The opportunity cost of being constrained by the type of assets: Bonds only or stocks only

I explore investors' welfare losses when they restrict themselves to invest in either stocks only or bonds only, but not in both. The restriction gives investors sub-optimal asset allocations that result in welfare losses. To measure these welfare losses I compare "only stock indices and Treasury bills" optimal portfolios and "only bond indices and Treasury bills" optimal portfolios with "stock and bond indices and Treasury bills" optimal portfolios using the concept of proportionate opportunity cost along with various CRRA utility functions. The original historical asset returns data set is used with a VAR in generating joint returns distributions for the portfolio formation period. I show that for investors with low levels of risk aversion welfare losses do not exceed 1.5% of initial wealth when they invest sub-optimally. For investors with medium and high levels of relative risk aversion, suboptimal portfolios of only one type of assets, stocks only or bonds only, along with Treasury bills, give expected utility about as high as optimal portfolios that include both types of assets, stocks and bonds.