Abstract

Empirical properties of stock prices have been studied extensively in financial econometrics and empirical asset pricing. While a great deal has been discovered about the statistical properties of individual assets at low and high frequencies, as well as cross-sectional properties in relation to macroeconomic, fundamental, and statistical factors, relatively little attention has been paid to macroscopic properties such as the capital distribution curve. Macroscopic properties of equity markets such as market diversity and intrinsic volatility are documented and exploited in stochastic portfolio theory, but a systematic empirical investigation is lacking. In the first part of this work, we present a detailed empirical study of these properties using CRSP data of the U.S. stock market. Then, we show how these properties are related to the relative performance of systematically rebalanced portfolios under proportional transaction costs.

It is often stated in stochastic portfolio theory that the capital distribution curve is “stable”. As the market may not be ergodic, we need an appropriate notion of stability. One interpretation of stability is that the shape of the curve can be reasonably fit by a low-dimensional model. To this end, we introduce a 3-parameter exponential family and find that it captures the main features of the capital curve except for the idiosyncratic fluctuations of the largest stocks. In particular, the calibrated exponential family accurately reproduces the realized market diversity. Market diversity, like market entropy, is a measure of the concentration of capital in the market. Using convex PCA we have shown previously that diversity corresponds to the first convex principal component and can be interpreted as the most efficient single-number summary of the capital curve.

Another fundamental object in stochastic portfolio theory is the intrinsic volatility of the market given by the excess growth rate which can be thought of as a measure of relative volatility among the stocks. Our investigation reveals that the unconditional distribution of the excess growth rate is highly skewed. We illustrate that the excess growth rate depends on the frequency at which the returns are measured and show that an ARCH effect is present. This suggests the possibility of forecasting the intrinsic volatility using GARCH-type models. We also introduce a decomposition of the excess growth rate that enables us to consider the distribution of contributions to intrinsic volatility by rank. On average, we find that large stocks contribute more than small stocks. As expected from our analysis of the capital curve, large stocks require separate attention.

Finally, in our study of portfolio performance, we simulate the performance of a diversity-weighted portfolio rebalanced at a fixed frequency with and without transaction costs. We use as a benchmark the buy-and-hold cap-weighted portfolio. Over a short horizon, we see that the relative performance is dominated by changes in market diversity and if changes in diversity are fixed, relative performance improves when the excess growth rate is large. Taken together, these insights have significant implications for portfolio management and shed light on the calibration of recently proposed models of the capital distribution.