Master Thesis 2012

Time-Varying Betas

A thesis submitted in partial fulfilment of the requirements for the degree of Master in Science

in Finance

Tilburg School of Economics and Management

Tilburg University

Name: Joep Hendriks

ANR: 479903

Date: July 30, 2012

Supervisor: Dr. Rik Frehen

Department: Finance

2

Abstract

Stock betas play an important role in modern finance but are notoriously difficult to estimate,

especially if we allow beta to vary over time. A particular estimator, the conditional estimator,

defines beta as a linear function of a set of conditioning variables. The conditional betas that result

from this estimator are priced cross-sectionally, contrary to most other betas. However, in the time-

series this conditional beta shows behavior that is economically cumbersome to interpret. The

observations in the time-series compared to the cross-section create a relevant puzzle which needs

further investigation. Useless factors, i.e. factors that do not relate to asset returns and hence, that

should not be priced can possibly solve this puzzle. It is shown by Kan and Zhang (1999) that in

misspecified models, useless factors are priced more often than the size of the asset-pricing test. I

will introduce a few (informal) tests aiming at detecting the presence of useless factors since the

presence of useless factors could possibly solve this puzzle. The power, in actually detecting useless

factors, of these tests is examined by adding “true” useless factors to the set of conditioning

variables. The two main findings of this thesis are first that most of the tests, including asset-pricing

tests, are not able to detect “true” useless factors. Therefore, the second finding is that not enough

evidence is obtained to claim that useless factors solve the puzzle.

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Table of contents

Abstract 2

Introduction 4

Literature 8

Data 9

Analysis 10

Estimation of beta 10

o OLS/Vasicek beta 10

o Conditional beta 11

Asset-Pricing tests 11

o OLS/Vasicek beta 11

o Conditional beta 12

Tests for useless factors 13

Results 17

Without any added specifications of “true” useless factors as conditioning variables 17

Including specifications of “true” useless factors as conditioning variables 19

Conclusions, limitations and further directions for research 24

Conclusions 24

Limitations 25

Further directions for research 25

References 27

4

Introduction

The beta of a stock measures the systematic risk of a stock. Hence, a stock’s beta provides important

information regarding the risk profile of a stock and thus the return a stock should yield, according to

the well-known CAPM first introduced by Sharpe (1964). Estimated betas and their predictions are

used in asset-pricing, cash flow valuation, risk management, making investment decisions, or simply

as a risk factor in multiple factor models. Estimating a stock’s beta correctly and precisely is therefore

of the utmost interest within the field of asset-pricing. We should allow beta to vary over time since

it is shown, by for instance Fabozzi and Francis (1978) and Chen and Lee (1982), that beta is not

constant over time. There are roughly two methods to introduce time-series variation in the beta

estimate: the rolling-window approach and conditioning variables. Avramov and Chordia (2006)

define a general specification of estimating beta, using conditional variables, Cosemans et al. (2011)

continue using this specification and compare it to various other methods of estimating beta.

In the method proposed by Avramov and Chordia (2006) the excess returns of a specific firm are

regressed on a set of factors, containing both firm characteristics and macro-economic variables. The

beta is thereafter defined as a linear function of this set of factors, the coefficients in this linear

function being the coefficients estimated in the regression. In the sequel of my thesis, the beta which

is estimated in this way will be called the conditional beta. It turns out that this conditional beta is

very volatile over time, see for instance figure 1 where the conditional beta for Ford Motor Company

is plotted against the Vasicek beta for Ford Motor Company, i.e. a rolling window OLS beta to which a

Vasicek shrinkage estimator has been applied.

Figure 1: Vasicek beta and conditional beta over time for stock Ford Motor Company

0 50 100 150 200 250 300 350 400 450 5000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Months

Beta

Vasicek beta

Conditional beta

5

To show that the time-series behavior, i.e. excessive time-series volatility, of the conditional beta, as

shown in figure 1, not only holds for the automotive industry or for Ford Motor Company two

additional examples are shown below and on the next page. Sara Lee is a large consumer utilities

firm and DuPont operates in the chemical industry.

Figure 2: Vasicek beta and conditional beta over time for stock Sara Lee

The conditional betas of each of these firms show very volatile behavior, far more volatile than their

Vasicek counterparts. The conditional beta even jumps from values as high as +1.5 to values as low as

+0.2 or from a value equal to +1.0 to +2.2, within a few months of time. It seems unreasonable to

assume that the risk profile of a large company like Ford Motor Company or DuPont changes so

dramatically within such a short time span.

For large and mature firms like Ford Motor Company, Sara Lee and DuPont these shocks and the high

volatility in the conditional beta make economically no sense. Beta is a measure of systematic risk.

Hence, frequently and severely changing betas imply that a company’s risk profile changes

drastically, even from pro-cyclical (positive betas) to anti-cyclical (negative betas), in short periods of

time. Taking into account the maturity and size of the above presented firms, it is economically

cumbersome to validate these changes in the conditional beta. The performance of the conditional

beta in the time-series is thus very poor.

However, the cross-sectional asset-pricing tests that are performed on this conditional beta by

Cosemans et al. (2011) show remarkably good performance. When tested, the conditional beta for

individual stocks returns a t-statistic, associated to , equal to 2,58 and a R-squared of 3,78%. The t-

statistic indicates that is significantly different from zero at the 5% level, making the conditional

beta being the only beta estimate returning a statistically significant . The R-squared is also

0 50 100 150 200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Months

Beta

Vasicek beta

Conditional beta

6

amongst the highest values. Hence, the conditional beta turns out to be priced in the cross-section,

contrary to betas estimated using other methods.

Figure 3: Vasicek beta and conditional beta over time for stock DuPont

The observation that cross-sectional performance of the conditional beta is very good, while in the

time-series the conditional beta exhibits very poor performance creates a puzzle. Solving this puzzle

has not yet been done in the existing literature and is definitely a relevant addition to the current

literature since beta is a very important notion in finance. The current consensus regarding beta is in

my opinion well reflected by Fama and French (1992, 1996): “Beta is dead” , since the betas

generated by CAPM do not relate to returns and hence, are not priced. A beta estimated in a

different manner, based on a set of conditioning variables, turns out to be very “alive” but it has no

economic interpretation. Are these results due to some technicalities that under certain

specifications the models return these results or can general ideas regarding beta and asset-pricing

tests not always be related to general economic ideas/principles? The aim of this thesis is to solve

this puzzle, i.e. answer the question why it is possible that betas that perform bad in the time-series,

turn out to be priced cross-sectionally.

In Kan and Zhang (1999) I find the first notion of useless factors. Ferson and Harvey (1999) later on

also use useless factors in the explanation of their results. A useless factor can be described as a

factor that is, in the extreme case, independent of the asset returns. If this is the case, the risk

premium associated to beta risk should be zero. However, when a model is misspecified, i.e. it

contains useless factors, Kan and Zhang (1999) show that beta risk is priced more often than the size

of the test. This effect does not decrease for larger samples. On the contrary, the effect even

increases when the sample grows larger. For misspecified models, asset-pricing tests return results

based upon which wrong conclusions might be drawn.

0 100 200 300 400 500 6000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Months

Beta

Vasicek beta

Conditional beta

7

Therefore the two-pass methodology, introduced by Fama and French (1973), as conducted in

Cosemans et al. (2011) is probably not justified. Useless factors could possibly solve the puzzle and

hence, be the explanation that when the two-pass methodology is applied nonetheless, this leads to

the contradictory observations as discussed earlier.

In this thesis, part of the results presented in the paper by Cosemans et al. (2011) will be reproduced.

Thereafter I will elaborate more on the phenomenon of useless factors, as defined by Kan and Zhang

(1999). Finally it is tried to establish whether these useless factors play a role in the (contradictory)

results obtained by Cosemans et al. (2011). I will do this based on a number of tests, unfortunately

neither of them being formal since in the literature no critical values, associated to the resulting

statistics, can be found. The tests used follow largely the approach proposed by Kan and Zhang

(1999) and will test amongst others the stability of asset-pricing tests over time on subsamples, the

hypothesis whether the average beta is different from zero and for every beta individually the null-

hypothesis that that specific beta is different from zero. Also different specifications of “true”

useless factors will be added to the set of factors based on which the conditional beta is estimated.

Using this approach it can be examined what happens with the results, stemming from the previously

mentioned tests whose aim are detecting useless factors, i.e. I can examine the power of these tests.

Basically two findings have been established in this thesis, the first finding being that not enough

proof is found to claim that useless factors solve the puzzle. The lack of proof relates to the second

finding, namely that tests aiming at detecting useless factors have very little power. Especially asset-

pricing tests are not powerful since adding “true” useless factors as conditioning variables, increases

the performance of the conditional beta in both the asset-pricing test on the whole sample and in

asset-pricing tests on subsamples. Hence, asset-pricing tests are probably not able to indicate the

presence of useless factors as conditioning variables. The test in which for every beta individually it is

tested whether this beta is statistically significant different from zero shows, in general, a declining

percentage of statistically significant betas when “true’’ useless factors are added. In the original

specification, i.e. the specification without any “true” useless factors as conditioning variables, the

percentage of statistically significant betas is pretty low. Although evidence is thin, based on this test

an explanation for the, in the cross-section, well priced conditional betas which lack economic

interpretation in the time-series, could be that the set of factors as proposed by Cosemans et al.

(2011) is to some extent useless.

The remainder of this thesis is organized as follows. I will continue with providing some background

information on why models aiming at estimating beta have transformed from the simple static CAPM

to the current, more extensive models. Thereafter the methodology will be discussed and empirical

results, of the tests performed, will be presented. Subsequently, I will draw some conclusions and

expound the statement that the set of factors, as proposed by Cosemans et al. (2011), might be

useless to some extent. This thesis is concluded by discussing some limitations and possible

directions for further research.

8

Literature

In this part of the thesis I will describe some of the literature and background on estimating beta and

explain that complex methods such as time-varying betas, shrinkage estimators and conditioning

variables are needed because it has been shown empirically that more simple models don’t price

beta.

The CAPM by Sharpe (1964) has long been a basic tenet in finance. However, subsequent work by

amongst others Fama and French (1992, 1996), Basu (1977) and Banz (1991) show that the CAPM

does not succeed in explaining cross-sectional differences in asset returns very well. These studies

show that average returns do not only depend on market risk, as suggested by the CAPM, but for

instance also depend on book-to-market ratio, size and prior returns.

Next to the above explanation for the failure of the CAPM, it has also been suggested that its static

nature is cumbersome. For example, Hansen and Richard (1987) show that even when a static CAPM

model fails completely, a dynamic version might be very valid. Models in which the book-to-market

ratio and a firm’s size depend on the market beta are developed by Gomes, Kogan and Zhang (2003).

A popular way to solve the problem concerning the static nature of the CAPM is to let the beta vary

over time. For example, this is done by Hansen et al. (1982), Jagannathan and Wang (1996) and Faffa

and Brooks (1998). In this last paper, the time-varying beta risk is measured during a long period

sample by considering a mean level of beta which is expected to decrease or increase in a number of

identifiable sub periods. These sub periods are mutually exclusive, i.e. different from the rolling-

window approach which is also commonly used. When using this rolling-window approach, it is

assumed that the future is similarly enough to the past otherwise the time-varying betas do still not

change quickly enough in order to adapt to changing market conditions. Studies, by for instance

Chiarella et al. (2010) and Nieto et al. (2011), show that the prior assumption usually not holds and

hence, a time-varying beta obtained by the rolling-window approach, still has little explanatory

power regarding differences in asset returns.

Another approach to let beta vary over time, is the approach taken by Avramov and Chordia (2006).

This is the approach I already shortly discussed in the introduction since it is the approach under

investigation in this thesis. The approach proposed by Avramov and Chordia (2006) can be split in

two parts. They first regress excess stock returns on asset-pricing factors that may vary cross

sectional and over time. Factors proposed by Avramov and Chordia (2006) include both stock specific

factors like size and book-to-market ratio as well as macro-economic variables like the default spread

whose aim is to capture the business cycle. The beta proposed by Avramov and Chordia (2006) is a

linear function of these factors. Thereafter, they run cross-sectional regressions, with risk-adjusted

returns as the dependent variable, on size, book-to-market ratio, turnover and variables related to

past return. Under the null-hypothesis of exact pricing, these characteristics should be insignificant.

In Cosemans et al. (2011) time-varying betas are estimated by a number of sophisticated methods.

One of these methods is the conditional estimator, the main focus of this thesis. This conditional

estimator is equivalent to the approach taken by Avramov and Chordia (2006).

By evolution, methods of estimating beta have become increasingly complex and sophisticated

because simple methods do not price beta in the cross-sectional asset-pricing tests.

9

Data

The firm data source is CRSP / Compustat and the data consists of monthly returns, book-to-market

ratio, size, financial leverage, market risk premium and the default spread for all NYSE and AMEX

listed stocks. The sample covers the period from August 1964 up to and including December 2011. A

given month for a given stock is included when it belongs to a consecutive period of at least 36

months for which all data is available in the current month (the return and market risk premium) or

in the previous month (the book-to-market ratio, size, market value of equity and default spread).

Following Fama and French (1992) and Cosemans et al. (2011) book-to-market ratio is measured by

the ratio of book- and market value of equity, size by the market value of equity and financial

leverage by the book value of assets over the market value of equity.

By applying the above mentioned restrictions on my initial dataset consisting of 11,535 stocks, I end

up with a dataset consisting of 10,539 stocks. For each month separately, I include on average 1,315

stocks in my dataset.

10

Analysis

Three methods will be used to estimate time-varying betas:

1) OLS: sample estimate based on rolling windows of monthly returns

2) Conditional: function of instruments estimated by time-series regression

3) Vasicek: shrinkage estimator applied to rolling windows of monthly returns

The Vasicek betas are based on the OLS betas, the difference being that the Vasicek betas are shrunk

towards a certain prior. Based on the two-pass methodology proposed by Fama and MacBeth (1973),

betas will be estimated first and thereafter the validity of the estimated betas will be tested using

asset-pricing tests.

Estimation of beta

OLS/Vasicek beta

For these betas I first calculate how many rolling windows I have for each stock based on the

number of data points for this stock given a certain minimum length of the rolling windows . To

obtain enough estimation precision I use rolling windows containing 36 months (3 years) of data.

Furthermore, my rolling windows are overlapping rather than mutually exclusive. When one rolling

window covers the period [ ] the subsequent window covers the period [ ]. This

approach only works under the assumption that the future is similar enough to the past because

betas will vary over time only gradually while changes in firm characteristics have an immediate

impact on a firm’s beta. In other words, this assumption comes down to implicitly assuming beta to

be constant within a specific window.

As a prior for the Vasicek beta I calculate for each stock, based on all data points available for this

stock, the static CAPM beta. Subsequently I calculate the mean and variance of the 10,539 estimated

betas which provides me with a prior for the Vasicek beta. Using the values suggested by Vasicek

(1973), and yields very similar results.

The Vasicek beta is now shrunk towards the prior:

( )

Where denotes the prior beta, denotes the OLS rolling window estimate of beta and

denotes the shrinkage estimator. is defined as:

Where denotes the prior variance of the Vasicek beta and where

represents the sampling

variance of which is defined as:

( )

11

In this last definition represents the estimate of the idiosyncratic volatility for stock during

rolling window and denotes the variance in market return during rolling window which has

length months.

Conditional beta

These betas are estimated using time-series regression without rolling-windows. The conditional

betas are a function of instruments / firm characteristics:

Where denotes the state of the business cycle, measured by means of the default spread, in the

previous month and contains the values of the other firm characteristics in the previous

month: book-to-market ratio, size and financial leverage, as they are proposed by Cosemans et al.

(2011).

So, based on all data points available for a specific stock , I estimate the following equation:

( )

Which provides me with estimates of all the parameters ,

Subsequently I can calculate for every month that data is available for stock , using the obtained

parameter estimates for stock the conditional beta according to the expression:

Asset-Pricing Tests

Asset-pricing tests are the second pass in the two-pass methodology proposed by Fama and MacBeth

(1973). The intention of the asset-pricing tests is to test whether estimated betas are priced cross-

sectionally, i.e. result in a positive market risk premium. This can be tested by estimating the

following equation for every month:

Where denotes the estimated Vasicek or conditional beta.

Since the Vasicek betas are estimated in a different way than the conditional betas, the approach

taken in the asset-pricing test of either beta is designed to capture the estimation error inherent to

as well as possible. Theoretically one would expect to be equal to zero and to be equal to

the average monthly market risk-premium, which is about 0,45%1.

OLS/Vasicek beta

For the Vasicek beta the above mentioned equation is estimated every month for which data

(returns and estimated betas) is available, so the first 36 months are excluded due to the rolling

window approach.

1 Rietz., T., 1988, The Equity Risk Premium: A Solution, Journal of Monetary Economics, 22 (1): 117-131

12

Once all regressions have been run, is defined as:

∑

Where denotes the number of months for which data is available and hence, also the number of

estimated . is defined analogously.

The variance associated to

is defined as:

∑(

)

The variance associated to

is defined analogously.

However, a correction with respect to and

has to be made. Because the independent variable

in the regression/asset-pricing test, , is not observed but estimated itself, the variance as defined

by and

would indicate too much estimation precision in

and , respectively. This effect is

called the Errors-In-Variables problem. In order to account for this effect I apply the Shanken

correction, proposed by Shanken (1992).

Let denote the matrix of factors based on which the betas are estimated, for the Vasicek beta this

comes down to a matrix containing the market risk premium. The Shanken correction ( ) is

then defined as:

√ ( )

Where ( ) denotes the variance of the matrix of factors based on which the betas, in this case

the Vasicek betas, are estimated.

can thereafter simply be multiplied by ( ) in order to account for the Errors-In-Variables

problem.

The test statistic, in the asset-pricing test is defined as:

√ ( )

The test statistic follows a student-t distribution.

Conditional beta

The asset-pricing test used for the conditional beta is very similar to the one used for the Vasicek

beta. As described earlier, in the first pass an equation is estimated:

( )

13

Using the estimated coefficients, for every month the conditional beta can be calculated by:

These estimated conditional betas would subsequently be used in the asset-pricing test:

Hence, is a linear combination of estimated coefficients.

To account for the estimation imprecision, due to the E rrors-In-Variables problem as discussed

earlier, I also apply the Shanken correction in the asset-pricing tests concerning the conditional

betas. In the working paper by Chordia et. al (2011) a very involving method is proposed to apply the

Shanken correction optimally in the asset-pricing tests for the conditional betas. However, since this

is still a working paper and the effect of the Shanken correction will be very small since I use a large

dataset, I restrain from the method proposed by Chordia et al. (2011) and choose to apply the same

Shanken correction to the asset-pricing tests regarding the conditional betas as I applied to the asset-

pricing tests for the Vasicek betas:

√ ( )

Where symbols are defined analogously to the Shanken correction in case of the Vasicek betas.

The asset-pricing tests for the conditional beta are very similar to the asset-pricing tests for the

Vasicek beta. Hence, I refer to the previous section where the asset-pricing tests involving the

Vasicek betas are discussed.

Tests for useless factors

The next step is to test for useless factors. As already discussed in the introduction, can a useless

factor be described as a factor that has zero, or almost zero, correlation with stock excess returns. In

this case, beta risk should not be priced. However, when a model is misspecified, i.e. it contains

useless factors, Kan and Zhang (1999) show that beta risk is priced more often than the size of the

test. For misspecified models, asset-pricing tests return results based upon which wrong conclusions

might be drawn. Since these useless factors could possibly solve the puzzle, defined in the

introduction, I would like to test the factors used to estimate the conditional betas on their

“usability”. Such a test should be capable of detecting useless factors within a set of factors, based

upon which betas are estimated.

It is very difficult to apply a formal test for useless factors, in the literature no such test is known. Kan

and Zhang (1999) suggest four (informal) tests to get an indication regarding the presence of useless

factors, I will use three of these tests. Furthermore I will add “true” useless factors, factors which are

randomly drawn from pre-specified distributions, to the factors on which the conditional beta is

based. This approach allows to examine how results change under the influence of “true” useless

factors and to test the power in detecting useless factors of the informal tests proposed by Kan and

Zhang (1999).

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Regress beta on a constant

The first test proposed by Kan and Zhang (1999) is to regress the betas resulting from either

estimation methodology, both Vasicek betas and conditional betas, on a constant, using standard

Ordinary Least Squares, to test the hypothesis:

It is useful to perform this test since you expect the average beta of all companies to be close to one.

If this is not the case, i.e. one cannot reject the above stated hypothesis, one should be concerned

about useless factors since in this case there would be excessive time-series variation.

Once I have an estimate for , the above stated hypothesis can easily be tested. However, it should

be noted that the power of this test will probably be very limited. The power of this test can be

examined by introducing “true” useless factors, i.e. randomly drawn vectors, to the equation. I will

elaborate more on this later.

Perform asset-pricing tests on subsamples

A second test that will be performed in order to test for useless factors is to perform asset-pricing

tests on subsamples, such that you also obtain ’s and on subsamples. According to Kan and

Zhang (1999) these ’s should be constant over time, with and , in the absence of

useless factors. I can rather easily calculate the estimated values for these ’s and their associated

variance on the subsamples by:

∑

( )

Where denotes the subsample and denotes the size of the subsamples. is defined

analogously.

The variance for the subsample is calculated by:

∑ ( )

( )

Where again denotes the subsample and denotes the size of the subsamples.

is defined

analogously.

The Shanken correction, proposed by Shanken (1992) and discussed previously, is also applied to

these asset-pricing tests on subsamples. I will run these asset-pricing tests on subsamples for both

the Vasicek and the conditional betas. I will not perform these tests regarding useless factors for the

OLS rolling window beta since in the asset-pricing tests, it turns out that this beta is even priced

worse cross-sectionally than the Vasicek beta. Hence, I choose the Vasicek betas as a reference for

the conditional betas.

The methodology as described in this section gives me another test/indication of the presence of

useless factors. However, although the power of this test is probably larger than the power of the

15

test in which the betas are regressed on a constant, the power will still be limited since ordinary

asset-pricing tests do not succeed in detecting useless factors. Asset-pricing tests on subsamples

won’t have much greater performance in this respect. Moreover, also this test faces the problem

that no critical values are available.

Test individual betas

A third, and final, test which might give some information regarding the presence of useless factors is

to test every beta individually with the null-hypothesis being:

I will perform these tests only for the conditional betas, since those betas are my main source of

interest.

In the first pass I have already estimated all ’s. Hence, I only have to calculate the variance of

every in order to be able to test the above mentioned null-hypothesis. Recall that the conditional

beta is defined as a function of factors which are assumed to be linearly independent:

Therefore the variance of the conditional beta is defined as:

( ) ( )

( ) ( ) ( ) ( )

( )

( )

( )

By combining the point estimate of and its variance I can make inferences regarding the

hypothesis stated above. I’m particularly interested in the percentage of

that are statistically

different from zero at the 5% level. The power of this test should be higher than the power of the

other two tests discussed in this section.

Add “true” useless factors

I will perform the previously discussed methods to test for useless factors on both the conditional

beta estimated using the factors size, book-to-market ratio, financial leverage, market risk premium

and the default spread as they are proposed by Cosemans et al. (2011). Subsequently, to this set of

factors (and ) I will also add different combinations/specifications of randomly drawn factors,

from different distributions, as conditioning variables. This approach yields me an opportunity to

examine the power of the above mentioned tests and provides me with some idea whether the

second pass, the asset-pricing test, works in the case of conditional betas. When I run the asset-

pricing tests on conditional betas which are generated using a certain combination of “true” useless

factors as conditioning variables, the performance of these betas in the asset-pricing tests should be

worse than the performance of conditional betas generated without “true” useless factors.

16

Since I measure the performance by means of the of the asset-pricing tests and the t-statistics

associated to the ’s resulting from the asset-pricing tests, one would expect the to decrease and

the t-statistic associated to and to increase and decrease, respectively, when more “true”

useless factors are added to the set of conditioning variables. The useless factors will be drawn from

uniform- and normal distributions for various different parameter values.

17

Results

Without any added specifications of “true” useless factors as conditioning variables

Hereby I present in table 1 some statistics regarding the estimated betas: OLS rolling-window betas,

Vasicek betas and the conditional betas.

OLS rolling-window Vasicek Conditional

Time series average 1,0726 0,9970 1,0493

Time series variance 0,6576 0,5936 1,0911

Time series kurtosis 6,5437 103 8,6197 103 27,5605 Table 1: summary statistics regarding the various different estimated betas in the time-series

From table 1 I see that all the betas are on average approximately equal to 1, which obviously should

be the case. Moreover, the variance of the conditional betas is much larger than the variance

associated to the Vasicek betas. For some stocks this observation was already mentioned and

illustrated during the introduction. The large variance associated to the conditional beta has no

economic interpretation. As already explained in the introduction it is economically cumbersome to

explain how the beta of a large and mature firm can vary drastically over time. Furthermore I observe

the presence of large differences in kurtosis when comparing the three methods of estimating betas

to each other. Especially the difference in kurtosis, between on the one hand the OLS/Vasicek beta

and on the other hand the conditional beta, is striking when relating it to the performance of the

different betas in asset-pricing tests. I will discuss this relationship after having shown the results of

the asset-pricing tests.

Subsequently I will present results of the asset-pricing tests for both the Vasicek betas and

conditional betas:

OLS rolling-window Vasicek Conditional

Value 0,7147 -0,0109 0,8925 -0,1881 0,0754 0,3286

T-statistic 5,5064 -0,0628 2,5611 -0,3616 1,4742 3,2711

15,36% 15,78% 17,98% Table 2: results of the cross-sectional asset-pricing test for each of the three different beta estimators. Firm characteristics

are not included in the asset-pricing tests.

The first conclusion I draw from table 2 is that the Vasicek beta is priced badly in the cross-section,

however, it is still priced better than the OLS rolling-window beta. Hence, in the sequel of the

analysis I will omit the OLS-rolling window beta.

Next, from table 2 it can be easily inferred that the conditional betas outperform the Vasicek betas in

the asset-pricing tests in the sense that the latter is priced worse, cross-sectionally. In the case of the

conditional betas, is statistically insignificant different from zero as it should be, contrary to the

result in the case of the Vasicek betas. Moreover, the , resulting from the test on the conditional

betas, is statistically significant different from zero, positive and approximately equal to the market

risk-premium which is 0,45. Again, this result is not obtained when the Vasicek betas are used in the

asset-pricing test. Finally, the higher indicates a better performance by the conditional betas.

Now I will elaborate more on the relationship between the difference in kurtosis and the results of

the asset-pricing tests. The interpretation of a larger kurtosis is that more of the variance stems from

18

infrequent large changes rather than frequent, modest in size, changes. This would imply that the

Vasicek beta is subject to infrequent large changes compared to the conditional beta. Hence, the

performance of the Vasicek beta in the asset-pricing test might be poor due to a number of extreme

values which hurt overall stability severely.

Next, I discuss some results regarding the tests for useless factors. When I run the asset-pricing tests

on subsamples I obtain the results as presented in table 3 below. I take mutually exclusive

subsamples of size equal to 20 months. Note that since I already estimated ’s per month in the

asset-pricing tests using the whole sample, the ’s on subsamples can be calculated directly from

these ’s. Hence, there is no need to run any new asset-pricing tests. The size of the subsamples is

therefore not really a relevant factor considering the amount of estimation precision, to increase the

power of the test subsamples should not be chosen too large.

Vasicek Conditional

Average 1,0314 0,1249

Average -0,4136 0,6019

Variance 4,2353 0,4196

Variance 8,6218 1,1070

Average / Variance 0,2435 0.2977

Average / Variance -0.0480 0.5437

Variance 2,2406 1,9036

Variance 1,4614 1,4020

% insignificant 88,00% 88,46%

% significant 4,00% 11,54% Table 3: results of the asset-pricing tests on subsamples. For subsamples of size equal to 20 months I performed for each

mutually exclusive subsample the asset-pricing test as described in the analysis. I omit the OLS rolling-window beta since in

table 2 I see that the beta obtained using this estimator is priced even worse in the cross-section than the Vasicek beta.

Reported averages of the ’s and t-statistics are concerning the average of the ’s and t-statistics obtained for each

subsample. Reported variances are defined analogously. The percentages of insignificant and significant denote

the percentage of subsamples of the total number of subsamples in which and are insignificant and significant,

respectively, at the 5% level.

When comparing the results for both betas I see that a number of statistics differ substantially. The

average and differ a lot across the different estimation methods, as well as their associated

variance. This is in line with what one would expect based on the performance of the conditional

beta in the ordinary asset-pricing test compared to the performance of the Vasicek beta. The fact

that the variance of is low in the case of the conditional beta strongly correlates with the good

performance of the latter in the asset-pricing test of which the results can be found in table 2 on the

previous page. The low variance associated to , in case of the conditional beta, also results in a

high ratio of the average divided by its variance. When looking at the t-statistics they seem to be

equally stable over time comparing both methods of estimation. The higher percentages of

insignificant and significant at the 5% level, resulting from the asset-pricing tests on the

conditional betas, underline the performance of these betas in asset-pricing tests regardless whether

these asset-pricing tests are performed on subsamples or not.

19

Next, I will test the hypothesis that the average beta is statistically significant different from zero:

In the previous section, Analysis, I explained a method involving a regression on a constant in order

to test the above stated hypothesis. Since I have a large number of betas this is practically

cumbersome, hence I will define my statistic as:

√

Where denotes the number of betas, denotes the sample mean of beta and denotes the

square root of the sample variance of beta. On the next page you find the statistic for both the

Vasicek beta and conditional beta.

Vasicek Conditional

Mean beta 0,9970 1,0493

Value statistic 1353,18 1214,82

Significant at 5% level? Yes Yes Table 4: results of the hypothesis for both the Vasicek beta and conditional beta, I omit the OLS rolling-window

beta on previously discussed grounds. The statistic

√ follows a student-t-distribution with degrees of freedom.

I see that both betas are highly, statistically significant, different from zero. This result is due to the

very large number of data points and a mean which is different from zero. This test has very little

power. Hence, I will omit the test for the case in which I add specifications of “true” useless factors as

conditioning variables.

The result of the tests in which the hypothesis:

is tested, can be presented in a single number: the percentage of ’s which is significantly different

from zero at the 5% level. As mentioned earlier, I only performed these tests for the conditional

betas. It turns out that 6,9984% of the individual conditional betas is significantly different from zero

at the 5% level. This low number combined with the good performance of the ’s in the asset-

pricing tests and the results presented in table 4, indicate that on average the ’s might perform

well but when tested individually, is far too noisy. Out of the three tests discussed, this test is

probably the most powerful.

Including specifications of “true” useless factors as conditioning variables

I perform the previously discussed method now for the conditional beta which is not solely

generated by the factors size, book-to-market ratio, financial leverage, market risk premium and the

default spread as they are proposed by Cosemans et al. (2011). I will use four different specifications

of randomly drawn factors which are added as conditioning variables to the set of factors proposed

by Cosemans et al. (2011).

20

In table 5 I present an overview of these specifications:

( ) ( ) ( ) ( ) ( ) ( )

Specification 1 X

Specification 2 X X

Specification 3 X X X X

Specification 4 X X X X X X Table 5: overview of different specifications of “true” useless factors which will be added as conditioning variables to the

set of already used conditioning variables - the set proposed by Cosemans et al. (2011) and discussed before - in each of the

four different specifications.

Due to the addition of “true” useless factors to the set of conditioning variables, the only beta that

changes is the conditional beta. Hence, the conditional beta generated by different specifications of

factors is the main source of interest in this section of my thesis.

In table 6 I present some statistics which provide information regarding the estimated conditional

betas:

Original specification

Specification 1 Specification 2 Specification 3 Specification 4

Time series average

1,0493 1,1317 1,1272 1,0838 1,1315

Time series variance

1,0911 11,9228 15,1684 18,4161 28,8969

Time series kurtosis

27,5605 14,7315 13,6174 14,8182 12,5836

Table 6: summary statistics regarding the various different estimated conditional betas in the time series.

When comparing the results presented in table 6 I conclude that the numbers do not change in terms

of magnitude. Neither can I observe a clear direction for most of the statistics when adding more

“true” useless factors as conditioning variables, except for the variance of the betas. It is also

intuitively correct that the variance of the conditional beta should increase when adding more

factors since the conditional beta is a function of these factors, hence the variance of the conditional

beta is the sum of the variances of the individual factors. Variances are always positive and therefore

the variance of the conditional beta will increase when adding additional factors.

Subsequently, I present the results from the asset-pricing tests when these tests are performed using

conditional betas which are generated on a set of variables containing amongst others “true” useless

factors. The results can be found below in table 6:

Original specification

Specification 1 Specification 2 Specification 3 Specification 4

Value 0,075 0,329 0,603 0,184 0,544 0,235 0,442 0,331 0,467 0,296

T-statistic 1,474 3,271 2,650 11,019 2,362 16,218 1,884 25,421 1,971 32,559

17,98% 14,33% 14,48% 15,01% 15,21% Table 7: results of the asset-pricing tests for each of the four specifications of “true” useless factors. The results for the

original specifications are copied from table 2. Firm characteristics are not taken into account in these asset-pricing tests.

21

When I look at table 7 I notice a striking result; when adding more “true” useless factors to the set of

conditioning variables I see that the conditional betas’ performance in the asset-pricing test increases

drastically. This increase in performance is best inferred from the increasing t-statistic associated to

but also follows from the increasing R-squared. If the cross-sectional asset-pricing test would be

capable of detecting useless factors, this R-squared should gradually move towards zero when more

and more “true” useless factors are added as conditioning variables. However, when comparing the

original specification to specification 1 in terms of R-squared I observe a drop. Since the

interpretation of the R-squared reads as the R-squared being the part of the variance in the

dependent variable that is explained by the independent variables in a linear model, it can be

concluded that when adding “true” useless factors, as conditioning variables, to the model without

any “true” useless factors less variance in the returns is explained by the conditional beta.

In some specifications also becomes statistically significant at the 5% level, but no clear trend can

be determined. In specification 1, in which “true” useless factors are added, I see the value of also

increase compared to the original specification. This is probably due to the fact that not all “true”

useless factors are drawn from a distribution with mean equal to zero.

Results from the asset-pricing tests on subsamples can be found in table 8, on the next page. As a

size of the subsamples I again take 20 months.

When comparing results for the different specifications, as presented in table 8, I observe a few

interesting trends. When adding “true” useless factors as conditioning variables, the average of the

’s doesn’t show a clear trend except for the previously discussed observation that in the

specifications including “true” useless factors, the average of the is higher than in the specification

without any “true” useless factors. Another interesting trend is that the variance of the ’s

decreases severely when more “true” useless factors are added as conditioning variables, i.e. the

become increasingly stable over time when “true” useless factors are added. This low variance

causes the ratio of the mean over the variance to rise steeply.

The variance of the t-statistic regarding the also increases when “true” useless factors are

added as conditional variables, however, this is largely due to the increase in mean of these t-

statistics since the ratio mean over variance doesn’t show a clear direction in magnitude. When

comparing the average of t-statistics resulting from the asset-pricing tests performed on subsamples

to the results presented in table 7, I see that the latter is higher for each of the five specifications.

This observation is probably due to the definition of the variance of the ’s in the asset-pricing test:

∑(

)

If we now also recall the definition of the variance of the ’s in the asset-pricing test on subsamples:

∑ ( )

( )

22

We see that the sum of squared errors is divided by the squared number of months taken into

account by the asset-pricing test. Due to this last operation, the transformation from the asset-

pricing test to the asset-pricing test on subsamples is linear in terms of the mean of but of second

order in terms of the associated variance Hence, t-statistics resulting from the asset-pricing test on

subsamples are lower than the t-statistics presented in table 7.

Better performance of the conditional beta in asset-pricing tests can also be inferred from the higher

percentages of insignificant ’s and significant ’s, at the 5% level, when additional “true”

useless factors are added as conditioning variables.

Original specification

Specification 1 Specification 2 Specification 3 Specification 4

Average 0,1249 0,5849 0,5259 0,4223 0,4469

Average 0,6019 0,1822 0,2330 0,3303 0,2957

Variance 0,4196 1,2078 1,2300 1,2952 1,3022

Variance 1,1070 0,0077 0,0064 0,0040 0,0033

Ratio Average over Variance

0.2977

0,4843

0,4276

0,3261

0,3432

Ratio Average over Variance

0.5437

23,6623

36,4063

82,5750

89,6081

Mean

0,3454 0,7320 0,6586 0,5516 0,5665

Mean

0,8354 2,5373 3,6411 5,8516 7,5994

Variance

1,9036

1,1171

1,0615

1,0475

1,0465

Variance

1,4020

1,9396

2,1356

3,8201

7,2901

Ratio Average over Variance

0,1814 0,6553 0,6204 0,5266 0,5413

Ratio Average over Variance

0,5959 1,3082 1,7050 1,5318 1,0424

% insignificant

88,46%

88,89%

88,89%

92,59%

96,30%

% significant

11,54%

66,67%

88,89%

100,00%

100,00%

Table 8: results of the asset-pricing tests on subsamples. For subsamples of size equal to 20 months I performed for each

mutually exclusive subsample the asset-pricing test as described in the analysis. Reported averages of the ’s and t-statistics

are concerning the average of the ’s and t-statistics obtained for each subsample. Reported variances are defined

analogously. The percentages of insignificant and significant denote the percentage of subsamples of the total

number of subsamples in which and are insignificant and significant, respectively, at the 5% level.

23

Also for the four different specifications I ran tests in order to test the hypothesis:

In table 9, on the next page, I present the result of this test for every different specification.

Original specification

Specification 1 Specification 2 Specification 3 Specification 4

Percentage of significant betas

6,9984% 11,4668% 10,6507% 8,5545% 10,1037%

Table 9: results of the hypothesis test . The percentages represent the part of betas which is statistically

significant different from zero and positive at the 5% level. The general trend that this percentage declines when more

“true” useless factors are added as conditioning variables, indicates that this test is to some extent capable of detecting

useless factors.

When comparing the results for the different specifications I see that in general the percentage of

statistically significant from zero, and positive betas, at the 5% level, decreases when more “true”

useless factors are added as conditioning variables. This would indicate that this test is capable of

detecting the presence of useless factors, contrary to the tests discussed previously. The percentage

of statistically significant betas in the original specification is pretty low, hence, based on this test

one could conclude that the factors used to estimate the conditional betas by Cosemans et al. (2011)

are to some extent useless.

However, I should be careful in calling factors like book-to-market and size useless since there is a lot

of literature that validates the use of these factors, see for instance Fama and French (1992) or Wang

(2004). But on the other hand is the set of conditioning variables not fully known, as already stated

by Roll (1977). What exactly is the market? A proxy portfolio is used but how good is this proxy?

Furthermore, in Cosemans et al. (2011) the default spread is a proxy for the business cycle but again

one can question how good this proxy is. Perhaps, not all factors are the correct ones and hence,

there is a possibility that some of them might be useless.

24

Conclusions, limitations and further directions for research

Conclusions

In work by Avramov and Chordia (2006) a method for estimating beta is proposed in which the beta

is a function of firm characteristics and macro-economic variables. Coefficients in this function are

obtained via regression of excess stock returns on these firm characteristics and macro-economic

variables. When the estimated, so-called, conditional betas are tested in asset-pricing tests it was

found that these betas are priced more often compared to betas found by other estimation methods,

for instance the method proposed by Vasicek (1973). This result in itself is obviously desirable,

however, the characteristics of the conditional beta have no economic interpretation in the time-

series. As showed in the introduction, the conditional beta for large and mature firms like Ford Motor

Company, Sara Lee and DuPont move all over the place in short periods of time. Within a few months

time we see the conditional beta for Ford Motor Company jump from +1.5 to +0.2 which would

indicate that Ford Motor Company has transformed from a pro-cyclical company to an almost anti-

cyclical company in a short time interval.

In this thesis I have conducted research to understand the above discussed, contradictory, results

that on the one hand the conditional beta performs very well in cross-sectional asset-pricing tests

but on the other hand this beta hardly has any economic interpretation in the time-series, since in

the time-series the conditional beta is subject to excessive volatility. I started with validating the

results found by Cosemans et al. (2011). An important notion to a possible explanation for the

contradictory results was found in Kan and Zhang (1999), the notion of useless factors. Recall that a

useless factor, in the extreme case, is characterized by independence of asset returns. A number of

tests were performed in order to detect the presence of useless factors. Unfortunately, these tests

are informal and the power of most of the tests is rather limited. To test for the power of these tests

and to see what happens in the asset-pricing tests when conditional betas are tested which are

subject to even more noise, I redid the whole analysis using conditional betas which are generated

based on the both the factors proposed by Cosemans et al. (2011) and on a number of “true” useless

factors, i.e. factors randomly drawn from a certain distribution, which are added to the set of

conditioning variables.

Basically two findings have been established in this thesis, the first finding being that not enough

proof is found to claim that useless factors solve the puzzle. The lack of proof relates to the second

finding, namely that tests aiming at detecting useless factors have very little power. Especially asset-

pricing tests are not powerful since adding “true” useless factors as conditioning variables, increases

the performance of the conditional beta in both the asset-pricing test on the whole sample and in

asset-pricing tests on subsamples. Hence, asset-pricing tests are probably not able to indicate the

presence of useless factors as conditioning variables. The test in which for every beta individually it is

tested whether this beta is statistically significant different from zero shows, in general, a declining

percentage of statistically significant betas when “true’’ useless factors are added. In the original

specification, i.e. the specification without any “true” useless factors as conditioning variables, the

percentage of statistically significant betas is pretty low. Although evidence is thin, based on this test

an explanation for the, in the cross-section, well priced conditional betas which lack economic

interpretation in the time-series, could be that the set of factors as proposed by Cosemans et al.

(2011) is to some extent useless.

25

However, as already discussed, I should be careful in calling factors like book-to-market and size

useless since there is a lot of literature that validates the use of these factors, see for instance Fama

and French (1992) or Wang (2004). But the set of conditioning variables is not fully known, and is

approximated by factors, see Roll (1977). Hence, some of these proxies might not be very good which

creates possibilities for factors being useless.

Limitations

A first note I want to make is that the rolling-window approach, as applied in this thesis to the OLS

and Vasicek betas, is subject to the limitation that estimated betas cannot vary quickly enough over

time. Since the rolling-windows are not mutually exclusive, the estimated betas cannot adapt quickly

enough to changing firm characteristics which have an immediate effect on the firm’s beta. Hence,

the rolling-window approach as applied, only works under the assumption that the future is similarly

enough to the past. It is shown empirically, by for instance Chiarella et al. (2010) and Nieto et al.

(2011), that this assumption often doesn’t hold.

The second limitation I would like to discuss concerns the tests aiming at detecting useless factors

and hence, the reliability of betas estimated by means of the conditional estimator. In this thesis a

puzzle has been investigated which has not yet been solved or discussed in the literature. Also the

phenomenon useless factors is not discussed extensively in the literature. On the one hand these are

nice aspects, we are exploring new parts of “Planet Finance”. On the other hand this means that not

much is available in terms of well-defined formal statistical tests, critical values and statistics. The

current literature regarding useless factors ceases at the point where these formal tests, critical

values, et cetera should be defined. Informal tests aiming at detecting the presence of useless factors

are discussed but these tests are very artificial, they relate only to limited behavior/symptoms of

some statistics. Moreover, as shown in this thesis, the power of these informal tests in detecting the

presence of useless factors is also rather limited. At this moment, the unavailability of formal

statistical tests regarding useless factors is a drawback in the analysis. Useless factors, as they are

extensively discussed by Kan and Zhang (1999), could very well solve the puzzle, i.e. the observation

that the beta obtained using the conditional estimator is priced cross-sectionally while in the time-

series this conditional beta has no economic interpretation, but we not yet have the tools to show

this in a formal statistical procedure.

To have the conditional estimator provide one with accurate and reliable estimates of a stock’s beta,

choosing the set of conditioning variables correctly is crucial since these variables will have great

influence on the estimate of beta. As discussed, it is currently not yet possible to formally test the

validity of the set of factors, which are used as a proxy for the set of conditioning variables, and

hence beta estimates obtained using the conditional estimator might be inaccurate. Caution with

regard to these estimates is therefore needed.

Further directions for research

The absence of formal statistical procedures to test the presence of useless factors in the set of

conditional variables is currently a major drawback. Hence, further research should focus on this

area. Developing powerful, formal statistical procedures and statistics, including their distributions,

to test for the presence of useless factors. In my view, testing the out-of-sample performance of the

conditional betas with regard to predicting asset returns is also of major interest, and hence, another

26

possible direction for further research. On the one hand, predictive power by beta with respect to

asset returns would be very useful in (long-term) portfolio optimization and investment decisions. On

the other hand, when out-of-sample conditional betas succeed in providing some information

regarding future returns, arbitrage opportunities might exist. In that case, the fundamental theorem

of asset-pricing, absence of arbitrage, would be violated.

27

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