Beta A Statistical Analysis of a Stock’s Volatility Courtney Wahlstrom Iowa State University, Master of School Mathematics Creative Component Fall 2008

A Statistical Model for beta Statistically, the beta value for a stock is thought of as the slope in the linear equation between the percent of change in the market or index fund the exp

A Statistical Analysis of a Stock’s Volatility

Courtney Wahlstrom Iowa State University, Master of School Mathematics

Creative Component

Fall 2008

Amy Froelich, Major Professor

Heather Bolles, Committee Member

Travis Sapp, Committee Member

Beta: A Statistical Analysis of a Stock’s Volatility

With recent economic uncertainty, discussion of the volatility of the stock market

is unavoidable Describing the current market, finance professor at New York University Robert Engle stated, "We have no idea where things are going That is what high

volatility means,” (Merle, 2008) The value β portrays the volatility of an individual stock compared to the market as a whole, usually represented by the S&P500 This β paints a picture of how much an investor can expect a particular stock to increase or decrease in price compared to the movement of the market as a whole A high β value does not mean that a stock is expected to increase in value; it means that it is much more volatile (likely

to increase or decrease at higher rates) than the market as a whole Therefore, if the market rises, one would expect a stock with a high β to rise at a higher rate Likewise, if the market falls, a stock with a high β should fall at a higher rate as well On the contrary,

a stock with a relatively small β value maintains a much steadier price over time than the market This can be summarized in the following chart describing the potential values of

β for any given stock:

Table 1 Summary of Values of beta and the relationship to market movements

market increases, the value of this stock is expected to decrease While this relationship theoretically exists, few stocks possess a negative beta One example of an investment with negative beta is gold

rises, this stock should also rise but not as drastically as the market; likewise

if the market falls, this stock is expected to be less volatile than the market

market rises, this stock should rise at a higher rate Likewise, a more severe

loss is anticipated in the event the market falls

1 A Statistical Model for beta

Statistically, the beta value for a stock is thought of as the slope in the linear equation between the percent of change in the market or index fund (the explanatory variable) to the percent of change in the price of the individual stock (the response variable) The relationship can be modeled with the following equation

i i

y    where

i

y : represents the percent change in the stock

 : represents the percent change in the stock if the percent change in the market

is zero

 : represents the rate of change in the stock compared to the change in the market

i

x : the percent of change in the market as a whole

i

e : the difference in the predicted compared to the observed for each coordinate

The values of alpha and beta can be estimated using the method of least squares This model can be expressed as

i

yˆ ˆˆ where

ˆi

y : represents the predicted value of the percent change in the stock given the

percent change in the market

ˆ

 : represents the y-intercept of the least squares regression line

ˆ

 : represents the slope of the least squares regression line

i

x : the percent of change in the market as a whole

Beta hat is then the estimate of the stock’s beta value In order to find this least squares regression line, the observed values are compared to the predicted values

determined by the model (regression line) This difference between the observed

each coordinate, i Therefore, the best line to represent the data is the line that minimizes

the sum of the square of these error terms (the square of the difference between the

2

ˆ ˆ

2 1

1

ˆ ˆ

ˆ

ˆ ˆ

2

n

i

n

i

 



 











And setting that derivative equal to 0 in order to minimize:

 

1

1

2 1

2

ˆ ˆ

ˆ

ˆ

ˆ

n

i

n

i

n

i i i i

i

 

 













2

2

1

ˆ And solving for :

ˆ ˆ

ˆ

ˆ

n

i

i

x















Similarly, taking the derivative with respect to ˆ :

2

1

1

ˆ

ˆ ˆ

ˆ

ˆ

ˆ

n

i

n

i



 

 

 

 











ˆ

So, solving for :

ˆ

ˆ

n











1 1

2 1

2 1

ˆ

ˆ

And multiplying numerator and denominator by n to simplify:

ˆ

ˆ

n i i

n i i

n x



















2

2

2 2

2 2

2

1

ˆ

i i i i

i

i







2 1

ˆ

i i

x







2 Implementation on TI Graphing Calculator

The task of turning this calculation process into an actual program on the

graphing calculator began with obtaining the historical prices of a stock and of the

S&P500 This data is easily obtained online, and I used the Yahoo! Finance page to download this data into a spreadsheet From there, data is copied, pasted, and saved into a list in the TI-Connect data software (Texas Instruments, 2008) This transfer of data to the calculator serves as the main input of the program The program simply asks the user for

the ticker symbol of the index (in order to pull up the correct list of prices) and the ticker symbol of the index (GSPC – for the S&P500) used to represent the market Lastly, the user inputs “n” the number of data points to be used

First, the program uses each list of prices to calculate the percent of change in

i

i n







1

i

i n

i







1

i

i n







1

, and

( )x i

i

i n

2

1





   



graphically displays the data as a scatter plot with the percent changes in the S&P500 along the x-axis and the changes in the stock price along the y-axis and draws the least squares line using the calculated β as its slope and the following calculated alpha as its

The following screen captures display the code written for this program to calculate beta:

3: 4:

3 An Example of Estimating Beta for Texas Instruments

The following displays an example through the program for the stock TXN (Texas Instruments) First, the daily data from a two year history has been downloaded from the Yahoo! Finance and stored into lists on the calculator:

Next, the opening screen obtaining the inputs, and then also displaying β:

Lastly, the calculator displays the scatter plot with the least squares line:

4 Discrepancies in the estimation of beta

In the example above using the TI calculator, the beta value of the stock is

estimated to be 0.91 This beta was estimated using market and stock daily values for a two year history of prices; however, a survey of different business sites shows

discrepancies in the estimation of beta For example, on September 6, 2008, the following websites cited the β values for the Texas Instruments, Inc (TXN) stock:

Table 2 Values for beta for Texas Instrument Stock

This chart clearly depicts the inconsistency associated with estimating β For a value that is commonly referred to in the investing world, there is no standard set of data

to be used for the estimation of β, so numerous possibilities exist such as daily prices for the past two years, monthly prices for the past two years, weekly prices for five years, etc According to the websites, Yahoo! uses monthly data compared to the S&P500 over a three year time period, and Reuters uses monthly data compared to the S&P500 over a five year time period (CNBC’s method for calculating β is not available) These

differences would cause the estimates of beta for a stock to be different

In order to more efficiently investigate this discrepancy, I used the statistical package JMP (SAS Institute, 2008) in order to estimate β values for comparison of multiple types of data for the same stock I began by downloading the historical daily values of the individual stock (TXN) and the market (GSPC) and then calculated the percentage change for each Once calculated, these data are easily transferred into JMP

Once in JMP, the least squares regression line is determined and the estimate of beta given Below is an example of the scatter plot of the percentage change in the index versus the percentage change in the Texas Instrument Stock using monthly prices for Texas Instruments, Inc over a five year period:

Figure 1 Scatter plot: Percent Change in S&P 500 to Percent Change in TXN

-0.2

-0.1

0

0.1

0.2

0.3

-0.1 -0.05 0 0.05

% Change in Index

The least squares regression line has the equation

(Predicted % Change in Stock) = -0.003842 + 1.7850076*(% Change in Index),

1.785 This suggests that the price of Texas Instruments, Inc stock is approximately 78.5% more volatile than the market as a whole The same process was used in order to determine the β value using weekly and daily prices and dating back one year, two years, and ten years for the Texas Instruments, Inc stock The following chart displays the resulting β values and corresponding R-Squared values for each type of data:

Table 3 Beta estimates for Texas Instrument Stock using monthly, weekly or daily values for a time period of 1, 2, 5 or 10 years

Monthly Weekly Daily

R 2 =0.260

0.869

R 2 =0.282

0.794

R 2 =0.262

R 2 =0.267

0.945

R 2 =0.262

0.861

R 2 =0.268

R 2 =0.331

1.343

R 2 =0.287

1.156

R 2 =0.266

R 2 =0.285

1.605

R 2 =0.258

0.815

R 2 =0.071

From the table, we see that the frequency and amount of data used to calculate β for a given stock greatly affects the value of β Most drastically, the β values from some sets of data (weekly and daily going back for one or two years; daily ten years) suggest that TXN is less volatile than the market as a whole while the β values calculated from the other sets of data suggest that TXN is more volatile than the market as whole Even beyond these drastically different results, even the β values that all suggest that TXN is more volatile than the market vary significantly in the level of volatility For instance, when the change in the market increases by 1%, the monthly data going back one year suggest that the individual stock would increase by 1.06%; however, for the same change

in the market, the monthly data going back ten years suggest that the individual stock, TXN would increase by 1.896%

One consideration for the type of data chosen to calculate β must be tradeoff between accuracy and relevance By using more data points (extending back more years), the accuracy of β may rise, however, the relevance of those data points must be

questioned Trends in data from ten years ago may have little bearing on trends seen today for the same stock Further, using daily or weekly price changes provides more data points (and generally more accuracy), but monthly data may provide increased relevance due to the nature of most investors’ needs (Deutsch, 2005)

The chart also includes the R-squared values for each of the corresponding

variability of observed

is diversifiable (not market-related) in nature While these values appear quite similar for each of the models used to calculate β, the highest by far exists when using monthly data that extends back five years

5 The Role of beta in Investment Decisions

β serves many valuable purposes in investing First, as previously mentioned, it assesses the risk involved with an individual stock This certainly plays a significant role for investors Some investors may be willing to invest in a stock with a large β with hopes

of a large payoff, understanding the larger risk involved with such a stock Other

investors may be more inclined to invest in a safer stock with a smaller β to put less of their investment at risk, understanding that their potential return will not be as high

In addition to assessing the risk involved with an individual stock, β can also be used in valuing a company (McClure, 2004) Determining the value of a company serves

in comparing its value to the price of the stock Quite simply, if the calculated value is less than the current stock price, the stock is most likely overpriced and should not be purchased In a similar fashion, if the calculated value is higher than the price, the stock should be a good investment β can be used in determining this value of the company as evidenced in the following formula:

V a l u e 

E x p e c t e d F u t u r e P a y o f f

The focus of this project is not on the other values involved in this calculation (they are easily obtained or estimated) but rather on the important role that β plays in investment decisions

A simpler method of valuing a stock would be to look at , the y-intercept of the

the asset is overpriced Another suggested use of beta is for an investor to calculate the weighted average beta of an entire portfolio As Tom Zwrilein, professor of finance at University of Colorado at Colorado Springs suggests, “Beta doesn’t work well on the individual stock level” (Pellet, 47, 2004) The average beta of the entire portfolio would provide the investor with a larger picture of the anticipated volatility compared to that of the market

6 The Role of Beta in Mathematics Education

This concept of beta is certainly something my students in my Algebra II classes could understand on an introductory level We briefly study linear regression in Algebra

II in the chapter dealing with linear relationships After reviewing writing equations of lines, students learn to estimate the line of best fit or trend line for a scatter plot and also

to use the TI Graphing calculators to perform linear regression Exploring the topic of beta provides an opportunity for my students to be able to explore a real world context with real world data that matters To further enrich, this lesson could become a

collaborative lesson with an economics class during an introductory unit to the stock market Such a lesson would include (see Appendix A):

(collaborated with economics teacher)

the market and the percent of change of a stock (See Appendix B)

plot

calculator, graph this line on the calculator and copy down the equation

of the line, focusing on beta (the slope)

in a discussion of slope (more/less steep) and the implications in this context

historical prices, calculate the percent of change, use their calculator to find the line (and therefore beta) and interpret

Furthermore, this project has again reminded me of the importance of cross-referencing and challenging information that is placed before me and presented as fact Obviously, this project is a perfect example of a case where the “facts” do not line up perfectly, and a truly mathematical approach seeks a reason why I want to teach my students to look at the world around them with open, inquisitive minds So often, mathematical education unfortunately seems to revolve around set processes that are presented with few thought-provoking lessons While some things in mathematics work this way, students need to learn to question why something works the way it does and if something is correct

without blindly accepting it as fact

From another perspective, I truly enjoyed this project because it revolved around a topic that interests me and that sparked curiosity I need to find ways to pull those types of projects and topics more into my classroom My students need to learn how to ask and answer mathematical questions in topics that matter to them While this presents a lofty goal in the classroom, it certainly creates for valuable, meaningful education