Tuesday, May 15, 2012

Illustration of Method Used to Calculate Beta

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Beta is an asset’s volatility relative to “the market.” An asset with a beta coefficient of 1.0


has tended to experience up and down movements of roughly the same magnitude as


the market. One with a beta of 1. has tended to gain roughly 0% more than the market


during rising periods, and has tended to experience declines 0% more severe than the


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market during periods of falling prices. The name “beta” refers to the “b” (the “slope”) in


the linear equation Y= a + bX.


CALCULATION METHODOLOGY


This method compares an asset’s volatility relative to “the market.” A formula is designed


to create a log-log regression of an asset. This is accomplished by using “log price


relatives” (the natural logarithms of the price relatives).


Factual Data


Time


Period “Market” % Fund %


1 0 0


0 -40


40 40


4 0 0


5 -10 -50


Step 1 Determine the Price Relative of each time period for both “the market” and the


fund using the following formula


Fund or Market Return/100 + 1 = Price Relative


Example


0/100 + 1 = 1. 0/100 + 1 = 1.0


0/100 + 1 = 1.0 -40/100 + 1 = 0.6


40/100 + 1 = 1.4 40/100 + 1 = 1.4


0/100 + 1 = 1. 0/100 + 1 = 1.


-10/100 + 1 = 0. -50/100 + 1 = 0.5


Step After you calculate the Price Relatives for all the time periods, you must convert


each value using “Log Price Relatives” (the natural logarithms of the price relatives). Do


this by using a financial calculator or a spreadsheet program with formulas.


Example From the previous example use the calculated price relatives to determine the


“Log Price Relatives”. Add the figures for each to calculate Sigma X and Sigma Y.


“ Market” (X) Fund (Y)


1. = 0.18 1.0 = 0.00000


1.0 = 0.00000 0.6 = -0.5108


1.4 = 0.647 1.4 = 0.647


1. = 0.66 1. = 0.66


0. = -0.1056 0.5 = -0.615


Sigma X = 0.67580 -0.60514 = Sigma Y


Step Multiply each log price relative against each other and then add the total together


to get Sigma XY.


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Example


XY


0.08 0.00000 = 0.00000


0.00000 -0.5108 = 0.00000


0.647 0.647 = 0.111


0.66 0.66 = 0.06884


-0.1.056 -0.615 = 0.070


Sigma XY = 0.5508


Step 4 Square both X and Y to calculate Sigma X squared and Sigma Y squared.


Example


0.18 0.18 = 0.04 0.00000 0.00000 = 0.00000


0.00000 0.00000 = 0.00000 -0.5108 -0.5180 = 0.604


0.647 0.647 = 0.111 0.647 0.647 = 0.111


0.66 0.66 = 0.06884 0.66 0.66 = 0.06884


-0.1056 -0.1056 = 0.0110 -0.615 -0.615 = 0.48045


Sigma X = 0.6 Sigma Y = 0.44


Step 5 Calculate “beta” using the following formula


Beta Coefficient =


n xy x y


n x x


-


-


å å å


å å ( )


Example


5 0.5508 - .67580 (-.60514)


Beta =


5 .6 - .67580 = .4445


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