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 img. 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, img. 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 ...

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