0:25

However, we have taken a little bit of a leap of faith

Â when constructing these portfolios because, in particular, we have not

Â limited the type of position we can take in assets.

Â We have said that the sum of the portfolio weight should sum up to one, but

Â we haven't restricted, for example the portfolio weights to be positive.

Â So, what does it mean for a portfolio weight to be negative?

Â How can you invest minus 20% of your wealth in a particular asset?

Â Well, this corresponds to what we call a short sale.

Â A short sale is a trading activity by which you

Â borrow a security from an investor who owns it, you sell it on the market

Â with a promise to buy it back and return it to its original owner.

Â When would you want to do something like that?

Â Well, if you have a negative view on a particular asset,

Â and you expect its return to be negative over a cumulative period of time,

Â you would want to sell this asset if you have it in your portfolio.

Â Well, if you don't have it and you want to benefit from this expected

Â negative return, one way of doing so is to enter into a short sale.

Â So, it's an agreement.

Â It is a contract between an investor, who sells the security, and

Â another, who makes it available for this particular investor to sell the security

Â with a promise, of course, to buy it back.

Â So these short-selling correspond to negative position

Â in the portfolio weights.

Â And when constructing the efficient frontier,

Â since we do not impose any restriction on the positivity of the invested weight,

Â we implicitly allow for these type of transactions.

Â 2:09

Usually, retail investors are not in a position

Â to perform this type of transaction.

Â They are restricted from short selling, so the type of portfolios they

Â can construct are those that only contain positive weight.

Â So, let's see how the efficient frontier is affected by this type of constraint.

Â 2:31

This is the original frontier we've constructed before.

Â We see that I have represented the three assets A, B, and C, and

Â the green line in the green envelope is the efficient frontier.

Â When we only include the three assets, the red line is the efficient frontier.

Â When you add the possibility of investing in the risk free security.

Â So let's first see by only considering the risky assets,

Â how the short sale constraint is going to effect the efficient frontier.

Â 3:02

In this next graph, we have two efficient frontiers being drawn here.

Â One in red and one in green.

Â The one in green corresponds to the upper part of the one that you've seen in

Â the previous slide.

Â So, this is the one that is completely unrestricted where we are trying to

Â obtain the minimum level of risk for any level of expected return, and

Â we do not impose any constraint on the portfolio weights.

Â The red line, however, is obtained by looking at all the portfolios that fulfill

Â the same objective, minimizing risk for a given level of expected return.

Â But with the added constraint that it is no longer possible to take

Â negative position.

Â And what do we see?

Â The red line is actually slightly to the right of the green line.

Â Meaning that, if we impose a short selling constraint, we reduce

Â the possibilities of diversification, and we obtain portfolios,

Â which for a given level of return have actually a slightly higher risk.

Â So the red frontier,

Â the red line here is obtained by solving this optimization problem.

Â Minimize risk with respect to an particular target of expected return,

Â with the added constraint that the weights are positive.

Â So, the short selling constraint is going to shift the efficient frontier

Â to the right.

Â 4:45

So, if we go back to this efficient frontier drone in red,

Â which includes the possibility of investing in the risk free security.

Â We see that again we haven't imposed any restriction on the portfolio weight for

Â the risk free asset.

Â 5:02

Positive investment in the risk free asset

Â corresponds to lending money to a risk free borrower.

Â But we could imagine also that we want to invest more than our available

Â wealth in the risky security, and we would like to leverage our position.

Â We can do this by borrowing at the risk-free rates and

Â investing more than 100% of our wealth in the risky asset.

Â 5:30

Typically, when we do that, when we borrow money,

Â we spend a little bit more than when we invest in a risk free security.

Â Think of the mortgage rate that you have on a house, it's going to be much

Â higher than the rate of return you will receive on a savings account.

Â So, when you borrow money, you actually pay more than when you

Â pay a higher rate than the rate you would receive when you invest in this asset.

Â So, how is this going to affect the shape of the efficient frontier

Â if you lend at a given rate, and borrow at a different rate?

Â So, this is going to be depicted in this next graph.

Â So, now you see that I've displayed here two possible level of the risk free rate.

Â One at 2% which is the return you would get by investing in the risk free asset.

Â And one slighter higher at 3%, which is the amount you

Â have to pay to borrow money at the risk free rate.

Â And you see that now we have two straight lines,

Â which corresponds to two sections of the efficient frontier.

Â And we have two points of tangency between the straight line and

Â the green efficient frontier, which is constituted only by the risky assets.

Â You can see that from the level of 2% up to the return generated by

Â the portfolio indicated by TG1 the first tangency portfolio.

Â This would follow the red line.

Â Here we are investing in the risky assets and the risk-free security.

Â The blue line will start to become the efficient frontier.

Â Just above the level of the second tangency portfolio.

Â 7:20

So, when you borrow at the risk free rate, at the higher rate.

Â Now you're going to over invest more than 100% in the risky asset, and reach return,

Â which are above the level attained by the second tangency portfolio in between.

Â So for all level of returns that are between the 6% roughly

Â of the tangency one and the 7% of the tangency two.

Â In between these two points the portfolio is absolutely not invested in the risk

Â free asset.

Â Neither borrowing or lending.

Â In between these two points the efficient frontier is actually the green curve.

Â So when there are restriction on borrowing and lending, so when you borrow and

Â lend at different rates, the efficient frontier has three sections.

Â The first one, the red line from

Â the risk-free rate of 2% till the point denoted by tangency one.

Â Then between tangency one and

Â tangency two, the efficient frontier is actually the green curve.

Â And, all the points above tangency two,

Â are related to the blue line, which intersect the Y axis at 3%.

Â And these correspond to the investment with leverage.

Â So this is an example of an other constraint, borrowing and

Â lending at different rates, which would have an impact on the official frontier.

Â [MUSIC]

Â