Relationships between risk aversion, prudence, and cautiousness

In this paper we investigate the relationships between risk aversion, prudence, and cautiousness, which have interpretations for investors' behaviour in different financial activities. We show if an investor is always more prudent then she is almost always more risk averse. Assuming investors' marginal utility of zero wealth is infinity, we show when an investor's wealth approaches zero or infinity, in limit cautiousness equals the inverse of relative risk aversion, if an investor's cautiousness is bounded from above (below) by a positive constant then his relative risk aversion is bounded below (above) by the inverse of the constant, and an investor who always has higher cautiousness is always less relative risk averse along her optimal risk-sharing rule in an economy. We also show that increasing (decreasing) cautiousness implies decreasing (increasing) relative risk aversion