Evaluation of Decision Rules in Robust Portfolio Modeling

Transcription

1 Mat Independent Research Proects in Applied Mathematics Evaluation of Decision Rules in Robust Portfolio Modeling Juha Saloheimo 57739V TKK, Department of Mathematics and System Analysis

2 1. Introduction Proect portfolio selection and allocation of resources are essential strategic decisions for example in public administration and industrial firms. Often the decision maker (DM) has limited resources to operate with and he/she is faced with a decision to choose a subset of existing proects to carry out. In many cases the DM also has to operate under incomplete information as it may be impossible to receive reliable information on how the proect will perform in the future or the DM is unwilling or unable to state exactly his preferences (Weber, 1987). Robust Portfolio Modeling (RPM) (Liesiö et al., 2007, 2008) is a framework for proect portfolio selection under incomplete information. RPM framework utilizes additive value model in which each proect is evaluated with regard to several criteria in order to calculate the additive of individual proects (Keeney and Raiffa, 1976). Finally, overall of feasible portfolios are computed as a of proects (e.g., Golabi et al., 1981; Golabi, 1987). Incomplete information is included in the framework by stating preferences between criteria through linear inequalities between criterion weights (e.g., Arbel, 1989; Park and Kim, 1997; Salo and Punkka, 2005), while intervals are used to model the performance of proect with regards to different criteria. A set of linear constraints can be used to model the scarcity of resources and proect interdependencies. In RPM framework a set of non-dominated portfolios is determined. This set includes all the proect portfolios that could be selected by a rational decision maker given the uncertainties in the preferences and proects future outcomes. In RPM framework a dynamic programming algorithm (Liesiö et al., 2007, 2008) is employed to calculate all the non-dominated feasible portfolios subect to incomplete information. The number of non-dominated portfolios can be quite high and thus the DM still has to conduct further research at the proect and portfolio-level in order to determine the best portfolio within the non-dominated portfolios. This study concentrates on evaluating the performance of eight different decision rules (Kouvelis and Yu, 1997; Salo and Hämäläinen, 2001) at the portfolio-level in RPM framework. The performance of the decision rules in selecting a portfolio within the non-dominated portfolios is evaluated through Monte Carlo simulations. The efficiency of the decision rules is determined in different cases varying the number of proects, the number of criteria and the level of budget in order to clarify possible interdependencies between these parameters and the performance of the 1

3 decision rules. The decision rules can be ranked analyzing the generated value distributions with chosen performance measures. The remainder of this study is organized as follows. In Section 2 the RPM-framework is shortly introduced. In Section 3 the decision rules are presented as well as the performance measures used to evaluate the decision rules. Section 4 holds the simulation framework and in Section 5 the results are presented. Conclusions are made in Section Robust Portfolio Modeling (RPM) 2.1 Additive Value In the RPM (Liesiö et al., 2007) framework m proect proposals X = { x 1,..., x m } are evaluated with regard to n criteria. The normalized criterion-specific of proect i = 1,...,n are contained in the s vector v = [ v,...,vn x with regard to the criteria 1 ]. The s vectors form the rows of the s matrix v m n R such that [] i v i. v = The overall value of proect x is obtained with an additive value function i.e., n V ( x ) = wivi, where the weight w i measures the relative importance of the ith criterion. The weights are normalized so that n i= 1 0 n w S : = { w R w 0, w = 1}. (1) w i i= 1 i A proect portfolio p X is a subset of all proects and the set of all possible portfolios is the X power set P := 2. The overall value of a portfolio is the of the overall of its proects. For a given s matrix v and criterion weights w, the overall value can be written n T V ( p, w, v) : = w v = z( p) vw, (2) x p i= 1 i i where z(p) is a biection z } m : P {0,1 such that ) 1 z ( p = if x p and z ( p) = 0 if x p. 2

4 2.2 Resource Constraints and Feasible Portfolios Normally only a subset of all possible portfolios meets the resource constraints. In the RPM framework B k represents the available amount of the kth resource type (k=1,,q) and the total B =,..., 1 B q R+ q budget vector is denoted by [ B ]. T A proect x uses resources according to its resource conption vector C ( x ) [ c,..., c ] T 1 q =.The total cost vector of a proect portfolio is formed by ming the cost vectors of its constituent proects, i.e, x p C ( p) = C( x ) and C ( 0) = 0. Consequently the set of feasible portfolios is P F = { p P C( p) B}, where the inequality holds componentwise Incomplete Information and Dominance In the RPM framework incomplete information is modeled by set inclusion. Instead of approximating exact for ss and weights, analysis is done by determining sets of feasible parameters that are consistent with the DM s preference statements. The criterion weights are modeled under incomplete information by the set of feasible weights, denoted by 0 S w S w, where 0 S w is given by (1). It is ased that the convex hull of the weight set S w is a polyhedron. The extreme points of this convex hull and the extreme point matrix are denoted by 1 t { w,..., w }: = ext( conv( S W ext : = 1 t n t [ w,..., w ] R. w )) Incomplete s information is modeled through s intervals [ v, v ] which contain the true ss v. In matrix form v and v denote lower and upper bounds of all s intervals and the set i i i m n of feasible ss is the set of matrixes S = { v R v v v}. v The information set is defined as the Cartesian product S : = S w S. By selecting different weights and ss from the feasible set v S, the overall of portfolios vary. According to overall of portfolios in the information set S, portfolios can be classified trough dominance concept. 3

5 Definition 1. Let p, p' P. Portfolio p dominates p ' with regard to information S, denoted by p f p', if V ( p, w, v) V ( p', w, v) for all ( w, v) S and V ( p, w, v) > V ( p', w, v) for some S ( w, v) S. For a given portfolio p the overall portfolio value for any w S w and Sv v belongs to the interval V ( p, w, v) minv ( p, w),maxv ( p, w), w S w w S w where the mappings V, V : P S R + are given by V ( p, w) = w i v i, x n i= 1 V ( p, w) = w v i. n x p i= 1 i w These upper and lower bounds on portfolio value are linear with regard to weights and dominance can be checked by comparing the portfolios at the extreme points of conv S ). When a DM maximizes the overall portfolio value, a dominated portfolio should never be chosen. Thus the DM can concentrate the analysis on the set of non-dominated portfolios P ( S) : = { p P / p' P s. t. p' f p}. N F F S ( w 2.4 Additional Information The number of non-dominated portfolios depends on the information set S and loose preference statements and wide s intervals typically result in a large number of non-dominated portfolios whereas point estimates give a unique optimal portfolio. Additional preference information on s ~ intervals and/or weight constraints reduce the initial information set S to S S. As the dominance relationships between portfolios are contingent on the information set, the shift from S to S ~ generally results in a different set of non-dominated portfolios. If we ase that the true parameter are contained in the interior of S, defined as int( S ) = { s S s' S δ > 0 s. t. s + ε ( s s') S ε [0, δ ]}, additional information may eliminate some portfolios from the set of non-dominated portfolios, but it cannot add any new portfolios in it. 4

6 Theorem 2 (Liesiö et al., 2007). Let S ~,S be information sets such that Then P ~ N ( S ) PN ( S). ~ S S ~ and int( S ) S 0/. 2.5 Robustness Measures for Proects It is possible to make certain classifications between individual proects appearing in the nondominated portfolios. For each proect the so called index can be calculated, which is defined as the share of non-dominated portfolios that include the proect. Definition 3. For a given information set S we can define of proect x : CI( x, S) { p PN ( S) x p} / PN ( S) =, Core proects: X ( S) = { x X CI( x, S) = 1} C, Borderline proects: X ( S) = { x X 0 < CI( x, S) < 1} B, Exterior proects: X ( S) = { x X CI( x, S) = 0 E. All proects should always be chosen as they are included in all of the non-dominated portfolios. Correspondingly, none of the exterior proects should be selected. Thus, the emphasis of further analysis on the proect-level should be directed to borderline proects. 3. Decision Rules In order to choose a single portfolio within the non-dominated portfolios, some decision recommendations at the portfolio-level must be provided. Non-dominated portfolios can be regarded as discrete decision alternatives whose performance can be analyzed through suitable robustness measures. This study focuses on this phase of the RPM framework as we try to find differences between performances of the decision rules in choosing a portfolio within the nondominated portfolios. The decision rules covered on this study can be classified into two different groups: 1. Decision rules that rank the portfolios based on their overall value V ( p, w, v) within the information set S (value-based decision rules). 2. Decision rules that rank the portfolios based on of the proect included in portfolios (Core-index based decision rules). 5

7 3.1 Value-based Decision Rules -rule recommends the portfolio for which the maximum of its overall portfolio value over the information set S is the highest. The recommended portfolio is therefore within the set p max arg max maxv ( p, w). p PN w S w -rule recommends the portfolio for which the minimum of its overall portfolio value over the information set S is highest. The recommended portfolio is in the set p min arg max minv ( p, w). p PN w S w rule recommends the portfolio for which the of the minimum and maximum of its overall portfolio value over the information set S is highest. The recommended portfolio is therefore within the set p central arg max[ minv ( p, w) + maxv ( p, w)]. p PN w S w w S w Minimax-regret rule recommends the portfolio for which the maximum regret defined as the greatest possible loss of value relative to some other portfolio over the information set is smallest. The recommended portfolio belong therefore in the set p mmr arg min max[ V ( p'\p, w) V ( p p PN p' PN, w S w \ p', w)]. 3.2 Core Index -based Decision Rules rule is an iterative process in which on every round a proect that has the highest index is added to the portfolio. The recommended portfolio following iteration: 0 1. k = 1, P ( S) P ( S). N = N p ccr can be found through the 2. Add proects x that have the highest index to the final portfolio. In the special case of having several proects with highest index, all of the proects are selected if there exists a portfolio within P ) k N ( S that all the proects belong to. If such a portfolio does not exist, 6

8 the proect to be selected is randomized (in this study the proect with the lowest index is selected). The set of portfolios under interest is updated: k P ( S) N = { p P N ( S) k 1 x p} 3. The are updated to reflect P ( S) N k k 4. If P ( S) > 1, k = k + 1, go to 2. Otherwise stop. N rule chooses the non-dominated portfolio p, for which the of the CI p m = = 1 z ( p) CI ( x ) is the highest. Thus the recommended portfolio is in the set p cis arg maxci. p PN p index rule recommends the non-dominated portfolio p for which the of the weighted by the costs of the proects CI p c = = 1 z ( p) c CI ( x ) is the highest. The recommended portfolio belongs thus in the set m p wcis arg max CI. p PN pc It should be noted that this decision rule can only be used in this form in the cases with one budget constraint. rule chooses a portfolio by selecting proects with high into the portfolio. Proects are picked to the portfolio in such a manner that on every round the proect with highest index is chosen to the portfolio without updating the. This is repeated until the budget constraint is reached. If the proect with the highest index cannot be chosen due to its costs, the proect with highest possible index with acceptable costs is chosen. The procedure is repeated until none of the proects can be added without exceeding the budget constraint. It should be noted that the resulting portfolio non-dominated portfolios. p hci does not necessarily belong to the 4. Simulation framework 4.1 Data The performance of the decision rules in RPM framework is evaluated through Monte Carlo simulations. 7

9 Weights of the criteria w i are defined through rank-ordering weight information w 1... wn and all of the weights fulfil also condition w i 1/[3n], where n is the number of criteria. For s vectors, point estimates are used which are determined by first generating for each proect the overall value V '( x ) from the [ 0, n ] - uniform distribution. The ss v i are finally generated in such a manner that the s vector v x ) = ( v,..., v ) is evenly distributed in the area ( 1 n n { v( x ) R vi [0,1], = v = V '( x )}. Only one budget constraint is used and the costs of the i i 1 n proects y c are generated as c e V '( x ) = where ~ N(0,ln(3) /(1,96 2)). Thus for 95 % of the y proect pairs (, k) the ratio y yk e / e is between [ 1/ 3,3]. The true weights * w are determined from the even distribution over the weight information set (Rubinstein, 1982). In the simulations, the number of criteria n, the number of proects m and the proportion of the m 1 budget of the total cost of the proects b = B / = c are being varied. 100 problem instances are generated according to the presented process for each combination of parameters n, m and b. The parameters take n = 3,5, 7, m = 40,50, 60 and b = B / = c = 0.30, 0.50, leading to 1 altogether 27 parameter combinations. For each case the non-dominated portfolios P N (S) are calculated. Based on the generated true * weights w, the for all non-dominated portfolios can be calculated. Thus the performance of the decision rules can be assessed based on the value distribution of the recommended portfolios in the 100 instances. The parameter combinations are evaluated separately as in addition to the introduced performance measures for the decision rules we aim to fathom out if there exist interdependencies between the parameters and the performance measures for decision rules. With the help of the true weight vector m * w, the value maximizing non-dominated portfolio * p can be calculated as well as the value minimizing non-dominated portfolio p for each problem instance. Values of non-dominated portfolios can be normalized between [ 0,1] through * V '( p) = [ V ( p) V ( p )]/[ V ( p ) V ( p )] making the results from different parameter combinations and problem instances better comparable. 8

10 4.2 Performance Measures for Decision Rules In this section some performance measures that are applied in evaluating the decision rules are introduced. Asing that each instance has an equal probability, the performance of the decision rules can be evaluated based on the value distributions of the selected portfolios. Expected (EV) can be determined from the value distributions of the portfolios recommended by the decision rules. Expected value is one possible way to rank the decision rules but not really good one, since some of the decision rules (the maximin rule, minimax-regret rule) aim to optimize the worst-case value. However, a DM that concentrates purely on value maximization would prefer a decision rule which yields the highest expected value. The aim is rather to identify decision rules that also minimize the risk of low value portfolio recommendations. A classical method to model risk aversion is expected utility theory (EUT) (von Neumann and Morgenstern, 1944) in which a DM is rather trying to optimize the overall utility than pure value of the portfolio. A utility function u (x) of an underlying asset x (money for instance) is increasing and its form depends on the risk attitude of the DM. For a risk-averse DM, the utility function is concave while a utility function of a risk-seeking DM is convex. In the special case of a risk-neutral DM, the utility function takes the form of a straight line. In this latter case maximization of the expected value would not differ from the maximization of the utility but for the two other risk profiles this would not be the case anymore. As decision makers tend to be riskaverse in decisions like portfolio selection, the more specific analysis can be concentrated on the class of concave utility functions. If ased that the DM is risk averse, the Second-degree Stochastic Dominance (SSD) can be employed. Decision rule dominates decision rule i in the sense of SSD if z { F ( x) F ( x)} dx 0 i for all z over x, where F i (x) is the cumulative distribution function of a decision rule i. If SSD holds and the utility function u (x) of a DM is increasing, concave and differentiable, then 9

11 u( x) = u'( x) = [ f ( x) f i ( x) ] dx = u'( x) [ Fi ( x) F ( x) ] u''( x) x [ Fi ( z) F ( z) ] dz u''( x) [ Fi ( z) F ( z) ] [ Fi ( z) F ( z) ] dz dx > 0. dx dz dx Thus, if a decision rule dominates a decision rule i in the sense of SSD, the expected utility E ( u( V ( p )) of a portfolio chosen by a decision rule is greater than the expected utility E( u( V ( p i )) of a portfolio chosen by a decision rule i for all the risk-averse utility functions. In other words, the decision rule i is generally more riskier than the decision rule. It should be noted that SSD also implies greater expected value as the equation holds also for the risk-neutral DM. In our setting (Section 4.1) let the normalized of the portfolios chosen by a decision rule A A A A A be v = v, v,..., v ). Thus the discrete cumulative distribution function is A ( F ( x) = P( V '( p A ) x) = A vi x and the integral of F A (x) at the point z is defined as z A z vi F A ( x) dx = ( ). 100 A vi z Correspondingly, if the normalized of portfolios from a decision rule B are B B B B v = v, v,..., v ), the decision rule A dominates the decision rule B in the sense of SSD if A v1 z ( z v ( 100 for all z. A 1 ) B vi z B z vi ( ) 100 In financial setting, risk measures are often used instead of expected utility theory. Among the measures Value-at-Risk (VaR); (Campbell et al., 2001) has won popularity. VaR is the maximum loss on a probability level p and in the case of a continuous distribution it is defined as VaR 1 p = f ( x) dx, 10

12 where f (x) is the density function of a variable x. A risk-averse DM prefers higher of VaR and is willing to compromise possible high returns from the proects in order to avoid the variance in the portfolio value. In our setting, as there are 100 instances, estimate of VaR on 90% probability level is defined as the 10th worst value of the recommended portfolios. 5. Results In this section the results from the simulation are presented. All the performance measures for the decision rules presented in Section 4.2 have been computed. In order to clarify interdependencies between the decision rules, sample cross correlations have also been calculated. All the results are presented in Appendix A. 5.1 Value Distributions In Appendix A Figures 1-27 present normalized value distributions of the recommended portfolios for each decision rule and each parameter combination. The expected (EV) of the value distribution of the recommended portfolios are shown on the graphs. For a reference point in the graphs, we use the expected value of all the non-dominated portfolios (EVA) in all 100 instances of the same parameter combination. The difference between EV and EVA illustrates the difference between choosing a portfolio according to a certain decision rule and choosing a portfolio randomly. All the expected from the different parameter combinations are marized in Table 2. According to the value distributions and expected, three decision rules the central rule, minimax-regret rule and weighted index rule seem to be performing much better than the other decision rules. Based on the value distributions, probability mass of recommended portfolios by these three decision rules seems to concentrate on the high end of the normalized value scale. In the sense of expected (Table 2) the central rule, minimax-regret rule and weighted index rule perform better than the other decision rules as EVs of these decision rules are for all the parameter combinations higher than the EVs of the other five decision rules. As far as the other decision rules are concerned the maximax-rule is clearly the worstperforming decision rule in the sense of EV. In all of the parametric combinations EV of the maximax-rule is lower than the expected value of a randomly chosen portfolio. Remaining decision 11

13 rules maximin, conditional, index and biggest rule s better than the maximax -rule but clearly worse than the top three decision rules. The performance of some of the decision rules seem to vary in the parameter space. First of all the performance of the index-based decision rules seem to enhance in sense of EV as the number of proects and criteria increases. It can also be seen that weighted index rule performs worst when b is 0.7. Of the value-based decision rules, only EVs of the minimax-regret rule seem to be positively correlated with the number of criteria. Comparing the EVs of the best-performing decision rules, the central -rule seem to s highest in a maority of parameter combinations with small number of proects and criteria. As the number of proects and criteria grows the performance of the weighted index rule enhances and it ss highest almost for all the parameter combinations when m = 60. According to the value distributions (figures 1-27), the minimax-regret rule appears to have the shortest tale of the three decision rules, but it chooses less portfolios with between 0.9 and 1 than the two other decision rules under scrutiny. The rule performs especially well in choosing a portfolio with value between 0.9 and 1 and in maority of the parameter combination the central rule has the highest number of portfolios in this value range. The positive correlation of the performance of the weighted index rule with the number of proects and criteria can also be seen in the distributions and as the number of proects and criteria grows the tail of the distribution becomes smaller and the probability mass concentrates better in the high end of the scale. 5.2 Value- at-risk Estimates Table 3 presents all the estimates of VaR on 90% probability level. VaR estimates are correlated in the parameter space in the same manner as the EV estimates. The performance of the index based decision rules improves as the number of proects and criteria grows and the performance of the minimax-regret rule is correlated with the number of criteria. According to Table 3, the minimax-regret and the weighted index rule have higher VaR estimates than the other decision rules in all of the parameter combinations. The minimax-regret rule performs best in maority of the combinations when the number of proects is 40, the performance of the two decision rules is quite similar as m = 50 and in the case of m = 60 the weighted index rule outperforms the minimax-regret rule in maority of the parameter combinations. The central rule outperforms the remaining five decision rules in all of the 12

14 parameter combination but two ( m = 50, n = 7, b = 0.3; m = 50, n = 7, b = 0.5), but there still exists a distinct gap in VaR estimates between the central rule and the minimax-regret rule/ weighted index rule. 5.3 Existence of Second-Degree Stochastic Dominance Tables 4-30 illustrate existence of second-degree stochastic dominance between the decision rules. A decision rule on row i dominates a decision rule on column in the sense of SSD, if cell (i, ) has value 1. Value 0 corresponds on the other hand to the inexistence of SSD. Tables 4-30 show that in all of the parameter combinations at least one of the decision rules central rule, minimax-regret rule and weighted index rule dominates the other remaining decision rules in the sense of SSD. On the other hand, the three decision rules are not dominated by the other decision rules in any combination. Thus a risk-averse (or risk-neutral) DM should always choose the central, minimax-regret or weighted index rule. The dominance relationships between the central rule, minimax-regret rule and weighted index rule alternate depending on the parameter combinations. As mentioned before, the performance of the weighted index rule improves as the number of proects and criteria grows while the performance of the minimax-regret rule is positively correlated with the number of criteria. The SSD relationships between these decision rules are gathered in Table 1. A decision rule on row i dominates a decision rule on column in the sense of SSD in all the parameter combinations that are illustrated in the corresponding cell. 13

15 Table 1. SSD relations between the central, minimax-regret and weighted index rule rule rule index rule rule Minimax regret rule index rule m = 40; n = 3; b = 0.3, 0.5, 0.7 m = 50; n = 3; b = 0.7 m = 40; n = 5; b = 0.3 m = 40; n = 3; b = 0.3, 0.5, 0.7 m = 40; n = 7; b = 0.3, 0.5, 0.7 m = 40; n = 5; b = 0.7 m = 50; n = 5; b = 0.3 m = 40; n = 7; b = 0.3, 0.7 m = 50; n = 7; b = 0.3, 0.5, 0.7 m = 50; n = 3; b = 0.3 m = 60; n = 3; b = 0.7 m = 50; n = 5; b = 0.7 m = 60; n = 5; b =0.5 m = 50; n = 7; b = 0.7 m = 60; n = 7; b = 0.3, 0.5, 0.7 m = 60; n = 7; b = 0.7 m = 40; n = 5; b = 0.3 m = 40; n = 7; b = 0.5 m = 50; n = 5; b = 0.3 m = 50; n = 7; b = 0.3, 0.5 m = 60; n = 3; b = 0.7 m = 60; n = 5; b = 0.3, 0.5, 0.7 m = 60; n = 7; b = 0.3, 0.5 m = 50; n = 5; b = 0.3 m = 50; n = 7; b = 0.5 m = 60; n = 3; b = 0.7 m = 60; n = 5; b = 0.3, 0.5, 0.7 m = 60; n = 7; b = 0.3 As far as dominance between the minimax-regret rule and the central rule is concerned the central rule does not dominate the minimax-regret rule in any parameter combination. On the contrary the minimax-regret rule dominates the central rule in sense of SSD always when n = 7 and in one third of the cases with n = 5. In the parameter combination with n = 3 SSD appears only in one of the cases. The central rule dominates in the sense of SSD the weighted index rule in the parameter combinations with m = 40 and n = 3 and in the case of m = 50, n = 3, b = 0.7. The weighted index rule dominates the central rule especially in the parameter combinations with high number of proects and criteria. The minimax-regret rule dominates the weighted index rule maorly in the parameter combinations with lower number of proects and in the combinations with b = 0.7. On the other hand the weighted index rule dominates the minimax-regret rule maorly in the combinations with higher number of proects. 5.4 Correlations Cross correlations are illustrated on Tables 31-57, where cell (i,) corresponds to the cross correlation between portfolio of decision rules on ith row and th column. 14

16 First of all it can be noted that the decision rules are mainly positively correlated suggesting that the rules tend to perform more or less equally. The minimax-regret rule, central rule and weighted index rule are also in all of the combinations positively correlated. The highest cross correlation estimates are between the conditional rule and index rule reaching in some cases over 0.7. The correlation estimates between the minimax-regret and central rule are also eye-catchingly high reaching at best over 0.5. The group of index-based decision rules are all positively correlated as they have already demonstrated similar correlation relationships with the parameters. 6. Conclusion According to the results, three decision rules the central, minimax-regret and weighted index rule umped out of the pack. They outperformed the other decision rules in terms of estimated EV, VaR and SSD. Thus, the results suggest that in RPM framework one of these decision rules should be employed. The central rule seems to be the weakest of the top three decision rules as it is much riskier than the other two decision rules according to the VaR estimates. It is also dominated in the sense of SSD in 15 of 27 parameter combination and it does not dominate the minimax-regret rule in any of the parameter combinations. Its performance is relatively poor in parameter combinations with higher number of proects and criteria. Thus the central rule could only be considered in the parameter combinations with lower number of proects and criteria and when the DM has a riskneutral or even risk-seeking attitude. It might be possible that the central rule would perform even better if the number of the proects were less than the minimum of this study. This is though out of the scope of this study. The minimax-regret rule proved to be one of the two strongest decision rules. Although in terms of estimated expected it was in general the worst decision rule of the three best ones and the value distributions also showed that it does not choose as frequently portfolios that s in the top tenth of the value scale as the other two decision rules. It showed however favorable characteristic for a risk-averse DM as it performed well in VaR estimates. For example, the minimax-regret rule dominates in the sense of VaR the central rule in all of the parameter combinations and the weighted index rule in maority of combinations with m = 40 or b = 0.7. The minimaxregret rule dominated (in the sense of SSD) the central rule especially in combinations with high number of proects and criteria and the weighted index rule in the 15

17 combinations with low number of proects. According to the results, the minimax-regret rule could be used by a risk-averse DM in the cases with relatively low number of proects and with looser budget constraints as the minimax-regret rule performs relatively better than the weighted index rule in the combinations with b = 0.7. The weighted index rule performed especially well in the parameter combinations with high number of proects. It is in general the best decision rule both in the estimated expected and the VaR estimates when m = 60. The performance of the weighted index rule is quite correlated with budget constraints, number of proects and number of criteria explaining partly why the selection of a best decision rule depends quite a lot on the parameters. As a result, the weighted index rule should be employed in the problems especially with high number of proects. According to the cross correlation matrixes, some of the decision rules seem to perform in a similar manner. This can be explained first of all by the interdependencies between the decision rules and budget constraints, number of proects and number of criteria. As noted, especially the performance of the index-based decision rules is positively correlated with the number of proects and criteria. It is also possible that the performance of the decision rules depends straightly on the number of non-dominated portfolios creating interdependencies between the decision rules. It should be remembered that the results are based on the simulations and thus too strong conclusions cannot be drawn from them. In this paper no estimates of inaccuracy have been conducted and it is left open for future studies. The accuracy of the results could be though improved by increasing the simulation rounds from 100. The results are also linked to the simulation framework and thus the results might not hold under different or more general asptions. References Arbel, A., Approximate Articulation of Preference and Priority Derivation. European Journal of Operational Research 43, Archer, N.P, Ghesemzadeh, F., An Integrated Framework for Proect Portfolio Selection, International Journal of Proect Management 17, Cambell, R., Huisman, R., Koedik, K, Optimal Portfolio Selection in a Value-at-Risk Framework, European Journal of Banking & Finance 25, Cooper, R.G., Edgett, S.J., Kleinschmidt, E.J., New Product Portfolio Management: Practices and Performance, Journal of Product Innovation Management 16,

18 Golabi, K., Selecting a Group of Dissimilar Proects for Funding, IEEE Transactions on Engineering Management 34, Golabi, K., Kirkwood, C. W., Sicherman, A., Selecting a Portfolio of Solar Energy Proects Using Multiattribute Preference Theory, Management Science 27, Hämäläinen, R.P., Reversing the Perspective on the Applications of Decision Analysis, Decision Analysis 1, Keefer, D.L., Kirkwoord, C.W., Corner, J.L., Perspective on Decision Analysis Applications, Decision Analysis 1, 4-22 Liesiö, J., Mild, P., Salo, A Preference Programming for Robust Portfolio Modeling and Proect Selection, European Journal of Operational research 181, Liesiö, J., Mild, P., Salo, A Robust Portfolio Modelling with Incomplete Cost Information and Proect Interdependencies, European Journal of Operational research 190, Von Neumann, J., Morgenstern, O Theory of Games and Economic Behavior, Princeston University Press Park, K.S., Kim, S.H., Tools for Interactive Decision Making with Incompletely Identified Information, European Journal of Operational Research 98, Rubinstein, R. (1982). Generating Random Vectors Uniformly Distributed inside and on the Surface of Different Regions, European Journal of Operational Research, 10, Salo, A., Hämäläinen, R. P Preference Ratios in Multiattribute Evaluation (PRIME) Elicitation and Decision Procedures under Incomplete Information, IEEE Transactions on Systems, Man, and Cybernetics Part A: Systems and Humans 31, Salo, A., Punkka, A Rank Inclusion in Criteria Hierarchies, European Journal of Operational Research, 163, Weber, M., Decision Making with Incomplete Information, European Journal of Operational Research 28,

19 Appendix A. Results Figure 1. Normalized of recommended portfolios when m = 40, n = 3 and b =

20 Figure 2. Normalized of recommended portfolios when m = 40, n = 3 and b =

21 Figure 3. Normalized of recommended portfolios when m = 40, n = 3 and b =

22 Figure 4. Normalized of recommended portfolios when m = 40, n = 5 and b =

23 Figure 5. Normalized of recommended portfolios when m = 40, n = 5 and b =

24 Figure 6. Normalized of recommended portfolios when m = 40, n = 5 and b =

25 Figure 7. Normalized of recommended portfolios when m = 40, n = 7 and b =

26 Figure 8. Normalized of recommended portfolios when m = 40, n = 7 and b =

27 Figure 9. Normalized of recommended portfolios when m = 40, n = 7 and b =

28 Figure 10. Normalized of recommended portfolios when m = 50, n = 3 and b =

29 Figure 11. Normalized of recommended portfolios when m = 50, n = 3 and b =

30 Figure 12. Normalized of recommended portfolios when m = 50, n = 3 and b =

31 Figure 13. Normalized of recommended portfolios when m = 50, n = 5 and b =

32 Figure 14. Normalized of recommended portfolios when m = 50, n = 5 and b =

33 Figure 15. Normalized of recommended portfolios when m = 50, n = 5 and b =

34 Figure 16. Normalized of recommended portfolios when m = 50, n = 7 and b =

35 Figure 17. Normalized of recommended portfolios when m = 50, n = 7 and b =

36 Figure 18. Normalized of recommended portfolios when m = 50, n = 7 and b =

37 Figure 19. Normalized of recommended portfolios when m = 60, n = 3 and b =

38 Figure 20. Normalized of recommended portfolios when m = 60, n = 3 and b =

39 Figure 21. Normalized of recommended portfolios when m = 60, n = 3 and b =

40 Figure 22. Normalized of recommended portfolios when m = 60, n = 5 and b =

41 Figure 23. Normalized of recommended portfolios when m = 60, n = 5 and b =

42 Figure 24. Normalized of recommended portfolios when m = 60, n = 5 and b =

43 Figure 25. Normalized of recommended portfolios when m = 60, n = 7 and b =

44 Figure 26. Normalized of recommended portfolios when m = 60, n = 7 and b =

45 Figure 27. Normalized of recommended portfolios when m = 60, n = 7 and b =

46 Table 2. Expected of all the parameter combinations Decision rules Core index index Parameters m=40, n=3, b=0.3 0,58 0,57 0,79 0,79 0,61 0,65 0,75 0,59 m=40, n=3, b=0.5 0,48 0,61 0,80 0,79 0,66 0,67 0,78 0,45 m=40, n=3, b=0.7 0,47 0,62 0,80 0,78 0,61 0,63 0,71 0,55 m=40, n=5, b=0.3 0,54 0,65 0,80 0,81 0,67 0,73 0,83 0,67 m=40, n=5, b=0.5 0,53 0,62 0,83 0,82 0,67 0,66 0,83 0,69 m=40, n=5, b=0.7 0,53 0,67 0,84 0,79 0,66 0,71 0,78 0,63 m=40, n=7, b=0.3 0,53 0,70 0,81 0,83 0,66 0,71 0,83 0,67 m=40, n=7, b=0.5 0,46 0,72 0,79 0,82 0,68 0,70 0,82 0,68 m=40, n=7, b=0.7 0,52 0,71 0,83 0,84 0,71 0,75 0,81 0,67 m=50, n=3, b=0.3 0,54 0,58 0,83 0,80 0,66 0,70 0,78 0,66 m=50, n=3, b=0.5 0,52 0,59 0,81 0,80 0,69 0,70 0,79 0,67 m=50, n=3, b=0.7 0,50 0,64 0,83 0,80 0,64 0,68 0,76 0,62 m=50, n=5, b=0.3 0,46 0,64 0,78 0,80 0,69 0,73 0,83 0,70 m=50, n=5, b=0.5 0,45 0,67 0,81 0,78 0,71 0,75 0,83 0,74 m=50, n=5, b=0.7 0,48 0,69 0,83 0,83 0,70 0,76 0,81 0,74 m=50, n=7, b=0.3 0,40 0,66 0,79 0,80 0,72 0,75 0,85 0,74 m=50, n=7, b=0.5 0,39 0,69 0,78 0,80 0,73 0,77 0,84 0,73 m=50, n=7, b=0.7 0,46 0,72 0,84 0,84 0,72 0,77 0,81 0,71 m=60, n=3, b=0.3 0,58 0,50 0,84 0,80 0,70 0,73 0,80 0,73 m=60, n=3, b=0.5 0,54 0,54 0,79 0,79 0,70 0,72 0,82 0,73 m=60, n=3, b=0.7 0,53 0,57 0,78 0,79 0,69 0,70 0,80 0,66 m=60, n=5, b=0.3 0,41 0,64 0,81 0,79 0,70 0,74 0,84 0,77 m=60, n=5, b=0.5 0,42 0,65 0,79 0,81 0,69 0,75 0,85 0,77 m=60, n=5, b=0.7 0,51 0,66 0,83 0,82 0,71 0,74 0,84 0,73 m=60, n=7, b=0.3 0,44 0,70 0,81 0,84 0,74 0,76 0,88 0,79 m=60, n=7, b=0.5 0,40 0,69 0,82 0,82 0,73 0,76 0,86 0,77 m=60, n=7, b=0.7 0,48 0,65 0,82 0,84 0,69 0,72 0,82 0,73 45

47 Parameters Table 3. Value-at-Risk on 90% probability level Decision rules Core index index m=40, n=3, b=0.3 0,00 0,00 0,40 0,57 0,13 0,23 0,42 0,17 m=40, n=3, b=0.5 0,00 0,04 0,51 0,55 0,23 0,25 0,50 0,30 m=40, n=3, b=0.7 0,00 0,03 0,46 0,51 0,21 0,19 0,30 0,15 m=40, n=5, b=0.3 0,00 0,21 0,50 0,58 0,32 0,41 0,58 0,35 m=40, n=5, b=0.5 0,00 0,13 0,52 0,65 0,39 0,32 0,69 0,38 m=40, n=5, b=0.7 0,00 0,16 0,52 0,52 0,29 0,37 0,39 0,23 m=40, n=7, b=0.3 0,00 0,23 0,46 0,61 0,32 0,44 0,59 0,27 m=40, n=7, b=0.5 0,00 0,36 0,46 0,61 0,34 0,44 0,62 0,42 m=40, n=7, b=0.7 0,00 0,27 0,49 0,67 0,34 0,47 0,55 0,26 m=50, n=3, b=0.3 0,00 0,04 0,61 0,62 0,29 0,44 0,54 0,36 m=50, n=3, b=0.5 0,00 0,00 0,57 0,60 0,33 0,40 0,54 0,31 m=50, n=3, b=0.7 0,00 0,11 0,52 0,59 0,25 0,28 0,49 0,30 m=50, n=5, b=0.3 0,00 0,26 0,56 0,63 0,35 0,47 0,65 0,43 m=50, n=5, b=0.5 0,00 0,14 0,54 0,63 0,41 0,43 0,64 0,44 m=50, n=5, b=0.7 0,00 0,14 0,55 0,62 0,41 0,47 0,55 0,47 m=50, n=7, b=0.3 0,00 0,25 0,50 0,64 0,47 0,56 0,67 0,40 m=50, n=7, b=0.5 0,00 0,29 0,50 0,62 0,43 0,53 0,66 0,49 m=50, n=7, b=0.7 0,00 0,39 0,59 0,67 0,40 0,50 0,54 0,44 m=60, n=3, b=0.3 0,00 0,01 0,61 0,64 0,43 0,44 0,65 0,45 m=60, n=3, b=0.5 0,00 0,00 0,46 0,59 0,39 0,44 0,63 0,47 m=60, n=3, b=0.7 0,00 0,00 0,42 0,60 0,31 0,34 0,57 0,36 m=60, n=5, b=0.3 0,00 0,27 0,57 0,64 0,46 0,53 0,71 0,56 m=60, n=5, b=0.5 0,00 0,16 0,51 0,67 0,42 0,49 0,69 0,59 m=60, n=5, b=0.7 0,00 0,17 0,58 0,64 0,30 0,40 0,64 0,48 m=60, n=7, b=0.3 0,00 0,35 0,56 0,69 0,54 0,55 0,74 0,60 m=60, n=7, b=0.5 0,00 0,39 0,60 0,65 0,44 0,51 0,71 0,56 m=60, n=7, b=0.7 0,00 0,21 0,52 0,67 0,29 0,41 0,66 0,41 46

48 Table 4. SSD of m = 40, n = 3, b = index index Table 5. SSD of m = 40, n = 3, b = 0.5 index Minimax-regret index Table 6. SSD of m = 40, n = 3, b = Minimax-regret index Minimax-regret index

49 Table 7. SSD of m = 40, n = 5, b = index index Table 8. SSD of m = 40, n = 5, b = 0.5 index Minimax-regret index Table 9. SSD of m = 40, n = 5, b = Minimax-regret index Minimax-regret index

50 Table 10. SSD of m = 40, n = 7, b = index index Table 11. SSD of m = 40, n = 7, b = 0.5 index Minimax-regret index Table 12. SSD of m = 40, n = 7, b = Minimax-regret index Minimax-regret index

51 Table 13. SSD of m = 50, n = 3, b = index index Table 14. SSD of m = 50, n = 3, b = 0.5 index Minimax-regret index Table 15. SSD of m = 50, n = 3, b = Minimax-regret index Minimax-regret index

52 Table 16. SSD of m = 50, n = 5, b = index index Table 17. SSD of m = 50, n = 5, b = 0.5 index Minimax-regret index Table 18. SSD of m = 50, n = 5, b = Minimax-regret index Minimax-regret index

53 Table 19. SSD of m = 50, n = 7, b = index index Table 20. SSD of m = 50, n = 7, b = 0.5 index Minimax-regret index Table 21. SSD of m = 50, n = 7, b = Minimax-regret index Minimax-regret index

54 Table 22. SSD of m = 60, n = 3, b = index index Table 23. SSD of m = 60, n = 3, b = 0.5 index Minimax-regret index Table 24. SSD of m = 60, n = 3, b = Minimax-regret index Minimax-regret index

55 Table 25. SSD of m = 60, n = 5, b = index index Table 26. SSD of m = 60, n = 5, b = 0.5 index Minimax-regret index Table 27. SSD of m = 60, n = 5, b = Minimax-regret index Minimax-regret index

56 Table 28. SSD of m = 60, n = 7, b = index index Table 29. SSD of m = 60, n = 7, b = 0.5 index Minimax-regret index Table 30. SSD of m = 60, n = 7, b = Minimax-regret index Minimax-regret index

57 Table 31. Cross correlation matrix of m = 40, n = 3, b = 0.3 index -0,14 0,34 0,27 0,22 0,60 1,00 0,56 0,67 index 0,00 0,30 0,36 0,37 0,44 0,56 1,00 0,66-0,02 0,25 0,21 0,23 0,51 0,67 0,66 1,00 Table 32. Cross correlation matrix of m = 40, n = 3, b = 0.5 index 1,00-0,51 0,19 0,04-0,15-0,16-0,04 0,07-0,51 1,00 0,04 0,11-0,01-0,05-0,03-0,15 0,19 0,04 1,00 0,40 0,08 0,23 0,24-0,07 Minimax-regret 0,04 0,11 0,40 1,00 0,11 0,14 0,10-0,11-0,15-0,01 0,08 0,11 1,00 0,58 0,16 0,20-0,16-0,05 0,23 0,14 0,58 1,00 0,33 0,27 index -0,04-0,03 0,24 0,10 0,16 0,33 1,00 0,34 0,07-0,15-0,07-0,11 0,20 0,27 0,34 1,00 Table 33. Cross correlation matrix of m = 40, n = 3, b = 0.7 1,00-0,43 0,14 0,12-0,05-0,14 0,00-0,02-0,43 1,00 0,14 0,04 0,30 0,34 0,30 0,25 0,14 0,14 1,00 0,56 0,27 0,27 0,36 0,21 Minimax-regret 0,12 0,04 0,56 1,00 0,31 0,22 0,37 0,23-0,05 0,30 0,27 0,31 1,00 0,60 0,44 0,51 index 1,00-0,45 0,08 0,13-0,02-0,13-0,09-0,02-0,45 1,00 0,23 0,16 0,04 0,12 0,03 0,04 0,08 0,23 1,00 0,68 0,03 0,30 0,09 0,07 Minimax-regret 0,13 0,16 0,68 1,00-0,07 0,21-0,02 0,05-0,02 0,04 0,03-0,07 1,00 0,42 0,33 0,36-0,13 0,12 0,30 0,21 0,42 1,00 0,26 0,33 index -0,09 0,03 0,09-0,02 0,33 0,26 1,00 0,44-0,02 0,04 0,07 0,05 0,36 0,33 0,44 1,00 56

58 Table 34. Cross correlation matrix of m = 40, n = 5, b = 0.3 index -0,01 0,28 0,15 0,22 0,55 1,00 0,33 0,40 index 0,09 0,12 0,11 0,10 0,25 0,33 1,00 0,42 0,21 0,11 0,11 0,17 0,37 0,40 0,42 1,00 Table 35. Cross correlation matrix of m = 40, n = 5, b = 0.5 index 1,00-0,21 0,10 0,08 0,04-0,02-0,12-0,08-0,21 1,00 0,31 0,00 0,38 0,25 0,38 0,40 0,10 0,31 1,00 0,40 0,20 0,15 0,27 0,02 Minimax-regret 0,08 0,00 0,40 1,00 0,19 0,04 0,16-0,03 0,04 0,38 0,20 0,19 1,00 0,40 0,36 0,34-0,02 0,25 0,15 0,04 0,40 1,00 0,01 0,29 index -0,12 0,38 0,27 0,16 0,36 0,01 1,00 0,24-0,08 0,40 0,02-0,03 0,34 0,29 0,24 1,00 Table 36. Cross correlation matrix of m = 40, n = 5, b = 0.7 1,00-0,30 0,21 0,10 0,02-0,01 0,09 0,21-0,30 1,00 0,18 0,06 0,28 0,28 0,12 0,11 0,21 0,18 1,00 0,37 0,26 0,15 0,11 0,11 Minimax-regret 0,10 0,06 0,37 1,00 0,00 0,22 0,10 0,17 0,02 0,28 0,26 0,00 1,00 0,55 0,25 0,37 index 1,00-0,02 0,24 0,26 0,21 0,28 0,00 0,07-0,02 1,00 0,16 0,11 0,22 0,25 0,18-0,02 0,24 0,16 1,00 0,25 0,12 0,10-0,06 0,07 Minimax-regret 0,26 0,11 0,25 1,00 0,21 0,31 0,15 0,03 0,21 0,22 0,12 0,21 1,00 0,55 0,15 0,25 0,28 0,25 0,10 0,31 0,55 1,00 0,20 0,28 index 0,00 0,18-0,06 0,15 0,15 0,20 1,00 0,16 0,07-0,02 0,07 0,03 0,25 0,28 0,16 1,00 57

59 Table 37. Cross correlation matrix of m = 40, n = 7, b = 0.3 index -0,03 0,17 0,00 0,08 0,33 1,00 0,31 0,13 index 0,02 0,17 0,16 0,15 0,14 0,31 1,00 0,34 0,08 0,11 0,02 0,07 0,42 0,13 0,34 1,00 Table 38. Cross correlation matrix of m = 40, n = 7, b = 0.5 index 1,00-0,15 0,26 0,06-0,01-0,15-0,08-0,08-0,15 1,00 0,29 0,09 0,04 0,20-0,07 0,01 0,26 0,29 1,00 0,35 0,02 0,02-0,01-0,01 Minimax-regret 0,06 0,09 0,35 1,00 0,15 0,10 0,30 0,09-0,01 0,04 0,02 0,15 1,00 0,42 0,23 0,35-0,15 0,20 0,02 0,10 0,42 1,00 0,03 0,48 index -0,08-0,07-0,01 0,30 0,23 0,03 1,00 0,24-0,08 0,01-0,01 0,09 0,35 0,48 0,24 1,00 Table 39. Cross correlation matrix of m = 40, n = 7, b = 0.7 1,00-0,23 0,32 0,07-0,16-0,03 0,02 0,08-0,23 1,00 0,04-0,18 0,15 0,17 0,17 0,11 0,32 0,04 1,00 0,24-0,13 0,00 0,16 0,02 Minimax-regret 0,07-0,18 0,24 1,00 0,00 0,08 0,15 0,07-0,16 0,15-0,13 0,00 1,00 0,33 0,14 0,42 index 1,00-0,22 0,00-0,12 0,01-0,29-0,03-0,06-0,22 1,00 0,33 0,14 0,18 0,23 0,08-0,03 0,00 0,33 1,00 0,42 0,30 0,07 0,19 0,14 Minimax-regret -0,12 0,14 0,42 1,00 0,28 0,15 0,33 0,13 0,01 0,18 0,30 0,28 1,00 0,30 0,15 0,29-0,29 0,23 0,07 0,15 0,30 1,00 0,04 0,31 index -0,03 0,08 0,19 0,33 0,15 0,04 1,00 0,20-0,06-0,03 0,14 0,13 0,29 0,31 0,20 1,00 58

60 Table 40. Cross correlation matrix of m = 50, n = 3, b = 0.3 index -0,09 0,20 0,00 0,06 0,30 1,00 0,32 0,52 index -0,09 0,09 0,07 0,15 0,34 0,32 1,00 0,36-0,10 0,21 0,20 0,26 0,36 0,52 0,36 1,00 Table 41. Cross correlation matrix of m = 50, n = 3, b = 0.5 index 1,00-0,64 0,03 0,10-0,14-0,31-0,10-0,26-0,64 1,00 0,00-0,09 0,17 0,34-0,04 0,23 0,03 0,00 1,00 0,40 0,19 0,40 0,27 0,29 Minimax-regret 0,10-0,09 0,40 1,00 0,20 0,33 0,58 0,20-0,14 0,17 0,19 0,20 1,00 0,59 0,35 0,21-0,31 0,34 0,40 0,33 0,59 1,00 0,41 0,45 index -0,10-0,04 0,27 0,58 0,35 0,41 1,00 0,30-0,26 0,23 0,29 0,20 0,21 0,45 0,30 1,00 Table 42. Cross correlation matrix of m = 50, n = 3, b = 0.7 1,00-0,45-0,03 0,12-0,05-0,09-0,09-0,10-0,45 1,00 0,24 0,15 0,22 0,20 0,09 0,21-0,03 0,24 1,00 0,42 0,11 0,00 0,07 0,20 Minimax-regret 0,12 0,15 0,42 1,00 0,18 0,06 0,15 0,26-0,05 0,22 0,11 0,18 1,00 0,30 0,34 0,36 index 1,00-0,48-0,03 0,01-0,07-0,24-0,13-0,08-0,48 1,00 0,13-0,03 0,29 0,38 0,10 0,05-0,03 0,13 1,00 0,40 0,01 0,12 0,05 0,26 Minimax-regret 0,01-0,03 0,40 1,00 0,09 0,27 0,20 0,22-0,07 0,29 0,01 0,09 1,00 0,43 0,33 0,17-0,24 0,38 0,12 0,27 0,43 1,00 0,17 0,11 index -0,13 0,10 0,05 0,20 0,33 0,17 1,00 0,39-0,08 0,05 0,26 0,22 0,17 0,11 0,39 1,00 59