2.1 Benchmark alphas

We have estimated the alphas of each benchmark using 4-, 5- and 6-factor models, in line with Carhart (1997) and Fama and French (2015, 2018). The 6-factor specification is written as:

where 𝑅𝑏_𝑡 − 𝑅𝑓_𝑡 is the excess return (over the risk-free rate) of the benchmark in period t and 𝑅𝑚_𝑡 − 𝑅𝑓_𝑡 is the excess return of a broader market index during the same period. 𝑆𝑀𝐵_𝑡, 𝐻𝑀𝐿_𝑡, 𝑅𝑀𝑊_𝑡 and 𝐶𝑀𝐴_𝑡 are the Fama and French (2015) size, book-to-market, profitability, and investment factors, respectively, and 𝑊𝑀𝐿_𝑡 is the Carhart (1997) momentum factor. The 5- factor alpha is estimated excluding the momentum factor (𝑊𝑀𝐿_𝑡 ) from equation [1], while the 4-factor alpha is estimated excluding the investment (𝑅𝑀𝑊_𝑡) and profitability (𝐶𝑀𝐴_𝑡) factors from the same equation3.

2.2 Unadjusted and benchmark-adjusted fund alphas

Unadjusted and benchmark-adjusted mutual fund alphas are also estimated with 4-, 5- and 6-factor models. The unadjusted (𝛼_𝑝) 6-factor alphas are based on the following equation:

where 𝑅𝑝_𝑡 − 𝑅𝑓_𝑡 is the excess return of fund p in period t and the remaining variables are as described previously. After this, we estimated the 6-factor benchmark-adjusted alphas ( ${\alpha }_p^{*}$ ) using the following formula:

where 𝑅_𝑝,𝑡 − 𝑅_𝑏,𝑡 is the excess return of fund p over its benchmark in period t. In equation [3], ${\alpha }_p^{\ast }$ is the difference between the alpha of the fund (𝛼_𝑝) and the alpha of the benchmark (𝛼_𝑏) and ${\beta }_1^{\ast }$ is the difference between the market beta of the fund (𝛽_1𝑝) and the market beta of the benchmark (𝛽 _1𝑏 ). Similar interpretations apply to the following parameters4.

2.3 Crisis and non-crisis phases

To investigate whether benchmark and mutual fund performance differ across different market conditions, we began by detecting the crisis / bear-market periods based on the econometric algorithm of Pagan and Sossounov (2003). The peaks / troughs of each benchmark index occur when they are the highest / lowest values in an eight-month window of surrounding values. Hence, with Pt representing the value of the index, a trough occurs at time t if 𝑙𝑛(𝑃_𝑡−8,..., 𝑃_𝑡−1) > 𝑙𝑛(𝑃_𝑡) < 𝑙𝑛(𝑃_𝑡+1,..., 𝑃_𝑡+8) and a peak occurs at time t if 𝑙𝑛(𝑃_𝑡−8,..., 𝑃_𝑡−1) < 𝑙𝑛(𝑃_𝑡) > 𝑙𝑛(𝑃_𝑡+1,..., 𝑃_𝑡+8). After this, all falls of at least 20% from peak to trough are classified as crisis / bear-market phases, whereas the remaining periods are considered non-crisis / bull-market phases.

We have used the PSI and the FTSE Eurofirst 100 as they are suitable stock market indices for our study. Over the period 2000-2020, we have identified six crisis periods for the PSI index and four for the FTSE Eurofirst 100 index, which are detailed in Table 1 below.

For both indices we can observe crisis periods in the early 2000s, related to the dot-com bubble crash, from the second half of 2007 to the beginning of 2009, associated with the global financial crisis, and during the first quarter of 2020, following the Covid-19 pandemic. The remaining periods (1 for the European index, in 2011, and 3 for the Portuguese index, covering from the last quarter of 2010 to the first half of 2016) are mostly linked to the Euro-area debt crisis, which also impacted the equity sector. As Portugal was one of the most affected countries, requiring an official bailout programme financed by the European Central Bank, the European Commission, and the International Monetary Fund, it may justify why the total length of the crisis period is considerably higher for the Portuguese index (100 months) than for the European index (63 months).

Following Leite and Cortez (2015), we have obtained alpha and beta estimates in the crisis and non-crisis phases separately by adding two dummy variables to all the previous equations. Based on the 6-factor model, we have obtained the benchmark alphas, the unadjusted fund alphas, and the benchmark-adjusted fund alphas through equations [4], [5] and [6], respectively:

where ${\mathit{\text{F}}}_{i,t}$ represents each of the six risk factors described earlier, ${\mathit{\text{D}}}_{\mathit{\text{NC}}}$ ( ${\mathit{\text{D}}}_{\mathit{\text{C}}}$ ) equals 1 for the non-crisis (crisis) phases and 0 otherwise. In equation [4], ${\alpha }_{b\mathit{,NC}}$ and ${\alpha }_{b\mathit{,C}}$ are the non-crisis and crisis period benchmark alphas, respectively. In equations [5] and [6], ${\alpha }_{p\mathit{,NC}}$ ( ${\alpha }_{p\mathit{,C}}$ ) and ${\alpha }_{p\mathit{,NC}}^{*}$ ( ${\alpha }_{p\mathit{,C}}^{*}$ ) are the non-adjusted and benchmark-adjusted fund alphas for the non-crisis (crisis) phases, respectively. The remaining parameters are straightforward to interpret5.

4.1 Benchmark alphas

Since benchmarks should be broad passive indices, they should not generate abnormal returns. However, if a positive (negative) performance is verified, it must be represented by a positive (negative) alpha. Figure 1 presents the alphas obtained for the domestic and European benchmarks, expressed in basis points per year, using 3-year moving averages and the 4-factor model of Carhart (1997), for the entire evaluation period12.

Figure 1.Evolution of Benchmark Alphas 

We have found evidence of consistently negative alphas for both benchmarks for most of the sample period. Comparing the domestic and European benchmarks, we can also see that these negative alphas were not only considerably more frequent, but also of a notably larger magnitude for the PSI than for the FTSE Eurofirst 100.

To explore this topic in greater detail, Table 3 shows performance (monthly and yearly alphas) and risk estimates for the domestic (PSI) and European (FTSE Eurofirst 100) benchmarks in the period 2000-2020. We used three alternative models: the 6-factor model (FF6) of equation [1]; a 5-factor variant (FF5) that omits the momentum factor; a 4-factor version (CH4) that excludes investment and profitability factors. Besides the total sample period, results are also presented for the crisis and non-crisis phases separately, following equation [4]13.

For the total sample period, the results show the existence of negative and statistically significant alphas for both benchmarks, reaching -143.53 basis points per year for the PSI index and between -111.03 and -124.33 basis points per year for the FTSE Eurofirst 100 index, in accordance with Cremers et al. (2013) and Mateus et al. (2016). Therefore, if mutual fund performance is evaluated against these benchmarks, fund alphas will have a downward bias. What is more, in comparison to the broader market index, the European (but not the domestic) benchmark exhibited clear large cap biases, as expected.

The results obtained separately for the market crisis and non-crisis phases show a substantial contrast between both benchmarks. Indeed, for the domestic benchmark, the alphas were significantly negative during the crisis periods, reaching -295.33 basis points per year, but became neutral during the non-crisis periods. For the European benchmark, the alphas were significantly negative during the non-crisis periods, varying between -142.11 and -167.54 basis points per year, but were neutral during the market crises in all the multifactor models used.

This prevalence of significantly negative benchmark alphas in the sampled period aligns with the results of Mateus et al. (2016). However, while these authors found that the performance of the domestic benchmark decreases during non-crisis periods, we have only observed the pattern occurring for the European (as opposed to the domestic) index. Hence, the evidence of fund underperformance may lie in model specification errors, which lead to negative reference alphas.

4.2 Unadjusted fund alphas

Table 4 shows the performance (monthly and yearly unadjusted alphas) and risk estimates for equally weighted portfolios of the domestic and European funds in the period 2000-2020 using standard / unadjusted performance evaluation models. We used three alternative models: the 6-factor model (FF6) of equation [2]; a 5-factor variant (FF5) that omits the momentum factor; a four-factor version (CH4) that excludes investment and profitability factors. Besides the total sample period, the results are also shown for the crisis and non-crisis phases separately, following equation [5].

The results show that the model has high explanatory power, with values of approximately 92% in both portfolios, which indicates that the right choice was made for the benchmarks. For the total period under evaluation, the unadjusted alpha of the domestic portfolio was negative (-55.78 basis points per year) but not statistically significant, indicating a neutral performance, in line with Leite and Cortez (2009), but unlike both Angelidis et al. (2013) and Mateus et al. (2016), who found negative and statistically significant unadjusted alphas for domestic funds domiciled in the US and UK markets. In contrast, the European funds exhibited significantly negative alphas in all cases, consistent with Leite and Cortez (2020), with values ranging between -463.49 basis points and -552.04 basis points per year. Additionally, in terms of investment styles, both fund portfolios exhibited significant exposure to small caps, but no clear value or growth tilts. The coefficient of the momentum factor was negative and statistically significant only for the domestic funds, suggesting contrarian strategies. For the European funds, there was also significant exposure to firms with low profitability.

Upon reviewing the alphas obtained for the crisis and non-crisis phases separately, we can see that, for both fund categories, unadjusted alphas were higher for the market crisis phases than for the market non-crisis phases, contrary to Mateus et al. (2016). However, while the domestic funds exhibited neutral performance during the two market phases, the European funds showed neutral performance for market crises and significant underperformance for the non-crisis phases. Consequently, the results for the European funds may be due to the lower benchmark alphas during the non-crisis phases, which underestimated fund alphas in the standard models. For the domestic funds, benchmark alphas were significantly lower during the crises, meaning that fund performance may be undervalued, especially during more turbulent times.

The factor coefficients of the domestic funds are similar for the market crisis and non-crisis phases, except for the significantly negative coefficient of the momentum factor, which was only found during the crises. For the European funds, we noted higher market betas, as well as higher exposure to small caps, weaker profitability and higher investment firms, for non-crisis phases than for crisis phases.

4.3 Benchmark-adjusted fund alphas

Table 5 shows performance estimates (yearly alphas, in basis points) before and after the benchmark adjustment procedure for equally weighted portfolios of domestic and European funds for the period 2000-2020. The benchmark-adjusted alphas were based on three model specifications: the 6-factor model (FF6) of equation [3]; a 5-factor variant (FF5) that omits the momentum factor; a four-factor version (CH4) that excludes the investment and profitability factors. The excess betas for each factor can also be seen. The unadjusted alphas were based on equation [2] and its variants. Besides the total sample period, the results are also shown for the crisis and non-crisis phases separately, following equation [6].

For the full sample period, we found a statistically significant increase in the alpha estimates (at the 1% level) when switching from unadjusted to benchmark-adjusted alphas, for both fund categories and regardless of the performance evaluation model applied. Although we have evidence of neutral performance for the domestic funds, the alphas increased by 144.70 basis points, soaring from -55.78 to 88.92 basis points per year. This increase of, approximately, 145 basis points per year was considerably higher than the 86 basis points registered by Angelidis et al. (2013) for US funds and the 127 basis points reported by Mateus et al. (2016) for UK funds. Therefore, the difference between pre- and post-adjustment alphas seems to be higher for funds domiciled in smaller markets. The European funds still exhibited significantly negative alphas, but these increased by between 106.49 and 120.41 basis points per year. This means that the differences between unadjusted and benchmark-adjusted alphas were higher for the funds investing in local stocks than for the ones investing abroad14.

With reference to the excess betas, we can see that both the domestic and European funds were significantly more exposed to small caps than the PSI or the FTSE Eurofirst indices, in line with Table 4. Additionally, compared to their benchmarks, the domestic funds show significantly lower loadings on the momentum factor, while the European ones exhibited significantly lower exposure to the profitability factor.

When comparing the results between the crisis and non-crisis phases, we uncovered some interesting results, especially concerning the contrast between fund categories. For the non-crisis periods, for both the domestic and the European funds, we can observe significant differences at the 1% level between pre- and post-adjustment alphas. In this case, the benchmark-adjusted alphas were significantly higher than the unadjusted alphas by 128.78 basis points per year for the domestic portfolio and between 134.62 and 161.60 basis points per year for the European portfolio. The higher differences for the European funds may be because only the European benchmark displayed a significantly negative alpha during the non-crisis phases, while the domestic benchmark exhibited a neutral alpha, thereby underestimating the unadjusted alphas by a greater margin. As a result, while the European funds exhibited higher differences in the alphas for the non-crisis phases than for the overall period, for the domestic funds, we found the opposite to be true.

During the crises, the benchmark-adjusted alpha of the domestic fund portfolio was significantly higher than the unadjusted alpha, at the 1% level, with the difference reaching a noteworthy 305.50 basis points per year. This result is undeniably related to the considerably lower (and statistically significant) benchmark alphas obtained for the domestic benchmark during the crisis periods. Conversely, for the European funds, there were no significant differences between unadjusted or adjusted alphas during the market crises in any of the model specifications used, in line with the neutral benchmark alphas obtained during these periods15.