Upside and Downside Beta  
 
 
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This time we will calculate Upside and downside Beta
Beta describes the relation of returns between the stock and the market. 
For example, if Beta equals 1.2, this means that when market goes up/down by 1%, the stock grows/falls by 1.2% (in theory of course)
Beta shows whether the stock grows/falls faster or slower than the market.
Upside Beta analyses the stock's behavior during the upward market's movements.
I.e. Upside Beta shows how the stock grows/falls when the market grows. 
Why do you need to calculate Upside Beta?
For example, you expect that the market will go up next week and you need to pick up the stock that Prices ($) Change (%) Upside changes (%) Downside changes (%)
will outpace the market. In this case, choose the stock with the highest Upside Beta AAPL C S&P AAPL C S&P AAPL C S&P AAPL C S&P
01.01.2010 210.73 3.31 1122.9      
Downside Beta on the opposite analyses the stock's behavior when the market falls 08.01.2010 211.98 3.59 1148 0.6% 8.5% 2.2% 0.6% 8.5% 2.2% 0.0% 0.0% 0.0%
For example, you expect the market to fall next week. Choose the stock with a lowest Downside Beta 15.01.2010 205.93 3.42 1137.6 -2.9% -4.7% -0.9% 0.0% 0.0% 0.0% -2.9% -4.7% -0.9%
If Downside Beta is less than 0, the stock will go up when the market goes down. 22.01.2010 197.75 3.25 1098.7 -4.0% -5.0% -3.4% 0.0% 0.0% 0.0% -4.0% -5.0% -3.4%
29.01.2010 192.06 3.32 1080.2 -2.9% 2.2% -1.7% 0.0% 0.0% 0.0% -2.9% 2.2% -1.7%
Next we will calculate simple, upside and downside Betas 05.02.2010 195.46 3.22 1064.5 1.8% -3.0% -1.5% 0.0% 0.0% 0.0% 1.8% -3.0% -1.5%
12.02.2010 200.38 3.18 1081.2 2.5% -1.2% 1.6% 2.5% -1.2% 1.6% 0.0% 0.0% 0.0%
STEP 1 CALCULATING SIMPLE BETA   19.02.2010 201.67 3.42 1112.3 0.6% 7.5% 2.9% 0.6% 7.5% 2.9% 0.0% 0.0% 0.0%
26.02.2010 204.62 3.4 1109.6 1.5% -0.6% -0.2% 0.0% 0.0% 0.0% 1.5% -0.6% -0.2%
First download the weekly values of the market index (S&P) and the stock prices 05.03.2010 218.95 3.5 1139.2 7.0% 2.9% 2.7% 7.0% 2.9% 2.7% 0.0% 0.0% 0.0%
We traditionally use the stocks of Apple (Ticker: AAPL) and Citi (Ticker: C) 12.03.2010 226.6 3.97 1147.3 3.5% 13.4% 0.7% 3.5% 13.4% 0.7% 0.0% 0.0% 0.0%
By the way, we use relatively small selection of values in this example. We recommend you to use more values 19.03.2010 222.25 3.9 1153.9 -1.9% -1.8% 0.6% -1.9% -1.8% 0.6% 0.0% 0.0% 0.0%
for more reliable results 26.03.2010 230.9 4.31 1170.4 3.9% 10.5% 1.4% 3.9% 10.5% 1.4% 0.0% 0.0% 0.0%
02.04.2010 235.97 4.18 1179.9 2.2% -3.0% 0.8% 2.2% -3.0% 0.8% 0.0% 0.0% 0.0%
We use weekly data, because we want to analyze the stock's performance over the week 09.04.2010 241.79 4.55 1195.7 2.5% 8.9% 1.3% 2.5% 8.9% 1.3% 0.0% 0.0% 0.0%
You can use daily, monthly, yearly data if you want. 16.04.2010 247.4 4.56 1189.3 2.3% 0.2% -0.5% 0.0% 0.0% 0.0% 2.3% 0.2% -0.5%
23.04.2010 270.83 4.86 1216.8 9.5% 6.6% 2.3% 9.5% 6.6% 2.3% 0.0% 0.0% 0.0%
After that calculate weekly returns for the index and for the stocks (to the right) 30.04.2010 261.09 4.37 1191.9 -3.6% -10.1% -2.0% 0.0% 0.0% 0.0% -3.6% -10.1% -2.0%
07.05.2010 235.86 4 1142.4 -9.7% -8.5% -4.2% 0.0% 0.0% 0.0% -9.7% -8.5% -4.2%
The formula for Beta is  14.05.2010 253.82 3.98 1136.5 7.6% -0.5% -0.5% 0.0% 0.0% 0.0% 7.6% -0.5% -0.5%
Beta = Cov (Rs; Rm) / Var (Rm)  21.05.2010 242.32 3.75 1085 -4.5% -5.8% -4.5% 0.0% 0.0% 0.0% -4.5% -5.8% -4.5%
28.05.2010 256.88 3.96 1085.7 6.0% 5.6% 0.1% 6.0% 5.6% 0.1% 0.0% 0.0% 0.0%
Cov (Rs; Rm) is the covariance of the stock's and market's returns  04.06.2010 255.97 3.79 1066.7 -0.4% -4.3% -1.8% 0.0% 0.0% 0.0% -0.4% -4.3% -1.8%
You can calculate covariance using COVAR function in excel (if you are interested you can also look at  11.06.2010 253.51 3.88 1096 -1.0% 2.4% 2.7% -1.0% 2.4% 2.7% 0.0% 0.0% 0.0%
the definition of covariance in excel).  18.06.2010 274.07 4.01 1124.5 8.1% 3.4% 2.6% 8.1% 3.4% 2.6% 0.0% 0.0% 0.0%
Var (Rm) is the dispersion of market returns. 25.06.2010 266.7 3.94 1078.2 -2.7% -1.7% -4.1% 0.0% 0.0% 0.0% -2.7% -1.7% -4.1%
You can calculate Var using VAR function in excel. 02.07.2010 246.94 3.79 1030.2 -7.4% -3.8% -4.5% 0.0% 0.0% 0.0% -7.4% -3.8% -4.5%
09.07.2010 259.62 4.04 1076.7 5.1% 6.6% 4.5% 5.1% 6.6% 4.5% 0.0% 0.0% 0.0%
Stock Cov (Rs; Rm) Var (Rm) Beta 16.07.2010 249.9 3.9 1067.5 -3.7% -3.5% -0.9% 0.0% 0.0% 0.0% -3.7% -3.5% -0.9%
AAPL 0.00065 0.00049 1.31 23.07.2010 259.94 4.02 1103.6 4.0% 3.1% 3.4% 4.0% 3.1% 3.4% 0.0% 0.0% 0.0%
C 0.00074 0.00049 1.51 30.07.2010 257.25 4.1 1108.7 -1.0% 2.0% 0.5% -1.0% 2.0% 0.5% 0.0% 0.0% 0.0%
06.08.2010 260.09 4.06 1123.6 1.1% -1.0% 1.3% 1.1% -1.0% 1.3% 0.0% 0.0% 0.0%
We can see that Citi's Beta is higher than Apple's which means that Citi's stocks grows and falls faster than the 13.08.2010 249.1 3.88 1077.5 -4.2% -4.4% -4.1% 0.0% 0.0% 0.0% -4.2% -4.4% -4.1%
market and than Apple's stocks. 20.08.2010 249.64 3.75 1074.2 0.2% -3.4% -0.3% 0.0% 0.0% 0.0% 0.2% -3.4% -0.3%
27.08.2010 241.62 3.76 1061.3 -3.2% 0.3% -1.2% 0.0% 0.0% 0.0% -3.2% 0.3% -1.2%
STEP 2 CALCULATING UPSIDE BETA   03.09.2010 258.77 3.91 1101.7 7.1% 4.0% 3.8% 7.1% 4.0% 3.8% 0.0% 0.0% 0.0%
q 10.09.2010 263.41 3.91 1113.4 1.8% 0.0% 1.1% 1.8% 0.0% 1.1% 0.0% 0.0% 0.0%
For the calculation of Upside Beta we need to exclude all negative values of market's returns because we analyze 17.09.2010 275.37 3.95 1127 4.5% 1.0% 1.2% 4.5% 1.0% 1.2% 0.0% 0.0% 0.0%
the stock's behavior on growing market. 24.09.2010 292.32 3.904 1148.6 6.2% -1.2% 1.9% 6.2% -1.2% 1.9% 0.0% 0.0% 0.0%
01.10.2010 282.52 4.09 1144.6 -3.4% 4.8% -0.4% 0.0% 0.0% 0.0% -3.4% 4.8% -0.4%
Use "if" excel function to choose only positive values of S&P returns first (see "Upside Changes" column to the right) 08.10.2010 294.07 4.19 1165.9 4.1% 2.4% 1.9% 4.1% 2.4% 1.9% 0.0% 0.0% 0.0%
 = if (S&P return > 0, value of S&P return, 0) 15.10.2010 314.74 3.95 1176.8 7.0% -5.7% 0.9% 7.0% -5.7% 0.9% 0.0% 0.0% 0.0%
22.10.2010 307.47 4.11 1186.6 -2.3% 4.1% 0.8% -2.3% 4.1% 0.8% 0.0% 0.0% 0.0%
Then use "if" function to choose only those return values of the stocks which correspond to the positive values of 29.10.2010 300.98 4.17 1187 -2.1% 1.5% 0.0% -2.1% 1.5% 0.0% 0.0% 0.0% 0.0%
S&P's returns 05.11.2010 317.13 4.49 1222.6 5.4% 7.7% 3.0% 5.4% 7.7% 3.0% 0.0% 0.0% 0.0%
 = if (S&P return > 0, value of the stock's return, 0) 12.11.2010 308.03 4.29 1202.4 -2.9% -4.5% -1.7% 0.0% 0.0% 0.0% -2.9% -4.5% -1.7%
19.11.2010 306.73 4.268 1195.3 -0.4% -0.5% -0.6% 0.0% 0.0% 0.0% -0.4% -0.5% -0.6%
Stock Cov (Rs; Rm) Var (Rm) Upside Beta 26.11.2010 315.76 4.11 1183.8 2.9% -3.7% -1.0% 0.0% 0.0% 0.0% 2.9% -3.7% -1.0%
AAPL 0.00021 0.00015 1.35 03.12.2010 317.44 4.45 1223.5 0.5% 8.3% 3.4% 0.5% 8.3% 3.4% 0.0% 0.0% 0.0%
C 0.00024 0.00015 1.58 10.12.2010 320.56 4.77 1243.4 1.0% 7.2% 1.6% 1.0% 7.2% 1.6% 0.0% 0.0% 0.0%
17.12.2010 320.61 4.7 1246.5 0.0% -1.5% 0.2% 0.0% -1.5% 0.2% 0.0% 0.0% 0.0%
As wee see, Citi and Apple grow faster than the market   24.12.2010 323.6 4.68 1253.6 0.9% -0.4% 0.6% 0.9% -0.4% 0.6% 0.0% 0.0% 0.0%
In theory for every 1% growth of S&P index Citi's and Apple's stocks should gain 1.58% and 1.35% respectively 31.12.2010 322.56 4.73 1262.9 -0.3% 1.1% 0.7% -0.3% 1.1% 0.7% 0.0% 0.0% 0.0%
STEP 3 CALCULATING DOWNSIDE BETA  
For the calculation of Downside Beta we need to exclude all positive values of returns because we analyze
the stock's behavior on falling market.
Use "if" excel function to choose only negative values of S&P returns (see "Downside Changes" column to the right)
 = if (S&P return < 0, value of S&P return, 0)
Then use "if" function to choose only those return values of the stocks which correspond to the negative values of
S&P's returns
 = if (S&P return < 0, value of the stock's return, 0)
Stock Cov (Rs; Rm) Var (Rm) Downside Beta
AAPL 0.00024 0.00018 1.30
C 0.00023 0.00018 1.28
Citi's stocks should fall slower than Apple's stocks when the market goes down. At the same time we should note than neither of these stocks can be considered as "defensive" stocks (the stocks that do not fall when 
the market falls).
So if the market falls by 1% Apple's and Citi's stocks should fall by 1.3% and 1.28% respectively.
We calculated the simplest Upside and Downside Betas to measure the stocks' performance on growing and falling market. But you can also calculate Betas relative to a risk free rate.
The formula for Downside Beta looks like this:
More information on that topic you can find here:
http://www.efmaefm.org/efma2006/papers/310329_full.pdf