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. 2 has tended to gain roughly 20% more than the market during rising periods, and has tended to experience declines 20% more severe than the 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 20 0 2 0 -40 3 40 40 4 30 30 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: 20/100 + 1 = 1. 2 0/100 + 1 = 1. 0 0/100 + 1 = 1.
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0 -40/100 + 1 = 0. 6 40/100 + 1 = 1. 4 40/100 + 1 = 1. 4 30/100 + 1 = 1. 3 30/100 + 1 = 1. 3 -10/100 + 1 = 0.
9 -50/100 + 1 = 0. 5 Step 2: 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. 2 = 0.
18232 1. 0 = 0. 00000 1. 0 = 0. 00000 0.
6 = -0. 51083 1. 4 = 0. 33647 1. 4 = 0. 33647 1.
3 = 0. 26236 1. 3 = 0. 26236 0.
9 = -0. 10536 0. 5 = -0. 69315 Sigma X = 0. 67580 -0. 60514 = Sigma Y Step 3: Multiply each log price relative against each other and then add the total together to get Sigma X Y.
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00000 -0. 51083 = 0. 00000 0. 33647 0. 33647 = 0. 11321 0.
26236 0. 26236 = 0. 06884 -0. 1. 0536 -0. 69315 = 0.
07303 Sigma X Y = 0. 25508 Step 4: Square both X and Y to calculate Sigma X squared and Sigma Y squared. Example: 0. 18232 0. 18232 = 0. 03324 0.
00000 0. 00000 = 0. 00000 0. 00000 0. 00000 = 0. 00000 -0.
51083 -0. 51803 = 0. 26904 0. 33647 0. 33647 = 0. 11321 0.
33647 0. 33647 = 0. 11321 0. 26236 0.
26236 = 0. 06884 0. 26236 0. 26236 = 0.
06884 -0. 10536 -0. 10536 = 0. 0110 -0. 69315 -0.
69315 = 0. 48045 Sigma X 2 = 0. 22639 Sigma Y 2 = 0. 92344 Step 5: Calculate “beta” using the following formula Beta Coefficient = n xy x y n x x – – a a a a a 2 2 () Example: 5 0. 25508 -. 67580 (-.
60514) Beta = 5 . 22639 -. 67580 2 = 2. 494435.
