, observed at the last day of each month June 2004 – July 2020. We compute geometric versions of these returns. From each such rate r, we obtain geometric total monthly returns for the next month $\ln (1 + r/1200)$. Then we compute equity premia $P_S, P_M, P_L$ for these funds. Regress the first two upon the third:

$$\begin{aligned} {\left\{ \begin{array}{ll} P_S(t) = \alpha _S + \beta _SP_L(t) + \varepsilon _S(t);\\ P_M(t) = \alpha _M + \beta _MP_L(t) + \varepsilon _M(t); \end{array}\right. } \quad \begin{bmatrix} \varepsilon _S(t)\\ \varepsilon _M(t) \end{bmatrix} \sim {\mathcal {N}}_2\left( \begin{bmatrix}0 \\ 0\end{bmatrix}, \varSigma \right) . \end{aligned}$$

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The quantile–quantile plots, Shapiro–Wilk and Jarque–Bera normality tests, and autocorrelation function plots allow us to assume that each series of residuals can be modeled by i.i.d. normal distribution. Thus we can apply standard Student tests for regression coefficients. The 95% confidence intervals for each of $\alpha _S$ and $\alpha _M$ contain zero. Thus we can assume that $\alpha _S = \alpha _M = 0$, but the confidence intervals for $\beta _S$ and $\beta _M$ do not contain 1. Point estimates of these coefficients are: $\beta _S = 1.27$, $\beta _M = 1.15$. Estimates for standard errors for residuals, and cross-correlation between residuals are: $\sigma _S = 0.026$, $\sigma _M = 0.019$, $\rho = 0.83$. The $R^2$ values for each regression are 87% and 80%. Thus we see that the CAPM works for actual traded size-based funds.