If you have a certain amount of money invested in the stock market for a moment of time, then there is an expected return on that investment (the stock market goes up on average), and a risk, a variance in that return (the stock market flucuates), both of which are proportional to the amount you have invested. As those moments of time are strung together, the expected returns for the different moments add, while the risks (since they are independent from one moment to the next) combine as square root of sum of squares. You have $10,000 to invest over one year. How should you allocate your investment over time to maximize your return/risk ratio?
Let there be n units of time.
For each unit of time let the expected return be E(r), (note that it is given in the problem that the returns can go up or down, but the expected return is positive, thus we deal with only expected returns).
Now, let us sum up the expected returns over n units of time, which is
n E(r). ...(1)
(As it is given in the problem that the expected returns for different moments add).
Now, we deal with the risk part.
The risk is the variance in the returns. Let us call it ...
How should you allocate your investment over time to maximize your return/risk ratio is determined.