It must therefore be the minimum-variance portfolio. Now consider the investor with a risk tolerance of 25. In effect, the investor takes 25 units of the rtswap. This, added to the initial portfolio, gives net holdings with relatively little cash, a substantial bond position and a smaller but still significant stock position. Moreover, the magnitude of the latter that an investor should take is proportional to his or her risk tolerance.
Optimal holdings are thus linear functions of risk tolerance.
In fact, all investors can be accommodated by any two optimal portfolios. Thus the higher the covariance with the optimal portfolio, the higher will be the expected return. Since this is true for every asset i, it must also be true for every combination of assets, including portfolio x. Bix This implies that there is a linear relationship between asset expected returns and their beta values with respect to the optimal portfolio. The Market Portfolio and the Security Market Line Now consider a world in which all investors solve the problem formulated here but differ in both risk tolerance and wealth.