تابع کمبود و انتخاب پورتفولیو: در برخی از الحاقات و موارد خاص

The shortage function has recently been introduced in portfolio selection theory for measuring efficiency. In this paper we focuss on the case of shortselling. We show that, in such a case, the shortage function can be computed in closed form. Some issues concerning duality are also analyzed. We also analyze the case of a riskless asset.

Distance functions, have been introduced by Shephard (1953) for efficiency measurement either in input or output orientation. At the same time, Markowitz, 1952 and Markowitz, 1959 has formulated the mean–variance model, a mathematical approach for determining the optimal risk-return trade-off for portfolio selection. This approach is based upon quadratic programming. However, its computational cost was very high. Hence, Sharpe (1963) had developed the simplified diagonal model and later formulated the capital asset pricing model (CAPM) with Lintner (1965). Markowitz (2008) criticized the relation between risk and excess returns described by the linear model due to Sharpe and Lintner. He argued that different expected returns might surely be obtained from the same risk structure. Nevertheless, the mean–variance approach is the cornerstone of portfolio management and risk assessment. The purpose of this paper is to consider some issue in the measurement of portfolio efficiency. This contribution extends the analysis proposed in Briec et al., 2004 and Briec et al., 2007 where a general framework was introduced that is based upon the shortage function a concept introduced by Luenberger (1995) in microeconomic analysis. Transposed in a portfolio optimization context, this function looks for possible simultaneous improvement of return and reduction of risk in the direction of a vector g. In this paper, we make other investigations about measures in the case where there is short-selling. In particular, it is shown that the shortage function can then be computed in closed form. Moreover, it is also established that one can obtain a duality result linking the indirect mean–variance utility and the shortage function. We also consider the case of a riskless asset and provide a computation of the shortage function in closed form in such a case. This paper is organized as follow. In Section 2, we succinctly present the basic tools of the portfolio management approach proposed in Briec et al. (2004). Section 3 focusses on the shortage function, duality properties and the indirect utility function under relaxed assumptions. In Section 4, we focuss on the Sharpe model (Sharpe, 1963 and Sharpe, 1964). A closed form for the return-oriented shortage function is established in presence of a riskless asset. In Section 5, the case of short selling is considered and we show that a closed form of the shortage function can be established using Lagrangian calculus. We also provide duality results in closed form under the presence of short selling. A concluding section outlines conclusions and possible extensions.

The objective of this paper was to propose a methodology to measure portfolio efficiency in the case of shortselling. One of the interesting features of this approach is that it yields a very simple solution under the simplifying hypotheses of Sharpe (1964). Another, perhaps promising area for further research might to be determine the composition of the tangent portfolio through our algorithmic approach, and to compare such an estimate with results obtained by traditional methods.