Problem: 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?