[MUSIC] In the previous videos, we have shown how to optimally diversify a portfolio by looking at the impact of correlation on the risk of a combination of assets embedded in a single portfolio. However, we have taken a little bit of a leap of faith when constructing these portfolios because, in particular, we have not limited the type of position we can take in assets. We have said that the sum of the portfolio weight should sum up to one, but we haven't restricted, for example the portfolio weights to be positive. So, what does it mean for a portfolio weight to be negative? How can you invest minus 20% of your wealth in a particular asset? Well, this corresponds to what we call a short sale. A short sale is a trading activity by which you borrow a security from an investor who owns it, you sell it on the market with a promise to buy it back and return it to its original owner. When would you want to do something like that? Well, if you have a negative view on a particular asset, and you expect its return to be negative over a cumulative period of time, you would want to sell this asset if you have it in your portfolio. Well, if you don't have it and you want to benefit from this expected negative return, one way of doing so is to enter into a short sale. So, it's an agreement. It is a contract between an investor, who sells the security, and another, who makes it available for this particular investor to sell the security with a promise, of course, to buy it back. So these short-selling correspond to negative position in the portfolio weights. And when constructing the efficient frontier, since we do not impose any restriction on the positivity of the invested weight, we implicitly allow for these type of transactions. Usually, retail investors are not in a position to perform this type of transaction. They are restricted from short selling, so the type of portfolios they can construct are those that only contain positive weight. So, let's see how the efficient frontier is affected by this type of constraint. This is the original frontier we've constructed before. We see that I have represented the three assets A, B, and C, and the green line in the green envelope is the efficient frontier. When we only include the three assets, the red line is the efficient frontier. When you add the possibility of investing in the risk free security. So let's first see by only considering the risky assets, how the short sale constraint is going to effect the efficient frontier. In this next graph, we have two efficient frontiers being drawn here. One in red and one in green. The one in green corresponds to the upper part of the one that you've seen in the previous slide. So, this is the one that is completely unrestricted where we are trying to obtain the minimum level of risk for any level of expected return, and we do not impose any constraint on the portfolio weights. The red line, however, is obtained by looking at all the portfolios that fulfill the same objective, minimizing risk for a given level of expected return. But with the added constraint that it is no longer possible to take negative position. And what do we see? The red line is actually slightly to the right of the green line. Meaning that, if we impose a short selling constraint, we reduce the possibilities of diversification, and we obtain portfolios, which for a given level of return have actually a slightly higher risk. So the red frontier, the red line here is obtained by solving this optimization problem. Minimize risk with respect to an particular target of expected return, with the added constraint that the weights are positive. So, the short selling constraint is going to shift the efficient frontier to the right. Another type of constraint that we may be facing when we want to construct our portfolio is now related to what we can do with the risk free asset. So, if we go back to this efficient frontier drone in red, which includes the possibility of investing in the risk free security. We see that again we haven't imposed any restriction on the portfolio weight for the risk free asset. Positive investment in the risk free asset corresponds to lending money to a risk free borrower. But we could imagine also that we want to invest more than our available wealth in the risky security, and we would like to leverage our position. We can do this by borrowing at the risk-free rates and investing more than 100% of our wealth in the risky asset. Typically, when we do that, when we borrow money, we spend a little bit more than when we invest in a risk free security. Think of the mortgage rate that you have on a house, it's going to be much higher than the rate of return you will receive on a savings account. So, when you borrow money, you actually pay more than when you pay a higher rate than the rate you would receive when you invest in this asset. So, how is this going to affect the shape of the efficient frontier if you lend at a given rate, and borrow at a different rate? So, this is going to be depicted in this next graph. So, now you see that I've displayed here two possible level of the risk free rate. One at 2% which is the return you would get by investing in the risk free asset. And one slighter higher at 3%, which is the amount you have to pay to borrow money at the risk free rate. And you see that now we have two straight lines, which corresponds to two sections of the efficient frontier. And we have two points of tangency between the straight line and the green efficient frontier, which is constituted only by the risky assets. You can see that from the level of 2% up to the return generated by the portfolio indicated by TG1 the first tangency portfolio. This would follow the red line. Here we are investing in the risky assets and the risk-free security. The blue line will start to become the efficient frontier. Just above the level of the second tangency portfolio. So, when you borrow at the risk free rate, at the higher rate. Now you're going to over invest more than 100% in the risky asset, and reach return, which are above the level attained by the second tangency portfolio in between. So for all level of returns that are between the 6% roughly of the tangency one and the 7% of the tangency two. In between these two points the portfolio is absolutely not invested in the risk free asset. Neither borrowing or lending. In between these two points the efficient frontier is actually the green curve. So when there are restriction on borrowing and lending, so when you borrow and lend at different rates, the efficient frontier has three sections. The first one, the red line from the risk-free rate of 2% till the point denoted by tangency one. Then between tangency one and tangency two, the efficient frontier is actually the green curve. And, all the points above tangency two, are related to the blue line, which intersect the Y axis at 3%. And these correspond to the investment with leverage. So this is an example of an other constraint, borrowing and lending at different rates, which would have an impact on the official frontier. [MUSIC]