Portfolio performance and risk-based assessment of the PORTRAIT tool

Abstract

In this paper we evaluate and extend the PORTRAIT tool. PORTRAIT applies an argumentation based methodology for composing fund portfolios. Argumentation allows for combining different contexts and preferences in a way that can be optimized. It allows for defining a set of different investment policy scenarios and supports the investor/portfolio manager in composing efficient portfolios that meet her/his profile. The performance and risk of the constructed portfolios is compared with portfolios based on a traditional performance index under different scenarios. This approach is applied on data of Greek domestic equity mutual funds over the period from January 2006 to December 2011 with positive results. The empirical results of our study showed that argumentation is well suited for this type of applications giving answers to two important questions, i.e. which mutual funds are the most suitable to invest in, and, what portion of the available capital should be invested in each of these funds.

Appendix

A contingency table (Table 5) is used in order to identify the frequency with which funds are defined as winners (W—a fund with returns above the median) and losers (L—a fund with returns below the median) over successive time periods. If a fund is a loser (L) for consecutive periods, it is defined as a loser–loser (LL). Similarly, a winner (W) in the first period that remains a winner (W) in the future period is defined as winner–winner (WW). A fund that shifts from loser (L) to winner (W) is a loser–winner (LW), while a fund that shifts from winner (W) to loser (L) is a winner–loser (WL). A fund that ceases operation and was a winner or loser during the previous year is defined as winner–gone (WG) or loser–gone (LG).

Table 5 Winner/loser contingency table

It is obvious that in the case where the number of funds in existence is the same in each period the definition of the funds is simple. However, if funds enter or leave the sample, between period t and t + 1, there is a difficulty in fund classification. Let examine the persistence of returns for N funds in period t, where M new funds are operating (K funds close) in period t + 1, thus M + N (M + N − K) funds are to be ranked. Funds that are new in period t + 1 are classified, but the contingency table includes only funds which operate over consecutive periods.

The three statistical tests used to examine the performance persistence of the examined funds for a twelve year period (2000–2011) are described below.

Malkiel’s Z-test (1995): $= ({\text{Y}} - {\text{np}})/\sqrt {({\text{np}}(1 - {\text{p}}))}$, which shows the proportion of repeat winners (WW) to winner–losers (WL), where Z is the statistic variable that follows a normal distribution (0, 1), Y is the number of winner funds in two consecutive periods, n is the sum of WW + WL, p is the probability of a winner fund in one period to repeat as a winner in the subsequent period. According to this criterion, a percentage of WW to WW + WL above 50 % and a Z-statistic above zero demonstrate performance persistence, while a percentage value below 50 % and Z-statistic above zero shows a reversal in performance. Malkiel’s Z-test is concentrated on only one quadrant of both repeat winners and repeat losers.

Brown and Goetzann Odds Ratio (OR) (1995): = (WW * LL)/(WL * LW). Using this ratio, the statistical significance of the OR is determined applying a Z-test to the following Z variable which follows a normal distribution (0, 1), ${\text{Z}} = \ln ({\text{OR}})/\sigma_{{\ln_{{({\text{OR}})}} }}$. An OR of one supports the hypothesis that the performance in one period is unrelated to that in another while an OR greater than one (below one) indicates persistence (reversals in performance dominate the sample). The OR ratio tests the persistence of both repeat winners and repeat losers.

Khan and Rudd Chi-square statistic (1995):

$$ x^{2} = \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {(O_{ij} - E_{ij} )^{{^{2} }} /E_{ij} } } $$

where O _ij and E _ij are the actual and expected frequency of the ith row and the jth column in the contingency table respectively. The associated p value is used in order to test for performance persistence. The Chi-square test is taking into account the persistence of the contingency table as a whole.

Due to the limited space, we present in Table 6, only the combined results of four tables (available upon request) which shows the contingency table of fund returns along with the results of the statistical tests of the null hypothesis of no performance persistence between consecutive periods. Precisely, Table 6, presents combined results of performance persistence with 1 (t to t + 1), 2 (t to t + 2), 3 (t to t + 3) and 4-year lag (t to t + 4).

Table 6 overall performance persistence

According to the results of this table, fifty-three percent (RW) of all winners in year 1 are winners in year 2, 179 (WW) of 336 (WW + WL). The percentage of RW for 2, 3 and 4 years are 52, 51 and 59 % respectively. However, the results of the examined tests show strong evidence of statistical significance performance persistence for 1-year and 4-years holding periods. More precisely, according to the first two tests, the percentage of RW is above 50 % while the Z-test is also above zero and statistical significant, thus is indicative of performance persistence. The same stands according to the OR ratio that is greater than one and the Z-stat which is also statistical significant. Thus, the results of the test statistics show that, at the overall level, there is evidence of performance persistence according to all three criteria for 1-year and 4-years holding periods. This evidence is in accordance with the results of other empirical studies which reveal that the relative performance of equity mutual funds persists from period to period (e.g. Hendricks et al. 1993; Gruber 1996; Vidal-García 2013). However, there is a series of empirical studies in support to the efficient markets hypothesis that past performance is no guide to future performance (e.g. Jensen 1969; Kahn and Rudd 1995).