# A Statistical Analysis of a Stock's Volatility

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```					                      Beta

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.

β <0        The stock moves contrary to the market in an inverse relationship. As the
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.
β =0        The stock’s returns are unrelated to market moves.

0< β <1     The stock is expected to move more slowly than the market. If the market
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.
β =1        The stock should move in a manner very similar to the market as a whole.

β >1        The stock has proven over time to be more volatile than the market. As the
market rises, this stock should rise at a higher rate. Likewise, a more severe

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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

y i = α + β x i + ei

where

yi : represents the percent change in the stock
α : represents the percent change in the stock if the percent change in the market
is zero
β : represents the rate of change in the stock compared to the change in the
market
xi : the percent of change in the market as a whole
ei : 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

ˆ ˆ
y i = α + βxi
ˆ

where

ˆ
yi : represents the predicted value of the percent change in the stock given the
percent change in the market
α : represents the y-intercept of the least squares regression line
ˆ
ˆ
β : represents the slope of the least squares regression line
xi : the percent of change in the market as a whole

3
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

coordinate and the predicted coordinate on the line is referred to as the error, or ei for

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

predicted and the observed). Let yi represent the observed percent of change in stock

ˆ ˆ                  (          )
value. Therefore, yi − yi = ei or yi − α + β xi = ei . Thus, in order to determine α and
ˆ                                                           ˆ

( (      ))
n          n                   2
βˆ
, the following must be minimized:                         ∑ ei2 = ∑ yi − α + βˆ xi
i =1       i =1
ˆ

ˆ
To minimize, first take the derivative with respect to β :

∂ n
( (                          ))
2

ˆ∑ i
ˆ ˆ
y − α + β xi
∂β i =1

( (                          )) ( − x )
n
= 2∑ yi − α + β xi
ˆ ˆ                                           i
i =1

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

( (                          )) = 0
n
−2∑ xi yi − α + β xi
ˆ ˆ
i =1

∑ x ( y − α − βˆ x ) = 0
n

i
ˆ           i            i
i =1

∑ ( x y − α x − βˆ x ) = 0
n
2
ˆ   i   i            i       i
i =1
n                         n                   n

∑ xi yi − α ∑ xi −βˆ ∑ xi 2 = 0
i =1
ˆ
i =1             i =1

4
ˆ
And solving for β :
n                                       n                          n

∑ x y −α ∑ x                                            = β ∑ xi
ˆ      2
ˆ i          i                              i
i =1                                    i =1                    i =1
n                                       n

∑ x y −α ∑ x
ˆ    i       i                               i
i =1                                    i =1
=βˆ
n

∑x
2
i
i =1

Similarly, taking the derivative with respect to α :
ˆ

∑(                                                          ))                 ( (        )) ( −1)
∂
(
n                                                           2       n
ˆ ˆ
yi − α + β xi                                    = 2∑ yi − α + β xi
ˆ ˆ
∂αˆ      i =1                                                                   i =1

( (                                           )) = 0
n
−2∑ yi − α + β xi
ˆ ˆ
i =1

∑ ( y − α − βˆ x ) = 0
n
ˆ       i                                  i
i =1
n                          n                             n

∑ y − ∑ α − βˆ ∑ x
i =1
ˆi
i =1                            i =1
i   =0
n                                                  n

∑ yi − nα − βˆ ∑ xi = 0
i =1
ˆ
i =1

So, solving for α :
ˆ
n                                  n

∑ y − βˆ ∑ x
i =1
i
i =1
i   = nα
ˆ
n                                  n

∑ yi − βˆ ∑ xi
i =1                            i =1
=α
ˆ
n

ˆ
Therefore, substituting this equation into α in the previously derived formula for β :
ˆ

5
⎛ n         n
⎞
n       ⎜ ∑ yi − βˆ ∑ xi                   ⎟ n
∑ xi yi − ⎜ i=1 n i=1
i =1      ⎜
⎟ ∑ xi
⎟ i =1
⎜                                  ⎟
⎝                                  ⎠       ˆ
=β
n

∑x
2
i
i =1

And multiplying numerator and denominator by n to simplify:
n
⎛ n             n
⎞ n
n∑ xi yi − ⎜ ∑ yi − β ∑ xi ⎟ ∑ xi
ˆ
i =1      ⎝ i =1        i =1 ⎠ i =1 = β
ˆ
n
n∑ xi
2

i =1

ˆ
Simplify to solve for β :

2
ˆ⎛       ⎞
n         n    n          n           n
n∑ xi yi − ∑ yi ∑ xi + β ⎜ ∑ xi ⎟ = β n∑ xi
ˆ       2

i =1      i =1 i =1     ⎝ i =1 ⎠      i =1
2
ˆ⎛       ⎞
n            n             n  n       n
n∑ xi yi − ∑ yi ∑ xi = β n∑ xi − β ⎜ ∑ xi ⎟
ˆ        2

i =1      i =1 i =1       i =1     ⎝ i =1 ⎠
⎛ n                2
⎞
ˆ ⎜n x 2 −⎛ x ⎞ ⎟
n         n    n                    n
n∑ xi yi − ∑ yi ∑ xi = β ∑ i ⎜ ∑ i ⎟
⎜ i =1    ⎝ i =1 ⎠ ⎟
i =1      i =1 i =1     ⎝                    ⎠
n           n                n
n∑ xi yi − ∑ yi ∑ xi
i =1        i =1           i =1        ˆ
=β
2
n
⎛      ⎞    n
n∑ xi − ⎜ ∑ xi ⎟
2

i =1   ⎝ i =1 ⎠

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

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

6
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

prices for each day (or month, depending on the data) by a simple “for loop” of (dayi+1-

dayi)/dayi. These percent changes are stored into lists. The program then calculates and
i=n           i=n                 i=n
stores the individual pieces of the linear regression:       ∑
i =1
xi ,   ∑
i =1
xi * y i ,   ∑y
i =1
i   , and

i=n

∑ (x )
i =1
i
2
. Then, putting these pieces together, the program computes the slope of the least

squares line: m =
n(∑ xy) − (∑ x)(∑ y)    and outputs this value as β. Next, the program
n( ∑ x ) − (∑ x )
2
2

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 y-

intercept: α =
(∑ y ) − β (∑ x ) .
n

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

1:                                         2:

3:                                         4:

7
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:

8
Table 2. Values for beta for Texas Instrument Stock

Financial Website           Beta Value

Yahoo                      1.41

Reuters                    1.77

CNBC                       1.13

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

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:

9
Figure 1. Scatter plot: Percent Change in S&P 500 to Percent Change in TXN

0.3

0.2
% Change in Stock

0.1

0

-0.1

-0.2
-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),

with an R-squared value of 33.0%. From the least squares equation, the estimate of β is

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.

10
Monthly                Weekly                  Daily

1 Year                   1.066                  0.869                   0.794

R2=0.260               R2=0.282                R2=0.262

2 Years                  1.153                  0.945                   0.861

R2=0.267               R2=0.262                R2=0.268

5 Years                  1.785                  1.343                   1.156

R2=0.331               R2=0.287                R2=0.266

10 Years                 1.896                  1.605                   0.815

R2=0.285               R2=0.258                R2=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

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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 predicted
models. In general, R 2 =                            gives the percentage of total risk that is
variability of observed

systematic (market-related) in nature, which 1 − R 2 gives the percentage of total risk that

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

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should be a good investment. β can be used in determining this value of the company as

evidenced in the following formula:

ExpectedFuturePayoff
Value =
(1 + R   F                  )
+ β (R M − R F )
t

Where RF = the risk-free rate of return (i.e., US Treasury bond)

And R M = the expected return on the market as a whole (i.e., S&P500)

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 α , the y-intercept of the

least squares line used to find beta. If α > 0 , the asset is underpriced, and if α < 0 , then

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

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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):

-         Brief introduction to stocks, volatility, beta, S&P 500/the market

(collaborated with economics teacher)

-         Review of calculating percent of change

-         Provide students with a table of data containing the percent of change of

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

-         Students will enter the data into graphing calculator and create a scatter

plot.

-         Find the least squares line using the linear regression feature on the

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

of the line, focusing on beta (the slope).

-         Graph the line y = x and compare to the least squares line. Lead the class

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

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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.

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Appendix A

Algebra II             Beta Activity          Name: ________________________

1: We must first turn our scatter plots ON on our calculators:
[2nd ] [y = ]   (STAT PLOT)

Choose Plot 1: Highlight “ON” and push [ENTER]
Choose the Scatter Plot TYPE
XList: L1
YList: L2

2:Enter the data:
[STAT] Option 1: Edit
Enter data into the columns for L1 and L2.
(if there is already data present, you can clear it out by highlighting L1 or L2 and pushing
the CLEAR button – do not push the DEL button to do this, it will delete the list, not
just the data – or you can simply type over the existing data)

Enter the % Change in the market (x) into L1 and the % change in the stock (y) into L2

3:Make a scatter plot:
To set the window, you can push the [ZOOM] and choose the ZOOM STAT option (you
must scroll down to it).

→Sketch the scatter plot here:

16
→As the % change in the market (x) increases, what is the trend in the % change in the
stock?

4:Find line of best fit:
[STAT]- go over to option CALC –
choose option 4 LinReg
Push [ENTER]

→ Write down the equation (round to the nearest thousandth):
→ What is the slope of the line (ie. the beta value for this stock)?

→What does this mean in the given context?
→Graph the line on the scatter plot (type the equation in Y=). Draw a sketch.

→Graph the line y = 1x in the same window. If this were the line of best fit, what
would it represent in the given context?

5: Another stock
Return to step #2 and repeat, changing the data in L2 to the General Motors data.

6: Comparison
Compare the beta values for TXN and GM. What expectations would you have for these
two stocks?

17
Appendix B

#1: TXN Data for Beta Activity
Date            S&P500      % Change (X) TXN        % Change(Y)
8-Nov        1
,005.75          0.04    19.09        -0.02
8-Oct         968.75         -0.17    19.56        -0.09
8-Sep        1 64.74
,1             -0.09     21 .5       -0.12
8-Aug        1
,282.83          0.01    24.51         0.01
8-Jul       1
,267.38         -0.01    24.38        -0.13
8-Jun        1
,280.00         -0.09    28.1 6       -0.13
8-May        1
,400.38          0.01    32.48         0.11
8-Apr        1
,385.59          0.05    29.1 6        0.03
8-Mar        1
,322.70         -0.01    28.27        -0.06
8-Feb        1
,330.63         -0.03    29.96        -0.03
8-Jan        1
,378.55         -0.06    30.95        -0.07
7-Dec        1
,468.36         -0.01     33.4         0.06
7-Nov         ,481 4
1 .1                        .57
31
#2: GM Data for Beta Activity

Date            S&P500       % Change (X) GM       % Change (Y)
8-Nov          1
,005.75           0.04    5.72       -0.01
8-Oct         968.75          -0.17    5.79       -0.39
8-Sep        1 64.74
,1              -0.09    9.45       -0.06
8-Aug          1
,282.83           0.01      10       -0.10
8-Jul        1
,267.38          -0.01   11.07       -0.04
8-Jun        1
,280.00          -0.09    11.5       -0.33
8-May          1
,400.38           0.01    17.1       -0.26
8-Apr        1
,385.59           0.05    23.2        0.22
8-Mar          1
,322.70          -0.01   19.05       -0.18
8-Feb        1
,330.63          -0.03   23.28       -0.17
8-Jan        1
,378.55          -0.06   28.21        0.13
7-Dec          1
,468.36          -0.01   24.89       -0.17
7-Nov            ,481 4
1 .1                    29.83

18
References

Babson. (Fall 2004). Data Finder: Finding Beta. Retrieved July 2007 from

Deutsch, J. (2005). Stock beta calculation. Retrieved October 10, 2008, from

http://www.statistics-help-online.com/node76.html.

Harrington, D. (1983). Stock prices, beta, and strategic planning. Harvard Business

Review. May-June 1983, 157-159.

McClure, B. July 2007. Beta: Know the Risk. Retrieved 30 November 2004, from

http://www.investopedia.com/articles/stocks/04/113004.asp .

Merle, R. October 2008. High levels of volatility show fear’s role in stock markets. The

Mercury News. Retrieved October 23, 2008, from

http://www.mercurynews.com/nationworld/ci_10779991.

Pellet, J. (2004). Beta on it. Money. March 2004, 47.

Stock Beta Calculation. (2006) Stock Beta Calculation. Retrieved October 10, 2008, from

http://www.stockbetacalculation.com/.

SAS Institute (2008). JMP Software (Version 7) [Software]. Available from

http://www.jmp.com/.

Texas Instruments (2008). TI Connect Software [Software]. Available from

http://education.ti.com/educationportal/sites/US/productDetail/us_ti_connect.html

Yahoo! Finance. (2008). Key Statistics. Retrieved September 2008 from

http://help.yahoo.com/l/us/yahoo/finance/tools/research-12.html.

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