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Roy Mehta was a Seeking Alpha editor until mid-2008.
  • “A Better Beta for Portfolio Managers"

    Introduction

    This paper will explore how best to predict beta in the Capital Asset Pricing Model (CAPM) over the next six months. In present day financial theory, we use the relationship between a stock and market in the past to figure out what that relationship might be in the future. That relationship is captured in the CAPM, but there are still nuances within the CAPM that can be altered for better or worse results. One of these attributes is how much data, in terms of time, is used to calculate the beta of a stock. Here, we firstly figure out what time frame from the stock’s past prices might be best to predict the stock’s future beta over the next six months.

                    Though the findings in this paper pinpoint what time horizon can be used to forecast the beta of a stock or ETF over the next six months, it still remains to be seen whether those betas were significantly similar. I tested this using three well-known stocks. The results were positive and show there is value in this exploration.  

     

     

    Background

                    Modern portfolio theory tells us there are two types of risk, systematic risk and unsystematic risk. Unsystematic risk can be reduced to zero using diversification, but systematic risk, in general, cannot be. Beta is a measure of this non-diversifiable risk. The beta tells the portfolio manager what will happen to the expected return of the asset if the market moves. It is a single risk factor that shows what a large run-up or drop could do to an asset’s value. A large beta means a larger return when the market increases, but a more negative return when the market decreases. It encompasses the direction and magnitude of the asset’s return based on the market. Higher beta stocks are more volatile, and should then be considered more risky.

                    With these ideas in mind, it is clear that knowing the beta of an asset, can be valuable. It can give risk managers an idea of how assets they manage will move and how much risk their portfolios actually have. Portfolio managers have views of the market in general, and can create portfolios to take advantage of their views by constructing an assortment of stocks with targeted betas. If portfolio manager X is bullish, he will want a portfolio that is at the very least positive, and maybe even above one to beat the market.

                    It should also be briefly noted that the CAPM does have its disadvantages, mainly the assumptions on which it is based. These assumptions include that investors hold diversified portfolios, single holding periods are used, anyone can borrow and lend at the same risk-free rate, all the securities in the market are valued efficiently, there are no taxes or transaction costs, and all asset returns can be plotted on the securities market line(SML). The SML shows the relationship between risk or beta and asset return. Though some of these assumptions are significant, they are made in order to focus closely on the relationship between risk and return.

    Analysis 1: Choosing a Time-Frame

                    Generally, people find the beta between the market and a stock over the last year, and consider that number to be the beta going forward. This paper challenges that notion, and instead tests whether we should use a different time-frame to increase the accuracy of that beta.

                    To get beta, you take the excess returns of a stock and index over a certain amount of time and use linear regression to find the coefficients in the CAPM. The risk-free is usually some relatively safe bond rate. In the calculations for this paper, I used the 10-year US Bond found on Yahoo Finance.com. The S&P500 is a popular candidate for the market portfolio, so I used that for my market returns. When running the regression (see Table 1 for code), we get two coefficients, the beta and alpha. The Alpha is the return that cannot be attributed to the market returns and beta. In regression terms, alpha is the intercept, while beta would be considered the slope.

    I tested three time frames — 6 months, 1 year, and 2 years. I assumed each year is about 250 days (the approximate amount of trading days in a year).  So, I took the log returns for nine different ETFs, the sector SPDRs, over about 10 years, from 4/12/2000 to 3/23/2010, and divided them into four parts with 625 days in each part. The prices were taken from Yahoo Finance as well and were adjusted for splits and dividends. In each part, I found the betas of the 375th to 500th day (six months), 250th to the 500th day(one year), and finally the 1st to 500th day(two year). This gave me three betas. Then, I found the beta over day 501 to 625. This gave me a “real beta” to use for comparison. The idea was to compare each of the three betas to the “real beta” and see which was the closest.  Below shows the results for each part. Part 1 shows data for the first 2.5 years or 625 days, part 2 shows the next 2.5 years, and so on.

    Part 1

                 
                   

     

    125

    %Dif

    250

    %Dif

    500

    %Dif

    Real Beta

    XLY

    1.0993

    9.76%

    0.9429

    5.86%

    0.9669

    3.46%

    1.0016

    XLP

    0.5050

    2.95%

    0.4851

    1.12%

    0.4912

    0.12%

    0.4906

    XLE

    0.7951

    39.51%

    1.2279

    6.58%

    1.2297

    6.45%

    1.3144

    XLF

    2.1765

    33.40%

    1.6093

    1.36%

    1.6537

    1.36%

    1.6316

    XLV

    0.5529

    7.32%

    0.6086

    2.01%

    0.5901

    1.08%

    0.5966

    XLI

    0.9893

    1.81%

    0.8806

    9.38%

    0.9365

    3.63%

    0.9718

    XLB

    0.7869

    38.99%

    0.9600

    25.56%

    1.0186

    21.02%

    1.2897

    XLK

    0.9260

    4.43%

    0.9153

    5.53%

    0.9112

    5.95%

    0.9689

    XLU

    0.6183

    6.20%

    0.7062

    21.30%

    0.6901

    18.54%

    0.5822

    Average

     

    16.04%

     

    8.75%

     

    6.85%

     

     


    Part 2

                 

     

                 

     

    125

    %Dif

    250

    %Dif

    500

    %Dif

    Real Beta

    XLY

    0.9709

    15.45%

    1.0002

    12.89%

    1.0143

    11.67%

    1.1483

    XLP

    0.6354

    14.87%

    0.6064

    18.76%

    0.5976

    19.93%

    0.7464

    XLE

    1.6386

    36.99%

    1.4689

    22.80%

    1.3081

    9.36%

    1.1962

    XLF

    0.9427

    1.65%

    0.9835

    2.61%

    1.1242

    17.29%

    0.9585

    XLV

    0.7096

    8.23%

    0.6887

    5.05%

    0.6880

    4.95%

    0.6556

    XLI

    0.8492

    19.59%

    0.9654

    8.59%

    0.9289

    12.04%

    1.0561

    XLB

    1.1239

    18.23%

    1.3120

    4.54%

    1.2239

    10.96%

    1.3744

    XLK

    1.0146

    4.71%

    1.0883

    12.32%

    0.9708

    0.19%

    0.9689

    XLU

    0.9258

    13.65%

    0.7796

    4.29%

    0.8589

    5.44%

    0.8146

    Average

     

    14.82%

     

    10.21%

     

    10.20%

     


    Part 3

                 
                   

     

    125

    %Dif

    250

    %Dif

    500

    %Dif

    Real Beta

    XLY

    1.0236

    6.58%

    1.0193

    6.97%

    1.0038

    8.39%

    1.0957

    XLP

    0.6851

    45.92%

    0.6376

    35.79%

    0.6453

    37.44%

    0.4695

    XLE

    0.4672

    22.44%

    0.5371

    10.84%

    0.6435

    6.84%

    0.6024

    XLF

    1.0839

    6.56%

    1.0428

    10.11%

    1.0163

    12.39%

    1.1600

    XLV

    0.8505

    39.61%

    0.8271

    35.77%

    0.8499

    39.51%

    0.6092

    XLI

    0.9695

    1.70%

    0.9752

    2.30%

    0.9899

    3.83%

    0.9533

    XLB

    0.9341

    8.16%

    0.9978

    1.89%

    1.0654

    4.75%

    1.0171

    XLK

    1.3248

    7.29%

    1.3770

    3.63%

    1.2847

    10.10%

    1.4289

    XLU

    0.5904

    12.27%

    0.5812

    13.65%

    0.6152

    8.59%

    0.6730

    Average

     

    16.73%

     

    13.44%

     

    14.65%

     


    Part 4

                 

     

                 

     

    125

    %Dif

    250

    %Dif

    500

    %Dif

    Real Beta

    XLY

    0.8245

    23.87%

    0.9424

    12.98%

    0.9637

    11.02%

    1.0831

    XLP

    0.4790

    0.01%

    0.4852

    1.30%

    0.4934

    3.00%

    0.4790

    XLE

    1.2632

    10.66%

    1.2374

    8.40%

    1.2390

    8.55%

    1.1415

    XLF

    1.3482

    18.42%

    1.6702

    1.06%

    1.6483

    0.26%

    1.6527

    XLV

    0.6429

    9.99%

    0.5882

    0.63%

    0.5912

    1.15%

    0.5845

    XLI

    0.8351

    14.65%

    0.9000

    8.01%

    0.9399

    3.93%

    0.9784

    XLB

    0.9458

    21.93%

    0.9731

    19.67%

    1.0211

    15.71%

    1.2115

    XLK

    0.9244

    8.90%

    0.9022

    11.09%

    0.9081

    10.51%

    1.0147

    XLU

    0.7945

    42.27%

    0.6858

    22.79%

    0.6913

    23.79%

    0.5585

    Average

     

    16.74%

     

    9.55%

     

    8.66%

     


    With this data, I could see which time-frame was best to use to predict beta over the next six month period, which was days 501 to 625 in each of the four parts. Figure 1 shows the betas of each of the nine SPDR sectors for each time frame and their real betas, and the percent difference between each of the predictor betas and the real beta. In three out of the four parts, the two year Beta produced the lowest percentage difference. Unsurprisingly, it also gave the lowest average percentage difference when all of the four parts were combined (below), which was about 10.09%.  For this reason, the two year beta was chosen as the best predictor, and was used in the next part of the analysis.

     

    Six Months

    One Year

    Two Years

    Part 1

    16.0396%

    8.7465%

    6.8453%

    Part 2

    14.8188%

    10.2063%

    10.2017%

    Part 3

    16.7258%

    13.4388%

    14.6482%

    Part 4

    16.7448%

    9.5487%

    8.6586%

    Averages

    16.0823%

    10.4851%

    10.0884%




    Analysis 2: Verification of Time-Frame Results

                    Getting the results from Analysis 1 was important, but those results should also be tested to see if they matter. Though the 500 day time-frame worked best, it is also worthwhile to test the results on a couple of single stocks to see how this 500 day beta performs against the real beta.

                    I looked at the excess log returns of three stocks, which were Ford, American Express, and Wal-Mart. For the market portfolio, I used the log returns for S&P500 Index and for the risk-free, I again used the 10 year T-Note. All prices were taken from Yahoo Finance and were adjusted for dividends and splits. The data went from 8/14/1980 to 5/10/2010, totaling 7500 daily closes for each stock. For each stock, I began by finding the beta of the first 500 days. I then compared that beta to the beta from day 501 to 625. I then found the beta from 501 to 1000 and compared that to the beta from 1001 to 1125. I repeated this process with all the data. This gave 14 sets of betas to compare.

    While using linear regression to find the betas, I also found the standard errors for each real beta. This standard error is basically a standard deviation for each beta and could be used to find confidence intervals for each of the real betas. All of the betas and confidence intervals for each stock can be seen in Figure 3. The Beta T-500 column shows the betas calculated using the days the first 500 days of each time frame, and the Beta T shows the beta for the next 125 days, and should be looked at as the beta I was trying to predict. Using the standard error of each of the Beta Ts, I found the 95% confidence interval of each of the real betas. The low confidence interval was found by subtracting the standard error multiplied by 1.96 from Beta T. The 1.96 comes from the fact that 95% of the area under a normal distribution is within 1.96 standard deviations from the mean. The high confidence interval was found by adding the standard error multiplied by 1.96 to the Beta T.

    Results

    The results were encouraging, but left room for improvement. In the area below, we can see how many of the predicted Betas fell within the 95% confidence interval of the real betas. If the interval is green, the Beta T-500 fell in the confidence interval. If the interval is red, the Beta T-500 was outside the 95% confidence interval range. About 60% of the Beta Ts fell within the intervals. In this case, the confidence interval represented the interval where 95% of the Beta Ts or real betas would fall in the long run.  The fact that most of the Betas T-500s were in the intervals is evidence that there might be some similarity between the betas, and therefore, probably could be used for a prediction in some cases. Still, 60% shows that perhaps this process may be worth exploring further to increase the prediction accuracy.

     

    Ford

             

    Time

    Beta T-500

    Beta T

    Standard Error

    Low Conf Interval

    High Conf Interval

    1

    1.1090

    1.471566

    0.1898

    1.0996

    1.8435

    2

    1.6381

    1.798416

    0.1737

    1.4580

    2.1388

    3

    1.5225

    1.06315

    0.1205

    0.8269

    1.2994

    4

    0.9694

    1.222246

    0.0854

    1.0548

    1.3897

    5

    1.1197

    0.9670847

    0.0924

    0.7859

    1.1483

    6

    1.1352

    1.248185

    0.2483

    0.7615

    1.7348

    7

    1.5017

    1.436097

    0.2300

    0.9853

    1.8869

    8

    1.3771

    0.8060476

    0.1640

    0.4845

    1.1276

    9

    0.7829

    0.8719964

    0.1914

    0.4968

    1.2472

    10

    1.0576

    0.3189331

    0.1400

    0.0444

    0.5934

    11

    0.7076

    0.8955937

    0.1350

    0.6311

    1.1601

    12

    1.2136

    1.537435

    0.1988

    1.1477

    1.9272

    13

    1.3313

    0.9338313

    0.2643

    0.4159

    1.4518

    14

    1.0867

    1.236589

    0.2120

    0.8211

    1.6521

               
               
               

    AXP

             

    Time

    Beta T-500

    Beta T

    Standard Error

    Low Conf Interval

    High Conf Interval

    1

    1.1665

    1.550724

    0.1248

    1.3060

    1.7954

    2

    1.4955

    1.916461

    0.1857

    1.5525

    2.2804

    3

    1.6657

    1.308973

    0.1105

    1.0924

    1.5255

    4

    1.3642

    1.404938

    0.1046

    1.1999

    1.6100

    5

    1.3838

    1.037827

    0.1561

    0.7319

    1.3438

    6

    1.5619

    1.086976

    0.2033

    0.6886

    1.4854

    7

    0.9622

    1.150741

    0.2092

    0.7407

    1.5608

    8

    1.1764

    1.172523

    0.1377

    0.9026

    1.4424

    9

    1.1545

    1.173122

    0.1602

    0.8592

    1.4871

    10

    1.4784

    1.114845

    0.1476

    0.8255

    1.4042

    11

    1.3602

    1.238425

    0.1053

    1.0320

    1.4448

    12

    1.3486

    0.8371283

    0.0872

    0.6662

    1.0081

    13

    0.9258

    1.064091

    0.0928

    0.8821

    1.2461

    14

    1.3640

    1.86013

    0.1235

    1.6181

    2.1021

               
               
               

    WMT

             

    Time

    Beta T-500

    Beta T

    Standard Error

    Low Conf Interval

    High Conf Interval

    1

    0.8846

    0.8105578

    0.2047

    0.4093

    1.2119

    2

    0.9375

    0.876469

    0.1666

    0.5500

    1.2030

    3

    1.1173

    1.602664

    0.1421

    1.3242

    1.8811

    4

    1.0030

    1.363929

    0.0943

    1.1790

    1.5489

    5

    1.3770

    1.73746

    0.1212

    1.4999

    1.9750

    6

    1.5526

    1.246138

    0.1337

    0.9840

    1.5083

    7

    1.2711

    1.548752

    0.2248

    1.1081

    1.9894

    8

    1.2624

    1.024404

    0.1682

    0.6947

    1.3541

    9

    0.9460

    1.159366

    0.1576

    0.8505

    1.4683

    10

    1.3153

    1.067866

    0.1879

    0.6996

    1.4361

    11

    0.9187

    0.8111939

    0.0932

    0.6286

    0.9938

    12

    0.8718

    0.737179

    0.1210

    0.5000

    0.9744

    13

    0.7412

    0.7796168

    0.1126

    0.5589

    1.0004

    14

    0.8580

    0.6597267

    0.0771

    0.5087

    0.8108

     

    Extensions

                    There is value in finding even more accurate ways to predict beta. It might be fruitful to explore more time-frames than just the three looked at in this study. Perhaps even longer time frames are better. It would also be interesting to explore other factors that might affect what time frame to use. High volatility stocks might lend themselves better to different time frames than lower volatility stocks. Finally, it would be worthwhile to look at betas during important market events and see how they act. For example, during market crashes, I would expect that the relationship between stocks and the market would increase in strength because investors are selling nearly anything. Seeing how this affects the beta of single stocks, and checking if it even matters, would be interesting and can change the way people really evaluate the risk of their portfolios.













     



     



    Disclosure: none
    Tags: Macro, CAPM
    May 14 2:47 PM | Link | 1 Comment
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