The Efficient Frontier

The following graph shows what the return and standard deviations of return would have been for 10,000 random portfolios from the 30 stocks in the Dow Industrials from 1986 to 1991, inclusive. (We assume that short selling and margin buying are not allowed.)

There may be multiple portfolios that have the same standard deviation. Modern portfolio theory assumes that for a specified standard deviation, a rational investor would choose the portfolio with the greatest return. Similarly, there may be multiple portfolios that have the same return and modern portfolio theory assumes that, for a specified level of return, a rational investor would choose the portfolio having the lowest standard deviation. A portfolio is said to be efficient if there is no portfolio having the same standard deviation with a greater expected return and there is no portfolio having the same return with a lesser standard deviation. The efficient frontier is the collection of all efficient portfolios. Below is the graph of the efficient frontier for the Dow Industrials from 1985 to 1991.

Let be the weight of the portfolio in asset i, and be the expected annual continuously compounded rate of return. Then the expected rate of return on the portfolio can easily be computed as:

Similarly, the variance of the return of a portfolio can be computed as:

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