Markowitz’s World in Continuous Time, and Beyond

Mean—variance portfolio selection is concerned with the allocation of wealth among a basket of securities so as to achieve a prescribed trade-off between the return of the investment and the associated risk. The model was first proposed and solved in the single-period setting by Harry Markowitz in his Novel-Prize winning work [38] (see also [39]). The most important contribution of Markowitz’s work is the introduction of quantitative and scientific approaches to risk analysis or risk management. This work has become the foundation of modern finance and has had tremendous impact on its further development. For example, in Markowitz’s world (i.e., the world where all the investors believes in the mean—variance theory), one of the important consequences of Markowitz’s results is the so-called mutual fund theorem, which asserts that two mutual funds (which are two efficient portfolios) can be established so that all investors will be indifferent in making portfolios from among the original set of assets or from these two funds. Moreover, if a risk-free asset (such as a back account) is available, then investors need only to make use of one specific risky fund — called the tangent fund — and the risk-free asset. Assuming that portfolios are not constrained, a logical consequence of this single-period mutual fund theorem is that the market portfolio turns out to be the tangent fund, which therefore must be efficient. This, in turn, leads to the elegant capital asset pricing model (CAPM).