I am trying to find an utility function that best describes a graph which has risk on x-axis and costs on y-axis. This is for evaluating different rebalancing strategies. The most optimal rebalancing strategy will have lower risk and lower costs. The parameters that describe costs can be low, medium and high. The parameters that describe risk can be low tolerance, medium tolerance and high tolerance. So, an optimal strategy will have low cost and high risk tolerance. I am thinking of the following functions - U = 1/(Costs * Risk)U = - (Costs) - ((Risk)^2)U = - (Costs) - (Risk)Any ideas will be greatly appreciated. Thanks,

The trouble with the first one is you can set your risk to zero by setting costs = ending market value (that is, give the portfolio to charity). Since it's always possible to reduce risk by at least $1 by giving up $1, (2) and (3) fail as well.It makes more sense to consider return - cost versus risk, so a function like:(A - Cost)/Risk