بودجه اصلی در تجزیه و تحلیل سبد سرمایه گذاری با اقدامات انحراف عمومی

Generalized measures of deviation are considered as substitutes for standard deviation in a framework like that of classical portfolio theory for coping with the uncertainty inherent in achieving rates of return beyond the risk-free rate. Such measures, derived for example from conditional value-at-risk and its variants, can reflect the different attitudes of different classes of investors. They lead nonetheless to generalized one-fund theorems in which a more customized version of portfolio optimization is the aim, rather than the idea that a single “master fund” might arise from market equilibrium and serve the interests of all investors. The results that are obtained cover discrete distributions along with continuous distributions. They are applicable therefore to portfolios involving derivatives, which create jumps in distribution functions at specific gain or loss values, well as to financial models involving finitely many scenarios. Furthermore, they deal rigorously with issues that come up at that level of generality, but have not received adequate attention, including possible lack of differentiability of the deviation expression with respect to the portfolio weights, and the potential nonuniqueness of optimal weights. The results also address in detail the phenomenon that if the risk-free rate lies above a certain threshold, the usually envisioned master fund must be replaced by one of alternative type, representing a “net short position” instead of a “net long position” in the risky instruments. For nonsymmetric deviation measures, the second type need not just be the reverse of the first type, and there can sometimes even be an interval for the risk-free rate in which no master fund of either type exists. A notion of basic fund, in place of master fund, is brought in to get around this difficulty and serve as a single guide to optimality regardless of such circumstances.

In classical portfolio theory, investors respond to the uncertainty of profits by selecting portfolios that minimize variance, or equivalently standard deviation, subject to achieving a specified level in expected gain (Markowitz, 1952, Kroll et al., 1984 and Markowitz, 1991). The well-known “one-fund theorem” (Tobin, 1958 and Sharpe, 1964) stipulates that this can be accomplished in terms of a single “master fund” portfolio by means of a formula which balances the amount invested in that portfolio with the amount invested at the current risk-free rate. Nowadays, other approaches to uncertainty have gained in popularity. Portfolios are being selected on the basis of percentile characteristics such as value-at-risk (VaR), conditional value-at-risk (CVaR), or other properties proposed for use in risk assessment; cf. ( Acerbi and Simonetti, 2002, Konno and Shirakawa, 1994 and Malevergne and Sornette, 2002) and earlier alternatives such as in Bawa and Lindenberg (1977). These measures have no pretension to being universal, however; VaR and CVaR depend, for instance, on the specification of a confidence level parameter, which could vary among investors. Instead, what is apparent in the alternative approaches currently being touted is a move toward a kind of partial customization of responses to risk, while still avoiding, as impractical, a reliance on specifying individual utility functions. A question in this evolving environment is the extent to which classical facts persist when the minimization of standard deviation is replaced by the minimization of some “nonstandard deviation.” Researchers have already looked into the possibilities in special cases under various limiting assumptions (recognized explicitly or imbedded implicitly). Our goal, in contrast, is to demonstrate that important parallels with classical results exist much more broadly, despite technical hurdles, and in this way to bring out features that have not completely been analyzed, or even perceived, in the past. We focus on the general deviation measures we developed axiomatically in Rockafellar et al. (2002a). Our idea is to substitute such a deviation measure for standard deviation in the setting of classical theory and investigate the consequences rigorously in detail. Furthermore, we aim at doing so, for the first time, in cases where the rates of return may have discrete distributions, or mixed discrete-continuous distributions (which can arise from derivatives, such as options), as well as cases where they have continuous distributions. The deviation measures we work with are paired one-to-one, through Rockafellar et al. (2002a), with risk measures in the sense of Artzner et al. (1999), but differ in partly relaxing their requirements while insisting on an additional property beyond theirs. A similar additional property was invoked by Ogryczak and Ruszczynski, 1999 and Ogryczak and Ruszczynski, 2002 for safety measures, which may be viewed as negatives of risk measures. Minimizing a deviation measure subject to a constraint on expected returns can anyway be different from minimizing the corresponding risk measure, since, as shown in Rockafellar et al. (2002a), the first problem always has a solution but the second problem can sometimes fail to have a solution, due to a phenomenon of “acceptably free lunches.” We are not, however, suggesting that deviation measures are better than risk measures. Both have their place, but deviation measures fit closer to the classical picture and therefore serve better the particular purposes of this paper. The axioms for deviation measures that we adopt from Rockafellar et al. (2002a) entail convexity. They cover numerous choices from classical type to CVaR type, but exclude the analogous expressions of VaR type, since those lack convexity. Convexity is essential for answering most of the harder questions that confront us. Its importance for sound applications in finance has already been recognized as well in connection with the coherency concept in Artzner et al. (1999). We do, however, try to indicate along the way the troubles that VaR type expressions would bring up. The first of our main results says that a one-fund theorem holds regardless of the particular choice of the deviation measure, but with certain modifications. The optimal risky portfolio need not always be unique, and it might not always be expressible by a “master fund” as traditionally conceived, even when only standard deviation is involved. An alternative concept of “basic fund” is introduced to fill the gap. The rest of our main results pin down precisely the degree to which basic funds can, or cannot, be rescaled into master funds. This turns out to require an understanding not only of an efficient frontier for risky portfolios at price 1, associated with “master funds of positive type,” but also of such a frontier for risky portfolios at price −1, associated with “master funds of negative type.” The magnitude of the risk-free rate of return plays a key role here. We prove that when it is below a certain threshold, the positive type prevails, but when it is above a certain threshold, the negative type has to be brought in. Moreover in special situations those thresholds can differ, leaving a gap filled by an interval of magnitudes of the risk-free rate for which neither type of master fund can replace a basic fund. We also explain how thresholds can be calculated by solving an auxiliary optimization problem. It deserves emphasis that, in contrast to much of the previous work in this area, our results are obtained without relying on the existence of densities for the statistical distributions that arise, or even on the continuity of the distribution functions, which would preclude applications to discrete random variables or effects tied to derivatives. We do not take for granted, or require, the differentiability of the deviation with respect to the parameters specifying the relative weights of the instruments in the portfolio. This is not merely for the sake of technical generality. An example provided in the final section of the paper illustrates how put and call options in portfolios can lead to nondifferentiability as well as to a threshold gap for the risk-free rate. Therefore no theory, unless it faces up to such troubles, can be regarded as fully applicable to portfolios involving derivative instruments. With standard calculus being inadequate for the problems at hand, we have had to rely instead on techniques of convex analysis (Rockafellar, 1970) while adhering strictly to the principles of optimization theory.2 The need for a “negative” efficient frontier referring to “net short positions,” along with the usual “positive” one for “net long positions,” is not surprising, in view of the diversity of measures that investors may be using. In line with their different opinions about risk, some investors may find the risk-free rate high enough to warrant borrowing from the market and investing that money risk-free, while others will prefer a fund in which the “longs” outweigh the “shorts.” An interesting analogy can be found in Sharpe (1991, p. 507) in terms of a stock index futures contract which might even consist entirely of short positions. The emergence of a variety of different master funds, optimal for different deviation measures, is an inescapable outcome of any theory which, like ours, attempts to cope with the current tendency toward customization in portfolio optimization. A master fund identified with respect to the wishes of one class of investors can no longer be proposed as obviously furnishing input for factor analysis of the market as a whole, because the financial markets react to the wishes of all investors. A master fund, in our general sense, can no longer be interpreted as associated with a sort of universal equilibrium. Whether some such master funds, individually or collectively, might nonetheless turn out to be valuable in factor analysis, is an issue outside the scope of this paper. CAPM-like covariance relations do indeed come out of the optimality conditions that characterize our master funds (as can be gleaned from the optimality prescriptions in Rockafellar et al., 2002a), but we reserve this development, requiring further elaboration of underpinnings in convex analysis, to a follow-up paper ( Rockafellar et al., forthcoming).

The replacement of standard deviation by other deviations, such as arise from conditional value-at-risk and other risk notions, in accordance with current trends, by no means causes the classical approach to optimization out-dated. Instead, it enriches that approach by making a degree of customization available. One-fund theorems still reign as a way of simplification, even though the designated funds, in their dependence on the deviation measure, can be different for different classes of investors. Nevertheless, mathematical complications created by instruments involving options, or models set up with only finitely many scenarios, require techniques beyond ordinary differential calculus to formulate and obtain results rigorously.