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The Efficient Frontier

Now, let's return to our two-asset portfolio. Let's call the expected return to X, r₁, and the expected return to Y, r₂. Let's also refer to the risks on the returns to X and Y as their standard deviations σ₁ and σ₂ with covariance σ₁₂. Suppose that the proportion of the portfolio held in asset Y is a fraction, w, and for X, this fraction is . Then, the return to the portfolio is:

The variance of these returns is

The standard deviation is the square root of this expression. Recall that the definition of the correlation coefficient ρ between the returns on X and Y requires ρ to be a number in absolute value between 0 and 1 and that by definition, . We can use this result to rewrite the variance formula then as a function of the correlation coefficient:

We will use this result to derive the efficient frontier, which is a set of portfolios (of X and Y) that yield the greatest return per unit of risk. Its shape will depend on the returns correlation. First, note we ...