A portfolio is simply a combination of assets. Portfolio theory shows how an investor can reach his optimal portfolio position. Portfolio theory is based on the assumption that the utility of the investor is a function of two factors: mean return and variance (or its square root, the standard deviation) of return. Hence it is also referred to as the mean-variance portfolio theory, or two-parameter portfolio theory. The investor is assumed to prefer a higher mean return to a lower one, and prefer a lower variance of return to a higher one.

The expected return on a portfolio is simply the weighted arithmetic average of the expected returns of the assets constituting the portfolio. The riskiness of a portfolio is measured by the standard deviation of the portfolio’s rate of return.

The investments available to an investor can be combined into any number of portfolios. Each possible portfolio may be described in terms of its expected rate of return and standard deviation of rate of return, and plotted on a two-dimensional graph. The investor should choose the portfolio that maximizes his utility function. This choice involves two steps: delineation of the set of efficiency portfolios, and selection of the optimal portfolio from the set of efficient portfolios. The efficient frontier is the same for all investors because the portfolio theory is based on the assumption that investors have homogeneous expectations.

We have merely defined what is meant by a set of efficient portfolios. How can this set be actually obtained from the innumerable portfolio possibilities that lie before the investor? The set of efficient portfolios may be determined with the help of graphical analysis, calculus analysis or quadratic programming analysis. The major advantage of graphical analysis is that it is easier to grasp. The disadvantage of graphical analysis is that it cannot handle portfolios containing more than three securities. The calculus method can handle portfolios containing any number of securities, since mathematical analysis can grapple with the n-dimensional space. Quadratic programming analysis can handle any number of securities and it can cope with inequalities as well. For practical purposes, the quadratic programming approach is the most useful approach.