Hard Math problem?

Question

As we all know, because of market psychology when stocks increase in price on one day, they tend to decrease in price the next day. A payoff matrix shows the pertinent percents.

Now here's the payoff matrix.

If today the stock price increases, then the likelihood it will increase tomorrow is 0.2 and the likelihood it will decrease tomorrow is 0.8.

If today the stock price decreases, then the likelihood it will increase tomorrow is 0.6 and the likelihood it will decrease tomorrow is 0.4.

So the question is, in the long run, what fraction of the time will the stock increase in price?

Answer 1

P(+l+) = 0.2
P(-l+) = 0.8
P(+l-) = 0.6
P(-l-) = 0.4

P(+)
= P(+l+)*P(+) + P(+l-)*P(-)

let P(+) = x,
x = 0.2x + 0.6(1-x)
= 0.6 - 0.4x
1.4x = 0.6
x = 0.6/1.4 = 3/7

answer = 3/7

Answer 2

I think this problem is quite complex to calculate , so the best thing to do would be to use logic.

First let us assume that the amount of increase is equal to the amount of decrease.

After the first day , one makes a profit , only if there has been an increase. The likelihood of this increase is just 20 %.

After 2 days , one makes a profit , only if there is an increase on both days , since with one increase and one decrease , there would be no profit. The likelihood of this happening is 0.2 * 0.2 = 4 %.

After 3 days , one makes a profit , only if there is an increase on at least 2 days. The chances of this happening are 0.2 * 0.2 * 0.2 + 0.2 * 0.6 , since we cannot multiply by 0.8 because that is the likelihood of a decrease ! So this adds up to 0.128 i.e. 12.8 %

We can go on like this for any number of days , but I can only conclude that in the long run , you are very likely to make a loss ! This is because the ratio of 0.2 : 0.8 is much higher than the ratio of 0.6 : 0.4.

Answer 3

This is, indeed, an interesting problem. I created a decision tree for the case where the stock starts out increasing and mapped out the probabilities out four days after the increase. I calculated the probability of an increase after that time as:

(.2)(.2)(.2)(.2) + (.2)(.2)(.8)(.6) + (.2)(.8)(.6)(.2)+ (.2)(.8)(.4)(.6) + (.8)(.6)(.2)(.2) + (.8)(.6)(.8)(.6) + (.8)(.4)(.6)(.2) + (.8)(.4)(.4)(.6) = .4432.

Now, a similar calculation could be done for the case where the stock starts out decreasing (I'm too tired to go through those machinations).

The point is that there is a method to do this for any number of days. Assuming the solutions for each side converge to the same number, I would estimate, from what I've done so far, that the long run fraction of the time the stock will increase is around 88 percent.

Answer 4

40%

.2 + .6 = .8/2 = .4 = 40%