Every investor knows that there is a risk-return tradeoff. In order to obtain greater returns on investments, the investor must be willing to take on greater risk. The only investments that are considered to be risk-free are U.S. Treasury bills, notes, and bonds. These investments yield a risk-free rate of return, r. (Note that r depends on the time to maturity of the investment.) Assuming riskless arbitrage opportunities do not exist, one cannot expect to have a return greater than the risk-free rate without taking on some risk.
Portfolio theory assumes that for a given level of risk, investors prefer higher returns to lower returns. Similarly, for a given level of expected return, investors prefer less risk to more risk. It is standard to measure risk in terms of the variance, or standard deviation, of return. We measure return as the average annual continuously compounded rate. Therefore, we can assume that investors would like to invest in an efficient portfolio, that is, one in which there is no other portfolio that offers a greater return with the same or less risk, or less risk with the same or greater expected return.
As many of us can imagine, all of the stocks traded in the stock market do not move together. In general the market has been moving up, but at the same time there are stocks that are losing value. There are some stocks that tend to move together, some that move in opposite directions, and others that seem to have no relation to one another. This tendency to move together or opposite can be measured by covariance (or if scaled, correlation). By using the covariance, we can measure the variability or risk in our portfolio. To reduce the volatility of the entire portfolio, it makes sense to include some stocks that move in opposite directions.
For the reader not familiar with the concept of efficient frontier, a more detailed explanation is available.