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Portfolio selection under strict uncertainty: A multi-criteria methodology and its application to the Frankfurt and Vienna Stock Exchanges
European Journal of Operational Research 181 (2007) 1476–1487 www.elsevier.com/locate/ejor
Portfolio selection under strict uncertainty: A multi-criteria methodology and its application to the Frankfurt and Vienna Stock Exchanges E. Ballestero a, M. Gu¨nther b, D. Pla-Santamaria a, C. Stummer a
b,*
Escuela Polite´cnica Superior de Alcoy, Universidad Politecnica de Valencia, Edificio Ferrandiz, 03801 Alcoy, Spain b School of Business, Economics, and Statistics, University of Vienna, Bruenner Str. 72, A-1210 Vienna, Austria Received 1 December 2004; accepted 1 November 2005 Available online 12 May 2006
Abstract In modern portfolio theory, it is common practice to first compute the risk-reward efficient frontier and then to support an individual investor in selecting a portfolio that meets his/her preferences for profitability and risk. Potential flaws include (a) the assumption that past data provide sufficient evidence for predicting the future performances of the securities under consideration and (b) the necessity to mathematically determine or approximate the investor’s utility function. In this paper, we propose a methodology whose initial phase filters portfolios that are inefficient from a historical perspective. While this is consistent with traditional approaches, the second phase differs from the standard approach as it uses a decision table constructed by considering multiple scenarios assuming strict uncertainty. The table cells measure consequences by a multi-criteria linear performance index of simulated future returns, which avoids difficulties with performance ratios. The real world applicability is illustrated through two studies based on data from the stock exchanges in Frankfurt and Vienna. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Multi-criteria performance indices; Portfolio analysis; Simulation; Strict uncertainty
1. Introduction Identifying the ‘‘best’’ portfolio of assets for an individual investor is one of the principal challenges in the world of finance. Based on the mean–variance
*
Corresponding author. Tel.: +43 1 4277 38146; fax: +43 1 4277 38144. E-mail address: [email protected] (C. Stummer).
(E-V) model of portfolio selection by Markowitz (1952, 1991) and its utilization for a capital market model (CAPM) by Tobin (1958), numerous researchers have contributed to the development of modern portfolio theory (cf. Constantinides and Malliaris, 1995; Elton and Gruber, 1999). As a result, nowadays it has become common practice to extend the classical economic model of financial investment to multi-criteria decision making for the purpose of supporting large-scale investors in setting up their portfolios with respect to (i) their
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.11.050
E. Ballestero et al. / European Journal of Operational Research 181 (2007) 1476–1487
preferences for profitability and risk and (ii) the uncertain development of stock markets. Finance has traditionally recognized the necessity to compute the risk-reward efficient frontier. However, the subsequent task of determining the utility function for an individual investor in order to select the most attractive ‘‘point’’ on that frontier has remained a critical issue. Accordingly, several approaches to multiple criteria decision making aim at eliciting the investor’s preferences and/or at proposing appropriate portfolios (for an overview cf. Steuer and Na, 2003). To this end, Ballestero and Romero (1996) introduced a compromise programming model (cf. Zeleny, 1982; Yu, 1985) to bound the ‘‘average’’ investor’s utility optimum between two close points on the efficient frontier. The underlying theorem was subsequently modified to approximate an individual investor’s optimal utility (Ballestero, 1998). This technique performed remarkably well for the common class of bi-attribute utility functions characterized by expected return and risk (or more precisely, an index of profitability and an index of safety). In contrast to attempts that simply apply standard methods of multi-criteria decision making to portfolio selection, Ballestero’s utility bounding approach is based on the strict principles of economics and finance. Unlike other studies in which bounding plays an essential role, this paper does not apply the bounding stage. Let us explain and motivate our new approach to portfolio selection that has not been published as yet; however, note that its mathematical foundation has been introduced by Ballestero (2002) and is useful not only in finance but in many other fields as well. This paper emphasizes portfolio selection based on future performance simulated under strict uncertainty instead of emphasizing choice entirely based on past data. While there are investors who absolutely believe in the predictive ability of historical information and also investors who completely refuse that idea, our paper is intended for neither of them, but for investors inbetween. These investors realize that past data should not be ignored when necessary but used in a limited way. Determining the E-V efficient frontier from historical information is a necessary preliminary step to filter inefficient portfolios that otherwise would considerably enlarge and thus complicate the decision table under strict uncertainty. Even for investors reluctant to rely on past data, this may be a major reason to use them. In the proposed selection process, the first phase is aimed to obtain
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the efficient frontier while in the second phase uncertainty from the stock market future development is taken into account. To this end, the efficient stock portfolios can be combined with risk-free assets in the second phase. A decision table is then set up with several potential scenarios of the market. In this table, each cell represents the simulated future performance of the ith pre-selected efficient portfolio (possibly combined with risk-free assets) when the jth scenario becomes true. As a means of measuring performance, we use a multi-criteria linear index reflecting the investor’s preferences for profitability and safety. This measure has advantages over standard performance ratios as (i) our performance index is a multi-criteria utility function while a performance ratio is not and (ii) no problem with zero denominators arises from the multi-criteria index. Through the decision table, the preselected efficient portfolios and their blends with risk-free assets are ranked. In short, the proposed approach provides a well-founded and straightforward methodology to identify the particular investment portfolio that best fits the notions of a pragmatic investor faced with an uncertain market. Note that pragmatism means neither working without past data – essential for the classic paradigm (Sharpe, 1997) – nor neglecting future market scenarios. The remainder of this paper is organized as follows. Section 2 provides background information on modern portfolio theory, while Section 3 focuses on the proposed multi-criteria portfolio selection methodology. By presenting two studies based on real data from the Frankfurt and Viennese stock markets, Section 4 starts out with a brief market description and then gives a step-by-step instruction of how an investor is guided towards his/her favorite portfolio and how that portfolio is constructed. Finally, Section 5 provides concluding remarks. 2. Background Modern portfolio theory is based on (i) analyzing risk by focusing on the investor’s portfolio instead of individual securities, and (ii) determining and exploiting the E-V efficient frontier, namely, minimizing risk (commonly measured in terms of variance) for every level of expected return. The latter has its analytical foundation in Von Neumann and Morgenstern’s (1944) utility theory under uncertainty. In the E-V framework, this theory implies the following assumptions: (i) risk aversion (Arrow,
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1965), which is estimated by absolute risk aversion (ARA) coefficients, (ii) normally distributed returns, or alternatively, quadratic utility, and (iii) efficient markets of securities. While the discussions of the late 1960s and the early 1970s focused on the paradoxes and shortcomings of the E-V paradigm (Borch, 1969; Feldstein, 1969; Tsiang, 1972; Levy, 1974), the model has more recently regained academic backing and has given rise to a variety of software applications (cf. Gauss, 2004, for an example). Nevertheless, practitioners often use other tools, although their inefficiency has been shown: for a discussion of the popular capitalization-weighted stock portfolios see Haugen and Baker (1991) and Haugen (1997) and for multi-objective and goal programming approaches see Arenas et al. (2001) or Leung et al. (2001). A further stream of portfolio analysis is based on the observation that measures of downside risk are more appropriate than the standard deviation (e.g., Nawrocki, 1999; for the construction of the corresponding mean–semivariance cf. Ballestero, 2005). Additional directions of research in the field include heuristics (Mansini and Speranza, 1999), and the investigation of stochastic dominance (Copeland and Weston, 1988) and mean–variance–skewness (Elton and Gruber, 1987) – the latter being based on empirical research showing that daily returns are not normally distributed on some stock markets (e.g., Tang, 1998). And finally, considerable attention is being placed on supporting an investor with specific preferences for profitability and safety in selecting the particular portfolio that provides the decision maker’s maximum utility. While Kallberg and Ziemba (1983) used data from the New York Stock Exchange to investigate different utility functions with respect to the question of whether or not functions with ‘‘similar’’ ARA always lead to ‘‘similar’’ optimal portfolios, a major step forward was made by Ballestero (1998), who formulated a bounding theorem and proved that the investor’s utility optimum lies between two bounds that can be easily determined on the E-V efficient frontier given that standard properties of ordinal bi-argument utility hold. When the predictive ability of past data is assumed, this theorem has advantages: (i) it assures both efficiency and solutions close to the investor’s utility optimum, (ii) it is remarkably easier to apply than approaches based on direct utility maximization and stochastic dominance, (iii) it permits the use of a bi-criteria utility function instead of a function depending only on wealth, and (iv) it does not
require testing hypotheses like in the method put forward by Kallberg and Ziemba. 3. Methodology The proposed methodology comprises the two phases of (i) computing the E-V frontier, which provides a number of ‘‘pre-selected efficient portfolios’’ (PEP), and (ii) simulating the future performance of the PEPs and their blends with risk-free assets (PEPRFA) as a way to rank them under uncertainty. Blending with risk-free assets involves de-leveraging the investment, but one can think that leveraged portfolios might be also attractive to the investor. If so, the latter can be also considered as alternatives to be ranked in the second phase. Regarding consistency between Phase 1 and Phase 2, there is a critical issue that pre-selection in the first phase might have the effect of rejecting portfolios that could become top investments in the second phase ranking. However, we will assume that no inefficient portfolio removed in Phase 1 can become optimal if evaluated in Phase 2. This assumption is plausible and assures compatibility between both phases. Since Phase 1 is a well-known standard, we will only cope with methodological aspects of Phase 2 in this section. Phase 2 aims at analyzing the future performance of PEPs and PEP-RFAs by considering several scenarios, each of them characterized by predictions on (a) the expected return and variance for the further development of market indices (e.g., the DAX on the Frankfurt Exchange) over the planning horizon, and (b) a potential change in the interest rate for risk-free assets over the same horizon. Briefly speaking, each scenario describes a potential change in the market trend/interest rate. In a first step, Sharpe’s (1964) beta regression equation is obtained for each asset with respect to both past returns and the market indicator. That allows us to simulate monthly series of returns on each PEP and PEP-RFA; afterwards, expected returns (E) and variances (V) are computed for each of the considered scenarios. The next step is to convert E and V into normalized indices h1 and h2 of profitability and safety, respectively. This is achieved through h1 ¼
E Emin Emax Emin
h2 ¼
V max V ; V max V min
and
ð1Þ ð2Þ
E. Ballestero et al. / European Journal of Operational Research 181 (2007) 1476–1487
where h1 and h2 range in the interval (0, 1), E and V are expected return and variance, Emax and Vmax are the highest mean value and variance in the set of values under consideration, and Emin and Vmin are the lowest mean value and variance in the same set. Now, from Eqs. (1) and (2), the following simulation-based multi-criteria performance measure for the ith alternative (either PEP or PEP-RFA) and the jth scenario is obtained: pij ¼ r0 h1ij þ h2ij ;
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the individual’s preferences. Indeed, the way to elicit r0 cannot be extended to groups (such as a board of directors) if a member of the group does not agree with the others about preferences for profitability and risk. Next, the following decision table or matrix is constructed:
ð3Þ
where r0 is a parameter defining the investor’s preferences (Ballestero, 1998); its elicitation is described below. Such a performance index pij has advantages over the Sharpe (1997) ratio, because it avoids problems with zero denominators and with negative values of the expected return. Note that risk-free assets involve zero (or close to zero) denominators while negative mean values appear in downturn scenarios. Unlike the Sharpe ratio, that has not the precise meaning of a standard utility function, the performance index of Eq. (3) is based on a multi-criteria linear utility function with profitability and safety as the two relevant criteria. In order to elicit parameter r0, we suggest asking the investor questions like: ‘‘Suppose you have a balanced profitability–safety portfolio. If you want to increase profitability, which percentage of safety are you willing to lose at most for increasing profitability by 10%? In contrast, if you want to increase safety, which percentage of profitability are you willing to lose at most for increasing safety by 10%?’’ Answers may be: ‘‘I would like to increase profitability but I am willing to lose as much as 18% of safety’’, or ‘‘I would like to increase safety but I am willing to lose as much as 18% of profitability.’’ The first one leads to r0 = 1.8 while the second answer results in r0 = 1/1.8 0.56. Remark 1. Note that in practice the above dialogue is intended for an individual investor who is sufficiently experienced to understand the questions and their implications. It is worth noting that the decision maker in such a dialogue is not a board of managers but an individual. Pension and mutual funds offer a wide range of portfolios, and then each individual investor decides which fund is the most appealing to him. Clerks of mutual funds, banks, etc. often advise a client on his ‘‘best’’ choice from the client’s particular preferences for profitability and risk. The above dialogue, or any other with the same purpose, can serve to elicit parameter r0 from
ð4Þ Matrix (4) holds several mutually exclusive scenarios, each represented by a column vj (j = 1, 2, . . . , n). As we assume strict uncertainty, their probabilities are unknown. The decision maker has to select a single portfolio from the so-called opportunity set of M alternatives. Values pij in the grid are simulation-based (multi-criteria) performance indices as described in Eq. (3), each specifying the consequence of taking the ith portfolio if the jth scenario becomes true. By means of a linear mathematical programming model (for a detailed description cf. Ballestero, 1999), it is easy to determine whether or not a portfolio is dominated by convex combinations of the other portfolios. Correspondingly, the M portfolios can be grouped in those that are non-dominated (i = 1, 2, . . . , m) and those that are dominated (i = m + 1, m + 2, . . . , M). For the purpose of ranking the non-dominated alternatives under strict uncertainty, we use a decision criterion for moderately pessimistic decision makers, who do not believe that the most favorable event to any alternative is likely to occur. This definition is a deviation from Wald’s (1950) extreme pessimism, where the worst scenario is assumed to occur. Theorem 1 (Ranking rule for moderately pessimistic decision makers; Ballestero, 2002). Let pj and pj
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be the highest and the smallest values in the jth column in the set of m non-dominated portfolios. Given that pj differs from pj in the non-dominated set (i.e., Dj ¼ pj pj > 0 for all columns in the decision matrix), a single weight system satisfying properties of moderate pessimism does exist to obtain the ranking of the non-dominated alternatives. The wj weight to be attached to the jth scenario is proportional to (1/Dj), which implies the ranking score Si ¼
n X
pij =ðKDj Þ;
ð5Þ
j¼1
where the proportionality factor K is K ¼1þ
n X
pj =Dj :
ð6Þ
j¼1
If pj coincides with pj for at least some jth scenario, Eqs. (5) and (6) – after solving the indeterminate form 0/0 caused by the zero Dj value – lead to the following ranking rule for non-dominated alternatives: first, remove those scenarios in which the difference Dj is zero, and second, apply Eq. (5). To make clear where Theorem 1 is heading to, let us couch the above ranking rule in intuitive comprehensible terms and compare it with other decision rules under uncertainty. (i) Laplace’s (1825) principle of insufficient reason is a consistent criterion since one and only one weight is attached to each scenario. Every weight is set to 1 as there is no reason for different weights. However, Laplace’s rule disregards pessimism in spite of the fact that most investors behave as moderately pessimistic decision makers. (ii) Wald’s (1950) maximin return was a breakthrough criterion in decision analysis under uncertainty in proposing the innovative notion of security/pessimism. However, Wald’s rule involves inconsistency in the attachment of weights, as a single weight system is not defined. Moreover, Wald’s pessimism is not moderate but extreme, which leads to quite biased solutions. (iii) Savage’s (1951) minimax regret also requires using inconsistent weights like in the case of Wald’s criterion. (iv) Hurwicz’s (1951) criterion has the same inconsistency disadvantage as in (ii) and (iii). Hurwicz’s middle course between pessimism and optimism is seemingly appealing as each decision maker can define his/ her own pessimism level by a coefficient a. However, Hurwicz’s decision rule (like in Wald’s maximin rule and other pessimism criteria) neglects most information contained in the decision table, which is a seri-
ous shortcoming of this criterion. Indeed, Hurwicz’s criterion is based on the weighted average H(ai) = a(row maximum) + (1 a)(row minimum) and therefore only two data in each row (namely, the row maximum and the row minimum) are used, while all the other data are not considered. Moreover, the logical principle of attaching one and only one weight to each column does not hold. Let us illustrate this by looking at two rows h and h 0 . For row h, the Hurwicz’s column weights are a, (1 a) and zero, corresponding to the maximum, the minimum and the remaining values of row h, respectively. The same holds for row h 0 but now the maximum and minimum row values generally correspond to columns different from those of the h case. Since the maximum and the minimum of row h do not generally coincide with the maximum and the minimum of row h 0 , the logical principle of consistency fails. In addition to these major flaws, a minor drawback is that applicability appears impossible in the common case that the decision is made by a board of managers, because the a coefficient can widely differ between the members of the board. (v) Moderate pessimism criterion. In order to overcome the above drawbacks, the decision rule from Theorem 1 combines both consistency and moderate pessimism. To illustrate this, suppose first that portfolio ah is dominated by a convex combination Ch of other portfolios in matrix (4). This means that Ch outperforms portfolio ah in all scenarios indicating that ah definitely should not be among the attractive portfolio alternatives. Suppose now that another portfolio ak is non-dominated. Then, we should define its ranking score Sk consistently by attaching one and only one weight to each scenario. From the moderate pessimism perspective, the appropriate weight is wj = 1/(KDj) for the jth scenario. An intuitive way to understand the problem can be given by the following example: Suppose Laplacian weighting (every column is ascribed a weight of 1) and also that value pj was very large as compared to value pj for some jth column. Then, this optimistic large amount of the highest pj value in the column would play a significant role to increase (in comparative terms) the ranking score of the corresponding alternative. Such a possibility is unacceptable to the moderately pessimistic investor, who does not believe that a highly favorable event to any alternative is likely to occur. Consequently, Theorem 1 modifies Laplacian weighting by attaching to the jth column a weight inversely proportional to the range Dj ¼ ðpj pj Þ so that
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columns where the highest value pj broadly exceeds the lowest value pj become less relevant to determine the ranking score. Moreover, notice that the special case pj ¼ pj implies that the jth column is irrelevant for ranking purposes, since all its values pij are equal. Conversely, if pj tends to infinity while pj remains bounded, the difference between the respective scores becomes Sðpj Þ Sðpj Þ ¼ ðpj pj Þ=KDj ¼ 1=K;
ð7Þ
assets until the blended portfolio reaches the desired risk level. Alternatively, if the investor was a strong profitability seeker, the ‘‘best’’ portfolio would be leveraged instead. This does not exactly conform with Tobin’s separation theorem but could be seen as a procedure ‘‘a` la Tobin’’, which has advantages (simplicity and more clarity to the user) and also disadvantages (future changes affecting risk-free assets are overlooked in the decision table).
where (1/K) remains bounded. From the moderate pessimism perspective, this result is much more appropriate than that from Laplace’s rule, in which the difference between both scores tends to infinity as pj tends to infinity and pj remains bounded. A similar advantage over Hurwicz’s rule is obtained. Note that standard decision making under strict uncertainty requires to completely ignore any probability distribution (even if some exist); i.e., in the strict uncertainty framework it is not eligible to, for example, assume a uniform distribution for the scenarios.
4. Case study
Remark 2. As noted above, a moderately pessimistic decision maker is defined as an individual who believes that the most favorable event will not occur (Ballestero, 2002). Accordingly, a decision maker can simultaneously behave as a moderate pessimist and as a profitability seeker with r0 = 2.25. Imagine, e.g., a lottery with one big prize and other less important prizes. While most lottery people may be unbridled optimists expecting to get the big prize, the moderate pessimist does not expect it; however, he/she may risk a lot of money in this lottery in an attempt to win less important prizes. Analogously, a moderately pessimistic investor may risk plenty of money in a stock portfolio although he/she does not think that capital gains are going to reach highly remarkable steady values.
4.1. Stock markets
Remark 3 (Separating the PEP-RFA blending from the PEP choice). Regardless of the fact that Tobin-Sharpe CAPM assumptions seem to be unrealistic and are not corroborated by empirical results (Fama and French, 1996), the investor can be tempted to separate the selection process into two parts: (a) select first from the decision table under strict uncertainty the ‘‘best’’ stock portfolio PEP for the unbiased/standard investor with r0 = 1, this task being accomplished by the simulation-based multi-criteria performance index; and (b) de-leverage this ‘‘best’’ PEP by blending it with risk-free
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We consider 26 stocks from the Frankfurt Exchange and 17 stocks from the Vienna Exchange. This selection represents all stocks that were constantly listed in the prime segments of these markets (i.e., the DAX and the ATX) during the four financial years 1999–2002. In the following, we will briefly describe the markets and then present the selection process and results of a numerical example in greater detail.
The Frankfurt Stock Exchange is one of the world’s largest trading centers for securities and the largest of Germany’s eight stock exchanges. Among the more than 320 corporations listed in total, the 30 largest German securities in terms of market capitalization and order book turnover together comprise the blue-chip index DAX (Deutscher Aktien Index). Considerably smaller than the Frankfurt Stock Exchange, the Vienna Exchange is the only stock market in Austria. The approximately 160 listings in the Austrian equity market are split into standard and prime markets. The blue chips traded in the latter form the foundation of the ATX Prime (Austrian Traded Index). To describe the relationship between ATX and DAX, a regression analysis–linear model leads to ATX = 0.0005677 + 0.299613 * DAX. The P-value in the analysis of the variance table is less than 0.01 and consequently there is a statistically significant relationship between ATX and DAX at the 99% confidence level. The R-squared statistic indicates that the model as fitted explains 23.4761% of the variability in ATX. The correlation coefficient equals 0.484522, indicating a relatively weak relationship between the variables. As a result of this analysis, we abandon the idea of selecting portfolios from a joint DAX–ATX opportunity set using the same
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decision table for the joint selection. Therefore, the model will be henceforward developed for the Frankfurt market, although we will also provide the results obtained for the Viennese market separately. 4.2. Frankfurt market: Selection process and results In this section, we provide a step-by-step description of how the proposed methodology can be applied to the Frankfurt market. Phase 1 comprises three steps that aim to determine the E-V efficient frontier.
Step 4.
Step 1. Derive the monthly returns on each stock based on data for prices and dividends over 1999–2002. Step 2. Calculate the mean values as well as the covariance matrix. Step 3. Compute the E-V efficient frontier of stock portfolios for the year 2003 according to the standard Markowitz’s parametric quadratic programming (using Lingo 8.0). Thus, 14 pre-selected efficient portfolios (PEP) are obtained in our numerical example (for mean values and variances see Table 1). No bullet shape arc appears in this table since the dominated arc typically found in determining E-V frontiers has been removed. Note that both expected return and variance monotonically increase from portfolio 1 to 14. While the lowest expectation is negative, the riskiest portfolio 14 reaches a monthly mean value of 3%.
Table 1 Stock portfolios on the E-V efficient frontier as specified by expected return and risk PEP code
Mean value
Variance
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0025 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 0.0225 0.0250 0.0275 0.0300
0.00290 0.00297 0.00320 0.00358 0.00419 0.00500 0.00607 0.00748 0.00967 0.01276 0.01762 0.02908 0.04794 0.07421
Step 5.
Step 6.
Step 7.
Step 8.
In Phase 2, we simulate the future performance of the 14 PEPs obtained in Phase 1. Therefore, instead of de-leveraging these 14 portfolios and bringing their PEP-RFA blends into the decision table under strict uncertainty, the ‘‘a` la Tobin’’ procedure (see Remark 3 in Section 3) will be used in what follows. Define a number of uncertain scenarios with respect to the year 2003 in terms of mean values and variances. Concerning mean values, we decide for the following monthly expected returns: (1%), 0%, 1%, 2% and 3% as five alternative forecasts for 2003. These levels are not arbitrarily chosen, as we have taken into account the historical yearly expected returns from DAX over 1991–2002. In equivalent monthly average terms, these expected returns for each year are: 0.28% (1991), (0.15%) (1992), 3.40% (1993), (0.50%) (1994), 0.65% (1995), 2.14% (1996), 3.56% (1997), 1.66% (1998), 3.02% (1999), (0.51%) (2000), (1.50%) (2001), (4.14%) (2002). As these monthly returns have a mean value of 0.66% with a standard deviation of 2.25%, we have focused on the interval between (0.66–2.25) = 1.59% and (0.66 + 2.25) = 2.91%, which advises us to choose the five alternative forecasts specified above. Concerning variances, we have proceeded analogously from the same period 1991–2002 which leads to choosing the three levels 0.00089, 0.00376 and 0.00663 as three alternative forecasts for 2003. By combining the five return forecasts with the three alternatives for the variances, 15 different scenarios are generated. By using Sharpe’s beta regression equation with historical data, specify returns on each stock as a linear function on the market index. For every scenario, simulate the monthly market returns over 2003 by assuming normal distribution of mean value and standard deviation defined in Step 4. For each scenario, compute the mean value and standard deviation for each PEP by using the results obtained in Step 5 and Step 6. Following Eqs. (1) and (2), normalize the mean values and standard deviations com-
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generated in Step 9 for r0 = 2.25 (top block), r0 = 1 (middle block) and r0 = 0.44 (bottom block). Step 11. In Table 4, top block, the 5 non-dominated PEPs are ordered by the score appearing in the last column. These scores have been calculated by Eq. (5) using the performance index of Eq. (3) with r0 = 2.25 as the appropriate parameter to profitability seekers. For this type of investor, the topranked PEP is portfolio 13, whose historical variance is the second highest in Table 1, while its simulation-based index of safety (averaged over the 15 scenarios) attains the second lowest level of 0.765 (see Table 3). By looking at Table 4, middle block, for unbiased/standard investors, the topranked PEP is portfolio 12 with the third highest historical variance and the third lowest simulated index of safety (on average). Finally, results in the bottom block indicate portfolio 11 as the ‘‘best’’ for safety seekers; notice that its historical variance in Table 1, although significantly lower than those of portfolios 13 and 12, is still high. With respect to the simulated
puted in Step 7. This leads to the index of profitability (h1) and the index of safety (h2) for each ith PEP and jth scenario, which are shown in Tables 2 and 3, respectively. Step 9. With data from Tables 2 and 3, use Eq. (3) to compute each simulation-based multicriteria performance index for the ith PEP and the jth scenario. In our case study, we use three different values for parameter r0, namely 2.25, 1.00, and 0.44. While r0 = 2.25 is the profitability/safety tradeoff tailored to strong profitability seekers, the trade-off for strong safety seekers is r0 = 1/2.25 = 0.44. An unbiased/standard investor, finally, is described through r0 = 1 (Ballestero and Pla-Santamaria, 2003). Step 10. Build Table 4, where rows refer to the nondominated PEPs and columns refer to the 15 scenarios. This table is divided into the top, middle and bottom blocks that correspond to the profitability seeker, the standard investor and the safety seeker, respectively. Each cell holds the numerical simulation-based performance measures
Table 2 Simulation-based indices of profitability h1 Scenarios
Pre-selected efficient portfolios (PEP) 1
2
3
4
5
6
7
8
9
10
11
12
13
14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.000 0.160 0.057 0.037 0.108 0.031 0.036 0.071 0.157 0.180 0.052 0.079 0.093 0.098 0.093
0.015 0.168 0.070 0.051 0.118 0.044 0.049 0.083 0.165 0.187 0.065 0.090 0.103 0.108 0.104
0.029 0.178 0.082 0.064 0.129 0.058 0.062 0.095 0.175 0.197 0.078 0.102 0.115 0.120 0.116
0.044 0.187 0.095 0.077 0.141 0.071 0.076 0.107 0.184 0.205 0.091 0.114 0.127 0.131 0.127
0.057 0.202 0.109 0.091 0.155 0.085 0.090 0.122 0.200 0.221 0.105 0.129 0.141 0.146 0.142
0.070 0.225 0.126 0.106 0.175 0.100 0.105 0.139 0.222 0.245 0.121 0.147 0.160 0.165 0.160
0.083 0.250 0.143 0.122 0.195 0.114 0.120 0.157 0.246 0.271 0.137 0.165 0.179 0.184 0.180
0.095 0.274 0.159 0.137 0.216 0.129 0.135 0.174 0.271 0.297 0.154 0.183 0.199 0.204 0.199
0.106 0.309 0.179 0.154 0.243 0.145 0.151 0.196 0.305 0.335 0.173 0.206 0.224 0.230 0.224
0.117 0.345 0.199 0.170 0.271 0.160 0.167 0.218 0.340 0.374 0.191 0.229 0.249 0.256 0.249
0.119 0.440 0.234 0.194 0.336 0.180 0.190 0.261 0.433 0.481 0.224 0.277 0.305 0.315 0.306
0.113 0.592 0.285 0.226 0.437 0.205 0.220 0.326 0.583 0.654 0.271 0.350 0.391 0.406 0.392
0.107 0.745 0.337 0.257 0.538 0.230 0.249 0.390 0.733 0.827 0.317 0.422 0.477 0.497 0.479
0.102 0.898 0.388 0.288 0.640 0.254 0.279 0.455 0.883 1.000 0.363 0.495 0.563 0.588 0.565
Average
0.083
0.095
0.107
0.119
0.133
0.151
0.170
0.188
0.212
0.236
0.286
0.363
0.440
0.517
Row descriptions: (1) mean value, (1%) and variance, 0.00663; (2) mean value, 0% and variance, 0.00663; (3) mean value, 1% and variance, 0.00663; (4) mean value, 2% and variance, 0.00663; (5) mean value, 3% and variance, 0.00663; (6) mean value, (1%) and variance, 0.00376; (7) mean value, 0% and variance, 0.00376; (8) mean value, 1% and variance, 0.00376; (9) mean value, 2% and variance, 0.00376; (10) mean value, 3% and variance, 0.00376; (11) mean value, (1%) and variance, 0.00089; (12) mean value, 0% and variance, 0.00089; (13) mean value, 1% and variance, 0.00089; (14) mean value, 2% and variance, 0.00089; (15) mean value, 3% and variance, 0.00089.
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Table 3 Simulation-based indices of safety h2 Scenarios
Pre-selected efficient portfolios (PEP) 1
2
3
4
5
6
7
8
9
10
11
12
13
14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.970 0.984 0.978 0.975 0.962 0.990 0.981 0.994 0.986 0.992 1.000 0.998 1.000 0.999 0.998
0.972 0.986 0.980 0.978 0.965 0.991 0.982 0.995 0.988 0.993 1.000 0.998 1.000 0.999 0.998
0.974 0.987 0.981 0.979 0.967 0.992 0.984 0.995 0.989 0.994 1.000 0.999 1.000 0.999 0.999
0.976 0.988 0.983 0.981 0.970 0.992 0.985 0.996 0.989 0.994 1.000 0.999 1.000 1.000 0.999
0.975 0.987 0.982 0.980 0.969 0.992 0.984 0.995 0.989 0.994 1.000 0.999 1.000 1.000 0.999
0.972 0.985 0.979 0.977 0.964 0.991 0.982 0.995 0.987 0.993 1.000 0.998 1.000 0.999 0.998
0.967 0.983 0.976 0.973 0.958 0.989 0.979 0.993 0.985 0.991 0.999 0.998 0.999 0.999 0.998
0.961 0.980 0.972 0.969 0.951 0.987 0.975 0.992 0.982 0.990 0.999 0.997 0.999 0.998 0.997
0.950 0.973 0.963 0.959 0.936 0.983 0.967 0.989 0.977 0.986 0.998 0.996 0.998 0.997 0.996
0.936 0.966 0.953 0.948 0.920 0.978 0.958 0.986 0.970 0.982 0.997 0.994 0.997 0.996 0.994
0.872 0.931 0.905 0.895 0.840 0.954 0.916 0.970 0.939 0.963 0.992 0.986 0.992 0.990 0.987
0.712 0.842 0.785 0.763 0.639 0.895 0.810 0.931 0.862 0.914 0.980 0.967 0.980 0.976 0.968
0.488 0.719 0.618 0.579 0.359 0.812 0.662 0.876 0.755 0.846 0.963 0.940 0.963 0.956 0.941
0.200 0.561 0.403 0.343 0.000 0.706 0.472 0.806 0.616 0.759 0.941 0.906 0.942 0.931 0.907
Average
0.987
0.988
0.989
0.990
0.990
0.988
0.986
0.983
0.978
0.972
0.942
0.868
0.765
0.633
Table 4 Simulation-based performance indices for three types of investors (with non-dominated PEPs) PEP
Scenarios 1
2
Score 5
6
7
8
9
10
11
12
13
14
15
Profitability seeker with r0 = 2.25 13 0.729 2.396 1.375 1.157 12 0.966 2.175 1.427 1.271 14 0.429 2.581 1.275 0.991 11 1.140 1.920 1.432 1.332 10 1.199 1.741 1.400 1.331
1.570 1.622 1.439 1.595 1.529
1.329 1.356 1.278 1.360 1.338
1.223 1.305 1.099 1.344 1.335
1.754 1.664 1.829 1.558 1.476
2.404 2.175 2.603 1.915 1.736
2.707 2.385 3.009 2.044 1.823
1.676 1.589 1.758 1.497 1.428
1.891 1.754 2.020 1.610 1.510
2.036 1.859 2.208 1.678 1.556
2.074 1.889 2.253 1.698 1.572
2.019 1.850 2.180 1.674 1.555
73.923 73.449 72.771 71.349 68.750
Unbiased/standard investor with r0 = 1 12 0.825 1.435 1.070 0.989 13 0.595 1.464 0.954 0.836 11 0.991 1.370 1.139 1.089 14 0.302 1.459 0.791 0.631 10 1.053 1.310 1.151 1.118 9 1.055 1.282 1.142 1.112 8 1.056 1.254 1.131 1.105
1.076 0.898 1.175 0.640 1.190 1.180 1.167
1.100 1.042 1.134 0.960 1.138 1.127 1.116
1.030 0.911 1.106 0.750 1.126 1.118 1.110
1.257 1.266 1.231 1.260 1.204 1.185 1.166
1.446 1.488 1.373 1.499 1.311 1.282 1.253
1.568 1.673 1.443 1.759 1.356 1.322 1.287
1.251 1.280 1.216 1.304 1.188 1.171 1.153
1.317 1.363 1.264 1.401 1.223 1.202 1.180
1.371 1.440 1.297 1.504 1.246 1.222 1.198
1.382 1.453 1.305 1.519 1.252 1.227 1.203
1.360 1.420 1.292 1.473 1.244 1.220 1.196
73.524 73.317 72.405 71.785 70.803 69.707 68.580
Safety 11 12 10 13 9 8 14
0.987 0.831 1.039 0.596 1.043 1.046 0.281
1.033 0.985 1.048 0.913 1.046 1.044 0.818
1.000 0.907 1.032 0.772 1.034 1.034 0.594
1.085 1.074 1.082 1.048 1.075 1.069 1.006
1.130 1.119 1.120 1.077 1.111 1.102 1.005
1.174 1.202 1.147 1.210 1.134 1.120 1.199
1.091 1.099 1.081 1.102 1.074 1.067 1.101
1.108 1.121 1.095 1.126 1.086 1.078 1.124
1.126 1.152 1.106 1.173 1.096 1.086 1.189
1.129 1.155 1.109 1.175 1.099 1.088 1.190
1.121 1.140 1.104 1.152 1.094 1.085 1.156
146.583 146.546 145.370 144.568 144.341 143.266 140.649
seeker 0.924 0.761 0.987 0.535 0.996 1.003 0.245
3
with r0 = 0.44 1.124 1.008 1.103 0.911 1.117 1.040 1.047 0.766 1.109 1.042 1.100 1.042 0.956 0.574
4
0.981 0.862 1.023 0.692 1.026 1.029 0.469
Frankfurt Exchange.
index of safety, portfolio 11 reaches an average value of 0.942, which is much closer to the value of portfolios 4 and 5 repre-
senting maximum safety (0.990) than to that of portfolio 14 representing minimum safety (0.633).
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Indeed, the two bottom portfolios are non-dominated alternatives, and three non-dominated portfolios are ranked after dominated alternatives. Focus now on the safety seeker. Here, the results are even less convincing as the three bottom portfolios are non-dominated alternatives, and four non-dominated portfolios are ranked after dominated alternatives. With a = 0.3 (also characterizing an individual more inclined to pessimism than to optimism), the shortcomings are similar. Moreover, for profitability seekers and safety seekers, the top alternative with a = 0.3 differs from the top one with a = 0.4. Concerning the proposed model, we have seen above that the dominated portfolios are excluded. However, what if such dominated portfolios were considered to be ranked by the moderate pessimism criterion together with the non-dominated portfolios? Then, all non-dominated portfolios are top ranked alternatives with only one exception affecting the safety seekers case, where the six top ranked portfolios are non-dominated followed by two dominated, one non-dominated and five dominated alternatives at the bottom. Therefore, the results from our model are much more consistent than those obtained by Hurwicz’s decision rule.
Step 12. Given the high variability just observed in Step 11, the selection process should be completed ‘‘a` la Tobin’’ (Remark 3) by blending risk-free assets with portfolio 12 being the ‘‘best’’ to unbiased/standard investors. The blend proportions should be adjusted to the investor’s preferences, which is a final step that can be easily accomplished.
4.3. Performance of Frankfurt results Over 2003 (this year being chosen from an anonymous referee’s precise suggestion), the performance analysis leads to the following results in terms of Sharpe’s mean to standard deviation ratio; (a) Profitability seekers’ top ranked portfolio (number 13), 0.1715; (b) unbiased/standard investors’ top ranked portfolio (number 12), 0.1928; (c) safety seekers’ top ranked portfolio (number 11), 0.1625; (d) DAX, 0.1352. Thus, the portfolios selected by our model significantly outperform the DAX index. 4.4. What if Hurwicz’s decision rule was applied to Frankfurt?
4.5. What if using Wald’s maximin rule?
Focus first on the unbiased/standard investor. From Hurwicz’s criterion with a = 0.4 (characterizing a behavior slightly more pessimistic than optimistic) the results are not quite convincing.
Results from this criterion are even less convincing that those obtained from Hurwicz’s rule. Focus
Table 5 Simulation-based performance indices for three types of investors (with non-dominated PEPs) PEP
Scenarios 1
Score 2
3
4
5
6
7
8
9
10
11
12
1.231 0.955 1.318
1.125 0.896 1.205
1.244 0.984 1.324
2.510 2.802 2.283
1.728 1.694 1.686
2.279 2.517 2.091
1.713 1.751 1.647
1.975 2.091 1.857
2.377 2.669 2.163
104.790 103.225 102.042
with r0 = 1 1.005 0.908 1.109 1.026 0.723 0.613
0.959 1.036 0.768
0.928 1.037 0.652
1.604 1.533 1.669
1.203 1.234 1.077
1.535 1.468 1.604
1.285 1.272 1.266
1.384 1.355 1.387
1.598 1.512 1.708
192.975 191.052 188.220
0.960 0.885 0.953 0.948 0.942 0.710
0.908 0.786 0.914 0.914 0.913 0.503
1.197 1.198 1.165 1.149 1.133 1.161
1.032 0.967 1.021 1.014 1.006 0.801
1.189 1.202 1.155 1.138 1.121 1.196
1.103 1.093 1.077 1.064 1.050 1.049
1.130 1.120 1.103 1.089 1.075 1.071
1.221 1.249 1.182 1.163 1.144 1.277
102.253 100.293 100.079 98.917 97.718 93.351
Profitability seeker with r0 = 2.25 6 0.731 2.500 1.505 7 0.185 2.708 1.304 5 0.957 2.303 1.540 Unbiased/standard investor 6 0.584 1.508 5 0.802 1.485 7 0.082 1.458
Safety seeker with r0 = 0.44 5 0.733 1.119 6 0.518 1.063 3 0.765 1.103 2 0.777 1.094 1 0.788 1.083 7 0.036 0.898
0.916 0.781 0.924 0.926 0.926 0.462
0.894 0.763 0.903 0.904 0.904 0.460
Vienna Exchange. Rankings in this table are as consistent as those from Frankfurt (see Table 4).
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After developing the step-by-step selection process in the same way as above for Frankfurt, the results are displayed in Table 5.
only few assumptions and comes with a low computational burden. Its applicability has been demonstrated by means of two studies on the Frankfurt and Vienna Stock Exchanges. From Subsections 4.4 and 4.5, there is empirical evidence that using Hurwicz’s or Wald’s criteria (i.e., the most representative decision rules involving pessimism under strict uncertainty) leads to unconvincing results in the portfolio selection problem. We are aware of the fact that the approach is currently limited to the risk–profitability criteria and particularly does not take into consideration additional criteria such as liquidity, the number of securities in the fund or perhaps a social responsibility quotient. Thus, the issue of how these objectives may be included in the proposed methodology will be the subject of further research.
5. Conclusion
Acknowledgement
In this paper, we have proposed a decision support approach for selecting portfolios with respect to an investor’s individual preferences for risk and profitability. It is designed to take into account the uncertainty inherent in stock markets and, to this end, it makes use of moderate pessimism decision tables and simulation-based multi-criteria performance indices, that are new tools in portfolio management and, particularly, are advantageous when compared with the popular Sharpe ratio. As a result, our methodology provides an attractive alternative to traditional approaches. Based on modern portfolio theory (MPT), the selection process requires two stages, the first being Markowitz’s efficient frontier, that assumes the predictive ability of past data. This assumption is classic in MPT and defended, among many authors, by Sharpe (1997). By and large in MPT, a second stage is needed to choose one or a few portfolios among the set of efficient funds. Therefore, in any MPT selection process, some prediction errors arise from both stages. For example, when fuzzy techniques are used in the second stage, then prediction errors come from the efficient frontier and the fuzzy application. Then, fuzzy errors generally surpass those expected from simulation with Sharpe’s betas (as we propose), because the fuzzy data (as applied to portfolio selection) are mostly beliefs about the future without objective support (see Arenas et al., 2001). In short, our procedure follows principles that are widely accepted not only in decision analysis but also in economics and finance. Moreover, it requires
Thanks are due to Paula Jorda for her assistance in the computational experiments.
first on the unbiased/standard investor. Indeed, the three bottom portfolios are non-dominated alternatives, and four non-dominated portfolios are ranked after dominated alternatives. Focus now on the safety seeker. Here, the four bottom portfolios are non-dominated alternatives, and six non-dominated portfolios are ranked after dominated alternatives. Finally, for profitability seekers, two bottom portfolios are non-dominated alternatives, and three nondominated portfolios are ranked after dominated alternatives. 4.6. Vienna market: Results
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Portfolio selection under strict uncertainty: A multi-criteria methodology and its application to the Frankfurt and Vienna Stock Exchanges E. Ballestero a, M. Gu¨nther b, D. Pla-Santamaria a, C. Stummer a
b,*
Escuela Polite´cnica Superior de Alcoy, Universidad Politecnica de Valencia, Edificio Ferrandiz, 03801 Alcoy, Spain b School of Business, Economics, and Statistics, University of Vienna, Bruenner Str. 72, A-1210 Vienna, Austria Received 1 December 2004; accepted 1 November 2005 Available online 12 May 2006
Abstract In modern portfolio theory, it is common practice to first compute the risk-reward efficient frontier and then to support an individual investor in selecting a portfolio that meets his/her preferences for profitability and risk. Potential flaws include (a) the assumption that past data provide sufficient evidence for predicting the future performances of the securities under consideration and (b) the necessity to mathematically determine or approximate the investor’s utility function. In this paper, we propose a methodology whose initial phase filters portfolios that are inefficient from a historical perspective. While this is consistent with traditional approaches, the second phase differs from the standard approach as it uses a decision table constructed by considering multiple scenarios assuming strict uncertainty. The table cells measure consequences by a multi-criteria linear performance index of simulated future returns, which avoids difficulties with performance ratios. The real world applicability is illustrated through two studies based on data from the stock exchanges in Frankfurt and Vienna. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Multi-criteria performance indices; Portfolio analysis; Simulation; Strict uncertainty
1. Introduction Identifying the ‘‘best’’ portfolio of assets for an individual investor is one of the principal challenges in the world of finance. Based on the mean–variance
*
Corresponding author. Tel.: +43 1 4277 38146; fax: +43 1 4277 38144. E-mail address: [email protected] (C. Stummer).
(E-V) model of portfolio selection by Markowitz (1952, 1991) and its utilization for a capital market model (CAPM) by Tobin (1958), numerous researchers have contributed to the development of modern portfolio theory (cf. Constantinides and Malliaris, 1995; Elton and Gruber, 1999). As a result, nowadays it has become common practice to extend the classical economic model of financial investment to multi-criteria decision making for the purpose of supporting large-scale investors in setting up their portfolios with respect to (i) their
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.11.050
E. Ballestero et al. / European Journal of Operational Research 181 (2007) 1476–1487
preferences for profitability and risk and (ii) the uncertain development of stock markets. Finance has traditionally recognized the necessity to compute the risk-reward efficient frontier. However, the subsequent task of determining the utility function for an individual investor in order to select the most attractive ‘‘point’’ on that frontier has remained a critical issue. Accordingly, several approaches to multiple criteria decision making aim at eliciting the investor’s preferences and/or at proposing appropriate portfolios (for an overview cf. Steuer and Na, 2003). To this end, Ballestero and Romero (1996) introduced a compromise programming model (cf. Zeleny, 1982; Yu, 1985) to bound the ‘‘average’’ investor’s utility optimum between two close points on the efficient frontier. The underlying theorem was subsequently modified to approximate an individual investor’s optimal utility (Ballestero, 1998). This technique performed remarkably well for the common class of bi-attribute utility functions characterized by expected return and risk (or more precisely, an index of profitability and an index of safety). In contrast to attempts that simply apply standard methods of multi-criteria decision making to portfolio selection, Ballestero’s utility bounding approach is based on the strict principles of economics and finance. Unlike other studies in which bounding plays an essential role, this paper does not apply the bounding stage. Let us explain and motivate our new approach to portfolio selection that has not been published as yet; however, note that its mathematical foundation has been introduced by Ballestero (2002) and is useful not only in finance but in many other fields as well. This paper emphasizes portfolio selection based on future performance simulated under strict uncertainty instead of emphasizing choice entirely based on past data. While there are investors who absolutely believe in the predictive ability of historical information and also investors who completely refuse that idea, our paper is intended for neither of them, but for investors inbetween. These investors realize that past data should not be ignored when necessary but used in a limited way. Determining the E-V efficient frontier from historical information is a necessary preliminary step to filter inefficient portfolios that otherwise would considerably enlarge and thus complicate the decision table under strict uncertainty. Even for investors reluctant to rely on past data, this may be a major reason to use them. In the proposed selection process, the first phase is aimed to obtain
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the efficient frontier while in the second phase uncertainty from the stock market future development is taken into account. To this end, the efficient stock portfolios can be combined with risk-free assets in the second phase. A decision table is then set up with several potential scenarios of the market. In this table, each cell represents the simulated future performance of the ith pre-selected efficient portfolio (possibly combined with risk-free assets) when the jth scenario becomes true. As a means of measuring performance, we use a multi-criteria linear index reflecting the investor’s preferences for profitability and safety. This measure has advantages over standard performance ratios as (i) our performance index is a multi-criteria utility function while a performance ratio is not and (ii) no problem with zero denominators arises from the multi-criteria index. Through the decision table, the preselected efficient portfolios and their blends with risk-free assets are ranked. In short, the proposed approach provides a well-founded and straightforward methodology to identify the particular investment portfolio that best fits the notions of a pragmatic investor faced with an uncertain market. Note that pragmatism means neither working without past data – essential for the classic paradigm (Sharpe, 1997) – nor neglecting future market scenarios. The remainder of this paper is organized as follows. Section 2 provides background information on modern portfolio theory, while Section 3 focuses on the proposed multi-criteria portfolio selection methodology. By presenting two studies based on real data from the Frankfurt and Viennese stock markets, Section 4 starts out with a brief market description and then gives a step-by-step instruction of how an investor is guided towards his/her favorite portfolio and how that portfolio is constructed. Finally, Section 5 provides concluding remarks. 2. Background Modern portfolio theory is based on (i) analyzing risk by focusing on the investor’s portfolio instead of individual securities, and (ii) determining and exploiting the E-V efficient frontier, namely, minimizing risk (commonly measured in terms of variance) for every level of expected return. The latter has its analytical foundation in Von Neumann and Morgenstern’s (1944) utility theory under uncertainty. In the E-V framework, this theory implies the following assumptions: (i) risk aversion (Arrow,
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1965), which is estimated by absolute risk aversion (ARA) coefficients, (ii) normally distributed returns, or alternatively, quadratic utility, and (iii) efficient markets of securities. While the discussions of the late 1960s and the early 1970s focused on the paradoxes and shortcomings of the E-V paradigm (Borch, 1969; Feldstein, 1969; Tsiang, 1972; Levy, 1974), the model has more recently regained academic backing and has given rise to a variety of software applications (cf. Gauss, 2004, for an example). Nevertheless, practitioners often use other tools, although their inefficiency has been shown: for a discussion of the popular capitalization-weighted stock portfolios see Haugen and Baker (1991) and Haugen (1997) and for multi-objective and goal programming approaches see Arenas et al. (2001) or Leung et al. (2001). A further stream of portfolio analysis is based on the observation that measures of downside risk are more appropriate than the standard deviation (e.g., Nawrocki, 1999; for the construction of the corresponding mean–semivariance cf. Ballestero, 2005). Additional directions of research in the field include heuristics (Mansini and Speranza, 1999), and the investigation of stochastic dominance (Copeland and Weston, 1988) and mean–variance–skewness (Elton and Gruber, 1987) – the latter being based on empirical research showing that daily returns are not normally distributed on some stock markets (e.g., Tang, 1998). And finally, considerable attention is being placed on supporting an investor with specific preferences for profitability and safety in selecting the particular portfolio that provides the decision maker’s maximum utility. While Kallberg and Ziemba (1983) used data from the New York Stock Exchange to investigate different utility functions with respect to the question of whether or not functions with ‘‘similar’’ ARA always lead to ‘‘similar’’ optimal portfolios, a major step forward was made by Ballestero (1998), who formulated a bounding theorem and proved that the investor’s utility optimum lies between two bounds that can be easily determined on the E-V efficient frontier given that standard properties of ordinal bi-argument utility hold. When the predictive ability of past data is assumed, this theorem has advantages: (i) it assures both efficiency and solutions close to the investor’s utility optimum, (ii) it is remarkably easier to apply than approaches based on direct utility maximization and stochastic dominance, (iii) it permits the use of a bi-criteria utility function instead of a function depending only on wealth, and (iv) it does not
require testing hypotheses like in the method put forward by Kallberg and Ziemba. 3. Methodology The proposed methodology comprises the two phases of (i) computing the E-V frontier, which provides a number of ‘‘pre-selected efficient portfolios’’ (PEP), and (ii) simulating the future performance of the PEPs and their blends with risk-free assets (PEPRFA) as a way to rank them under uncertainty. Blending with risk-free assets involves de-leveraging the investment, but one can think that leveraged portfolios might be also attractive to the investor. If so, the latter can be also considered as alternatives to be ranked in the second phase. Regarding consistency between Phase 1 and Phase 2, there is a critical issue that pre-selection in the first phase might have the effect of rejecting portfolios that could become top investments in the second phase ranking. However, we will assume that no inefficient portfolio removed in Phase 1 can become optimal if evaluated in Phase 2. This assumption is plausible and assures compatibility between both phases. Since Phase 1 is a well-known standard, we will only cope with methodological aspects of Phase 2 in this section. Phase 2 aims at analyzing the future performance of PEPs and PEP-RFAs by considering several scenarios, each of them characterized by predictions on (a) the expected return and variance for the further development of market indices (e.g., the DAX on the Frankfurt Exchange) over the planning horizon, and (b) a potential change in the interest rate for risk-free assets over the same horizon. Briefly speaking, each scenario describes a potential change in the market trend/interest rate. In a first step, Sharpe’s (1964) beta regression equation is obtained for each asset with respect to both past returns and the market indicator. That allows us to simulate monthly series of returns on each PEP and PEP-RFA; afterwards, expected returns (E) and variances (V) are computed for each of the considered scenarios. The next step is to convert E and V into normalized indices h1 and h2 of profitability and safety, respectively. This is achieved through h1 ¼
E Emin Emax Emin
h2 ¼
V max V ; V max V min
and
ð1Þ ð2Þ
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where h1 and h2 range in the interval (0, 1), E and V are expected return and variance, Emax and Vmax are the highest mean value and variance in the set of values under consideration, and Emin and Vmin are the lowest mean value and variance in the same set. Now, from Eqs. (1) and (2), the following simulation-based multi-criteria performance measure for the ith alternative (either PEP or PEP-RFA) and the jth scenario is obtained: pij ¼ r0 h1ij þ h2ij ;
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the individual’s preferences. Indeed, the way to elicit r0 cannot be extended to groups (such as a board of directors) if a member of the group does not agree with the others about preferences for profitability and risk. Next, the following decision table or matrix is constructed:
ð3Þ
where r0 is a parameter defining the investor’s preferences (Ballestero, 1998); its elicitation is described below. Such a performance index pij has advantages over the Sharpe (1997) ratio, because it avoids problems with zero denominators and with negative values of the expected return. Note that risk-free assets involve zero (or close to zero) denominators while negative mean values appear in downturn scenarios. Unlike the Sharpe ratio, that has not the precise meaning of a standard utility function, the performance index of Eq. (3) is based on a multi-criteria linear utility function with profitability and safety as the two relevant criteria. In order to elicit parameter r0, we suggest asking the investor questions like: ‘‘Suppose you have a balanced profitability–safety portfolio. If you want to increase profitability, which percentage of safety are you willing to lose at most for increasing profitability by 10%? In contrast, if you want to increase safety, which percentage of profitability are you willing to lose at most for increasing safety by 10%?’’ Answers may be: ‘‘I would like to increase profitability but I am willing to lose as much as 18% of safety’’, or ‘‘I would like to increase safety but I am willing to lose as much as 18% of profitability.’’ The first one leads to r0 = 1.8 while the second answer results in r0 = 1/1.8 0.56. Remark 1. Note that in practice the above dialogue is intended for an individual investor who is sufficiently experienced to understand the questions and their implications. It is worth noting that the decision maker in such a dialogue is not a board of managers but an individual. Pension and mutual funds offer a wide range of portfolios, and then each individual investor decides which fund is the most appealing to him. Clerks of mutual funds, banks, etc. often advise a client on his ‘‘best’’ choice from the client’s particular preferences for profitability and risk. The above dialogue, or any other with the same purpose, can serve to elicit parameter r0 from
ð4Þ Matrix (4) holds several mutually exclusive scenarios, each represented by a column vj (j = 1, 2, . . . , n). As we assume strict uncertainty, their probabilities are unknown. The decision maker has to select a single portfolio from the so-called opportunity set of M alternatives. Values pij in the grid are simulation-based (multi-criteria) performance indices as described in Eq. (3), each specifying the consequence of taking the ith portfolio if the jth scenario becomes true. By means of a linear mathematical programming model (for a detailed description cf. Ballestero, 1999), it is easy to determine whether or not a portfolio is dominated by convex combinations of the other portfolios. Correspondingly, the M portfolios can be grouped in those that are non-dominated (i = 1, 2, . . . , m) and those that are dominated (i = m + 1, m + 2, . . . , M). For the purpose of ranking the non-dominated alternatives under strict uncertainty, we use a decision criterion for moderately pessimistic decision makers, who do not believe that the most favorable event to any alternative is likely to occur. This definition is a deviation from Wald’s (1950) extreme pessimism, where the worst scenario is assumed to occur. Theorem 1 (Ranking rule for moderately pessimistic decision makers; Ballestero, 2002). Let pj and pj
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be the highest and the smallest values in the jth column in the set of m non-dominated portfolios. Given that pj differs from pj in the non-dominated set (i.e., Dj ¼ pj pj > 0 for all columns in the decision matrix), a single weight system satisfying properties of moderate pessimism does exist to obtain the ranking of the non-dominated alternatives. The wj weight to be attached to the jth scenario is proportional to (1/Dj), which implies the ranking score Si ¼
n X
pij =ðKDj Þ;
ð5Þ
j¼1
where the proportionality factor K is K ¼1þ
n X
pj =Dj :
ð6Þ
j¼1
If pj coincides with pj for at least some jth scenario, Eqs. (5) and (6) – after solving the indeterminate form 0/0 caused by the zero Dj value – lead to the following ranking rule for non-dominated alternatives: first, remove those scenarios in which the difference Dj is zero, and second, apply Eq. (5). To make clear where Theorem 1 is heading to, let us couch the above ranking rule in intuitive comprehensible terms and compare it with other decision rules under uncertainty. (i) Laplace’s (1825) principle of insufficient reason is a consistent criterion since one and only one weight is attached to each scenario. Every weight is set to 1 as there is no reason for different weights. However, Laplace’s rule disregards pessimism in spite of the fact that most investors behave as moderately pessimistic decision makers. (ii) Wald’s (1950) maximin return was a breakthrough criterion in decision analysis under uncertainty in proposing the innovative notion of security/pessimism. However, Wald’s rule involves inconsistency in the attachment of weights, as a single weight system is not defined. Moreover, Wald’s pessimism is not moderate but extreme, which leads to quite biased solutions. (iii) Savage’s (1951) minimax regret also requires using inconsistent weights like in the case of Wald’s criterion. (iv) Hurwicz’s (1951) criterion has the same inconsistency disadvantage as in (ii) and (iii). Hurwicz’s middle course between pessimism and optimism is seemingly appealing as each decision maker can define his/ her own pessimism level by a coefficient a. However, Hurwicz’s decision rule (like in Wald’s maximin rule and other pessimism criteria) neglects most information contained in the decision table, which is a seri-
ous shortcoming of this criterion. Indeed, Hurwicz’s criterion is based on the weighted average H(ai) = a(row maximum) + (1 a)(row minimum) and therefore only two data in each row (namely, the row maximum and the row minimum) are used, while all the other data are not considered. Moreover, the logical principle of attaching one and only one weight to each column does not hold. Let us illustrate this by looking at two rows h and h 0 . For row h, the Hurwicz’s column weights are a, (1 a) and zero, corresponding to the maximum, the minimum and the remaining values of row h, respectively. The same holds for row h 0 but now the maximum and minimum row values generally correspond to columns different from those of the h case. Since the maximum and the minimum of row h do not generally coincide with the maximum and the minimum of row h 0 , the logical principle of consistency fails. In addition to these major flaws, a minor drawback is that applicability appears impossible in the common case that the decision is made by a board of managers, because the a coefficient can widely differ between the members of the board. (v) Moderate pessimism criterion. In order to overcome the above drawbacks, the decision rule from Theorem 1 combines both consistency and moderate pessimism. To illustrate this, suppose first that portfolio ah is dominated by a convex combination Ch of other portfolios in matrix (4). This means that Ch outperforms portfolio ah in all scenarios indicating that ah definitely should not be among the attractive portfolio alternatives. Suppose now that another portfolio ak is non-dominated. Then, we should define its ranking score Sk consistently by attaching one and only one weight to each scenario. From the moderate pessimism perspective, the appropriate weight is wj = 1/(KDj) for the jth scenario. An intuitive way to understand the problem can be given by the following example: Suppose Laplacian weighting (every column is ascribed a weight of 1) and also that value pj was very large as compared to value pj for some jth column. Then, this optimistic large amount of the highest pj value in the column would play a significant role to increase (in comparative terms) the ranking score of the corresponding alternative. Such a possibility is unacceptable to the moderately pessimistic investor, who does not believe that a highly favorable event to any alternative is likely to occur. Consequently, Theorem 1 modifies Laplacian weighting by attaching to the jth column a weight inversely proportional to the range Dj ¼ ðpj pj Þ so that
E. Ballestero et al. / European Journal of Operational Research 181 (2007) 1476–1487
columns where the highest value pj broadly exceeds the lowest value pj become less relevant to determine the ranking score. Moreover, notice that the special case pj ¼ pj implies that the jth column is irrelevant for ranking purposes, since all its values pij are equal. Conversely, if pj tends to infinity while pj remains bounded, the difference between the respective scores becomes Sðpj Þ Sðpj Þ ¼ ðpj pj Þ=KDj ¼ 1=K;
ð7Þ
assets until the blended portfolio reaches the desired risk level. Alternatively, if the investor was a strong profitability seeker, the ‘‘best’’ portfolio would be leveraged instead. This does not exactly conform with Tobin’s separation theorem but could be seen as a procedure ‘‘a` la Tobin’’, which has advantages (simplicity and more clarity to the user) and also disadvantages (future changes affecting risk-free assets are overlooked in the decision table).
where (1/K) remains bounded. From the moderate pessimism perspective, this result is much more appropriate than that from Laplace’s rule, in which the difference between both scores tends to infinity as pj tends to infinity and pj remains bounded. A similar advantage over Hurwicz’s rule is obtained. Note that standard decision making under strict uncertainty requires to completely ignore any probability distribution (even if some exist); i.e., in the strict uncertainty framework it is not eligible to, for example, assume a uniform distribution for the scenarios.
4. Case study
Remark 2. As noted above, a moderately pessimistic decision maker is defined as an individual who believes that the most favorable event will not occur (Ballestero, 2002). Accordingly, a decision maker can simultaneously behave as a moderate pessimist and as a profitability seeker with r0 = 2.25. Imagine, e.g., a lottery with one big prize and other less important prizes. While most lottery people may be unbridled optimists expecting to get the big prize, the moderate pessimist does not expect it; however, he/she may risk a lot of money in this lottery in an attempt to win less important prizes. Analogously, a moderately pessimistic investor may risk plenty of money in a stock portfolio although he/she does not think that capital gains are going to reach highly remarkable steady values.
4.1. Stock markets
Remark 3 (Separating the PEP-RFA blending from the PEP choice). Regardless of the fact that Tobin-Sharpe CAPM assumptions seem to be unrealistic and are not corroborated by empirical results (Fama and French, 1996), the investor can be tempted to separate the selection process into two parts: (a) select first from the decision table under strict uncertainty the ‘‘best’’ stock portfolio PEP for the unbiased/standard investor with r0 = 1, this task being accomplished by the simulation-based multi-criteria performance index; and (b) de-leverage this ‘‘best’’ PEP by blending it with risk-free
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We consider 26 stocks from the Frankfurt Exchange and 17 stocks from the Vienna Exchange. This selection represents all stocks that were constantly listed in the prime segments of these markets (i.e., the DAX and the ATX) during the four financial years 1999–2002. In the following, we will briefly describe the markets and then present the selection process and results of a numerical example in greater detail.
The Frankfurt Stock Exchange is one of the world’s largest trading centers for securities and the largest of Germany’s eight stock exchanges. Among the more than 320 corporations listed in total, the 30 largest German securities in terms of market capitalization and order book turnover together comprise the blue-chip index DAX (Deutscher Aktien Index). Considerably smaller than the Frankfurt Stock Exchange, the Vienna Exchange is the only stock market in Austria. The approximately 160 listings in the Austrian equity market are split into standard and prime markets. The blue chips traded in the latter form the foundation of the ATX Prime (Austrian Traded Index). To describe the relationship between ATX and DAX, a regression analysis–linear model leads to ATX = 0.0005677 + 0.299613 * DAX. The P-value in the analysis of the variance table is less than 0.01 and consequently there is a statistically significant relationship between ATX and DAX at the 99% confidence level. The R-squared statistic indicates that the model as fitted explains 23.4761% of the variability in ATX. The correlation coefficient equals 0.484522, indicating a relatively weak relationship between the variables. As a result of this analysis, we abandon the idea of selecting portfolios from a joint DAX–ATX opportunity set using the same
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decision table for the joint selection. Therefore, the model will be henceforward developed for the Frankfurt market, although we will also provide the results obtained for the Viennese market separately. 4.2. Frankfurt market: Selection process and results In this section, we provide a step-by-step description of how the proposed methodology can be applied to the Frankfurt market. Phase 1 comprises three steps that aim to determine the E-V efficient frontier.
Step 4.
Step 1. Derive the monthly returns on each stock based on data for prices and dividends over 1999–2002. Step 2. Calculate the mean values as well as the covariance matrix. Step 3. Compute the E-V efficient frontier of stock portfolios for the year 2003 according to the standard Markowitz’s parametric quadratic programming (using Lingo 8.0). Thus, 14 pre-selected efficient portfolios (PEP) are obtained in our numerical example (for mean values and variances see Table 1). No bullet shape arc appears in this table since the dominated arc typically found in determining E-V frontiers has been removed. Note that both expected return and variance monotonically increase from portfolio 1 to 14. While the lowest expectation is negative, the riskiest portfolio 14 reaches a monthly mean value of 3%.
Table 1 Stock portfolios on the E-V efficient frontier as specified by expected return and risk PEP code
Mean value
Variance
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0025 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 0.0225 0.0250 0.0275 0.0300
0.00290 0.00297 0.00320 0.00358 0.00419 0.00500 0.00607 0.00748 0.00967 0.01276 0.01762 0.02908 0.04794 0.07421
Step 5.
Step 6.
Step 7.
Step 8.
In Phase 2, we simulate the future performance of the 14 PEPs obtained in Phase 1. Therefore, instead of de-leveraging these 14 portfolios and bringing their PEP-RFA blends into the decision table under strict uncertainty, the ‘‘a` la Tobin’’ procedure (see Remark 3 in Section 3) will be used in what follows. Define a number of uncertain scenarios with respect to the year 2003 in terms of mean values and variances. Concerning mean values, we decide for the following monthly expected returns: (1%), 0%, 1%, 2% and 3% as five alternative forecasts for 2003. These levels are not arbitrarily chosen, as we have taken into account the historical yearly expected returns from DAX over 1991–2002. In equivalent monthly average terms, these expected returns for each year are: 0.28% (1991), (0.15%) (1992), 3.40% (1993), (0.50%) (1994), 0.65% (1995), 2.14% (1996), 3.56% (1997), 1.66% (1998), 3.02% (1999), (0.51%) (2000), (1.50%) (2001), (4.14%) (2002). As these monthly returns have a mean value of 0.66% with a standard deviation of 2.25%, we have focused on the interval between (0.66–2.25) = 1.59% and (0.66 + 2.25) = 2.91%, which advises us to choose the five alternative forecasts specified above. Concerning variances, we have proceeded analogously from the same period 1991–2002 which leads to choosing the three levels 0.00089, 0.00376 and 0.00663 as three alternative forecasts for 2003. By combining the five return forecasts with the three alternatives for the variances, 15 different scenarios are generated. By using Sharpe’s beta regression equation with historical data, specify returns on each stock as a linear function on the market index. For every scenario, simulate the monthly market returns over 2003 by assuming normal distribution of mean value and standard deviation defined in Step 4. For each scenario, compute the mean value and standard deviation for each PEP by using the results obtained in Step 5 and Step 6. Following Eqs. (1) and (2), normalize the mean values and standard deviations com-
E. Ballestero et al. / European Journal of Operational Research 181 (2007) 1476–1487
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generated in Step 9 for r0 = 2.25 (top block), r0 = 1 (middle block) and r0 = 0.44 (bottom block). Step 11. In Table 4, top block, the 5 non-dominated PEPs are ordered by the score appearing in the last column. These scores have been calculated by Eq. (5) using the performance index of Eq. (3) with r0 = 2.25 as the appropriate parameter to profitability seekers. For this type of investor, the topranked PEP is portfolio 13, whose historical variance is the second highest in Table 1, while its simulation-based index of safety (averaged over the 15 scenarios) attains the second lowest level of 0.765 (see Table 3). By looking at Table 4, middle block, for unbiased/standard investors, the topranked PEP is portfolio 12 with the third highest historical variance and the third lowest simulated index of safety (on average). Finally, results in the bottom block indicate portfolio 11 as the ‘‘best’’ for safety seekers; notice that its historical variance in Table 1, although significantly lower than those of portfolios 13 and 12, is still high. With respect to the simulated
puted in Step 7. This leads to the index of profitability (h1) and the index of safety (h2) for each ith PEP and jth scenario, which are shown in Tables 2 and 3, respectively. Step 9. With data from Tables 2 and 3, use Eq. (3) to compute each simulation-based multicriteria performance index for the ith PEP and the jth scenario. In our case study, we use three different values for parameter r0, namely 2.25, 1.00, and 0.44. While r0 = 2.25 is the profitability/safety tradeoff tailored to strong profitability seekers, the trade-off for strong safety seekers is r0 = 1/2.25 = 0.44. An unbiased/standard investor, finally, is described through r0 = 1 (Ballestero and Pla-Santamaria, 2003). Step 10. Build Table 4, where rows refer to the nondominated PEPs and columns refer to the 15 scenarios. This table is divided into the top, middle and bottom blocks that correspond to the profitability seeker, the standard investor and the safety seeker, respectively. Each cell holds the numerical simulation-based performance measures
Table 2 Simulation-based indices of profitability h1 Scenarios
Pre-selected efficient portfolios (PEP) 1
2
3
4
5
6
7
8
9
10
11
12
13
14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.000 0.160 0.057 0.037 0.108 0.031 0.036 0.071 0.157 0.180 0.052 0.079 0.093 0.098 0.093
0.015 0.168 0.070 0.051 0.118 0.044 0.049 0.083 0.165 0.187 0.065 0.090 0.103 0.108 0.104
0.029 0.178 0.082 0.064 0.129 0.058 0.062 0.095 0.175 0.197 0.078 0.102 0.115 0.120 0.116
0.044 0.187 0.095 0.077 0.141 0.071 0.076 0.107 0.184 0.205 0.091 0.114 0.127 0.131 0.127
0.057 0.202 0.109 0.091 0.155 0.085 0.090 0.122 0.200 0.221 0.105 0.129 0.141 0.146 0.142
0.070 0.225 0.126 0.106 0.175 0.100 0.105 0.139 0.222 0.245 0.121 0.147 0.160 0.165 0.160
0.083 0.250 0.143 0.122 0.195 0.114 0.120 0.157 0.246 0.271 0.137 0.165 0.179 0.184 0.180
0.095 0.274 0.159 0.137 0.216 0.129 0.135 0.174 0.271 0.297 0.154 0.183 0.199 0.204 0.199
0.106 0.309 0.179 0.154 0.243 0.145 0.151 0.196 0.305 0.335 0.173 0.206 0.224 0.230 0.224
0.117 0.345 0.199 0.170 0.271 0.160 0.167 0.218 0.340 0.374 0.191 0.229 0.249 0.256 0.249
0.119 0.440 0.234 0.194 0.336 0.180 0.190 0.261 0.433 0.481 0.224 0.277 0.305 0.315 0.306
0.113 0.592 0.285 0.226 0.437 0.205 0.220 0.326 0.583 0.654 0.271 0.350 0.391 0.406 0.392
0.107 0.745 0.337 0.257 0.538 0.230 0.249 0.390 0.733 0.827 0.317 0.422 0.477 0.497 0.479
0.102 0.898 0.388 0.288 0.640 0.254 0.279 0.455 0.883 1.000 0.363 0.495 0.563 0.588 0.565
Average
0.083
0.095
0.107
0.119
0.133
0.151
0.170
0.188
0.212
0.236
0.286
0.363
0.440
0.517
Row descriptions: (1) mean value, (1%) and variance, 0.00663; (2) mean value, 0% and variance, 0.00663; (3) mean value, 1% and variance, 0.00663; (4) mean value, 2% and variance, 0.00663; (5) mean value, 3% and variance, 0.00663; (6) mean value, (1%) and variance, 0.00376; (7) mean value, 0% and variance, 0.00376; (8) mean value, 1% and variance, 0.00376; (9) mean value, 2% and variance, 0.00376; (10) mean value, 3% and variance, 0.00376; (11) mean value, (1%) and variance, 0.00089; (12) mean value, 0% and variance, 0.00089; (13) mean value, 1% and variance, 0.00089; (14) mean value, 2% and variance, 0.00089; (15) mean value, 3% and variance, 0.00089.
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Table 3 Simulation-based indices of safety h2 Scenarios
Pre-selected efficient portfolios (PEP) 1
2
3
4
5
6
7
8
9
10
11
12
13
14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.970 0.984 0.978 0.975 0.962 0.990 0.981 0.994 0.986 0.992 1.000 0.998 1.000 0.999 0.998
0.972 0.986 0.980 0.978 0.965 0.991 0.982 0.995 0.988 0.993 1.000 0.998 1.000 0.999 0.998
0.974 0.987 0.981 0.979 0.967 0.992 0.984 0.995 0.989 0.994 1.000 0.999 1.000 0.999 0.999
0.976 0.988 0.983 0.981 0.970 0.992 0.985 0.996 0.989 0.994 1.000 0.999 1.000 1.000 0.999
0.975 0.987 0.982 0.980 0.969 0.992 0.984 0.995 0.989 0.994 1.000 0.999 1.000 1.000 0.999
0.972 0.985 0.979 0.977 0.964 0.991 0.982 0.995 0.987 0.993 1.000 0.998 1.000 0.999 0.998
0.967 0.983 0.976 0.973 0.958 0.989 0.979 0.993 0.985 0.991 0.999 0.998 0.999 0.999 0.998
0.961 0.980 0.972 0.969 0.951 0.987 0.975 0.992 0.982 0.990 0.999 0.997 0.999 0.998 0.997
0.950 0.973 0.963 0.959 0.936 0.983 0.967 0.989 0.977 0.986 0.998 0.996 0.998 0.997 0.996
0.936 0.966 0.953 0.948 0.920 0.978 0.958 0.986 0.970 0.982 0.997 0.994 0.997 0.996 0.994
0.872 0.931 0.905 0.895 0.840 0.954 0.916 0.970 0.939 0.963 0.992 0.986 0.992 0.990 0.987
0.712 0.842 0.785 0.763 0.639 0.895 0.810 0.931 0.862 0.914 0.980 0.967 0.980 0.976 0.968
0.488 0.719 0.618 0.579 0.359 0.812 0.662 0.876 0.755 0.846 0.963 0.940 0.963 0.956 0.941
0.200 0.561 0.403 0.343 0.000 0.706 0.472 0.806 0.616 0.759 0.941 0.906 0.942 0.931 0.907
Average
0.987
0.988
0.989
0.990
0.990
0.988
0.986
0.983
0.978
0.972
0.942
0.868
0.765
0.633
Table 4 Simulation-based performance indices for three types of investors (with non-dominated PEPs) PEP
Scenarios 1
2
Score 5
6
7
8
9
10
11
12
13
14
15
Profitability seeker with r0 = 2.25 13 0.729 2.396 1.375 1.157 12 0.966 2.175 1.427 1.271 14 0.429 2.581 1.275 0.991 11 1.140 1.920 1.432 1.332 10 1.199 1.741 1.400 1.331
1.570 1.622 1.439 1.595 1.529
1.329 1.356 1.278 1.360 1.338
1.223 1.305 1.099 1.344 1.335
1.754 1.664 1.829 1.558 1.476
2.404 2.175 2.603 1.915 1.736
2.707 2.385 3.009 2.044 1.823
1.676 1.589 1.758 1.497 1.428
1.891 1.754 2.020 1.610 1.510
2.036 1.859 2.208 1.678 1.556
2.074 1.889 2.253 1.698 1.572
2.019 1.850 2.180 1.674 1.555
73.923 73.449 72.771 71.349 68.750
Unbiased/standard investor with r0 = 1 12 0.825 1.435 1.070 0.989 13 0.595 1.464 0.954 0.836 11 0.991 1.370 1.139 1.089 14 0.302 1.459 0.791 0.631 10 1.053 1.310 1.151 1.118 9 1.055 1.282 1.142 1.112 8 1.056 1.254 1.131 1.105
1.076 0.898 1.175 0.640 1.190 1.180 1.167
1.100 1.042 1.134 0.960 1.138 1.127 1.116
1.030 0.911 1.106 0.750 1.126 1.118 1.110
1.257 1.266 1.231 1.260 1.204 1.185 1.166
1.446 1.488 1.373 1.499 1.311 1.282 1.253
1.568 1.673 1.443 1.759 1.356 1.322 1.287
1.251 1.280 1.216 1.304 1.188 1.171 1.153
1.317 1.363 1.264 1.401 1.223 1.202 1.180
1.371 1.440 1.297 1.504 1.246 1.222 1.198
1.382 1.453 1.305 1.519 1.252 1.227 1.203
1.360 1.420 1.292 1.473 1.244 1.220 1.196
73.524 73.317 72.405 71.785 70.803 69.707 68.580
Safety 11 12 10 13 9 8 14
0.987 0.831 1.039 0.596 1.043 1.046 0.281
1.033 0.985 1.048 0.913 1.046 1.044 0.818
1.000 0.907 1.032 0.772 1.034 1.034 0.594
1.085 1.074 1.082 1.048 1.075 1.069 1.006
1.130 1.119 1.120 1.077 1.111 1.102 1.005
1.174 1.202 1.147 1.210 1.134 1.120 1.199
1.091 1.099 1.081 1.102 1.074 1.067 1.101
1.108 1.121 1.095 1.126 1.086 1.078 1.124
1.126 1.152 1.106 1.173 1.096 1.086 1.189
1.129 1.155 1.109 1.175 1.099 1.088 1.190
1.121 1.140 1.104 1.152 1.094 1.085 1.156
146.583 146.546 145.370 144.568 144.341 143.266 140.649
seeker 0.924 0.761 0.987 0.535 0.996 1.003 0.245
3
with r0 = 0.44 1.124 1.008 1.103 0.911 1.117 1.040 1.047 0.766 1.109 1.042 1.100 1.042 0.956 0.574
4
0.981 0.862 1.023 0.692 1.026 1.029 0.469
Frankfurt Exchange.
index of safety, portfolio 11 reaches an average value of 0.942, which is much closer to the value of portfolios 4 and 5 repre-
senting maximum safety (0.990) than to that of portfolio 14 representing minimum safety (0.633).
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Indeed, the two bottom portfolios are non-dominated alternatives, and three non-dominated portfolios are ranked after dominated alternatives. Focus now on the safety seeker. Here, the results are even less convincing as the three bottom portfolios are non-dominated alternatives, and four non-dominated portfolios are ranked after dominated alternatives. With a = 0.3 (also characterizing an individual more inclined to pessimism than to optimism), the shortcomings are similar. Moreover, for profitability seekers and safety seekers, the top alternative with a = 0.3 differs from the top one with a = 0.4. Concerning the proposed model, we have seen above that the dominated portfolios are excluded. However, what if such dominated portfolios were considered to be ranked by the moderate pessimism criterion together with the non-dominated portfolios? Then, all non-dominated portfolios are top ranked alternatives with only one exception affecting the safety seekers case, where the six top ranked portfolios are non-dominated followed by two dominated, one non-dominated and five dominated alternatives at the bottom. Therefore, the results from our model are much more consistent than those obtained by Hurwicz’s decision rule.
Step 12. Given the high variability just observed in Step 11, the selection process should be completed ‘‘a` la Tobin’’ (Remark 3) by blending risk-free assets with portfolio 12 being the ‘‘best’’ to unbiased/standard investors. The blend proportions should be adjusted to the investor’s preferences, which is a final step that can be easily accomplished.
4.3. Performance of Frankfurt results Over 2003 (this year being chosen from an anonymous referee’s precise suggestion), the performance analysis leads to the following results in terms of Sharpe’s mean to standard deviation ratio; (a) Profitability seekers’ top ranked portfolio (number 13), 0.1715; (b) unbiased/standard investors’ top ranked portfolio (number 12), 0.1928; (c) safety seekers’ top ranked portfolio (number 11), 0.1625; (d) DAX, 0.1352. Thus, the portfolios selected by our model significantly outperform the DAX index. 4.4. What if Hurwicz’s decision rule was applied to Frankfurt?
4.5. What if using Wald’s maximin rule?
Focus first on the unbiased/standard investor. From Hurwicz’s criterion with a = 0.4 (characterizing a behavior slightly more pessimistic than optimistic) the results are not quite convincing.
Results from this criterion are even less convincing that those obtained from Hurwicz’s rule. Focus
Table 5 Simulation-based performance indices for three types of investors (with non-dominated PEPs) PEP
Scenarios 1
Score 2
3
4
5
6
7
8
9
10
11
12
1.231 0.955 1.318
1.125 0.896 1.205
1.244 0.984 1.324
2.510 2.802 2.283
1.728 1.694 1.686
2.279 2.517 2.091
1.713 1.751 1.647
1.975 2.091 1.857
2.377 2.669 2.163
104.790 103.225 102.042
with r0 = 1 1.005 0.908 1.109 1.026 0.723 0.613
0.959 1.036 0.768
0.928 1.037 0.652
1.604 1.533 1.669
1.203 1.234 1.077
1.535 1.468 1.604
1.285 1.272 1.266
1.384 1.355 1.387
1.598 1.512 1.708
192.975 191.052 188.220
0.960 0.885 0.953 0.948 0.942 0.710
0.908 0.786 0.914 0.914 0.913 0.503
1.197 1.198 1.165 1.149 1.133 1.161
1.032 0.967 1.021 1.014 1.006 0.801
1.189 1.202 1.155 1.138 1.121 1.196
1.103 1.093 1.077 1.064 1.050 1.049
1.130 1.120 1.103 1.089 1.075 1.071
1.221 1.249 1.182 1.163 1.144 1.277
102.253 100.293 100.079 98.917 97.718 93.351
Profitability seeker with r0 = 2.25 6 0.731 2.500 1.505 7 0.185 2.708 1.304 5 0.957 2.303 1.540 Unbiased/standard investor 6 0.584 1.508 5 0.802 1.485 7 0.082 1.458
Safety seeker with r0 = 0.44 5 0.733 1.119 6 0.518 1.063 3 0.765 1.103 2 0.777 1.094 1 0.788 1.083 7 0.036 0.898
0.916 0.781 0.924 0.926 0.926 0.462
0.894 0.763 0.903 0.904 0.904 0.460
Vienna Exchange. Rankings in this table are as consistent as those from Frankfurt (see Table 4).
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After developing the step-by-step selection process in the same way as above for Frankfurt, the results are displayed in Table 5.
only few assumptions and comes with a low computational burden. Its applicability has been demonstrated by means of two studies on the Frankfurt and Vienna Stock Exchanges. From Subsections 4.4 and 4.5, there is empirical evidence that using Hurwicz’s or Wald’s criteria (i.e., the most representative decision rules involving pessimism under strict uncertainty) leads to unconvincing results in the portfolio selection problem. We are aware of the fact that the approach is currently limited to the risk–profitability criteria and particularly does not take into consideration additional criteria such as liquidity, the number of securities in the fund or perhaps a social responsibility quotient. Thus, the issue of how these objectives may be included in the proposed methodology will be the subject of further research.
5. Conclusion
Acknowledgement
In this paper, we have proposed a decision support approach for selecting portfolios with respect to an investor’s individual preferences for risk and profitability. It is designed to take into account the uncertainty inherent in stock markets and, to this end, it makes use of moderate pessimism decision tables and simulation-based multi-criteria performance indices, that are new tools in portfolio management and, particularly, are advantageous when compared with the popular Sharpe ratio. As a result, our methodology provides an attractive alternative to traditional approaches. Based on modern portfolio theory (MPT), the selection process requires two stages, the first being Markowitz’s efficient frontier, that assumes the predictive ability of past data. This assumption is classic in MPT and defended, among many authors, by Sharpe (1997). By and large in MPT, a second stage is needed to choose one or a few portfolios among the set of efficient funds. Therefore, in any MPT selection process, some prediction errors arise from both stages. For example, when fuzzy techniques are used in the second stage, then prediction errors come from the efficient frontier and the fuzzy application. Then, fuzzy errors generally surpass those expected from simulation with Sharpe’s betas (as we propose), because the fuzzy data (as applied to portfolio selection) are mostly beliefs about the future without objective support (see Arenas et al., 2001). In short, our procedure follows principles that are widely accepted not only in decision analysis but also in economics and finance. Moreover, it requires
Thanks are due to Paula Jorda for her assistance in the computational experiments.
first on the unbiased/standard investor. Indeed, the three bottom portfolios are non-dominated alternatives, and four non-dominated portfolios are ranked after dominated alternatives. Focus now on the safety seeker. Here, the four bottom portfolios are non-dominated alternatives, and six non-dominated portfolios are ranked after dominated alternatives. Finally, for profitability seekers, two bottom portfolios are non-dominated alternatives, and three nondominated portfolios are ranked after dominated alternatives. 4.6. Vienna market: Results
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