Learning outcomes. After watching this video, you will be able to calculate the optimal weights of three risky assets in the mean-variance efficient portfolio using Excel, discuss what it means to have a negative weight for a risky asset in a portfolio. Investment opportunity set with three risky assets and a risk-free asset. How do we interpret situations where the rate of the risky asset or the risk-free asset is negative? That is the focus of this video. As a first step let's introduce the third risky asset z. Remember, the expected returns and standard deviation of x are 10% and 7% respectively and those of y are 20% and 10% respectively. Now asset Z has expected returns of 15% and standard deviation of 12%. The correlation between X and Y is 0.10. That between Y and Z is 0.90, and that between X and Z is 0. You can set up the Excel spreadsheet as you can see. Cell B5 contains the sum of the rates of the three risky assets in the MVE portfolio. The formula for the MVE portfolio's expected return is in cell C7 and it's standard deviation is in cell D7. The sharp ratio for the MVE portfolio is in cell C8. You want to maximize the shop ratio using sulfur subject to the cell B5 being equal to 1. That is the weights must add to 1. Once you unsolved it you will see that the weights of X, Y and Z in the MVE portfolio are 0.2274 1.7793 and -1.0067 respectively. The Sharpe ratio of the MVE portfolio is 1.94. What does the Sharpe ratio of 1.94 mean? For every 1% of risk the MVE portfolio gives an excess return of 1.94%. Given X, Y and Z and risk free rate of return of 5% the maximum Sharpe ratio you achieve is 1.94. The MVE portfolio is represented by the red dot in the figure that you see. It's expected return is 22.76% and it's standard deviation of returns is 9.16%. If you look at the rates you can see that the rate of Z in the MVE portfolio is negative. What does it mean to have a negative rate? It says that you sell asset z and invest the proceeds of that sell in assets x and y, but what if you do not own asset z, how do you sell it? In this case you short sell asset z. Short selling means that you sell something that you do not own. You can achieve this by borrowing asset z from another investor and selling it along with an obligation to replace the borrowed asset at a future time. How do we know interpret the weights? For every $1,000 you have, you short sell $1,006.70 worth of asset Z. This gives you a total of $2006.70 in cash. Out of this, use $227.40 to buy asset X and the balance $1779.30 to buy asset Y. The capital allocation line, from the risk free asset, to the MVE portfolio and beyond is now the investment opportunity set. Without the risk free rate, the investment opportunity set is simply the efficient front here, which is a curve. Adding a risk free asset changes the shape of the investment opportunity set and increases the range of feasible risk and return combinations. An investor can invest in any combination of the MVE portfolio and the risk-free asset. If you're at the point of the vertical axis, all your wealth is in the risk-free asset. If you're at the red dot, all your wealth is in the MVE portfolio. Any point between the risk free asset and MVE portfolio, denotes a fraction of your wealth in rested in each. Points on the capital allocation line to the right of the MVE portfolio. The note points where the rate in the MVE portfolio is greater then one. And that in the risk free asset is negative. We have discussed this in an earlier video. Remember, these are points where you borrowed the risk free rate of return and invest your wealth plus what you've borrowed in the MVE portfolio. For example, let's say that the rate in the MVE portfolio is 1.20, the rate in the risk free asset is a -0.20, and you have $100 in cash. The weights imply that you borrow $20 at the risk-free rate of return. You now have a total of $120 which you invest in the MVE portfolio. Next time we will discuss how to make the optimal allocation between risky and risk-free assets.