Counting Humps in Motzkin paths

In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order $n$ with $k$ peaks is the Narayana number. By double counting super Schr\"{o}der paths, we also get an identity involving products of binomial coefficients.Comment: 8 pages, 2 Figure

Does exchange rate risk matter for asset pricing? Working paper series--10-01

Following Adler and Dumas (1983), it is a common practice in the exchange rate literature to use the contemporaneous exchange rate change as the relevant risk factor. However, if exchange rate fluctuations affect cash flows of firms as Stulz (1984), among others, suggest and future not current cash flows matter for asset pricing, we should focus on future not contemporaneous exchange rate changes. Furthermore, if as Starks and Wei (2005) suggest exchange rate fluctuations can push a firm into financial distress, the exchange rate risk is a distress factor and should behave like the value factor in the three-factor model and carry a positive risk premium. We test these conjectures in this paper and find supporting evidence in the post Plaza-Accord period

Momentum in weekly industry portfolio returns: Working paper series--08-13

Contrary to Lehmann (1990) and Jegadeesh (1990), Gutierrez and Kelley (2008) recently find a long-lasting momentum in weekly individual stock returns. We extend Gutierrez and Kelley (2008) and examine momentum in weekly industry portfolio returns. We find that where momentum in six-month returns is mainly explained by cross-serial correlations as in Lewellen (2002), momentum in weekly returns is largely due to serial correlations, and that momentum does not always exhibit reversals in the long run. Our findings present a challenge to the popular behavioral models

Momentum and behavioral finance: Working paper series--10-16

Bulkley and Nawosah (2009) recently find that momentum in individual stock returns is explained by dispersion in unconditional mean returns, which suggests a risk-based explanation for momentum. However, they do not decompose the momentum component due to time-series dependence into the component due to serial correlation and that due to cross-serial correlation. Therefore, their approach may underestimate the importance of time-serial dependence if the serial correlation component has opposite sign from the cross-serial correlation component. Motivated by this observation, we apply the Du and Watkins (2007) methodology, which enables an unbiased decomposition of momentum profits into three relevant components. Different from Bulkley and Nawosah (2009), we find that momentum is due to both dispersion in unconditional mean returns and time-series dependence. Our findings therefore suggest that risk or behavioral biases in isolation may not be sufficient to explain momentum

Value premium and investor sentiment: Working paper series--11-02

There are two competing explanations for the value premium. One suggests that value premium is a compensation for risk, while the other implies that it is driven by investor sentiment. Previous empirical studies typically test these two competing explanations of value premium in isolation, which may lead to incorrect inferences. In this paper, we extend the literature by testing these two competing explanations of value premium in a joint fashion. We find that while value premium is correlated with investor sentiment, it shows very little correlation with the state of the economy. Based on this evidence, it is very difficult to argue that value premium is due to risk