4
To look at this probability, bring up an Excel worksheet. Click on Insert in the toolbar, and pick Function. A box appears on the screen. Under Cate- gory, pick Statistical, and then, under Function, click on BINOMDIST.
In the first box, enter the Number of successes (the number of times the stock price went up), which is 8. The second box asks for the Number of trials (in our example, this is how many years are in our sample), which is 9. The third box asks for the probability of a success (the stock price going up) on each trial, which is 0.5.
The fourth box asks if you want the cumulative probability, which would be the sum of the chances of a stock going up zero times out of nine, going up one time out of nine, two times out of nine, three out of nine . . . all the way up to eight out of nine. We don’t want the cumulative probability because all we are interested in is the probability of it going up exactly eight out of nine times. So, we enter False (which means we don’t want the cumulative probability). You then get the answer, which is p⫽0.01758, or less than 2 percent.
This is the probability of a stock going up eight out of nine years if its probability of going up or down each year were equal.
We just found the likelihood of the stock going up exactlyeight out of nine times. However, what we really want to know is the likelihood of a stock going up at
TABLE 4.1 GE Real Price Growth in the 1990s
Year GE without Inflation 2006 $
1990 4.93
1991 6.56
1992 7.33
1993 8.98
1994 8.76
1995 12.38
1996 16.81
1997 24.94
1998 34.60
1999 51.71
Source:Data from http://finance.yahoo.com and www.inflationdata.com/inflation/Inflation_Rate/
HistoricalInflation.aspx
Cumulative Probability The summed probabilities of multiple outcomes
leasteight out of nine times. This would also include the likelihood of the stock going up nine out of nine times. You solve for nine out of nine the same way we just did, except the number of successes in the first BINOMDIST box now becomes nine. Solving this, you find that the probability of nine out of nine times is 0.00195. Adding that to the probability of eight out of nine times, we get 0.01758 ⫹0.00195 ⫽0.01953, which still is a probability less than 2 percent.
Using Excel’s Cumulative Function
Now, adding the probabilities for eight and nine suc- cesses in our example was easy. But what if you wanted to know the probability of a stock randomly going up 40 times in the last 60 trading days, again assuming that the probability of going up on any given day is random, or p⫽0.5. As we previously stated, what we really want to know is the chance of a stock going up 40 or more times out of 60. The chance of hitting one specific num- ber (like 40 in this case) is always low when there are a large number of data points, so hitting exactly 40 is not what we are interested in. We couldcalculate the chance of 40 or more in the same way we just did for eight or more out of nine trials, but do we really want to calcu- late and add together the chances of 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, and 60 out of 60? That would certainly be possible, but it would also be te- dious. Instead, we will use the cumulative function in Excel to calculate this with less fuss.
First, we need one basic logic understanding: The sum of the probabilities of all possible outcomes of an event ⫽1.
For example, the sum of the probabilities of all possible ways to toss a coin, a head or a tail, is 0.5 ⫹0.5 ⫽1. The sum of the probabilities of all pos- sible outcomes of one toss of a die (getting a 1, 2, 3, 4, 5, or 6) are 1⁄6⫹1⁄6⫹
1⁄6 ⫹ 1⁄6 ⫹ 1⁄6 ⫹ 1⁄6 ⫽ 1. We can use this knowledge to use the cumulative function in Excel.
Cumulative Function A specific function within Excel that sums the prob- abilities of something happening a given amount of times or less
Key Point
The sum of the probabilities of all possible outcomes of an event = 1.
We can use this knowledge to use the cumulative function in Excel.
Just to try a simple example, let’s redo the earlier problem of eight or more successes in nine trials, assuming that the probability on each trial is 0.5. From
our earlier logic statement, if a stock has an equal chance of going up or down on each day, the probability of a stock going up seven times or lessout of nine times, plus the probability of it going up eight times or moreout of nine, must be equal to 1! This is because we have included all possible outcomes, and we said that the sum of the probabilities of all possible outcomes of an event ⫽1.
So, using simple algebra: 1.0 minus the probability of a stock going up seven times or less out of nine must equal the probability of the stock going up eight times or more out of nine, which is what we desire to know. Since we can use the cumulative function in Excel to find the likelihood of something going up seven or fewer times out of nine, we can then subtract that result from 1 to determine what we want to know: the probability of the stock going up eight or more times out of nine. Let’s do this for our example.
Bring up an Excel worksheet. Click on Insert in the toolbar and pick Function. A box appears on the screen. Under Category, pick Statistical, and then, under Function, click on BINOMDIST.
In the first box, enter the Number of successes (the number of times the stock price went up), which is 7. The second box asks for the Number of trials (in our example, this is how many years), which is 9. The third box asks for the probability of a success (the likelihood of the stock price going up) on each trial, which is 0.5. The fourth box asks if you want the cumulative probability, which would be the sum of the chances of a stock going up seven or fewer times out of nine. Since this is what we want, enter True. The probability then comes up: p⫽0.98047. But as we just discussed, we have to subtract this from 1: 1.0 ⫺0.98047 ⫽0.01953. This p⫽0.01953 matches the earlier answer we got by adding the probabilities of eight out of nine and nine out of nine.
Let’s go back to the earlier example of wanting to know the random chance of a stock going up 40 or more times out of 60 samples. From our earlier logic statement, if a stock has an equal chance of going up or down on each day, or a p⫽0.5, the probability of a stock going up 39 times or less out of 60 times, plus the probability of it going up 40 times or more out of 60, must be equal to 1! This is because we have included all possible out- comes, and we said that the sum of the probabilities of all possible outcomes of an event ⫽ 1. So, using simple algebra: 1.0 minus the probability of a stock going up 39 times or less out of 60 must equal the probability of the stock going up 40 times or more out of 60, which is what we desire to know.
Since we can use the cumulative function in Excel to find the likelihood of something going up 39 or less times out of 60, we can then subtract that re- sult from 1. Let’s do this.
Bring up an Excel worksheet. Click on Insert in the toolbar and pick Function. A box appears on the screen. Under Category, pick Statistical, and then, under Function, click on BINOMDIST.
In the first box, enter the Number of successes (the number of times the stock price goes up), which is 39 (because we want 39 times or less). The
second box asks for the Number of trials, which is 60.
The third box asks for the probability of a success (the random probability of the stock price going up) on each trial, which is 0.5.
The fourth box asks if you want the cumulative probability, which would be the sum of all the individual probabilities of the stock going up 39 times or fewer times out of 60. Since this is what we want, we type in True. Doing this, we find that the cumulative probabil- ity of 39 or less is 0.99326. We need to subtract this from 1 to get the chance of going up 40 or more times out of 60: 1 ⫺0.99326 ⫽0.00674. So, the chance of a stock randomly going up 40 or more times out of 60 times ⫽ 0.00674. Note that if you had solved for the probability of a stock going up exactly 40 out of 60 times, you would have gotten p = 0.00363, which is about half the probability you calculated when consider- ing 40 or moretimes.