مدیریت پرتفولیو تحت عدم قطعیت معرفتی با استفاده از تسلط تصادفی و تئوری اطلاعات-شکاف

Portfolio management in finance is more than a mathematical problem of optimizing performance under risk constraints. A critical factor in practical portfolio problems is severe uncertainty – ignorance – due to model uncertainty. In this paper, we show how to find the best portfolios by adapting the standard risk-return criterion for portfolio selection to the case of severe uncertainty, such as might result from limited available data. This original approach is based on the combination of two commonly conflicting portfolio investment goals: (1) Obtaining high expected portfolio return, and (2) controlling risk. The two goals conflict if a portfolio has both higher expected return and higher risk than competing portfolio(s). They can also conflict if a reference curve characterizing a minimally tolerable portfolio is difficult to beat. To find the best portfolio in this situation, we first generate a set of optimal portfolios. This set is populated according to a standard mean-risk approach. Then we search the set using stochastic dominance (SSD) and Information-Gap Theory to identify the preferred one. This approach permits analysis of the problem even under severe uncertainty, a situation that we address because it occurs often, yet needs new advances to solve. SSD is attracting attention in the portfolio analysis community because any rational, risk-averse investor will prefer portfolio y1 to portfolio y2 if y1 has SSD over y2. The player’s utility function is not relevant to this preference as long as it is risk averse, which most investors are.

A portfolio consists of a set of segments, each of which is predefined as a particular asset category, such as stocks, bonds, commodities, etc. Solving the selection problem means determining the best proportion each segment should be of the total investment. The portfolio selection problem is the subject of a vast body of work. The process can be divided into two phases. The first is asset allocation, in which investor philosophy, including risk position, is used to choose the best percentage of the portfolio to place in each segment. The second, rebalancing, responds to changes in asset values by adjusting the percentages so that the portfolio continues to accurately reflect the investment philosophy. This work focuses on allocation. The well known CAPM leads to various allocation strategies, including for example BIRR and BARRA (search the Web for further information about these). The correct treatment of the risk-reward problem addressed by Markowitz ([12]) is fundamental to such modern methods, and its extension to problems characterized by severe uncertainty motivates this report. Little has been done to determine portfolio allocation when dependency relationships, such as correlations, among portfolio segment return distributions are unknown. We address this problem with a novel application of Information-Gap Theory (Ben-Haim [2]), using it together with the concept of second-order stochastic dominance (SSD) to help choose among portfolio allocations. SSD holds between two distributions r1 and r2 when the curves of their integrals do not cross. The slower rising curve is then said to have second-order stochastic dominance over the other curve. If r2 has SSD over r1, then we write r2  2 r1. Analogously, FSD (first-order stochastic dominance, 1) applies if the distribution curves themselves do not cross. However, most investors are risk averse, and if r2  2r1 then any risk averse player will prefer r2 (e.g. Perny et al. [13]). FSD is thus an unnecessarily strong (and therefore undesirable) constraint for the risk averse player. We build on a standard approach to finding optimal portfolios based on mean and risk and parameterized by amount of risk aversion, arising originally from Markowitz ([12]). The mean is the expectation (i.e. average) for a distribution describing the investment return, while the risk expresses the danger of loss or low returns and is typically a measure of the spread of the return distribution. Optimal portfolios are identified by finding the weights of the portfolio segments such that a mean-risk objective function is maximized (e.g. Elton et al. [8]). ‘‘Mean-risk’’ refers to a tradeoff between seeking a return distribution with a high mean, which is good, while minimizing the higher risk that tends to be associated with a high mean return, which is bad. Formally, the problem to be considered is to find such a portfolio given the constraints r ¼ Xs i¼1 wiri2eR ð1Þ Xs i¼1 wi ¼ 1 ð2Þ where r is a portfolio return distribution, i is one of the s segments in the portfolio, wi is the weight of segment i, and ri is the return distribution of segment i. eR is a given reference curve representing a minimally tolerable bound (the ‘‘risk limit’’) that the return distribution should not cross. As an additional constraint set (Eq. (2)), segment weights sum to 1 because each weight is a proportion of the whole. Each weight may be required to be within some interval in order to enforce a balance across segments, as might be specified by a company’s business model constraints and investment policies. In current practice optimization would typically be done without the 2 constraint, but perhaps with other constraints present such as Value-at-Risk (VaR), which is known to be mathematically inconsistent, or Conditional VaR (CVAR) which is less intuitive but without VaR’s consistency problem [3]. A given portfolio’s return distribution can be tested for compliance with an SSD constraint using numerical integration. Numerical integration can be done straightforwardly by summing areas of trapezoids under the curve. The size and number of trapezoids to sum is determined by the step size chosen for the integration process. Given a step size, SSD is considered to hold if the summed areas of all trapezoids to the left of any given value xi are lower for the return distribution r of a candidate portfolio than for reference curve eR. A set of representative return values x1, x2, x3, . . ., xn that are possible sample value of r should be checked. These points should cover low and high values of return, as well as a reasonable number of intermediate points (e.g. m = 10 or 20).An optimal portfolio might or might not comply with an additional requirement that it have stochastic dominance over a reference curve eR. A second-order stochastic dominance relation (‘‘2’’) between two distributions ensures that the dominant portfolio is preferred by any risk averse player (De Giorgi [7]). Risk aversion implies that, given two return distributions with the same expected return, the one with less spread (e.g. variance) is preferred. Define robustness as the amount by which a portfolio dominates by SSD a reference curve (robustness could be negative if it does not dominate). By testing various optimal portfolios for robustness, one with the highest robustness available can be identified. Alternatively, one with the highest expected return that also meets the SSD constraint could be found. In either case, the strategy is to search among a set of optimal portfolios provided, even if barely, by an under-constrained optimization problem for the one that is best according to a second criterion. This is discussed next.

This paper introduces an approach, and specific variations (Table 2), to determining the best possible investment plan given the two standard conflicting portfolio investment goals of mean return and risk. On the one hand we seek a high expected (mean) return. On the other we seek to control risk. To manage risk we seek to guarantee that the portfolio model has second-order stochastic dominance (SSD) over a minimum tolerable reference curve, because it has been shown that (1) if an SSD relationship exists between two return distributions, any risk-averse investor will prefer the dominant one, and (2) this constraint is weaker than the FSD relationship, which is unnecessarily strong. Strong constraints are undesirable because they reduce the space of allowable portfolios, tending to limit investment choices. We find the best portfolio by first generating a set of optimal portfolios. Then we search the set using stochastic dominance and Information-Gap Theory to identify the best one. The traditional approach to portfolio optimization using Markowitz theory is challenged when correlations or other dependencies among portfolio segments are hard to provide, return distribution shapes are uncertain, there is a lack of price data, or various other fundamental data are unavailable. The analyses shown in this paper address the first two of these challenges, thereby showing how rational portfolio choice is possible even under severe (epistemic) uncertainty.