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The leverage effect refers to the generally negative correlation between the return of an asset and the changes in its volatility. There is broad agreement in the literature that the effect should be present, and it has been consistently found in empirical work. However, a few papers have pointed out a puzzle: the return distribution of many assets do not appear to be affected by the leverage effect. We analyze the determinants of the return distribution and find that it is driven primarily by an interaction effect involving both the leverage effect and the mean-reversion effect. When the leverage effect is large and the mean-reversion effect is small, then the interaction exerts a strong effect on the return distribution. However, if the mean-reversion effect is large, even a large leverage effect has little effect on the return distribution. To better understand the impact of the interaction effect, we propose an indirect method to measure it. We apply our methodology to empirical data and find that the S&P500 data exhibits a weak interaction effect, and consequently its returns distribution is little impacted by the leverage effect. Furthermore, the interaction effect is closely related to the size factor: small firms tend to have a strong interaction effect and large firms tend to have a weak interaction effect.
Stochastic gradient descent (SGD) is almost ubiquitously used in training non-convex optimization tasks. Recently, a hypothesis by Keskar et al. (2017) that large batch SGD tends to converge to sharp minima has received increasing attention. We justify this hypothesis by providing new properties of SGD in both finite-time and asymptotic regimes, using tools from Partial Differential Equations. In particular, we give an explicit escaping time of SGD from a local minimum in the finite-time regime. We prove that SGD tends to converge to flatter minima in the asymptotic regime (although it may take exponential time to converge) regardless of the batch size. We also find that SGD with a larger learning rate to batch size ratio tends to converge to a flat minimum faster. However, its generalization performance could be worse than the SGD with a smaller learning rate to batch size ratio.
This paper provides a comprehensive analysis of portfolios of active mutual funds, ETFs and hedge funds through the lens of risk (anomaly) factors. We show that these funds do not systematically tilt their portfolios towards profitable factors, such as high book-to-market (BM) ratios, high momentum, small size, high profitability and low investment growth. Strikingly, there are almost no high-BM funds in our sample while there are many low-BM “growth” funds. Portfolios of “growth” funds are concentrated in low BM-stocks but “value” funds hold stocks across the entire BM spectrum. In fact, most “value” funds hold a higher proportion of their portfolios in low-BM (“growth”) stocks than in high-BM (“value”) stocks. While there are some micro/small/mid-cap funds, the vast majority of mutual funds hold very large stocks. But the distributions of mutual fund momentum, profitability and investment growth are concentrated around market average with little variation across funds. The characteristics distributions of ETFs and hedge funds do not differ significantly from the those of mutual funds. We conclude that the characteristics of mutual fund portfolios raises a number of questions about why funds do not exploit well-known return premia and how their portfolio choices affects asset prices in equilibrium.
Beginners first learn to price stock options with a simple binomial tree model for random price changes. It is well known that this classical one-dimensional random walk converges weakly to Brownian motion in the proper space-time scaling limit. Actual stock prices changes occur not at regular times but at random times according to the order flow in an electronic limit order book (LOB), and these are observed to have heteroscedastic and self-exciting characteristics.
In this talk, we consider random walks in which jumps occur at random times described by an independent general point process, which could be a self-exciting process such as a Hawkes process. We show that in the correct scaling limit, this converges to a time-changed Brownian motion, where the time change is the compensator of the original point process. The resulting stock price process can exhibit many of the stylized properties of observed stock prices. We establish a familiar formula for the price of an option for this model, forming a connection between models of LOB dynamics and financial derivative pricing. (This paper is joint work with Navid Salehy and Nima Salehy.)
Mutual funds that invest in private securities value those securities at stale prices. Prices change on average every 2.5 quarters, vary across fund families, and are revised upward dramatically at follow-on funding events. The infrequent, but dramatic price changes yield predictably large fund returns. Fund investors can exploit the stale pricing by buying (selling) before (after) the follow-on funding events (though we find little evidence of this behavior to date). Fund families can opportunistically save up and unleash dry powder (unused markup of private securities) when doing so helps their high-priority funds get to the top of league tables at year ends. Consistent with these incentives, funds near the top of league tables increase private valuations more around fourth quarter follow-on funding events than funds ranked lower.
ABSTRACT: We describe two problems – omitted variable bias and measurement error – that arise when a ratio is the dependent variable in a linear regression. First, we show how bias can arise from the omission of two variables based on a ratio’s denominator, and we describe tests for the degree of bias. As an example, we show that the familiar “inverse U” relationship between managerial ownership and Tobin’s Q is reversed when omitted variables are included. Second, we show how measurement error in the ratio denominator can lead to bias. We urge caution about using ratios as dependent variables.