4. The consistency of the size premium with the ICAPM
4.2. The SMB* as a possible state variable
4.2.1. The SMB* in the different BM states
Different from the previous results using BMt as a state variable, there is uncondi-tional evidence supporting the intertemporal risk explanation of the size premium consider-ing SM B∗ as a state variable. But after conditioning on the BM state of the economy the evidence usually disappears. This happens because the positive relation betweenSM B∗ and the investment opportunities tends to be restricted to the low and medium BM states as we saw in details in Table 5. These are the BM states where the size premium is insignificant.
Consistent with the data, the unconditional ICAPM predicts a positive size premium in the full 1926-2014 sample and after 1960, as we see in Table 8. The unconditional results in the first column of Table 8 show that the covariance risk price with the innovations in SM B∗, γz in Eq. (7), is positive in the full sample and in each subsample. The covariance of the innovations in SM B∗ and the return on the SMB portfolio in Eq. (21) is significantly positive in the 1926-2014 sample and before 1960. Hence, the ICAPM predicts a positive size premium consistent with the data in those samples. Before 1960 the unconditional covariance in Eq. (21) is insignificant, so the ICAPM predicts no size premium, which is again in line with the data.
[Place Table 8 about here]
Conditionally, the risk price γz in Eq. (7) is significantly positive in high BM states and especially in low BM states in the full 1926-2014 sample, as we see in Table 8. Therefore, the positive covariance of the return on the SMB portfolio and the innovations in SM B∗ in Eq. (21) implies a positive size premium in high BM states and especially in low BM states according to the ICAPM. But while this prediction is consistent with the positive size premium in high BM states, it is inconsistent with the insignificant size premium in low BM states. Again in line with the data, the ICAPM predicts no size premium in medium BM states because the (apparently positive) relation between SM B∗ and the future investment
opportunities is not significant.
Before 1960 the ICAPM predicts no size premium in any of the BM states, as we see in Table 8. The relation betweenSM B∗ and the investment opportunities is insignificant in the high BM states. Accordingly, the risk price, γz, in Eq. (7) is zero and the ICAPM predicts no size premium in the high BM states. On the other hand, the relation betweenSM B∗ and the investment opportunities is significantly positive in low and medium BM states, so the risk price γz in Eq. (7) is positive. But the insignificant covariance between the return on the SMB portfolio and the innovations inSM B∗ in low and medium BM states implies that the size premium should be zero according to the ICAPM. The insignificant size premium is consistent with the ICAPM in the low and medium BM states, but the ICAPM cannot explain the significantly positive size premium in the high BM states.
After 1960, the ICAPM predicts no size premium in low and medium BM states (con-sistent with the data) and a negative premium in high BM states (not con(con-sistent with the data). The relation between SM B∗ and the investment opportunities in low, medium, and high BM states is, respectively, insignificant, positive, and negative. So the risk price γz in Eq. (7) is, respectively, zero, positive, and negative in low, medium, and high BM states.
Given γz = 0, the ICAPM predicts no size premium in low BM states. The insignificant co-variance between the return on the SMB portfolio and the innovations inSM B∗ in medium BM states also implies no size premium in medium BM states according to the ICAPM.
And the positive covariance between the return on the SMB portfolio and the innovations in SM B∗ in high BM states implies a negative size premium. Again, the insignificant size premium is consistent with the ICAPM in the low and medium BM states. But the ICAPM cannot explain the insignificant size premium in high BM states, especially considering its positive point estimation.
4.2.2. The SMB* as a state variable conditioning on the SMB* states
The relation between the size premium and SM B∗ is not similar to the relation between the size premium and BMt. So conditioning the analysis on the SM B∗ should not be as informative as conditioning on theBMt. But we can still investigate the properties ofSM B∗ as a state variable conditioning the analysis on the low, medium, and highSM B∗ years. In Table 9 we see the same unconditional results that were presented in the previous section:
Consistent with the data, the ICAPM predicts a positive size premium in the full 1926-2014 sample and after 1960, and no premium before 1960. The difference is in the conditional results.
[Place Table 9 about here]
The conditional results in the full 1926-2014 sample show that the covariance between the SM B∗ and the investment opportunities is, respectively, positive, marginally positive, and negative in low, medium, and highSM B∗years. Accordingly, the risk priceγz in Eq. (7) is positive in low and medium SM B∗ years and negative in high SM B∗ years. In addition, the covariance between the return on the SMB portfolio and the innovations in SM B∗ is positive in all periods. Thus, the ICAPM predicts a positive size premium in mediumSM B∗ years that is consistent with the data. But the ICAPM also predicts a positive size premium in low SM B∗ years and a negative premium in high SM B∗ years that are not consistent with the data.
Before 1960, the covariance between the return on the SMB portfolio and the innovations inSM B∗ is insignificant in low and medium SM B∗ years. So consistent with the data, the ICAPM predicts no size premium in these periods. This happens even if the risk price γz in Eq. (7) is positive in low SM B∗ years given the positive covariance between SM B∗ and the investment opportunities in these years. But in high SM B∗ years the risk price γz in Eq. (7) is zero, so again the ICAPM predicts no size premium instead of the large premium that we observe in these periods.
After 1960, the covariance between the return on the SMB portfolio and the innovations in SM B∗ is positive in all periods. Hence, the size premium predicted by the ICAPM has the same sign as the risk price γz in Eq. (7), given by the relation between SM B∗ and the investment opportunities. The relation between SM B∗ and the investment opportunities is, respectively, positive, insignificant, and negative in low, medium, and high SM B∗ years.
Therefore, the insignificant size premium in medium SM B∗ years is consistent with the ICAPM predictions. But the insignificant premiums in low and high SM B∗ years are not consistent with the ICAPM.
5. Summary
We learn two broad new facts from this paper: First, the existence of the size premium is restricted to the states with good investment opportunities. Second, the intertemporal risk exposures of small stocks do not in general explain the existence of the size premium within an ICAPM framework.
The paper provides a more detailed description of the size premium, contributing to the original documentation in Banz (1981). I show that the size premium is pervasive but conditional. This conditional premium is in fact around three times larger than previously estimated, and it is approximately 9% per year. But there is no evidence that the premium exists around 70% to 90% of the time. This explains the lack of evidence about the size premium in several studies and suggests the inclusion of a conditional size factor in empirical asset pricing models.
I also show that the ICAPM explanation for the size premium has little empirical support both in its unconditional and, especially, conditional forms. The unconditional results are mixed: The results based on theSM B∗variable of Maio and Santa-Clara (2012) in fact imply that the unconditional version of the ICAPM is consistent with the positive size premium.
But considering the BMt of Pontiff and Schall (1998) as the state variable implies that
even the unconditional version of the ICAPM is inconsistent with the size premium: The ICAPM predicts a negative unconditional premium in this case. The conditional version of the ICAPM is also usually inconsistent with the size premium because it often predicts a premium with a sign different from the one observed in the data. This conditional analysis is crucial because the risk premiums are time varying and the ICAPM does not condition down.
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-1-.50.511.5BM
1920 1940 1960 1980 2000 2020
Date
US 1926-2014
Fig. 1. Book-to-market time series. The panel plots the time series of BMt between 1926-2014. I use these values to classify the states of the economy according to their “good”
or “bad” investment opportunities.
Table1:Summarystatisticsforselectedvariablesinthefullsampleandindifferentstatesoftheeconomy (BMterciles).Thetabledescribesthe1926-2014sampleandtwosubsamples1926-1960,and1960-2014.Thepanelsplits horizontallyintofourparts:Allstates,LowBMstates,MediumBMstates,andHighBMstates.“Allstates”containsthe resultsfortheentiresampleperiod.Low,Medium,andHighBMstatescorrespondtothethreestatesoftheeconomy(BMt terciles):Inagivensample,theyearsinwhichBMtisinthelowesttercileacrossalltheyearsare“Low”,theonesinthehighest tercileare“High”,andtheremainingonesare“Medium”.TheMarket(premium)isthevalue-weightedreturnonthemarket minustheriskfreerate.ThereturnontheSMBportfolioistheaveragereturnonthethreesmallFama/Frenchportfolios minustheaveragereturnonthethreebigFama/Frenchportfolios.Ireportthemean,standarddeviation(Stddev)andthe t−Mean,whichistheratioofmeantoitsstandarderrorofeachvariable.ThevariableBMtisdeterminedinJulyofyeart, andallthereturnsarefromJulyofyearttotheendofJuneint+1. Thesizepremiumaccordingtothestateoftheeconomy(medianBM) AllstatesLowBMstatesMediumBMstatesHighBMstates MarketSMBBMMarketSMBBMMarketSMBBMMarketSMBBM 1926-2014 Mean9.062.780.005.161.18-0.394.62-1.29-0.0517.278.390.44 Stddev2.721.290.043.552.010.022.911.980.026.542.350.06 t-Mean3.342.150.001.460.59-16.871.59-0.65-2.132.643.567.75 1926-1960 Mean13.342.950.284.58-2.16-0.0511.530.990.2424.8710.650.69 Stddev5.902.190.065.863.230.045.272.180.0216.754.970.11 t-Mean2.261.344.400.78-0.67-1.432.190.4613.651.482.146.23 1960-2014 Mean6.462.64-0.183.240.01-0.477.753.68-0.218.334.170.17 Stddev2.271.570.043.942.310.023.872.930.014.112.890.05 t-Mean2.841.68-4.430.820.00-21.632.001.26-14.792.031.443.04
Table 2: Returns on the SMB portfolio when I vary the number of BM quantiles to group the years with similar states: Mean and (t−M ean), the ratio of the mean return to its standard error, of the return on the SMB portfolio in 1926-2014 and subsamples 1926-1960 and 1960-2014.
I split each sample into (1), 2, 3, 5, 7, or 10 quantiles based on theBMtstate variable and report the results for all of these groups of years. Each row corresponds to a given number of quantiles.
The number of quantiles is in the first column: All years (i.e., the whole sample), 2, 3, 5, 7, or 10.
The number of years in each group is in brackets. The next 10 columns contain the results for each respective group of years, from 1 to 10, depending on the number of quantiles considered. The last column, “Ex top”, displays the results considering all the years except the ones in the highest BM quantile. The variable BMt is determined in July of year t, and all the returns are from July of year tto the end of June int+ 1.
Return on the SMB portfolio in each state (BM quantile)
Bottom 2 3 4 5 6 7 8 9 Top Ex top
(-0.77) (2.42) (-0.27) (-1.37) (-0.59) (3.07) (2.58) (1.13)
10 (9) -5.20 3.18 5.63 -0.48 -3.63 -1.91 0.41 10.48 5.22 12.83 1.65
(-1.61) (-0.96) (0.53) (0.22) (1.17) (2.35) (1.81) (0.26)
10 (4) -10.20 1.46 -0.01 -0.70 -1.01 4.18 2.88 2.35 7.64 28.01 0.60
Table 3: Returns on the size portfolio based on a double sort on betas and size when I vary the number of BM quantiles to group the years with similar states: Mean and (t−M ean), the ratio of the mean return to its standard error, of the return on the SMB portfolio in 1963-2014.
I split each sample into (1), 2, 3, 5, 7, or 10 quantiles based on theBMtstate variable and report the results for all of these groups of years. Each row corresponds to a given number of quantiles.
The number of quantiles is in the first column: All years (i.e., the whole sample), 2, 3, 5, 7, or 10.
The number of years in each group is in brackets. The next 10 columns contain the results for each respective group of years, from 1 to 10, depending on the number of quantiles considered. The last column, “Ex top”, displays the results considering all the years except the ones in the highest BM quantile. The variable BMt is determined in July of year t, and all the returns are from July of year tto the end of June int+ 1.
Return on the Size portfolio double sorted on size and betas in each state (BM quantile)
Bottom 2 3 4 5 6 7 8 9 Top Ex top
1963-2014
All (51 obs.) 3.23 (2.21)
2 (26) 1.64 4.76 1.64
(0.86) (2.17) (0.86)
3 (17) -1.50 6.62 4.57 2.56
(-0.78) (2.49) (1.73) (1.45)
5 (10) -2.69 4.83 4.96 -2.65 10.78 1.39
(-1.02) (1.42) (1.52) (-1.24) (3.71) (0.89)
7 (7) -6.18 3.23 7.70 1.94 0.61 3.98 11.08 1.98
(-2.24) (1.09) (2.1) (0.85) (0.12) (1.06) (3.61) (1.28)
10 (5) -6.85 1.46 0.35 9.31 3.92 5.70 -4.34 -1.30 12.37 9.19 2.58
(-1.74) (0.55) (0.08) (1.88) (1.21) (1.08) (-0.94) (-0.83) (2.73) (2.29) (1.67)
Table 4: Predictive regressions for 1 or 5 years for the excess return on the market, the market variance, and the derivative of the pseudo-conditional Sharpe ratio with respect to BMt. I show the results in the full sample and in different states of the economy (BM terciles) in the periods 1926-2014, 1926-1960, and 1960-2014.
The row Rm displays the BMt coefficients, βq, from Eq. (17): Rm,t→t+q =αq+βqBMt+t→t+q
followed by the respective adjusted coefficient of determination (in %). The rowSV ARm displays the BMt coefficients, βq, from Eq. (18): SV ARm,t→t+q = αq+βqBMt+t→t+q followed by the respective adjusted coefficient of determination (in %). The row ∂BM∂SR
t displays the average of the partial derivatives of the pseudo-conditional Sharpe ratios in Eq. (20) with respect to BMt. The t-statistics of the results are in brackets. The left panel reports the results for the future q = 1 (year) values, and the right panel reports the results for the future q = 5 (years) in the equations above. Each panel splits horizontally into four parts: All, Low BM, Med BM, and High BM states.
“All” contains the results for the entire sample period. Low, Med, and High BM states correspond to the three states of the economy (BMt terciles): In a given sample, the years in whichBMtis in the lowest tercile across all the years are “Low”, the ones in the highest tercile are “High”, and the remaining ones are “Med”.
q = 1 year q = 5 years
All Low BM Med BM High BM All Low BM Med BM High BM
1926-2014
Rm 19.90 24.13 15.27 35.92 66.16 158.51 75.95 79.98
(3.02) (0.83) (0.61) (1.74) (4.27) (1.87) (1.24) (1.81)
R2 8.5 -1.1 -2.2 6.5 0.2 0.1 0.0 0.1
(18.47) (121.59) (14.74) (-2.22) (68.83) (65.1) (41.53) (42.28) 1926-1960
Rm 38.94 120.84 135.15 52.21 100.60 97.72 344.90 71.14
(2.65) (3.92) (1.56) (1.11) (2.9) (0.74) (0.89) (0.91)
∂BMt 2.57 172.26 10.03 0.93 1.82 2.27 21.08 -0.01
(3.92) (1.11) (17.58) (12.72) (43.45) (18.03) (3.31) (-0.16) 1960-2014
Rm 4.56 22.90 -83.98 -3.51 29.20 277.83 -358.88 -46.99
(0.59) (0.52) (-1.34) (-0.19) (1.38) (2.35) (-2.11) (-1.23)
R2 -1.2 -4.5 4.2 -6.0 1.8 24.4 16.9 3.0
(18.8) (76.28) (-24.18) (-77.13) (6.16) (16.16) (-410.8) (-44.7)
Table 5: Predictive regressions for 1 or 5 years for the excess return on the market, the market variance, and the derivative of the pseudo-conditional Sharpe ratio with respect to SM B∗t. I show the results in the full sample and in different states of the economy (BM terciles) in the periods 1926-2014, 1926-1960, and 1960-2014.
The rowRmdisplays theSM B∗t coefficients,βq, from Eq. (17): Rm,t→t+q=αq+βqSM Bt∗+t→t+q
followed by the respective adjusted coefficient of determination (in %). The rowSV ARm displays theSM Bt∗ coefficients,βq, from Eq. (18): SV ARm,t→t+q=αq+βqSM B∗t+t→t+qfollowed by the respective adjusted coefficient of determination (in %). The row ∂SM B∂SR∗
t displays the average of the partial derivatives of the pseudo-conditional Sharpe ratios in Eq. (20) with respect toSM Bt∗. The t-statistics of the results are in brackets. The left panel reports the results for the future q = 1 (year) values, and the right panel reports the results for the future q = 5 (years) in the equations above. Each panel splits horizontally into four parts: All, Low BM, Med BM, and High BM states.
“All” contains the results for the entire sample period. Low, Med, and High BM states correspond to the three states of the economy (BMt terciles): In a given sample, the years in whichBMtis in the lowest tercile across all the years are “Low”, the ones in the highest tercile are “High”, and the remaining ones are “Med”.
q = 1 year q = 5 years
All Low BM Med BM High BM All Low BM Med BM High BM
1926-2014
Rm 17.28 22.26 0.24 32.52 72.45 85.50 40.98 33.68
(1.52) (1.95) (0.02) (0.68) (2.67) (2.61) (1.39) (0.33)
R2 1.5 9.0 -3.6 -1.9 0.1 0.2 0.0 0.0
(40.3) (12.28) (40.24) (55.63) (30.28) (23.3) (23.07) (7.77) 1926-1960
Rm 71.41 42.72 103.71 63.77 251.94 172.43 201.69 251.74
(1.88) (1.34) (2.77) (0.32) (3.05) (2.51) (1.01) (0.79)
R2 6.9 6.8 37.8 -9.9 21.6 39.8 0.1 -3.9
(92.08) (3.91) (2.24) (2.16) (9.83) (1.9) (2.62) (308.06)
1960-2014
Rm 10.68 16.85 21.82 -35.79 53.95 45.99 75.60 -24.89
(1.29) (1.19) (1.6) (-1.44) (2.48) (1.07) (1.98) (-0.45)
R2 1.2 2.4 8.0 5.9 9.3 1.0 14.7 -4.9
SV ARm -0.02 -0.04 0.00 -0.02 -0.01 0.06 -0.05 0.11
(-1.29) (-0.95) (0.06) (-1.61) (-0.27) (0.58) (-1.27) (1.68)
R2 1.2 -0.7 -5.9 8.5 -1.9 -5.0 3.6 9.6
∂SR
∂SM B∗t 0.87 1.06 1.54 -2.28 1.53 0.98 2.89 -1.51
(34.02) (15.83) (269.23) (-18.23) (328.47) (53.42) (28.84) (-19.49)
Table 6: Predictive regressions for 1 or 5 years for the excess return on the market, the market variance, and the derivative of the pseudo-conditional Sharpe ratio with respect to SM B∗t. I show the results in the full sample and for different values of SM B∗ (in terciles) in the periods 1926-2014, 1926-1960, and 1960-2014.
The rowRmdisplays theSM B∗t coefficients,βq, from Eq. (17): Rm,t→t+q=αq+βqSM Bt∗+t→t+q
followed by the respective adjusted coefficient of determination (in %). The rowSV ARm displays theSM Bt∗ coefficients,βq, from Eq. (18): SV ARm,t→t+q=αq+βqSM B∗t+t→t+qfollowed by the respective adjusted coefficient of determination (in %). The row ∂SM B∂SR∗
t displays the average of the partial derivatives of the pseudo-conditional Sharpe ratios in Eq. (20) with respect toSM Bt∗. The t-statistics of the results are in brackets. The left panel reports the results for the future q = 1 (year) values, and the right panel reports the results for the future q = 5 (years) in the equations above. Each panel splits horizontally into four parts: All, Low SMB*, Med SMB*, and High SMB*.
“All” contains the results for the entire sample period. Low, Med, and High SMB* correspond to the three SM B∗t terciles: In a given sample, the years in which SM Bt∗ is in the lowest tercile across all the years are “Low”, the ones in the highest tercile are “High”, and the remaining ones are “Med”.
q = 1 year q = 5 years
All Low SMB* Med SMB* High SMB* All Low SMB* Med SMB* High SMB*
1926-2014
Rm 17.28 44.07 145.34 -100.17 72.45 124.91 -199.32 -155.50
(1.52) (2.38) (1.54) (-2.01) (2.67) (3.55) (-0.77) (-1.31)
R2 1.5 13.9 4.5 9.8 0.1 0.3 0.0 0.0
(40.3) (19.05) (7.43) (-3.74) (30.28) (19.34) (-79.78) (-49.63)
1926-1960
Rm 71.41 64.40 -24.94 58.40 251.94 180.83 -82.71 517.82
(1.88) (1.42) (-0.09) (0.14) (3.05) (2.4) (-0.18) (0.54)
R2 6.9 8.4 -9.9 -10.9 21.6 34.6 -10.7 -8.5
SV ARm -0.02 -0.16 0.31 1.04 -0.76 -1.41 0.52 0.99
(-0.29) (-1.74) (0.45) (1.46) (-2.95) (-2.78) (0.25) (0.59)
R2 -2.8 15.6 -7.9 10.1 18.5 37.9 -9.3 -7.0
∂SR
∂SM Bt∗ 3.40 3.85 -2.22 -21.67 9.52 4.21 -8.04 7.49
(92.08) (9.52) (-10.69) (-3.6) (9.83) (7.46) (-22.09) (30.88)
1960-2014
Rm 10.68 24.38 9.76 -92.72 53.95 69.55 79.91 0.24
(1.29) (1.21) (0.21) (-2.15) (2.48) (2.06) (0.3) (0)
R2 1.2 2.6 -6.0 17.5 9.3 15.2 -6.5 -7.1
SV ARm -0.02 0.00 -0.02 -0.06 -0.01 -0.10 -0.35 -0.19
(-1.29) (0.07) (-0.27) (-0.98) (-0.27) (-1.22) (-0.94) (-0.89)
R2 1.2 -5.9 -5.8 -0.2 -1.9 2.7 -0.7 -1.5
∂SR
∂SM Bt∗ 0.87 1.29 1.03 -5.92 1.53 2.15 5.10 0.61
(34.02) (207.32) (70.01) (-9.28) (328.47) (27.75) (15.73) (16.35)
Table 7: The 1-year returns on the SMB portfolio compared with the ICAPM predictions, considering the validity ofBMt as a state variable and its properties.
For each sample 1926-2014, 1926-1960, and 1960-2014 I reportSM B, the average return on the SMB portfolio;cov(SM B,∆BM), the covariance between the return on the SMB portfolio and the 1-year changes inBMt; cov(M P1y, BM) andcov(M P5y, BM), respectively the covariance between the future 1- or 5-years returns on the market andBMt;cov(svar1y, BM) andcov(svar5y, BM), respectively the covariance between
For each sample 1926-2014, 1926-1960, and 1960-2014 I reportSM B, the average return on the SMB portfolio;cov(SM B,∆BM), the covariance between the return on the SMB portfolio and the 1-year changes inBMt; cov(M P1y, BM) andcov(M P5y, BM), respectively the covariance between the future 1- or 5-years returns on the market andBMt;cov(svar1y, BM) andcov(svar5y, BM), respectively the covariance between