Cautiousness in the Small and in the Large
27 Pages Posted: 27 Oct 2012 Last revised: 17 Dec 2012
Date Written: October 11, 2012
In this paper we propose a new downside risk aversion measure, which is called cautiousness in the literature. Using a simple portfolio problem with a risk-free bond, a stock, and an option on the stock, we show that, an agent has higher cautiousness (i) if and only if she is always more likely to buy the option, (ii) if and only if she always demands more options per share. As an option's payoff is a convex function of the underlying stock price, increasing positions in the option increases the convexity of a portfolio's payoff and results in a strong increase in skewness which is defined by Van Zwet (Van Zwet, W. R. (1964). Convex Transformations of Random Variables, Mathematical Centre Tracts 7, Mathematisch Centrum, Amsterdam) and Chiu (Chiu, W. H. (2005). Skewness Preference, Risk Aversion, and the Precedence Relations on Stochastic Changes, Management Science 12, 1816-1828).
Keywords: cautiousness, convex transformation of random variables, demand for options, downside risk aversion, strong increases in skewness
JEL Classification: D81, G11
Suggested Citation: Suggested Citation