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.

我们引入了一个新的随机微分方程系统，该系统模拟了市场贝塔系数和非系统性风险对以市值衡量的规模的依赖性。我们使用来自 Kenneth French 数据库的大小十分位数数据来拟合我们的模型。该模型有点类似于广义波动率稳定模型。我们工作的新颖性是双重的。首先，我们考虑价格和总回报之间的差异（换句话说，市场规模和财富过程之间的差异）。其次，我们使用实际的市场数据。我们研究了这个方程组的长期特性，并重现了观察到的资本分配曲线的线性。在“附录”中，我们分析了基于规模的现实世界指数基金。