There is a wall street banker. The banker invests in a kind of share called as options. The main features of this share is as follows:
- You make a bet with a specified amount of information as to whether the stock will go high or be brought down low
- If your prediction is right , then you get back the total money you invested along with a profit on the money you invested. Ex: lets say the profit percentage is
g
Determine the amount the banker should invest to double his money everyday(risk should be less than 0.003) and the rate 'r' which he should invest after every subsequent bet which fails to be executed correctly? The banker makes a right call with a probability of 0.52. You also need to find the minimum capital to make a profit of 100$ everyday. Also derive a generic formula if he wants to make xUSD everyday.
I have paraphrased the question above so that guys dont have to read the entire question. The main features are:
- Banker makes right call with probability of 0.52
- The profit percentage on a successful transaction is
g - The rate I use(the multiplication factor each time money has to be invested is
r
So far I have done the following. I have only targetted the initial part of the problem and not the final part which asks for the total capital needed to make 100$ everyday.
Let the rate be r
$$x(1+r+r^2+r^3+r^4+...)={x(r^n-1)\over(r-1)}$$
Total expenditure = Total investment = $$x(r^n-1)\over(r-1)$$
Since (.52)^9 = 0.0027, and this is a low enough probability, I used this as the starting point. So n = 9. If he wants to double his money , his total profit should be equal to the money he invested last. Once the banker makes his money, I reset the entire process and start from scratch again. One question I have is is this allowed? So assuming this is okay, Profits = $${x(r^n-1)\over(r-1)} -{g*x*r^{n-1}} = x*r^{n-1} $$
Assuming n=10, I solved for r to get some value. Now I need to target the next part of the question which asks me the investment he should have to make 100$ everyday. But what should I assume 'x' as? How do I proceed? If I end up taking the initial amount as 100USD then the total capital which he has to have turns out to be very high. What is an optimum value of x which I should take?
Also for the first part, if the process is terminated when the banker makes a right call, that is when n<9, then it appears that although the banker makes a correct call, he would not make sufficient money, Ex: Making a right call everytime means he only doubles 1$, assuming he starts with 1$. However, if he makes the wrong call upto the 9th time, then makes the right call, he ends up making more money with a lower risk. Can somebody explain this?