For a given stock, you are certain that for the next 100 days, it will move either 10% up or 10% down each day. You can invest now, but if you choose to do so, you have to hold it for the entire 100 days. Would you do so?
Introduction:
Let’s try using our intuition.
If we have 100 days, and 50-50 odds, the most likely outcome would, intuitively, be the one in which we have 50 winning days and 50 losing days.
In that scenario, starting with 100 dollars, after 100 days you’d end up having less than what you started with. This is due to the following equation that tells us that losing 10% in 50 days and gaining 10% in the other days yields us less than 100 dollars
HOWEVER:
This doesn’t actually tell us if the game is worth playing.
“We have to use the expected value!
Solution:
Let’s first look at two particular cases:
If we consider the easier problem, where we only have two days, the total number of winning days could either be 0, 1, or 2. Considering that the possible scenarios are LL, LW, WL or WW, we can compute the probabilities as the number of combinations that result in these.
As we expect, for the most likely outcome we end up losing money.
However, If we look at the expected value formula, we are actually expecting to retain our initial 100!
Similarly, we compute our scenario values for the case of 3 trading days. Now we can have up to 3 wins, resulting in 4 different outcomes distributed across 8 equally likely scenarios.
Interestingly, we arrive at the same result.
Given what we’ve learned, we think that we are going to get the same expected value for 100 days.
Let’s consider a random variable X that represents the profit after 100 days of playing the game. We want to find the expected value of this variable.
For a given number of days k:
The probability that we have exactly k winning days is the number of scenarios with exactly k wins divided by the total number of scenarios, which further equals 100 choose k divided by 2 to the power of 100.
The value of X for exactly k winning days is the one on the screen.
Putting it all together by summing X(k) times the probability of k over all possible values of k between 0 and 100, we get a formula that resembles the Binomial identity. Applying that we get that our expectation is equal to our starting value.
Conclusion:
Let’s now go back to our initial question: should we play the game?
Depending on your risk appetite you can choose to play or not to play, but in the long run, they are no differences.
However, if you consider this as a real-life scenario, you need to subtract the inflation from your expected returns.
Risk-free government bonds are, at every level, a better option than playing this game.