International Review of Financial Analysis 17 (2008) 1110–1122
Dynamic betas for Canadian sector portfolios☆
Zhongzhi (Lawrence) He a,⁎, Lawrence Kryzanowski b,1
a Faculty of Business, Brock University, 500 Glenridge Ave., St. Catharines, ON, Canada L2S 3A1b GM300-77, John Molson School of Business, Concordia University, 1455 De Maisonneuve Blvd. West,
Montreal, QC, Canada H3G 1M8
Received 28 September 2006; received in revised form 19 July 2007; accepted 9 August 2007Available online 19 August 2007
The market beta of a given asset is a widely used measure to determine the systematic risk ofthe asset, to calculate the cost of equity, and to evaluate the performance of managed investmentfunds. Early studies such as Sunder (1980), Kryzanowski and To (1984), Rahman, Kryzanowski,and Sim (1987) and others find considerable evidence that asset betas are unstable over time. Toaccommodate time-variation in betas, the literature has proposed a variety of approachesincluding the rolling regressions of Fama and MacBeth (1973), the GARCH-type conditional
☆ Financial support from the Concordia University Research Chair in Finance, SSHRC, SSQRC_CIRPÉE and IFM2 isgratefully acknowledged.⁎ Corresponding author. Tel.: +1 905 688 5550x4540; fax: +1 905 688 9779.E-mail addresses: [email protected] (Z.(L.) He), [email protected] (L. Kryzanowski).
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betas of Schwert and Seguin (1990), and the Kalman filter applied by Wells (1994) and others.Among these various approaches, recent literature finds overwhelming evidence that the Kalmanfilter is the best approach to capture beta dynamics.2
In this paper, we estimate the dynamic betas for ten Canadian sector portfolios using the Kalmanfilter approach. We make two contributions to the existing literature. First, previous studies haveestimated beta dynamics for the U.S. (e.g., Jostova & Philipov, 2005), Australian (e.g., Faff, Hillier,& Hillier, 2000), and many European (e.g., Mergner & Bulla, 2005) stock markets. Compared to theU.S. and European markets where the various economic sectors are well represented, the Canadianmarket has a much larger proportional representation of resource and financial firms.3 To ourknowledge, similar work of estimating beta dynamics for the Canadian stock market is still missing,and this paper attempts to fill this void using the unique Canadian sample. Furthermore, given theproliferation of sector mutual funds in the Canadian stock market, our focus on the betas of sectorportfolios is of particular interest to individual and institutional investors who practice passive andactive sector-based investing. Since the risk characteristics of sector funds exhibit very differenttime-varying behaviors (as our results show), estimating the beta dynamics for each sector is crucialfor fund managers to make asset allocations, to implement active management strategies (such assector rotations), to evaluate portfolio performance, and to alter or hedge the market risk of sectorinvestments. To illustrate the merits of estimating the beta dynamics, we describe a practicalapplication of dynamic betas to hedge the market risk of a sector portfolio.
Second and more important, although the existing literature generally favors the Kalman filterapproach, little agreement exists on what type of stochastic process is the most appropriate for thebeta of a given portfolio. Specifically, while Adrian and Franzoni (2005) and Jostova and Philipov(2005) support a stationary mean-reverting process of asset betas, other studies such as Fama andFrench (1997) andMergner and Bulla (2005) find that the betas of (at least) some industry portfoliosfollow a non-stationary random walk. As Fama and French (1997) clearly suggest, the supply anddemand conditions of a particular industry group may be subject to permanent shocks, such aschanges inmonetary or regulatory policies, new information or production technology, or changes inconsumer tastes, which permanently shift the risk characteristics of the sector over the long run.
In this paper, we do not take a stand onwhether the beta of a sector portfolio should be consideredas a randomwalk or a mean-reverting process. Instead, we consider a more general process in whichthe beta is modeled as a combination of a trend (random walk) and a cyclical (mean-reverting)component. In other words, the beta process is allowed to revert to a stochastic trend that is itselfvarying over time. By decomposing the beta process into a trend and a cycle, our dynamic betamodelembraces the existing beta models as special cases, so that the empirical estimates of modelparameters will determine what mixture of the trend and cycle is more appropriate for the beta of aparticular portfolio. Furthermore, the relative importance of the trend and cycle components can beempirically examined by their respective contributions to the time-variation of sector betas.
We estimate the dynamic beta model using the Kalman filter and extract the trend and cyclecomponents, which are combined to form the time series of sector betas.We provide strong evidenceof time-variation of sector betas for the time period of 1991 to 2004. To assess the relative importanceof the two components, we calculate the trend-to-cycle ratios. Based on these ratios, the sector betas
2 A partial list of the studies that support the Kalman filter approach (or more generally, the Bayesian learning model)include: Adrian and Franzoni (2005), Jostova and Philipov (2005) for U.S. stocks and portfolios; Brooks, Faff, andMckenzie (1998) and Faff et al. (2000) for Australian industry portfolios; and Mergner and Bulla (2005) for the Pan-European industry portfolios.3 At the end of 2004, the Energy, Materials, and Financials sectors, respectively, represent 20%, 18% and 32% of the
S&P/TSX index, whereas the Health sector represents less than 5% of the index.
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are classified into two broad categories: 1) those primarily driven by cycles; and 2) those primarilydriven by trends. We find that the sectors in the first category are typical cyclical sectors (e.g.,Consumer Discretionary, Industrials), and those in the second category are typical non-cyclicalsectors (e.g., Consumer Staples, Utilities). This finding suggests that the risk characteristics of sectorportfolios are closely related to different sensitivities of the corresponding sectors to business cycles.
We calculate both in-sample R-square values and out-of-sample hedging errors to evaluate theperformance of different beta specifications. For both the in-sample and out-of-samplemeasures, thebeta process specified as a mix of the trend (random walk) and cycle (mean reverting) components,namely the RWMR, significantly outperforms constant or rolling betas. Furthermore, the RWMRperforms at least as well as the randomwalk (RW) or themean reversion (MR) specification for eachsector. These results support the RWMR as the best beta model for the Canadian sector portfoliosoverall, and in particular for the Utilities sector that has the most significant increase in R-square andthe largest decrease in hedging errors.
The rest of the paper is organized as follows. Section 2 describes the data for ten Canadian sectorportfolios. Section 3 introduces the specification of the dynamic beta model, with the focus on thetrend and cycle components in the beta process. Section 4 discusses the time-series behaviors, andthe classification of sector betas. Section 5 compares the in-sample R-square values and the out-of-sample hedging errors for various beta specifications. In Section 6, we conclude the paper anddiscuss some practical implications of our model and findings.
2. Data
We obtain monthly closing values (including distributions) for ten S&P/TSX sector indices fromthe 2004 Canadian Financial Markets Research Center (CFMRC) monthly index file. Monthly datafor S&P/TSX sectors begin on Dec. 1987 for the ten economic sectors based on the Global IndustryClassification Standard (GICS).4 Continuously compounded returns are calculated by first taking thelog differences of index values between two consecutive months, and then subtracting the 30-dayreturn onCanadian T-bills to calculate excess returns on each sector. Themarket index is taken as theS&P/TSX Composite index value and excess returns are calculated in a similar way.5 The wholesample covers the 204 monthly observations from 1988:01 to 2004:12. Table 1 reports the teneconomic sectors, their corresponding industry groups, and the mean, standard deviation, minimum,25% percentile, median, 75% percentile, and maximum values of excess returns for each sector overthe studied period. The Financials sector is the top performer with an average monthly excess returnof 0.70%, followed by Consumer Staples (0.67%). In terms of risk, the IT sector has a much highervolatility than the other sectors, whereas the Utilities sector has the lowest volatility.
3. Dynamic beta model
The decomposition of the sector beta into trend and cyclical components is motivated by thebroad GICS classification into cyclical and non-cyclical sectors. In general, the productions ofcyclical sectors such as the Consumer Discretionary (e.g., automobiles, leisure) and Industrials (e.g.,transportation) are highly sensitive to the business cycles. So we expect the betas of these sectors to
4 The GICS was jointly developed by Morgan Stanley Capital International (MSCI) and Standard & Poor's (S&P).Their classification has been widely used in sector-based investing. See http://www.msci.com/equity/gics.html for details.5 We also use the excess returns on the CFMRC value-weighted index as a proxy for the market portfolio. Empirical
results are very similar, given that the returns on the S&P/TSX index and those on the CFMRC index have a correlationof 0.97.
This table reports ten economic sectors according to the global industry classification standard (GICS), the correspondingindustry groups, the means, standard deviations, minimum, 25% percentile, median, 75% percentile, and maximum valuesof excess returns for each sector. The sector data are taken from the Canadian FinancialMarkets Research Center (CFMRC)sector indices, and the sample period is from 1988:01 to 2004:12.
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be primarily driven by the cyclical component. On the other hand, typical non-cyclical sectors, suchas the Consumer Staples (e.g., food, tobacco products) and Utilities, are largely independent of thebusiness cycles, so the trend is expected to be the major component of the betas of these relativelyinsensitive sectors. The dynamic beta model developed herein captures both components, and itallows us to empirically examine the relative importance of the two components for each sector.
Let rp,t and rm,t be the excess returns on sector portfolio p and market portfolio m. For eachportfolio p, the dynamic beta model is given by the following equations:
2 ) is the error term of the random walk process;vp,t∼N(0, σvp
2 ) is the error term of the AR(1) process; andεp,t, wp,t and vp,t are uncorrelated to each other at all leads and lags.
Eq. (1) is the standard market model with a time-varying beta βp,t whose dynamics arespecified by Eq. (2), which provides a decomposition of βp,t into two additive components, trendBp,t and cycle Cp,t. The trend (permanent) component is modeled as a random walk as in Eq. (3).Eq. (4) specifies the cycle (transitory) component as an AR(1) process in a demeaned form. Thus,βp,t is assumed to revert to its time-varying mean Bp,t, and the speed of this mean-reversion isgiven by the persistence parameter ϕp.
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The dynamic beta model encompasses a wide range of beta processes in the literature.6 Withσwp
2 =0, our model reduces to the smooth mean-reverting process of Adrian and Franzoni (2005)or the constant mean-reverting process of Jostova and Philipov (2005).7 If the mean-revertingprocess is also non-persistent (ϕp=0), then our model becomes the random coefficient modelproposed by Fabozzi and Francis (1978) and others. On the other hand, if the transitory variationaround the mean does not exist (σvp
2 =0), then the model reduces to the random walk processproposed by numerous studies such as Engle and Watson (1987).8 Finally, if both σwp
2 and σvp2 are
zero, then Eq. (1) reduces to the standard market model whose constant beta can be estimated byOLS. Thus, given the general model specification, the relative importance between the trend andcyclical components for the beta of a given sector portfolio can be empirically assessed by theirrespective contribution to the time-variation of the betas.
The dynamic beta model is estimated using the Kalman filter where the trend and cycle aretreated as unobserved latent variables whose values are taken as the contemporaneous conditionalexpectations at each time t, i.e., Bp,t|t and Cp,t|t.
9 The key feature about the Kalman filter is that itgives us insight into how a rational agent would revise his estimates of the trends and cycles in aBayesian type of learning fashion when new information becomes available. The appendix givesthe details on the Kalman filter.
4. The dynamic behaviors of the sector betas
The parameter estimates of the dynamic beta model are reported in the first four columns ofTable 2. These parameter estimates are used to extract the real-time conditional expectations ofthe trend component (Bp,t|t) and the cycle component (Cp,t|t), which are combined to form thedynamic betas for each sector, i.e., βp,t|t=Bp,t|t+Cp,t|t. To assess the relative importance betweenthe two components for each sector, we report, in the last two columns of Table 2, the standarddeviations of Bp,t|t and Cp,t|t where t=1991:01–2004:12.10 We then calculate the trend-to-cycleratio B/C=σ(Bp,t|t) /σ(Bp,t|t) to measure the relative importance of the two components, using B/C=1 as the benchmark for equal importance. The time series of βp,t|t (the dashed line) and Bp,t|t
(the solid line) for each sector from 1991:01 to 2004:12 are depicted in Fig. 1 in ascending orderof the B/C ratios.11
In Fig. 1, we find strong time-variation of sector betas. For example, the Industrials betas rangefrom0.12 to 2.19, and the IT betas range from 0.55 to 3.55 during the past 14 years. TheUtilities betasare mostly negative from 1999 to 2003, suggesting that the sector negatively co-varies with (i.e.,
6 The dynamic betamodel is in the spirit ofWells' (1994)movingmean betamodel where the beta is assumed to revert to atime-varyingmean that is also driven by a randomwalk.Moreover, ourmodel is formulated as a structural time-series modelproposed by Harvey (1989). According to Harvey (1989), the structural form provides a natural interpretation of the twobeta components as the trend and the cycle.7 Given σwp
2 =0, whether the beta reverts to a smooth mean or constant mean depends on the prior belief of betauncertainty, i.e., Var(B1|0). Adrian and Franzoni (2005) assume a large number for Var(B1|0), so Bt|t is time-varying butvery smooth in their model; whereas Jostova and Philipov (2005) assume that Var(B1|0)=0, so Bt|t is a constant in theirmodel. For the purpose of this study, we adopt the smooth mean-reverting process. This can be seen from Fig. 1, wherethe trends vary smoothly for those sectors with σwp
2 =0 (i.e., Energy, Financials, Consumer Discretionary, Health).8 With σvp
2 =0 and the prior C1|0=0 (see the appendix), the cycle component plays no role in the beta process.9 The use of Gauss routines described in Kim and Nelson (1999) is gratefully acknowledged.
10 The Kalman filter generates some outliers in the initial stage of data extraction. Thus, the first three years of data areexcluded in the analysis and graphs.11 As a robustness check, we also perform the same analysis for the first-half of the studied time period (1988:01–1996:06)and for the second-half of the studied time period (1996:07–2004:12), and obtain qualitatively similar results.
Table 2Parameter estimates of the dynamic beta model and volatilities of the trend and the cycle
The four columns under Parameter Values report the maximum-likelihood estimates of parameters for the dynamic betamodel as in Eqs. (1)–(4). The Kalman filter, as described in the appendix, is used to estimate the model parameters for thesample for the time period of 1988:01 to 2004:12. The last two columns report the volatilities (standard deviations) of thereal-time conditional expectations of the trend (Bp,t|t) and the cycle (Cp,t|t) extracted from the Kalman filter for the timeperiod of 1991:01 to 2004:12.
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hedges against) the market index during that period. More important, the B/C ratios in each graphclearly suggest that we can classify the dynamics of sector betas into the following two categories.
4.1. Financials, energy, industrials, consumer discretionary and telecom sectors (B/Cb1)
For these sector betas, the cyclical components (Cp,t|t) are more important than the trendcomponents (Bp,t|t), with the B/C ratios ranging from 0.36 (Financials) to 0.63 (Telecom). This isbest visualized by the fact that βp,t (the dashed lines) wander widely around their relatively stablemeans Bp,t (the solid lines). The results are intuitive. By the GICS definition, the ConsumerDiscretionary sector encompasses those industries (e.g., automobiles, leisure) that tend to be themost sensitive to the business cycles. Major industrial groups in the Industrials and Energy sectors(e.g., transportation, fuel production) are also cyclical in nature. In addition, the Financials sectoris the most sensitive to interest rate movements that are generally thought to follow a highlypersistent mean-reverting process. This is reflected in the result that the Financials beta has thehighest persistency parameter of 0.95 among all the ten sectors.
4.2. Utilities, materials, health, consumer staples, and IT sectors (B/CN1)
For these sector betas, the B/C ratios range from 2.37 (Utilities) to 7.09 (IT). This is bestvisualized by the fact that the movement of Bp,t|t essentially drives the time-variation of βp,t|t. It isintuitive that transitory shocks have only minor effects on the betas of these sectors. For example,by the GICS definition, the Consumer Staples sector contains those industries (e.g., food) that arethe least sensitive to business cycles. The supply and demand conditions of the other four sectorsare largely non-cyclical by nature. The IT sector beta exhibits the strongest trending effect, whichincreases (almost) monotonically during the whole time period, with the fastest rate of increaseoccurring at the turn of the millennium. The trend essentially reflects the effect of the permanenttechnology shocks on the risk characteristics of the IT sector.
In summary, we show that our dynamic beta model is able to accommodate various betaprocesses, and that the beta dynamics for the Canadian sector portfolios are described by a mix of
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the trend and cycle components. Our findings of the relative importance of the two componentsfor each sector are generally consistent with the broad GICS classification of cyclical sectors andnon-cyclical sectors. This implies that the risk characteristics of sector portfolios are closelyrelated to different sensitivities of the corresponding sectors to business cycles.
5. Model performance
In this section, we evaluate the relative performance for four different beta processes withinthe dynamic beta framework of Section 3. The most general process is the full specification of
Fig. 1. Trends and betas for the Canadian sector portfolios (1991:01–2004:12). Each figure plots the trend Bp,t|t (solid line)and the beta βp,t|t=Bp,t|t+Cp,t|t (dashed line) for each sector. Bp,t|t and βp,t|t are the real-time conditional expectations of Bp,t
and βp,t and are extracted from the Kalman filter for the dynamic beta model as in Eqs. (1)–(4). The appendix gives detailson the Kalman filter. In each graph, B/C is the trend-to-cycle ratio, defined as the standard deviation of the trend componentto the standard deviation of the cycle component, i.e., B/C=σ(Bp,t|t) /σ(Cp,t|t), where t=1991:01 to 2004:12. The values ofσ(Bp,t|t) and σ(Cp,t|t) for each sector are reported in Table 2. The graphs are depicted in ascending order of the B/C ratios.
Fig. 1 (continued ).
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Eqs. (1)–(4), which is termed as the RWMR (random walk plus mean reversion) betas. With thediffuse priors given by the appendix, the RWMR reduces to the RW (random walk) betas bysetting σvp
2 =0, to the MR (mean reversion) betas by setting σwp
2 =0, and to the OLS betas bysetting both σvp
2 and σwp
2 to zeros.
5.1. In-sample R-squares
The first measure of model performance is the proportional variance explained by the marketmodel (i.e., R-square) whose value is calculated as R2 =1−σ2
εp /σp2, where σp is the standard
deviation of excess returns in Table 1, and σεp is the idiosyncratic standard deviation of excessreturns (Table 2 reports σεp
2 for the RWMR betas). The four sets of R-square values for each sectorare presented in Table 3 for both the full time period and the half sub-periods.
First, the R-square values increase significantly for any version of the dynamic beta models, ascompared to the constant OLS model. This highlights the importance of modeling time-varyingbetas instead of assuming constant betas. Dynamic betas are especially important for the Utilities
Table 3Proportional variances explained by the beta models (R-squares)
This table reports the R-square values for four beta processes. For the OLS, the R-squares are obtained from the standardOLS regressions. For the RW, the dynamic beta model has only the trend component, i.e., no Eq. (4). For the MR, thedynamic beta model has only the cycle component, i.e., no Eq. (3). For the RWMR, the dynamic beta model has the fullspecification as in Eqs. (1)–(4). The R-square values are calculated as 1−σε
2 /σp2, where σp is the standard deviation of
sector p's excess returns, and σε is the idiosyncratic standard deviation. The results are reported for the full studied timeperiod (1988:01–2004:12), the first-half studied time period (1988:01–1996:06), and the second-half studied time period(1996:07–2004:12).
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sector whose R-square value is only 7% with constant betas, but increases to nearly 50% with theRWMR betas for the full time period, suggesting that the time-variation in betas is the mostimportant source of variation in the market model for this sector.
Second, the R-square values are quantitatively close to each other for the three versions of betaprocesses, which demonstrate a consistent pattern. For the full time period, the RWMR betasperform at least as well as the MR betas, which in turn outperform the RW betas for every sector.Specifically, the RWMR betas outperform the MR betas for the Energy, Consumer Staples, IT,and (especially) Utilities sectors, all of which have predominant trend components (i.e., trend-to-cycle ratios B/C greater than one), suggesting that the mix of RWand MR provides a better modelfit than either MR or RW process alone. Finally, all the four beta processes produce similarly lowR-squares for the Health sector. This is not surprising because with σwp
2 =0 and σvp2 =0 (Table 2),
the RWMR reduces to a constant beta for the Health sector. The results for the half sub-periods aregenerally consistent with those for the full time period.
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5.2. Out-of-sample dynamic hedging strategies
For the second method of model evaluation, we compare the out-of-sample pricing errors forthe four beta processes. This method can be motivated by a practical application of dynamichedging strategies where a portfolio manager hedges the time-varying market risk of a sector fundusing dynamic betas. Specifically, for each dollar invested in fund p at time t, the fund managerhedges the market risk by short selling βp,t+1|t units of the market portfolio, where
bp;tþ1jtuEt½bp;tþ1� ¼ Bp;t þ /pCp;t ð5Þ
is the optimal one-period-ahead forecast of the fund's beta, as implied from Eq. (2). Thecomposition of the next-period's beta forecast is very intuitive. It is the sum of the current trend(Bp,t) plus the current deviation from the trend (ϕpCp,t) whose forecasting effect depends on thepersistence parameter ϕp. To conduct the out-of-sample beta forecasts, the fund manager uses 120data observations of rp,t and rm,t (i.e., from t−120 to t) to estimate the model parameters, uponwhich the hedge ratio βp,t+1|t is obtained from Eq. (5). At time t+1, the error of this hedgedposition is calculated by:
ep;tþ1 ¼ rp;tþ1 � bp;tþ1jtrm;tþ1 ð6Þ
The dynamic hedging strategy is implemented for each time t from 1997:12 to 2004:11,thereby generating 84 out-of-sample hedging errors ep,t+1 over the last 7-year period in oursample.12 The effectiveness of the hedging strategy is evaluated by two measures: mean absoluteerror (MAE) and root mean squared error (RMSE), which are respectively calculated as:
MAEpuP
t jep;tþ1jT
and RMSEpu
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPt e
2p;tþ1
T;
s
where t+1 is from 1998:01 to 2004:12 and T=84. The MAE and RMSE account for differentaspects of the hedging errors, with the MAE capturing the average level, and the RMSE capturingthe variability of hedging errors over time. The more effective hedging strategy should producesmaller MAE and RMSE.
To compare model performance, we implement the above hedging strategy for the RW, MR,and RWMR beta processes, and also for the rolling OLS (ROLS) betas using the 60-month rollingregression procedure of Fama and MacBeth (1973) to forecast the one-period-ahead hedgeratios.13 Table 4 summarizes the MAE and RMSE for the four beta forecasting procedures. Justlike any other out-of-sample model comparisons (e.g., Fama & French, 1997), the improvementfrom one model to another is small relative to the large hedging errors. However, some consistentpatterns emerge across the four models. First, compared to the ROLS betas, there is a noticeabledecrease in both MAE and RMSE for the RW, MR, and RWMR processes. On average, theRWMR reduces the MAE (RMSE) by over 20 (nearly 30) basis points over the ROLS, suggestingthat statistical modeling of dynamic betas indeed improves the beta forecasts over the ad hoc
12 Due to data limitations, out-of-sample comparisons of model performance based on dynamic betas are not amenableto examination for the two half sub-periods.13 We also arrive at similar inferences when rolling betas are estimated using various window lengths, for example, 120-dayrolling windows.
This table reports the mean absolute error (MAE) and the root mean squared error (RMSE) for the hedging errors from theout-of-sample dynamic hedging strategies. For the ROLS, the rolling regression procedure of Fama andMacBeth (1973) isused to estimate the hedge ratio. For the RWMR, the dynamic beta model as in Eqs. (1)–(4) is used to estimate the one-period-ahead beta forecast as the hedge ratio. For the RW, the dynamic beta model has only the trend component, i.e., noEq. (4). For the MR, the dynamic beta model has only the cycle component, i.e., no Eq. (3). The hedging strategy isimplemented for each time t from 1998:01 to 2004:12. The values of the MAE and RMSE are in percentages.
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rolling regressions. Second, among the three stochastic beta processes, the RW and MR producevery similar hedging errors, whereas the RWMR performs at least as well as either RWor MR forevery sector. The RWMR performs particularly well for the Utilities sector, for which the increasein R-squares is also the largest. In summary, the out-of-sample model comparison producesconsistent patterns with our in-sample analysis. Both tests support the RWMR as the best modelthat can further improve the beta estimate and risk management of sector funds.
6. Conclusion
This paper examines the dynamic processes of market betas for ten Canadian sector portfoliosusing the Kalman filter approach, and finds that the beta dynamics are best described as a mix ofthe random walk (trend) and mean reverting (cycle) processes. The relative importance of thetrend and cycle components of sector betas is found to be consistent with the GICS classificationof cyclical and non-cyclical sectors. Incorporating the beta dynamics into the standard marketmodel greatly increases its explanatory power and produces smaller pricing or hedging errors thanthe alternative specifications of beta processes.
In addition to the dynamic hedging strategies described in the paper, the dynamic beta modelcarries other practical implications such as the fundamental analysis of individual stocks using thediscounted cash flow (DCF) approach. The key component of the DCF model is the appropriatediscount rate for a given stock. It is well known that the discount rate at the individual stock levelis very difficult to measure, so the common practice in asset management is to use the betasestimated from the appropriate economic sector or industry group to replace the stock beta. Thus,the enhanced accuracy of the beta forecasts emanating from our model should provide betterestimates of the discount rate in the application of the DCF model.
Furthermore, the dynamic beta model carries implications for establishing industry costs ofcapital, and especially for the Utilities sector at Canadian regulatory hearings. In a related study, Heand Kryzanowski (2007) estimate the cost of equity for the Canadian sectors, and suggest that
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higher importance should be given to estimating the dynamics of betas for the Canadian sectors. Inparticular, time-variation in the betas is the most important source of variation in the market modelfor the Canadian Utilities sector. The dynamic beta methodology described in this paper deliveredsuperior performance over traditional approaches, thereby leading to more precise estimates ofindustry costs of equity. For the Canadian Utilities sector in particular, our approach couldpotentially alleviate some of the difficulties in rendering decisions in utility rate-of-return hearings.
Finally, although the Kalman filter approach was favored by this paper and others, some cautionshould still be exercised in practice. As an econometric technique that optimally extracts time seriesdata, the Kalman filter typically generates more volatile beta estimates than the traditional approachsuch as rolling regressions. For Canadian fundmanagers, this implies higher asset turnovers (whichmeans higher transaction costs) when implementing dynamic hedging strategies. Managers shouldoptimally balance the potential benefits and costs of using the Kalman filter approach for estimatingdynamic betas when measuring and controlling the market risks of their investment portfolios.
Appendix A. Details on the Kalman filter
For each sector portfolio, we define the following matrices (subscript p is dropped):
ntuBt
Ct
� �Uu
1 00 /
� �Rm;tu½rm;t rm;t� gt ¼ wt
vt
� �X ¼ r2w 0
0 r2v
� �
With this notation, Eqs. (1)–(4) are expressed in the following state-space form.Observation equation: rt=Rm,tξt+εtState equation: ξt=Φξt−1+ηtIn the above equations, ξt is a vector of latent variables, which contain the trend (Bt) and cycle
(Ct) components of the beta at each time t. ξt is unobserved, so its conditional expectations (ξt|t)are extracted from the Kalman filter using the following iterative procedure.
Beta forecast: ξt|t−1=Φξt−1|t−1, with covariance matrix: Rt|t−1=ΦRt−1|t−1Φ′+Ω;Forecast error: et|t−1= rt−Rm,tξt|t−1, with covariance matrix: ft|t−1=Rm,tRt|t−1Rm,t′ +σε
2
Beta update: ξt|t=ξt|t−1+Ktet|t−1, with covariance matrix: Rt|t=Rt|t−1−KtRm,tRt|t−1, whereKt=Rt|t−1Rm,t′ ft|t−1
−1 is the gain matrix.For the initial values of ξt and Σt, the following diffuse priors are used:
n1j0 ¼ bOLS0
� �and R1j0 ¼ l 0
0 r2v=ð1� /2Þ� �
where
βOLS is the constant beta estimated from the OLS;∞ is an arbitrary large number; andσv2 / (1−ϕ2) is the unconditional variance of the cyclical component of the beta process.
For further details on the Kalman filter, see Harvey (1989).
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