Classical mean-variance model revisited: pseudo efficiency

Investigating the inverse problem of the classical Markowitz mean‐variance formulation: Given a mean‐variance pair, find initial investment levels and their corresponding portfolio policies such that the given mean‐variance pair can be realized, we reveal that any mean‐variance pair inside the reachable region can be achieved by multiple portfolio policies associated with different initial investment levels. Therefore, in the mean‐variance world for a market of all risky assets, the common belief of monotonicity: ‘The larger you invest, the larger expected future wealth you can expect for a given risk (variance) level’ does not hold, which stimulates us to extend the classical two‐objective mean‐variance framework to an expanded three‐objective framework: to maximize the mean and minimize the variance of the final wealth as well as to minimize the initial investment level. As a result, we eliminate from the policy candidate list the set of pseudo efficient policies that are efficient in the original mean‐variance space, but inefficient in this newly introduced three‐dimensional objective space.