Greetings. This is simply the binomial probability model with parameters p=2/3, n=3 for x=0,1,2,3 successes. By success, we mean a tripling of the price. Thus,
Prob(X=0) = 1/27, as you suggest
Prob(X=1) = 2/9
Prob(X=2) = 4/9
Prob(X=3) = 8/27
The mean is 2 and the standard deviation is sqrt(n*p*(1-p))=.82
Thus, 4 out of 9 times, one would expect to have exactly two of the investments tripling and one going to zero. This illustrates the gains from spreading your risk. It also assumes that the events are independent. If they are correlated, that is an entirely different calculation.
Now let's look at the expected change in wealth assuming that $1000 is placed in each stock. If each goes to zero, you have a -3000 change in wealth. With one success and two failures you have a (+2000 -1000 -1000)or zero change in wealth. With two successes and one failure you have (+2000 + 2000 -1000) or $3000 change in wealth; with three successes a +$6000 change in wealth. The expected change in wealth (Sum Wi*P(Xi)) works out to be exactly $3000, given your parameters and equal investments. Thus you have an expectation of exactly a 100% gain.
I hope this helps, but recognize the key assumption of independence of outcomes, as in a coin toss.
Cheers! |
