Richard C. Stapleton’s research while affiliated with University of Manchester and other places

We search for a definition of the downside risk premium analogous to the Pratt–Arrow definition of the risk premium. However, even in the local analysis difficulties arise. To overcome these, we propose a definition based on the difference between two gambles. Further, a global analysis reveals that higher-order terms affect the downside risk premium and these cannot be ignored. We show that all five measures of the intensity of downside risk aversion that have been suggested are invalid in the case of the global analysis.

We add an independent unfair background risk to higher-order risk-taking models in the current literature and examine its interaction with the main risk under consideration. Parallel to the well-known concept of risk vulnerability, which is defined by Gollier and Pratt (Econometrica 64:1109–1123, 1996), an agent is said to have a type of higher-order risk vulnerability if adding an independent unfair background risk to wealth raises his level of this type of higher-order risk aversion. We derive necessary and sufficient conditions for all types of higher-order risk vulnerabilities and explain their behavioral implications. We find that as in the case of risk vulnerability, all familiar HARA utility functions have all types of higher-order risk vulnerabilities except for a type of third-order risk vulnerability corresponding to a downside risk aversion measure called the Schwarzian derivative.

We show that the utility premium of Friedman and Savage can be used to explain comparative risk aversion and comparative prudence. More precisely, we show that the greater the risk aversion measure, the greater a risk’s utility premium normalized by the marginal utility and that the greater the prudence measure, the greater the utility premium for disaggregating a certain loss of wealth and a zero-mean risk normalized by the utility function’s second derivative.

In this article we establish cautiousness as a new downside risk aversion measure, using a portfolio problem with a risk-free bond, a stock, and an option. We show that, an investor has higher cautiousness (i) if and only if she is always more likely to buy the option; and (ii) if and only if she always demands more options per share. As an option’s payoff is a convex function, increasing positions in the option increases the convexity of a portfolio, which leads to an increase in skewness. Thus the results in this article establish the link between cautiousness and skewness preference.

We define a class of risk-taking-neutral (RTN) background risks. These background risks have the property that they will not alter decisions made with respect to another risk, for individuals with HARA utility. If we wish to compare a decision made with and without some exogenous background risk, it is often easier to compare the decision made to one made with a RTN background risk. We use this methodology to prove and extend a well-known theorem about dynamic investment strategy, due to Mossin (1968a). We also use this methodology to analyze investment behavior in the presence of an income tax as well as to analyze investment behavior in the presence of particular types of background risks.

Huang and Stapleton (2012) characterize cautiousness as a measure of skewness preferences using a simple portfolio problem with a risk-free bond, a stock, and an option on the stock. In this paper, we explain the link between cautiousness and skewness preferences in a more general context where agents face two risks with one of them being a convex (or concave) function of the other.

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).

Should you buy a stock or a corporate bond? A common belief is that the Pratt-Arrow risk aversion measure gives the answer: a more risk averse investor will prefer more a corporate bond to a stock. However, this is not always true. In a simple portfolio problem with a riskless bond, a stock and a corporate bond from a firm, we show that, it is not the risk aversion measure but a downside risk aversion measure called cautiousness which gives the answer to the question: a more cautious investor will prefer more a stock to a corporate bond. While in some cases downside risk aversion agrees with risk aversion, in some other cases it does not.

In this paper, we assume that investors have the same information, buttrade due to the evolution of their non-market wealth. In ourformulation, investors rebalance their portfolios in response to changesin their expected non-market wealth, and hence trade. We assume anincomplete market in which risky non-market wealth is non-hedgeable andindependent of market risk, and thus represents an additive backgroundrisk. Investors who experience positive shocks to their expected wealthbuy more stocks from those who experience less positive shocks. Theextent of trading depends on the heterogeneity of the shocks to theexpected background risk across the agents. The demands of the twoagents are convex or concave in the state of the economy, whichjustifies trading in the aggregate assets and contingent claims.