Learning Outcomes. After watching this video, you will be able to explain the two-fund separation theorem. Solve for optimal weights of a portfolio using Solver in Excel. Find an efficient portfolio given two efficient portfolios. Two-fund separation. In the last video, we defined the efficient frontier. In this video, we will use two-fund separation to help us draw the efficient frontier when we have three risky assets. So what is two-fund separation? It says that any efficient portfolio may be obtained by combining two efficient portfolios. To plot the efficient frontier, let's start by identifying two portfolios on the efficient frontier. Remember, the expected returns and standard deviation of x are 10% and 7%, respectively. Those of y are 20% and 10%, respectively, and those of z are 15% and 12%, respectively. Correlation between x and y is 0.1, that between y and z is 0.9, and that between x and z is 0. Let's pick two expected returns, 16% and 22%. We want to identify portfolios of x, y, and z that have returns of 16% and 22% and have the least risk at these levels of expected returns. In other words, we want to minimize sigma sub b squared, which = Wx squared x 0.07 squared + Wy squared x 0.1 squared + Wz squared x 0.12 squared + 2 times Wx, Wy x 0.1 x 0.7 x 0.1 + 2 x Wy, Wz x 0.9 x 0.1 x 0.12 + 2 x Wx, Wz x 0 x 0.07 x 0.12. Subject to the portfolio expected returns Sub P which is Wx x 0.1 + Wy x 0.2 + Wz x 0.15 be = 0.16 and Wx + Wy + Wz be = 1. This is a difficult problem to solve by hand, but Solver in Excel makes it easy to solve for Wx, Wy, and Wz. Set up your Excel spreadsheet exactly as you see on the screen. Use the SUMPRODUCT function to calculate the portfolio expected return. Fill in the formula, the long one, for portfolio variance as you see on your screen. Then click on Solver in the data tab on your Excel screen. Your objective is to minimize the portfolio variance. So set objective should be linked to cell B7, and you want to minimize that. You want to minimize the variance by changing the weights of x, y, and z. So by changing variable cells, should solve B2:B4. The two constraints are that the expected return should be 0.16 and the weights should add to 1. We'll add those constraints, as you can see. Once everything is set up, you can have Excel solve for the weights. We find that Wx is 0.54, Wy is 0.74, and Wz is -0.28. With these weights, the lowest possible variance of 0.004118 is achieved at an expected return of 16%. You can verify that the weights add to 1. The negative weight for z says that you need to sell z, whereas the positive weights on x and y say that you need to buy them. We can similarly find the efficient portfolio with an expected return of 22%. You can verify that the weights are 0.2625, 1.6625, and -0.925 for x, y, and z, respectively. Every other portfolio on the efficient frontier can be determined as a combination of these two efficient portfolios. Let's say you want to identify the efficient portfolio with an expected return of 20%. We need Sub b to be equal to 0.12, which is equal to w x 0.16 + (1- w) x 0.22. Solving this for w, we get w = one-third. So what are the weights of x, y, and z in the efficient portfolio with an expected return of 20%? Wx equals one-third times 0.54 plus two-thirds times 0.2625, which equals 0.355. Wy equals one-third times 0.74 plus two-third times 1.6625, which equals 1.355. Wz equals one-third times -0.28 plus two-thirds times -0.925, which equals -0.71. Given these weights, you can now calculate the variance of this efficient portfolio. You can verify that this is 0.0783. You can see where this lies on the efficient frontier. In this manner, you can identify an efficient portfolio at each level of expected return and then join them to draw the efficient frontier. All you need is just two efficient portfolios to start with. This is the idea behind two-fund separation. Next time, we will revisit the concept of diversification and see how much risk can actually be diversified away.
