| Since Markowitz published his seminal work on mean-variance portfolio selection in 1952, almost all literatures in the past half century adhere their investigation to a binding budget spending assumption in static problem settings and a self financing assumption in dynamic settings. In the mean-variance world for a market of all risky assets, however, the common belief of monotonicity does not hold, i.e., not the larger amount you invest, the larger expected future wealth you can expect for a given risk (variance) level. We introduce in this thesis the concept of pseudo efficiency to remove from the candidates such efficient mean-variance policies which can be achieved by less initial investment level. By relaxing the binding budget spending restriction in investment, we derive an optimal scheme in managing initial wealth which dominates the traditional mean-variance efficient frontier. Moreover, as the general dynamic mean-variance portfolio selection formulation does not satisfy the principle of optimality of dynamic programming, phenomena of time inconsistency occur, i.e., investors may have incentives to deviate from the pre-committed optimal mean-variance portfolio policy during the investment process under certain circumstances. By introducing the concept of time inconsistency in efficiency and defining the induced trade-off, we further demonstrate in this thesis that investors behave irrationally under the pre-committed optimal mean-variance portfolio policy when their wealth is above certain threshold during the investment process. By relaxing the self-financing restriction to allow withdrawal of money out of the market, we develop a revised dynamic mean-variance policy for a market with a riskless asset which dominates the pre-committed optimal mean-variance portfolio policy in the sense that, while the two achieve the same mean-variance pair of the terminal wealth, the revised policy enables the investor to receive a free cash flow stream (FCFS) during the investment process. We further apply the concept of pseudo efficiency to a dynamic market of all risky assets and explore (better) revised dynamic mean-variance policies. By including the free cash flow stream in the total wealth, our proposed policy dominates the pre-committed optimal mean-variance portfolio policy in the sense that while both achieve the same total mean, the revised policy generates a smaller total variance. We reveal in this thesis that the time consistency in efficiency is closely related to the completeness of the market. We further discuss the relationship between time consistency in efficiency and the variance-optimal signed martingale measure (VSMM) of the market. Finally we show that time inconsistency in efficiency can be eliminated by enforcing no-shorting constraint for some market setting. |
