8. Results
8.5 Multifactor Model
62 Norge Verdi. The findings of the passive funds implies that the fund manager’s contribution to the value is zero.
The estimated beta-values varies from 0.7999 to 1.0231. Only four funds have an estimated beta value over 1, whereas the remaining funds have a value below, indicating that the funds overall have a lower degree of systematic risk, known as market risk, compared to the market.
On the other hand, not all of these estimations are statistically significantly different from 1.
Of the 22 funds with beta estimations below 1, there are 13 funds significantly different from 1, whereas 11 on a 1% significance level. The low beta-values can be seen in parallel with not having the possibility of loans to strengthen their position or with the sufficient amount of cash the funds must hold to cover potential redemption shares. A position in foreign stocks also plays a part to mislead the result.
The r-squared illustrates to what degree the funds’ return can be explained by the marked return and is defined as one minus explained variation over total variation. A large r-squared is symbolic to a diversified fund, as a value close to one would indicate low unsystematic risk.
As seen from table 15, the interval of our funds r-squared is spread from 0.7879 to 0.9793.
Nordea Kapital is the active fund with the highest r-squared, 0.9739, justifying their approach to invest broadly in the Norwegian stock market. The lowest r-squared belongs to Pareto Aksje Norge B with 0.7879. This fund has a specified stock picking strategy, which means less diversification, matching their low r-squared.
The results obtained by the Single Index model reflect the findings of prior Norwegian, European and American studies, as the majority of our funds lack the ability to realize significantly positive, or negative, alpha estimates, indicating no superior performance contributed by the fund managers (Otten et al, 2002) (Fama & French, 2010).
63 running the regression connected to Carhart’s 4-factor model, we are to test the robustness of the underlying data.
8.5.1 Test of Robustness
The results from the heteroscedasticity, autocorrelation and normality tests are presented in table 16. Additionally, the data sample is tested for multicollinearity, as the regression model includes multiple independent variables. As for the Single Index model, the p-values and critical values are based on the null hypothesis of “no autocorrelation”, “homoscedasticity”, and “normally distributed error term”.
Table 16. Robustness test of Carhart’s 4-factor model (own contribution based on test results) To investigate for heteroscedasticity, the Spearman Ranks correlation test is applied. The test instructs us to look for heteroscedasticity between each of the variables set up against the residuals. It appears from the test result that there are four indications of heteroscedasticity on a 5% significant level. As for the Single Index model, it will not be considered a problem for our multifactor-model, as the test results are fairly close to homoscedasticity.
Fund name Active funds:
Alfred Berg Aktiv 0.7578 0.7351 0.0234 * 0.3783 1.5864 * 0.6321
Alfred Berg Gambak 0.5779 0.5661 0.1697 0.4649 1.8709 0.0389*
Alfred Berg Humanfond 0.0329 * 0.3911 0.8531 0.1107 1.9991 0.0337*
Alfred Berg Norge Classic 0.9864 0.8178 0.3245 0.3166 1.7085 0.4830
C WorldWide Norge 0.7795 0.7346 0.6997 0.8132 1.6544 0.3781
Danske Invest Norge I 0.2590 0.6440 0.5562 0.9635 2.1914 0.3387
Delphi Norge A 0.6755 0.4379 0.4097 0.8051 2.0019 0.0550
DNB Norge (IV) 0.9956 0.1843 0.0232 * 0.5197 2.0620 0.0258*
Eika Norge 0.5783 0.8417 0.6763 0.2231 1.9768 0.0031**
Fondsfinans Norge 0.5790 0.6176 0.0564 0.8938 2.1680 0.0789
Handelsbanken Norge 0.7530 0.2972 0.0739 0.3212 1.6065 * 0.1818
Holberg Norge 0.6129 0.6728 0.6172 0.2690 1.7874 0.5795
KLP AksjeNorge 0.5116 0.6700 0.6300 0.0660 1.7875 0.1035
Nordea Avkastning 0.4739 0.7006 0.4123 0.1846 1.9147 0.6730
Nordea Kapital 0.7757 0.8421 0.8357 0.8054 2.0379 0.1430
Nordea Norge Verdi 0.8452 0.2809 0.0986 0.7294 2.1178 0.3592
ODIN Norge C 0.1298 0.4068 0.5132 0.8147 1.9192 0.1046
Pareto Aksje Norge B 0.2271 0.9294 0.6577 0.8347 2.0499 0.8892
Pareto Investment Fund A 0.1617 0.7398 0.0241 * 0.4200 1.8174 0.0429*
Storebrand Norge 0.7288 0.3584 0.9814 0.8269 1.9075 0.0000**
Index funds:
Alfred Berg Index Classic 0.9011 0.8745 0.1031 0.4130 1.8423 0.0211 *
DNB Norge Indeks 0.2351 0.8715 0.4293 0.5089 1.6636 0.2401
KLP AksjeNorge Indeks II 0.7768 0.9058 0.0524 0.8609 1.7912 0.0501
Nordnet Superfondet Norge 0.9823 0.3325 0.5767 0.4708 2.3014 0.8503
PLUSS Indeks 0.3627 0.7149 0.5435 0.8383 2.0129 0.0065 **
Storebrand Indeks - Norge A 0.9120 0.8523 0.7813 0.7130 2.1299 0.6845
H0 rejected on a 5 % significance level * H0 rejected on a 1 % significance level **
Heteroscedasticity Autocorrelation Normality
p market p SMB p HML p UMD DW p-value
64 The appearance of autocorrelation is also tested through the Durbin-Watson test in our multifactor model. From three indications in the Single Index model to two indications when including more variables. Handelsbanken Norge increasing its DW value from 1.5967 to 1.6065 but is still below the limit of 1.654. Alfred Berg Aktiv decreased its DW value from 1.591 to 1.5864. The appearance of two indications, relatively close to the line of inconclusive autocorrelation and accepted by the rule of thumb, will not lead to any corrections.
Normal distribution among the residuals is tested through the Skewness/Kurtosis test. Having a total of 12 funds rejecting the hypothesis of normal distribution in the Single Index model, whereas half on a 1% significant level, the amount has decreased in the multifactor model. The test results show that eight funds reject the null hypothesis of a normal distribution, whereas only three on a 1% significance level. However, we assume that the number of observations is sufficiently large to provide reasonably approximations, despite the indications of non-normal distributions, in congruence with the Central Limit Theorem. Conversely, Storebrand Norge is left out of when analyzing the results, as the indications of non-normal distribution are above a confidence level of 99% in both the Single Index model and Carhart’s 4-factor model.
Storebrand Norge’s distribution is presented in figure 7, which demonstrates significant deviations from the normal distribution curve.
Figure 7. The distribution of Storebrand Norge (own contribution using Stata)
The variance inflation factor (VIF) is applied to evaluate the appearance of multicollinearity in our model. The maximum acceptable VIF value is argued to be 5 (Ringle, Wende, & Becker,
01020304050Density
-.1 -.05 0 .05 .1
Resid20
Kernel density estimate Normal density
Kernel density estimate
65 2015) and the minimum tolerance (TOL) is 0.20. Table 17 presents the results of the calculations made in Stata:
Table 17. Values to test for multicollinearity (own contribution based on calculations in Stata) The test results demonstrate that all estimated values for our factors are below the VIF limit of 5, and thereof, below the TOL limit 0.20, indicating that the independent variables in our model are unaffected by multicollinearity.
8.5.2 Test of Regression
Table 18 presents the estimated coefficient of the regression based on Carhart’s 4-factor model, where OSEFX is used as a proxy for the market and where the SMB-, HML- and UMD-factor expresses respectively, size premium, value premium, and momentum premium. Additionally, the table includes the p-value corresponding to the estimated alpha values, whereas the p-value is tested on the null hypothesis of 𝛼𝛼 equals zero. A significant positive alpha value indicates that the asset manager adds value beyond allocating investments to the four factors.
Variable VIF TOL
Marketr 1.65 0.6073
SMB 1.52 0.6573
HML 1.01 0.9868
UMD 1.11 0.8974
Mean 1.32 0.7872
66 Table 18. Regression of Carhart's 4-factor model: monthly estimates (own contribution based on test results)
An advantage of using Carhart’s 4-factor model approach is that the model has a higher degree of explanatory power, as it includes a higher number of independent variables. The adjusted r-squared is considered as the appropriate measure of the explanatory power of the multifactor model, as it adjusts for the number of independent variables. The value of the average r-squared generated from the single index model is 0.9155, compared to the average adjusted measure of 0.9229 delivered by the 4-factor regression model. The multifactor regression model has increased the explanatory power for the majority of the mutual funds, except for three active funds. Overall, these findings justify the statement of higher explanatory power.
Further, the number of funds that deliver positive alpha values are equivalent to the Single Index model. For two of the active funds, the alpha estimate changes from positive to negative, while the estimate for two of the passive funds changes from negative to positive. However, the alpha estimates provided by the multiple regression model are reduced for all active funds, except for DNB Norge (IV). A potential cause is that positive alpha estimates assessed by the
Fund name α p-value βMarket SMBi HMLi UMDi R2Adj
Active funds:
Alfred Berg Aktiv 0.0004 0.7278 1.0494 0.1399 -0.0035 0.0772 0.9358
Alfred Berg Gambak 0.0012 0.4702 1.0145 0.2074 -0.0054 0.1354 0.8630
Alfred Berg Humanfond -0.0012 0.2140 0.9816 0.0479 0.0097 0.0666 0.9512
Alfred Berg Norge Classic 0.0009 0.2000 0.9932 0.0640 0.0015 0.0478 0.9715
C WorldWide Norge -0.0007 0.4269 0.9966 -0.0019 -0.0371 0.0441 0.9662
Danske Invest Norge I 0.0002 0.8116 0.9895 0.0434 -0.0187 0.0128 0.9631
Delphi Norge A -0.0012 0.4789 1.0785 0.2179 0.0015 0.0607 0.8648
DNB Norge (IV) -0.0004 0.6599 0.9428 0.0010 -0.0214 -0.0233 0.9585
Eika Norge -0.0029 0.0307 * 1.0912 0.2129 0.0482 0.0105 0.9184
Fondsfinans Norge 0.0005 0.7870 1.0863 0.2277 0.0477 -0.0315 0.8699
Handelsbanken Norge 0.0000 0.9766 1.0666 0.1547 -0.0143 0.0728 0.8889
Holberg Norge -0.0032 0.0483 * 1.0337 0.2561 0.0422 0.0123 0.8718
KLP AksjeNorge -0.0006 0.5257 1.0578 0.0905 0.0024 -0.0207 0.9628
Nordea Avkastning 0.0001 0.9284 1.0321 0.0677 -0.0057 0.0184 0.9700
Nordea Kapital 0.0005 0.4635 1.0130 0.0528 -0.0179 0.0145 0.9746
Nordea Norge Verdi 0.0028 0.0585 0.8718 0.1588 0.1023 -0.0031 0.8544
ODIN Norge C -0.0026 0.1359 0.9477 0.2113 0.0491 0.0197 0.8338
Pareto Aksje Norge B -0.0010 0.5854 0.9326 0.2342 0.0413 -0.0508 0.8099
Pareto Investment Fund A 0.0003 0.8781 1.0471 0.0544 -0.0767 0.0640 0.8664
Storebrand Norge 0.0005 0.7048 0.9511 0.0150 -0.0150 0.0626 0.8935
Index funds:
Alfred Berg Index Classic 0.0007 0.2807 0.9033 -0.0412 0.0232 -0.0097 0.9793
DNB Norge Indeks* 0.0005 0.4179 0.9356 -0.0288 0.0492 -0.0142 0.9777
KLP AksjeNorge Indeks II 0.0006 0.3203 0.9095 -0.0418 0.0243 -0.0126 0.9800 Nordnet Superfondet Norge* 0.0002 0.8983 1.0350 0.0053 0.0986 -0.0153 0.9361 PLUSS Indeks (Fondsforvaltning) 0.0007 0.4054 0.8914 -0.0833 0.0106 -0.0427 0.9582 Storebrand Indeks - Norge A* -0.0001 0.9347 1.0237 0.0179 0.0713 -0.0037 0.9753
* p < 0.05, **p < 0.01
67 Single Index model, can be explained by other factors that are not taken into consideration by the model. By running the multifactor regression approach, the average active return obtained per month has been reduced from 0.04% to -0.01%.
Additionally, the results show that there are only four active and two passive funds that show no significant exposure to neither of the added factors in the 4-factor model. These findings imply that parts of the active return reported by the Single Index model may be a result of exposure to either SMB, HML, UMD or a combination including the market risk. The active fund Alfred Berg Gambak is an example of this incident, where the estimated alpha value has been reduced from 0.0035 (Single Index) to 0.0012 (Carhart’s 4-factor), and the estimated coefficients of the SMB and UMD factors are statistically significantly different from zero on a 1% significance level. As for the passive funds, the active return has increased, according to the 4-factor model. The only exception is Storebrand Indeks – Norge A, where the estimated active return is slightly reduced. Despite these results, Eika Norge and Holberg Norge are the only funds that deliver statistically significant alpha values, on a 5% significance level. As these estimates are negative, they imply that the fund managers destroy value rather than creating it.
The estimated beta coefficient has increased for 21 the funds, whereas the estimates vary from 0.8752 to 1.0912. This signifies a higher systematic risk than the one estimated by the Single Index Model. However, the majority of the beta estimates are not significantly different from 1, on neither a 1% nor 5% significance level shown in Appendix 2. A potential explanation is that some of the return assigned to the market, are instead caused by exposure to the SMB, HML or UMD factors. Therefore, it is reasonable to assume that the beta coefficient is close to 1, which supports the low number of statistically significant estimates.
Assessing the exposure of the added variables to the active funds, we need to see the p-values.
68 Table 19. P-values of the funds' exposure to the factor variables (own contribution)
The results in table 19 correspond to the findings of Næs et al. (2007), suggesting little evidence of HML in the Norwegian market. Only Nordea Norge Verdi in our sample has significant exposure to the HML factor on a 5% significance level. Value is specified as the fund’s “route”
and indicates a portfolio consisting of undervalued stocks in price, believed to eventually see their worth recognized by the market (Morningstar, 2019). Regarding the UMD factor, we find stronger evidence. The UMD is a significant factor used in all Alfred Berg’s funds, where growth is central, and stocks believed to grow faster than the market. The SMB factor in a portfolio indicates long positions in small companies and short positions in large companies (Næs et al., 2007), which seem to be the strategy for nearly all of our funds.
The overall results of the multifactor model are that the active return, measured by the alpha estimate, is not statistically significantly different from zero. These findings apply to both active and passive funds. Therefore, we can conclude that fund managers do not deliver abnormal returns through active management. The results are similar to the studies of Sørensen (2009), Otten et al. (2002) and Kosowski et al. (2006), respectively for the Norwegian, the European and the American mutual funds market.
Fund name Morningstar
Style Box
Alfred Berg Aktiv 0.0002 ** 0.9133 0.0031 ** Med/Grow
Alfred Berg Gambak 0.0001 ** 0.9053 0.0003 ** Med/Grow
Alfred Berg Humanfond 0.1232 0.7216 0.0024 ** Med/Grow
Alfred Berg Norge Classic 0.0076 ** 0.9435 0.0043 ** Med/Grow
C WorldWide Norge 0.9431 0.1160 0.0189 * Med/Grow
Danske Invest Norge I 0.1160 0.4389 0.5010 Med/Mix
Delphi Norge A 0.0001 ** 0.9762 0.1183 Med/Grow
DNB Norge (IV) 0.9734 0.3961 0.2443 Med/Grow
Eika Norge 0.0000 ** 0.2064 0.7261 Med/Mix
Fondsfinans Norge 0.0001 ** 0.3349 0.4203 Med/Value
Handelsbanken Norge 0.0025 ** 0.7459 0.0389 * Med/Mix
Holberg Norge 0.0000 ** 0.3521 0.7308 Small/Mix
KLP AksjeNorge 0.0024 ** 0.9256 0.3085 Med/Mix
Nordea Avkastning 0.0087 ** 0.7983 0.2987 Med/Mix
Nordea Kapital 0.0237 * 0.3800 0.3687 Med/Mix
Nordea Norge Verdi 0.0013 ** 0.0174 * 0.9262 Small/Value
ODIN Norge C 0.0002 ** 0.3135 0.6077 Med/Mix
Pareto Aksje Norge B 0.0002 ** 0.4337 0.2243 Med/Value
Pareto Investment Fund A 0.3431 0.1296 0.1096 Small/Mix
Storebrand Norge 0.7460 0.7136 0.0541 Med/Grow
* p < 0.05, **p < 0.01
SMB HML UMD
69