Hello. The first thing I want to talk about in modern portfolio theory is this notion of expected returns, and what does that really mean? So, the expected return of one security is the amount of profit or loss an investor can make or anticipate on an investment, given the historical returns. So, the way you calculate the expected returns of a security is to look at the historical average. Note that this part of the theory does not take into account risk. So to calculate the expected return of a security, it's really simple. All you have to do is take the log difference. Now that we have calculated the expected return of an individual security, let's look at how we calculate the expected returns for a portfolio of investments. So, the expected return for a portfolio investment is the weighted average of the expected returns of each of the components. The weight of a security is the ratio of the investment in the security to the total investment of the portfolio. So what do I mean by that? We can look at the formula this way: E(r) is the expected return, and it's a weighted average. Here, we see the weight of asset a, and here is the return of asset a, here is the weight of asset b, and here is the return of asset b. One thing to note is that the sum of the weight should equal one. So, if you add up all the weights, they should equal one. So, one way to think about it if you have two assets like in this example, you could put 60% of your money in asset a and 40% of your money in asset b, whatever those weights are. For this class, we're going to assume that all the weights are positive and that should help simplify the results. For those of you who have more experience in finance or modern portfolio theory, what I'm about to discuss is easily extensible into those areas. Another way to think about expected returns of a portfolio, this is the same formula. Just written that as you take the weight of A times the expected return of A, the weight of B times expected return of B, the weight of C times the expected return of C, and remember the weights also sum up to a 100%. So, let's look at an example of expected returns of a portfolio. So, let's consider an investment portfolio of two assets. Asset A has a 12% expected return with a 20% standard deviation. Asset B has a 16% return with a 35% standard deviation. Then given that you have, let's say, $10,000 that you're going to invest, you choose to invest $6,000 in Asset A and $4,000 in Asset B. So, how do we do that? First, we calculate the weights. The weights are here and here, and the first way it is really $6,000 divided by $10,000, which is the same as 60 percent or 6 over 10. Similarly, the 4 over 10 represents the $4,000 divided by the $10,000, so you have to spend total. So this is really equal to 0.6 and 0.4. Your expected return in the example was given to be 12 percent and 16 percent. Then once you multiply out and add up the products, you have 7.2 and 4.8 or an expected return of 12 percent. That's how you calculate the expected return of a portfolio.