Hi. In this video, I'd like to drill down a little more into the details of a Minimum Variance Portfolio. So the minimum variance portfolio is a portfolio comprised of the risky assets and we're trying to minimize the amount of risk. It can be traced out on that efficient frontier that we saw in the last video. One thing to note is that the optimal minimum variance portfolio will decrease the overall volatility when you add another risky asset to it, even if that asset is volatile in nature. We'll see that a little later on. Now that we have drawn the efficient frontier, we have a bunch of choices of portfolios with different weights, all of asset A versus all of asset B, or some mix of the two, and that's what that line traces out. But one thing investors often do is also combine the risky assets with something that's risk-free like a government bond and that helps to reduce the overall risk. So a complete portfolio is defined as a combination of risky assets and risk-free assets, with returns of R sub B and R sub F respectively. So the optimal risky portfolio is found at a point where this capital allocation line is tangent to the efficient frontier. Before I show you a graph of that, let's look at the equation for the capital allocation line. The expected return of the complete portfolio denoted r_c, is the expected return, is equal to r_f, that's the risk-free return times S_p, which is the Sharpe ratio times the standard deviation which is the risk. So recall the graph of the efficient frontier risk which is measured in sigma and the expected return is on the y-axis. So we are saying that if you recall from your high school algebra that y equals mx plus b, y is your expected return, r_f is your intercept point, that's your risk-free return, and then you have these two terms here. C is your x-variable here and then S_p is the slope of this line. So something like that. Recall that the S_p is the Sharpe ratio which is the expected return minus the risk-free rate over your risk. It's a ratio of return to risk. The standard deviation of the complete portfolio, denoted sigma sub c, is given by this equation here, which is the risk or weight of the risk of the portfolio and the risk-free rate. Since the risk free rate has a zero standard deviation, it's risk-free, there is no risk. We have this equation here, where basically the risk of the complete portfolio is equal to the weight of the risky portfolio times it's risk. So now that we have an understanding of what the capital allocation line is. Next we want to understand where that line touches the efficient frontier and that's known as the tangency portfolio. That is the point on the efficient frontier which has the greatest Sharpe ratio. Recall that the Sharpe ratio is the ratio of the expected returns over the risk. So we want more returns, less risk, and the bigger that number the better. It's geometrically represented by the slope of that line. The point of tangency between the capital allocation line and the efficient frontier is the tangency portfolio. The capital allocation line makes up a combination of risk-free assets and a risky portfolio. Note that portfolio may only have one asset in there, so that's a straight line. The point of tangency represents the point where the investor may completely invest in a risky portfolio. So let's look at a geometric representation. Here we have a graph of the efficient frontier. It's this dark line here, that's the efficient frontier. Then we also have a tendency line right there. That's the capital allocation line. The capital allocation line as you recall has the equation of expected return, is equal to the risk-free rate, r_f, plus the Sharpe ratio and multiplied by sigma of the complete portfolio c. We want a point where this Sharpe ratio is the biggest. This point is fixed, right here, that's the risk-free rate. We want to find a Sharpe ratio that's the biggest that touches this line, and that tangency point is there, right there. Here, on this end is the minimum variance portfolio. So that's the portfolio with the least amount of risk but the returns are low, and here is the point where they are in different between the capital allocation line and this line here, the Efficient Frontier. So let's look at some asset weight from this example. Here, this Efficient Frontier represents a portfolio, makes a five Tech Stocks, Apple, Intel Cisco, Nvidia, and Texas Instruments. Here are the weights in the minimum variance portfolio, this is right here. In order to get that point, you would invest about 22 percent into Apple, 14 percent into Intel, Cisco would be about 43 percent of your portfolio, nothing in Nvidia, and about 20 percent in Texas Instrument. But if you want to get to this point here, that's your tangency portfolio, you can see the weights are slightly different. You would not invest in anything in Apple or Intel, but most of your money into Cisco, 45 percent into Nvidia, and about four percent into Texas Instruments. Now, one question you might ask is why not pick one of these portfolios down here? Recall that you're always looking for something to the left and and upwards. So from here, the only portfolios that would do better are portfolio mixes that you'd find in this blue square here. These little dots here are either below that expected return or have more risk to the right. This graph represents various frontier weights, different portfolios with these five stocks. At an extreme case, we can have all of our money into Apple, or all of our money into Texas Instrument, or some mix of stocks and that's what this represents. So given some target expected returns, what are the portfolios that give us these minimum levels of risk or minimum variance? We can see that on the y-axis, it goes from zero to one. So zero is no money in from your wealth and one is 100 percent of your investment amount. In this first line, it's hard to see but you're investing mostly in Apple and little in Intel, and then as we go to the right, we can see mostly Nvidia, it looks like, and none of Apple or Intel. Here we have a couple of bar charts of optimal weights. On the left, we have the minimum variance portfolio. So recall that's this point here, and the second bar chart represents that point there. So in the minimum variance portfolio which is not our optimal returns, but it's just the minimum variance, we have about 22 percent invested in Apple, 43 percent invested in Cisco, 14 in Intel and 20 percent in Texas Instrument. In the tangency portfolio, recall that's the point where the capital allocation line is tangent to the efficient frontier, we have a different set of weights. In this case, we have Apple at zero and Intel at zero, and we're going to invest more than half our money in Cisco and 45 percent in Nvidia and a small 4.34 percent in Texas Instruments. That's the portfolio where the Sharpe ratio is the biggest, where it's maximized given our constraints. That wraps up this section where we described the capital allocation line and its relationship to the efficient frontier, and the description of that tangency point being the optimal weights of a portfolio.
