One-fund theorem states that an efficient portfolio in a Mean-Variance (M-V) portfolioselection problem for a set of some risky assets and a riskless asset can be representedby a combination of a unique risky fund (tangency portfolio) and the riskless asset. Inthis paper, we introduce a method for which the tangency portfolio can be produced as acorner portfolio. So, the tangency portfolio can be computed easily and fast by anyalgorithm designed for tracing out the M-V efficient frontier via computing the cornerportfolios. Moreover, we show that how this method can be used for tracing out the M-Vefficient frontier when problem contains a riskless asset in which the borrowing is notallowed.

A risk measure in a portfolio selection problem is linear programming (LP) solvable, ifit has a linear formulation when the asset returns are represented by discrete randomvariables, i.e., they are defined by their realizations under specifiedscenarios. The efficient frontier corresponding to an LP solvable model is a piecewiselinear curve. In this paper we describe a method which realizes and produces a tangencyportfolio as a by-product during the procedure of tracing out of the efficient frontier ofrisky assets in an LP solvable model, when our portfolio contains some risky assets and ariskless asset, using nonsmooth optimization methods. We show that the test of finding thetangency portfolio can be limited only for two portfolios. Also, we describe that how thismethod can be employed to trace out the efficient frontier corresponding to a portfolioselection problem in the presence of a riskless asset.