The paper investigates different measures of downside risk aversion and relations among them. The authors begin with the relation between risk preference \(u\) and risk neutrality \(\eta\) expressed as \(u = \phi(\eta)\) and its inverse \(\eta = \psi(u)\). These two relations expressing the absolute aversion to downside risk are regarded as extremes along a spectrum of other definitions of increasingly restrictive definitions of absolute aversion to downside risk. Further utility function \(v(y)\) over \(u(y)\) are considered, where \(y\) denotes an income and consider the relation \(v(y) = \phi(u(y))\) with the condition \(\phi'' 0\). A comparison with the Arrow-Pratt measure of risk aversion depending on income is presented. A novel downside risk aversion measure is obtained