Persistence of Performance

3.6 Methodology

3.6.4 Persistence of Performance

73

study utilizes the classic market timing regression of Treynor and Mazuy (1966), which expresses the regression in a quadratic form as below;

𝑅_𝑖𝑡 = 𝛼_𝑖 + 𝛽_1𝑖(𝑅𝑀_𝑡− 𝑅_𝑓𝑡) + Ƴ(𝑅𝑀_𝑡− 𝑅_𝑓𝑡)²+ 𝜀_𝑖𝑡 (10)

Where;

𝑅_𝑖𝑡 = return of fund 𝑖 at time t in excess of the risk free rate 𝛼_𝑖 = abnormal return of the fund

𝛽_1𝑖 = beta of the fund 𝑅𝑀_𝑡 = return of the market 𝑅_𝑓𝑡 = the risk free rate

Ƴ = market timing coefficient 𝜀_𝑖𝑡 = the error term.

The sign on the estimated coefficient Ƴ of the quadratic term, and whether it is statistically different from zero, evaluates market-timing ability. If it is significantly positive, then it represents a convex upward sloping regression line and indicates evidence of successful market timing by the fund manager. Thus, the coefficient will be positive if the manager increases beta when receiving a positive signal about the market. The hypothesis of no timing ability, implies that the coefficient Ƴ on the quadratic term is zero or negative. The market proxy for this model is the funds’

specific equity style benchmarks.

74

relationship between rankings of performance on those of initial periods over subsequent periods (Carhart, 1997, Huij and Lansdorp, 2012).

Performance persistence is therefore very important in portfolio management, since it differentiates the winners from the losers over a given time period, a key element in explaining the flow of funds from underachieving to skilled fund managers (Barberis and Shleifer, 2003). The study of persistence is to determine whether managers can systematically beat the market over time. To do so, after choosing and applying the performance measurement methods, one has to classify or rank the funds. Using statistical tools, it suffices to study the distribution of these rankings to reach a conclusion about the persistence of this performance (El Khamlichi et al., 2014b). To analyse the persistence of performance, two types of tests are conventionally used:

1. Parametric tests: using time series (Goetzmann and Ibbotson, 1994) or regression (Christopherson et al., 1998); and

2. Nonparametric tests (based on contingency tables) which are proposed for use in this analysis. This method has been widely used, throughout relevant literature, to assess performance persistence of portfolios, for example, Firer et al. (2001), Fletcher and Forbes (2002) and also Clark (2013).

3.6.4.1 Contingency Table

Formally, the contingency table approach is defined as a method used to establish the frequency with which funds are described as winners and losers over consecutive time periods (Thomas, 2012). For each classification of the funds respectively, contingency tables are applied on the basis of performance assessment results, or alphas, to determine the degree of persistence. The funds are apportioned into two classes, Winner (W) or Loser (L), based on the median abnormal return over the relevant ranking period. Over two consecutive time periods P1 and P2, a two by two table is formed so a fund can have one of four outcomes or quartiles, (WW), (WL), (LW) or (LL).The contingency table displays the probability of a portfolio in one quartile being in the exact same quartile in the subsequent period (Eling, 2009).

75

Supposing pure random performance, one would envisage these probabilities to be a quarter, that is, 25%. This means that, there is an identical prospect of a top quartile portfolio winding up in any of the four quartiles in the following investment period. Such occurrence is based on the premise that the previous evaluation period does not have an effect on the future period (Hereil et al., 2010). Winner (or loser) designation mainly defines a fund that achieved a rate of return, across the calendar year, that exceeds (or falls short of) the median fund return. Therefore W (winner) represents returns above the median abnormal return, whereas L (Loser) represents returns below the median abnormal return (Clark, 2013). WW refers to a fund being a winner this period and the next; LL is a loser fund this period and the next period;

WL is a winner this period followed by being a loser the next period and LW is a loser this period, then a winner next period. For two subsequent sub-periods (P1 and P2), a contingency table like the following one is obtained:

TABLE 3-2: Contingency table for testing performance persistence

P2

P1

Performance above the median value

Performance below the median value

Tests to be conducted

Performance above the median value

Winners funds (WW)

Variable Performance (WL)

Chi-square

Performance below the median value

Performance variable

(LW)

Losers Funds

(LL)

CPR, Z-test and Chi-square

Source: (Brooks, 2014) pp 287

76

To analyse the robustness of the phenomenon of persistence, several statistical tests are used. The following two statistical procedures are most commonly found in the literature used, together with contingency tables, to test for performance persistence and robustness of the contingency table method. They are the Cross Product Ratio and the Chi-squared test (Norma et al., 2012).

3.6.4.2 Cross Product Ratio

The Cross Product Ratio (CPR), also known as Odds ratio, is a non-parametric method established by Brown and Goetzmann (1995). The fundamental idea is based on performance evaluation; hence the CPR outlines the odds ratio of the number of repeat performers against those that do not repeat. In detail, the Cross Product Ratio calculates the ratio of ‘Persistence’ (WW & LL) versus ‘reversal’ (WL &

LW) using the formulae:

𝐶𝑃𝑅 = (𝑊𝑊 × 𝐿𝐿)

(𝑊𝐿 × 𝐿𝑊) (11)

Where;

𝑊𝑊 = number of winner funds in both formation and holding periods 𝐿𝐿 = number of losers in both periods

𝑊𝐿 = number of winners then losers 𝐿𝑊 = number of losers then winners.

The significance of the deviations of Cross Product Ratio from unity is then tested. If the test statistic is significantly positive, then it provides evidence of persistence in performance. A significantly negative test statistic provides evidence of reversals in performance. In other words, the study observes whether the CPR is above or below one. If the CPR is significantly higher than one (equivalent to a positive t-statistic), it indicates persistence, that is, winners followed by winners, or losers followed by losers (Joaquim and Moura, 2011). Conversely, a CPR lower than one (equivalent to a negative t-statistic) indicates a reversal, that is, winners followed by losers, or

77

losers followed by winners. A reversal, in essence, refers to a ‘return reversal’

situation where WW*LL is less than WL*LW in Equation (11) above.

Therefore, a Cross Product Ratio above one signifies evidence of persistence, a CPR of one means no evidence of persistence is observed and a CPR below one signifies reversals of performance. The study, hence, tests the null hypothesis that there is no significant persistence, which must be equivalent to a CPR of one. This is on the basis that under the null hypothesis, the probability of winning or losing in each period equals one-half and does not depend on the return horizon (Liwei and Peng, 2012). In that sense, the four quartiles; Winner-Winner (WW), Loser-Loser (LL), Winner-Loser (WL) and Loser-Winner (LW), each has 25% of the funds.

Basically the test is as follows:

𝐻₀: 𝐶𝑃𝑅 = 1 Or ln 𝐶𝑃𝑅 = 0 (12) The statistical significance of the Cross Product ratio was tested using the Z-test, which follows a standard normal distribution. This test allows the significance of the deviations of Cross Product Ratio from unity to be tested. A Z-statistic test can be implemented as outlined:

𝑍 =

ln (𝐶𝑃𝑅) 𝜎_{ln (𝐶𝑃𝑅)}

√𝑛 ~ 𝑁 (0,1) (13)

Where the standard error of the natural logarithm of the CPR is given by:

𝜎_{ln(𝐶𝑃𝑅)} = ( 1

𝑊𝑊+ 1

𝑊𝐿+ 1 𝐿𝑊+ 1

𝐿𝐿)¹² (14)

A Z-statistic of 1.96 corresponds to a 5 percent significance level, that is, when the Z-statistic is higher than 1.96, the null hypothesis of no persistence is rejected at the 5 percent significance level. If the Z- test statistic is significantly positive, then it

78

provides evidence of persistence in performance. A significantly negative test statistic provides evidence of reversals in performance.

3.6.4.3 Chi-Square Test

In fulfilment of the last objective of the study, which tests for performance persistence, the study further tests the predictability of future returns based on past performances. The chi-squared test will be used for this purpose. Since the study follows the non-parametric approach of employing contingency tables, similar to the studies of Yu (2008), Clark (2013), it will conduct the chi-squared test with 1 degree of freedom as follows:

CHI =

(WW- ^N

4)²+ (WL- ^N

4)²+ (LW- ^N

4)²+ (LL- ^N

4)² (^N

4)

(15)

Where N represents the number of observations in the analysis. A positive, statistically significant chi-squared statistic supports the hypothesis that abnormal past performance can be used to predict future abnormal performance.

Contrastingly, a negative, or statistically insignificant statistic, suggests that future returns cannot be predicted from past performances. Prediction of future performances is of interest to portfolio managers and investors alike, since they anticipate future market movement in order to earn positive returns, hence the result from the chi-square statistic is of paramount importance. It should be noted that the chi- squared statistic is premised on the persistence of performances and thus the chi-squared test completes the study’s last objective.