Adapting the Capital Asset Pricing Model
to capture the influence of macroeconomic conditions
V.I. Biemond
Master Thesis
Rijksuniversiteit Groningen Faculty of Economics Master of Finance
Under supervision of dr. ing. N. Brunia Iwan Biemond
Van Heemskerckstraat 3/30 9726 GB Groningen
1. Introduction
The Capital Asset Pricing Model (CAPM) as described by Sharpe (1964) and Lintner (1965) still plays a crucial role in financial research and practice. Practitioners use the CAPM to calculate the cost of capital, to design portfolios and to evaluate the performance of asset managers. Economic researchers rely on the CAPM for capital asset pricing models, testing relative risks, etc. The CAPM has been developed to understand the pricing of risk. It posits a stable linear relationship between an asset’s systematic risk and its expected return. However, empirical research has found weak or no evidence for this relationship (e.g. Fama French 1992, 1993). Fama and French argue that the model’s problems reflect weaknesses in the theory or in its empirical implementation. According to Fama and French (1992) the failure of the CAPM in empirical tests implies that most applications of the model are invalid.
Several researchers aimed already to develop alternatives for the CAPM. Their research can be divided in two lines. One line focuses on macroeconomic variables to explain expected returns, like the Arbitrage Pricing Theory (Ross 1976). The Arbitrage Pricing Theory (APT) postulates that a security’s expected return is influenced by a variety of factors, in contrast to just the single market index of the CAPM. However, in contrast to the CAPM, the APT is more complex and therefore less convenient in common practice. Instead of working with one risk factor (systematic risk), multiple risk factors have to be estimated, which potentially decrease the out-of-sample performance of the model relative to the original CAPM.
dual-beta CAPM approach, as originally proposed by Fabozzi and Francis (1977), allowing for different betas in a bull and bear market. Although up and down-markets are evidently caused by macroeconomic conditions, the direct effect of macroeconomic variables on the beta is never investigated with a dual-beta approach. This paper seeks to fill this void in the existing literature.
The objective of this paper is to adjust the CAPM in a way that it increases performance and keeps it usable for financial practice. I will define the key concepts, performance and usability, as follows. Better performance means a more stable linear relationship between the systematic risk of a portfolio and its expected return, in-sample and out-sample. Usability means using a small number of explanatory variables. The main challenge is to find the balance between in-sample performance, out-in-sample performance and usability. The use of more explanatory variables increases in-sample performance, but normally decreases the performance out-of-sample (Campbell, Thompson (2007)). In this paper I will propose a dual beta CAPM (an asymmetric CAPM), allowing for macroeconomic variables, which are widely used in previous research on the Arbitrage Pricing Theory.
I propose in this paper an adjustment to the CAPM. Using a dual-beta strategy, the systematic risk is allowed to differ from several macroeconomic conditions. Using a dual beta approach allowing for economic variables can give us more insight in the influence of macro-economic conditions to the CAPM-beta and can potentially outperform the explanatory power of the static CAPM and still keeps (some of) the usability of the original CAPM.
2. Theoretical background
2.1 Failure of the CAPM
The capital asset pricing model of William Sharpe (1964) and John Lintner (1965) is developed to understand the pricing of risk. Now almost half a century later the CAPM is still widely used in financial practice and research, for example to estimate the cost of capital of firms. The basic logic behind this model is that assets with the same systematic risk should have the same expected return. This elementary economic rule is called the “law of one price”.
The CAPM simply states that the systematic difference in security returns can be explained by a single measure of risk, beta. According to the CAPM, the expected return on any risky security or portfolio of risky securities is measured by the risk-free rate and the expected market risk premium multiplied by the beta coefficient. Despite some early evidence in favor of the CAPM (Black et al., 1973: Fama and Macbeth, 1973), more recent tests fail to give a strong basis for evaluating beta as a reliable measure of systematic risk (e.g. Fama and French, 1992). Empirical tests of the CAPM are often questionable due to many obstacles, like inadequate proxy of the market portfolio, problems associated with unobservable expected returns and non-stationarity of the beta (Fama and French, 2004).
2.2. Solutions provided in previous research
Many in finance are looking for alternatives for the CAPM. The APT, first introduced by Ross (1976) is such alternative. The APT is a multifactor asset pricing model, which allows multiple measures of systematic risk. This is in contrast to the CAPM, where only one measure of systematic risk is allowed. Chen, Roll and Ross (1986) examined the ability of macro-economic variables to explain US stock returns. They investigated the sensitivity of size-portfolio returns to 7 macro-series: term structure, industrial production, risk premium, inflation, market return, consumption and oil prices. They assumed that all changes are unexpected and that the underlying variables are not correlated. They found that industrial production, changes in the risk premium and twists in the yield curve are significant in explaining expected stock returns.
The APT has the potential to overcome CAPM weaknesses: it requires less and more realistic assumptions to be generated by a simple arbitrage argument and its explanatory power is potentially better since it is a multifactor model. However, the power and the generality of the APT encompass its main strength and weakness: the APT permits the analyst to choose whatever factors provide the best explanation for the data but it cannot explain variation in asset return in terms of a limited number of easily identifiable factors. An enormous amount of literature has been written on the two models. Although empirical research shows that the APT can outperform the CAPM in terms of within-sample explanatory power, the out-of-sample performance is worse (e.g. Groenewold and Fraser (1997)). Therefore, the model is seldom used in financial practice.
To address the problem of the time-varying nature of asset betas, Jagannathan and Wang (1996) developed the CCAPM as a model of the cross-section of returns in which the value of a firm’s beta is conditional on the state of the economy. Their ultimately derived model is described by the following equation:
E(Rit)= c0+ cvwβvw + cprem βprem + clabourβlabour
Where βvw is the market beta based on a value-weighted portfolio of all stocks, βprem captures the systematic changes in a firm’s beta with variation in the market premium and βlabour is the beta for the human capital market. Whereas the conventional CAPM prices assets to compensate only for levels of systematic risk, the addition of the extra variable postulates that asset prices will be determined not only by an asset’s systematic risk, but also by the predictable component of the security’s change in systematic risk when there are shifts in the state of the economy.
Jagannathan and Wang found that for the US stock market the CCAPM outperforms the single-beta CAPM and even the APT model of Chen, Roll and Ross (1986). However, the out-of-sample performance of the CCAPM is poor, as is highlighted by Ghysels (1998). The reason for this is the fact that the conditional CAPM fails to capture the time-variation of the beta The beta changes slowly through time and the CCAPM has the tendency to overstate the time variation (Ghysels (1998)). Ghysels (1998) showed that because the beta risk is misspecified by the CCAPM, pricing errors out-of sample with constant traditional beta models are smaller than with the CCAPM. He also concluded that if we succeed in capturing the time-variation of the beta, we are sure to outperform the original CAPM.
based portfolios (Bhardwaj and Brooks (1993), Howton and Peterson (1998)). Most of those studies found evidence that beta varies with market conditions. Hence investors could improve the performance of their portfolios by using up and down market betas in their asset selection practice.
2.3 Proposed model: adapting the CAPM to capture macroeconomic influences
In this paper I propose a new alternative for the CAPM, based on three conclusions in previous literature, which were reviewed in paragraph 2.3. First the statement of Ghysels (1998) that if we succeed in capturing the time-variation of the beta, we are sure to outperform the original CAPM. Second, the conclusion that allowing different betas between bull and bear markets improves in-sample performance compared to the original CAPM and out-sample performance compared to the CCAPM (e.g. Bhardwaj and Brooks (1993)). And third, the conclusion that macroeconomic conditions have a direct effect on portfolio returns (Chen, Roll, Ross (1986)).
Combining the above stated conclusions leads to a void in the existing literature. Available knowledge indicates that the Dual-Beta CAPM has the potential to outperform (in-sample and possibly out-sample) the original CAPM, and that up and down-markets are evidently the consequence of macroeconomic variables. However surprisingly information on the direct effect of macroeconomic variables on the beta with a dual-beta approach is lacking. To fill this gap I propose an asymmetric dual-beta model, allowing for three macroeconomic conditions identified by Chen, Roll and Ross (1986).
In order to achieve this goal I will measure the influence of macro-economic conditions on the CAPM-beta and will test whether the proposed model can potentially outperform the static CAPM while most of the usability of the original CAPM is preserved.
portfolios will provide insight in the influence of macro-economic conditions to the CAPM-beta. To implement the model in financial practice, further research (with industry-based portfolios or with individual firms) will be needed.
I will modify the macroeconomic variables into threshold-variables by using the methodology of Hansen (2000). To test the accuracy of the new model, the performance of this new model will be compared with the original CAPM and the dual CAPM of Fabozzi and Francis (1977). In-sample performance is tested by the goodness of fit and the significance of the betas. Out-sample performance is analyzed by the Mean Squared Error (MSE) and the Mean Average Error (MAE), where I compare the actual betas with the predicted betas.
3. Data
Table 1. Construction of macroeconomic variables
Symbol Factor Origin of data / measurement
IP Monthly growth in industrial production Federal Reserve Bank of St. Louis BAA Monthly return on low-grade bonds Federal Reserve Bank of St. Louis LTGB Monthly return on long term government bonds Federal Reserve Bank of St. Louis TB Montly return on short term riskless portfolio Federal Reserve Bank of St. Louis
UIP
Unexpected monthly growth in industrial
production IP (t) = α + β ln (IP(t-1)) - ln(IP(t - 2) + εt, where UIP = εt
RP Unexpected monthly changes in risk premium RP(t) = BAA(t) – LTGB (t)
TS Unexpected monthly changes in term structure of
interest rates TS = LTGB (t) – TB(t)
3.1 Construction of macroeconomic variables
The three macroeconomic variables used for analysis are those which have been widely used in previous literature, namely unexpected monthly growth in industrial production, risk premium and term structure of interest rates (Chan, Chen and Hsieh, 1985; Chen Roll and Ross, 1986; Burnmeister and Wall, 1986; Chang and Pinegar (1990; Kryzanowski and Zhang, 1992; Chen and Jordan, 1993; Altay, 2001). The rational expectations and the efficient market assumptions require the identification of unexpected changes in the series.
a) Unexpected monthly growth in industrial production (UIP)
The following estimator of unexpected monthly growth in industrial production is used in previous literature (e.g. Chen, Roll and Ross, 1986).
UIP (t) = ln (IP(t)) - ln(IP(t - 1))
IP (t) = α + β ln (IP(t-1)) - ln(IP(t - 2) + εt
Here, the unexpected growth in industrial production (UIP) equals εt.
b) Risk premium (RP)
The risk premium is defined as the impact of unexpected changes in risk premia on equity returns, captured by the difference between the return on government bonds and that of low grade bond portfolios (Chen, Roll and Ross, 1986).
RP(t) = BAA return(t) – LTGB return(t), (3)
BAA stands for the return on a low grade bond portfolio and LTGB for the return on a high grade (long term government bond (LTGB)) portfolio. A change in RP represents a shift in the degree of risk aversion (Günsel, Çukur (2007)).
c) Term structure of interest rates (TS)
TS = LTGB (t) – TB(t) (4)
is the difference in returns on a long-term government bond portfolio (LTGB) and short-term riskless portfolio (TB).
d) Bull-Bear variable
4. Empirical framework
4.1 The original CAPM
Empirically, the beta is defined as the systematic risk of a portfolio:
Rit = αi + βi (Rmt – Rft ) + uit (1)
where Rit, Rmt and Rf stand for the total return of the portfolio, market index and the risk
free-rate in period t. ui is defined as the unsystematic risk. αi and βi are constants and the beta can be
estimated by an OLS time-series regression. The time period (1988-2007) is divided in four subperiods of five years each. For each period the beta coefficient is estimated.
To provide an insight in the CAPM-beta, I will test the influence of three macroeconomic conditions. This will be done by analysing the influence of three macroeconomic variables with a dual-beta strategy.
4.2 Dual beta CAPM
For the dual beta strategy I use the following model.
Rit = αi,1D+ βi,1 (Rmt – Rft ) D + αi,2(1-D)+ βi2 (Rmt – Rft ) (1-D) + uit, (5)
where D is a dummy variable. If the macroeconomic variable is above the threshold level, D is equal to 1. If the macroeconomic variable is below the threshold level, D equals 0. In the case of the bull-bear variable, D equals 1 in a bull market and 0 in a bear market. The same model will be tested for the three different macro-economic variables and will be compared with the results of the bull-bear dual beta CAPM and the original CAPM.
4.3 Threshold estimation
To implement the macroeconomic variables into the dual beta CAPM, I will generate for each macroeconomic variable an OLS-estimate of the threshold value. This threshold is needed to create a dummy variable for the dual beta strategy. For calculating purposes the threshold is calculated by pooled estimation using the methodology of Hansen (2000). So, all portfolios have the same threshold. For this the Matlab-procedure provided by Hansen will be used and the procedure will be adjusted to the specifics of my data. For further mathematical details of the Hansen-methodology see the paper of Hansen (2000). The following theoretical basis of his theory will serve the purpose of my paper.
As been specified in paragraph 4.2 the following asymmetric beta model is used:
Rit = αi,1D+ βi1 (Rmt – Rft ) D + αi,2(1-D)+ βi2 (Rmt – Rft ) (1-D) + uit, (5)
Here the value of D is dependent on a macroeconomic variable. D is equal to one is the macroeconomic variable is above the threshold value and otherwise D is equal to zero.
This threshold value is estimated by minimizing the sum of squared error of equation (5).
4.4 In sample performance
a) Estimating beta risks
b) Relevant restrictions
In addition I will test whether the betas estimated below and above the threshold value are significantly different from zero for each of the portfolios in each period. The static CAPM (model (1)) is termed as the restricted version of the unrestricted dual beta model (DBM, model (5)). I will test for the following restrictions: αi,up = αi,down and βi,up = βi,. Using the Wald
F-test I test the H0: αi,up = αi,down , βi,up = βi,. The associated p-values will indicate whether the
restrictions are statistically relevant (like Faff, 2001).
c) Goodness of fit
Differences between the original static CAPM (model (1)) and the dual beta CAPM (model (5)) is also reflected in the R-squared statistics. R-squared measures the percentage of the portfolio’s return volatility that is explained by volatility in the overall market return. A higher R-squared value means a better fit of the tested model. A student T-test is performed to test whether the possible differences between the R-squared statistics of the different models are statistically relevant.
4.5 Out of sample performance
The out of sample performance is defined as the degree of forecast error of the beta coefficients, expressed as the mean squared error of the beta prediction. After obtaining the historical betas, the predictive power of those betas can be tested. I will use the mean squared error (MSE). The MSE is calculated by:
MSE =
(
)
n
∑
β1−β2 2The MSE allows to test whether the proposed adjustment to the CAPM is an improvement of the static CAPM. It will provide information whether the dual-CAPM betas have more predictive power compared with the static CAPM.
A potential problem with the use of the MSE is it places a heavy penalty on outliers. An alternative approach is the mean absolute error (MAE) approach.
MAE =
(
)
n
2 1 β
β −
The MAE measure weighs all errors equally. Accordingly, using these two measures of forecasting error I will compare the out-of-sample performance of the original CAPM, the macroeconomic Dual CAPMs and the bull-bear Dual CAPM.
I calculate the MSE and MAE values of the macroeconomic dual-beta CAPM twice. First I calculate the forecast errors with the calculated threshold value in period t. However this calculation cannot hold in practice. In reality, the threshold value in period t is unknown in period t-1. So, for the beta-coefficient in period t-1 to be a fair prediction, a forecast of the new threshold value has to be made. Although, research about the predicting of threshold value is beyond the scope of this paper, I can calculate the MSE and MAE between the beta-coefficient in period t-1 (as a prediction of the beta-coefficient in period t) and the actual beta-coefficient in period if the threshold value remains constant. In period t the same threshold is used as in period t-1.
5. Results
5.1 Threshold estimation
of observations below and above the threshold value are given for the three macroeconomic variables and the bull bear model as originally proposed by Fabozzi and Francis (1977).
As you can see, the data points above and under the threshold are not always equally divided. For example, there are in the last period only 10 data points under the threshold value for the term structure model. This can have a negative influence on the reliability of the OLS-regression of this model.
Table 4. Threshold estimation of the macroeconomic variables
Data points above and
below threshold value threshold value above under 1988-1992 Industrial Production -0.01 37 23 Risk Premium 2.06 15 45 Term Structure 1.74 32 28 Bull Bear Rmt – Rft = 0 36 24 1993-1997 Industrial Production 0.01 49 11 Risk Premium 1.50 44 16 Term Structure 1.17 44 16 Bull Bear Rmt – Rft = 0 43 17 1998-2002 Industrial Production -0.01 35 25 Risk Premium 1.91 49 11 Term Structure 0.79 32 28 Bull Bear Rmt – Rft = 0 31 29 2003-2007 Industrial Production 0.00 40 20 Risk Premium 2.25 16 44 Term Structure 0.02 50 10 Bull Bear Rmt – Rft = 0 39 21
5.2 Estimation beta risks
Together with the threshold values (table 2), the static CAPM (model (1)) and the dual beta CAPM (model (5)) are estimated. In table 5 (see appendix), beta coefficients, including t-statistics, are provided for the static CAPM, the dual beta CAPM with bull bear estimation and the dual beta CAPM with the three different macro-economic variables used as a threshold.
If the beta is significantly different from zero, the regression model can be used to predict the dependent variable for any value of the independent variable. As shown in table 6 all beta coefficients of the static CAPM differ significantly (α =0.01) from zero in all four periods.
In terms of significance of beta-coefficients the macro-economic dual beta models perform less compared to the original CAPM. This is in contrast to the expectation that adding more explanatory variables will improve in-sample performance. This is not due to the small point of data points as mentioned in section 5.1. In the periods that there are a small amount of data points the macro-economic dual beta models perform well (near 100% significance).
Table 6. Summary of t-statistics of beta coefficients of the CAPM and Dual Beta CAPMs in all periods F-value significant at 1% level F-value significant at 5% level F-value significant at 10% level Threshold Number of portfolios Percentage Number of portfolios Percentage Number of portfolios Percentage CAPM 40 100% 40 100% 40 100%
Industrial Production beta1 35 88% 35 88% 37 93%
beta2 34 85% 35 88% 37 93%
Risk Premium beta1 40 100% 40 100% 40 100%
beta2 37 93% 38 95% 38 95%
Term Structure beta1 33 83% 37 93% 39 98%
beta2 39 98% 40 100% 40 100%
Bull Bear beta1 36 90% 40 100% 40 100%
beta2 32 80% 35 88% 37 93%
beta. Although in some cases the dual beta models have no significant slope, the beta-coefficients of all dual beta models are in general significant, and therefore the dual beta models can in general be used to predict the dependent variable (portfolio return) for any value of the independent variable (excess market return).
5.3 Relevant restrictions
The estimation of the unrestricted model (equation 5) and the restricted model (equation 1) has been performed for the 10 portfolios. All F-statistics are provided in table 7 (see appendix). Table 8 shows the percentages of all significant F-statistics for the different dual beta models.
Table 8. Significant F-values of restrictions Dual-Beta CAPMs
F-value significant at 1% level F-value significant at 5% level F-value significant at 10% level Threshold Number of portfolios Percentage Number of portfolios Percentage Number of portfolios Percentage Industrial Production 0 0% 0 0% 2 5% Risk Premium 16 40% 17 43% 20 50% Term Structure 8 20% 10 25% 18 45% Bull Bear 1 3% 1 3% 7 18%
If we compare these results with the results of the dual beta CAPM with a bull bear threshold the dual beta model with a risk premium threshold performs remarkably better. With α =0.01 only 3% of the portfolios are significant for different betas in bull and bear markets. The dual beta model with term structure as threshold performs slightly worse than the risk premium threshold. Percentages vary between 20% (α =0.01) and 45% (α =0.1), which is still higher than the original bull bear model.
5.4 Goodness of fit
Another measure to compare the in-sample performance the different models is the goodness of fit, expressed by the R-squared of the OLS regression. R-squared measures the percentage of the portfolio’s return volatility that is explained by volatility in the overall market return. A higher R-squared value results in a better fit of the tested model.
Table 9 (see appendix) provides the R-squared values of all models for each portfolio and period. The static CAPM performs in terms of R-squared worse than all dual beta models. In all periods and with all portfolios the R-squared values of the macroeconomic dual beta model exceed those of the static CAPM and even the R-squared values of the Bull Bear dual beta CAPM. However, none of those differences are significant and therefore the results are only provided in the appendix (table 10 and 11).
5.5 Out of sample performance
In period 2 and period 3 the industrial production dual beta model has the highest forecast error. This means that the beta coefficients in period 1 are not a good prediction of the beta coefficients in period 2. The risk premium and the term structure models perform better in terms of mean squared error than the industrial production model. In period 3 the MSE of the risk premium and the term structure model are lower than the MSE of the Bull-Bear model. In the last period both the risk premium and the term structure model perform better than the static CAPM and the Bull Bear model. In that period the historical beta of the RP and TS model is a better forecasting of their future beta compared to the beta prediction of the static CAPM.
However it is remarkable that in the last period the RP and TS model outperforms the original CAPM out-of-sample, because in the same period the in-sample performance of those two models are lower than the original CAPM. If we recall the Wald F-statistics of the RP and TS model (section 5.3), almost none of F-statistics were significant. The unrestricted version (the dual beta CAPM) is in those periods inappropriate to use in-sample. In the period 1 and 2, where the dual beta model (RP and TS) is appropriate to use in-sample, the original CAPM clearly outperforms the dual beta CAPMs out-of-sample.
Table 12. MSE values of the predicted and actual values of the betas of all models
Dual CAPM
Static
CAPM Bull Bear
Industrial
Production Risk Premium Term Structure
beta beta 1 beta 2 beta 1 beta 2 beta 1 beta 2 beta 1 beta 2
average average average Average
1993-1997 0.01 0.08 0.01 0.01 0.81 0.04 0.09 0.16 0.02 0.04 0.41 0.06 0.09 1998-2002 0.01 0.01 0.28 0.84 0.79 0.02 0,.2 0,13 0,01 0.14 0.81 0.02 0.07 2003-2007 0.11 0.21 0.76 0.72 0.18 0.03 0,06 0,03 0,11 0.48 0.45 0.04 0.07
beta-coefficient in period t. This true beta-coefficient is estimated ex post, with the right threshold value in that period. However, the threshold value in period t is unknown in period t-1. So, in table 13 data are shown following re-estimation of the MSE values assuming the threshold values remain constant between period t-1 and period t.
table 13. MSE values – constant threshold
Dual CAPM
Static
CAPM Bull Bear
Industrial
Production Risk Premium Term Structure
beta beta 1 beta 2 beta 1 beta 2 beta 1 beta 2 beta 1 beta 2
average average average average
1993-1997 0.01 0.076 0.011 8.915 0.008 0.007 0.378 0.017 0.008 0.043 4.462 0.192 0.012 1998-2002 0.01 0.010 0.278 0.003 0.709 0.181 0.048 0.193 0.022 0.144 0.356 0.115 0.108 2003-2007 0.11 0.210 0.759 0.781 0.141 0.021 0.143 0.078 0.106 0.484 0.461 0.082 0.092
Conclusion
In order to test whether adapting the CAPM to capture macroeconomic influences improves the performance of the CAPM I studied a dual-beta strategy with size-based US-stock portfolios and allowed the systematic risk to differ from three macroeconomic conditions as identified in the APT by Chen, Roll and Ross (1986) (industrial production, risk premium and term structure of interest). Performance of the macroeconomic Dual Beta CAPMs is compared to the performance of the Bull Bear Dual Beta CAPM and the original CAPM.
My research showed that the Risk Premium and the Term Structure models perform remarkably better than the Bull Bear CAPM. The betas of the Risk Premium and the Term Structure model estimated below and above the threshold value are significantly (α = 0.1) different from zero for almost half of the portfolios in all periods. This indicates that for almost half of the portfolios, the possibility of an asymmetric beta is significant. The performance of the Bull Bear CAPM and the Industrial Production model are worse. Only for 18% (Bull Bear) and 5% (Industrial Production) of the portfolios, the possibility of an asymmetric beta is significant. The Dual Beta models seems to have a better fit in terms of R-squared than the original CAPM. However, none of those differences are significant.
Assuming that the threshold values remain constant between periods, the out of sample performance of the macroeconomic dual-beta models deteriorates. It is likely that the varying threshold plays a role in the out of sample performance of the macroeconomic dual beta models.
We can conclude that the risk premium and the term structure dual beta model outperforms the bull bear dual beta model as originally proposed by Fabozzi and Francis (1977). The results of the out of sample test are mixed. The industrial production dual beta model performs remarkably less the static CAPM. The risk premium and the term structure models perform in two of the three periods better than the bull bear model and in one period better than the static CAPM. However, because the original CAPM clearly outperforms (out-of-sample) the dual beta models in the first two periods, and the fact that in the last periods using the dual beta CAPMs in-sample is inappropriate (section 5.3), further research is not recommended.
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Appendix
Table 2. Descriptive statistics of monthly returns of size-based portfolio
and market return
Table 3. Descriptive statistics of macroeconomic variables
Table 5. Beta estimation and t-statistics of beta coefficients of the CAPM and Dual Beta CAPMs in all periods
Table 8. Wald F-statistics probabilities of relevance asymmetry in Dual Beta CAPMs
Table 9. R-Squared statistics of CAPM and Dual CAPMs in all periods
Table 10. T-statistics differences in goodness of fit between Dual-beta CAPMs and CAPM
Table 11. T-statistics differences in goodness of fit between Macroeconomic and Bull Bear CAPMs
Table 14. MAE values of beta coefficients of the CAPM and Dual Beta CAPMs (assuming a time-varying threshold)
Table 2. Descriptive statistics of monthly returns of size-based portfolio and market return
smallest decile 2 3 4 5 6 7 8 9 Largest decile market Mean 1.22 1.17 1.18 1.06 1.17 1.09 1.21 1.15 1.16 0.99 0.66 Standard Error 0.37 0.39 0.35 0.34 0.33 0.30 0.29 0.30 0.26 0.26 0.26 Median 1.38 1.38 1.74 1.58 1.78 1.53 1.42 1.40 1.69 1.18 1.19 Mode 4.28 -0.37 3.61 0.23 4.92 0.98 -0.60 0.01 4.86 2.96 -2.39 Standard Deviation 5.71 5.97 5.45 5.26 5.13 4.65 4.45 4.57 4.05 3.98 3.99 Sample Variance 32.62 35.69 29.66 27.63 26.29 21.62 19.76 20.92 16.42 15.84 15.89 Kurtosis 3.48 2.32 1.23 1.09 1.05 1.29 1.34 1.24 0.80 1.03 1.12 Skewness 0.29 0.04 -0.52 -0.48 -0.54 -0.66 -0.50 -0.44 -0.47 -0.43 -0.62 Range 50.26 48.80 39.50 36.51 35.15 31.30 32.13 30.33 26.47 26.10 26.81 Minimum -20.85 -22.78 -21.09 -19.52 -20.19 -19.95 -18.68 -17.11 -14.81 -14.43 -15.99 Maximum 29.41 26.02 18.41 16.99 14.96 11.35 13.45 13.22 11.66 11.67 10.82 Count 240.00 240.00 240.00 240.00 240.00 240.00 240.00 240.00 240.00 240.00 240.00
Table 3. Descriptive statistics of macroeconomic variables
Table 5. Beta estimation and t-statistics of beta coefficients of the CAPM and Dual Beta CAPMs in all periods (sign. 0.01(***),0.05(**),0.1(*) Static Dual CAPM
CAPM Industrial production Risk premium Term structure Bull bear
Table 9. R-Squared statistics of CAPM and Dual CAPMs in all periods
Dual Beta CAPM
Table 10. T-statistics differences in goodness of fit between Dual-beta CAPMs and CAPM
H0: R2_model = R2_static CAPM
model Industrial production Risk premium Term Structure Bull Bear 1988-1992 0.010 *** 0.005 *** 0.026 ** 0.016 ** 1993-1997 0.000 *** 0.001 *** 0.002 *** 0.030 ** 1998-2002 0.027 ** 0.001 *** 0.002 *** 0.004 *** 2003-2007 0.000 *** 0.114 0.000 *** 0.005 *** all periods 0.000 *** 0.000 *** 0.000 *** 0.000 ***
Table 11. T-statistics differences in goodness of fit between Macroeconomic and Bull Bear CAPMs
H0: R2_model = R2_Bull Bear DBM
model Industrial production Risk Premium Term Structure 1988-1992 0.068 * 0.004 *** 0.073 *** 1993-1997 0.000 *** 0.001 *** 0.002 *** 1998-2002 0.439 0.034 ** 0.304 2003-2007 0.000 *** 0.168 0.000 *** All periods 0.005 *** 0.000 *** 0.000 ***
Table 14. MAE values of beta coefficients of the CAPM and Dual Beta CAPMs (assuming a time-varying threshold)
Static CAPM
Dual CAPM
Bull Bear Industrial Production Risk Premium Term Structure beta beta 1 beta 2 beta 1 beta 2 beta 1 beta 2 beta 1 beta 2
average average average average
Table 15. MAE values of beta coefficients of the CAPM and Dual Beta CAPMs (assuming a constant threshold)
Static CAPM
Dual CAPM
Bull Bear Industrial Production Risk Premium Term Structure beta beta 1 beta 2 beta 1 beta 2 beta 1 beta 2 beta 1 beta 2
average average average average